Smoothness and frequency content m dash The smoothness of the signal determines

Question

Smoothness and frequency content m dash The smoothness of the signal determines the frequency content of its spectrum. Consider the signals x(t) = u(t + 0.5) - u(t - 0.5), y(t) = (1 + cos(Pit))[u(t + 0.5) - u(t - 0.5)] Plot these signals. Can you tell which one is smoother? Find X( 2)and carefully plot its magnitude |X(Ohm) | vs frequency Ohm Find (use the Fourier transform properties) and carefully plot its magnitude |y(Ohm)| vs frequency Ohm. Which one of these two signals has higher frequencies? Can you now tell which of the signals is smoother? Use MATLAB to decide. Make x(t) and y(t) have unit energy. Plot 20 log_10 |y(Ohm)|and20 log_10 |X(Ohm)| using MATLAB and see which of the spectra shows lower frequencies. Use Fourier function to compute Fourier transforms.

Explanation / Answer

Plot the given signals and their frequency spectrums.

Software used:

Algorithm:

Matlab programs:

%Program to generate Exponential Fouries series of a periodic Fullwave rectified signal

clc;close all;clear all;

t=-3*pi:0.001:3*pi;

% periodic rectangular pulse

A=20;

w=1;

y=A*sin(w*t);

figure; plot(t,abs(y));

title('signal')

% number of harmonics in the series

N=25;

%generate fourier coefficients and basis functions

for i=1:N

    %y1(0)=2*A/pi;

    cp(i)=2*A/(pi*(1-4*(i-1)^2));

    cn(i)=2*A/(pi*(1-4*(-i)^2));

    cn1=fliplr(cn);

    y2(i,:)=cp(i)*(exp(j*2*w*(i-1)*t));

    y3(i,:)=cn(i)*(exp(-j*2*w*(i)*t));

end

%plotting of Discrete fourier spectrum

    figure;stem([-N:N-1],[cn1,cp]);

    title('Discrete fourier spectrum Cn')

% reconstruction of the signal from coefficients

    y4(1,:)=sum(y2)+sum(y3);

    figure;plot(t,y4)

    title('reconstructed rectangular signal from coefficients')

Output:

% Fourier transform of a signal

clc;close all;clear all;

syms t w

% input signal x(t)

x=exp(-2*t).*heaviside(t);

subplot(3,1,1);ezplot(x);

title('input signal')

% fourier transform

disp('the fourier transform of x(t) is')

X=fourier(x); X=simplify(X);

% frequency response

subplot(3,1,2);ezplot(abs(X));

title('Magnitude response of Fourier transform ')

subplot(3,1,3);ezplot(atan((imag(X))/(real(X))))

title('phase response of Fourier transform ')

Output:

   

Result:

*Exponential Fourier series of fullwave rectified signal have been studied and estimated.

* Fourier transform of the given signal along with frequency spectrum was studied.

Note : fourier transform is used to conver the timedomain continious signal in to the frequency domain.

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