Universiti Malaysia Perlis EKT430: DIGITAL SIGNAL PROCESSING LAB ASSIGNMENT 5: DIGITAL FILTER STRUCTURES

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1 Universiti Malaysia Perlis EKT430: DIGITAL SIGAL PROCESSIG LAB ASSIGMET 5: DIGITAL FILTER STRUCTURES Pusat Pengajian Kejuruteraan Komputer Dan Perhubungan Universiti Malaysia Perlis

2 Digital Filter Structures ITRODUCTIO For the design of digital filters, the system function H(z) or the impulse response h(n) must be specified. Then the digital filter structure can be implemented or synthesized in hardware/software form by its difference equation obtained directly from H(z) or h(n). Each difference equation or computational algorithm can be implemented by using a digital computer or special purpose digital hardware or special programmable integrated circuit. A chosen structure determines a computational algorithm. Generally, different structure determines a computational algorithm. Generally, different structures give different results. The most common methods for realizing digital linear systems are direct, cascade and parallel forms, and state variable realizations. 2.1 Basic Realization Block Diagram The output of a finite order linear time invariant system at time n can be expressed as a linear combination of the inputs and outputs, y[ n] = ak y[ n k] + bk x[ n k] k = 1 k = 0 (5.1) Where a k and b k are constants with a 0 0 and M. The current output y[n] is equal to the sum of past outputs, from y[n-1] to y[n-], which are scale by the delay-dependent feedback coefficient a k, plus the sum of future, present and past inputs, which are scaled by the delay-dependent feed forward coefficient b k. After taking z-transform of the output sequence y[n] given in Eq. (5.1) and changing the order of summation, it becomes H ( z) Y ( z) X ( z) 1 M k k = 0 = = b k = 1 The computational algorithm of an LTI digital filter can be represented as a block diagram using basic building blocks which are the unit delay (z z a k k -1 z k (5.2) ) or storage element, the multiplier and the adder. For example, the difference equation of the first-order digital system may be written as y[ n] = a y[ n ] + x[ n] + b x[ n ] 1 1 (5.3) where y[n] and x[n] are the output and input sequences, respectively. 2

3 2.2 Direct Form Realization of FIR and IIR Systems Eq. (5.2) is the standard form of the system transfer function. By inspection of this equation, the block diagram representation can be drawn directly for the direct form realization. The multipliers in the feed forward paths are the numerator coefficients and the multipliers in the feedback paths are the negatives of the denominator coefficients Direct Form of FIR System A causal FIR filter of length M is characterized by a transfer function H(z): H ( z) = M k = 0 h[ k]. z k (5.4) which is a polynomial in z 1 of degreem 1. In the time domain the input-output relation of the above FIR filter is given by y [ n] = M k = 0 h[ k] x[ n k] (5.5) A direct form realization of an FIR filter can be readily developed from Eq. (5.5) as indicated in Figure 1(a) for M = 5. Its transpose, as sketched in Figure 1(b), is the second direct form structure. An FIR filter of length M is characterized by M coefficients and, in general, requires M multipliers and (M 1) two-input adders for implementation. Figure 1 Direct Form FIR structures 3

4 2.2.2 Cascade Form of FIR System A higher-order FIR transfer function can also be realized as a cascade of FIR sections with each section characterized by either a first-order or a second-order transfer function. To this end, the FIR transfer function H(z) of Eq. (5.4) is expressed in a factored form as H ( z) = h[0] (1 + β z + β z ) 1k 2k (5.6) k where for a first-order factor β 2k = 0. A realization of Eq. (5.6) is shown in Figure 2 for a cascade of three second-order sections. Each second-order stage in Figure 2, of course, can be realized also in the transposed direct form. The cascade form realization also requires, in general, (M ) two-input adders and M multipliers for an FIR transfer function of length M. Figure 2 Cascade form for FIR structure for a length-7 of FIR filter Linear-phase FIR Filter A length-m linear-phase FIR filter is characterized by either a symmetric impulse response h[n] = h[m n] or an anti-symmetric impulse response h[n] = h[m n]. The symmetry (or the anti-symmetry) property of a linear-phase FIR filter can be exploited to reduce the total number of multipliers to half of those needed in the direct form implementations of the transfer function. For example, Figure 3(a) shows the realization of a length-7 Type 1 FIR transfer function with a symmetric impulse response and Figure 3(b) shows the realization of a length-8 Type 2 FIR transfer function with a symmetric impulse response. 4

