Why Kaiser window is better than fixed windows

A Kaiser window is w(n) = I₀(β√(1 − ((n − α)/α)²)) / I₀(β), 0 ≤ n ≤ M, where I₀ is the zeroth-order modified Bessel function and α = M/2. It is better than the fixed windows for these reasons:

• Adjustable shape. The parameter β controls the trade-off between main-lobe width and side-lobe level. β = 0 gives the rectangular window, and a larger β gives lower side lobes and a wider main lobe. Fixed windows (Hamming, Hanning, Blackman) each have one fixed trade-off.
• Any attenuation can be met. The attenuation A = −20 log₁₀ δ can be set exactly through β = 0.1102(A − 8.7) for A > 50, and β = 0.5842(A − 21)^0.4 + 0.07886(A − 21) for 21 ≤ A ≤ 50. Fixed windows jump in steps (21, 25, 44, 53, 74 dB), so they often over-satisfy the specification.
• Minimum length. The order M = (A − 8) / (2.285 Δω) is about the smallest that meets the specification, so the filter is shorter and cheaper than one using a fixed window.
• Near-optimal. Among windows it is almost optimum in the sense of having the maximum energy in the main lobe for a given main-lobe width (it approximates the prolate spheroidal sequence).
• Direct design formulas for both ripple and transition width make the design routine.