Digital Filter Structures 1.Block Diagram Representation Chapter 8 Input-output relation of an LTI system can be Block Diagram Representation realized using different computational ●●● algorithms Equivalent Structures .Basic realization forms of FIR and IIR digital Digital Filter Structures Basic FIR Digital Filter Structures filters are considered Basic IIR Digital Filter Structures Mitra's book covers also various more sophisticated realizations of digital filters,e.g. lattice structures,allpass sections,and state space structures,not discussed in this course 1.Block Diagram Representation 1.Block Diagram Representation 1.Block Diagram Representation The convolution sum description of an LTI In the time domain,the input-output relations To illustrate what we mean by a discrete-time system can,in principle,be used of an LTI digital filter is given by the computational algorithm,consider the causal to implement the system. convolution sum or,by the linear constant first-order LTI digital filter shown below For an IIR finite-dimensional system,this coefficient difference equation approach is not practical as here the impulse For the implementation of an LTI digital filter, response is of infinite length. the input-output relationship must be However,a direct implementation of the IIR described by a valid computational algorithm. finite-dimensional system is practical w(n)=-dv(n-1)+px(n)+px(n-1)
Chapter 8 Digital Filter Structures Digital Filter Structures Block Diagram Representation Equivalent Structures Equivalent Structures Basic FIR Digital Filter Structures Basic FIR Digital Filter Structures Basic IIR Digital Filter Structures Basic IIR Digital Filter Structures 1. Block Diagram Representation Input-output relation of an LTI system can be realized using different computational algorithms Basic realization forms of FIR and IIR digital filters are considered Mitra’s book covers also various more sophisticated realizations of digital filters, e.g. lattice structures, allpass sections, and state space structures, not discussed in this course 1. Block Diagram Representation The convolution sum description of an LTI discrete-time system can, in principle, be used to implement the system. For an IIR finite-dimensional system, this approach is not practical as here the impulse impulse response is of infinite length. However, a direct implementation of the IIR finite-dimensional system is practical 1. Block Diagram Representation In the time domain, the input-output relations of an LTI digital filter is given by the convolution sum or, by the linear constant linear constant coefficient difference equation For the implementation of an LTI digital filter, the input-output relationship must be described by a valid computational algorithm. 1. Block Diagram Representation To illustrate what we mean by a computational algorithm, consider the causal first-order LTI digital filter order LTI digital filter shown below 1 z 1 z 0 p 1 p 1 d x( ) n y( ) n 1 pxn( 1) 0 pxn( ) 1 dyn( 1) 1 01 yn dyn pxn pxn ( ) ( 1) ( ) ( 1)
1.Block Diagram Representation 1.1 Basic Building Blocks 1.1 Basic Building Blocks Using the above equation we can compute y(n) The computational algorithm of an LTI digital filter can be conveniently represented in block The corresponding signal flow charts are for n0 knowing the initial condition -1) diagram form using the basic building blocks shown on the right-hand side and the inputx(n)for n-1 shown below →☒+一0 y(O)=-dy-1)+Pwx(0)+Ax(-1) )=-d(O)+Px()+P,x(0) 一甲一ww一一m sH() -a(n) y(2)=-4)+Px(2)+Px1) Multiplior We can continue this calculation for any value of n we desire (by iterative computation) 网一回一 Uait dday PicLoff nade(Bruncling (n) 1.1 Basic Building Blocks 1.1 Basic Building Blocks 1.2 Analysis of Block Diagrams Steps of Analyzing Block Diagrams Advantages of block diagram/signal flow 3 Easy to manipulate a block diagram to derive Carried out by writing down the expressions chart representation other"equivalent"block diagrams yielding for the output signals of each adder as a sum 1 Easy to write down the computational different computational algorithms. of its input signals,and developing a set of algorithm by inspection. 