1、1ch8 Digital Filter Structures ky nh k x nk2Introductionl llR system cant be implemented using the convolution sum,because the impulse response is of infinite lengthl FIR system can be implemented using the convolution sum which is a finite sum of productsNkknxkhny0MkkNkkknxpknydny013Introductionl H
2、owever,a direct implementation of the llR finite-dimensional system is practicall Forms of Implementation:The actual implementation of an LTI digital filter can be either in software or hardware form,depending on applicationsl Note That:In either case,the signal variables and the filter coefficients
3、 cannot be represented with infinite precision.4 So,a direct implementation of a digital filter based on either the difference equation or the finite convolution sum may not provide satisfactory performance due to the finite precision arithmetic:It is thus of practical interest to develop alternate
4、realizations and choose the structure that provides satisfactory performance under finite precision arithmetic5The Importance of the structural representation:-the first step in the hardware or software implementation of an LTI digital filter The structural representation provides the key relations
5、between some pertinent internal variables with the input and output that in turn provides the key to the implementation.68.1 Block Diagram Representation1)The representation of the input-output relation with analytical expressionl convolution sumkknxkhnyLinear constant coefficient difference equatio
6、nMkkNkkknxpknydny0172)The implementation of an LTI filter -a Valid computational algorithmTo illustrate what we mean by a computational algorithm,consider the causal first-order LTI digital filter shown below8.1 Block Diagram Representation8 The filter is described by the differenceequationyn=-d1yn-
7、1+p0 xn+p1xn-1 Using the above equation we can compute yn for n 0 knowing the initial condition yn-1 and the input xn for n -1y0=-d1y-1+p0 x0+p1x-1y1=-d1y0+p0 x1+p1x0y2=-d1y1+p0 x2+p1x1 .8.1 Block Diagram Representation9We can continue this calculation for any value of the time index n we desireAs a
8、 result,the first order difference equation can be interpreted as a valid computational algorithm8.1 Block Diagram Representation108.1.1 Basic Building Blocks The computational algorithm of an LTIdigital filter can be conveniently represented in block diagram form using the basic building blocks sho
9、wn belowxnynwnAxnynyn1zxnxny2ny1nAdderUnit delayMultiplierPick-off node11Advantages of block diagram representation (l)Easy to write down the computational algorithm by inspection (2)Easy to analyze the block diagram to determine the explicit relation between the output and input8.1.1 Basic Building
10、 Blocks12(3)Easy to manipulate a block diagram to derive other equivalent,block diagrams yielding different computational algorithms(4)Easy to determine the hardware requirements(5)Easy to develop block diagram representations from the transfer function directly8.1.1 Basic Building Blocks138.1.2 Ana
11、lysis of Block Diagramsl 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 signalsl Eliminating the unwanted internal variables then re
12、sults in the expression for the output signal as a function of the input signal and the filter parameters that are the multiplier coefficientsAnalysis Method14Example(1)Consider the shown below single-loop feedback StructurelThe output E(z)of the adder is E(z)=X(z)+G2(z)Y(z)lBut from the figure Y(z)
13、=G1(z)E(z)8.1.2 Analysis of Block Diagrams15(2)Analyze the cascaded lattice structure shown below where the z-dependence of signal variables are not shown for brevitylEliminating E(z)from the previous two equations we arrive at 1-G1(z)G2(z)Y(z)=G1(z)X(z)which leads to )()(1)()()()(211zGzGzGzXzYzH8.1
14、.2 Analysis of Block Diagrams16lThe output signals are given by W1=X-S2 W2=W1-S1 W3=S1-W2 Y=W1-S2lFrom the figure we observe S2=z-1 W3 S1=z-1 W28.1.2 Analysis of Block Diagrams172121)(1)()(zzzzXYzHlEliminating W1,W2,W3,S1and S2 we finally arrive at8.1.2 Analysis of Block Diagrams188.1.3 The Delay-fr
15、ee Loop ProbIeml To illustrate the delay-free loop problem consider the structure belowFor physical realizability of the digital filter structure,it is necessary that the block diagram representation contains no delay-free loops(contain delay loops)19l Analysis of this structure yieldsun=wn+ynyn=B(v
