Question

3a. Let s(t ) = 20 sinc( 20t ) i. Predict the appearance of S(f) for this time-domain sinc function. ii. Confirm your answer in MATLAB using s = 20*sinc(20*t);

b. Let s(t ) = 20 sinc2 ( 20t ) i. Predict the appearance of S(f) for this time-domain squared sinc function. ii. Confirm your answer in MATLAB using s = 20*sinc(20*t).^2;

c.. Let s(t ) = 20 e − j 30π t rect( 20t ) i. What general property of the Fourier transform describes e − j 30π t s(t ) ↔ ? ii. Predict the appearance of S(f) for this rectangle multiplied by e − j 30π t . iii. Give an exact expression for S(f) by inspection.

Answer 1

a)

for a sinc function the fourier transform is a rectangular function

by the properties of fourier transform

and

with b=20 , a=20

t=-1:1e-2:1;
x=20.*sinc(20.*t);
X=fft(x,1024);
X=[X(513:1024) X(1:512)];
f=linspace(-50,50,1024);
subplot(211);
plot(f,abs(X));
xlabel('frequency');
ylabel('magnitude');
subplot(212);
plot(f,angle(X));
xlabel('frequency');
ylabel('Phase');

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b)

Multiplication in time domain results in convolution in frequency domain.

Convolution of rectangular function results in TRIANGLUAR FUNCTION.

--

t=-1:1e-2:1;
x=20.*sinc(20.*t).^2;
X=fft(x,1024);
X=[X(513:1024) X(1:512)];
f=linspace(-50,50,1024);
subplot(211);
plot(f,abs(X));
xlabel('frequency');
ylabel('magnitude');
subplot(212);
plot(f,angle(X));
xlabel('frequency');
ylabel('Phase');

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c)

where

we have

with a=20, b=20