4.2.4. Convolution in Z-domain

\(\newcommand{\op}[1]{\mathsf #1}\) \(\newcommand{\ztarrow}{\stackrel{\op Z}{\longrightarrow}}\)

Consider a LTI system characterized with the impulse response function \(h[n]\). Given an input signal \(x[n]\) the output of the system equals \(y[n]=x[n]\ast h[n]\) with:

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

The Zt{} of the output is defined as:

\[\begin{split}Y(z) &= \sum_{n=-\infty}^{\infty} y[n] z^{-n}\\ &= \sum_{n=-\infty}^{\infty} \left(\sum_{k=-\infty}^{\infty} x[n-k] h[k]\right) z^{-n}\\ &= \sum_{k=-\infty}^{\infty} h[k] \sum_{n=-\infty}^{\infty} x[n-k] z^{-n}\\ &= X(z) \sum_{k=-\infty}^{\infty} h[k] z^{-k}\\ &= X(z) H(z)\end{split}\]

Therefore:

\[x[n]\ast h[n] \ztarrow X(z) H(z)\]