4Easy to determine the hardware requirements. equations relating the filter input and output 2 Easy to analyze the block diagram to 5 Easier to develop block diagram signals in terms of all internal signals determine the explicit relation between the representations from the transfer function Eliminating the unwanted internal variables output and input. directly. then results in the expression for the output signal as a function of the input signal and the filter parameters that are the multiplier coefficients
1. Block Diagram Representation Using the above equation we can compute y(n) for nı0 knowing the initial condition y(ˉ1) and the input x(n) for nıˉ1 We can continue this calculation for any value of n we desire (by iterative computation) 1 01 y dy px px (0) ( 1) (0) ( 1) 1 01 y(1) (0) (1) (0) d y px px 10 1 y(2) (1) (2) (1) d y px px 1.1 Basic Building Blocks The computational algorithm of an LTI digital filter can be conveniently represented in block diagram form using the basic building blocks basic building blocks shown below 1 z x n( ) y n( ) x n( ) y n( ) x n( ) y n( ) x n( ) x n( ) x n( ) w n( ) The corresponding signal flow charts signal flow charts are shown on the right-hand side z 1 1 z a x(n) x(n1) x(n) ax(n) a x1(n) x2(n) x1(n)+x2(n) x(n) x(n1) x(n) ax(n) x1(n) x1(n)+x2(n) x2(n) 1.1 Basic Building Blocks Advantages of block diagram/signal flow Advantages of block diagram/signal flow chart representation ķ Easy to write down the computational algorithm by inspection. ĸ Easy to analyze the block diagram to determine the explicit relation between the output and input. 1.1 Basic Building Blocks 1.1 Basic Building Blocks Ĺ Easy to manipulate a block diagram to derive other “equivalent equivalent” block diagrams yielding different computational algorithms. ĺ Easy to determine the hardware requirements. Ļ Easier to develop block diagram representations from the transfer function directly. 1.2 Analysis of Block Diagrams Steps of Analyzing Block Diagrams Steps of Analyzing Block Diagrams Carried out by writing down the expressions for the output signals of each adder as a sum of its input signals, and developing a set of equations relating the filter input and output signals in terms of all internal signals Eliminating the unwanted internal variables then results in the expression for the output signal as a function of the input signal and the filter parameters that are the multiplier coefficients
1.2 Analysis of Block Diagrams 1.2 Analysis of Block Diagrams 1.3 Canonic and Noncanonic Structures Example Eliminating E(z)from the previous two A digital filter structure is said to be canonic Consider the single-loop feedback structure equations we arrive at if the number of delays in the block diagram shown below [1-G,(z)G2(e)]Y(z)=G(2)X(2) representation is equal to the order of the transfer function which leads to Otherwise,it is a noncanonic structure G国 Y(2) H(2)= G() The structure shown in the next slide is The output E(z)of the adder is X(2)1-G(2)G(2) noncanonic as it employs two delays to E(z)=X(2)+G,(2)Y(z) realize a first-order difference equation But from the figure Y(2)=G(z)E(z) 1.3 Canonic and Noncanonic Structures 2.Equivalent Structures 2.Equivalent Structures .Two digital filter structures are defined to be Transpose Operation y(n)=-d(n-1)+px(n)+px(n-1) equivalent if they have the same transfer (1)Reverse all paths function (2)Replace pick-off nodes by adders,and vice We describe next a number of methods for the versa generation of equivalent structures (3)Interchange the input and output nodes However,a fairly simple way to generate an .All other methods for developing equivalent equivalent structure from a given realization structures are based on a specific algorithm is via the transpose operation for each structure
