FREQUENCY RESPONSE FUNCTION AND DISCRETE TIME FOURIER TRANSFORM (DTFT)

DIGITAL SIGNALING PROCESSING

FREQUENCY RESPONSE FUNCTION AND DISCRETE TIME FOURIER TRANSFORM (DTFT)

 THEORY:

 Frequency Response Function:-The frequency response function of a discrete time system is found by substituting  z = e^jw in the pulse transfer function of the system.

Mathematically,

 The following example demonstrates how MATLAB computes and plots the frequency response function.

 Example 1: Sketch the normalized frequency response function of the system having the pulse transfer function

 Solution:

Write the following MATLAB code.

 num=[1 0.95];

den = [1 –1.8 0.81];

w =- pi : pi/255 : pi % frequency vector

h=freqz(num,den,w); % compute frequency response of the system.

h1 = abs(h); % magnitude response

 h2 = h1/(max(h1)); % normalization

 h3=20*log10(h2); % normalized magnitude shown in dB scale

plot(w,h3)

Discrete Time Fourier Transform (DTFT):

We have already studied the Discrete-Time Fourier Transform (DTFT) in the class. There are several ways to plot DTFT using MATLAB. Consider the following example:

Example 2: Plot the spectrum (DTFT) of the following system:

Solution:

Write the following MATLAB code.

k = 256; % frequency points

num=[ 0.008 -0.033 0.05 -0.033 0.008];

 den=[1 2.37 2.7 1.6 0.41];

w=0:pi/k:pi;

h=freqz(num,den,w);

subplot(221); plot(w/pi,real(h)),

grid ;

title('Real Part')

 xlabel('Normalized angular frequency')

ylabel('Amplitude')

 subplot(222)

plot(w/pi,imag(h));

grid

title('imaginary part')

xlabel('Normalized angular frequency')

 ylabel('Amplitude')

 subplot(223)

plot(w/pi,abs(h)),

grid

title('Magnitude Spectrum')

xlabel('Normalized angular frequency')

ylabel('Magnitude')

subplot(224)

plot(w/pi,angle(h));

grid title('Phase Spectrum')

xlabel('Normalized angular frequency')

ylabel('Phase, radians')

EXERCISE

 Repeat example1 for the following system:

num=[1 1];

den = [1 0.1 -0.2];

w =- pi : pi/255 : pi % frequency vector

h=freqz(num,den,w); % compute frequency response of the system.

h1 = abs(h); % magnitude response

 h2 = h1/(max(h1)); % normalization

 h3=20*log10(h2); % normalized magnitude shown in dB scale

plot(w,h3)

Sketch the magnitude and phase response of the system described by the system function.

k = 256; % frequency points

num=[1];

 den=[1 -0.8];

w=0:pi/k:pi;

h=freqz(num,den,w);

subplot(221); plot(w/pi,real(h)),

grid ;

title('Real Part')

 xlabel('Normalized angular frequency')

ylabel('Amplitude')

 subplot(222)

plot(w/pi,imag(h));

grid

title('imaginary part')

xlabel('Normalized angular frequency')

 ylabel('Amplitude')

 subplot(223)

plot(w/pi,abs(h)),

grid

title('Magnitude Spectrum')

xlabel('Normalized angular frequency')

ylabel('Magnitude')

subplot(224)

plot(w/pi,angle(h));

grid; title('Phase Spectrum')

xlabel('Normalized angular frequency')

ylabel('Phase, radians')