Prony's method and matrix pencil method performance on determining the complex natural resonance frequencies of a linear system

Revision as of 20:13, 21 February 2022

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Abstract

Characterization of a physical system is an important issue to approach some applied physics and engineering problems. The complex natural resonance frequencies of the system which are included in its impulsive response are characteristic of such system and are part of its description. Few works written in English language show a comparisson among discrete methods that extract natural complex frequencies from a system impulsive response. Much less common is to find works written in Spanish language about this important research topic. Given this situation, the most important discrete numeric methods to estimate the complex natural resonance frecuencies of a system through its impulsive response are described, tested and compared in different simulation scenarios in this document. According to the obtained results, the matrix pencil method with a SVD filter is the less sensitive method to the noise, while the Prony method and its different versions are the fastest ones. Scenarios that could be more suitable for each method are discussed.

Keywords:Complex Natural Resonances (CNRs), Matrix Pencil (MP) Method, Prony's Method, Singular Value Decomposition (SVD), Total Least Square (TLS) Method.

Resumen

La caracterización de un sistema físico es un asunto de suma importancia para abordar algunos problemas relacionados con la física aplicada y la ingeniería. Las frecuencias complejas de resonancia de un sistema, las cuales están incluidas en su respuesta impulsiva son parte característica de dicho sistema y ayudan a describirlo. Son pocos los trabajos escritos en idioma inglés que muestran una comparación entre los métodos discretos que extraen las frecuencias complejas de resonancia a través del estudio de la respuesta impulsiva del sistema que se esté estudiando. Mucho menos común es encontrar este tipo de estudios comparativos escritos en idioma castellano. Dada esta situación, en este trabajo de investigación se describen, prueban y comparan los métodos numéricos discretos más importantes para estimar las frecuencias naturales complejas de resonancia de un sistema, a través del estudio de su respuesta impulsiva en diferentes escenarios de simulación. De acuerdo con los resultados obtenidos, el método del haz de matrices con filtro SVD es el método menos sensible al ruido, mientras que el método Prony y sus diferentes versiones son los más eficientes desde el punto de vista computacional. Se discuten los escenarios de simulación que podrían ser más adecuados para cada método.

Palabras Clave: Resonancias Naturales Complejas, Método del Haz de Matrices, Método de Prony, Descomposición en valores singulares, Método de Mínimos cuadrados totales.

1. Introducción

El estudio de un sistema físico a partir de su respuesta impulsiva es un asunto relevante y vigente en el mundo de la física aplicada y la ingeniería. La respuesta impulsiva de un sistema lineal invariante en el tiempo contiene las denominadas frecuencias naturales complejas de resonancia (FCNR), distintivas de dicho sistema, a partir de las cuales el sistema puede ser identificado sin necesidad de conocer a priori su constitución interna.

Entre las técnicas más utilizadas para estimar las FCNR de un sistema dado (en el dominio del tiempo) se reconocen dos grandes métodos, a saber: el método de Prony y el método de haz de matrices, los cuales son algoritmos con un fundamento teórico que no está basado en el análisis de Fourier. Vale la pena mencionar que en el dominio de la frecuencia existen otros métodos, ver por ejemplo[1], los cuales, sin embargo, no serán motivo de estudio en este trabajo. La versión original del método de Prony fue propuesta por Gaspard Riche de Prony en 1795 [2-3] y la del método del haz de matrices por Y. Hua y T.K. Sarkar en 1989 [4]. A partir de entonces y hasta la fecha de hoy, se han planteado nuevas versiones de ambos métodos, algunas de las cuales han demostrado ser más robustas en presencia de ruido [5-9].

Si bien el método de Prony fue concebido originalmente para tratar de modelar el fenómeno de expansión de los gases y el método del haz de matrices recibió al principio una atención especial en el campo de la tecnología de radar para la identificación de blancos no amigables en un contexto militar, en el proceso de revisión bibliográfica realizada durante el desarrollo de este trabajo se encontró una gran utilidad práctica de las distintas versiones de ambos métodos, para aplicaciones tan disímiles como las siguientes: determinación de las frecuencias naturales de oscilación de una red eléctrica [10], detección de fallas y monitoreo de armónicos de la red de suministro de potencia eléctrica [11-13], monitoreo de armónicos de convertidores electrónicos de sistemas de potencia eléctrica [14], estudio de los sistemas de control [15], procesamiento de señales de radar[16-21], detección de armas de mano escondidas en la ropa [22-23], diseño y análisis de metamateriales [24-25], diseño de líneas de transmisión y análisis de antenas de microondas [26-27], monitoreo y diagnóstico biomédico [28-32], procesamiento de imágenes de resonancia magnética nuclear [33-35], microscopía de superresolución [36], medición no invasiva de glucosa en sangre mediante microondas[37], así como análisis y procesamiento de señales provenientes de sistemas de audio[38], telescopios astronómicos [39], detectores de ondas gravitacionales [40], sonar [41-43], sismógrafos [44] y medidores de fuerzas de Coriolis [45].

Vale la pena mencionar las aplicaciones particulares que realizan el proceso de detección e identificación de un objeto en calidad de dispersor electromagnético a partir de su respuesta impulsiva [16-25]. En este caso, el conjunto de elementos físicos conformados por el objeto dispersor, la señal impulsiva que lo excita y su respectiva respuesta es lo que de ahora en adelante llamaremos sistema lineal. Visto el dispersor como un sistema, su respuesta impulsiva tiene como rasgo distintivo estar compuesta por un conjunto de FCNR dadas por Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): s_n=-\sigma _n+\jmath \omega _n , las cuales permiten, entre otras cosas, identificar unívocamente al propio objeto y en conjunto con cierto conocimiento previo, caracterizarlo desde un punto de vista físico. Las FCNR propias de un objeto dispersor electromagnético se correlacionan directamente con su geometría y constitución material y se distribuyen sobre el plano complejo formando “ramas” simétricas respecto del eje real con cierto orden y separación que es también característico del objeto. Los objetos “altamente” resonantes, como por ejemplo un dipolo metálico, poseen un cojunto de FCNR que se alojan en ramas muy próximas al eje imaginario, alejándose de este a medida que aumenta la frecuencia. Las FCNR de los objetos “suavemente” resonantes, en cambio, como por ejemplo una esfera metálica, se alojan en ramas que se separan mucho más notablemente del eje imaginario [46]. La extracción de las componentes imaginarias Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \omega _n

de estas FCNR no puede realizarse con precisión usando el análisis de Fourier por que al seccionar la función del sistema Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): H(s)
con el plano Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \sigma=0
para obtener a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): H(\omega )

, las Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \omega _n

pueden quedar “escondidas” o aliased, lo cual se observa en mayor medida para sistemas suavemente resonantes en comparación con los altamente resonantes [29].

También vale la pena agregar que por lo general el conjunto de valores de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \omega _n}

no son múltiplos de una frecuencia armónica natural fundamental como suele indicar la teoría de Fourier para señales periódicas, sino que más bien están distribuidas de manera arbitraria en el espectro, donde su valor es característico del sistema que se esté estudiando.

Aún dado el gran auge del uso del método de Prony, del método del haz de matrices y de sus variantes, no abundan las publicaciones arbitradas acerca del desempeño comparativo de dichos métodos en idioma castellano. Durante el proceso de revisión bibliográfica, en idioma castellano solo se consiguió el contexto histórico en el cual se crea el método de Prony, su descripción y su comparación con el método de la descomposición armónica de Pisarenko [47]. También se encontraron en idioma castellano algunas publicaciones en las que se aplica el método de Prony para modelar una falla en una central termonuclear[48], para la determinación de las frecuencias naturales de oscilación de una red eléctrica de potencia[49-50], para el procesamiento de imágenes de alta resolución [51] y para la creación de un software para la predicción del valor de acciones financieras [52].

