






 Over to the engineers...
To a hardware engineer, a waveform with only two levels looks very much like
a digital signal, and straight away ideas begin to form about how to do
analysis and synthesis in hardware, viz:
 Analysis involves integrating the products of the signal with
a set of Walsh functions. Integration is really just a big summation,
and with Walsh functions being +1 or 1 the multiplication becomes addition
and subtraction.
 Synthesis is almost identical  sum the products of a set of
Walsh functions and their coefficients. Again, the multiplication (by +1 or 1)
can be turned into addition or subtraction.
 The Coefficient Multiplier
Right, we know that to synthesize a waveform using Walsh functions we need
some sort of adder/subtractor, to be controlled by a digital signal. And,
just to add some interest to the sound we create, we also want to dynamically
change the value of the coefficient as we play.
All of this can be done with an analogue switch and an opamp:
With one of these for each coefficient, we then sum all of the outputs together
to create the final waveform, ready for further processing by our beloved
VCFs and VCAs.
 The Digital Revolution
The advent of cheap digital technology now makes Walsh function synthesis even
more exciting. Instead of doing all the maths in the analogue world it can all
be done in the nice, clean, antiseptic digital world, where noise is a thing
of the past.
 Actually doing it
Most synthesizer builders know a bit about additive synthesis: creating a new
waveshape from basic building blocks. Well, so far the only mainstream technique
for this was to use Fourier, based on sinewaves. Nice in theory, but hell in
practice.
Now, lets look at the implementation of a Walsh function synthesizer. In
particular I will only consider the generation of the audio waveform, and not
further signal processing modules (filters, envelopes, amplifiers, etc).
Also, the circuits presented in this discussion are conceptual circuits meant
to inspire designers to take these ideas and create proper working circuits from
them. I make no apology if they seem a bit crude; they are for illustration
purposes only. Right, with that out of the way, let's begin.
As I see it there are four main stages in creating waveforms with Walsh functions:
 generate clock signal from pitch data
 generate Walsh functions from clock signal
 multiply Walsh functions by coefficients
 sum all products to create resultant signal
These stages will be described in a different order to the one above, and once you
have read them through, you will understand why.
 Generate the Walsh functions
The core of the Walsh function synthesizer is the Walsh function generator. We use
the clock signal to generate all of the required Walsh functions using a mass of
logic gates and counters. But first, a bit of theory.
Since we cannot create a synthesizer which used all Walsh functions, we must
limit ourselves to a workable subset. Obviously, the more Walsh functions you
use the more accurately you can synthesize a waveform, but the more hardware and/or
software you need  it's a tradeoff. Bear in mind the number of Walsh functions used
must be a power of 2 (2, 4, 8, 16, 32, 64, etc) and you can quickly use lots
of hardware for creating waveforms. You gain about 6dB SNR for each doubling
of the number of Walsh functions you use.
So, as an illustration we might consider an 8term Walsh function synthesizer. The
theory discussed below applies equally well to a 16term, 32term , 64term, etc
synthesizer, it just makes the diagrams a bit easier to read. The first new bit
of information to absorb is the Rademacher function. Not as frightening as it
sounds, this is actually a square wave, where each next member of the set has
a frequency double that of its predecessor:
The relationship between Rademacher functions and Walsh functions is:
WAL(2^{n}1,t) = R(n,t)
For example, R(4,t) is the same as WAL(15,t)
Now, I bet the engineers amongst you see this not as mathematics, but the output
waveforms of a counter. Yes indeed, a binary counter makes a great Rademacher
function generator. The next step is to go from Rademacher functions to Walsh
functions.
We have seen from the above that some Walsh functions can be derived directly
from Rademacher functions. The others require a bit of modulo2 mathematics 
well, okay, I really mean EXOR gates. Back to theory:
for N = P (+) Q
WAL(N,t) = R(P,t) (+) R(Q,t)
where (+) signifies modulo2 addition
So, what does this mean in practice? In modulo2 mathematics, addition is the
same as subtraction, and cannot be negative. In effect we are using EXOR gates
to do the maths.
So, how do we go about it? Let's make some more of the Walsh functions and you
will soon see. Start with WAL(2,t). We already have WAL(1,t) and WAL(3,t), and
we know, I hope, that 3  1 = 2, so WAL(2,t) = WAL(1,t) (+) WAL(3,t).
Likewise, to make WAL(4,t) we know that 7  3 = 4, so WAL(4,t) = WAL(7,t) (+)
WAL(3,t).
Putting all this together, we have something like:
 Clock Generation
To the left of the above diagram is a binary counter. This generates the basic
clock signals for the Walsh function generator, and is itself driven by a
clock generator.
The fundamental frequency of the final waveform is defined by the frequency of
WAL(1,t). Since this is a fraction of the driving clock signal we need to generate
a clock signal that is some multiple of the desired pitch. Using a synchronous
counter, which ensures all its outputs change state at the same time, we will need
a clock signal with a frequency eight times that of the fundamental, WAL(1,t).
There are two main approaches to generating the clock signal: analogue and digital.
Both approaches have their advantages and disadvantages, and I don't really want to
venture into such a politically sensitive area as the analoguedigital debate.
However, he says as diplomatically as possible, here are a couple of views of both
approaches:
 With an analogue clock source, it is easily interfaced to other analogue
synth components, as it could be driven by the familiar 1V/octave scale. But,
for any reasonable number of Walsh functions, eg. 32, you would need an oscillator
running at 32 times the fundamental.
 A digital clock source can be easily controlled by a microcontroller.
And, it offers greater stability than an analogue oscillator. But, interfacing
to analogue synths is much more difficult, as some micro would be involved. Also,
any modulation scheme would be more complex to design, requiring some involved
programming of said micro.
 Multiply Walsh coefficients
OK, now we have our set of Walsh functions oscillating up and down. The next stage
is to multiply these waveforms by their respective coefficients. A simple approach
to this has already been shown above in the Coefficient Multiplier, using an opamp
and analogue switch. I'm sure there are many other ways of doing this.
 Summing the terms
The final stage in this process is to sum together all the Walsh terms (the products
of the Walsh functions ans their coefficients) together. The easiest way of doing
this is with a multipleinput summing amplifier. Circuits abound for this sort of
thing, so I won't embarass myself and draw such a circuit here.
 Putting it all together
We have covered all the main sections needed to build a basic Walsh function
waveform generator. The clock generator takes information from the keyboard
and creates a clock signal which is some multiple of the required pitch. This
feeds into the Walsh function generator which generates a large set of Walsh
functions. These in turn are multiplied by their respective coefficients to
create a set of Walsh terms, which are all summed together to make the final
waveform.
Once you have the synthesized waveform it is then ready to be further processed
by all manner of signal processing functions: filters, amplifiers, modulators,
whatever  the list is endless.
The diagram below brings together some of the above points, as well as a couple
of refinements for improved signal quality. The latch after the EXOR matrix
resolves logic path timing differences (direct functions, like WAL(3) will settle
before a processed function, like WAL(13) due to the delays introduced by the
EXOR gates). The inverter before the latch makes it operate on the oppposite
edge to the counter, allowing half a clock period for the gates to settle down.
The Sample & Hold on the output allows the analogue coefficient multipliers to
settle down from the abrupt changes due to the Walsh functions.
Some people may choose to skip the latch and run the S/H in antiphase  it is
just a matter of personal choice, and actually building some hardware and
seeing exactly how well it behaves.





Copyright © 20012018 Neil Johnson


  
