2 e jωt, 0 ωt < 2π 2.1 2.1.1 x(n) X(z) = x(n)z n (2.1) z z z = e σ+jωt = re jωt, r = e σ (2.2) ωt 0 2π σ e σ+jωt = re jωt r = e σ 2.1 0 <r<1(σ <0) 1 <r(0 <σ) r =1(σ =0) 2.1.2
2.1 29 2.1 X(z) = x(n)z n = x(n)r n e jωnt (2.3) x(n)r n S = x(n)r n < (2.4) x(n) x(n) r z z 1, n 0 x(n) =u(n) = 0, n < 0 (2.5) x(n) S = x(n)r n 1 r N = lim N 1 r 1 (2.6) r 1 < 1 S z e jωnt =1 (2.7) z 1 < 1 z > 1 1
30 2. 2.2 X(z) = x(n)z n 1 =, z > 1 (2.8) 1 z 1 2.2 2.1.3 X(z) = z X(z) =0 z x(n) = a n u(n) (2.9) X(z) = x(n)z n = a n z n = (az 1 ) n (2.10) az 1 < 1 X(z) = 1 1 az 1 = z z a (2.11) z > a a z =0 z = a 2.3
2.1 31 2.3 2.1.4 1 x(n) X(z) = n 2 n=n 1 x(n)z n (2.12) n 1 < 0 z p, p > 0 z = n 2 > 0 z p, p < 0 z =0 2 X(z) = x(n)z n, n 1 < (2.13) n=n 1 z = z 1 S = x(n)z1 n < (2.14) n=n 1 n 1 0 z > z 1 z S = x(n)z n < x(n)z1 n < (2.15) n=n 1 n=n 1
32 2. n 1 < 0 (2.13) 1 X(z) = x(n)z n + x(n)z n, n=n 1 n 1 < (2.16) z = z > z 1 a 2.2 2.2.1 1. x(n) X(z) z 2. x(n) x(n) < 3. x(n) X(z) z = e jωt x(n) x(n) 4. e jωt 2.2.2 1
2.2 33 h(n) S = h(n) < (2.17) h(n) h(n) 2.1.3 S < a < 1 z > a X(z) a 2.4 2.4 2 h(n)
34 2. h(n) h(n) =a n,n 0 az 1 < 1 h(n) 2.5 a > 1 2.5 2.2.3 1. 2. 3. 4. 2.3 x(n) Z[x(n)]
2.3 35 2.3.1 z p z > max{ z p } 2.6 2.6 2.3.2 1 X(z) =Z[x(n)], R x < z (2.18) Y (z) = Z[y(n)], R y < z (2.19) a b Z[ax(n) +by(n)] = (ax(n)+by(n))z n = a x(n)z n + b y(n)z n = ax(z)+by (z), max{r x,r y } < z (2.20)
36 2. 2 X(z) =Z[x(n)] x(n + n 0 ) Z[x(n + n 0 )] = x(n + n 0 )z n = x(m)z (m n0) = z n0 [ m= m= x(m)z m ](2.21) Z[x(n + n 0 )] = z n0 X(z) (2.22) n 0 > 0 x(n + n 0 ) z =0, z = 2.3.3 1 Z[a n x(n)] = X(a 1 z), a R x < z (2.23) X(z) z p = r p e jωp z z = r z e jωz (2.24) (2.25) X(a 1 z) a = a e jφ (2.26) z pa = a r p e j(ωp+φ) (2.27) z za = a r z e jωz+φ (2.28) 2 X(z) x(n) z
dx(z) dz = = z 1 x(n) dz n dz = 2.3 37 nx(n)z n 1 nx(n)z n (2.29) zdx(z)/dz nx(n) 3 x(n) x (n) x (n) ( Z[x (n)] = = x (n)z n ) x(n)(z ) n = X(z ) (2.30) Z[x (n)] = X (z ) (2.31) 2.3.4 1 y(n) = h(k)x(n k) (2.32) k= Y (z) =H(z)X(z) (2.33) 2 y(n) =h(n)x(n) (2.34) Y (z) = h(n)x(n)z n (2.35)
