![Drawing2.jpg](data:image/jpeg;base64,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)
by Ljiljana Milic
Supplemental material for Chapter I:
1. Single-Rate Signals and Systems: Background Review
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 Example 1.1
In this example, the spectrum of the triangular signal
is computed using function freqz from the Signal Processing Toolbox. Introducing the test signal ![](data:image/png;base64,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)
L = length(x); % length of the sequence
N = 256; % number of DTFT points
Computing Discrete Time Fourier Transform
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 =
xlabel('Time index n'), ylabel('x[n]')
title('Magnitude Spectrum')
xlabel('Normalized frequency \omega/\pi'), ylabel('|X(e^{j\omega})|')
plot(w/pi,unwrap(phase)),
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):
, ![](data:image/png;base64,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)
where ![](data:image/png;base64,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)
With substitution
DFT is presented in the form:
, ![](data:image/png;base64,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)
Example 1.2
In this example, we use function fft to compute DFT sequence
for the test signal
composed of three sinusoidal components . 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);
xlabel('Time index n'), ylabel('x[n]')
Computing Discrete Fourier Trnsform X[k]
X = fft(x); % Computation of DFT
k = n; % Frequency index k
xlabel('Frequency index k'), ylabel('|(X[k])|')
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 ![](data:image/png;base64,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)
Frequency response
Magnitude response: ![](data:image/png;base64,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)
Phase response ![](data:image/png;base64,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)
Group delay ![](data:image/png;base64,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)
Example 1.3
LTI system example: Design and analysis of the 5th order Chebyshev filter by using function cheby1 for design.
Designing the transfer function ![](data:image/png;base64,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)
[B,A] = cheby1(5,1,0.4); % 5th order Chebyshev filter
Computing frequency response ![](data:image/png;base64,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)
[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
zplane(B,A),text(1,0.9,'(a)') % Pole-zaro plot
title('Pole-zero locations')
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')
plot(f,Gd), axis([0,1,0,15])
xlabel('\omega/\pi'), ylabel('Group delay, samples'),text(0.8,13,'(d)')
disp('END OF EXAMPLE 1.3')
disp(' END OF CHAPTER I')