For math challenged - what is the Walsh-Hadamard transform?

Randy Yates wrote:
> Richard Owlett <rowlett@atlascomm.net> writes:
> 
>>[...]
> 
> 
> Richard,
> 
> Take what you read in this thread with a grain of salt. You
> are treading on ivory-tower feet. 

I'm not sure what to salt ;)

> 
> Everything that's being said can be broken down into common
> sense, but I see few people here doing so. Granted, it may
> take a bit of explaining as well.

There's terms I don't understand. When I look them up I come across 
terms (associative, commutative, eigen... etc) that I've not had cause 
to use in 40+ years.

Hi.

> Unlikely.
> Rader put it nicely in his bookreview ( IEEE ASP August 1984 )
> on the book by Elliott, Rao " Fast Transforms ..." Academic Press 1982:
>
> "With the next chapter begins the description of the more exotic
> species of transform. The first of these is the Walsh transform.
> As with real biological species it is well to beware.
> New species are often named by biologists, unaware that they have
> renamed existing species. Thus we have the Walsh transform, the
> Hadamard transform and the BIFORE transform which are all identical.
> This reviewer prefers to think of all these transforms as a special
> case of the multidimensional DFT. With log2N dimensions and two points
> per dimension the DFT is a Walsh transform. With four points per
> dimension we find the generalized transforms of the following chapter.
> The authors prefer to introduce all these transforms as discretized
> versions of continous transforms using some rather contrived basic
> functions. The principal deficiency of all these transforms has been
> their poor fit to the real-world application areas. All the
> transforms avoid multiplications but the price paid is that such
> natural waveform properties as bandwidth, time invariance and
> frequency are replaced by analogous but unnatural properties
> appropriate to the contrived basis functions.
> In spite of the dearth of ready applications the books extensive
> bibliographie should permit the reader to locate Walsh transform
> applications in his own interest area."

The Hadamard transform made a fantastic difference to a reverb algorithm I 
worked on, and I believe it's also used for image recognition in neural 
networks. But it's cool uses seem to be in CDMA digital wireless 
technologies, giving each phone user a different matrix, see here: 
http://www.patentstorm.us/patents/6829289-description.html

VC

Hello,

CDMA in a nutshell:

Take several orthogonal sequences, for example

11111111
11110000
11001100
..

multiply each with a number (the information) and add.
Transmit the sum through the wireless channel.

At the receiver side I can reconstruct the information "numbers" by
matching up (correlating) with the known sequences.

Now that's too easy... in reality, for example WCDMA specifications are
measured in shelf meters. So there is plenty of fungus on the leaves...

Cheers

Markus