by Ljiljana Milic
Supplemental material for Chapter I:
1. Single-Rate Signals and Systems: Background Review
Table of Contents
1.1 Discrete Time Fourier Transform (DTFT)
Remainder
Definition of the discrete-time Fourier transform (DTFT):
DTFT for the test signal 
of a finite length L
of a finite length L
Example 1.1
In this example, the spectrum of the triangular signal is computed using function freqz from the Signal Processing Toolbox.
is computed using function freqz from the Signal Processing Toolbox.clear all, close all
Introducing the test signal
L = 13;
x = triang(13);
L = length(x); % length of the sequence
N = 256; % number of DTFT points
Computing Discrete Time Fourier Transform 
using Matlab function freqz
using Matlab function freqz[X,w] = freqz(x,1,N); % computation of the signal spectrum
mag = abs(X); % magnitude spectrum
phase = angle(X); % phase spectrum
Displaying the results =
figure
stem(0:L-1,x),
title('Signal {x[n]}')
xlabel('Time index n'), ylabel('x[n]')
figure
plot(w/pi,mag),
title('Magnitude Spectrum')
xlabel('Normalized frequency \omega/\pi'), ylabel('|X(e^{j\omega})|')
figure
plot(w/pi,unwrap(phase)),
title('Phase Spectrum')
xlabel('Normalized frequency \omega/\pi'), ylabel('\phi(\omega)')
disp('END OF EXAMPLE 1.1')
1.2 Discrete Fourier Transform (DFT)
Remainder
Definition of the discrete Fourier transform (DFT):
, 
where 
With substitution 
DFT is presented in the form:
DFT is presented in the form:
, Example 1.2
In this example, we use function fft to compute DFT sequence 
for the test signal composed of three sinusoidal components .
for the test signal composed of three sinusoidal components .clear all close all
n = 0:63; % Time index n
Introducing test signal x[n]
x=sin(2*pi*4*n/64)+0.6*sin(2*pi*8*n/64) + 0.8*sin(2*pi*18.5*n/64);
figure
stem(n,x)
xlabel('Time index n'), ylabel('x[n]')
title('Signal x[n]')
axis([0,63,-3,3])
Computing Discrete Fourier Trnsform X[k]
X = fft(x); % Computation of DFT
k = n; % Frequency index k
figure
stem(k,abs(X))
xlabel('Frequency index k'), ylabel('|(X[k])|')
title('DFT sequence')
axis([0,63,0,40])
disp('END OF EXAMPLE 1.2')
1.3 Linear Time-Invariant (LTI) System
Computation of frequency response and pole-zero plot
Remainder
Transfer function of an LTI system 
Frequency response
Magnitude response: 
Phase response 
Group delay 
Example 1.3
LTI system example: Design and analysis of the 5th order Chebyshev filter by using function cheby1 for design.
clear all, close all
Designing the transfer function
[B,A] = cheby1(5,1,0.4); % 5th order Chebyshev filter
Computing frequency response 
[H,f] = freqz(B,A,250,2); % Frequency response
Mag=abs(H); % Magnitude response
Phase = unwrap(angle(H)); % Phase response
[Gd,f] = grpdelay(B,A,250,2); % Group delay
Displaying the results
figure
zplane(B,A),text(1,0.9,'(a)') % Pole-zaro plot
title('Pole-zero locations')
figure
plot(f,Mag), axis([0,1,0,1.1]),text(0.8,0.97,'(b)')
xlabel('\omega/\pi'), ylabel('Magnitude')
title('Magnitude response')
plot(f,Phase), axis([0,1,-8,0]),text(0.8,-1,'(c)')
xlabel(' \omega/\pi'), ylabel('Phase, rad')
title('Phase response')
plot(f,Gd), axis([0,1,0,15])
xlabel('\omega/\pi'), ylabel('Group delay, samples'),text(0.8,13,'(d)')
title('Group delay')
disp('END OF EXAMPLE 1.3')
disp(' END OF CHAPTER I')