1.2 Analysis of Block Diagrams Example Example Consider the single-loop feedback structure shown below The output E(z) of the adder is But from the figure 1 G z( ) 2 G z( ) X ( )z Y z( ) E z( ) 2 Ez X z G zY z () () () () 1 Yz G zEz () () () 1.2 Analysis of Block Diagrams Eliminating E(z) from the previous two equations we arrive at which leads to 12 1 1 () () () () () G zG z Y z G zX z 1 1 2 ( ) ( ) ( ) () 1 () () Y z G z H z X z G zG z 1.3 Canonic and Noncanonic Structures A digital filter structure is said to be canonic canonic if the number of delays in the block diagram representation is equal to the order of the transfer function Otherwise, it is a noncanonic noncanonic structure The structure shown in the next slide is noncanonic as it employs two delays to realize a first-order difference equation 1.3 Canonic and Noncanonic Structures 1 z 0 p 1 p 1 d x( ) n y( ) n 1 z 1 01 yn dyn pxn pxn ( ) ( 1) ( ) ( 1) 2. Equivalent Structures Two digital filter structures are defined to be equivalent equivalent if they have the same transfer function We describe next a number of methods for the generation of equivalent structures However, a fairly simple way to generate an equivalent structure from a given realization is via the transpose operation 2. Equivalent Structures Transpose Operation (1) Reverse all paths (2) Replace pick-off nodes by adders, and vice versa (3) Interchange the input and output nodes All other methods for developing equivalent structures are based on a specific algorithm for each structure
2.Equivalent Structures 2.Equivalent Structures 2.Equivalent Structures There are literally an infinite number of Under infinite precision arithmetic any given Hence,it is important to choose a structure equivalent structures realizing the same realization of a digital filter behaves that has the least quantization effects when transfer function identically to any other equivalent structure implemented using finite precision arithmetic It is thus impossible to develop all equivalent However,in practice,due to the finite One way to arrive at such a structure is to realizations wordlength limitations,a specific realization determine a large number of equivalent In this course we restrict our attention to a behaves totally differently from its other structures,analyze the finite wordlength discussion of some commonly used structures equivalent realizations effects in each case,and select the one showing the least effects 2.Equivalent Structures 3.FIR Digital Filter Structures 3.FIR Digital Filter Structures .In certain cases,it is possible to develop a ◆Direct Form .A causal FIR filter of order N-I is structure that by construction has the least characterized by a transfer function A(z) quantization effects ◆Cascade Form given by We defer the review of these structures after a Linear-phase Structure H(2)=∑hk)z discussion of the analysis of quantization which is a polynomial in effects (not included in Kuo's revised book) In the time-domain the input-output relation Here,we review some simple realizations that of the above FIR filter is given by in many applications are quite adequate m=2Aka-k利 -0
2. Equivalent Structures There are literally an infinite number of equivalent structures realizing the same transfer function It is thus impossible to develop all equivalent realizations In this course we restrict our attention to a discussion of some commonly used structures 2. Equivalent Structures Under infinite infinite precision arithmetic any given realization of a digital filter behaves identically to any other equivalent structure However, in practice, due to the finite wordlength limitations limitations, a specific realization behaves totally differently from its other equivalent realizations 2. Equivalent Structures Hence, it is important to choose a structure that has the least quantization effects least quantization effects when implemented using finite precision arithmetic finite precision arithmetic One way to arrive at such a structure is to determine