Laboratory Exercise DIGITAL FILTER STRUCTURES 6.1 REALIZATION OF FIR TRANSFER FUNCTIONS Project 6.1 Cascade Realization A copy of Program P6_1 is given below: % Program P6_1 % Conversion of a rational transfer function % to its factored form num = input('Numerator coefficient vector = '); den = input('Denominator coefficient vector = '); [z,p,k] = tf2zp(num,den); sos = zp2sos(z,p,k) Answers: Chương trình khơng chạy ( ma trân khơng cho khơng phù hợp) Q6.1 By running Program P6_1 with num = [2 10 23 34 31 16 4] and den = [1] we arrive at the following second-order factors: The block-diagram of the cascade realization obtained from these factors is given below: H1(z) is a _-phase transfer function Q6.2 By running Program P6_1 with num = [6 31 74 102 74 31 6] and den = [1] we arrive at the following second-order factors: The block-diagram of the cascade realization obtained from these factors is given below: H2(z) is a _-phase transfer function The block-diagram of the cascade realization of H 2(z) with only multipliers is shown below: 6.2 REALIZATION OF IIR TRANSFER FUNCTIONS Project 6.2 Cascade Realization Answers: Q6.3 By running Program P6_1 with num = [3 12 –2] and den = [16 24 24 14 5] we arrive at the following second-order factors: Parallel Form I Parallel Form II Residues are Residues are -0.0006 + 0.0069i -0.0056 - 0.0037i -0.0006 - 0.0069i -0.0056 + 0.0037i 0.1612 -0.1497 0.2138 - 0.2683i 0.1899 + 0.0710i 0.2138 + 0.2683i 0.1899 - 0.0710i Poles are at Poles are at -0.4686 + 0.8626i -0.4686 + 0.8626i -0.4686 - 0.8626i -0.4686 - 0.8626i -0.9290 -0.9290 0.1831 + 0.5617i 0.1831 + 0.5617i 0.1831 - 0.5617i 0.1831 - 0.5617i Constant value Constant value -0.4000 0.1875 The block-diagram of the cascade realization obtained from these factors is given below: Q6.4 By running Program P6_1 with num = [2 10 23 34 31 16 4] and den = [36 78 87 59 26 1] we arrive at the following second-order factors: Parallel Form I -0.5556 - 2.2785i Residues are -0.5556 + 2.2785i -0.5952 - 0.7561i -0.8214 + 4.3920i -0.5952 + 0.7561i -0.8214 - 4.3920i Poles are at 1.2593 + 0.4976i -0.5000 + 0.2887i 1.2593 - 0.4976i -0.5000 - 0.2887i -1.6964 - 1.4537i -0.3333 + 0.4714i -1.6964 + 1.4537i -0.3333 - 0.4714i Poles are at -0.2500 + 0.4330i -0.5000 + 0.2887i -0.2500 - 0.4330i -0.5000 - 0.2887i Constant value -0.3333 + 0.4714i -0.3333 - 0.4714i Parallel Form II -0.2500 + 0.4330i Residues are -0.2500 - 0.4330i 0.5159 + 0.2062i Constant value 0.5159 - 0.2062i 0.0556 The block-diagram of the cascade realization obtained from these factors is given below: A copy of Program P6_2 is given below: % Program P6_2 disp('Parallel Form I') % Parallel Form Realizations of an disp('Residues are');disp(r1); IIR Transfer disp('Poles are at');disp(p1); num = input('Numerator coefficient disp('Constant value');disp(k1); vector = '); disp('Parallel Form II') den = input('Denominator coefficient disp('Residues are');disp(r2); vector = '); disp('Poles are at');disp(p2); [r1,p1,k1] = residuez(num,den); disp('Constant value');disp(k2); [r2,p2,k2] = residue(num,den); Project 6.3 Parallel Realization Answers: Q6.5 By running Program P6_2 with num = [3 12 –2] and den = [16 24 24 14 5] we arrive at the partial-fraction expansion of H 1(z) in z–1 given by: Lattice parameters are: 0.547525053143030 0.880465268192976 0.741562180019706 -0.173160173160173 0.312500000000000 and the partial-fraction expansion of H 1(z) in z given by: Feedforward multipliers are: -0.125000000000000 0.139339826839827 0.200149183265393 0.021257212268448 0.312500000000000 -0.036775866451382 The block-diagram of the parallel-form I realization of H 1(z) is thus as indicated below: The block-diagram of the parallel-form II realization of H 1(z) is thus as indicated below: Q6.6 By running Program P6_2 with num = [2 10 23 34 31 16 4] and den = [36 78 87 59 26 1] we arrive at the partial-fraction expansion of H 2(z) in z–1 given by: Lattice parameters are: 0.810935846413523 0.771127725064015 0.592151877699842 0.371690524785502 0.134362934362934 0.027777777777778 and the partial-fraction expansion of H 2(z) in z given by: Feedforward multipliers are: 0.111111111111111 -0.047392657732535 -0.011120370334857 0.203703703703704 -0.014564520383792 0.151994851994852 0.023453136625124 The block-diagram of the parallel-form I realization of H 2(z) is thus as indicated below: The block-diagram of the parallel-form II realization of H 2(z) is thus as indicated below: