z-Transform Properties

The previous sections focused on the unilateral z- transform in discussing properties, but as it turns out most of the properties are similar for the two cases. Since they are easier and more general for the bilateral case, we'll focus on those. We discuss the most important ones here:

• Linearity:
  
  $$ax[n]+by[n] \longleftrightarrow aX(z)+bY(z)$$
  
  where the new ROC $R^\prime \supset R_x \cap R_y$.
  
  
  
• Time shift:
  
  $$x[n-n_0] \longleftrightarrow z^{-n_0}X(z)$$
  
  
  where the new ROC is the same as $R_x$ with the possible addition or deletion of the origin or infinity.
  
  

• Convolution:
  
  $$y[n] = x[n]*h[n] \longleftrightarrow X(z)H(z)$$
  
  
  where the new ROC $R_y \supset R_x \cap R_h$.

Linearity and the time shift property will be useful for LCCDE systems, and the convolution property lets us avoid discrete-time convolutions. We'll use these properties a lot.

Convolution in Time

$$y[n]=x[n] \ast h[n] \leftrightarrow \sum_{n=-\infty}^{\infty}[ \sum_{k=-\infty}^{\infty} x[k]h[n-k]] z^{-n}$$

because we have a z-transform. Switching the order of the summations (OK except for pathological cases), we get:

$$= \sum_{k=-\infty}^{\infty} x[k] \sum_{n=-\infty}^{\infty} h[n-k] z^{-n}$$

Now, let $m = (n-k)$ and we get:

$$= \sum_{k=-\infty}^{\infty} x[k] [ \sum_{m=-\infty}^{\infty} h[m] z^{-(m+k)}]$$

$$= \sum_{k=-\infty}^{\infty} x[k]z^{-k} \sum_{m=-\infty}^{\infty} h[m] z^{-m} = X(z) H(z)$$

The new ROC will depend on both the poles in $X(z)$ and $H(z)$, giving $R_x \cap R_h$ since the ROC cannot include poles. However, if one transform has a zero that cancels a pole of the other then the ROC can be bigger, hence $R_y^\prime \supset R_x \cap R_h$.