Digital Signal Processing - Degital Filter Realizations
• b multiplier terms are feeding forward and x
dependent. (non-recursive terms)
• a multiplier terms are feeding back and y
dependent (recursive terms)
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12/15/2011 1 DIGITAL SIGNAL PROCESSING DIGITAL FILTER REALIZATIONS Lectured by: Assoc. Prof. Dr. Thuong Le-Tien National Distinguished Lecturer September, 2011 1 Digital filter realizations 2 2 1 10 2 2 1 10 )( )( )( zazaa zbzbb zD zN zH 221102211 nnnnùnn xbxbxbyayay 1. Direct form realization • b multiplier terms are feeding forward and x dependent. (non-recursive terms) • a multiplier terms are feeding back and y dependent (recursive terms) 12/15/2011 2 M M L L zazazaa zbzbzbb zD zN zH 2 2 1 10 2 2 1 10 )( )( )( Second order IIR digital filter ** General case of a IIR digital filter 12/15/2011 3 z -1 z -1 x(n) v (n)0 v (n)1 v (n)2 b2 b1 b0 w (n)0 w (n)1 w (n)2 y(n) z -1 z -1 -a1 -a2 LnLnnnMnMnnn xbxbxbxbyayayay 221102211 12/15/2011 4 z -1 z -1 x(n) v0 v1 v2 b2 b1 b0 w0 w1 w2 y(n) z -1 z -1 -a1 -a2 z -1 vL z -1 wM -aMbL 421 31 5.03.02.01 432 )( zzz zz zH 31421 432503020 nnnnnnn xxxyyyy ... Example: 12/15/2011 5 z -1 z -1 x(n) v0 v1 v2 -3 2 w0 w1 w2 y(n) z -1 z -1 -0,2 0,3 z -1 w4 -0,5 z -1 v3 4 w3 )()( 221122110 nnnnnn yayaxbxbxby )z(N )z(D )z(H 1 2. Canonical or Direct form 2 of IIR digital filters 12/15/2011 6 z -1 z -1 N(z) 1/D(z) z -1 z -1 x(n) y(n) v (n)0 v (n)1 v (n)2 b2 b1 b0 w (n)0 w (n)1 w (n)2 -a1 -a2 * Interchanging N(z) and 1/D(z) 12/15/2011 7 z -1 z -1 N(z)1/D(z) z -1 z -1 x(n) y(n) b2 b1 b0w (n)0 w (n)1 w (n)2 -a1 -a2 w(n) w (n)0 w (n)1 w (n)2 w(n) •Merging blocks to build the canonical form of the second order section (SOS) digital filter 12/15/2011 8 z -1 x(n) y(n) w (n)o w (n)1 b1 bo -a1 -a2 w(n) = w (n)o z -1 w (n)2 b2 * M order IIR filter realization 12/15/2011 9 z -1 x(n) y(n) wo w1 b1 bo -a1 -a2 w(n) z -1 w2 b2 z -1 -aM wM bM 3. Cascade The cascade realization form of a general transfer function assumes that the transfer function is the product of such second order sections 1 0 1 0 21 210 21 21 1 )()( K i K i ii iii i zaza zbzbb zHzH 12/15/2011 10 12/15/2011 11 z -1 z -1 z -1 z -1 z -1 z -1 z -1 z -1 x = x0 x(n) w (n)0 w00 w01 w02 -a01 -a02 b00 b01 b02 x = y1 0 w (n)1 b10 b11 b12 -a11 -a12 w10 w11 w12 w (n)2 w20 w21 w22 -a21 -a22 b20 b21 b22 x = y3 2 w (n)3 w30 w31 w32 -a31 -a32 b30 b31 b32 y = y3 y(n) x = y2 1 H (z)0 H (z)1 H (z)2 H (z)3 42 42 1021 21 21 21 25.084.01 449 )()( 5.04.01 243 5.04.01 243 )( zz zz zHzH zz zz zz zz zH Example 12/15/2011 12 z -1 x(n) y(n) w (n)0 w2 9 -0,84 w(n) z -1 w4 z -1 z -1 -0,25 w3 4 -4 The roots of Numerator: z = 0.9, z = -0.5 ± 0.7j, z = 0.8 ± 0.4j The roots of Denominator: p = -0.8, p = - 0.7 ± 0.4j 321 54321 52.077.12.21 5328.09376.033.048.05.11 )( zzz zzzzz zH 2111 2111 1 806114050140801 74017050170501 901 z.z.z)j..(z)j..( z.zz)j..(z)j..( )z.( 2111 1 6504114070140701 801 z.z.z)j..(z)j..( )z.( Example: 21 21 21 1 1 80611 650411 7401 801 901 z.z.. z.z. z.z . z. z. )z(H 12/15/2011 13 88 1 ( ) 1 0.0625 z H z z )k(jjkjj eeeezz 12288 101 710812 ,,,k,ez /)k(jk 43526170 z,z,z,z,z,z,z,z Example: Complex conjugate pairs 21211413 21211 5 1 2 21211 6 1 1 21211 7 1 0 847811 8 7 2111 765401 8 5 2111 765401 8 3 2111 847811 8 2111 zz.zzcoszzzz zz.zzcoszzzz zz.zzcoszzzz zz.zzcoszzzz 12/15/2011 14 43526170 p,p,p,p,p,p,p,p 21211413 2211 5 1 2 21211 6 1 1 2111 7 1 0 50150 8 6 2111 50150 8 4 2111 50150 8 2 2111 50150150111 z.zz.zcoszpzp z.z.zcoszpzp z.zz.zcoszpzp z.z.z.zpzp . z.z zz. . z. zz. . . z.z zz. . z. zz. )z(H 21 21 2 21 21 21 2 21 501 847811 501 765401 501 765401 501 847811 )12(4288 )5.0(0625.00625.000625.01 kjjk eezz 71050 82 ,,,k,e.p /jkk 12/15/2011 15 12/15/2011 16 z3 z2 z1 z0 z4 z5 z6 z7 p5 p7 p6 p0p4 p3 p1 p2 = poles = zeros Unit circle /8 Magnitude response Pole frequencie s = (2k) /8 zero frequencie s = (2k+1) /8 3 2 1 0 0 1 2 3 4 5 6 7 8 in units of /8 |H ( )| z -8 x y w0 0,0625 w8
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