WOLA Processing Steps

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

1. Extract the th windowed frame of data , (assuming a length causal window and hop size ).

2. Take an FFT of the th frame translated to time zero,
, to produce the th spectral frame
, .

3. Process as desired to produce .

4. Inverse FFT to produce , .

5. Apply a synthesis window to to yield a weighted output frame , .

6. Translate the th output frame to time as and add to the accumulated output signal .

(The overlap-add method discussed previously is obtained from the above procedure by deleting step 5.)

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.

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Overlap-Add (OLA) Interpretation of the STFT
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Length L FIR Frame Filters