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

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. In the table, 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.^10.4For robustness in the presence of spectral modifications, it is advisable to keep . For compression, it is common to use together with a ``post-window'' (synthesis filter) in a weighted overlap-add (WOLA) scheme (see §8.7).

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.