Spectral Rotation of Real Signals

Note that if we rotate the spectrum of a real signal by half a bin, we obtain positive-frequency samples and negative-frequency samples, with no sample at dc or at the Nyquist limit. This is typically desirable for audio signals because dc is inaudible and the Nyquist limit is a degenerate point of the spectrum that, for example, cannot have a phase other than 0 or . If is a power of 2, then so is , and the octave-band partitioning of the previous subsection can be applied separately to each half of the spectrum:

N = 32;  x = randn(1,N); % Specific example
LN = round(log2(N)); % number of octave bands
shifter = exp(-j*pi*[0:N-1]/N); % half-bin
xs = x .* shifter; % apply spectral shift
X = fft(xs,N); % project xs onto rotated basis
XP = X(1:N/2); % positive-frequency components
XN = X(N:-1:N/2+1); % neg.-frequency components
YP = dcells(XP); % partition to octave bands
YN = dcells(XN); % ditto for neg. frequencies
YPe = dcells2spec(YP); % unpack "dyadic cells"
YNe = dcells2spec(YN); % unpack neg. freqs
YNeflr = fliplr(YNe);  % undo former flip
ys = ifft([YPe,YNeflr],N,2);
y = real(ones(LN,1)*conj(shifter) .* ys);
% = octave filter-bank signals (real)
yr = sum(y); % filter-bank sum (should equal x)
yrerr = x - yr;
disp(sprintf(...
  'Total filter-bank sum L2 error = %0.2f %%',...
   100*norm(yrerr)/norm(x)));

In the above listing, the function dcells^10.16simply forms a cell array in matlab containing the spectral partition (``dyadic cells''). The function dcells2spec^10.17is the inverse of dcells, taking a spectral partition and unpacking it to produce a usual spectrum vector.

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Next: Improving the Octave Band Filters

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.