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na.numerical evaluation – Deconvolution utilizing the discrete Fourier remodel Answer

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na.numerical evaluation – Deconvolution utilizing the discrete Fourier remodel

It is a well-known property of convolution, that convolution of two steady features is a multiplication within the frequency province i.e., $y$(t)=$a$*$b$ turning into a multiplication after Fourier remodel: F(y(t))= F($a$)F($b$). This was well-known by early 1900s and clearly talked about in 1941 as mentioned in earlier questions right here.

In sensible functions, deconvolution is too fascinating. In deconvolution, two features are divided within the Fourier province to regain the unique duty, say $a$, if $y$(t) and $b$(t) are recognized. For instance, if we want to regain $a$, we are able to divide F($y$(t)) by F($b$) and do an inverse remodel to get $a$. It could not breathe a mathematically rigorous route, however it’s a well-liked approach in spectroscopy from an empirical perspective.

In majority of the instrumental functions, the duty is a discrete and downside is to execute deconvolution utilizing Discrete Fourier Transform. Now the purpose is that within the DFT, the multiplication of the DFT of two features relent a round convolution, not a linear one.

However, one can extract the undoubted linear convolution by making certain that the size of the DFT equal to 2N-1, the place N is the size of the information factors contained within the discrete duty.

Illustration Suppose $x_{1}=[underline{1}, 2,3]$ and $x_{2}=[underline{1}, 1,1] .$ We can compute the linear convolution as
$$
x_{3}[n]=x_{1}[n] * x_{2}[n]=**[underline{1}, 3,6,5,3]**
$$

If we as an alternative compute
$$
x_{3}[n]=operatorname{IDFT}_{N}left(operatorname{DFT}_{N}left(x_{1}[n]privilege) cdot operatorname{DFT}_{N}left(x_{2}[n]privilege)privilege)
$$

we get
$$
x_{3}[n]=left{commence{array}{ll}
{[underline{6}, 6,6]} & N=3
{[underline{4}, 3,6,5]} & N=4
**{[underline{1}, 3,6,5,3]} & N=5**
{[underline{1}, 3,6,5,3,0]} & N=6
aim{array}privilege.
$$

Linear convolution and DFT convolution match when the size of the DFT is not less than 2N-1, which is 5 on this illustration.

Question: Do we necessity to use the identical size restriction of 2N-1 dimension of information when want to divide within the frequency province or not, in different phrases attempting to do a linear deconvolution utilizing DFT?

I searched loads of locations, books and papers however couldn’t discover a limpid requirement in regards to the size of DFT within the illustration of division/deconvolution. Thanks

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