Power-of-Cosine

$$w(n)=w_R(n) \cos^P\left( \frac{ \pi n}{M} \right), \hspace{1cm} n \in \left[ -\frac{M-1}{2} ,\frac{M-1}{2} \right]$$

• $P = 0, 1, 2, \ldots$
• first $P$ terms of its Taylor expansion, evaluated at the endpoints (1/2 sample beyond last sample) are 0
• roll-off rate $\approx 6(P+1)$ dB/octave
• $P=0 \Rightarrow$ Rectangular window
• $P=1 \Rightarrow$ MLT sine window (shifted to zero-phase)
• $P=2 \Rightarrow$ Hann window (``raised cosine'' = ``$\cos^2$ '')
• $P=4 \Rightarrow$ Alternate Blackman (max roll-off rate in Blackman family)
• $\cdots$

Thus, $\cos^P$ windows parametrize $L$ th-order Blackman-Harris windows configured to use all degrees of freedom to maximize roll-off rate ($L=P/2+1$ )