The frequency domain analysis Z-transform

1. The frequency domain analysisZ-transform The z-transform is a useful tool for the frequency analysis of discrete-time signals and systems. The z-transform may be used to solve constant coefficient difference equations, evaluate the response of a linear time-invariant system to a given input, and design linear filters.

2. The z-transform Unilateral (one side z-transform) bilateral (two side z-transform) This type of z-transform is defined as This type of z-transform is defined as Where z is a complex variable. We can see that X(z) is not concerned with the history of x(n) prior to n=0 The main disadvantages is its convergence conditions . Its the mathematical conditions for the transform to exist and converge

3. Example : Find the Z-transform of the finite duration signal Solution:- Homework :Find the Z-transform of the signal

4. The properties of Z-transform • Linearity • Shifting Property • Convolution Theorem

5. Homework 2 : Consider the two sequences as below , Find the z-transform of the convolution of x(n) with h(n) • Final value theorem:

6. The long division method Partial fraction method Inverse z-transform In which x(z) as a power series of In which table of z-transform pairs is needed to look up the function Example 11-4: Find and sketch the signal corresponding to the z-transform function

7. Solution: • Example : A signal has the z-transform , use the long division to recover the signal

8. The coefficient of X(z) are equal to the sample value of x(n) .x(0)=0 , x(1)=0, x(2)=0, x(3)=0.5 , x(1)=3/4, and so on

9. poles and zeros on the z-plane and stability The z-transform used to described a real signal or any LTI system , it is always a rational function of the frequency variable z . it can be written as the ratio of numerator and denominator polynomials in z

10. The constants z1 ,z2 ,z3 … are called the zero of X(z), because they are the values of z for which X(z) is zero. Conversely p1,p2, … are known as the poles of X(z), giving values of z for which X(z) tends to infinity. • A very useful representation of a z-transform is obtained by plotting its poles and zeros in the complex plane referred as a z-plane. Note that a zero is shown as an open circular symbol, and a pole as a cross.

11. The z-plane represent as a circle of unit radius centered at the z-plane origin as shown in Figure below

12. Example:- plot the z-plane poles and zeros of the following z-transforms