# How do you use Fourier transforms to perform a deconvolution

I've been trying to teach myself about deconvolution through Fourier transforms, but I seem to be missing something simple, as my results are garbage.

I start by defining a test function and a window which I will convolve with it

Window[width_, x_] = UnitStep[x + width/2] UnitStep[width/2 - x];
Test[x_] = UnitStep[x] UnitStep[count - x] Sin[0.1 x]^2;


I then sample the convolution across a set of data points

count=100;

MakeData[width_] =
Simplify[Table[
Integrate[Window[width, x - i] Test[x], {x, -Infinity, Infinity},
Assumptions -> {count > width > 0}], {i, 0, count}]];

tempData = MakeData;


I make a similar list for my sample window

tempWindow = Table[Window[20, x], {x, -count/2, count/2}];


I use the convolution theorem to deconvolve my window from my data

deconvolved = InverseFourier[Fourier[tempData]/Fourier[tempWindow]];


I was expect that this will return the original function

expected = Table[Test[x], {x, 0, count}];


However, the results are completely different. There's obviously something I'm doing wrong in my convolution or deconvolution, but I don't see what it is.

• For starters, you're generating the data by sampling the theoretical result of the convolution but trying to deconvolve some other sampled window... Deconvolutions (noisy) are generally unstable. See this message of mine for an example demonstrating the convolution theorem (a and b are random vectors). – rm -rf Jan 30 '14 at 0:36
• I'm not sure if this question is related to MMA. Anyway, when your window has zeros, as usually happens, the result is not unique and unstable. You can try making your smallest values of Fourier@tempData be really zero, and add some small constant value to the denominator to get zero out of those 0/0. Furthermore, your convolution formula only works because it is even. Otherwise it should say Window[width, x - i]. Furthermore, given that you didn't rotate your window samples, and sampled from -count/2, your expected should be rotated right count/2 – Rojo Jan 30 '14 at 1:19

While it's tempting to attribute the errors you're observing to floating-point errors due to zeros in the DFT of the window, this is actually not the case here:

Window[width_, x_] := UnitStep[x + width/2] UnitStep[width/2 - x];
Test[x_] := UnitStep[x] UnitStep[count - x] Sin[1/10 x]^2;
tempWindow = Table[Window[20, x], {x, -count/2, count/2}];
{Min[Abs[Fourier[tempWindow]]],Max[Abs[Fourier[tempWindow]]]}

(*Out: {0.00455694, 2.08958}*)


Instead, the primary issue is a misuse of the convolution theorem. The inverse DFT of the product of the DFT's of the samples of two functions is NOT the samples of the convolution of the two functions; rather, it is the cyclic convolution of the samples of the two functions, which is an important distinction.

Your code samples the convolution of the two functions (I modified it to be a bit cleaner):

count = 100;
expr=Integrate[Window[width, x - i] Test[x], {x, -Infinity, Infinity}];
tempData = N@Table[expr/.width->20,{i,0,count}];


It then samples the window function, and then attempts to deconvolve the sampled window from the samples of the convolution of the two functions:

deconvolved = Abs[InverseFourier[Fourier[tempData]/Fourier[tempWindow]]];


The result looks like nonsense because tempData is not actually a cyclic convolution of tempWindow with the samples of Test, so you have no reason to expect that the cyclic deconvolution of tempData with the samples of Window will give you the samples of Test. It's a bit of a subtle error, but it's actually very important.

For reference, to demonstrate the cyclic convolution and deconvolution, try this:

tempTest = Table[Test[x], {x, -count/2, count/2}];
tempConv = Re@InverseFourier[Fourier[tempTest] Fourier[RotateLeft[tempWindow,count/2]]];
tempDeconv = Re@InverseFourier[Fourier[tempConv]/Fourier[RotateLeft[tempWindow, count/2]]];
ListPlot[tempTest]
ListPlot[tempConv]
ListPlot[tempDeconv - tempTest]   • What a nice answer & explanation. +1 – ciao Jan 30 '14 at 7:59
• That was an incredibly clear explanation. Thank you. – rprospero Jan 30 '14 at 19:50