Incluso en idioma inglés, son escasos los estudios comparativos publicados en revistas arbitradas acerca del desempeño de cada una de las variantes del método de Prony y del método del haz de matrices en diferentes escenarios de simulación. Entre los estudios comparativos encontrados durante el trabajo de revisión pueden citarse [30,40,53-55], en los cuales se examinan solamente alguna de las versiones de los métodos de Prony y del haz de matrices en algunos escenarios de simulación particulares. Entre estos estudios, uno de los más detallados que se encontró fue el descrito en [30], el cual muestra el desempeño de varios métodos generalmente utilizados en sistemas de análisis de señales bio-médicas, pero probando solo una de las versiones del método de Prony y del método del haz de matrices, sin tomar en cuenta el tiempo de cálculo de cada algoritmo, sin variar la frecuencia de muestreo y además variando los niveles de ruido en un rango no tan amplio como en el trabajo de investigación realizado en nuestro artículo. Otro trabajo de relevancia, de alta rigurosidad y enfoque estadístico, en el cual se muestra el desempeño de varias variantes del método de Prony y una variante del haz de matrices para diferentes números de muestras, sin variar la frecuencia de muestreo y tampoco el nivel de ruido presente en las señales que se estudian, se reporta en [40].

Vale la pena agregar que también durante el proceso de revisión bibliográfica se encontró un importante estudio analítico no comparativo del desempeño particular del método de Prony [56], en el cual se estudia la precisión teórica de dicho método en presencia de ruido.

Con base en el contexto anterior, con la intención de realizar un pequeño aporte al conocimiento y documentarlo en idioma castellano nos planteamos describir, analizar y comparar mediante simulación el desempeño de cinco métodos derivados de las teorías originales de Prony y del haz de matrices, a saber: el método de Prony clásico, el método de Prony con filtrado a través de la descomposición de valores singulares (DVS), el método de Prony usando mínimos cuadrados totales, el método del haz de matrices sin filtro y el método del haz de matrices con filtrado usando DVS. Para ello ponemos a prueba cada método en dos escenarios distintos de simulación, uno en el que se usan distintos niveles de ruido añadido y otro en el que se prueban distintas frecuencias de muestreo, ambos para una ventana de observación constante. En cada caso se registran los tiempos de cómputo de cada método. Luego, a partir de este trabajo comparativo, se determinan los escenarios más favorables para cada método, con los fines de proveer información valiosa que puede servir para desarrollar aplicaciones en los campos de la física y la ingeniería en general.

2 Breve descripción dcite-5e los métodos estudiados.

En este apartado describiremos brevemente las prescripciones de los métodos de Prony clásico, de haz de matrices y sus derivados. Específicamente revisaremos los fundamentos matemáticos de los métodos de Prony, de Prony con filtrado de valores singulares, de Prony con mínimos cuadrados, de haz de matrices y de haz de matrices con filtrado de valores singulares. Todos estos métodos se implementan en el dominio del tiempo y permiten estimar las frecuencias naturales complejas de resonancia de un sistema lineal invariante en el tiempo (LIT). Las frecuencias naturales complejas de un sistema LIT coinciden con los polos de la respuesta Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H(s)}

del sistema.

La respuesta impulsiva Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): h(t)

de un sistema lineal está compuesta por una combinación lineal de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): N'
cosenos amortiguados de la forma

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {eq:h} h(t)=\sum _{n=1}^{N'} |r_n|e^{-\sigma _nt}\cos (\omega _nt+\phi _n)=\sum _{n=1}^{N'}\Re \left\{r_ne^{s_nt} \right\}

(1)

y forma un par transformado con la función Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): H(s)

del sistema:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): h(t)=\sum _{n=1}^{N'} \Re \left\{r_ne^{s_nt} \right\}\Leftrightarrow ~~~~~~~~~~~~~~~~~

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): ~~~~~~~~~~~~~~~~\sum _{n=1}^{N'}\frac{1}{2}\left(\frac{r_n}{s-s_n}+\frac{r_n^\ast }{s-s_n^\ast }\right)=H(s)

(2)

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): N'=N/2 , siendo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): N

el número par de frecuencias naturales complejas de resonancia del sistema, donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r_n}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r_n^\ast }

, con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r_n=|r_n|e^{\jmath \phi _n}} , son residuos complejos, y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle s_n}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle s_n^\ast }

, con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): s_n=-\sigma _n+\jmath \omega _n

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \sigma _n>0

, donde el símbolo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \ast }

denota conjugación,  son los polos complejos del sistema también llamados frecuencias complejas naturales de resonancia.

Los métodos de Prony y de haz de matrices y todas sus versiones, son métodos de estimación mediante ajuste numérico que parten de un número finito, dígase Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle M} , de muestras de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h(t)}

tal cual se muestra en la Fig.  fig:2.

Figure 1: Sistema de reconstrucción de la respuesta impulsiva de un sistema lineal.

En la Fig. fig:2 el sistema lineal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): H(s)

 es estimulado con un impulso Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \delta (t)
ideal. Como resultado, el sistema responde produciendo a la salida una respuesta impulsiva de la forma dada por la Ec. eq:h. Luego, la respuesta Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): h(t)
del sistema es muestreada a una tasa Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): T_s
uniforme: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): h_{m}=h(mTs)

. Las muestras Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): h_{m} , contaminadas con ruido Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): n_m

en parte proveniente del propio sistema y en parte de tipo numérico, dan lugar a la señal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): y_m

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {eq:ym} y_m = y(mT_s) = \displaystyle \sum _{n=1}^{N} r_n e^{s_n mT_s } + n_m

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): = \displaystyle \sum _{n=1}^{N} r_n {z_n}^{m} + n_m

(3)

con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle m=0,1,2,\ldots ,M-1} ,y donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle z_n=e^{s_n T_s}}

son los polos de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H(z)}

, y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H(z)}

es la representación discreta del sistema Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H(s)}

.

El conjunto de muestras Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \{ y_m\}

alimenta al algoritmo de estimación o método de ajuste objeto de estudio en este trabajo, el cual permite  estimar las frecuencias complejas naturales de resonancia Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \hat{s}_n
a partir  de las cuales finalmente se construye una estimación de la respuesta impulsiva que se denominará de aquí en adelante Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \hat{h}(t)

.

2.1 Método de Prony clásico.

Gaspard Riche de Prony creó el método que hoy lleva su nombre en la última parte del siglo XVIII [2] para tratar de modelar el fenómeno de expansión de los gases. Prony asumió originalmente que el número Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N}

de frecuencias complejas naturales era conocido y la señal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y(t)}
bajo estudio tenía un ruido añadido despreciable, de modo que la condición Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h(t) \approx y(t)}
se cumplía.

El primer paso del algoritmo de Prony consiste en organizar las muestras Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h_m}

obtenidas a partir del muestreo uniforme de la señal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y(t)}

, con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle m=0,1,2,\ldots , M-1} , en un vector columna Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {h}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {eq:hvector} {\boldsymbol {h}}=\left(h_0,h_1,\ldots ,h_{M-1}\right)^{\mathit{H}}

(4)

donde el superíndice Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathit{H}}

denota conjugación transpuesta. Luego, tomando en cuenta que en ausencia de ruido Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y_m=h_m=h(mTs)}

, el valor de cada muestra puede expandirse a partir de la Ec. eq:ym de la forma

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {eq:he} \begin{array}{rlcl}h_0~=&r_1&+~...~+&r_N\\ h_1~=&r_1e^{s_1T_s} &+~...~+&r_Ne^{s_NT_s}\\ \vdots ~= & \\ h_m~=&r_1e^{s_1mTs} &+~...~+&r_Ne^{s_NmT_s }\\ \vdots ~= & \\ h_{M-1}~=&r_1e^{s_1 (M-1)T_s}&+~...~+&r_Ne^{s_N(M-1)T_s } \end{array}

(5)

o de forma más compacta

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {eq:heb} \begin{array}{rclcccl}h_0 & = & r_1 & + & \ldots & + & r_N\\ h_1 & = & r_1z_1 & + & \ldots & + & r_Nz_N\\ \vdots & = & \\ h_m & = & r_1z_1^m & + & \ldots & + & r_Nz_N^m\\ \vdots & = & \\ h_{M-1} & = & r_1z_1^{(M-1)} & + & \ldots & + & r_Nz_N^{(M-1)} \end{array}

(6)

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \{ z_n\} } , con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n=1,2,\ldots ,N} , y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle z_n=e^{s_nTs}}

son los polos de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H(z)}

. Por otra parte, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H(z)} , es la representación discreta del sistema original Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H(s)} .

