4. CONTENTS:
1. Z-Transform
2. Commands in MATLAB
3. Convergence Region of Z-transform
4. The Inverse Z-Transform
5. Methods of finding Inverse Z-transform
6. Z-transform pairs
7. Properties of the z-Transform
8. z-Transform Using MATLAB
5. Z-Transform
The z transform is a mathematical tool commonly used
for the analysis and synthesis of discrete-time control
systems.
For discrete-time systems, z-transforms play the same
role as Laplace transforms do in continuous-time
systems
11. Convergence
Region of Z-
transform
Region of convergence (ROC)
Since the z-transform can be interpreted as the Fourier
transform of the product of the original sequence x[n]
and the exponential sequence r-n, it is possible for the z-
transform to converge even if the Fourier transform does
not.
Because
X(z) is convergent (i.e. bounded) i.e., Σx[n]r-n <∞, if x[n]
is absolutely summable.
n
n
n
n
znxznxzX
12. Continue..
Eg., x[n] = u[n] is absolutely summable if r>1. This
means that the z-transform for the unit step exists with
ROC |z|>1.
In fact, convergence of the power series X(z) depends
only on |z|.
If some value of z, say z = z1, is in the ROC, then all
values of z on the circle defined by |z|=| z1| will also be
in the ROC.
Thus the ROC will consist of a ring in the z-plane.
n
n
znx
14. The Inverse Z-
Transform
Inverse z-transform is opposite of z-transform.
For discrete-time systems, Inverse z-transforms play the
same role as Inverse Laplace transforms do in
continuous-time systems.
16. Methods of
finding Inverse
Z-transform
There are generally three methods of inverse z-
transform
I. Synthetic division method
II. Partial fraction expansion
III. Power series expansion
22. Advantages of
z-transform
Stability of LTI system can be determined using z-
transform.
By calculating z-transform of a given signal, Discrete
Fourier Transform (DFT) and Fourier Transform (FT)
can be determined.
The solution of differential equations can be simplified
using z-transform.