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Eigenfunciones de los Sistemas LTI

Module by: Justin Romberg. E-mail the authorTranslated By: Fara Meza, Erika Jackson

Based on: Eigenfunctions of LTI Systems by Justin Romberg

Summary: Una introducción a los eigenvalores y eigenfunciones para un Sistema Lineal Invariente en el Tiempo.

Introducción

Ahora que ya esta familiarizado con la noción de eigenvector de una “matriz de sistema”, si no lo esta de un pequeño repaso a lasgeneralidades de eigenvectores y eigenvalores . También podemos convertir las mismas ideas para sistemas LTI actuando en señales. Un sistema lineal invariante en el tiempo (LTI) ℋ ℋ operando en una salida continua f⁢t f t para producir una salida continua en el tiempo y⁢t y t

ℋ⁢f⁢t=y⁢t ℋ f t y t
(1)

Figura 1: ℋ⁢f⁢t=y⁢t ℋ f t y t . f f y t t son señales de tiempo continuo(CT) y ℋ ℋ es un operador LTI.
Figura 1 (transfn.png)

La matemática es análoga a una matriz A A de NNxNN operando en un vector x∈ ℂ N x ℂ N para producir otro vector b∈ ℂ N b ℂ N (véase matrices y sistemas LTI para una descripción).

A⁢x=b A x b
(2)

Figura 2: A⁢x=b A x b donde x x y b b estan en ℂ N ℂ N y A A es una matriz de N N x N N .
Figura 2 (transfn2.png)

Solo como un eigenvector de A A es v∈ ℂ N v ℂ N tal que A⁢v=λ⁢v A v λ v , λ∈C λ ,

Figura 3: A⁢v=λ⁢v A v λ v donde v∈ ℂ N v ℂ N es un eigenvector de A A.
Figura 3 (transfn3.png)
podemos definir una eigenfunción (o eigenseñal) de un sistema LTI ℋ ℋ para ser una señal f⁢t f t tal que
ℋ⁢f⁢t=λ⁢f⁢t  ,   λ∈C    λ λ ℋ f t λ f t
(3)

Figura 4: ℋ⁢f⁢t=λ⁢f⁢t ℋ f t λ f t donde f f es una eigenfunción de ℋ ℋ.
Figura 4 (transfn4.png)

Las Eiegenfunciones son las señales mas simples possibles para ℋ ℋ π para operar en ellas: para calcular la salida, simplemente multiplicamos la entrada por un número complejo λ λ.

Eigenfunciones para cualquier sistema LTI

La clase de sistemas LTI tiene un conjunto de eigenfunciones en común: el exponencial complejo es⁢t s t , s∈C s son eigenfunciones para todo sistema LTI.

ℋ⁢es⁢t= λ s ⁢es⁢t ℋ s t λ s s t
(4)

Figura 5: ℋ⁢es⁢t= λ s ⁢es⁢t ℋ s t λ s s t donde ℋ ℋ es un sistema LTI.
Figura 5 (transfn5.png)

Nota:

Mientras que es⁢t  ,   s∈C    s s s t siempre son eigenfunciones para todo sistema LTI, estas no son necesariamente las únicas eigenfunciones.

Podemos probar la ecuación 4 expresando la salida como una convolución de la entrada es⁢t s t y de la respuesta al impulso h⁢t h t de ℋ ℋ:

ℋ⁢es⁢t=∫−∞∞h⁢τ⁢es⁢(t−τ)d τ =∫−∞∞h⁢τ⁢es⁢t⁢e−(s⁢τ)d τ =es⁢t⁢∫−∞∞h⁢τ⁢e−(s⁢τ)d τ ℋ s t τ h τ s t τ τ h τ s t s τ s t τ h τ s τ
(5)
Ya que la expresión de la derecha no depende de t t, es una constante λ s λ s ; Por lo tanto
ℋ⁢es⁢t= λ s ⁢es⁢t ℋ s t λ s s t
(6)
El eigenvalor λ s λ s es un número complejo que depende del exponente s s y por supuesto, el sistema ℋ ℋ. Para hacer esta dependencia explicita, vamos a usar la notación H⁢s≡ λ s H s λ s .

Figura 6: es⁢t s t es la eigenfunción y H⁢s H s son eigenvalores.
Figura 6 (transfn6.png)

Ya que la acción del operador LTI en esta eigenfunción es⁢t s t es fácil de calcular y de interpretar, es conveniente representar una señal arbitraria f⁢t f t como una combinación lineal de exponentes complejos. Las Series de Fourier nos dan la representación para una señal periódica continua en el tiempo, mientras que (poco más complicada) transformada de Fourier nos deja expandir señales arbitrarias de tiempo continuo.

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