El sistema de ecuaciones dado por eq:heb contiene Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 2N}

incógnitas: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N}
residuos complejos desconocidos y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N}
polos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \{ z_n\} }
desconocidos. Por esa razón se requiere un número Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle M}
de muestras mayor o igual a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 2N}
 para resolver dicho conjunto de ecuaciones para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \{ r_n\} }
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \{ z_n\} }

.

Luego, los valores de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \{ r_n\} }

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \{ z_n\} }
se estiman tal que, mediante la técnica de mínimos cuadrados, la Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hat{h}(t)}
obtenida minimice el error Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \xi }
dado por

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \xi =\left[\sum _{m=0}^{M-1}\left(\hat{h}_m-h_m\right)^2\right]^{\frac{1}{2}}

Posteriormente, el método de Prony prosigue con los tres pasos siguientes:

La solución de un sistema lineal algebraico de ecuaciones de por lo menos orden Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N} , para los coeficientes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \{ a_n\} } del sistema Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H(z)}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): H(z)=\frac{b_0z^K+b_1z^{K-1}+\ldots +b_{K-1}z+b_K}{a_0z^N+a_1z^{N-1}+\ldots + a_{N-1}z+a_N}=\frac{N(z)}{D(z)}

donde se asume que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle a_0=1}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle K<N}

. Observe que los ceros del polinomio en el denominador son los polos o frecuencias naturales de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H(z)} .

El cálculo de las Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N} raíces Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \{ z_n\} } (las frecuencias naturales del sistema Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H(z)} ) del polinomio característico

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {eq:cp} \begin{array}{rl} D(z)=a_0z^N+a_1z^{N-1}+\ldots +a_N = & 0\\ a_0(z-z_N)(z-z_{N-1})\times \ldots \times (z-z_1)= & 0\\ a_0\Pi _{n=1}^N(z-z_n)= & 0 \end{array}

(7)

La solución de un sistema de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N} ecuaciones algebraicas para los residuos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \{ r_n\} } .

Estos pasos se explican de manera detallada a continuación.

Para comenzar la estimación de coeficientes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \{ a_n\} } , el sistema dado por la Ec. eq:heb se multiplica sucesivamente por Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (a_N,a_{N-1},\ldots ,a_1,a_0)^H}

como se indica a continuación:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {eq:heb2} \begin{array}{crlll}{a_N}~\times & (h_0 & = r_1 & +\ldots &r_N)\\ {\vdots }~~~~\times & (h_1 & = r_1z_1 & +\ldots &r_Nz_N)\\ {a_0}~~\times & \vdots & & & {\vdots } \\ & (h_m & = r_1z_1^m & +\ldots &r_Nz_N^m)\\ & \vdots & = & & {\vdots } \\ & (h_{M-1} & = r_1z_1^{(M-1)} & +\ldots &r_Nz_N^{(M-1)}) \end{array}

(8)

entonces, también se puede realizar el proceso iniciándolo en la fila siguiente

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {eq:heb3} \begin{array}{crlll}&(h_0 & = r_1 & +\ldots &r_N)\\ {a_N}~\times & (h_1 & = r_1z_1 & +\ldots &r_Nz_N)\\ {\vdots }~~~~\times & \vdots & & & {\vdots } \\ {a_0}~~\times & (h_m & = r_1z_1^m & +\ldots &r_Nz_N^m)\\ & \vdots & = & & {\vdots } \\ & (h_{M-1} & = r_1z_1^{(M-1)} & +\ldots &r_Nz_N^{(M-1)}) \end{array}

(9)

y así sucesivamente, hasta la última fila como se observa a continuación

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {eq:heb4} \begin{array}{ccrll}&&(h_0 & = r_1 & +...+r_N)\\ & & (h_1 & = r_1z_1 & +...+r_Nz_N)\\ & & \vdots & = & \\ {a_N} & \times & (h_m & = r_1z_1^m & +...+r_Nz_N^m)\\ {\vdots } & \times & \vdots & = & \\ {a_0} & \times & (h_{M-1} & = r_1z_1^{(M-1)} & +...+r_Nz_N^{(M-1)}) \end{array}

(10)

Si nos centramos ahora en el sistema que resulta de completar la operación de multiplicación de la Ec. eq:heb2 y se suman los términos en cada lado de las Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N+1}

primeras ecuaciones, se obtiene:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): a_Nh_0+\ldots a_0h_N =

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): a_N(r_1 + \ldots +r_N) + \ldots +a_0(z_1^N+\ldots r_Nz_N^N) {eq:heb5}

(11)

la cual puede ser reordenada como

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {eq:heb6} a_Nh_0+\ldots a_0h_N =

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): (a_Nr_1 + \ldots +a_0 r_1 z_1^N)+

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): (a_Nr_2 + \ldots +a_0 r_2 z_2^N)+

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \ldots +(a_Nr_N+\ldots +a_0r_Nz_N^N)

(12)

Luego de notar que los términos del lado derecho de la Ecuación eq:heb6 son todas versiones del polinomio característico eq:cp, puede escribirse que

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {eq:heb7} a_Nh_0+\ldots a_0h_N = 0

(13)

Luego, esta ecuación puede generalizarse mediante la repetición de un proceso similar aplicado a todos los sistemas lineales que van desde eq:heb2 hasta eq:heb4, para obtener:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): a_Nh_m+a_{N-1}h_{m+1}+\ldots a_0h_{m+N}=0

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): para \quad m=0,1,\ldots , M-N-1 {eq:heb8}

(14)

por eso, un sistema de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle M-N}

ecuaciones lineales con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N-1}
incógnitas puede escribirse:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {eq:coef} \begin{array}{rl}a_Nh_0+\ldots a_0h_{N}= & 0\\ a_Nh_1+\ldots a_0h_{N+1}= & 0\\ \vdots & \\ a_Nh_{M-N-1}+\ldots a_0h_{M-1}= & 0 \end{array}

(15)

Luego, recordando que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle a_0=1} , el sistema de ecuaciones eq:coef puede ser escrito nuevamente, con la forma

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {eq:coef2} \begin{array}{rl}a_Nh_0+\ldots a_1h_{N-1}~= &-h_{N}\\ a_Nh_1+\ldots a_1h_{N}~= &-h_{N+1}\\ \vdots & \\ a_Nh_{M-N-1}+\ldots a_1h_{M-2}~= &-h_{M-1} \end{array}

(16)

Y este a su vez, puede ser reescrito en la forma matricial siguiente

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {eq:coef3} \underbrace{\begin{pmatrix} h_0 & h_1 & \ldots & h_{N-1}\\ h_1 & h_2 & \ldots & h_{N}\\ \vdots & \vdots & \vdots & \vdots \\ h_{M-N-1} & h_{M-N-2} &\ldots & h_{M-2} \end{pmatrix}}_{\mathbf{H}} \underbrace{\begin{pmatrix} a_N\\ a_{N-1}\\ \vdots \\ a_1 \end{pmatrix}}_{\mathbf{a}}=

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \underbrace{\begin{pmatrix} -h_{N}\\ -h_{N+1}\\ \vdots \\ -h_{M-1} \end{pmatrix}}_{\mathbf{b}}

(17)

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {H}}

es una matriz de Hankel de orden Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (M-N)\times N}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {b}}

es el vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (M-N)\times 1}
de valores conocidos, ambos construidos a partir de las muestras Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h_m}
de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h(t)}

, y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {a}}

es el vector incógnita con los coeficientes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle a_n}

, con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n=1,2,\ldots , N}

 del polinomio característico de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H(z)}

.

