DSP - Z-Transform Introduction
Discrete Time Fourier Transform(DTFT) exists for energy and power signals. Z-transform also exists for neither energy nor Power (NENP) type signal, up to a certain extent only. The replacement $z=e^{jw}$ is used for Z-transform to DTFT conversion only for absolutely summable signal.
So, the Z-transform of the discrete time signal x(n) in a power series can be written as −
$$X(z) = \sum_{n-\infty}^\infty x(n)Z^{-n}$$
The above equation represents a two-sided Z-transform equation.
Generally, when a signal is Z-transformed, it can be represented as −
$$X(Z) = Z[x(n)]$$
Or $x(n) \longleftrightarrow X(Z)$
If it is a continuous time signal, then Z-transforms are not needed because Laplace transformations are used. However, Discrete time signals can be analyzed through Z-transforms only.
Region of Convergence
Region of Convergence is the range of complex variable Z in the Z-plane. The Z- transformation of the signal is finite or convergent. So, ROC represents those set of values of Z, for which X(Z) has a finite value.
Properties of ROC
- ROC does not include any pole.
- For right-sided signal, ROC will be outside the circle in Z-plane.
- For left sided signal, ROC will be inside the circle in Z-plane.
- For stability, ROC includes unit circle in Z-plane.
- For Both sided signal, ROC is a ring in Z-plane.
- For finite-duration signal, ROC is entire Z-plane.
The Z-transform is uniquely characterized by −
- Expression of X(Z)
- ROC of X(Z)
Signals and their ROC
| x(n) |
X(Z) |
ROC |
| $\delta(n)$ |
$1$ |
Entire Z plane |
| $U(n)$ |
$1/(1-Z^{-1})$ |
Mod(Z)>1 |
| $a^nu(n)$ |
$1/(1-aZ^{-1})$ |
Mod(Z)>Mod(a) |
| $-a^nu(-n-1)$ |
$1/(1-aZ^{-1})$ |
Mod(Z)<Mod(a) |
| $na^nu(n)$ |
$aZ^{-1}/(1-aZ^{-1})^2$ |
Mod(Z)>Mod(a) |
| $-a^nu(-n-1)$ |
$aZ^{-1}/(1-aZ^{-1})^2$ |
Mod(Z)<Mod(a) |
| $U(n)\cos \omega n$ |
$(Z^2-Z\cos \omega)/(Z^2-2Z \cos \omega +1)$ |
Mod(Z)>1 |
| $U(n)\sin \omega n$ |
$(Z\sin \omega)/(Z^2-2Z \cos \omega +1)$ |
Mod(Z)>1 |
Example
Let us find the Z-transform and the ROC of a signal given as $x(n) = \lbrace 7,3,4,9,5\rbrace$, where origin of the series is at 3.
Solution − Applying the formula we have −
$X(z) = \sum_{n=-\infty}^\infty x(n)Z^{-n}$
$= \sum_{n=-1}^3 x(n)Z^{-n}$
$= x(-1)Z+x(0)+x(1)Z^{-1}+x(2)Z^{-2}+x(3)Z^{-3}$
$= 7Z+3+4Z^{-1}+9Z^{-2}+5Z^{-3}$
ROC is the entire Z-plane excluding Z = 0, ∞, -∞