38 2. = = 1 2πj [ 1 h(n) 2πj [ c c c ] X(v)v n 1 dv z n (2.36) h(n)( z v ) n ] v 1 X(v)dv = 1 H( z 2πj c v )X(v)v 1 dv (2.37) = 1 H(v)X( z 2πj v )v 1 dv (2.38) (2.36) x(n) 2.4 (2.38) x(n) h(n) z = re jφ v = ρe jθ (2.39) (2.40) v θ v c θ [ π, π] dv dθ = jρejθ = jv (2.41) dv jv = dθ (2.42) π Y (re jφ )= 1 H(ρe jθ )X( r 2π π ρ ej(φ θ) )dθ (2.43) Y φ H X θ 2.3.5 (2.35) (2.38) z =1
2.4 39 h(n)x (n) = 1 2πj H(v)X ( 1 v )v 1 dv (2.44) c h(n) x(n) v = e jωt (2.45) dv dωt = jejωt = jv (2.46) dv = dωt (2.47) jv h(n)x (n) = 1 π H(e jω )X (e jω )dωt (2.48) 2π π h(n) =x(n) (2.49) x(n) 2 = 1 π X(e jω ) 2 dωt (2.50) 2π π 2.4 X(z) x(n) 2.4.1 X(z) z x(n)
40 2. X(z) = N 1 i=0 a iz i M 1 i=0 b iz = i N 1 i=0 a i z i = M 1 i=0 b i z i x(n)z n, b 0 =1 (2.51) x(n)z n (2.52) z z n a 0 = x(0) a 1 = b 1 x(0) + x(1). (2.53) 0 = b M 1 x(n M +1)+b M 2 x(n M)+ + x(n) x(0) x(1) x(2) x(n) 2.4.2 z N 1 i=0 X(z) = a iz i M 1 i=0 b iz i L M 1 = α i z i β j + 1 p i=0 j=1 j z 1 (2.54) L = N M 0 M 1 X(z) b i z i =0 p j i=0 β j 1 p j z 1 = β j (p j z 1 ) n = β j p n j z n (2.55)
2.4 41 M 1 x(n) =α n + β j p n j, 0 n L (2.56) x(n) = M 1 j=1 j=1 β j p n j, L < n (2.57) N = M =2 X(z) = a 0 + a 1 z 1 + a 2 z 2 1+b 1 z 1 + b 2 z 2 = α β 1 0 + 1 p 1 z 1 + β1 1 p 1 z 1 = α 0 + β 1 (p 1 z 1 ) n + β1 (p 1z 1 ) n (2.58) x(0) = α 0 +2R[β 1 ] x(n) =β 1 p n 1 + β 1 p n 1 =2R[β 1 p n 1 ], n 1 (2.59) 2.4.3 X(z) = x(n)z n (2.60) z k 1 C 1 X(z)z k 1 x(n) dz = z n+k 1 dz (2.61) 2πj 2πj c 1 z k 1 1, k =0 dz = 2πj c 0, k 0 c (2.62) (2.61) n = k 1 n k (2.61) x(k)
42 2. x(k) = 1 X(z)z k 1 dz (2.63) 2πj c C 2.7 Z Im 1 C Re C 1 2.7 1 F (z)dz C (2.64) 2πj c F (z) F (z) K ψ k (z) F (z) = (z z k ) r (2.65) k k=1 ψ k (z) =0 z k z k r k F (z) K [ 1 d r k 1 ] ψ k (z) Res[F (z)] = (r k 1)! dz r (2.66) k 1 z=z k k=1 (2.63) F (z) =X(z)z k 1 (2.67) n 0
2.5 43 1 X(z) =, z > a (2.68) 1 az 1 x(n) = 1 z n 1 dz (2.69) 2πj c 1 az 1 = 1 z n dz (2.70) 2πj c z a n 0 C z = a 1 F (z) = zn z a (2.66) [ 1 d 0 z n ] x(n) = (1 1)! dz 0 z=a (2.71) = a n, n 0 (2.72) 2.5 2.5.1 H(z) h(n) (2.33) x(n) y(n) Y (z)/x(z) H(z) = h(n)z n (2.73) H(z) = Y (z) X(z) M 1 j=0 b j y(n j) = N 1 i=0 (2.74) a i x(n i) (2.75) x(n) y(n) z 2.3.2