Como la Ec. eq:coef3 representa un sistema de ecuaciones linealmente independientes con más ecuaciones que incógnitas no hay manera de encontrar una solución exacta para el vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {a}} . La mejor estimación del vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {a}}

se obtiene, sin embargo, mediante la técnica de aproximación por mínimos cuadrados [57] de la forma:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {eq:mH} \mathbf{a}=\mathbf{H}^+\mathbf{b}

(18)

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{H}^+=({\mathbf{H}^{H}}\mathbf{H})^{-1}\mathbf{H}^H}

es la inversa de Moore-Penrose, o pseudo-inversa, de la matriz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{H}}

.

Una vez que los valores de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \{ a_n\} }

son obtenidos, las frecuencias naturales (discretas) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \{ z_n\} }
del sistema Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H(z)}
podrían estimarse encontrando las raíces del polinomio característico dado por eq:cp.

A partir de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \{ z_n\} }

las frecuencias complejas naturales de resonancia Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \{ s_n\} }
pueden estimarse como sigue. Como Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle z_n=e^{s_nTs}=e^{-\sigma _nTs}e^{\pm \jmath \omega _n T_s}}

, y además se cumple que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle |z_n|=e^{-\sigma _nT_s}}

y que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \angle z_n=\pm \omega _nTs}

, entonces

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): s_n = \frac{\ln{z_n}}{T_s}

(19)

Finalmente, de ser necesario, el sistema de ecuaciones de la Ec. eq:heb puede ser utilizado para estimar los Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N}

residuos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \{ r_n\} }
y construir la señal estimada Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hat{h}(t)}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {eq:r} \underbrace{\begin{pmatrix} 1 & 1 & \ldots & 1\\ z_1 & z_2 & \ldots & z_N\\ \vdots & \vdots & \vdots & \vdots \\ z_1^m & z_2^m & \vdots & z_N^m\\ \vdots & \vdots & \vdots & \vdots \\ z_1^{M-1} & z_2^{M-1} & \ldots & z_N^{M-1}\\ \end{pmatrix}}_{\mathbf{Z}} \underbrace{\begin{pmatrix} r_1\\ r_2\\ \vdots \\ r_N \end{pmatrix}}_{\mathbf{r}}= \underbrace{\begin{pmatrix} h_0\\ h_1\\ \vdots \\ h_m\\ \vdots \\ h_{M-1} \end{pmatrix}}_{\mathbf{h}}

(20)

Y de aquí

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {eq:r2} \mathbf{r}=\mathbf{Z}^+\mathbf{h}

(21)

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{Z}^+}

es la matriz inversa de Moore-Penrose de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{Z}}

, que a su vez es una matriz de Vandermonde transpuesta.

2.2 Método de Prony clásico utilizando filtrado con descomposición en valores singulares (DVS).sec:2

Cuando el número exacto de frecuencias naturales complejas de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h_m}

se desconoce y además el ruido Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_m}
no se puede despreciar, el método clásico de Prony en general falla. Para remediar este inconveniente se puede proceder, suponiendo un valor máximo de polos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_\mathrm{MAX}}
inicial, a filtrar el ruido mediante una descomposición en valores singulares de la matriz de Hankel del sistema antes de aplicar el método de Prony clásico [5]. Una versión similar a esta estrategia se conoce con el  nombre de método de Kumaresan-Tufts [6,40].

En este método, así como en el clásico de Prony, se procede primero a la creación de una matriz de Hankel como Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {H}}

en la Ec. eq:coef3, pero que esta vez designaremos como Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {Y}}
debido a la presencia del ruido, la cual tiene la siguiente  estructura

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\boldsymbol {Y}}={\begin{pmatrix} y_{0} &y_{1} &...&y_{N_\mathrm{MAX}-1}\\ y_{1} &y_{2} &... &...\\ y_{2} &y_{3} &... &...\\ . & . &... &...\\ . & . &... &y_{M-3}\\ y_{M-N_\mathrm{MAX}-1} & . &... &y_{M-2}\\ \end{pmatrix}} {eq:Yp}

de dimensiones Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (M-N_\mathrm{MAX}) \times N_\mathrm{MAX}}

tal que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_\mathrm{MAX} > N}

.

Luego, la matriz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {Y}}

es factorizada mediante DVS de la forma

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {eq:svdProny} \mathbf{Y}=\mathbf{U}\boldsymbol{\Sigma } \mathbf{V^{\mathit{H}}}

(22)

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{U}}

es una matriz unitaria de dimensiones Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (M-N_\mathrm{MAX})\times (M-N_\mathrm{MAX})}
cuyas columnas son los autovectores de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{Y}\mathbf{Y}^{\mathit{H}}}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{V}}

es una matriz de dimensiones Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (N_\mathrm{MAX}+1) \times (N_\mathrm{MAX}+1)}
cuyas columnas son los autovectores de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{Y}^{\mathit{H}}\mathbf{Y}}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{\Sigma }}
es una matriz diagonal de dimensiones Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (M-N_\mathrm{MAX}) \times (N_\mathrm{MAX}+1)}
la cual contiene los valores singulares de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{Y}}
(raíz cuadrada de los autovalores de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathbf{Y}}^{\mathit{H}}{\mathbf{Y}}}
) ordenados en forma  descendente tal que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _1 > \sigma _2 > ... > \sigma _{N_\mathrm{MAX}+1}}
[9,58].

Debido a la presencia de ruido, el número de valores singulares no nulos en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{\Sigma }}

se incrementa más allá del orden Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N}
del sistema, el cual no se conoce a priori. Los valores singulares originales de la matriz “limpia” Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{H}}
son perturbados por el ruido en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{Y}}
y además otros valores singulares adicionales aparecen en el proceso. Dada esta situación, los errores debidos al ruido pueden suprimirse por medio de la eliminación de los valores singulares espurios de la matriz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{\Sigma } }
[6,40].

Con base en este principio y tomando en cuenta que los valores singulares asociados al ruido poseen menor potencia, se establece una tolerancia Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle T}

relativa  al máximo valor singular Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _{1}}
a partir de un umbral Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _T}
 de acuerdo a  la siguiente ecuación

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {eq:th} \frac{\sigma _T}{\sigma _{1}}=T\approx 10^{-p}

(23)

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p}

es el número de dígitos significativos en los datos debido a la resolución esperada, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle T}
es la tolerancia relativa admitida,  y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _T}
es el umbral de filtrado. Es natural esperar que el número  de valores singulares Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _n}
que cumplen con la condición Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _n > \sigma _T}
coincida con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N}
en condiciones ideales. Por otra parte, los valores singulares que cumplen con la condición Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _n < \sigma _T}
se consideran espurios. Un ejemplo de aplicación de este procedimiento para determinar el número verdadero de polos del sistema analizado se muestra en la Fig. 2, donde ya se ha calculado el valor de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _T}

, umbral con el cual se determina el número de polos del sistema, que en este caso particular mostrado en la figura resulta ser Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N=8} .

Seguidamente, utilizando este criterio los valores singulares de la matriz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {\Sigma }}

con una tolerancia relativa inferior a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle T}
se reemplazan con  ceros, para generar una matriz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {\Sigma ^\prime }}

, la cual podría interpretarse como una matriz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {\Sigma }}

de bajo nivel de ruido o “limpia” [5]. Con esta nueva matriz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {\Sigma ^\prime }}
se crea luego  una matriz de Hankel Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {Y^\prime }}
de bajo nivel de ruido o “limpia” a partir de la expresión

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol {Y^\prime } ={\boldsymbol {U}}{{\boldsymbol {\Sigma ^\prime }}}{\boldsymbol {V}^H} {eq:YSVDlimipia}

(24)

Usando la matriz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {Y^\prime }}

en lugar de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {H}}
en la Ec. eq:mH se obtiene  el vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{a}}
con los coeficientes del polinomio característico de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H(z)}
de la forma:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{a}=\boldsymbol {Y^\prime }^+\mathbf{b}

a partir de lo cual se siguen todos los pasos subsiguientes del método clásico de Prony.

