Kaiser low-pass FIR, 0.19π and 0.21π

Specifications

Pass band: 0.99 ≤ |H| ≤ 1.01 for 0 ≤ ω ≤ 0.19π, so δ₁ = 0.01. Stop band: |H| ≤ 0.01 for 0.21π ≤ ω ≤ π, so δ₂ = 0.01. Hence ωp = 0.19π, ωs = 0.21π.

Kaiser parameters

δ = 0.01, A = −20 log₁₀(0.01) = 40 dB.

Δω = ωs − ωp = 0.02π

β = 0.5842(19)^0.4 + 0.07886(19) = 3.395

M = (A − 8)/(2.285 Δω) = 32/(2.285 × 0.02π) = 222.9, so M ≥ 223. Take M = 224 (even, so α = M/2 = 112 is an integer and the filter is Type I); length 225.

Cut-off

ωc = (0.19π + 0.21π)/2 = 0.2π

Ideal response

h_d(n) = sin(0.2π(n − 112)) / (π(n − 112)), n ≠ 112; h_d(112) = 0.2

Kaiser window

w(n) = I₀(3.395 √(1 − ((n − 112)/112)²)) / I₀(3.395), 0 ≤ n ≤ 224

Impulse response

h(n) = h_d(n) w(n) = h(224 − n):

Numerical check of the 225-tap filter: the stop-band peak is 0.0097, so the stop band is met. The pass band stays between 0.9899 and 1.0097; its lowest value, right at the edge 0.19π, is a hair below 0.99, because the Kaiser length formula is only an estimate. Taking M = 226 (227 taps) gives a pass band of 0.9912 to 1.0093 and a stop-band peak of 0.0094, meeting both tolerances strictly.

Answer: A = 40 dB, β = 3.395, M = 224 (formula minimum 223; M = 226 meets the pass band strictly), ωc = 0.2π, h(n) = h_d(n)·w(n).