Low-pass FIR window design, αs = 40 dB

Choice of window

Required stop-band attenuation: 40 dB. The minimum attenuation of the standard windows: rectangular 21 dB, Bartlett 25 dB, Hanning 44 dB, Hamming 53 dB, Blackman 74 dB. The simplest window meeting 40 dB is the Hanning window (its main lobe is 8π/M wide).

Length

Transition width: Δω = ωs − ωp = 0.5π − 0.3π = 0.2π.

M ≥ 8π/Δω = 8π/0.2π = 40. Take M = 41 (odd, so the filter is Type I with an integer delay α = (M−1)/2 = 20).

Cut-off

ωc = (ωp + ωs)/2 = (0.3π + 0.5π)/2 = 0.4π

Ideal and window sequences

h_d(n) = sin(0.4π(n − 20)) / (π(n − 20)), n ≠ 20; h_d(20) = ωc/π = 0.4

w(n) = 0.5 [1 − cos(2πn/40)], 0 ≤ n ≤ 40

h(n) = h_d(n) · w(n)

Coefficients (h(n) = h(40 − n))

Impulse response: h(n) = {0, −0.0001, −0.0003, 0.0006, 0.0018, 0, −0.0045, −0.0039, 0.0054, 0.0116, 0, −0.0194, −0.0153, 0.0194, 0.0401, 0, −0.0685, −0.0590, 0.0913, 0.3009, 0.4000, 0.3009, … (symmetric) …, 0}.

The transfer function is H(z) = Σ (n = 0 to 40) h(n) z⁻ⁿ, with linear phase θ(ω) = −20ω.

Check

Evaluating H(e^jω) numerically: the pass-band ripple is under 0.7 %, and the largest stop-band peak is −43.9 dB, which satisfies the 40 dB requirement.

Answer: Hanning window, M = 41, ωc = 0.4π, h(n) as in the table (h(20) = 0.4).