Figure 2: Caso típico de los valores singulares de la diagonal de la matriz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol {\Sigma }

en función de la Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): n-

ésima posición en la diagonal de dicha matriz, para una respuesta transitoria Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): y(t)

que proviene de un sistema lineal de ocho polos.

2.3 Método de Prony usando mínimos cuadrados totales.

El uso del método de mínimos cuadrados totales en conjunto con el método de Prony, descrito en la presente sección, presupone que el número Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N}

de frecuencias naturales se conoce a prori y que la señal bajo análisis está contaminada de ruido. En este método se aplica una descomposición DVS a la matriz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol {Y}}}
ampliada de la forma Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle [\boldsymbol {Y},\boldsymbol {b}]}
[7,8].

La modificación sustancial planteada surge cuando no se propone resolver directamente el sistema de ecuaciones Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {Y}\boldsymbol {a} = \boldsymbol {b}} , sino el sistema Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle [{\boldsymbol {Y}},{\boldsymbol {b}}][{\boldsymbol {a}}^{\mathit{H}} ~~-1]^{^{\mathit{H}}}=0} .

Como la matriz aumentada Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle [\mathbf{Y},\mathbf{b}]}

contiene muestras contaminadas con ruido, se la expande mediante una descomposición en valores singulares de la forma [59]:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): [\boldsymbol {Y},\boldsymbol {b}] = U{\Sigma }{V}^{\mathit{H}} {eq:TLSDVS}

(25)

Luego, comprobamos que la matriz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {V}}

de eq:TLSDVS puede representarse como

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \boldsymbol {V}=\begin{pmatrix}\boldsymbol {V_{pp}} &~\boldsymbol {v_{pq}} \\ \boldsymbol {v_{qp}} &~v_{qq} \\ \end{pmatrix}

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {v_{pq}}}

es el último vector columna de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {V}}
sin su último elemento, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {v_{qp}}}
es el último vector fila de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {V}}
sin su último elemento, y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle v_{qq}}
es el elemento faltante de ambos vectores. Y el vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle -v_{qq}^{-1}{\boldsymbol {v_{pq}}}}
contiene los coeficientes del polinomio característico de Prony dado en la Ec. eq:cp  [8]:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\boldsymbol {a}}\approx -v_{qq}^{-1}{\boldsymbol {v_{pq}}}

A partir de aquí se aplican los pasos subsiguientes del método de Prony clásico.

2.4 Método de haz de matrices en ausencia de ruido

El método de haz de matrices o de haz de funciones generalizado (GPOF) es un método de un solo paso de cálculo, el cual permite estimar los polos discretos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \{ z_n\} }

del sistema Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H(z)}
 como los autovalores generalizados de un haz lineal de matrices de la forma:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {eq:mp0} \mathbf{H_2}-\lambda \mathbf{H_1}

(26)

donde las matrices Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{H_1}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{H_2}}
se derivan de la matriz aumentada de Hankel Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle [\mathbf{H},-\boldsymbol {b}]}

, de dimensiones Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (M-N)\times (N+1)}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \centering {[\boldsymbol {H},-\boldsymbol {b}]}=\begin{pmatrix}h_{0} &h_{1}&..&h_{N}\\ h_{1} &h_{2}&..& h_{N+1}\\ . & . &..& .\\ . & . &..& .\\ . & . &..& .\\ h_{M-N-1}& &..&h_{M-1}\\ \end{pmatrix}_{(M-N)\times ({N+1})} {eq:Hmatrixpencil}

Las matrices Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{H_1}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{H_2}}
se construyen a partir de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle [\mathbf{H},-\boldsymbol {b}]}
eliminando de esta la segunda y la primera columna, respectivamente:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\boldsymbol {H_1}}= \begin{pmatrix}h_{0} &h_{1}& ...&h_{N-1}\\ h_{1} &h_{2}& ... &...\\ h_{2} &h_{3}& ... &...\\ . & . & ... &...\\ . & . & ... &h_{M-3}\\ h_{M-N-1}& . & ... &h_{M-2}\\ \end{pmatrix}_{(M-N) \times N} {eq:H1matrix}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\boldsymbol {H_2}}= \begin{pmatrix}h_{1} &h_{2} &...&h_{N}\\ h_{2} &h_{3} &... &...\\ h_{3} &h_{4} &... &...\\ . & . &... &...\\ . & . &... &h_{M-2}\\ h_{M-N}& . &... &h_{M-1}\\ \end{pmatrix}_{(M-N) \times N} {eq:H2matrix}

Fácilmente se comprueba que las matrices Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{H_1}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{H_2}}
 pueden ser  factorizadas  de la forma [4]:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \centering \mathbf{H_1}= \mathbf{Z_1RZ_2} {eq:H1factorizado}

(27)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \centering \mathbf{H_2}= \mathbf{Z_1RZ_0Z_2} {eq:H2factorizado}

(28)

donde

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{Z_1}=\begin{pmatrix}1 & 1 & \ldots &1\\ z_1 & z_2 & \ldots & z_N\\ \vdots & \vdots & \ldots & \vdots \\ z_1^{M-N-1} & z_2^{M-N} & \ldots & z_N^{M-1} \end{pmatrix}_{(M-N)\times N}

(29)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{R}=\begin{pmatrix}r_1 & 0 & \ldots & 0\\ 0 & r_2 & \ldots & 0\\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \vdots & r_N \end{pmatrix}_{N\times N}

(30)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{Z_0}=\begin{pmatrix}z_1 & 0 & \ldots & 0\\ 0 & z_2 & \ldots & 0\\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \vdots & z_N \end{pmatrix}_{N\times N} {eq:Z0}

(31)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{Z_2}= \begin{pmatrix}1 & z_1 & \ldots & z_1^{N-1}\\ 1 & z_2 & \ldots & z_2^{N-1}\\ \vdots & \vdots & \ldots & \vdots \\ 1 & z_N & \ldots & z_N^{N-1} \end{pmatrix}_{N\times N}

(32)

Ahora bien, combinando apropiadamente las ecuaciones eq:H1factorizado y eq:H2factorizado se obtiene:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{H_1^+}\mathbf{H_2} = \mathbf{Z_2^+}\mathbf{Z_0}\mathbf{Z_2} {eq:H1masH2}

(33)

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{H_1^+}}

es la pseudoinversa de Moore-Penrose de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{H_1}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{H_1^+}=\left({\mathbf{H_1^\mathit{H}}}\mathbf{H_1}\right)^{-1}\mathbf{H_1^\mathit{H}}

(34)

De la inspección de la Ec. eq:H1masH2, puede concluirse que los autovalores de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{H_1^+}\mathbf{H_2}}

son los autovalores de la matriz diagonal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{Z_0}}
dado que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{Z_2^+}\mathbf{Z_0}\mathbf{Z_2}}
es una forma de diagonalización de la matriz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{H_1^+}\mathbf{H_2}}

. Por tanto, los autovalores de la matriz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{H_1^+}\mathbf{H_2}}

son iguales a los elementos de la matriz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{Z_0}}
que de acuerdo a la Ec. eq:Z0 son  las raíces Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle z_n}
que se desea estimar.

Una vez obtenidas las raíces Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle z_n} , las frecuencias naturales complejas Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle s_n}

pueden determinarse como Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle s_n =\ln (z_n)/\Delta t}

.