a large number of equivalent structures, analyze the finite wordlength effects effects in each case, and select the one showing the least effects 2. Equivalent Structures In certain cases, it is possible to develop a structure that by construction has the least quantization effects We defer the review of these structures after a discussion of the analysis of quantization effects (not included in Kuo’s revised book) Here, we review some simple realizations that in many applications are quite adequate 3. FIR Digital Filter Structures Direct Form Cascade Form Linear-phase Structure phase Structure 3. FIR Digital Filter Structures A causal FIR filter of order Nˉ1 is characterized by a transfer function H(z) given by which is a polynomial in In the time-domain the input-output relation of the above FIR filter is given by 1 0 () () N k k H z hkz 1 z 1 0 () ()( ) N k y n hkxn k
3.1 Direct Form FIR Digital Filter 3.1 Direct Form FIR Digital Filter 3.1 Direct Form FIR Digital Filter Structures Structures Structures .An FIR filter of order N-lis characterized by A direct form realization of an FIR filter can An analysis of this structure yields N coefficients and,in general,require N be readily developed from the convolution yn)=h0)x(n)）+hI)x(n-1)+h(2)x(n-2) multipliers and N-1 two-input adders sum description as indicated below for N=5 +h(3)x(n-3)+h4)x(n-4) Structures in which the multiplier coefficients which is precisely of the form of the are precisely the coefficients of the transfer convolution sum description function are called direct form structures The direct form structure shown on the previous slide is also known as a tapped delay line or a transversal filter. 3.1 Direct Form FIR Digital Filter 3.2 Cascade Form FIR Digital Filter 3.2 Cascade Form FIR Digital Filter Structures Structures Structures .A cascade realization for N=6 is shown below A higher-order FIR transfer function can also be realized as a cascade of second order FIR 2 M4 sections and possibly a first-order section To this end we express H(z)as General Form xdaH He)=ho1+A2+B2） -2 where k=N/2 if N is even,andk=(N+1)/2 if Nis odd,with =0
3.1 Direct Form FIR Digital Filter Structures An FIR filter of order Nˉ1is characterized by N coefficients and, in general, require N multipliers and Nˉ1 two-input adders Structures in which the multiplier coefficients are precisely the coefficients of the transfer function are called direct form structures 3.1 Direct Form FIR Digital Filter Structures A direct form realization of an FIR filter can be readily developed from the convolution sum description as indicated below for N =5 1 x n( ) z x n( 1) x n( 2) x n( 3) x n( 4) h(0) h(1) h(2) h(3) h(4) y n( ) 1 z 1 z 1 z 3.1 Direct Form FIR Digital Filter Structures An analysis of this structure yields which is precisely of the form of the convolution sum description The direct form structure shown on the previous slide is also known as a tapped delay tapped delay line or a transversal transversal filter. ( ) (0) ( ) (1) ( 1) (2) ( 2) (3) ( 3) (4) ( 4) yn h xn h xn h xn h xn h xn 3.1 Direct Form FIR Digital Filter Structures x n 1 z h 0 h 2 1 z h 3 1 z h 4 1 z h 1 y n x n 1 x n 2 x n 3 x n 4 x n 1 z h 0 h 2 1 z h N 2 h N 1 1 z h 1 y n General Form 3.2 Cascade Form FIR Digital Filter Structures A higher-order FIR transfer function can also be realized as a cascade of second order FIR cascade of second order FIR sections sections and possibly a first-order section To this end we express H(z) as where if N is even, and if N is odd, with 1 2 1 2 1 ( ) (0) 1 K k k k Hz h z z k N / 2 k N 1 /2 2 0 k 3.2 Cascade Form FIR Digital Filter Structures A cascade realization for N = 6 is shown below 11 21 1 z 12 22 13 23 h(0) 1 z 1 z 1 z 1 z 1 z h(0) 11 x n 21 12 22 13 23 y n
