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Walsh Functions - Introduction

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  • Getting Started

    To get you started, check out some of the maths associated with Walsh functions at The CRC Concise Encyclopedia of Mathematics page on Walsh Functions. Please note that the page on Hadamard matrices has some errors on it, namely the pictures of Hadamard matrices are wrong. The diagrams with numbers are, however, correct. Also note on the Walsh function page the definitions of Cal and Sal are incorrect, they should be:
    Cal(n,k) = W(n,2k)
    Sal(n,k) = W(n,2k-1)

  • What are Walsh Functions?

    We all know (I hope) that we can both describe and synthesize any real, finite waveform using a set of sinewaves (Well, to be more precise, closely approximate since in reality we limit ourselves to a finite set of functions). This is basically what Fourier proposed in his (in)famous Fourier analysis and series. What makes this analysis and synthesis possible is that each member of the series is orthogonal to the other - each one is mathematically unique to each other in the set.

    Well, in 1923 a mathematician called J.L.Walsh formally documented another set of orthogonal functions, which have since been named after him. Again, this set of orthogonal functions can be used for both analysis and synthesis of signals. But this time, the individual Walsh functions look very much like digital signals, as they only occupy two values: +1 and -1.

    Now, mathematically Walsh functions are discrete valued, the upshot of this being that there are discontinuities - where the value suddenly jumps from +1 to -1 or vice versa. This makes the mathematics a bit hairy if you're used to the neat operations you can do with sine waves.

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