additive

[Chris Stecker]

>>well, thats not what they are teaching in engineering school these days!
>>They say that (loosely quoted from textbook) "Any sound can be broken up
>>into its harmonic series using fourier analysis. Once these basic
>>frequencies and amplitudes are known, then, all that needs to be done to
>>replicate the original sound is to do a time domain analysis of the
>>amplitude changes of each of the various harmonics."
>
>Yeah, that's what they said when I was in engineering school 20 years ago,
>too. This is the kind of oversimplification that is rampant in engineering
>texts - as exemplified by the statement "any sound can be broken into its
>harmonic series"... As has been discussed on the list, not all sounds
>consist only of integer harmonics. Furthermore, no musical instrument
>produces a single sound - there is an incredible range of variability in
>the sound produced by any single instrument.
>

I'm almost totally certain that the textbook in question says "Any _periodic_
signal can be represented by a set of basis waveforms (sin or not) at
integer multiple of the fundamental period," or something like that.  The
Fourier transform is a linear transformation from one form of signal
representation (time-domain waveform) to another (frequency-domain spectrum).
The math may be complex, but the solution is exact; inverting the fourier
transform gives you your input signal back, unchanged.  This is not an
oversimplification, it's just math.  Adding these frequency components
together,
with the proper phase and amplitude relationships, reconstructs the original
signal.

"Now," you may be saying, "that only holds true for periodic signals, when
sounds in the world are aperiodic."  Good point.  In fact, the resynthesis
I just described could only really work for steady-state (non-varying) signals.
Most signals change over time in non-repeating patterns, which makes them
aperiodic.  However, an aperiodic signal can still be analyzed using
Fourier techniques.  It can even be resynthesized.   The difference is in
the
frequencies which are present in the sound.  While periodic sounds have spectra
with energy found only at integer multiples (harmonics) of the fundamental
frequency, aperiodic sounds possess energy at many more frequencies, some
harmonically related to each, and some not.   Resynthesis becomes a much
more difficult task for these signals.  With an infinite number of
oscillators sounding for an infinite duration and aligning in an exact
phase relationship at one specific moment in all of time, the signal can be
recreated exactly.
Of course, if we had the power to do all that, we'd have more important things
to think about than analog synthesizers.  It's simply not a possibility.

Digital techniques can be used to simulate hundreds or thousands of
oscillators, which, although not an infinite number, can sound very
realistic.  Add to this the ability to do short, overlapping FFT's to
generate time-varying spectral data, and you come close to the quality you
would achieve using a sampler (although you've gained a whole lot of
flexibility).  The technology to do
this kind of thing, IN REAL TIME, is available in relatively inexpensive form
right now!  I find this very exciting.  If you have a PowerMac, go to
www.cycling74.com and find your way to "Download MSP" and download a copy
of "MSP Runtime."  This is a runtime-only version of the MSP programming
language.
When you run the application, open the file "Forbidden Planet Demo" and
check it out.  Real-time FFT analysis and resynthesis, with arbitrary
(analog junkies read as "highly resonant and tweakable") filtering performed
by simple multiplication in the frequency domain.  If you're not floored,
you should be.   And soon you'll start thinking of prototyping all your
synth modules in software, because you can, (again) IN REAL TIME.  Neat.

PS - Martin Czech's comments about temporal and frequency resolution
trading off are absolutely true.  You can't have perfect signal
reconstruction (of aperiodic signals) with window lengths shorter than
eternity.  But you can get surprisingly close (please don't write back
saying "close" doesn't count...you may have to listen yourself to be
convinced), and like many computational problems, this one yields pretty
well when you get enough horsepower behind it.

-----

Okay, I started this message as "know-it-all prick" and ended it like an
advertisement.  I'm embarassed,

-Chris


>And to top it off - the statement had to do with modular and FM synths, not
>Fourier (additive) synthesis. Yes, we can talk about modulars that can do

Oh, yeah...you're right.  Ooops.