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(Xi); % 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:

• A window $w(n)$ other than the rectangular window can be applied to the data frame, so that $W(\ejo )$ can be used as an interpolation kernel in place of the aliased sinc function. If the window side lobes are sufficiently small, they can be neglected. An example would be $w(n)$ set to a Hann window, with 50% overlap between frames, as commonly used in the STFT.
• In pure frequency-warping applications, windows can be used for both analysis and reconstruction, as is done in apparently all perceptual audio coding methods such as MPEG [16,273]. For example, the square root of the Hann window can be applied to the analysis frame as well as the final overlap-add frame. This scheme has the advantage of tapering any discontinuities at the frame boundary more gracefully to zero, thus reducing the audibility of artifacts.
• Specially optimized narrow-support kernels can be designed by a number of methods [72,218]. If they are constrained to be nonnegative, their square root can be used twice like the Hann window in the previous point. Asymmetric factorizations are also possible when the kernel changes sign.