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Non-Parametric Frequency Warping

Figure F.1: Matlab function specinterp for performing ideal spectral interpolation.

 
function [si] = specinterp(spec,indices);
%SPECINTERP     
%          [si] = specinterp(spec,indices);
%
%          Returns spectrum sampled at indices (from 1).
%          Assumes a causal rectangular window was used.
%          The spec array should be the WHOLE spectrum,
%          not just 0 to half the sampling rate.
%          An "index" is a "bin number" + 1.
% EXAMPLE:
%          N = 128;
%          x = sin(2*pi*0.1*[0:N-1]);
%          X = fft(x);
%          Xi = specinterp(X,[1:0.5:N]); % upsample by 2
%          xi = ifft(x); % should now see 2x zero-padding

N = length(spec);
ibins = indices - 1;
wki = ibins*2*pi/N;

M = length(indices);
si = zeros(1,M);
wk = [0:N-1]*2*pi/N;
for i=1:M
  index = indices(i);
  ri = round(index);
  if abs(index - ri) < 2*eps
    si(i) = spec(ri);
  else
    w = wki(i);
    % ideal interp kernel:
    wts = (1-exp(j*(wk-w)*N)) ./ (N*(1-exp(j*(wk-w))));
    si(i) = sum(wts .* spec);
  end
end

Note that resampling a spectrum using the ideal interpolation kernel requires $ {\cal O}(N^2)$ operations, This computational burden can be reduced in a number of ways:


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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2016-07-18 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA