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I have performed IFFT for converting frequency to time domain and back to frequency domain. I am getting same results also. However in time domain plot, my response should start with zero and I am getting slighly higher value at time zero.

Here is my code:

clear all

close all

clc

a=0.5;eta=0.3;beta=0.025;wb=14.25;

% Define time and frequency axis variables

fs =400; % samples/s

N = 1024; % number of points

dt = 1 / fs; % s, time step

t = (0:N-1)*dt; % s, time axis

df = 1 / N / dt; % Hz, frequency step

f = (-N/2:N/2-1)*df; % Hz, frequency axis

% Define function

y=sin(pi.*a).*eta.*(1+exp(-1i.*pi.*f./eta))./((-f.^2+(2.*1i).*f.*beta+1).*(eta.^2-f.^2));

%plot original signal

figure, plot(f,y),title('Frequency response');ylabel('Response of beam');xlabel('Frequency'); % plot initial time domain signal

%conversion to time domain

y2 = ifft(ifftshift(y));

figure, plot(t,(y2)),title('Time response');ylabel('Response of beam');xlabel('Time');

axis([0 2.5 -0.05 0.05]); % time domain signal after IFFT

%back to original signal

%y3=fftshift(fft(y2));

%figure,plot(f,y3),title('Frequency response');ylabel('Response of beam');xlabel('Frequency');

Any Help is highly appreciated.

Thanks

David Goodmanson
on 31 Mar 2021

Edited: David Goodmanson
on 31 Mar 2021

Hi Susmita,

A couple of things going on here. First, it's good to use more points. Use a lot more points. There is no need to go with something as small as 1024 and there is also no need to go with a power of 2, the importance of which, in my opinion, is a myth. I used a nice even million points in the code below.

Second, in the expression for y there is a factor of (eta^2-f.^2) in the denominator. That puts poles at +-eta on the real frequency axis. The path of integration runs right over them, so by implication you are taking the principal value of the integral. The p.v. is the same as the average value of two functions (y3 and y4 below) where the poles are moved slightly off of the real axis in either the plus or minus i direction. The small parameter gamma accomplishes that.

clear all

close all

clc

a=0.5;eta=0.3;beta=0.025;wb=14.25;gamma = 1e-10; % introduce gamma

% Define time and frequency axis variables

fs =400; % samples/s

N = 1e6; % number of points

dt = 1 / fs; % s, time step

t = (0:N-1)*dt; % s, time axis

df = 1 / N / dt; % Hz, frequency step

f = (-N/2:N/2-1)*df; % Hz, frequency axis

% Define function

y3=sin(pi.*a).*eta.*(1+exp(-1i.*pi.*f./eta))./((-f.^2+(2.*1i).*f.*beta+1).*(eta.^2-i*f*gamma-f.^2));

y4=sin(pi.*a).*eta.*(1+exp(-1i.*pi.*f./eta))./((-f.^2+(2.*1i).*f.*beta+1).*(eta.^2+i*f*gamma-f.^2));

y = (y3+y4)/2;

%plot original signal

figure(1)

% plot initial time domain signal

plot(f,y),title('Frequency response');ylabel('Response of beam');xlabel('Frequency');

grid on

%conversion to time domain

y2 = ifft(ifftshift(y));

figure(2)

plot(t,(y2)),title('Time response');ylabel('Response of beam');xlabel('Time');

axis([0 2.5 -0.05 0.05]); % time domain signal after IFFT

grid on

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