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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

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

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

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 ft f t para producir una salida continua en el tiempo yt y t

ℋft=yt ℋ f t y t (1)

Figura 1: ℋft=yt 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).

Ax=b A x b (2)

Figura 2: Ax=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 Av=λv A v λ v , λ∈ℂ λ ,

Figura 3: Av=λv 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 ft f t tal que
∀λ,λ∈ℂ:ℋft=λft λ λ ℋ f t λ f t (3)

Figura 4: ℋft=λft 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 ⅇst s t , s∈ℂ s son eigenfunciones para todo sistema LTI.

ℋⅇst= λ s ⅇst ℋ s t λ s s t (4)

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

Nota:

Mientras que ∀s,s∈ℂ:ⅇst 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 ⅇst s t y de la respuesta al impulso ht h t de ℋ ℋ:

ℋⅇst=∫-∞∞hτⅇst−τdτ=∫-∞∞hτⅇstⅇ-sτdτ=ⅇst∫-∞∞hτⅇ-sτdτ 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
ℋⅇst= λ s ⅇst 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 Hs≡ λ s H s λ s .

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

Ya que la acción del operador LTI en esta eigenfunción ⅇst s t es fácil de calcular y de interpretar, es conveniente representar una señal arbitraria ft 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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