6 FIR FIR FIR FIR 6.1 FIR 6.1.1 H(e jω ) H(e jω )= H(e jω ) e jθ(ω) = H(e jω ) (cos θ(ω)+jsin θ(ω)) (6.1) H(e jω ) θ(ω) θ(ω) = KωT, K > 0 (6.2) 6.1.2 6.1
6.1 FIR 123 6.1 H(e jω 1, ω <ω c ) = (6.3) 0, ω ω c θ(ω) = KωT, K > 0 (6.4) X(e jω ) =0, ω ω c (6.5) Y (e jω )=X(e jω )e jkωt (6.6) y(n) y(n) =x(n K) (6.7) x(n) K 6.2 IIR
124 6. FIR 6.2 x(n) ω 1 y(n) Kω 1 T x(n) = cos(ω 1 nt + φ) (6.8) y(n) = cos(ω 1 nt Kω 1 T + φ) = cos(ω 1 (n K)T + φ) = x(n K) (6.9) x(n) K τ(ω) = dθ(ω) dω τ(ω) = d( KωT) dω (6.10) = KT (6.11)
6.1 FIR 125 6.1.3 FIR h(n) = h(n 1 n) (6.12) N [ ] ω()t j e H(e jω 2 h( 2 )= )+ (N 3)/2 2h(n) cos(ω(n 2 (6.13) )) N [ ω()t (N/2 1) ] j e 2 2h(n) cos(ω(n 2 )) h(n) = h(n 1 n) (6.14) N [ ] ω()t j e H(e jω 2 h( 2 )= )+ (N 3)/2 j2h(n) sin(ω(n 2 (6.15) )) N [ ω()t (N/2 1) ] j e 2 j2h(n) sin(ω(n 2 )) h( N 1 )=0 (6.16) 2 [ ] e jωt()/2 [] j 90 6.1.4 FIR 6.3 z 1 z 1 z 2 z =1 z 3 z 3 z 1 H 1 (z) =1 2r cos θ 1 z 1 + r 2 z 2 (6.17)
126 6. FIR z z 1 ' z 2 z 1 z 3 6.3 z 1 H 1 (z) =1 2 1 r cos θ 1z 1 + 1 r 2 z 2 (6.18) z 2 H 2 (z) =1 2 cos θ 2 z 1 + z 2 (6.19) z 3 H 3 (z) =1 z 1 (6.20) 6.4 z 1,z 1,z 2 H(z) =H 1 (z)h 1 (z)h 2(z) (6.21) z 1,z 1,z 3 H(z) =H 1 (z)h 1 (z)h 3(z) (6.22)
6.1 FIR 127 6.4 1 6.5 6.5 FIR
128 6. FIR 6.1.5 FIR A/D FIR 6.6 6.7 A/D 6.6 0 FIR 0 f sh /2 FIR f sh 0 fs /2 f s 2fs 3f s 6.7
6.2 FIR 129 6.2 FIR FIR 6.2.1 6.8 6.8 FIR
130 6. FIR 1. H d (e jω ) 6.8 H d (e jω 1, ω ω c = (6.23) 0, ω >ω c θ(ω) = 0 (6.24) 2. H d (e jω ) h d (n) h d (n) = 1 2π π π H d (e jω )e jωnt dωt (6.25) 6.8 h d (n) n =0 3. w(n) 4. h d (n) w(n) FIR h(n) h(n) =w(n)h d (n), N 1 2 n N 1 2 (6.26) Gibbs 5. H(z) H(z) = H(e jω )= h(n)z n (6.27) h(n)e jωnt (6.28) h(n), 0 n N 1 (6.26) h(n) (N 1)/2 e jω()t/2 H d (e jω ) H(e jω ) jω(n 1)T/2
6.2 FIR 131 6.2.2 (6.29) 6.9 H(e jω )= 1 π H d (e jθ )W (e j(ωt θ) )dθ (6.29) 2π π 6.9 H d (e jω ) W (e j(ωit θ) ) ω = ω i H(e jω ) W (e jω ) 6.10 W (e jω ) W (e jω ) H(e jω ) H d (e jω ) W (e jω ) w(n) =0.54 0.46 cos( 2πn ), 0 n N 1 (6.30) N 1
132 6. FIR 6.10 N N 6.3 H d (e jω ) H d (k), k=0, 1,, IDFT FIR h(n),, 1,, N N 6.3.1
6.3 133 6.11 1. H d (e jω ) 6.11 2. H d (e jω ) 0 ω<2π/t N H d (k) N =16 H d (k) =H d (e j 2πk N ), 0 k N 1 (6.31) 3. H d (k) IDFT FIR h(n) h(n) = 1 H d (k)e j 2πkn N, 0 n N 1 (6.32) N k=0 4. H d (k) N H(z) = H(e jω )= h(n)z n (6.33) h(n)e jωnt (6.34) h(n) [ (N 1)/2, (N 1)/2] N h(n),, 1,, (N 1)/2 e jω()t/2 5. ω p < 2πk/NT < ω s k 6.3.2
134 6. FIR 6.11 FIR
