4.2.2. \(\ZT\)-Transform Properties

Time Domain	Z-domain
Synthesis (Inverse Z=Transform) \[x[n] = \frac{1}{2\pi j}\oint_C X(z)z^{n-1}dz\]	Analysis (Z-Transform) \[X(z) = \sum_{n=-\infty}^{\infty} x[n] z^{-n}\]
Complex Conjugate \[x[n]^\ast\]	\[X^\ast(z^\ast)\]
Real Signal \[x[n]\in\setR\]	\[X(z^\star) = X^\star(z)\]
Time Shift \[x[n-n_0]\]	‘Phase’ factor \[z^{-n_0} X(z)\]
Time difference \[x[n] - x[n-1]\]	\[(1-z^{-1}) X(z)\]
Convolution \[x[n] \ast y[n]\]	Multiplication \[X(z) Y(z)\]

The properties in the above table are easy to prove. Some proofs are given below.

Complex Conjugate

Let calculate the Z-transform of \(x[n]^\ast\):

\[\begin{split}\sum_{n=-\infty}^{\infty} x[n]^\ast z^{-n} &= \sum_{n=-\infty}^{\infty} \left(x[n] (z^{-n})^\ast \right)^\ast \\ &= \left(\sum_{n=-\infty}^{\infty} x[n] (z^{-n})^\ast \right)^\ast \\ &= \left(\sum_{n=-\infty}^{\infty} x[n] (z^\ast)^{-1} \right)^\ast \\ &= \left( X(z^\ast) \right)^\ast = X^\ast(z^\ast)\end{split}\]

Time Shift

The Z-transform of \(x[n-n_0]\) is given by

\[\begin{split}\sum_{m=-\infty}^{\infty} x[n-n_0] z^{-n} &= \sum_{k=-\infty}^{\infty} x[k] z^{-(k+n_0)}\\ &= \sum_{k=-\infty}^{\infty} x[k] z^{-k} z^{-n_0}\\ &= z^{-n_0} \sum_{k=-\infty}^{\infty} x[k] z^{-k}\\ &= z^{-n_0} X(z)\end{split}\]

Convolution Property

The definition of the convolution is:

\[x[n] \ast y[n] = \sum_{m=-\infty}^{\infty} x[n-m] y[m]\]

Its Z-transform is:

\[\begin{split}\sum_{n=-\infty}^{\infty} \left(x[n]\ast y[n]\right)z^{-n} &= \sum_{n=-\infty}^{\infty} \left(\sum_{m=-\infty}^{\infty} x[n-m] y[m]\right) z^{-n}\\ &= \sum_{m=-\infty}^{\infty} y[m] \sum_{n=-\infty}^{\infty} x[n-m] z^{-n}\\\end{split}\]

In the last summation we recognize the Z-transform of the shifted signal \(x\) so:

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

Summarizing:

\[x[n]\ast y[n] \ZTright X(z) Y(z)\]