[SciPy-Dev] Fast Walsh–Hadamard transform for SciPy ?

[Chaman Agrawal]

Issue #8590 https://github.com/scipy/scipy/issues/8590

Currently there is no implementation of Fast Walsh–Hadamard transform in
SciPy. Although it seems that FWHT is not as general as FFT but it is
pretty ubiquitous ,it is there is other maths and science softwares like
MATLAB etc. .I would like to contribute towards it. Following are the
details about it.
Fast Walsh–Hadamard transform

Hadamard transform is an example of a generalized class of Fourier
transforms. It performs an orthogonal, symmetric, involutive, linear
operation on 2^(m) numbers. It is equivalent to a multidimensional DFT.
Time Complexity:

O(nlogn) with Fast Walsh-Hadamard transform (FWHT)
Comparison with FFT:

FWHT is very useful for reducing bandwidth storage requirements and
spread-spectrum analysis. Compared to the FFT, the FWHT requires less
storage space and is faster to calculate because it uses only real
additions and subtractions, while the FFT requires complex values. The FWHT
is able to represent signals with sharp discontinuities more accurately
using fewer coefficients than the FFT.
Some Usage examples:

The Hadamard transform is used in data encryption, as well as many signal
processing and data compression algorithms, such as JPEG XR and MPEG-4 AVC.
It is also a crucial part of Grover's algorithm and Shor's algorithm in
quantum computing. The Hadamard transform is also applied in scientific
methods such as NMR, mass spectroscopy and crystallography.
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