UNIT – V
Z-Transform
Analysis of continuous time LTI systems can be done using z-transforms. It is a powerful
mathematical tool to convert differential equations into algebraic equations.
The bilateral (two sided) z-transform of a discrete time signal x(n) is given as
The unilateral (one sided) z-transform of a discrete time signal x(n) is given as
Z-transform may exist for some signals for which Discrete Time Fourier Transform (DTFT) does
not exist.
Concept of Z-Transform and Inverse Z-Transform
Z-transform of a discrete time signal x(n) can be represented with X(Z), and it is defined as
The above equation represents the relation between Fourier transform and Z-transform
Inverse Z-transform:
Z-Transform Properties:
Z-Transform has following properties:
Linearity Property:
Time Shifting Property:
Multiplication by Exponential Sequence Property
Time Reversal Property
Differentiation in Z-Domain OR Multiplication by n Property
Convolution Property
Correlation Property
Initial Value and Final Value Theorems
Initial value and final value theorems of z-transform are defined for causal signal.
Initial Value Theorem
For a causal signal x(n), the initial value theorem states that
This is used to find the initial value of the signal without taking inverse z-transform
Final Value Theorem
For a causal signal x(n), the final value theorem states that
This is used to find the final value of the signal without taking inverse z-transform
Region of Convergence (ROC) of Z-Transform
The range of variation of z for which z-transform converges is called region of convergence of z-
transform.
Properties of ROC of Z-Transforms
ROC of z-transform is indicated with circle in z-plane.
ROC does not contain anypoles.
If x(n) is a finite duration causal sequence or right sided sequence, then the ROC is
entire z-plane except at z =0.
If x(n) is a finite duration anti-causal sequence or left sided sequence, then the ROC is entire z-plane
except at z =∞.
If x(n) is a infinite duration causal sequence, ROC is exterior of the circle with radius a.
i.e. |z| > a.
If x(n) is a infinite duration anti-causal sequence, ROC is interior of the circle with radius
a. i.e. |z| < a.
If x(n) is a finite duration two sided sequence, then the ROC is entire z-plane except at z
= 0 & z = ∞.
The concept of ROC can be explained by the following example:
−n
Example 1: Find z-transform and ROC of a n u[n]+a u[−n−1] anu[n]+a−nu[−n−1]
The plot of ROC has two conditions as a > 1 and a < 1, as we do not know a.
In this case, there is no combination ROC.
Here, the combination of ROC is from a<|z|<1/a
Hence for this problem, z-transform is possible when a < 1.
Causality and Stability
Causality condition for discrete time LTI systems is as follows:
A discrete time LTI system is causal when
ROC is outside the outermost pole.
In The transfer function H[Z], the order of numerator cannot be greater than the order
ofdenominator.
Stability Condition for Discrete Time LTI Systems
A discrete time LTI system is stable when
its system function H[Z] include unit circle|=1.
all poles of the transfer function lay inside the unit circle|=1.
Z-Transform of Basic Signals
Some Properties of the Z- Transform:
Inverse Z transform:
Three different methods are:
1. Partial fractionmethod
2. Power seriesmethod
3. Long divisionmethod
Example: A finite sequence x [ n ] is defined as
Find X(z) and its ROC.
Sol: We know that
For z not equal to zero or infinity, each term in X(z) will be finite and consequently X(z) will
converge. Note that X ( z ) includes both positive powers of z and negative powers of z. Thus,
from the result we conclude that the ROC of X ( z ) is 0 <lzl< m.
Example: Consider the sequence
Find X ( z ) and plot the poles and zeros of X(z).
Sol:
From the above equation we see that there is a pole of ( N- 1)thorder at z = 0 and a pole at z = a .
Since x[n] is a finite sequence and is zero for n < 0, the ROC is IzI>0. The N roots of the
numerator polynomial are at
The root at k = 0 cancels the pole at z = a. The remaining zeros of X ( z ) are at
The pole-zero plot is shown in the following figure with N=8
