Lecture 4&5 MATLAB applications In Signal Processing · 2014. 4. 19. · MATLAB applications In...
Transcript of Lecture 4&5 MATLAB applications In Signal Processing · 2014. 4. 19. · MATLAB applications In...
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Lecture 4&5
MATLAB applications
In
Signal Processing
Dr. Bedir Yousif
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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
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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
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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).
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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')
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Fourier Series Solution
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Fourier Series Solution
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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
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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
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Fourier Series
If an input signal xn(t) written in complex F.S
the response at the output of the system is
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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.
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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);
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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')
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Fourier Series Spectrum Result
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Fourier Series Result
dc value of g(t)
ans =
107.5344
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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.
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Fourier Series
Solution
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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)')
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Fourier Series Solution
Approximation of g(t) .
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Fourier Series Fourier Series for Several Periodic Signals
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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
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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
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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
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Properties of Fourier transform
7- Integration in the time domain
8- Multiplication in the time domain
9- Convolution in the time domain
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Properties of Fourier transform
7- Integration in the time domain
8- Multiplication in the time domain
9- Convolution in the time domain
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Thanks