**Pole-Zero Diagrams: Z-Transform**

**Logic and Matlab Functions Used:**

In general z- transform of a function can be represented as, x(z) =P(z)/Q(z), where P(z) and Q(z) are polynomials in z. The value of z for which (z)=0are called "zeros" of x(z). The values of z for which X(z) is =infinity are called as poles of X(z).The plot of these values of poles and zeroes on the z-plane is known as pole-zero plot.

The MATLAB function "zplane" allows the computation and display of the pole-zero diagrams. The syntax for the command is "zplane(b,a)" where b and a are coefficient vectors of the numerator and denominator polynomials, p(z)/Q(z). In this command format, the command first finds the locations of zeros and poles (i.e. roots of P(z) and Q(z), respectively) and then plots the z -plane diagram.

The locations of the poles and zeros can be directly found (without drawing z-plane diagram) using "roots(X)" command where X is the numerator or denominator coefficients.

Thus if the locations of the poles and zeroes are known, these can be used as the inputs to the "zplane" command. The syntax of the command in this case is "zplane(z,p)" where z and p are the zeroes and poles respectively.

**Algorithm:**

1. Input the numerator coefficients (increasing -ve powers of z)

2. Input the denominator coefficients (increasing -ve powers of z)

3. Find zeroes using "roots" command.

4. Find poles using "roots" command.

5. Plot pole zero diagram using "zplane" command.

**Program:**

**b=input('enter the numerator coefficients');**

**a=input('enter the denominator coefficients');**

**z=roots(*b);**%find zeros

**p=roots(a);**%find poles

**zplane(z,p);**%plot poles zero diagram

**disp(z);**%display the zeros

**disp(p);**%display the poles

Z-transform comes in mathematics and signal processing, the Z-transform converts a sequence of real or complex numbers into a complex frequency-domain representation.It can be considered as a discrete-time equivalent of the Laplace transform.

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