PPT On Fourier Series Analysis And It's Properties

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Fourier Series Analysis And It's Properties Presentation Transcript:
1.Signals and Systems (Lab)

2.Fourier series Analysis an its Properties.

3.In Previous Lab…
What is meant by Impulse Response
How to Compute Continuous-Time Convolutions
Analytical way of Computing the convolution
Direct Method
Computation of Discrete-Time Convolution
How to Preserve Indices Information using “conv”
How to construct input or impulse response of a System using De-convolution

4.Introduction to Fourier Series
Complex Exponential Fourier Series Representation
Trigonometric Fourier series Representation
Properties of Fourier series
Linearity
Time Shifting
Time Reversal
Signal Multiplication
Parseval’s Identity
Summary

5.Complex Exponential Fourier Series Representation

6.    Using the complex exponential Fourier series representation, calculate the Fourier Series Coefficients of the periodic signal shown below. Approximate by using 5, 11 and 61 terms of FS coefficients, and compare the results.
    x(t) in one period is given by :

7.Solution:
t1=0:.01:1;
t2=1.01:.01:2;
x1=ones(size(t1));
x2=zeros(size(t2));
x=[x1 x2];

xp=repmat(x,1,5);
t=linspace(0,10,length(xp));
plot(t,xp)
ylim([-0.2 1.2])

8.Approximation with 5 terms
syms t
x= heaviside(t)- 2*heaviside (t-1);

k=-2:2;
t0 =0;  T=2;
w=2*pi/T;

a=(1/T)*int(x*exp(-j*k*w*t),t,t0,t0+T)

xx=sum(a.*exp(j*k*w*t))

ezplot(xx,[0 10])
title('approximation with 5 terms')

9.Approximation with 11 terms

k=-5:5;

a=(1/T)*int(x*exp(-j*k*w*t),t,t0,t0+T);

xx=sum(a.*exp(j*k*w*t));

ezplot(xx,[0 10])

title('approximation with 11 terms')

10.
k=-30:30;

a=(1/T)*int(x*exp(-j*k*w*t),t,t0,t0+T);

xx=sum(a.*exp(j*k*w*t));

ezplot(xx,[0 10])

title('approximation with 61 terms')

11.Trigonometric Fourier series Representation

12.    Using the Trigonometric Fourier series representation, calculate the Fourier Series Coefficients of the periodic signal shown below. Approximate by using 3, 6 and 41 terms of FS coefficients, and compare the results.
    x(t) in one period is given by :

13.
syms t
x= heaviside(t)- 2*heaviside (t-1);

a0=(1/T)*int(x,t0,t0+T);
n=1:2;

b=(2/T)*int(x*cos(n*w*t),t,t0,t0+T);
c=(2/T)*int(x*sin(n*w*t),t,t0,t0+T);

xx=a0+sum(b.*cos(n*w*t))+sum(c.*sin(n*w*t));

ezplot(xx,[0 10])
title('approximation with 3 terms')

14.
n=1:5;
b=(2/T)*int(x*cos(n*w*t),t,t0,t0+T);
c=(2/T)*int(x*sin(n*w*t),t,t0,t0+T);
xx=a0+sum(b.*cos(n*w*t))+sum(c.*sin(n*w*t});
ezplot(xx,[0 10])
title(' approximation with 6 terms')
n=1:40;
subplot(3,1,3)
b=(2/T)*int(x*cos(n*w*t),t,t0,t0+T);
c=(2/T)*int(x*sin(n*w*t),t,t0,t0+T);
xx=a0+sum(b.*cos(n*w*t))+sum(c.*sin(n*w*t));
ezplot(xx,[0 10])
title('approximation with 41 terms')

15. To verify the linearity property , we consider the periodic signals  x(t)=cos(t) and y(t)=sin(2t) and the scalars  and z1=2+3i and z2=2.