Oscillator Bank resynthesis question

[Paul Masri]

Hi all,

A correction of one point here:

On Tue, 17 Mar 1998 11:11:57 -0800 Jens Johansson <jens at panix.com> wrote:
> (FFT will be a lot faster than an oscillator bank, BTW.. but the
> frequencies will be equally spaced of course.)
                      ^^^^^^^^^^^^^^ 

This is a phase vocoder way of thinking.  There is no actual restriction on the 
frequency spacing for synthesis using the FFT.  To prove this from a reverse 
direction...

If you take the FFT of a sinewave, including an incomplete number of 
cycles in the FFT window, then the corresponding magnitude peak will not lie on 
a frequency bin and you will end up with spectral leakage... but the sinewave 
would still be represented.  True, it cannot be represented by a single FFT 
bin, but several bins together can accurately describe any frequency.

The reverse is true for synthesis: if you want to synthesise a sinewave of any 
frequency below the Nyquist limit, calculate its profile in the FFT (by 
sampling points along the sinc function, for a rectangular time window) and 
place those points into the FFT array.  When you do the IFFT, voila, you get a 
sinewave at the desired frequency.

This principle has been applied in Rodet et al's IFFT Synthesis method.  It has 
also been extended for non-stationary (linear FM) frequency sinusoids, using 
the same principle.

Full details can be found on IFFT Synthesis from the following references:

Basic method (a good introduction):
Rodet,X;Depalle,Ph. 1992. "A new additive synthesis method using inverse 
Fourier transform and spectral envelopes". International Computer Music 
Conference (ICMC), pp410-411.

The start of non-stationary synthesis:
Goodwin,M;Rodet,X. 1994.  "Efficient Fourier synthesis of nonstationary 
sinusoids".  ICMC.  pp333-334.

Full linear FM synthesis:
Goodwin,M;Kogon,A. 1995.  "Overlap-add synthesis of nonstationary sinusoids".  
ICMC. pp355-356.

Hope the above is helpful.
Paul.
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Dr. Paul Masri
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