Lecture 4&5 MATLAB applications In Signal Processing. Dr. Bedir Yousif

Transcription

1 Lecture 4&5 MATLAB applications In Signal Processing Dr. Bedir Yousif

2 Signal Analysis Fourier Series and Fourier Transform Trigonometric Fourier Series Where w0=2*pi/tp and the Fourier coefficients a n and b n are determined by the following equations

3 Fourier Series Another Form of trigonometric Fourier Series a 0 /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

4 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 f 0 = 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).

5 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')

6 Fourier Series Solution

7 Fourier Series Solution

8 Exponential Fourier Series Fourier Series The coefficient c n is related to the coefficients a n and b n In addition, cn relates to A n and φ n of Equations

9 Fourier Series The plot of c n 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 x n (t) passes through a system with transfer function H(w), then the output of the system y n (t) is

10 Fourier Series If an input signal x n (t) written in complex F.S the response at the output of the system is

11 Fourier Series Example 2 For the full-wave rectifier waveform shown in Figure, the period is 1/60 s and the amplitude is 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.

12 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);

13 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')

14 Result Fourier Series Spectrum

15 Result Fourier Series dc value of g(t) ans =

16 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.

17 Solution Fourier Series

18 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)')

19 Solution Fourier Series Approximation of g(t).

20 Fourier Series Fourier Series for Several Periodic Signals

21 Fourier Transform formula: Fourier transform 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

22 Fourier transform Complex exponential Fourier coefficient: Properties of Fourier transform 1- Linearity Ag 1 (t) +bg 2 (t) ag 1 (f) + bg 2 (f) Where a and b are constants 2- Time scaling

23 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 f c t)g(t) G(f -f c ) 6- Differentiation in the time domain

24 Properties of Fourier transform 7- Integration in the time domain 8- Multiplication in the time domain 9- Convolution in the time domain

25 Properties of Fourier transform 7- Integration in the time domain 8- Multiplication in the time domain 9- Convolution in the time domain

26 Thanks