6.3 135 FIR H d (k) H d (k) H(z) = h(n)z n (6.35) h(n) (6.32) [ ] 1 H(z) = H d (k)e j2πkn/n N = 1 N = 1 N k=0 k=0 = 1 z N N k=0 H d (k) z n (e j2πk/n z 1) n 1 z N H d (k) 1 e j2πk/n z 1 k=0 H d (k) 1 e j2πk/n (6.36) z 1 1 z N N 6.12 (6.36) 6.13 H d (k) H d (k) H d (k) N 1 j2π H d (k) = H d (k) e 2 k (6.37) ( 1 e jθ = e jθ/2 e jθ/2 e jθ/2) = j2e jθ/2 sin(θ/2) (6.38) z N =(e j2πk/n e jω ) N (6.39) H d (e jω )
136 6. FIR 6.12 6.13
H(e jω )= e jω()t/2 N 6.4 137 k=0 sin(n(ωt 2πk/N)/2) H d (k) sin((ωt 2πk/N)/2) (6.40) (6.36) (6.40) 6.3.3 (6.36) (6.40) H(e jω ) H k, k Φ H(e jω )= a k H k + b (6.41) k Φ a k b H k E = M 1 i=0 H d (e jωi ) H(e jωi ) 2 (6.42) ω i H k H k E =0,k Φ (6.43) H k H k 6.3.4 6.4 FIR h(n) H(z) = h(n)z n (6.44)
138 6. FIR h(n) =h(n 1 n) (6.45) H(e jω )= M 1 α(n) cos nω (6.46) M N/2, N (N 1)/2, N α(n) h(n) 2h(n) e j2π()t/2 ω E(e jω )=W (e jω ) [ H d (e jω ) H(e jω ) ] (6.47) W (e jω ) ω 6.14 6.14 ω i α(n) E(e jωi )=W(e jωi ) [ H d (e jωit ) H(e jωit ) ] =( 1) i δ (6.48) α(n) δ α(n +1) δ 6.14 ω i
6.5 FIR 139 E(e jω ) 6.15 6.15 6.5 FIR 6.5.1 6.16 FIR re ±jθ H in (z) r 1 e ±jθ H out (z) H out (e jω ) = 1 2r 1 cos θe jωt + r 2 e j2ωt = r 2 r 2 e jωt 2r cos θ + e jωt (6.49) H in (e jω ) = 1 2r cos θe jωt + r 2 e j2ωt = e jωt 2r cos θ + r 2 e jωt (6.50) H out (e jω ) = r 2 H in (e jω ) (6.51)
140 6. FIR 6.16 FIR
6.6 141 (6.52) ω =0 H out (z) H in (z) r 2 6.5.2 FIR FIR FIR 6.6 1. h(n) FIR h(0) = 0.5,h(1) = 1,h(2) = 0.5,h(n) =0,n 0, 1, 2 (6.53) (a) (b) (c) H(z) H(e jω ) ω H(z) x 1 (n),x 2 (n) y 1 (n),y 2 (n) f s =1/T =8kHz x 1 (n) = cos(ω 1 nt ),ω 1 =2πf 1,f 1 =1kHz (6.54) x 2 (n) = cos(ω 2 nt ),ω 2 =2πf 2,f 2 =2kHz (6.55)
142 6. FIR i. y 1 (n),y 2 (n) n =0 8 x 1 (n) =0,x 2 (n) =0,n<0 x 1 (n),y 1 (n),x 2 (n),y 2 (n) n =0 8 x i (n) y i (n) (b) ii. y 1 (n),y 2 (n) H(e jω ) x i (n) y i (n) (b) (d) y 1 (n) y 2 (n) 2. FIR (a) H d (e jω ) FIR H(z) N =8 (6.56) ωt =0,π/4, π/2, 3π/4, π,5π/4, 3π/2, 7π/4 (6.57) H d (e jω )=1, 1, 0, 0, 0, 0, 0, 1 (6.58) i. IDFT h(n) ii. n = 7 7 h(n) 3.5 h(n) h(n 3.5) h(n),n =0 7 3.5 h(n) H(z) (b) H d (e jω ) FIR i. H 1 (z),h 2 (z),h 3 (z)
6.6 143 ωt = π/2, 3π/4,π,5π/4, 3π/2 (6.59) H d (e jω )=0, 0, 0, 0, 0 (6.60) ii. iii. H s (z) =H 1 (z)h 2 (z)h 3 (z) z 1 H(z) h(n) N =8 H p (z) =h p (0) + h p (1)z 1 + h p (2)z 2 H(z) =H p (z)h s (z) h p (0),h p (1),h p (2) ωt =0,π/4, 7π/4 (6.61) H d (e jω )=1, 1, 1 (6.62) iv. H(z) =H p (z)h s (z) H(z) h(n) (c) H(z) h(n) H(z) h(n) (a) 3.5