3.3 Linear-Phase FIR Digital Filter 3.3 Linear-Phase FIR Digital Filter 3.3 Linear-Phase FIR Digital Filter Structures Structures Structures Linear-phase FIR filter of length N is characterized by the symmetric impulse ·Length Nis odd(=7) response h(n)=h(N-1-n) H(z)=h0)+h)z-+h(2)z2+h3)zd An antisymmetric impulse response condition h(m)=-h(N-1-n) +h2)z4+h(①)z5+M0)z6 results in a constant group delay and"linear- =h(01+z)+h1)(2+z-5) phase"property Symmetry of the impulse response +h2)(22+2)+h3)z3 coefficients can be used to reduce the number of multiplications 3.3 Linear-Phase FIR Digital Filter 3.3 Linear-Phase FIR Digital Filter 3.3 Linear-Phase FIR Digital Filter Structures Structures Structures General Form ·Length Nis even(=8) Nis oen Type 1 and 3 c(n) H(E)=h0)+h0)z+h2)z-2+h(3)z-3 Nn2 multipliers +3)z4+h(2)z5+h1)z+h(0)z7 A(2 N2-W Direct Form =h0)1+z)+h1)(21+z6) N is odc Type 2 and 4 needs N s(n) multipliers +h2)z2+25)+h(3)(2-3+z) W+1)2 multipliers
3.3 Linear-Phase FIR Digital Filter Structures Linear-phase FIR filter of length N is characterized by the symmetric impulse symmetric impulse response response An antisymmetric antisymmetric impulse response impulse response condition results in a constant group delay and “linearphase” property Symmetry of the impulse response Symmetry of the impulse response coefficients can be used to reduce the number coefficients can be used to reduce the number of multiplications of multiplications hn hN n () ( 1 ) hn hN n () ( 1 ) 3.3 Linear-Phase FIR Digital Filter Structures Length N is odd ( N=7 ) 123 45 6 6 15 24 3 ( ) (0) (1) (2) (3) (2) (1) (0) (0)(1 ) (1)( ) (2)( ) (3) Hz h h z h z h z h z hz h z h z hz z h z z hz 3.3 Linear-Phase FIR Digital Filter Structures h(2) h(3) 1 z h(0) h(1) 1 z 1 z 1 z 1 z 1 z x(n) y(n) 1 z 1 z 1 z 1 z 1 z 1 z h(0) h(1) h(2) h(3) 3.3 Linear-Phase FIR Digital Filter Structures Length N is even ( N=8) 123 4 56 7 7 16 25 34 ( ) (0) (1) (2) (3) (3) (2) (1) (0) (0)(1 ) (1)( ) (2)( ) (3)( ) Hz h h z h z h z hz h z hz h z h z hz z hz z hz z 3.3 Linear-Phase FIR Digital Filter Structures 1 z 1 z 1 z 1 z h(0) h(1) h(2) h(3) 1 z 1 z 1 z h(2) h(3) 1 z h(0) h(1) 1 z 1 z 1 z 1 z 1 z 1 z 3.3 Linear-Phase FIR Digital Filter Structures 1 z 1 z 1 z 1 z x n( ) 1 z 1 z 1 z y n( ) 1 z 1 z 1 z 1 z 1 z 1 z x n( ) y n( ) h(0) h(1) h(2) h N( / 2 1) h(0) h(1) h(2) 1 2 N h General Form Type 1 and 3 (N+1) /2 multipliers N/2 multipliers Type 2 and 4 f1 f1 f1 f1 f1 f1 f1 f1 f1 Direct Form needs N multipliers
4.1 Direct Form IIR Digital Filter 4.1 Direct Form IIR Digital Filter 4.IIR Digital Filter Structures Structures Structures The causal IIR digital filters we are concerned Direct forms--Coefficients are directly the ◆Direct Form transfer function coefficients with in this course are characterized by a real ◆Cascade Form rational transfer function of zor, Consider for simplicity a 3rd-order IIR filter ◆Parallel Form equivalently by a constant coefficient with a transfer function (assuming do=1) difference equation. H)=P) From the difference equation representation,it D2)1+dz+d,22+d23 can be seen that the realization of the causal We can implement H(z)as a cascade of two IIR digital filters requires some form of filter sections as shown below feedback. 4.1 Direct Form IIR Digital Filter 4.1 Direct Form IIR Digital Filter 4.1 Direct Form lIR Digital Filter Structures Structures Structures .where Hi(z)=P(z)=po+p+p+pz The time-domain representation of (z)is Considering the basic cascade realization given by results in Direct form I: H,(z)=1/D(z) n)=w(n)-dyn-1)-d2yn-2)-dy(i-3) The filter section H (z)can be seen to be an H()=P(a).-1 (z)】 FIR filter and can be realized as shown below Realization of H,(z)follows from the above equation and is shown below in) Pe P m-2 -d -d M-3) zeros poles
