#### 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 is both the number of constant-plus-cosine terms in the window definition (§3.3) and the half-main-lobe width in units of side-lobe 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 Blackman-Harris | M/2L | M/L |

*any*suppresses aliasing well.

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*.^{10.4}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 overlap-add (WOLA)
scheme (§8.6).

**Next Section:**

Hop Sizes for WOLA

**Previous Section:**

Filter Bank Reconstruction