### WOLA Processing Steps

The sequence of operations in a WOLA processor can be expressed as follows:

- Extract the
th windowed frame of data
,
(assuming a length
causal window
and hop
size
).
- Take an FFT of the
th frame translated to time zero,

, to produce the th spectral frame

, . - Process
as desired to produce
.
- Inverse FFT
to produce
,
.
- Apply a
*synthesis window*to to yield a*weighted*output frame , . - Translate the
th output frame to time
as
and add to the accumulated output signal
.

To obtain perfect reconstruction in the absence of spectral modifications, we require

which is true if and only if

(9.44) |

#### Choice of WOLA Window

The synthesis (output) window in weighted overlap-add is typically chosen to be the same as the analysis (input) window, in which case the COLA constraint becomes

(9.45) |

We can say that -shifts of the window in the time domain are

*power complementary*, whereas for OLA they were

*amplitude complementary*.

A trivial way to construct useful windows for WOLA is to take the
*square root* of any good OLA window. This works for all non-negative
OLA windows (which covers essentially all windows in Chapter 3
other than Portnoff windows). For example, the
``root-Hann window'' can be defined for odd
by

Notice that the root-Hann window is the same thing as the ``MLT Sine Window'' described in §3.2.6. We can similarly define the ``root-Hamming'', ``root-Blackman'', and so on, all of which give perfect reconstruction in the weighted overlap-add context.

**Next Section:**

Overlap-Add (OLA) Interpretation of the STFT

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Length L FIR Frame Filters