4. IIR Digital Filter Structures Direct Form Cascade Form Parallel Form 4.1 Direct Form IIR Digital Filter Structures The causal IIR digital filters we are concerned with in this course are characterized by a real rational transfer function of or, equivalently by a constant coefficient difference equation. From the difference equation representation, it can be seen that the realization of the causal IIR digital filters requires some form of feedback. 1 z 4.1 Direct Form IIR Digital Filter Structures Direct forms Direct forms -- Coefficients are directly the transfer function coefficients Consider for simplicity a 3rd-order IIR filter with a transfer function (assuming ) We can implement H(z) as a cascade of two filter sections as shown below 123 01 2 3 123 123 ( ) ( ) () 1 P z p pz pz pz H z D z dz dz dz 0 d 1 2 H z( ) W z( ) 1 X z( ) H z( ) Y z( ) 4.1 Direct Form IIR Digital Filter Structures where The filter section H1(z) can be seen to be an FIR filter and can be realized as shown below 123 1 01 2 3 2 () () ( ) 1/ ( ) H z P z p pz pz pz H z Dz p1 1 z p2 1 z p3 1 z p0 x n( ) w n( ) 4.1 Direct Form IIR Digital Filter Structures The time-domain representation of H2(z) is given by Realization of H2(z) follows from the above equation and is shown below 12 3 yn wn dyn d yn dyn ( ) ( ) ( 1) ( 2) ( 3) 1 d 1 z 2 d 1 z d3 1 z w n( ) y n( ) y n( 1) y n( 2) y n( 3) 4.1 Direct Form IIR Digital Filter Structures Considering the basic cascade realization results in Direct form Direct form I : 1 () () ( ) Hz Pz D z p1 1 z p2 1 z 3 p 1 z p0 x n( ) 1 d 1 z d2 1 z d3 1 z y n( ) zeros poles
4.1 Direct Form IIR Digital Filter 4.1 Direct Form IIR Digital Filter 4.1 Direct Form IIR Digital Filter Structures Structures Structures .Changing the order of blocks in cascade Observe in the direct form structure shown .Sharing of all delays reduces the total number results in Direct formⅡ： below,the signal variable at nodes①and① of delays to 3 resulting in a canonic H()=P().-1 are the same,and hence the two top delays realization shown below along with its P(z) (2)D(2) can be shared transpose structure. 。Likewise,.the signal variables at nodes② P and are the same,permitting the sharing of 100 the middle two delays d Following the same argument,the bottom two delays can be shared P poles zeros 4.2 Cascade Realizations 4.2 Cascade Realizations 4.2 Cascade Realizations By expressing the numerator and the Examples of cascade realizations obtained by .There are altogether a total of 36(P2P2) denominator polynomials of the transfer different pole-zero pairings are shown below different cascade realizations of function as a product of polynomials of lower degree,a digital filter can be realized as a H(z)= Pe)E(2)E(e)E(2) cascade of low-order filter sections (often sos) D(2)D(zD,(2)D(z) .Consider,for example,H(z)=P(zD(z) 岛岛儡岛岛岛 based on pole-zero-pairings and ordering expressed as H(2)= P(2)P(z)B(z)B(2) Due to finite wordlength effects,each such 圖圆岛岛岛岛 cascade realization behaves differently from D(=)D.(E)D.(2)D,(=) Others
4.1 Direct Form IIR Digital Filter Structures Changing the order of blocks in cascade results in Direct form Direct form II : 1 1 () () () () () Hz Pz Pz Dz Dz poles zeros p0 x( ) n d1 1 z d2 1 z d3 1 z p1 1 z 2 p 1 z 3 p 1 z 1 y n( ) 1' 2 3 2' 3' 4.1 Direct Form IIR Digital Filter Structures Observe in the direct form structure shown below, the signal variable at nodes and are the same, and hence the two top delays can be shared Likewise, the signal variables at nodes and are the same, permitting the sharing of the middle two delays Following the same argument, the bottom two delays can be shared 1 1’ 2 2’ 4.1 Direct Form IIR Digital Filter Structures Sharing of all delays reduces the total number of delays to 3 resulting in a canonic canonic realization shown below along with its transpose transpose structure structure. 