12.5 3 ISBN4-7856-1194-4 C3055 12.5.1 x(t) =A sin(t) rad/sec T s =1=F s t = nt s n x(nt s )=Asin(nT s ) F s! =T s x(nt s )! x(n) x(n) =A sin(nt s )=Asin(!n)! rad { F (Hz) =2F { f! =2f f {! f F! = T s ==F s (1) f =!=(2) =F=F s (2) 12.5.2 2 1 x(0);x(1);x(2);::: 3 2 x(0);x(1);x(2);::: y(n) = 1 (x(n)+x(n 0 1)) 2 3 2 1
4 3 y(n) = 1 (x(n)+x(n 0 1) + x(n 0 2)) 3 3 x(n) [] y(n) () 1: 2: 3: 2 4: 3 5 6 5 7 { ( ) { ( ) {? {? 2
5: 5 6: 7 12.5.3 sin(!n) cos(!n) e i!n = cos(!n)+i sin(!n) i 1 + e cos(!n) = ei!n 0i!n 2 0 e sin(!n) = ei!n 0i!n 2 (3) u(n)( 7(a)) ( 1 n 0 u(n) = 0 n<0 ( )(n)( 7(b)) ( 1 n =0 (n) = 0 n 6= 0 7: 3
8: (n) 12.5.4 ) (n 0 2) n =2 1 ( 8 (a)) 8 (b) x(01) = 01; x(0) = 2; x(2) = 1 0 x(n) x(n) = x(n) = x(01)(n +1) +x(0)(n) +x(1)(n 0 1) + x(2)(n 0 2) x(n) x(n) = k=01 x(k)(n 0 k) (4) 12.5.5 x(n) y(n) y(n) =T[x(n)] ( ) (time-invariant system) x(n) y(n) k y(n 0 k) =T [x(n 0 k)] 9(a)(b) 1 j 4
9: (linear system) x 1 (n);x 2 (n) y 1 (n);y 2 (n) a; b T [ax 1 (n)+bx 2 (n)] = at [x 1 (n)] + bt[x 2 (n)] = ay 1 (n)+by 2 (n) ) 9(c) (linear timeinvariant system) (causal system) n 0 y(n 0 ) n n 0 x(n) ) 5
12.5.6 (impulse response) h(n) h(n) =T [(n)] h(n) x(n) ( ) y(n) = m = n 0 k y(n) = k=01 x(k)h(n 0 k) (5) 0 m=1 x(n 0 m)h(m) = h(k)x(n 0 k) (6) k=01 x(n) h(n) (convolution) y(n) =x(n)?h(n) 3 h(0) = h(1) = h(2) = 1 3 (5) y(n) =T [x(n)] (4) y(n) =T[x(n)] = T [ y(n) = = k=01 k=01 k=01 x(k)(n 0 k)] T [x(k)(n 0 k)] (7) x(k)t [(n 0 k)] (8) P (7) (8) x(k) h(n) =T [(n)] h(n 0 k) =T [(n 0 k)] (8) (5) ) 10 6
10: 12.5.7 (5) ( ) 11 11 y(n) =x(n)+2x(n 0 2) 12.5.8 y(n) =x(n)+by(n 0 1) y 12 n =0 1;b;b 2 ;b 3 ; 111 IIR FIR IIR (innite impulse response) FIR(nite impulse response) FIR IIR IIR (6) ( x(n) y(n)) 7
11: 12: 8
h(k) =b k ; (k = 0; 1; 2;:::) y(n) = IIR k=0 b k x(n 0 k) 12 jbj > 1 12.5.9! ( ) x(n) =cos(!n) y(n) =A(!)cos(!n + (!)) { (!) { (A(!)) ((!)) { (A(!)) ((!)) (!) (frequency characteristics) { ( ) A(!) (amplitude characteristics) { (!) (phase characteristics) e i!n = cos(!n)+isin(!n) h(n) (6) 9
y(n) = k=01 h(k)ei!(n0k) = e i!n k=01 h(k)e0i!k (9) = e i!n H(e i! ) (10) H (9) 6 H(e i! )= k=01 h(k)e0i!k (11) H e i! H H(e i! )=A(!)e i(!) (12) A(!) (!) (10) y(n) = e i!n H(e i! )=A(!)e i(!n+(!)) { { (11) (12) 3 y(n) = 1 (x(n)+x(n 0 1) + x(n 0 2)) 3 13: 3 10
e i! = e i(!+2) 0 (!) < (!) 01 (!) < 1 3 13 {! = A(!)! = F =! F 2 p i s! = F = F s =2 { 12.5.10 z H(e i! ) (11) (12) IIR z H(e i! ) z x(n) z X(z) X(z) = (13) n=01 x(n)z0n z x(n) z X(z) x(n) $ z X(z) Z[x(n)] = X(z) (x(n)) (X(z)) (n) z 14(a) (n) z 1 11
14(b) 2(n 0 2) z c(n 0 k) z cz 0k c ) 14(c) x(n) z x(n) ( (4)) z z 14: z z z z { 2 x 1 (n);x 2 (n) z X 1 (z) =Z[x 1 (n)];x 2 (z) = Z[x 2 (n)] Z[ax 1 (n)+bx 2 (n)] = az[x 1 (n)] + bz[x 2 (n)] = ax 1 (z)+bx 2 (z) a b { x(n) z X(z) =Z[x(n)] Z[x(n 0 k)] = X(z)z 0k ; (k : ) 12
