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, $w^2(n)=w(n)$ , and $W\ast W = W$ (since $W(\omega_k)$ is a sinc function which reduces to $\delta(\omega_k)$ when $\omega_k = 2\pi k / M$ , and $\delta\ast \delta = \delta$ .

• For the Hamming window, the critically sampled window transform has three nonzero samples (where the rectangular-window transform has one). Therefore, $W\ast W$ has $3+3-1=5$ nonzero samples at critical sampling. Measuring main-lobe width from zero-crossing to zero-crossing as usual, we get $6\cdot 2\pi/M$ radians per sample, or ``6 side lobes'', for the width of $W\ast W$ .

• The squared-Blackman window transform width is $(5+5-1)+1=10$ .

• The square of a length $M$ $L$ -term Blackman-Harris-family window (where rect is $L=1$ , Hann is $L=2$ , etc.) has a main lobe of width $(2(2L-1)-1)+1=4L-2$ , measured from zero-crossing to zero-crossing in ``side-lobe units'' ($2\pi/M$ ). This is up from $(2L-1)+1=2L$ for the original $L$ -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 $2\pi/M$ ).