5 Figure 3 Linear-phase FIR structures: (a) Type 1 and (b) Type Direct Form I of IIR System The digital system structure determined directly from either Eq. (5.1) or (5.2) is called the direct form I. In this case, the system function is divided into two parts connected in cascade, the first part containing only the zeros, and the second part containing only the poles. An intermediate sequence w[n] is introduced. The equation represents direct form I is as followed, Y ( z) = 1 W ( z) k = 1 a k z k (5.7) The direct form I realization is shown in Figure 4. 5

6 Figure 4 Direct Form I structure Direct Form II of IIR System An intermediate sequence u[n] is introduced to obtain the output of the filter, u[ n] = ak u[ n k] + x[ n] k = 1 (5.8) and y[ n] = bku[ n k] k = 0 (5.9) The direct form II realization requires only the larger of M or storage elements. When the addition is performed sequentially, the direct form II needs two adders instead of one adder required for the direct form II. Figure 5 shows structure of direct form II with combine common delays. Figure 5 Direct Form II structure 6

7 2.2 Cascade Realization of IIR Systems In cascade realization, the transfer function H(z) is broken into a product ofransfer functions H 1 (z), H 2 (z),.,h k (z). There are several cascade realizations for the factored form, based on pole-zero pairings and ordering. A possible realization of a third-order transfer function is shown in Figure 6. Figure 6 Cascade realization of a third-order IIR transfer function 2.3 Parallel Realization of IIR Systems By using the partial fraction expansion, the transfer function of an IIR system can be realized in a parallel form. Parallel structure of third-order IIR transfer function is shown in Figure 7. 7

8 Figure 7 Parallel realization of a third-order IIR transfer function Realization of FIR Transfer Functions Experiment 1: Cascade Realization The factored form of a causal FIR transfer function H(z) of order M-1, as given in Eq. (5.6) can be determined from its polynomial form representation given by Eq. (5.4) which can be utilized to realize H(z) in a cascade form. To this end, a modified form of Program P5_1 that uses the function zp2sos can be employed. MATLAB Code % Program P5_1_MATLAB % Conversion of a rational transfer function to its factored form num = input( umerator coefficient vector = ); den = input( Denominator coefficient vector = ); [A, B] = eqtflength(num, den); [z,p,k] = tf2zp(a, B); sos = zp2sos(z,p,k); SCILAB Code % Program P5_1_MATLAB % Conversion of a rational transfer function to its factored form z = %z; num = [ ];sample of num den = [ ];sample of den Gz = syslin('d',num/den); [sos,g]=tf2sos(gz); Report: Q4.1 Using Program P5_1, develop a cascade realization of the following FIR transfer function: H ( z) = z + 23z + 34z + 31z + 16z + 4z 1 (5.10) 8

9 Sketch the block diagram of the cascade realization. Is H 1 (z) a linear-phase transfer function? Q4.2 Using Program P5_1, develop a cascade realization of the following FIR transfer function: H ( z) = z + 74z + 102z + 74z + 31z + 6z 2 (5.11) Sketch the block diagram of a cascade realization. Is H 2 (z) a linear-phase transfer function? Develop a cascade realization of H 2 (z) with only 4 multipliers. Show the block diagram of the new cascade structure. Realization of IIR Transfer Functions Experiment 2: Cascade and Parallel Realizations The factored form of a causal IIR transfer function H(z) of order can be determined from its rational form representation given in Eq. (5.2) which can be used to realize H(z) in a cascade form. To this end, Program P5_1 can be employed. Report: Q4.1 Using Program P5_1, develop a cascade realization of the following IIR transfer function: z + 12z + 7z + 2z z H 1( z) = (5.12) z + 24z + 14z + 5z + z Sketch the block diagram of the cascade realization. Is H 1 (z) a linear-phase transfer function? Q4.2 Using Program P5_1, develop a cascade realization of the following IIR transfer function: z + 23z + 34z + 31z + 16z + 4z H 1( z) = (5.13) z + 87z + 59z + 26z + 7z + z Sketch the block diagram of a cascade realization. Parallel-form realization of a causal IIR transfer function is based on its partial-fraction expansion z -1, which can be obtained using SCILAB function residuez. Program P5_2 develops parallel realization. 9