z 2 x 1 (n);x 2 (n) z X 1 (z);x 2 (z) ( (5)) x 1 (n)?x 2 (n) = n=01 x 1(n)x 2 (n 0 k) $ z X 1 (z)x 2 (z) (14) y(n) = k=01 x 1(k)x 2 (n 0 k) z Z[y(n)] = = = n=01 y(n)z0n n=01 ( k=01 x 1(k)( k=01 x 1(k)x 2 (n 0 k))z 0n n=01 x 2(n 0 k)z 0n ) n 0 k = p z 0n = z 0p z 0k = = = k=01 x 1(k)( p=01 x 2(p)z 0p z 0k ) k=01 x 1(k)z 0k ( p=01 x 2(p)z 0p ) k=01 x 1(k)z 0k X 2 (z) = X 1 (z)x 2 (z) IIR h(n) z H(z) H(z) =Z[h(n)] x(n) h(n) y(n) y(n) =x(n)?h(n) = k=01 h(k)x(n 0 k) z z Y (z) =H(z)X(z) Y (z) =Z[y(n)];H(z) =Z[h(n)];X(z) =Z[x(n)] H(z) 13
H(z) z H(z) = Y (z) X(z) 3 y(n) = 1 (x(n)+x(n 0 1) + x(n 0 2)) 3 h(n) = 1 ((n)+(n 0 1) + (n 0 2)) 3 h(n) z H(z) = 1 3 (1 + z01 + z 02 ) 2 z Z[y(n)] = Y (z) Z[x(n)] = X(z) Z[x(n 0 1)] = X(z)z 01 ;Z[x(n 0 2)] = X(z)z 02 Y (z) = 1 3 (X(z)+X(z)z01 + X(z)z 02 ) = 1 3 (1 + z01 + z 02 )X(z) H(z) = Y (z) X(z) = 1 3 (1 + z01 + z 02 ) (6) 2 y(n) = N01 X k=0 h(k)x(n 0 k); (N : ) 3 Y (z) = N01 X k=0 X N01 h(k)x(z)z 0k =( X k=0 H(z) = Y (z) N01 X(z) = h(k)z 0k k=0 h(k)z 0k )X(z) z z (order) N 0 1 z H(z) X(z) H(z)X(z) x(n) 2 k<0 0 0 k<0 h(k) =0 FIR N>0 0 k>n h(k) =0 14
) x(0) = 2; x(1) = 02; x(2) = 2 x(n) =0 2 y(n) = 1 (x(n)+x(n 0 1)) 2 z z y(n) =x(n)+by(n 0 1) z Y (z) =X(z)+bY (z)z 01 Y (z) X(z) H(z) Y (z)(1 0 bz 01 )=X(z) H(z) = Y (z) X(z) = 1 1 0 bz01 H(z) z e i! H(e i! )=H(z)j z=e i! x(n) z ( (13)) X(z) = n=01 x(n)z0n h(n) z H(z) x(n) h(n) H(z) H(z) = z e i! H(e i! )= n=01 h(n)z0n n=01 h(n)e0i!n (11) z e i! (11) Z 15 15
y(n)=t[x(n)] z h(n) y(n) = h(k)x(n-k) x(n) = exp(i n) H(z) = Y[z]/X[z] H(exp(i )) z = exp(i )) 15: z 12.5.11 16
16: LPF (Low Pass Filter), BPF (Band Pass Filter), HPF (High Pass Filter), BRF (Band Reject Filter) 17
// lpf1k.java // // // 8kHz 8 PCM // (C) 1999/9/13 static void convert(string infn, String outfn) { FileInputStream fin; FileOutputStream fout; // // : 60.000 [db] // : 100 // : 8.00000 [khz] // : 1.00000 [khz] double h[] ={ 0.00012980, 0.00012571, 0.00000000,-0.00021187,-0.00037517, -0.00032645, 0.00000000, 0.00047473, 0.00079661, 0.00066250, 0.00000000,-0.00089617,-0.00146002,-0.00118280, 0.00000000, 0.00153030, 0.00244607, 0.00194765, 0.00000000,-0.00244520, -0.00385770,-0.00303525, 0.00000000, 0.00373275, 0.00583714, 0.00455654, 0.00000000,-0.00553114,-0.00860537,-0.00668972, 0.00000000, 0.00807882, 0.01255811, 0.00976658, 0.00000000, -0.01185753,-0.01853088,-0.01452100, 0.00000000, 0.01805053, 0.02870358, 0.02300258, 0.00000000,-0.03057276,-0.05112438, -0.04387509, 0.00000000, 0.07433807, 0.15850612, 0.22485221, 0.25000417, 0.22485221, 0.15850612, 0.07433807, 0.00000000, -0.04387509,-0.05112438,-0.03057276, 0.00000000, 0.02300258, 0.02870358, 0.01805053, 0.00000000,-0.01452100,-0.01853088, -0.01185753, 0.00000000, 0.00976658, 0.01255811, 0.00807882, 0.00000000,-0.00668972,-0.00860537,-0.00553114, 0.00000000, 0.00455654, 0.00583714, 0.00373275, 0.00000000,-0.00303525, -0.00385770,-0.00244520, 0.00000000, 0.00194765, 0.00244607, 0.00153030, 0.00000000,-0.00118280,-0.00146002,-0.00089617, 0.00000000, 0.00066250, 0.00079661, 0.00047473, 0.00000000, -0.00032645,-0.00037517,-0.00021187, 0.00000000, 0.00012571, 0.00012980 }; for (i = 0; i < size; i++) { byte temp; buf[0] = (byte)fin.read(); o = 0.0; for (m = 0; m<=nf; m++) { o += h[m]*((double)buf[m]); } fout.write((byte)o); for (m=nf; m>0; m--) { buf[m] = buf[m-1]; } } 18