The \(z\)-transform plays the same role for discrete-time signals and systems as does the Laplace transform for continuous-time signals and systems.
The two-sided \(\pmb{z}\)-transform \(X(z)\) of a discrete signal \(x[n]\) is defined as
\(\boxed{X(z) = \mathcal{Z}\left\{x[n]\right\} = \sum_{k = -\infty}^{\infty} x[k]\,z^{-k}}\quad\textrm{(two-sided z-transform)}\)
where \(n\) is an integer index and \(z\) is a complex number.
The complex quantity \(z\) generalizes the concept of digital frequency \(F\) or digital angular frequency \(\Omega\) to the complex domain and is usually described in polar form as
\(\boxed{z = re^{\,j2\pi F} = re^{\,j\Omega} = r\left[\cos(\Omega) + j\sin(\Omega)\right]}\)
where \(r = |z|\) is the magnitude, \(F\) is the digital frequency, and \(\Omega = 2\pi F\) is the digital angular frequency.
The values of \(z\) can be plotted on a complex plane called the \(\pmb{z}\)-plane.
Since the defining relation for \(X(z)\) describes a power series (Laurent series), it may not converge for all \(z\). The values of \(z\) for which it does converge define the region of convergence (ROC) for \(X(z)\). Two completely different sequences may produce the same two-sided \(z\)-transform \(X(z)\).
It is important (unlike Laplace transform) that the ROC associated with each \(X(z)\) is specified, especially when dealing with the two-sided \(z\)-transform.
Properties of Two-Sided \(z\)-Transform
| Property | \(\displaystyle x[n] = \mathcal{Z}^{-1}\left\{X(z)\right\}\) | \(\displaystyle X(z) = \mathcal{Z}\left\{x[n]\right\}\) | Comments |
|---|---|---|---|
| Linearity | \(\displaystyle a_1 x_{1}[n] + a_2 x_{2}[n]\) | \(\displaystyle a_1 X_{1}(z) + a_2 X_{2}(z)\) | |
| Shifting | \(\displaystyle x[n - N]\) | \(\displaystyle z^{-N}X(z)\) | |
| Reflection | \(\displaystyle x[-n]\) | \(\displaystyle X\left(\frac{1}{z}\right)\) | |
| Anti-causal | \(\displaystyle x[-n]\,u[-n - 1]\) | \(\displaystyle X\left(\frac{1}{z}\right) - x[0]\) | for causal \(x[n]\) |
| Scaling | \(\displaystyle \alpha^{n}x[n]\) | \(\displaystyle X\left(\frac{z}{\alpha}\right)\) | |
| Times-\(n\) | \(\displaystyle n\,x[n]\) | \(\displaystyle -z\frac{dX(z)}{dz}\) | |
| Times-\(\cos\) | \(\displaystyle \cos(n\Omega)\,x[n]\) | \(\displaystyle \frac{1}{2}\left[X\left(ze^{\,j\Omega}\right) + X\left(ze^{-j\Omega}\right)\right]\) | |
| Times-\(\sin\) | \(\displaystyle \sin(n\Omega)\,x[n]\) | \(\displaystyle j\frac{1}{2}\left[X\left(ze^{\,j\Omega}\right) - X\left(ze^{-j\Omega}\right)\right]\) | |
| Convolution | \(\displaystyle x[n] \ast h[n]\) | \(\displaystyle X(z)\,H(z)\) | |
| Accumulation | \(\displaystyle \sum_{k = -\infty}^{n} x[k]\) | \(\displaystyle \frac{1}{1 - z^{-1}}X(z)\) |
See: Properties of z-Transforms
\(z\)-Transform Pairs
| \(\displaystyle x[n] = \mathcal{Z}^{-1}\left\{X(z)\right\}\) | \(\displaystyle X(z) = \mathcal{Z}\left\{x[n]\right\}\) | ROC | |
|---|---|---|---|
| unit impulse | \(\displaystyle \delta[n]\) | \(\displaystyle 1\) | |
| unit step | \(\displaystyle u[n]\) | \(\displaystyle \frac{z}{z - 1}\) | |
| unit ramp | \(\displaystyle n\,u[n] = r[n]\) | \(\displaystyle \frac{z}{(z - 1)^2}\) | |
| \(\displaystyle u[n] - u[n - N]\) | \(\displaystyle \frac{1 - z^{-N}}{1 - z^{-1}}\) | \(\displaystyle |z| \neq 1\) |