






 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,2k1)
 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.





Copyright © 20012021 Neil Johnson


  
