Discrete Systems and Digital Signal Processing with MATLAB, 2nd Edition

4.8    x(n), h(n), y(n), and the z-Transform

Throughout this book, we have been using x(n) to represent the input, h(n) to represent the impulse response, and y(n) to represent the output. In discrete real time, the output is given using the convolution.

$y\left(n\right)=x\left(n\right)*h\left(n\right)$

But as we saw earlier, convolution is multiplication is the transform domain. Thus, we write Y(z) = X(z) H(z). We then can use Tables 4.1 and 4.2 to get back to the real-time signal y(n).

Example 4.16

If x(n) = u(n) is an input to the system with h(n) = (.5)ⁿ u(n), what is the output y(n)?

Solution

The output in the z-domain is

$Y\left(z\right)=X\left(z\right)H\left(z\right)$

The input in the z-domain is X(z) = z/(z − 1) and the impulse response is H(z) = z/(z −.5). Thus, using the convolution property of the z-transform, ...