# Division in the Fourier Domain (Deconvolution) - How to Handle Lengths of the Signals

In order to avoid circular convolution $$y(t)$$ of two functions say $$u(t)$$ and $$v(t)$$ in Fourier transforms, the data length must be at least (length $$u(t)$$)+length($$v(t)$$)$$-$$1. If we are interested in deconvolution using the Fourier transform, which may require division of Fourier transform of $$y(t)$$ by the Fourier transform of $$v(t)$$ to obtain $$u(t)$$, do we need to maintain the same length requirements in a mathematically rigorous sense?

All the papers surveyed so far on deconvolution by Fourier methods in the field of spectroscopy never talk about it or mention the issue of circular convolution. Understandably, they don't care about the edge-effects and experimentally there are always excessive data points near the edges which can be discarded. I wanted to confirm that theoretical requirement of data length for deconvolution (division in the Fourier domain) is the same as convolution or not. Thanks.

• Division is same as another kind of multiplication by the following logic : $Y(\omega)./H(\omega) = Y(\omega).*(1/H(\omega))$. So I think rules are the same. – jithin Mar 27 at 15:01
• I was thinking along these lines but I cannot see this discussed anywhere. If we are multiplying by a reciprocal 1/H(w), what will correspond in the time domain? How should we notate undoing convolution in the time domain? – M. Farooq Mar 27 at 16:11
• By not going into too much details but usually the inverse of FIR filter (Convolution) is IIR Filter. hence to get the exact inverse Filter in FIR form might be unreachable. Regarding your concern about Edge effects - You're right. Hence it is better to solve inverse imaging problem in Spatial Domain or do some tricks in Frequency Domain (Which will make it slower hence no point). – Royi Mar 27 at 16:36
• I recommend reading Chapter 6 in R. de Levie, "Advanced Excel for scientific data analysis", 3rd Ed., Atlantic Academic LLC, Maine, USA, ©2012. Lots of discussion of the pitfalls of deconvolution (even without noise) and ways to deal with them, including Wiener and von Hann filtering, iterative van Cittert deconvolution, and lots more. I have only skimmed the chapter, but it is worth a close read. – Ed V Mar 27 at 17:19
• @EdV Thank you. I was looking a simple version of van Cittert method as well. Good to know it is there. Will check this book. – M. Farooq Mar 27 at 17:22