Periodic-Hamming OLA from Poisson Summation Formula

Matlab code:

ff = 1/R; % frame rate (fs=1)
N = 6*M;  % no. samples to look at OLA
sp = ones(N,1)*sum(w)/R; % dc term (COLA term)
ubound = sp(1);  % try easy-to-compute upper bound
lbound = ubound; % and lower bound
n = (0:N-1)';
for (k=1:R-1) % traverse frame-rate harmonics
  f=ff*k;
  csin = exp(j*2*pi*f*n); % frame-rate harmonic
  % find exact window transform at frequency f
  Wf = w' * conj(csin(1:M));
  hum = Wf*csin;   % contribution to OLA "hum"
  sp = sp + hum/R; % "Poisson summation" into OLA
  % Update lower and upper bounds:
  Wfb = abs(Wf);
  ubound = ubound + Wfb/R; % build upper bound
  lbound = lbound - Wfb/R; % build lower bound
end

In this example, the overlap-add is theoretically a perfect constant (equal to $ 1.08$ ) because the frame rate and all its harmonics coincide with nulls in the window transform (see Fig.9.24). A plot of the steady-state overlap-add and that computed using the Poisson Summation Formula (not shown) is constant to within numerical precision. The difference between the actual overlap-add and that computed using the PSF is shown in Fig.9.23. We verify that the difference is on the order of $ 10^{-15}$ , which is close enough to zero in double-precision (64-bit) floating-point computations. We thus verify that the overlap-add of a length $ 33$ Hamming window using a hop size of $ R = (M-1)/2 = 16$ samples is constant to within machine precision.

Figure 9.23: Periodic-Hamming Poisson summation formula test.
\includegraphics[width=\textwidth ]{eps/olassmmpHammingC}

Figure 9.24 shows the zero-padded DFT of the modified Hamming window we're using ( $ w(M)\leftarrow 0$ ) with the frame-rate harmonics marked. In this example ($ R=M/2$ ), the upper half of the main lobe aliases into the lower half of the main lobe. (In fact, all energy above the folding frequency $ \pi/R$ aliases into the lower half of the main lobe.) While this window and hop size still give perfect reconstruction under the STFT, spectral modifications will disturb the aliasing cancellation during reconstruction. This ``undersampled'' configuration is suitable as a basis for compression applications.

Figure 9.24: Hamming window transform and frame-rate.
\includegraphics[width=\textwidth ]{eps/windowTransformHammingC}

Note that if we were to cut $ R$ in half to $ R=M/4$ , then the folding frequency in Fig.9.24 would coincide with the first null in the window transform. Since the frame rate and all its harmonics continue to land on nulls in the window transform, overlap-add is still exact. At this reduced hop size, however, the STFT becomes much more robust to spectral modifications, because all aliasing in the effective downsampled filter bank is now weighted by the side lobes of the window transform, with no aliasing components coming from within the main lobe. This is the central result of [9].


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Kaiser Overlap-Add Example
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Hamming Overlap-Add Example