## Weighted Overlap Add

In the*weighted overlap add*(WOLA) method, we apply a

*second window*after the inverse DFT [49] and prior to the final overlap-add to create the output signal. Such a window can be called a ``synthesis window,'' ``postwindow,'' or simply ``output window.''

Output windows are important in

*audio compression*applications for minimizing ``blocking effects.'' The synthesis window ``fades out'' any spectral coding error at the frame boundaries, thereby suppressing audible discontinuities. Output windows are

*not*used in simple FFT convolution processors because the input frames are

*supposed*to be expanded by the convolution, and a synthesis window would ``pinch off'' the ``filter ringing'' from each block, yielding incorrect results. Output windows can always be used in conjunction with spectral modifications made by means of the ``filter bank summation'' (FBS) method, which is the subject of the next chapter. The WOLA method is most useful for nonlinear ``instantaneous'' FFT processors such as

- perceptual audio coders,
- time-scale modification, or
- pitch-shifters.

*artifacts*caused by nonlinear spectral modifications. Another common factor in these applications is that filtering effects are not desired, so no provision for ``ringing'' in the time domain is necessary. In other words, WOLA is good for ``instantaneous nonlinear spectral processing.''

### 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 .

(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

**Next Section:**

Review of Zero Padding

**Previous Section:**

Time Varying OLA Modifications