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La Transformada Z: Definición

Module by: Benjamin Fite. E-mail the authorTranslated By: Fara Meza, Erika Jackson

Based on: The Z Transform: Definition by Benjamin Fite

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Summary: Una definicion breve de la transformada-z, que explica su relacion con la transformada de Fourier y su region de convergencia,ROC.

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Definición Básica de la Transformada-Z

La transformada-z de una secuencia es definida por

Xz=∑n=-∞∞xnz-n Xz n x n z n (1)
Algunas veces esta ecuación es conocida como la transformada-z bilateral. En veces la transformada-z es definida por
Xz=∑n=0∞xnz-n X z n 0 x n z n (2)
la cual es conocida como la transformada-z unilateral.

Hay una relación cercana entre la transformada-z y la transformada de Fourier de una señal discreta, la cual es definida como

Xⅇⅈω=∑n=-∞∞xnⅇ-ⅈωn X ω n x n ω n (3)
Note que cuando z-n z n es remplazada con ⅇ-ⅈωn ω n la transformada-z se convierte en la transformada de Fourier. Cuando la transformada de Fourier existe, z=ⅇⅈω z ω , la cual debe de tener la magnitud unitaria para zz.

El Plano Complejo

Para entender la relación entre la transformada de fourier y la transformada-z uno tiene que ver el plano complejo o el plano-z. echemos un vistazo al plano complejo:

Figura 1
Plano-Z
Plano-Z (zplane1.jpg)

El plano-Z es un plano complejo con un eje imaginario y real que se reflejen eso se refiere a la variable compleja zz. . La posición en el plano complejo es dada por rⅇⅈω r ω , y el ángulo se toma del eje real positive al rededor del plano y es dado por ωω. XzXz es definida en todos los lados del plano. Xⅇⅈω Xω es definida solo donde |z|=1 z1 , la cual se refiere al circulo unitario. Por ejemplo, ω=1ω1 en z=1z1 y ω=πω en z=-1z-1. Esto ayuda por que, al representar la transformada de fourier como una transformada-z en el círculo unitario, se puede ver muy fácilmente la periodicidad de la transformada de Fourier.

Región de Convergencia

La región de convergencia, también conocida como ROC, es importante entender por que define la región donde la transformada-z existe. La ROC para una xn x n , is defined as the range of z z es definida como el rango de z para la cual la transformada-z converge. Ya que la transformada –z es una serie de potencia, converge cuando xnz-n x n z n es absolutamente sumable. En otras palabras,

∑n=-∞∞|xnz-n|<∞ n x n z n (4)
Se tiene que satisface para la convergencia. Esto se explica mejor al ver las diferentes ROC de las transformadas-z de αnun α n u n y αnun−1 α n u n 1 .

Ejemplo 1

Para

xn=αnun x n α n u n (5)

Figura 2: xn=αnun x n α n u n donde α=0.5α0.5.
Figura 2 (sig1.png)

Xz=∑n=-∞∞xnz-n=∑n=-∞∞αnunz-n=∑n=0∞αnz-n=∑n=0∞αz-1n Xz n x n z n n α n u n z n n 0 α n z n n 0 α z 1 n (6)
Esta secuencia es un ejemplo de una exponencial del lado derecho por que tiene un valor de no cero para n≥0 n 0 . Solo converge cuando |αz-1|<1 α z 1 . Cuando converge,
Xz=11−αz-1=zz−α Xz 1 1 α z z z α (7)
Si |αz-1|≥1 α z 1 , entonces las series, ∑n=0∞αz-1n n 0 α z n no convergen. Así que el ROC es el rango de valores cuando
|αz-1|<1 α z 1 (8)
o, equivalentemente,
|z|>|α| z α (9)

Figura 3: ROC para xn=αnun x n α n u n donde α=0.5 α 0.5
Figura 3 (ROC1.jpg)

Ejemplo 2

Para

xn=-αnu-n−1 x n α n u n 1 (10)

Figura 4: xn=-αnu-n−1 x n α n u n 1 donde α=0.5α0.5.
Figura 4 (sig2_2.png)

Xz=∑n=-∞∞xnz-n=∑n=-∞∞-αnu-n−1z-n=-∑n=-∞-1αnz-n=-∑n=-∞-1α-1z-n=-∑n=1∞α-1zn=1−∑n=0∞α-1zn Xz n x n z n n α n u -n 1 z n n -1 α n z n n -1 α -1 z n n 1 α -1 z n 1 n 0 α -1 z n (11)
En este caso la ROC es en el rango de valores donde
|α-1z|<1 α -1 z 1 (12)
o, equivalentemente
|z|<|α| z α (13)
Si la ROC se satisface, entonces
Xz=1−11−α-1z=zz−α Xz 1 1 1 α -1 z z z α (14)

Figura 5: ROC para xn=-αnu-n−1 x n α n u n 1
Figura 5 (ROC2.jpg)

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