[sdiy] Re: Walsh Generator Release!!!

[Magnus Danielson]

From: "Maciej Bartkowiak" <mbartkow at et.put.poznan.pl>
Subject: Re: [sdiy] Re: Walsh Generator Release!!!
Date: Thu, 4 Apr 2002 00:37:58 +0200

> Magnus,

Maciej,

> Are you talking about short-time Fourier transform 
> on windowed data? If so, how could you compare it
> to the continuous integral-based Laplace transform on
> infinite support? It's unfair.

Indeed, that would be unfair. Unfortunately we tend to use the term
Fourier analysis or Fourier transform for both the continous time and
discrete time variants of analysis. We also need to separate between
theoretical analysis (Calculus) and that of practical analysis.

To make an interesting distinction, one usually only refer to the
LaPlace transform for existing in continous time where as the discrete
time variant is called the Z-transform.

So, in reality there exist 4 different transforms, and their
respective inverse transforms. All are naturally part of a much larger
set of transforms called linear transforms, in which Walsh transforms
also fall. Also, for all these transforms you can state that they are
non-lossy, in that they will not loose information. However, this does
not prohibit them from making certain information obstructed such that
it effectively becomes more or less unavailable, and then acts as
distrorsion or noise on the other data.

Now, this is actually part of what I am targeting at, I am stating
that Fourier transform does not have sufficient guts to describe the
amplitude change properties of a signal. The information is there, but
obstructed to an unsuiting way.

There is also the aspect of numerical precission if we talk about
analysis of actual data, may it be in form of a time continous or time
discrete measurement. I think I will leave that out of the discussion
for the moment, since it would mostly add to the confusion I think.

> To make things clear, I am talking about the analytical
> integral-based Fourier transformation which is fully revertible,
> which means all the information of the signal is there, nothing
> is lost in the Fourier domain, and the input signal may be 
> obtained back through the inverse transformation.

Indeed.

> Of course such transformation is not a tool aplicable for
> practical measurements, due to infinite integration time
> (the Laplace transformation shares the same drawback).
> It is perfect for theoretical analysis of signals, though.

Um, not if I do want to capture the non-zero sigma values, ie. time
dependent amplitude. I may not loose any actual data, but I will not
get the information I want in a format I want, and thus the transform
is not well suited anyway. I think my point still stands. If you use
the wrong transform method in relation to what type of information you
are after, you might not be better off than doing without that
transformation. Now, if you do the math correctly you may combine a
time-exponential weighing of a signal with a Fourier transform to
achieve the same result as a LaPlace transform, but then you are doing
the LaPlace transform according to the book, and then the
argumentation is constructed.

> Both Laplace and Fourier transforms are complementary,
> no one is a generalization or simplification of the other. The
> domain of Laplace transform is a complex halfplane, but it
> is limited to signals starting from zero, so you are basically
> limited to the analysis of transients.

Actually, it is not limited to zero, this is just the assumption that
comes with the assumption of the transient. It all comes down to
causuality and this just must assume for the linear system you
analyse. Basically, if you have a system which gets an input signal at
time t, then due to causuality there can be no output of the system
correlating to that event prior to t, i.e. that output must be assumed
to be 0 up to the time t. The time integral from minus eternity to
(but not including) time t can thus be safely assumed to be exactly 0,
and we does not have to integrate over that time and thus we can start
at the time t. For practical reasons the time t has been selected to
be 0. If we do not assume the impulse, but rather views a continous
signal, then we must integrate from minus eternity to eternity, since
it all applies. This happends to rule out the amplitude change in
practice, so we end up with only frequency terms, which comes out to
be the Fourier transform case.

Also, the domain for the LaPlace transform is not a complex halfplane,
it is the full complex plane s, often refered to the s-plane. You can
make the LaPlace analysis for any point in this complex
plane. However, for a stable linear system, you may not place any
poles on the right half-plane, but you may place zeros on any point in
the s-plane. The stability aspects of a linear system does however not
set a limit for the LaPlace analysis itself.

> Fourier is applicable to
> all signals that are absolute integrable, and some others, too.
> But its domain is only the imaginary axis. One cannot argue
> any of them to be better than the other one.

As I have stated, for both you will not loose information, but the
LaPlace is better on analysing the transient case and bringing out
important data than I consider the Fourier transform to be. This does
not at all mean that I *need* to do LaPlace all over the place, you
will find me do Fourier most of the time anyway, but with it comes a
healthy scepsis about the data interpretation from what comes out of
the Fourier analysis.

When I state that the LaPlace transform has stronger guts, I simply
mean that it provides me with a better analytical tool than Fourier,
and when I use the Fourier tool, I must be aware of the limits it
actually have.

> Certainly, the usefullness of any transformation when applied
> to signal analysis depends on our ability to understand and
> properly interpret its output. In case of Fourier, one cannot 
> neglect the phase spectra. I think this is a common mistake
> in engineering practice, and your arguments come from this
> bad experience.

I've allways looked carefully on phase response but also group delay
and dispersion. However, part of my argumentation is also against the
overbeleif in the strength of the Fourier analysis and the underlying
assumption that you can analyse anything with pure sine/cosine waves.
My point is that there is much more than "Frequency response".

I want to know where I have my poles and zeros. When I know that
(assuming a linear system here) everything else comes out of that.
Whenever I deal with audio/acoustics I think of zeros and poles, I
think of zeros and poles in equalizer operations, in the obstruction
of transient sounds, I think of them as in speaker design, I think of
them in speaker couplings, I think of them when I do my RF stuff.

I love the power which Fourier analysis gives me, I have a combined
Network Analyser and Spectrum Analyser here at home to prove it. But
as with any tool, it is useless to me for quality measurements unless
I understand the fundamental limits of the method at hand. May it be
as a physical measurement instrument, theoretical instrument, or
theoretical model of analysis. So, for me it goes beyond the paper and
pencil aspects, it has to do with the full insight and engineering of
any relevant problem and measurement.

Best regards,
Magnus