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∀n : (h (n) = ha (nT )) |z| > |e skT | Ha (s) = b0 + b1s + b2s2 + ... + bpsp 1 + a1s + a2s2 A1 = + + ... + apsp s − s1 A2 + ... + s − s2 Ap s − sp ha (t) = A1e s1t + A2e s2t + ... + Ape spt u (t) h (n) = ha (nT ) = A1e s1nT + A2e s2nT + ... + Ape spnT u (n) Ake (skT )n u (n) ≡ Akz z − e skT H (z) = p k=1 Ak z z − eskT skT |z| > maxk |e | z −1 = g z −1 e −(iω1) = g e −(iω) = |g (ω) |e i∠(g(ω)) |g e −(iω) | = 1 g z −1 = p k=1 −1 z − αk 1 − αkz−1
|αK| < 1 z1 −1 = z−1 − a 1 − az −1 ωc ω ′ c a = sin 1 2 (ωc − ω ′ c) sin 1 2 (ωc + ω ′ c) z1 −1 = z−1 + a 1 + az −1 ωc ω ′ c a = cos 1 2 (ωc − ω ′ c) cos 1 2 (ωc + ω ′ c) H (z) h (n) = 0 n < 0 h = 0 h (0) = b0 H (z) = b0 1 + M k=1 (akz −k ) M h (n) = − (akh (n − k)) + b0δ (n) k=1 hd (n) ɛ 2 = ∞ n=0 (|hd (n) − h (n) |) 2 H (z) (|desired − actual|) 2
Page 1: Digital Filter Design By: Douglas L
Page 4 and 5: Digital Filter Design By: Douglas L
Page 7: L ∞
Page 10 and 11: H (ω) = M−1 −(iωn) h=0 h (n)
Page 12 and 13: H (ω) = A (ω) e−(iθ0) M−1
Page 14 and 15: ∀n, 0 ≤ n ≤ M − 1 : (h [n])
Page 16 and 17: h [n] h [n] = h [−1] M
Page 18 and 19: ∞ M W (ω)
Page 20 and 21: x = cos (ω) E (ω) = W
Page 22 and 23: L + 2 ⎛ ⎜ ⎝ 1 cos (ω0) c
Page 24 and 25: O 16L 2 O L 3 L
Page 26 and 27: M L∞ E (ω) = W (ω
Page 28 and 29: H (s) = ∞ ha (t) e −∞ −(st
Page 30 and 31: H (s) = LHP H (s) = √ 2
Page 32 and 33: B λ 2 1 = 1 + ɛ2CM 2 (λ) CM 2
Page 34 and 35: L ∞ Ha (s)
Page 36 and 37: L ∞ L ∞ L
Page 40 and 41: n > 0 M h (n) = − (akh (n
Page 42 and 43: opt = V H V −1 V H h
Page 44 and 45: ∂ 2 ɛ ∂a = 2r + 2Ra R
Page 46 and 47: ∂ 2 ɛ ∂a = 2r + 2Ra R
Page 48 and 49: ∂ 2 ɛ ∂a = 2r + 2Ra R

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