Downsampling with Anti-Aliasing
In OLA, the hop size
is governed by the COLA constraint
![]() |
(10.26) |
In FBS,




Properly Anti-Aliasing Window Transforms
For simplicity, define window-transform bandlimits at first
zero-crossings 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 Blackman-Harris family, where


L | Window Type (Length ![]() |
![]() |
![]() |
1 | Rectangular | M/2 | M |
2 | Generalized Hamming | M/4 | M/2 |
3 | Blackman Family | M/6 | M/3 |
L | ![]() |
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 FFT-bin 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.10.4For 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 overlap-add (WOLA)
scheme (§8.6).
Hop Sizes for WOLA
In the weighted overlap-add method, with the synthesis (output) window equal to the analysis (input) window, we have the following modification of the recommended maximum hop-size 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 | ![]() |
M/(4L-2) | M/(2L-1) |
-
is equal to
divided by the main-lobe width in ``side lobes'', while
-
is
divided by the first notch frequency in the window transform (lowest available frame rate at which all frame-rate harmonics are notched).
- For windows in the Blackman-Harris families, and
with main-lobe widths defined from zero-crossing to zero-crossing,
.
Next Section:
Constant-Overlap-Add (COLA) Cases
Previous Section:
Downsampled STFT Filter Bank