44 2. M N b j z j Y (z) = a i z i X(z) (2.76) j=0 i=0 H(z) = Y (z) X(z) = N 1 i=0 a iz i M 1 j=0 b jz j (2.77) 2.8 2.8 IIR (1) y(n) =a 0 x(n)+a 1 x(n 1) + a 2 x(n 2) b 1 y(n 1) b 2 y(n 2) (2.78) z H(z) =Y (z)/x(z) Y (z) = (a 0 + a 1 z 1 + a 2 z 2 )X(z) (b 1 z 1 + b 2 z 2 )Y (z) (2.79) H(z) = Y (z) X(z) = a 0 + a 1 z 1 + a 2 z 2 1+b 1 z 1 + b 2 z 2 (2.80) 2.8 2.9 2 z 1,z 2 2 2 a 0,a 1,a 2,b 1,b 2 5 2 4 w(n)
2.5 45 2.9 IIR (2) w(n) =x(n) b 1 w(n 1) b 2 w(n 2) (2.81) y(n) = a 0 w(n)+a 1 w(n 1) + a 2 w(n 2) (2.82) z W (z) (2.83) W (z) =X(z) b 1 z 1 W (z) b 2 z 2 W (z) (2.84) Y (z) = a 0 w(z)+a 1 z 1 W (z)+a 2 z 2 W (z) (2.85) H(z) = Y (z) X(z) = a 0 + a 1 z 1 + a 2 z 2 1+b 1 z 1 + b 2 z 2 (2.86) 2 2.5.2 H(z) =0 H(z) = z h(n) =0,n<0 n 0 z =0 z = z (2.77) z
46 2. N i=1 H(z) =a (1 z iz 1 ) M j=1 (1 p jz 1 ) (2.87) z i p j 2.5.3 h(n) h(n) z = e jωt z H(z) H(z) = 1 az 1 1 bz 1 = z a z b (2.88) a b b < 1 H(z) H(e jω ) = ejωt a e jωt b (2.89) z z 2.10 ω 1 T 1 < H(e jω ) ω 2 T H(e jω ) < 1 2.11( ) 0.8 ±π/4 ±π/2 π
2.6 47 Z Im e 2 jω T e 1 jω T a 0 b 1 Re 2.10 z = e jωt H(z) = (1 ejπ z 1 )(1 e jπ/2 z 1 )(1 e jπ/2 z 1 ) (1 0.8e jπ/4 z 1 )(1 0.8e jπ/4 z 1 ) = (2.90) (1 + z 1 )(1 + z 2 ) 1 1.6 cos(π/4)z 1 +0.64z 2 (2.91) 2.11( ) ωt = π/4, 7π/4,f = f s /8, 7f s /8 ωt = π/2,π,3π/2,f = f s /4,f s /2, 3f s /4 ωt =2π f = f s ωt =0 π f =0 f s /2 2.6 2.12 1 h(n) =0, n < 0
48 2. Z Im 0 π/4 π/4 1 Re j H( e ω T ) 0 π/ 2 π 3π / 2 0 fs /4 f S /2 f / 4 3 S 2π f S ωt ωt f 2.11 h(0) = a 0 (2.92) h(n) =a 0 ( b) n + a 1 ( b) n 1, n 1 H(z) h(n) H(z) = h(n)z n = a 0 + [a 0 ( b) n + a 1 ( b) n 1 ]z n n=1 = a 0 + a 0bz 1 1+bz 1 + a 1z 1 1+bz 1, bz 1 < 1
2.6 49 n =- h (n ) < [8] [5] z = e [6] [1] [4] jωt [7] [2] [3] [9] 2.12 H(z) =a 0 = a 0 + a 1 z 1 1+bz 1 (2.93) = ( b) n z n + a 1 n=1 ( b) n z n a 0 1+bz 1 + a 1z 1 1+bz 1 = a 0 + a 1 z 1 1+bz 1 (2.94) H(z) = a 0 + a 1 z 1 1+b 1 z 1 = a 0 1+b 1 z 1 + a 1z 1 1+b 1 z 1 (2.95) 1 2 a 0 1+b 1 z 1 = a 0[1+( bz 1 )+(b 1 z 1 ) 2 + ] (2.96) a 1 z 1 1+b 1 z 1 = a 1z 1 [1+( bz 1 )+(b 1 z 1 ) 2 + ] (2.97) z 1 H(z) =a 0 +(a 0 ( b 1 )+a 1 )z 1