0 p 1 d 1 z d2 1 z 3 d 1 z p1 p2 3 p 0 p d1 d2 3 d p1 1 z 2 p 1 z p3 1 z 4.2 Cascade Realizations By expressing the numerator and the denominator polynomials of the transfer function as a product of polynomials of lower degree, a digital filter can be realized as a cascade of low-order filter sections (often sos) Consider, for example, H(z)=P(z)/D(z) expressed as 123 123 ( ) () () () ( ) () () () () P z PzP zP z H z D z D zD zD z 4.2 Cascade Realizations Examples of cascade realizations obtained by different pole-zero pairings are shown below 1 1 ( ) ( ) P z D z 2 2 ( ) ( ) P z D z 3 3 ( ) ( ) P z D z 1 3 ( ) ( ) P z D z 2 1 ( ) ( ) P z D z 3 2 ( ) ( ) P z D z 1 1 ( ) ( ) P z D z 2 3 ( ) ( ) P z D z 3 2 ( ) ( ) P z D z 1 2 ( ) ( ) P z D z 2 3 ( ) ( ) P z D z 3 1 ( ) ( ) P z D z 1 2 ( ) ( ) P z D z 2 1 ( ) ( ) P z D z 3 3 ( ) ( ) P z D z 1 3 ( ) ( ) P z D z 2 2 ( ) ( ) P z D z 3 1 ( ) ( ) P z D z 4.2 Cascade Realizations There are altogether a total of 36 different cascade realizations of based on pole-zero-pairings and ordering Due to finite wordlength effects, each such cascade realization behaves differently from Others 123 123 ( ) () () () ( ) () () () () P z PzP zP z H z D z D zD zD z 2 2 3 3 ( ) P P
4.2 Cascade Realizations 4.2 Cascade Realizations 4.2 Cascade Realizations x(n) wn) xn) Realizing complex conjugate poles and zeros -1) with second order blocks results in real coefficients n-2) m-2) Example R(z) D() Usually,the polynomials are factored into a product of Ist-order and 2nd-order(sos) Third order transfer function polynomials: 1+B2+B22 H(2)= P(2) (1+-1++B ()-Po u D(z) 1+g2 1+a22+a2 D() (a) for a first-order factor==0 4.2 Cascade Realizations 4.3 Parallel Realizations 4.3 Parallel Realizations One possible realization is shown below Parallel realizations are obtained by making The two basic parallel realizations of a 3rd use of the partial fraction expansion of the order IIR transfer function are shown below transfer function Parallel form 1: Yos +ruz ●General structure: H回=%+∑ Parallel form II: 1+au2+a2z H(日 H() () He)=d+∑ +2 1+aa2+a2 Parallel FormI Parallei Form
4.2 Cascade Realizations x n( ) y n( ) x n( 1) x n( 2) y n( 1) y n( 2) 0 b 1b 2 b 1 z 1 z 1 z 1 z 1 a 2 a 1 z 1 z 1 a 2 a 0 b 1b 2 b 1 z 1 z 1P z( ) 1 D z( ) w1 w2 x n( ) y n( ) 1 D z( ) 1P z( ) 4.2 Cascade Realizations x( ) n y n( ) a1 a2 b0 b1 b2 1 z 1 z Usually, the polynomials are factored into a product of 1st-order and 2nd-order (sos) polynomials: for a first-order factor 1 2 1 2 0 1 2 1 2 1 ( ) 1 k k k k k z z Hz p z z 2 2 0 k k 4.2 Cascade Realizations Realizing complex conjugate poles and zeros with second order blocks results in real coefficients Example Example Third order transfer function 1 12 11 12 22 0 1 12 11 12 22 ( ) 1 1 ( ) () 1 1 P z z z z Hz p D z z zz 4.2 Cascade Realizations One possible realization is shown below General structure: 0 p 12 1 z 22 1 z 12 22 11 1 z 11 H1(z) H2(z) HN/2(z) 4.3 Parallel Realizations Parallel realizations are obtained by making use of the partial fraction expansion of the transfer function Parallel form I: Parallel form II: 1 0 1 0 1 2 1 2 ( ) 1 k k k k k z H z z z 1 2 1 2 0 1 2 1 2 ( ) 1 k k k k k z z H z z z 4.3 Parallel Realizations The two basic parallel realizations of a 3rd order IIR transfer function are shown below 11 12 22 0 11 12 22 1 z 12 22 01 12 11 02 0 x( ) n x n( ) y n( ) y n( ) 1 z 1 z 1 z 1 z 1 z Parallel Form I Parallel Form II
4.3 Parallel Realizations 4.3 Parallel Realizations 4.3 Parallel Realizations ●General structure: Example Their realizations are shown below He) .A partial-fraction expansion of -01 0.44+0.362z2+0.00225 0.6 H Φ H(z)= 1+0.4z+0.18z2-0.2z inzyields 04 040 -0.5-0.221 Hvn() He)=-0.1+_0.6 +1-0.42+1+0.82+0.52 =0.2 ·Easy to realize: Likewise,a partial-fraction expansion of H(z) No choices in section ordering and in z yields -05 No choices in pole and zero pairing 0.2424,0.22-1+0.25z2 H(z)= 1-0.42+1+0.82+0.52 Parallel Form I Parallel For ll
4.3 Parallel Realizations General structure: Easy to realize: No choices in section ordering and No choices in pole and zero pairing H1(z) H2(z) HN/2(z) 4.3 Parallel Realizations Example Example A partial-fraction expansion of in yields Likewise, a partial-fraction expansion of H(z) in z yields 2 3 1 23 0.44 0.362 0.002 ( ) 1 0.4 0.18 0.2 z z H z z z z 1 z 1 1 12 0.6 0.5 0.2 ( ) 0.1 1 0.4 1 0.8 0.5z H z z z z 1 12 1 12 0.24 0.2 0.25 ( ) 1 0.4 1 0.8 0.5 z z z H z z z z 4.3 Parallel Realizations Their realizations are shown below 1 z 0.8 1 z 0.5 1 z 0.6 0.2 0.4 0.5 0.1 1 z 1 z 1 z 0.24 0.2 0.25 0.8 0.5 0.4 Parallel Form I Parallel Form II