2.5 Método de haz de matrices en presencia de ruido

La estrategia que describiremos a continuación también se conoce como método de haz de matrices con mínimos cuadrados totales [58], y se utiliza cuando las muestras Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h_m}

están contaminadas de ruido y además el número Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N}
de frecuencias naturales complejas del sistema bajo estudio se desconoce.  Con este método los polos discretos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle z_n}
del sistema Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H(z)}
se estiman también en un solo paso como solución del problema generalizado de autovalores del haz lineal de matrices de la forma:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {eq:mp1} \mathbf{Y_2}-\lambda \mathbf{Y_1}

(35)

donde las matrices Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{Y_1}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{Y_2}}
se derivan de la matriz de Hankel Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{Y}}
 de dimensiones Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (M-L)\times (L+1)}
construida a partir de las muestras Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y_m=h_m+n_m}
(ver Fig. fig:2):

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\boldsymbol {Y}}= \begin{pmatrix}y_{0} & y_{1}&.. &y_{L}\\ y_{1} & y_{2}&.. & y_{(L+1)}\\ . & . &.. & .\\ . & . &.. & .\\ y_{M-L-2} & . &.. & y_{M-2}\\ y_{M-L-1}& &.. &y_{M-1}\\ \end{pmatrix}_{(M-L) \times ({L+1})} {eq:YmatrixSVD}

(36)

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L}

se conoce como  parámetro de  haz, y se utiliza para eliminar algunos de los efectos del ruido presente en los datos.  Los valores recomendados de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L}
se ubican en el rango entre Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle M/3}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle M/2}

. Para estos valores de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L}

 la varianza de los valores estimados de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle z_n}
debida al ruido ha resultado mínima [9].

La matriz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{Y_1}}

se construye por medio de la eliminación de la última columna de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{Y}}

, mientras que la matriz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{Y_2}}

se construye  mediante la eliminación de la primera columna de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{Y}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\boldsymbol {Y_1}}= \begin{pmatrix}y_{0} &y_{1}& ...&y_{L-1}\\ y_{1} &y_{2}& ... &y_{L}\\ y_{2} &y_{3}& ... &y_{L+1}\\ . & . & ... &...\\ y_{M-L-2} & . & ... &y_{M-3}\\ y_{M-L-1}& . & ... &y_{M-2}\\ \end{pmatrix}_{(M-L) \times L} {eq:Y1matrix}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\boldsymbol {Y_2}}= \begin{pmatrix}y_{1} &y_{2} &...&y_{L}\\ y_{2} &y_{3} &... &y_{L+1}\\ y_{3} &y_{4} &... &y_{L+2}\\ . & . &... &...\\ y_{M-L-1} & . &... &y_{M-2}\\ y_{M-L}& . &... &y_{M-1}\\ \end{pmatrix}_{(M-L) \times L} {eq:Y2matrix}

Debido a la presencia de ruido en las muestras, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L-N}

polos  matemáticos falsos aparecen, acompañando las Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N}
frecuencias naturales complejas propias del sistema. Una descomposición DVS de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {Y}}
permite  discriminar las frecuencias naturales de los polos espurios. En efecto:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {eq:svdPencil} \mathbf{Y}=\mathbf{U}\boldsymbol{\Sigma } \mathbf{V^\mathit{H}}

(37)

donde vale la pena recordar que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol{\Sigma }}

es una matriz diagonal de dimensiones Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (M-L) \times (L+1)}
la cual contiene  los valores singulares de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{Y}}
ordenados de forma descendente tal que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _1 > \sigma _2 > ... > \sigma _{L+1}}
[9,58].

Definiendo una tolerancia Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle T}

 tal como se hizo en la Sección sec:2 (Ec. eq:th), se construye la matriz diagonal cuadrada Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {\Sigma }_\mathit{T}}
de dimensiones Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle T\times T}
con los Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle T}
valores singulares más significativos de la matriz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{Y}}

, y luego las matrices rectangulares truncadas Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathbf{U}_\mathit{T}}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathbf{V}_\mathit{T}}}

, por medio de la supresión de todas las columnas más allá de la Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle T} -ésima columna de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{U}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{V}}

, respectivamente.

Por medio de la eliminación de la última fila de la matriz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{V}_\mathit{T}}

se crea la matriz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{V_1}}
y mediante la eliminación de la primera fila de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{V}_\mathit{T}}
se crea la matriz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{V_2}}

. Las matrices obtenidas Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{V_1}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{V_2}}
cumplen con la relación

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{Y_1}=\,\boldsymbol {U}_\mathit{T}\boldsymbol {\Sigma }_\mathit{T}\boldsymbol {V_1^\mathit{H}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{Y_2}=\,\boldsymbol {U}_\mathit{T}\boldsymbol {\Sigma }_\mathit{T}\boldsymbol {V_2^\mathit{H}}

Luego, tomando en cuenta que el problema de autovalores generalizado Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (L\times T)\Rightarrow (T\times T) }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{Y_2}-\lambda \mathbf{Y_1}\Rightarrow \mathbf{Y_1^+}\mathbf{Y_2}-\lambda \mathbf{I}

(38)

es equivalente al problema ordinario de autovalores dado por

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{V_2^\mathit{H}}-\lambda \mathbf{V_1^\mathit{H}}\Rightarrow \mathbf{V_1^{{\mathit{H}}^+}}\mathbf{V_2}^\mathit{H}-\lambda \mathbf{I}

(39)

los polos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle z_n}

del sistema Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H(z)}
se estiman como los autovalores de la matriz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathbf{V_1^{{\mathit{H}}^+}}}{\mathbf{V_2}}^{\mathit{H}}}
[9].

3 Resultados.

Con la intención de comparar su desempeño, a continuación se presentan los resultados obtenidos a partir de la programación en MATLAB de los métodos descritos en la sección anterior en diferentes escenarios de simulación. Como señal de prueba se usó una compuesta de la suma de cuatro cosenos amortiguados de la forma:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {eq:ytexperimental} y(t) = r_1e^{-\sigma _1 t}\cos (\omega _1 t)+r_2e^{-\sigma _2 t}\cos (\omega _2 t)+\ldots

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \dots + r_3e^{-\sigma _3 t}\cos (\omega _3 t)+ r_4e^{-\sigma _4 t}\cos (\omega _4 t)+n(t)

(40)

donde tanto la potencia de ruido blanco Gausiano Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n(t)}

así como los valores de las frecuencias naturales complejas Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle s_n=-\sigma _n\pm \jmath \omega _n}

, con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _n>0} , y sus residuos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r_n}

asociados fueron definidos ad hoc para cada experimento.

3.1 Comparación del desempeño de cada algoritmo en presencia de diferentes niveles de ruido añadido.

Este primer experimento fue realizado con la idea de examinar la precisión de cada uno de los métodos numéricos presentados en la Sección II en la estimación de las frecuencias naturales complejas asociadas al sistema lineal bajo estudio en presencia de diferentes niveles de ruido blanco Gausiano añadido. Para realizar este experimento, se seleccionó una señal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y(t)}

de la forma dada por la Ec. eq:ytexperimental con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r_1=10}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r_2=5} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r_3=r_4=0} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _1=1.1} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \omega _1=2\pi } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _2=1.4}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \omega _2=4\pi }

. Nótese que dado que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r_3=r_4=0} , cada algoritmo, de ser exitoso, solo debe estimar las frecuencias naturales complejas Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle s_1} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle s_2} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle s_5}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle s_6}

.

En este experimento se utilizó un formato numérico de doble precisión para la estimación de los valores de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle s_1} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle s_2} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle s_5}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle s_6}

. La señal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y(t)}

fue muestreada desde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t_0=0}
hasta Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t_f=M t_s}

, con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle M=}

80, con una frecuencia de muestreo de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f_s=11}

, correspondiente a un tiempo de muestreo de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t_s = 1/f_s = 1/11} .

Para los métodos de haz de matrices con filtro DVS, y de Prony con filtro DVS, se utilizó un número máximo de componentes de la matriz diagonal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol {Z}_0}}

de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_{\mathrm{MAX}}=50}

. El factor de truncamiento Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle T}

se fijó en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle T=4}

, mientras que el parámetro del haz de matrices Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L}

se fijó en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L=[5M/12]=33}
donde la utilización de los corchetes, en este caso, simbolizan la función parte entera.

Por otra parte, para los métodos de Prony clásico y con mínimos cuadrados totales, así como para el de haz de matrices sin filtro DVS, se fijó el número de frecuencias naturales complejas a calcular en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N=T=4} .

