Post on 01-Mar-2021
Lecture 4&5
MATLAB applications
In
Signal Processing
Dr. Bedir Yousif
Signal Analysis
Fourier Series and Fourier Transform
• Trigonometric Fourier Series
Where w0=2*pi/Tp and the Fourier coefficients an and bn are
determined by the following equations
Fourier Series Another Form of trigonometric Fourier Series
a0/2 is the dc component of the series and is the average value
of g(t) over a period.
The total power in g(t) is given by the Parseval’s equation:
And
Fourier Series • Example 1:
• Using Fourier series expansion, a square wave with a period of 2 ms, peak-to peak value of 2 volts and average value of zero volt can be expressed as
Where f0 = 500 Hz if a(t) is given as
Write a MATLAB program to plot a(t) from 0 to 4 ms at intervals of
0.05 ms and to show that a(t) is a good approximation of g(t).
Fourier Series
Solution • clear all
• f = 500; c = 4/pi; w0 = 2*pi*f;
• t=0:0.05e-3:4e-3;
• s=zeros(1,length(t));
• for n = 1: 12
• s = s+c*(1/(2*n - 1))*sin((2*n - 1)*w0*t);
• end
• plot(t,s)
• xlabel('Time, s')
• ylabel('Amplitude, V')
• title('Fourier series expansion')
Fourier Series Solution
Fourier Series Solution
Fourier Series Exponential Fourier Series
The coefficient cn is related to the coefficients an and bn
In addition, cn relates to An and φn of Equations
Fourier Series The plot of |cn| versus frequency is termed the
discrete amplitude spectrum or the line spectrum.
A similar plot of ∠cn versus frequency is called the
discrete phase spectrum
If an input signal xn(t)
passes through a system with transfer function H(w), then the
output of the system yn(t) is
Fourier Series
If an input signal xn(t) written in complex F.S
the response at the output of the system is
Fourier Series Example 2
For the full-wave rectifier waveform shown in Figure,
the period is 1/60 s and the amplitude is 169.71 Volts.
(a) Write a MATLAB program to obtain the exponential
(b) Fourier series coefficients cn for n = 0,1, 2, .. , 19
(b) Find the dc value.
(c) Plot the amplitude and phase spectrum.
Fourier Series Solution
% generate the full-wave rectifier waveform
f1 = 60;
inv = 1/f1; inc = 1/(80*f1); tnum = 3*inv;
t = 0:inc:tnum;
g1 = 120*sqrt(2)*sin(2*pi*f1*t);
g = abs(g1);
N = length(g);
% obtain the exponential Fourier series coefficients
num = 20;
for i = 1:num
for m = 1:N
cint(m) = exp(-j*2*pi*(i-1)*m/N)*g(m);
Fourier Series Solution end
c(i) = sum(cint)/N;
end
cmag = abs(c); cphase = angle(c);
%print dc value
disp('dc value of g(t)'); cmag(1)% c0
% plot the magnitude and phase spectrum
f = (0:num-1)*60;
subplot(121), stem(f(1:5),cmag(1:5))
title('Amplitude spectrum')
xlabel('Frequency, Hz')
subplot(122), stem(f(1:5),cphase(1:5))
title('Phase spectrum')
xlabel('Frequency, Hz')
Fourier Series Spectrum Result
Fourier Series Result
dc value of g(t)
ans =
107.5344
Fourier Series
Example .3
The periodic signal shown in Figure
(i) Show that its exponential Fourier series expansion can be
expressed as
(ii) Using a MATLAB program, synthesize g(t) using 20 terms, i.
Fourier Series
Solution
Fourier Series Solution % synthesis of g(t) using exponential Fourier series expansion
dt = 0.05;
tpts = 8.0/dt +1;% No. of points on time axis
cst = exp(2) - exp(-2);
for n = -10:10
for m = 1:tpts
g1(n+11,m) = ((0.5*cst*((-1)^n))/(2+j*n*pi))*(exp(j*n*pi*dt*(m-1)));
end
end
for m = 1: tpts
g2 = g1(:,m);
g3(m) = sum(g2);
end
g = g3';
t = -4:0.05:4.0;
plot(t,g)
xlabel('Time, s')
ylabel('Amplitude'); title('Approximation of g(t)')
Fourier Series Solution
Approximation of g(t) .
Fourier Series Fourier Series for Several Periodic Signals
Fourier transform
Fourier Transform formula:
Inverse Fourier Transform formula:
If g(t) is continuous and nonperiodic, then G(f) will be continuous
and periodic. However, if g(t) is continuous and periodic,
then G(f) will discrete and nonperiodic; that is
Fourier transform
Complex exponential Fourier coefficient:
Properties of Fourier transform
1- Linearity
Ag1 (t) +bg2 (t) ⇔ aG1(f) + bG2(f)
Where a and b are constants
2- Time scaling
Properties of Fourier transform
3- Duality
G(t) ⇔ g(−f )
4- Time shifting
g(t−t )⇔ G( f ) exp(−j2Π ft )
5- Frequency Shifting
exp(j2 fc t)g(t) ⇔ G(f -fc )
6- Differentiation in the time domain
Properties of Fourier transform
7- Integration in the time domain
8- Multiplication in the time domain
9- Convolution in the time domain
Properties of Fourier transform
7- Integration in the time domain
8- Multiplication in the time domain
9- Convolution in the time domain
Thanks