ODP: AW: Harmonix content formula

[mbartkow at ET.PUT.Poznan.PL]

Roman Sowa wrote:

> > Now I have dangerously silly idea - DCT in analog hardware:
> > (Discrete Cosine Transform - simplifed version of Discrete Fourier
> > Transform)

In fact, DCT is not a simplified Fourier Transform. DCT can be vieved 
as Discrete Fourier Transform of a signal, which has been modified
by repeating twice the same sequence of samples with the second 
replica reversed in time. This results in mirror symmetry of the signal,
thus its spectrum is real and it is not neccessary to perform complex
math. However, calculating DCT requires exactly the same number of
operations as DFT. Moreover, DCT is just a transformation which is
useful for signal compresion based on quantization of DCT coefficients.
DCT is not a spectrum analysis, since what you get is not a spectrum
of the original wave. E.g. DCT calculated from a saw wave is essentially
the same as DFT calculated from a triangle - see, what I mean ?

> > say, 32 sine oscilators of harmonic frequencies (or not harmonic - why stick
> > to the original?), 32 multipliers (ringmods) multiplying input signal with
> > each of
> > 32 sine oscs (or one muxed), 32 envelope followers and we have harmonic
> > content output.

Martin Czech wrote:

> 
> My understanding is that FT looks like convolution with sine waves,
> not multiplying. That would be too easy.
> 
> If you see it this way, FT is like filtering and the filter impulse
> response is a sin/cos wave.

What Roman has described is a spectrum analyser based on a modulated
filter bank. In fact, DFT *can* be viewed as a filter bank. The multiplication
of the wave by the transform core (the complex exp term) is responsible
to appropriate shifting of the spectrum, integration (or summation in a 
discrete case) is the lowpass filtering.
Such devices exist for years. The other possibility (the so called
wobbulator based design) is to employ a VCO quadrature sweeped by 
the time base signal of the scope, then modulate the signal by the 
VCO, integrate and put into the Y input of the scope - 
voila' an instant realtime spectrum analyser.


BTW, I teach Fourier Transform here, at the University.
If you have any questions or need to analyse a particular waveform,
do not hesitate to ask.

m.b.


--

Maciej Bartkowiak
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