Una vez definidos los valores de entrada, cada algoritmo fue corrido en presencia de ruido un número de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle K=200}

veces para 20 valores distintos de la relación señal a ruido, desde SNR=10 hasta SNR=153. Con los valores estimados de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hat{s}_1}
(Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hat{\sigma }_1}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hat{\omega }_1}

) se estimaron, a su vez, las varianzas normalizadas Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{Var}(\hat{\sigma }_1) / \sigma _1^2}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{Var}(\hat{\omega }_1) / \omega _1^2}
para cada valor de SNR a partir de las ecuaciones:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathrm{Var}(\hat{\sigma }_1)=\dfrac{\displaystyle \sum _{k=1}^{K}(\hat{\sigma }_{1,k}-\bar{\sigma }_1)^2}{K-1} {eq:Varianzasigman}

(41)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathrm{Var}(\hat{\omega }_1)=\dfrac{\displaystyle \sum _{k=1}^{K}(\hat{\omega }_{1,k}-\bar{\omega }_1)^2}{K-1} {eq:Varianzaomegan}

(42)

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hat{\sigma }_{1,k}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hat{\omega }_{1,k}}
son los valores estimados de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _{1}}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \omega _{1}}
en la Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k}

-ésima corrida realizada, mientras que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \bar{\sigma }_1}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \bar{\omega }_1}
son los respectivos promedios aritméticos de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hat{\sigma }_1}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hat{\omega }_1}

, estimados en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle K=200}

realizaciones.

Los resultados obtenidos se presentan en las Figs. Fig3 y Fig4. En tales figuras se han trazado las curvas de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left(\mathrm{Var}(\hat{\sigma }_1) / \sigma _1^2\right)^{-1}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left(\mathrm{Var}(\hat{\omega }_1) / \omega _1^2\right)^{-1}}

, respectivamente, en ambos casos como una función de la relación señal a ruido (SNR), para cada uno de los métodos analizados, de manera similar a la realizada en [4].

Figure 3: Valor de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left[\mathrm{Var}(\hat{\sigma }_1) / \sigma _1^2\right]^{-1}

en decibelios versus la relación señal a ruido (SNR) en decibelios, para cada uno de los métodos analizados.

Figure 4: Valor de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left[\mathrm{Var}(\hat{\omega }_1) / \omega _1^2\right]^{-1}

en decibelios versus la relación señal a ruido (SNR) en decibelios, para cada uno de los métodos estudiados.

3.2 Comparación del desempeño de cada algoritmo para distintas frecuencias de muestreo.

Este segundo experimento fue realizado con la idea de examinar la precisión de cada uno de los métodos numéricos presentados en la Sección II cuando se estiman las frecuencias naturales complejas de un sistema en presencia de un nivel de ruido añadido fijo, al utilizar distintas frecuencias de muestreo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f_s} . Para realizar este experimento, se utilizó una señal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y(t)}

tal que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r_1=10}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r_2=r_3=r_4=0} , con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _1=1.1}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \omega _1=2\pi }

. En este caso cada algoritmo debe estimar las frecuencias complejas Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle s_1}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle s_1^\ast }
del sistema lineal. Ruido  del tipo Gausiano blanco Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n(t)}
con una potencia relativa apropiada para proveer una relación señal a ruido resultante de SNR=60 dB  fue utilizado en este experimento.

Para todas las simulaciones realizadas en este experimento se utilizó un formato numérico de doble precisión. La señal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y(t)}

fue muestreada en una ventana de observación que va desde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t_0=0}
hasta Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t_f=4}

, para un rango de frecuencias de muestreo (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f_s} ) que va desde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {f_s}_{MIN}=2.5}

hasta Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {f_s}_\mathrm{MAX}=27}

. Para lograr esta condición hubo que cambiar para cada simulación el número de muestras Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle M} , tal que se cumpliese Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle M=[{f_s}(t_f-t_0)]}

con la idea de mantener la ventana de observación constante para cada valor de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f_s}
utilizado en cada simulación.

Para las simulaciones realizadas en este experimento con el método de Prony con filtro DVS y para el haz de matrices con filtro DVS, se fijó el número máximo de componentes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_\mathrm{MAX}}

de la matriz diagonal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol {Z}_0}}
en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_\mathrm{MAX}=20}

, mientras que el parámetro del haz de matrices Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L}

se fijó en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L=[5M/12]=33}

, para un factor de truncamiento Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle T=2}

pues son dos las frecuencias complejas naturales a calcular.

Por otra parte, para las simulaciones realizadas con el método de Prony clásico, con el método de Prony con mínimos cuadrados totales y con el método del haz de matrices sin filtro DVS, se utilizó un número de frecuencias naturales complejas Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N=2} .

Luego, para cincuenta valores de frecuencia de muestreo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f_s}

equiespaciadas, desde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f_s=2.5}
hasta Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f_s=27.5}
se muestra el inverso de las varianzas normalizadas Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{Var}(\hat{\sigma }_1) / \sigma _1^2}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{Var}(\hat{\omega }_1) / \omega _1^2}

, donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{Var}(\hat{\sigma }_1)}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathrm{Var}(\hat{\omega }_1)}
están definidas  por las ecuaciones  eq:Varianzasigman y eq:Varianzaomegan para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle K=40}
realizaciones. Como resultado, en las Figs. Fig5 y Fig6 se muestran las gráficas de los valores Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \left(\mathrm{Var}(\hat{\sigma }_1) / \sigma _1^2\right)^{-1}}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (\mathrm{Var}\left(\hat{\omega }_1) / \omega _1^2\right)^{-1}}
en función de la frecuencia Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f_s}
de muestreo, para cada algoritmo bajo estudio.

Figure 5: Valor de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left[\mathrm{Var}(\hat{\sigma }_1) / \sigma _1^2\right]^{-1}

 en decibelios versus la frecuencia de muestreo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): f_s

, para cada uno de los métodos estudiados.

Figure 6: Valor de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left[\mathrm{Var}(\hat{\omega }_1) / \omega _1^2\right]^{-1}

en decibelios versus la frecuencia de muestreo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): f_s

, para cada uno de los métodos estudiados.

3.3 Comparación de los tiempos de cómputo que cada algoritmo necesita para determinar las frecuencias naturales complejas de un sistema dado.

Este experimento fue realizado con el objetivo de comparar el tiempo de cómputo de cada algoritmo para estimar las ocho frecuencias naturales complejas que forman parte de la señal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y(t)}

dada por la Ec. eq:ytexperimental variando la ventana temporal de observación de la señal y manteniendo constante la tasa de muestreo.

En este experimento la señal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y(t)}

fue construida artificialmente usando los siguientes valores: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r_1=10}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r_2=7} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r_3=3} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r_4=1} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _1=1.1} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _2=1.4} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _3=2} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _4=3} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \omega _1=2\pi } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \omega _2=4\pi } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \omega _3=6\pi }

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \omega _4=10\pi }
usando un formato numérico de doble precisión para todos los cómputos.

Las muestras de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y(t)}

fueron tomadas durante cuatro ventanas temporales de duración distinta, desde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t_0=0}
hasta Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t_f=M t_s}

, con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle M=\{ 100,200,300,400\} } , usando en todos los casos la misma frecuencia de muestreo de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f_s=11} , correspondiente a un tiempo de muestreo de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t_s = 1/f_s = 1/11} .

Para las simulaciones realizadas en este experimento con el haz de matrices con filtro DVS y con el método de Prony con filtro DVS, se utilizó un número máximo de componentes de la matriz diagonal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\boldsymbol {Z}_0}}

de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_\mathrm{MAX}=20}

. El parámetro del haz de matrices se fijó para cada caso en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L=[5M/12]}

y el factor de truncamiento se fijó en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle T=8}

.

Por otra parte, para las estimaciones realizadas con los métodos de Prony clásico, de Prony con mínimos cuadrados totales y de haz de matrices sin filtro DVS, se fijó un número de frecuencias naturales complejas a estimar de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N=T=8} .

En la Tabla 1 se muestra el tiempo de cómputo en milisegundos empleado por cada algoritmo durante la estimación de la frecuencias naturales complejas Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle s_n}

según el número de muestras Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle M}
de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y(t)}
utilizadas.

