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1 5 IIR IIR z IIR(Infinite Impulse Response) FIR(Finite Impulse Response) IIR FIR

2 IIR 5.1 IIR FIR IIR FIR H(z) = a 0 +a 1 z 1 +a 2 z 2 1+b 1 z 1 +b 2 z 2 H(z) =h 0 + h 1z 1 + h 2z 2 + z 1 z = h 3z 3, z 1 0 z =0 z =0 y(n) = b 1y(n 1) b 2y(n 2)+ a 0x(n)+a 1x(n 1) + a 2x(n 2) y(n) = h 0x(n) +h 1x(n 1) + h 2x(n 2) + h 3x 3(n 3) 5.1 IIR

3 FIR N (Low Pass Filter: LPF) H a (jω) 2 1 = 1+(jω/jω c ) 2N (5.1) ω =2πf (5.2) 5.3 ω = ω c 1/2 3dB H a (jω) 2 ω =0 2N 1 ω =0 N ω c 5.4 ω c N 2 H a (jω) 2 jω s

4 IIR N ( ) 2N s 1+ =0 (5.3) jω c s 1 2N N N = s k = ω c e j2πk/2n, k =0, 1,..., 2N 1 (5.4) N = s k = ω c e jπ/2n e j2πk/2n, k =0, 1,..., 2N 1 (5.5) s k ( ) 2N sk = 1 (5.6) jω c N =2 N = H a (s) 2 H a (s) s N =

6 IIR 5.7 H a (s) 5.7 H a(s) 2 H a(s) H a (s) = a k = b k, R[s k ] < 0 (5.7) s s k s s k k Φ k Φ Φ s = H a (s) =0 s = N H a (jω) 2 1 = 1+ɛ 2 VN 2 (jω/jω (5.8) c) V N (x) = cos(ncos 1 x) (5.9) V N (x) N 5.8

7 ɛ ω c N V 0 (x) = 1 (5.10) V 1 (x) = cos(cos 1 x)=x (5.11) V 2 (x) = cos(2 cos 1 x) = 2 cos 2 (cos 1 x)) 1 =2x 2 1 (5.12). V n+1 =2xV n (x) V n 1 (x) (5.13) (5.9) V N (x) (5.13) cos θ = x, cos 1 x = θ (5.14) V n+1 = cos(n +1)θ = cos nθ cos θ sin nθ sin θ (5.15) sin nθ sin θ = 1 [cos(n +1)θ cos(n 1)θ] (5.16) 2 cos(n +1)θ = 2 cos θ cos nθ cos(n 1)θ (5.17) (5.13) 5.9 Vn 2 (x) 0 x 1 [0, 1]

8 IIR 1 <x x (5.8) H a (jω) 2 0 ω ω c 1/ 1+ɛ 2 H a (jω) 2 1 ω <ω c N =3 H a (jω) H a(jω) 2

9 y bω c aω c bω c N ɛ ω c α = ɛ ɛ 2 (5.18) a = 1 2 (α1/n α 1/N ) (5.19) b = 1 2 (α1/n + α 1/N ) (5.20) x 2 (aω c ) 2 + y2 =1 (5.21) (bω c ) 2 θ k = kπ N,k=0, 1,, 2N 1 N (5.22) π 2N + kπ N,k=0, 1,, 2N 1 N y k = bω c sin θ k (5.23) x k = ±aω c 1 y2 k (bω c ) 2 (5.24) = ±aω c 1 sin 2 θ k (5.25) H a (s) s 1 H a (s) =h 0, R[s k ] < 0 (5.26) s s k k Φ h 0 f =0 1 N 1/ 1+ɛ 2 N

10 IIR [1, 1 δ 1 ] [0,δ 2 ] ω p ω s H a (jω) 2 1 = 1+ɛ 2 UN 2 (ω) (5.27) U(ω) H a (s) h a (t) h a (nt ) 5.12

11 ω s H a (jω) =0, ω ω s /2 (5.28) Low-pass Filter: LPF Band-pass Filter: BPF High-pass Filter: HPF Bandelimination Filter: BEF All-pass Filter: APF (5.28) ω ω s / H a (s) h a (t) h(n) H a (s)

12 IIR N a k H a (s) =, R[s k ] < 0 (5.29) s s k k=1 H a (s) h a (t) N h a (t) = a k e skt,t 0 (5.30) k=1 h a (t) T N h(n) =h a (nt )= a k e sknt,n 0 (5.31) k=1 h(n) H(z) = [ N ] h(n)z n = a k e s knt z n n=0 = = n=0 k=1 [ N ] a k (e skt z 1 ) n k=1 N k=1 n=0 a k 1 e s kt z 1, e s k T z 1 < 1 (5.32) N a k N a k H a (s) = H(z) = s s k 1 e s kt z 1 (5.33) k=1 k=1 H a (s) s k H(z) e skt H a (s) R[s k ] < 0 e s kt < 1 H(z) s z s R[s k ] < 0 e skt < 1 (5.34) 3.5.3

13 R[s k ]=0 e skt =1 (5.35) e j(ωt+2nπ) = e jωt (5.36) ( π +2nπ)/T ω (π +2nπ)/T, n = e jπ e jπ (a) (b) 5.3.4

14 IIR 5.4 ω π/t ω π/t Ω ω s-z s z s = f(z) f(e jωt )= (5.37) f() s = f(z) = 2 T 1 z 1 1+z 1 (5.38) jω =f(e jωt )=j 2 T tan(ωt 2 ) (5.39) jω e jωt Ω= 2 T tan(ωt/2) (5.40) Ω π/t ω π/t 5.14

16 IIR (5.38) s-z (pre-warping) (5.38) z s = σ + jω z = 1+ T 2 s 1 T 2 s = 1+ T 2 σ + j T 2 Ω 1 T 2 σ j T 2 Ω (5.41) σ>0 z > 1 (5.42) σ =0 z =1 (5.43) σ<0 z < 1 (5.44) (5.40) 0 ω ω 0 0 ω π/t π/t ω

17 Ω= 2 T tan(ωt 2 ) (5.45) 2. H a (s) 3. H(z) =H a ( 2 1 z 1 ) (5.46) T 1+z 1 H(e jω ) LPF BPF HPF BEF 3. LPF

18 IIR H a (s) = N 1 k=0 s k s s k (5.47) H a (s) = N 1 k=0 c k s s k (5.48) H a (s) H a (s) h a (t) h a (t) = N 1 k=0 c k e s kt s k s k = δ k + jω k (5.49) (5.50) (5.49) h a (t) = = N 1 k=0 N 1 k=0 c k e δ k+jω k c k e δ kt e jω kt (5.51) e jω kt =1 h a (t) t δ k δ k > 0 (5.52) δ k =0 (5.53) δ k < 0 (5.54) δ k < 0 s

1 (Berry,1975) 2-6 p (S πr 2 )p πr 2 p 2πRγ p p = 2γ R (2.5).1-1 : : : : ( ).2 α, β α, β () X S = X X α X β (.1) 1 2

1 (Berry,1975) 2-6 p (S πr 2 )p πr 2 p 2πRγ p p = 2γ R (2.5).1-1 : : : : ( ).2 α, β α, β () X S = X X α X β (.1) 1 2 2005 9/8-11 2 2.2 ( 2-5) γ ( ) γ cos θ 2πr πρhr 2 g h = 2γ cos θ ρgr (2.1) γ = ρgrh (2.2) 2 cos θ θ cos θ = 1 (2.2) γ = 1 ρgrh (2.) 2 2. p p ρgh p ( ) p p = p ρgh (2.) h p p = 2γ r 1 1 (Berry,1975) 2-6