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Can a non-periodic function be padded at the boundaries and deconvolved with inverse FFT?

Since a Toeplitz matrix can be embedded in a circulant matrix to perform the deconvolution, is there an analogy with the inverse FFT?

Clarification

Not sure if there are other types of deconvolutions but expressed in the frequency ($\omega$) domain, would be the product of the input function $G$ and response function $F$ to get convolved function $H$,

\begin{equation} G(\omega) = \frac{H(\omega)}{F(\omega)} \end{equation}

It is assumed that $G$ and $F$ are periodic, but I am wondering about the case when they are not.

Rodrigo de Azevedo
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hatmatrix
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  • You might want to explain a bit more about what you mean by "deconvolution". – Igor Rivin Jan 12 '14 at 00:53
  • Sorry, wasn't aware there were many definitions, but I added clarification. – hatmatrix Jan 12 '14 at 01:09
  • I still don't understand what you mean, what is the input, two finite sequences $h,f$, and you want a 3rd one such that $h\approx f \ast g$ where $\ast$ is the non-circular convolution ? – reuns Dec 21 '19 at 05:22
  • Imagine one of the functions is a sinc function $f(\vec{x})=\text{sinc}(\vec{x})$ so its Fourier Transform is going to be the circular function $F(\vec{\rho})=\text{circ}(\vec{\rho})$... How you could treat the fraction $\frac{H(\vec{\rho})}{F(\vec{\rho})}$ on the points where the circular function is zero? – Joako Jun 06 '22 at 22:10
  • Yes you can do it like that. But it will be noise sensitive. Also you will need to avoid to divide by zeroes. – mathreadler Feb 20 '24 at 05:26

2 Answers2

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The discrete Fourier transform assumes a discrete and periodic signal. For a discrete non-periodic signal, the inverse of the Fourier series pertains (the transform is a periodic, continuous function). Then you can probably use the second convolution theorem. https://en.wikipedia.org/wiki/Convolution_theorem#Convolution_theorem_for_Fourier_series_coefficients

But I am not sure that the FFT algorithm remains applicable.

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I believe that this is one of the standard methods, but there are stability issues. Check out the Wikipedia article on Wiener deconvolution.

Igor Rivin
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  • Wiener deconvolution creates an inverse filter in the frequency domain when there is noise in the measurements, but doesn't make any explicit statement about periodicity (but canonically assumes periodicity according to Fourier transforms). – hatmatrix Jan 12 '14 at 01:19
  • @crippledlambda Wiener deconvolution makes no statement about periodicity, which is precisely why I mentioned it: I thought you did not want to make any periodicity assumptions. – Igor Rivin Jan 12 '14 at 01:21
  • if you don't make any assumptions, by default you will be working with the same assumption as ordinary DFT/FFT. – hatmatrix Jan 12 '14 at 01:30