Table. 1 Tiempo de cómputo en milisegundos empleado por cada algoritmo durante la estimación de la frecuencias Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): s_n según el número de muestras Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): M de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): y(t) utilizadas.
Algoritmo	M=100	M=200	M=300	M=400
Prony clásico	0.00029	0.00033	0.00039	0.00045
Prony con DVS	0.00165	0.00218	0.00490	0.00427
Prony con
mínimos cuadrados	0.00167	0.00434	0.01150	0.02474
totales
Haz de matrices	0.00616	0.04337	0.13920	0.27570
Haz de matrices	0.00460	0.02660	0.07272	0.17300
con DVS

tabla:tabla1

4 Análisis de Resultados.

Del análisis comparativo de los resultados observados de todas las simulaciones realizadas durante el desarrollo de este trabajo, puede concluirse que en presencia de bajos niveles de ruido, todos los métodos estudiados son capaces de estimar las frecuencias naturales de resonancia del sistema bajo estudio. Sin embargo, del análisis comparativo de los resultados expuestos en la Sección III.A, en las Figs. Fig3 y Fig4, puede concluirse que el método convencional más preciso y robusto cuando la señal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y(t)}

está contaminada con ruido es el método de haz de matrices convencional con descomposición DVS. Este resultado es consistente con los estudios reportados en [4], en los cuales se observa que la precisión de los resultados provenientes de este método está muy cerca del límite de Cramer-Rao, el cual es el límite teórico de máxima precisión posible para este tipo de algoritmos estudiados.

Del estudio realizado en la misma Sección III.A llama la atención el buen desempeño que tiene el método de Prony cuando utiliza como insumo una matriz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {Y}}

que se ha limpiado mediante una descomposición DVS. También es importante resaltar la significativa diferencia de tolerancia al ruido que tiene el método de  haz de Matrices con y sin filtrado DVS, pues al no utilizar la descomposición DVS, el método de  haz de Matrices pierde capacidad de tolerar al ruido presente en las muestras.

De los resultados mostrados en la Sección III.B, particularmente en las Figs. Fig5 y Fig6, pudo constatarse que en la medida en que la frecuencia de muestreo aumenta dentro de una misma ventana de observación, la precisión que brinda cada una de las versiones de los métodos de Prony y de haz de matrices aumenta, con la excepción del método de Prony clásico que más bien parece mejorar su desempeño en la medida en que la frecuencia de muestreo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f_s}

se aproxima a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 6\omega _1/(2\pi )}
(ver Fig. Fig6). De hecho, de acuerdo a la información presentada en estas gráficas, el método de Prony clásico puede brindar una mejor precisión que varios de los métodos utilizados si es pequeña la cantidad de muestras que se dispone dentro de una ventana de observación.

Otro aspecto interesante que puede observarse comparando los resultados de las Figs. 3 y 4, y también comparando los resultados de las Figs. 5 y 6, es que en general, la precisión con la que los métodos analizados calculan el factor de amortiguamiento Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _1}

es menor que la con la que calculan la frecuencia angular de resonancia Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \omega _1}

. También llama la atención el desempeño del método de Prony clásico cuando, en presencia de poco ruido añadido, casi iguala la precisión del método de haz de matrices con SVD para el cálculo de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \omega _1} , mientras que el mismo método de Prony clásico es el menos preciso para el cálculo de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _1}

bajo las mismas condiciones. La mayoría de estas observaciones son consistentes con las realizadas en el estudio comparativo presentado en [54].

De los resultados mostrados en la Sección III.C, pudo constatarse a través de los resultados expuestos en la Tabla I, que las distintas versiones basadas en el método de Prony son más rápidas en el cálculo que los métodos que utilizan la teoría del haz de Matrices. Esta diferencia en la velocidad de cálculo se incrementa en la medida en que el número de muestras utilizadas se hace más grande. De acuerdo al análisis realizado al código del programa que se elaboró, este fenómeno se debe a que las rutinas de MATLAB que calculan las raíces del polinomio característico de Prony son más rápidas que las rutinas que calculan los autovalores de las matrices dadas para el haz de matrices. Estos resultados son congruentes con el estudio presentado en [38], en el cual se establece la dificultad de utilizar el método del haz de matrices en la medida en que se incrementa el número de muestras Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle M}

tomadas a la señal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y(t)}

.

5 Conclusiones y Recomendaciones.

La respuesta impulsiva de un sistema lineal tiene como rasgo distintivo un conjunto de frecuencias naturales complejas de resonancia las cuales le son propias. Durante el desarrollo de este trabajo, se estudiaron y probaron en distintos escenarios de simulación, algunos de los más importantes algoritmos de cálculo numérico que existen para estimar tales frecuencias complejas naturales del sistema. Entre estos algoritmos, se probó el método de Prony clásico, el método de Prony con filtrado usando descomposición en valores singulares, el método de Prony usando mínimos cuadrados totales, el método de haz de matrices sin filtro y el método de haz de matrices con filtrado mediante descomposición en valores singulares. También durante el desarrollo de este trabajo, se realizó una revisión bibliográfica que evidenció la importancia que tienen estos métodos en diferentes aplicaciones de la física y de la ingeniería.

Se evaluó y comparó mediante simulación la precisión obtenida en la estimación de las frecuencias naturales complejas Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle s_n}

de cada uno de los algoritmos en presencia de ruido. Pudo concluirse que el método de haz de matrices con descomposición y filtrado DVS es el más robusto en presencia ruido de los métodos convencionales estudiados, tal y como sugiere la bibliografía revisada. De hecho, es difícil encontrar un algoritmo que supere este desempeño pues su precisión está muy cerca del límite teórico de precisión de Cramer-Rao [4] para algoritmos de este tipo. Por tanto, el método de  haz de matrices puede utilizarse en aplicaciones en que el tiempo de cálculo no sea una variable crítica a tomar en cuenta, debido a su excelente precisión en ambientes de medición ruidosos.

El notable desempeño en presencia de ruido del método de haz de matrices con descomposición DVS se debe, justamente, a la descomposición y filtrado DVS, la cual permite discriminar y eliminar los valores singulares asociados al ruido presente en la matriz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {Y}} . De hecho, el método pierde su robustez cuando no se utiliza la descomposición DVS. En este mismo sentido, cuando se utiliza filtrado a partir de una descomposición DVS en el método de Prony clásico en presencia de ruido se obtiene un desempeño comparable al del método haz de matrices.

La frecuencia de muestreo y el número de muestras utilizadas para crear la matriz Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \boldsymbol {Y}}

son variables importantes cuando se quiere aumentar la precisión de la estimación de cada frecuencia natural compleja con ruido presente. A excepción del método de Prony clásico, los métodos estudiados en los escenarios dados aumentan su precisión cuando la frecuencia de muestreo aumenta.

Por otra parte, el método clásico de Prony y sus variantes presentaron mayor velocidad de cálculo que los métodos basados en un haz de matrices, debido a que las rutinas utilizadas para el cálculo de las raíces del polinomio característico de Prony son más rápidas que las que estiman los autovalores de las matrices definidas para el método de haz de matrices. Esto podría indicar que si la relación señal a ruido fuese lo suficientemente alta y si se tiene una aplicación que necesite realizar el cálculo de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle s_n}

tan rápido como sea posible,  el método de Prony y sus variantes pudiesen ser más apropiados que el método del Haz de Matrices y sus variantes.

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@@ Line 976: / Line 976: @@
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-'''[[#citeF-1|[1]]]''' W. Lee, T. K. Sarkar, H. Moon and M. Salazar-Palma. (2012) "Computation of the Natural Poles of an Object in the Frequency Domain Using the Cauchy Method", Volume 11. IEEE Antennas and Wireless Propagation Letters 1137-1140
+'''[[#citeF-1|[1]]]''' Lee W., Sarkar T. K., Moon H., Salazar-Palma M. Computation of the natural poles of an object in the frequency domain using the Cauchy method. IEEE Antennas and Wireless Propagation Letters, 11:1137-1140, 2012.
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