### Constant-Overlap-Add (COLA) Cases

**Weak COLA:***Window transform has zeros at frame-rate harmonics:*- Perfect OLA reconstruction
- Relies on
*aliasing cancellation*in frequency domain - Aliasing cancellation is disturbed by spectral modifications
- See Portnoff for further details

**Strong COLA:**Window transform is bandlimited consistent with downsampling by the frame rate:- Perfect OLA reconstruction
- No aliasing
- better for spectral modifications
- Time-domain window infinitely long in ideal case

#### Hamming Overlap-Add Example

**Matlab code:**

M = 33; % window length w = hamming(M); R = (M-1)/2; % maximum hop size w(M) = 0; % 'periodic Hamming' (for COLA) %w(M) = w(M)/2; % another solution, %w(1) = w(1)/2; % interesting to compare

#### 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
) 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
, which is close enough to
zero in double-precision (64-bit) floating-point computations. We
thus verify that the overlap-add of a length
Hamming window using
a hop size of
samples is constant to within machine
precision.

Figure 9.24 shows the zero-padded DFT of the
modified Hamming window we're using (
) with the
frame-rate harmonics marked. In this example (
), the upper
half of the main lobe *aliases* into the lower half of the main
lobe. (In fact, *all* energy above the folding frequency
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.

Note that if we were to cut
in half to
, 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].

#### 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.

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.

Finally, Fig.9.28 shows the Kaiser window transform, with marks indicating the folding frequency at the chosen hop size , 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 dB (height of highest side lobe in Fig.9.28). We conclude that this example--a length 33 Kaiser window with and hop-size -- 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.

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

**Next Section:**

FBS Fixed Modifications

**Previous Section:**

Downsampling with Anti-Aliasing