Downsampling with AntiAliasing
In OLA, the hop size is governed by the COLA constraint
(10.26) 
In FBS, is the downsampling factor in each of the filterbank channels, and thus the window serves as the antialiasing filter (see Fig.9.19). We see that to avoid aliasing, must be bandlimited to , as illustrated schematically in Fig.9.20.
Properly AntiAliasing Window Transforms
For simplicity, define windowtransform bandlimits at first zerocrossings about the main lobe. Given the first zero of at , we obtain
(10.27) 
The following table gives maximum hop sizes for various window types in the BlackmanHarris family, where is both the number of constantpluscosine terms in the window definition (§3.3) and the halfmainlobe width in units of sidelobe widths . Also shown in the table is the maximum COLA hop size we determined in Chapter 8.
L  Window Type (Length )  
1  Rectangular  M/2  M 
2  Generalized Hamming  M/4  M/2 
3  Blackman Family  M/6  M/3 
L  term BlackmanHarris  M/2L  M/L 
It is interesting to note that the maximum COLA hop size is double the maximum downsampling factor which avoids aliasing of the main lobe of the window transform in FFTbin signals . Since the COLA constraint is a sufficient condition for perfect reconstruction, this aliasing is quite heavy (see Fig.9.21), yet it is all canceled in the reconstruction. The general theory of aliasing cancellation in perfect reconstruction filter banks will be taken up in Chapter 11.
It is important to realize that aliasing cancellation is disturbed by FBS spectral modifications.^{10.4}For robustness in the presence of spectral modifications, it is advisable to keep . For compression, it is common to use together with a ``synthesis window'' in a weighted overlapadd (WOLA) scheme (§8.6).
Hop Sizes for WOLA
In the weighted overlapadd method, with the synthesis (output) window equal to the analysis (input) window, we have the following modification of the recommended maximum hopsize table:
L  In and Out Window (Length )  
1  Rectangular ( )  M/2  M 
2  Generalized Hamming ( )  M/6  M/3 
3  Blackman Family ( )  M/10  M/5 
L  term BlackmanHarris  M/(4L2)  M/(2L1) 

is equal to
divided by the mainlobe width
in ``side lobes'', while

is
divided by the first notch
frequency in the window transform (lowest available frame rate at
which all framerate harmonics are notched).
 For windows in the BlackmanHarris families, and
with mainlobe widths defined from zerocrossing to zerocrossing,
.
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ConstantOverlapAdd (COLA) Cases
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Downsampled STFT Filter Bank