50 2. +(a 0 ( b 1 ) 2 + a 1 ( b 1 ))z 2 + (2.98) = h(0) + h(1)z 1 + h(2)z 2 + (2.99) (2.92) h(n) 2 H(z) H(z) = Y (z) X(z) = a 0 + a 1 z 1 1+bz 1 (2.100) (1 + bz 1 )Y (z) =(a 0 + a 1 z 1 )X(z) (2.101) y(n)+by(n 1) = a 0 x(n)+a 1 x(n 1) (2.102) y(n) =a 0 x(n)+a 1 x(n 1) by(n 1) (2.103) 3 2.13(a) w(n) a 0 x (n) w (n) y(n) + + b T a 1 (a) x(n) a 0 + y(n) T T a 1 b x ( n 1) y( n 1) (b) 2.13 x(n) y(n) w(n) w(n) =x(n) bw(n 1) (2.104) y(n) = a 0 w(n)+a 1 w(n 1) (2.105)
x(n) w(n) w(n 1) = b x(n) w(n) y(n) = a 0 w(n)+a 1 2.6 51 b (2.106) (2.107) w(n) a1 y(n) b = a 0 a1 b (2.108) w(n) (2.105) (2.103) y(n) y(n) a1 b y(n) =a y(n 1) a1 b x(n 1) 0 + a a 0 a1 1 b a 0 a1 b (2.109) = a 0 x(n)+a 1 x(n 1) by(n 1) (2.110) (2.104) (2.105) z W (z) X(z) Y (z) z (2.104) (2.105) 2.13(b) x(n) y(n) y(n) =a 0 x(n)+a 1 x(n 1) by(n 1) (2.111) (2.104) (2.105) w(n) w(n 1) W (z) Y (z)/x(z) W (z) =X(z) bz 1 W (z) (2.112) Y (z) = a 0 W (z)+a 1 z 1 W (z) (2.113) W (z) = X(z) 1+bz 1 (2.114) Y (z) = (a 0 + a 1 z 1 )W (z) = a 0 + a 1 z 1 X(z) (2.115) 1+bz 1
52 2. H(z) = Y (z) X(z) = a 0 + a 1 z 1 1+bz 1 (2.116) (2.110) w(n) w(n 1) W (z) 4 h(n) y(n) = n h(m)x(n m) (2.117) m=0 (2.104) (2.105) n =0, 1, 2, x(n) w(n 1) w(n) y(n) 5 H(e jω )= h(n)e jωnt (2.118) h(n) = 1 2π π π H(e jω )e jωnt dωt (2.119) 6 z = e jωt 7 (2.93) 2.13
2.6 53 8 h(n) h(n) < (2.120) z p z p < 1 z p =1 z p > 1 9 1. (a) (b) (c) (d)
54 2. (e) (f) z =( ) 2. h(n) z H(z) h(n) 2r n cos(ω 0 nt ), 0 n h(n) = (2.121) 0, n < 0 (a) h(n) z H(z) z 1 (b) (c) (d) (e) H(z) r =0.9 ω 0 T = π/2 H(z) ωt =0,π/4,π/2,π/4,π x(n) y(n) x(n) =ae j(ωnt+φ), a =0.7,ωT = π/4,φ=0.3[rad](2.122) 3. (a) H(z) = 1 0.3z 1 (1 + 0.5z 1 )(1 0.8z 1 ) H(z) h(n) (2.123) (b) (c) h(n) H(z) H(z) h(n) H(z)
2.6 55 (d) (e) (f) (g) (h) (i) (j) h(n) e jωt H(z) e jωt H(z) x(n) y(n) H(z) w(n) x(n) y(n) 1, n =0, 1, 2 x(n) = 0, n 0, 1, 2 (2.124) h(n) x(n) y(n) 4. 2.14 (a) (b) (c) (d) (e) (f) y(n) x(n) w(n) H(z) =Y (z)/x(z) H(z) H(z) h(n) 2.15 x(n) x(n) =a cos(ωnt + φ) a ωt φ x(n) y(n)() () b =0.5 a 0 =1 a 1 =1y(n) θ tan θ =
56 2. x (n) w(n) y(n) + + a 0 b T a 1 2.14 IIR 1 0 x(n) ~ 2 1 3 4 n -1 2.15 x(n)
2.6 57 1 z k 1 dz (2.125) 2πj c C C e jθ z = e jθ dz dθ = jejθ dz = je jθ dθ (2.126) (2.127) (2.128) C θ 0 2π (2.125) 1 2πj 2π 0 2π e jθ(k 1) je jθ dθ = 1 e jθk dθ 2π 0 1, k =0 = 0, k 0 (2.129) 1 1 dz (2.130) 2πj c (z z p ) k 1 (z z p ) k 1 dz (2.131) 2πj c C z p