MATLAB: FFT and IFFT Problem (numerically and analytically result is not same)

convolutiondeconvolutionfftfilterifftMATLABnoise

Hello every one i stuck in one problem i have two signal first one is h(t)=exp(-a*t) and other one is x(t)=1-exp(-a*t). i convoluted these two function and estimated y(t)= h(t) convolution x(t) or simply multiplication in frequency domain. my numerical estimation of y(t) is not same as i got analytical expression of these two function convolution can any one help me what is problem in numerically estimation in matlab. is it matlab fft and ifft problem? my code is also attached with this
ts=1e-4;
point=2000;
t=ts*(0:point-1);
fs=1/ts;
f1=fs*(0:point-1)/point;
a=4e2;
h_t=a*exp(-a*t);
b=3e2;
x_t=1-exp(-b*t);
fft_x=fft(x_t)*ts;
fft_h=fft(h_t)*ts;
y_t=real(ifft(fft_h.*fft_x)/ts);
figure(1) 
subplot(3,1,1)
plot(t,x_t)
subplot(3,1,2)
plot(t,h_t)
subplot(3,1,3)
plot(t,y_t)
and analytical expression of y(t) is 1+1/(-b+a)*(-exp(-b*t)*a+b*exp(-a*t));

Best Answer

This is caused by side effects of digital domain
  1. You are calculating circular convolution instead of normal one. To make convolution via fft/ifft you have to use zero padding.
  2. After convolution you will get vector longer than those first two vectors. Because your vector x(t)=1-exp(-a*t) should contain values near 1 when we are counting towards infinity and we used zero padding to be able to convolve via fft it is wise to abandon some of this data.
  3. No need to multiply and then divide by ts.
ts=1e-4;
point=2000;
t=ts*(0:point-1);
fs=1/ts;
f1=fs*(0:point-1)/point;
a=4e2;
h_t=a*exp(-a*t);
b=3e2;
x_t=1-exp(-b*t);
%%fft convolution
n = 2*point-1; 
y_t = real(ifft(fft(x_t, n).*fft(h_t, n)));
y_t = y_t(1:point)*ts;
figure(1) 
subplot(3,1,1)
plot(t,x_t)
subplot(3,1,2)
plot(t,h_t)
subplot(3,1,3)
plot(t,y_t)
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