Hey guys can anyone explain to me what the Walsh/Hadamard/Fourier Transform actually does and how and when do I use it? Can you also recommend me some textbooks that I can use to help me understand it better? It's impossible to find a clear explanation online. Thanks!

  • $\begingroup$ You might want to look up Arndt's "Matters Computational" or Bracewell's "The Fourier Transform and Its Applications". $\endgroup$ – J. M. ain't a mathematician Apr 8 '13 at 17:53
  • $\begingroup$ Will these books help me to understand how Fourier transform is used in coding theory? $\endgroup$ – user71407 Apr 9 '13 at 6:35

The Fourier transform is a special type of integral transform most commonly defined as \begin{equation*} \hat{f}(\xi)=\int^{\infty}_{-\infty}f(x)e^{-2\pi ix\cdot\xi}dx,~\xi\in\mathbb{R} \end{equation*} that takes a function from real space into Fourier space, and vice-versa. It also changes various operations, such as transforming multiplication into convolution. The Fourier transform $\hat{f}$ is one of the most important operators in mathematics and is used everywhere in analysis. It was derived from the original study of Fourier series when the period blows up to infinity. One important purpose is to solve PDEs, such as the wave equation $u_{tt}=c^2\Delta _{xx} u$.

The Hadamard transform $H_n$ can be considered a generalised class/variant of discrete Fourier transforms. It is a matrix transform, rather than an integral transform that performs various operations (such as involutional operations) on $2^n$ numbers for an $2^n-$dimensional vector space. It is used in quantum computing and data encryption.

Try the following books:

  • "Hadamard Transforms" by S. Agaian, H. Sarukhanyan, D. Fenton and J. Astola

  • "Fourier Series and Integral Transforms" by A.Pinkus and S.Zafrany

  • "Fourier Series " by G. Tolstov

  • $\begingroup$ I want to add to this helpful answer (it would be difficult to be more helpful without writing pages of text) that the Pinkus/Zafrany and Tolstov books do not discuss Walsh-Hadamard transforms, afaics. (Amazon's "search inside" shows no results for "Hadamard", "Walsh", or "Sylvester".) The first book does, obviously. $\endgroup$ – Mars Feb 8 at 20:20

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