additive question

[Martin Czech]

> 
> 	>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.   
> 
> No more discrete frequencies for non periodic signals.
> Continuous spectrum.
> 
> OTAH, a sine wave with an *envelope* applied isn't a  discrete
> frequency either. 
> 
> (Academical) Question: 
> Imagine a sound that was created *only* by a number of periodic 
> waveforms, a linear (no overdrive) VCF with envelope, and a linear 
> VCA with envelope. Is it possible to resynthesize such a sound 
> with a *finite* number of *fixed* frequency oscillators and individual
> VCAs / envelopes ?
> I know this is far from real life sounds, but I just wonder if this can work
>  or not.
> 

AFAIK it is not neccessary to map the odd frequencies, ie. inharmonic
partials exactly. If you're in the digital domain you have only the
discrete fourier transform, and this works only for periodic signals
and spectra. But: by clever windowing/chopping you can transform any
input wave into a stream of spectral chunks. Ie. you get a series of
harmonic waveforms which make any input signal if they are chained
together. Or in other words:  An inharmonic input gives ever changing
spectra, a harmonic input which is synchronous to sampling rate and has
an integer freqeuency ratio to the window size is a very seldom special
case, and  gives a constant spectrum. 

Don't think too much about the meaning of these spectra. You can
manipulate them and transform back, but you have to be carefull at the
window boundarys.  I mean, isn't this the way that mpeg audio
compression works ?  AFAIK the data is windowed, transformed (phase
information thrown away), "unneccessary" spectral information is
dropped, the stuff is then coded (compressed). On the reciever side the
data is uncompressed and time domain information is obtained via revers
transformation. And then something must be done at the window boundarys
to avoid artefacts. This works, as you can hear.

Or think about fast convolution algorithms. Again the input stream is
chopped into windows, transformed, the convolution becomes
multiplication in the frequency domain, and then the whole stuff is
back transformed and reasembled into the now convoluted (filtered) data
stream.  This gives exactly the same result as if you do the
convolution in the time domain, you need not worry about inharmonic
input data, it works for every input signal.

So academical answer: I'd say the above case could be done, the
question is however how manipulations in the time domain map into the
frequency domain.  It's not quite straight forward, because one has
always to keep the windowing in mind, and this requires additional
steps to be taken.

m.c.