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Kaiser Overlap-Add Example

Matlab code:

M = 33;    % Window length
beta = 8;
w = kaiser(M,beta);
R = floor(1.7*(M-1)/(beta+1)); % ROUGH estimate (gives R=6)

Figure 9.25 plots the overlap-added Kaiser windows, and Fig.9.26 shows the steady-state overlap-add (a time segment sometime after the first 30 samples). The ``predicted'' OLA is computed using the Poisson Summation Formula using the same matlab code as before. Note that the Poisson summation formula gives exact results to within numerical precision. The upper (lower) bound was computed by summing (subtracting) the window-transform magnitudes at all frame-rate harmonics to (from) the dc gain of the window. This is one example of how the PSF can be used to estimate upper and lower bounds on OLA error.

Figure 9.25: Kaiser OLA waveforms.
\includegraphics[width=\textwidth ]{eps/olakaiserC}

Figure 9.26: Kaiser OLA, steady state.
\includegraphics[width=\textwidth ]{eps/olasskaiserC}

The difference between measured steady-state overlap-add and that computed using the Poisson summation formula is shown in Fig.9.27. Again the two methods agree to within numerical precision.

Figure 9.27: Kaiser OLA from Poisson summation formula minus computed OLA.
\includegraphics[width=\textwidth ]{eps/olassmmpkaiserC}

Finally, Fig.9.28 shows the Kaiser window transform, with marks indicating the folding frequency at the chosen hop size $ R$ , as well as the frame-rate and twice the frame rate. We see that the frame rate (hop size) has been well chosen for this window, as the folding frequency lies very close to what would be called the ``stop band'' of the Kaiser window transform. The ``stop-band rejection'' can be seen to be approximately $ 58$ dB (height of highest side lobe in Fig.9.28). We conclude that this example--a length 33 Kaiser window with $ \beta=8$ and hop-size $ R=6$ -- represents a reasonably high-quality audio STFT that will be robust in the presence of spectral modifications. We expect such robustness whenever the folding frequency lies above the main lobe of the window transform.

Figure 9.28: Kaiser window transform and frame-rate.
\includegraphics[width=\textwidth ]{eps/windowTransformkaiserC}

Remember that, for robustness in the presence of spectral modifications, the frame rate should be more than twice the highest main-lobe frequency.

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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University