Blackman Window Family

When $L=3$ in (3.26), we obtain the Blackman family:

$$w_{B}(n)= w_R(n)\left[\alpha_0 + \alpha_1 \cos(\Omega_M n) + \alpha_2 \cos(2\Omega_M n)\right]$$ (4.28)

Relative to the generalized Hamming family (§3.2), we have added one more cosine weighted by $\alpha_2$ . We now therefore have three degrees of freedom to work with instead of two. In the Hamming family, we used one degree of freedom to normalize the window amplitude and the second was used either to maximize roll-off rate (Hann) or side-lobe rejection (Hamming). Now we can use two remaining degrees of freedom (after normalization) to optimize these objectives, or we can use one for each, resulting in three subtypes within the Blackman window family.