16、n+Aun)which when combined results inyn=B(vn+A(wn+yn)l The determination of the current value of yn requires the knowledge of the same value yn 8.1.3 The Delay-free Loop ProbIem20l However,this is physically impossible toachieve due to the finite time required tocarry out all arithmetic operations on
17、 adigital machinel Method exists to detect the presence ofdelay-free loops in an arbitrary structure,along with methods to locate and removethese loops without the overall input-outputrelation8.1.3 The Delay-free Loop ProbIem21lFigure below shows such a realization of the example structure described
18、 earlier8.1.3 The Delay-free Loop ProbIem228.1.4 Canonic and Noncanonic StructuresDefinition:A digital filter structure is said to be 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 structure The structur
19、e shown below is noncanonic as it employs two delays to realize a first-order difference equationyn=-d1yn-1+p0 xn+p1xn-18.3 Basic FIR Digital Filter Structure NnnznhzH0)(Nkknxkhny023Expression of FIR Filter with Transfer Function and Convolution Suml A causal FIR filter of order N is characterized b
20、y a transfer function H(z)given by which is a polynomial in z-1lIn the time-domain the input-output relation of the above FIR filter is given by8.3.1 Direct Form FIR Digital Filter Structures24Definitionl An FIR filter of order N is characterized byN+1 coefficients and,in general,requireN+1 multipli
21、ers and N two-input addersl Structures in which the multipliercoefficients are precisely the coefficients ofthe transfer function are called direct formstructures25 A direct form realization of an FIR filter canbe readily developed from the convolutionsum description as indicated below for N=44433nx
22、hnxh22110nxhnxhnxhny4433nxhnxh An analysis of this structure yieldsconvolution sum description8.3.1 Direct Form FIR Digital Filter Structures26 The transpose of the direct form structureshown earlier is indicated below Both direct form structures are canonic with respect to delays8.3.1 Direct Form F
23、IR Digital Filter Structuresxnh0ynz1z1z1h1h2h3278.3.2 Cascade Form FIR Digital Filter Structuresl A higher-order FIR transfer function canalso be realized as a cascade of second orderFIR sections and possibly a first-ordersectionl To this end we express H(z)asKkkkzzhzH1221110)()(where:k=N/2 if N is
24、even,k=(N+1)/2 if N is odd,with 2k=028l A cascade realization for N=6 is shown belowl Each second-order section in the above structure can also be realized in the transposed direct form8.3.2 Cascade Form FIR Digital Filter Structuresxnynz1z1h011 21z1z1z1z112 2213 23298.3.4 Linear-Phase FIR Structure
25、s IntroductionType I:hn=hN-n,N is evenThe symmetry(or antisymmetry)property of a linear-phase FIR filter can be exploited to reduce the number of multipliers into almost half of that in the direct form implementations Consider a length-7 Type 1 FIR transferfunction with a symmetric impulse response:
26、3213210zhzhzhhzH)(654012zhzhzh30lRewriting H(z)in the form)()()(516110zzhzhzH34232zhzzh)(we obtain the realization shown below8.3.4 Linear-Phase FIR Structures xnh0ynh1h2z1z1z1z1h3z1z131Type 2:hn=hN-n,N is oddThe corresponding realization is shown as right For example,a length-8 Type 2 FIR transferf
27、unction can be expressed as)()()(617110zzhzhzH)()(435232zzhzzh8.3.4 Linear-Phase FIR Structures 328.3.4 Linear-Phase FIR Structures xnh0ynh1h2z1z1z1z1h3z1z1z133 Note:The Type 1 linear-phase structure fora length-7 FIR filter requires 4 multipliers,whereas a direct form realization requires 7multipli
28、ers Note:The Type 2 linear-phase structure for a length-8 FIR filter requires 4 multipliers,whereas a direct form realization requires 8 multipliers Similar savings occurs in the realization of Type 3(hn=-hN-n,N is odd)and Type 4(hn=-hN-n,N is even)linear-phase FIR filters with antisymmetric impulse
29、 responses8.3.4 Linear-Phase FIR Structures 8.4 Basic IIR digital filter StructureNNNNMMMMzdzdzddzpzpzppzH)1(1110)1(1110)(0010knxdpknyddnyMkkNkk34l Description of IID Digital Filter with Transfer Function and Difference Equation l From the difference equation representation,it can be seen that the r
