Deconvolution using FFT - a classical problem

Hello friends, I am new to signal processing and I am trying to achive deconvolution using FFT. I have an input step function u(t) applied to an impulse response given by . The output function is . I am trying to convolve g and u to get y as well as deconvolve y and g to get u. However, I quite cannot get the right answers. I understand that the deconvolution process is ill-posed and I have to use some kind of normalization process but I am lost. I also apply zero padding to twice the length of the input signals. Any sort of guidance will be appreciated.

After using deconvolution in the fourier domain:

Y = fft(y)

G = fft(g)

X = Y./G

x = ifft(X)

I am getting an output shown below:

Which is not the expected outcome. Can someone shead light on what is happening here? Thank you.

Answer 1

[Matt J]

dt=0.001;
N=20/dt;

t= ( (0:N-1)-ceil((N-1)/2) )*dt; %t-axis

u=(t>=0);
g=3*exp(-t).*u;
y=conv(g,u,'same')*dt;

Y = fft(y);
G = fft(g);
X = Y./G;
x = fftshift(ifft(X,'symmetric')/dt);

figure;
sub=1:0.3/dt:N;
plot(t,3*(1-exp(-t)).*u,'r.' , t(sub), y(sub),'-bo'); 
xlabel t
legend Theoretical Numeric Location northwest
title 'Output y(t)'

figure;
plot(t, u,'r.' , t(sub), x(sub),'-bo'); ylim([-1,4])
title Deconvolution
xlabel t
legend Theoretical Numeric Location northwest

[Vijayananda]

Thank you so much Matt. This was insightfull. When I just change the time axis to t= (0:N-1)*dt this, my whole result changes. I want the time to start from 0 since negative time is not physical. However, the results should not vary right? What could be the reason?I changed nothing else other than the time.

I also pad my data to twice the length of the array and time interval dt is aorund e-6. Will this create any issues?

[Matt J]

FFTs and IFFTs inherently integrate over both positive and negative times and frequencies. So, no, you cannot define the t-axis arbitrarily. But you can just delete the unwanted time points from the final result if you want.

As an aside, it is strange that you would be processing causal signals with FFTs. Usually FFTs are for non-causal, finite duration signal processing, whereas infinite, causal signals are usually processed with z-transforms.

[Vijayananda]

You are right. I am trying to deconvolve a heat transfer problem. The time is finite from 0 to 1 with an interval of 1 microseconds. I also zeropad the impulse response and the output data to twice their length to negate the periodicity of teh FFT.

[Matt J]

I also zeropad the impulse response and the output data to twice their length to negate the periodicity of teh FFT.

Zero-pad the impulse response and the input data, I think you meant to say.

That is effectively the same as my original answer, although I am zero-padding on the left, instead of the right. In other words, you can interpret the zeros for t<=0 that you say you don't want as an alternative implementation of zero-padding.