10 MATLAB Code % Program P5_2 % Parallel Form Realizations of an IIR Transfer Function num = input( umerator coefficient vector = ); den = input( Denominator coefficient vector = ); [r1,p1,k1] = residuez(num,den); disp( Parallel Form ) disp( Residues are );disp(r1); disp( Poles are at );disp(p1); disp( Constant value );disp(k1); SCILAB Code Program P5_2 Parallel Form Realizations of an IIR Transfer Function num=[ ];example num value den=[ ];example den value [r1,p1,k1] = residuez(num,den); disp(r1) disp(p1) disp(k1) Report: Q4.3 Using Program P5_2, develop parallel form realization of the causal IIR transfer function of Eq. (5.12). Sketch the block diagram of realization. Q4.3 Using Program P5_2, develop parallel form realization of the causal IIR transfer function of Eq. (5.13). Sketch the block diagram of realization. Experiment 3: The transfer function of a third-order IIR system, in factored form, is given by 1( z) 2 ( z) H 1( z) = (5.14) D ( z) D ( z) 1 2 where ( z) = z + z 1 ( z) = 1 + z 2 D ( z) = z z 1 D ( z) = z 2 10

11 An example of MATLAB/SCILAB commands to find the partial fraction expansion of X(z) is shown below. MATLAB Code SCILAB Code % Program P5_3_MATLAB Program P5_3_MATLAB % Conversion of a factored transfer function to Conversion of a factored transfer function to its rational form its rational form 1 = [ ]; 1 = [ ]; 2 = [1 1 0]; 2 = [1 1 0]; D1 = [ ]; D1 = [ ]; D2 = [ ]; D2 = [ ]; sos = [1 D1;2 D2]; sos = [1 D1;2 D2] [b, a] = sos2tf(sos); [B, A] = sos2tf(sos) [r,p,k] = residuez(b,a); [r,p,k] = residuez(b,a); disp( Parallel Form ) disp('parallel Form ') disp( Residues are );disp(r); disp('residues are') disp( Poles are at );disp(p); disp(r) disp( Constant value );disp(k); disp('poles are at') disp(p) disp('constant value') disp(r) Report: Q4.1 Sketch the block diagram of realization for the system, for a cascade structure, using a second-order and one first-order section. Q4.2 Using Program P5_3, convert the cascade form to direct Form II realization of the IIR transfer function. Sketch the block diagram of realization. Q4.3 Using Program P5_3, convert the cascade form to parallel form realization of the IIR transfer function. Sketch the block diagram of realization. Q4.4 Plot and display the zero-pole diagram for this system by adding command in Program P4_3. Q4.5 Is H 1 (z) in Eq is a stable transfer function? Justify your answer. Q4.6 Compare the computational complexity of the direct Form II, cascade and parallel form realizations of the IIR transfer function. Hint: The computational complexity of a digital filter is given by the total number of multipliers and the total number of two-input adders required for its implementation. #Function file for [B, A] = sos2tf(sos) Copyright (C) 2005 Julius O. Smith III 11

12 This program is free software; you can redistribute it and/or modify it under the terms of the GU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT AY WARRATY; without even the implied warranty of MERCHATABILITY or FITESS FOR A PARTICULAR PURPOSE. See the GU General Public License for more details. You should have received a copy of the GU General Public License along with this program; If not, see < -*- texinfo {Function File} =} sos2tf Convert series second-order sections to direct = @var{sos} = matrix of series second-order sections, one per @var{a}.'], b1 b2] a1 a2]} for section 1, etc.@* b0 must be nonzero for each section.@* for documentation of the second-order direct-form filter is an overall gain factor that effectively scales the vector (or any one of the itemize are vectors specifying the digital = B(z)/A(z)}. for further zp2sos sos2pz zp2tf deftypefn function [B,A] = sos2tf(sos,bscale) nargin = argn(2); if nargin<2, Bscale=1; end [,M] = size(sos); if M~=6, error('sos2tf: sos matrix should be by 6'); end 12

13 A = 1; B = 1; for i=1: B = mtlb_conv(b, sos(i,1:3)); A = mtlb_conv(a, sos(i,4:6)); end nb = length(b); while nb & B(nB)==0, B=B(1:nB-1); nb=length(b); end na = length(a); while na & A(nA)==0, A=A(1:nA-1); na=length(a); end B = B * Bscale; endfunction test B=[1 1]; A=[1 0.5]; [sos,g] = tf2sos(b,a); [Bh,Ah] = sos2tf(sos,g); assert({bh,ah},{b,a},10*eps); 13