30、ealization of the causal IIR digital filters requires some form of feedback35l An N-th order IIR digital transfer function is characterized by 2N+1 unique coefficients,and in general,requires 2N+1 multipliers and 2N two-input adders for implementation8.4 Basic IIR digital filter Structure368.4.1 Dir
31、ect Forml Direct Form IIR filters:Filter structures in which the multiplier coefficients are precisely the coefficients of the transfer functionl Example:Consider for simplicity a 3rd-order IIR filter with a transfer function33221133221101zdzdzdzpzpzppzDzPzH)()()(We can implement H(z)as a cascade of
32、 two filter sections as shown on the next37Note:The direct form I structure is noncanonic as it employs 6 delays to realize a 3rd-order transfer function8.4.1 Direct FormDirect form 138Various other noncanonic direct form structures can be derived by simple block diagram manipulations as shown below
33、The order of the cascade linear system can be changed8.4.1 Direct Form3911Observe in the direct form structure shown right,the signal variable at nodes and are the same,and hence the two top delays can be shared22 Likewise,the signal variables at nodes and are the same,permitting the sharing of the
34、middle two delays8.4.1 Direct Form40l Following the same argument,the bottom two delays can be sharedl Sharing of all delays reduces the total number of delays to 3 resulting in a canonic realization shown on the next along with its transpose structureCanonic realization(Direct Form II)8.4.1 Direct
35、Form41Direct Form II(Left)Direct Form II(right)8.4.1 Direct Form428.4.2 Cascade Form IIR Filter Realizationsl The Analytical Expression FormBy expressing the numerator and the denominator polynomials of the transfer function as a product of polynomials of lower degree,a digital filter can be realize
36、d as a cascade of low-order filter sections)()()()()()()()()(321321zDzDzDzPzPzPzDzPzHlConsider,for example,H(z)=P(z)/D(z)expressed as43 Examples of cascade realizations obtainedby different pole-zero pairings are shown below There are altogether a total of 36 different cascade realizations of H(z)ba
37、sed on pole-zero-pairings and ordering8.4.2 Cascade Form IIR Filter Realizations44Due to finite wordlength effects,each such cascade realization behaves differently from othersl Wordlength EffectsUsually,the polynomials are factored into a product of 1st-order and 2nd-order polynomialsl polynomial f
38、actorIn the above,for a first-order factorkkkkkzzzzpzH22112211011)(022kk8.4.2 Cascade Form IIR Filter Realizations45l Consider the 3rd-order transfer function22211222211211111111110zzpzHzzzz)(l One possible realization is shown below8.4.2 Cascade Form IIR Filter Realizations46Example:321321201804010
39、203620440zzzzzzzH.)(112121401508010203620440zzzzzz.are shown on the nextDirect form IICascade form8.4.2 Cascade Form IIR Filter Realizations478.4.3 Parallel Form IIR Digital Filter Structuresl The Analytical Expression FormA partial-fraction expansion of the transfer function in z-1 leads to the par
40、allel form I structure l Assuming simple poles,the transfer function H(z)can be expressed askzzzkkkkzH221111010)(012kkl In the above for a real pole48l A direct partial-fraction expansion of the transfer function in z leads to the parallel form II structurel Assuming simple poles,the transfer functi
41、on H(z)can be expressed askzzzzkkkkzH2211221110)(220kkl In the above for a real pole8.4.3 Parallel Form IIR Digital Filter Structures49 The two basic parallel realizations of a 3rd-order IIR transfer function are shown belowParallel form IParallel form II8.4.3 Parallel Form IIR Digital Filter Struct
42、ures50321321201804010203620440zzzzzzzH.)(21115.08.012.05.04.016.01.0)(zzzzzHThe corresponding parallel form I realization is shown belowExample:8.4.3 Parallel Form IIR Digital Filter Structures51 Likewise,a partial-fraction expansion ofH(z)in z yields212115.08.0125.02.04.0124.0)(zzzzzzH The correspo
43、ndingparallel form IIrealization is shown on the right8.4.3 Parallel Form IIR Digital Filter Structures52Exercise:8.3(stable problem),8.7(transfer function),8.14(structure of symmetric FIR),8.15(structure of symmetric FIR),8.19(canonic and transpose configuration),8.20(a)(cascade canonic structure),8.24(Transfer function,difference equation)realization,impulse response,input and output),M8.1,M8.2
