### Example COLA Windows for WOLA

In a weighted overlap-add system, the following windows can be used to satisfy the constant-overlap-add condition:

- For the rectangular window,
, and
(since
is a sinc function which reduces to
when
, and
.
- For the Hamming window, the critically sampled window transform
has three nonzero samples (where the rectangular-window transform has
one). Therefore,
has
nonzero samples at critical
sampling. Measuring main-lobe width from zero-crossing to
zero-crossing as usual, we get
radians per sample, or
``6 side lobes'', for the width of
.
- The squared-Blackman window transform width is
.
- The square of a length
-term Blackman-Harris-family window
(where rect is
, Hann is
, etc.) has a main lobe of width
, measured from zero-crossing to zero-crossing in
``side-lobe units'' (
). This is up from
for the
original
-term window.
- The width of the main lobe can be used to determine the
*hop size*in the STFT, as will be discussed further in Chapter 9.

Note that we need only find the *first* zero-crossing in the
window transform for any member of the Blackman-Harris window family
(Chapter 3), since nulls at all harmonics of
that frequency will always be present (at multiples of
).

**Next Section:**

Block Diagram Interpretation of Time-Varying STFT Modifications

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PSF and Weighted Overlap Add