Example COLA Windows for WOLA

In a weighted overlap-add system, the following windows can be used to satisfy the constant-overlap-add condition:

• For the rectangular window, , and (since is a sinc function which reduces to when , and .

• For the Hamming window, the critically sampled window transform has three nonzero samples (where the rectangular-window transform has one). Therefore, has nonzero samples at critical sampling. Measuring main-lobe width from zero-crossing to zero-crossing as usual, we get radians per sample, or ``6 sidelobes'', for the width of .

• The squared-Blackman window transform width is .

• The square of a length -term Blackman-Harris-family window (where rect is , Hann is , etc.) has a main lobe of width , measured from zero-crossing to zero-crossing in ``sidelobe units'' (). This is up from for the original -term window.

• The width of the main lobe can be used to determine the hop size in the STFT, as will be discussed further in Chapter 9.

Note that we need only find the first zero-crossing in the window transform for any member of the Blackman-Harris window family (Chapter 3), since nulls at all harmonics of that frequency will always be present (at multiples of ).

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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.