数字信号处理教学课件：Chapter8 Digital Filter Structures

1、Chapter 8Digital Filter StructuresThe Main Contents:The representation of structures of digital filters 8.1,8.2Basic structures of FIR filter1. Direct Forms 2. Cascade Form 3.Polyphase Realization 4.Linear-Phase Forms. 8.3Basic structures of IIR filter1.Direct Forms 2.Cascade Realizations 3.Parallel

2、 Realizations. 8.4Summary The n-th output of a digital filter can be represented as:Where hn is the impulse response of the filter. Here we discuss two kinds filters: 1.IIR LTI digital filter with following equation:2.FIR LTI digital filter with following equation:The implementation of a digital fil

3、ter:1. Software: using program to perform.2. hardware: using special DSP devices.The question in implementation:Unsatisfactory perform due to the finite precision arithmetic.Given an implementation of a FIR digital filter:Examples of Implementation of a Digital Filterloop:movmovCall addincincdectest

4、jnz*r0, x0*r1, h0mpy(x0, h0, a)a, br0r1ctrctrloop1.software:Memory arrange is in right-side: 2.Hardware:(1)Hardware multiply-addition unit(2)High speed parallel implementation(3)High speed memory access LARAR3, #X0;the beginning of samplingZAC;set acc to zeroRPT#15;loopMAC#H0, *+;multiply and additi

5、onSACHRESULT;save result8.1 The Representation of Structures of Digital Filters Basic arithmetic units of a digital filter: adder , constant multiplier , unit delayTwo representation of these units:1.Block forms: 2.Signal flow graph: unit delayadderAconstant multiplierAExamples of a Digital Filters

6、Two Representations Eg. Given a 2nd-order digital filter is:Based on above representations of basic units, we can get its block form and its signal flow chart.Examples of a Digital Filters Two Representations Example Its block diagram is as below: The advantages of representation in block diagram fo

7、rm:(1) Easy to write down the computational algorithm.(2) Easy to determination the relation between the input and output.(3) Easy to derive other “equivalent” block diagram yielding different algorithms.(4) Easy to determine the hardware requirements.Examples of a Digital Filters Two Representation

8、s Some Questions in Digital Filter Structure Design:The delay-free loop problemFor physical realization of the digital filter structure, it is necessary that the block diagram representation contains no delay-free loops.Some Questions in Digital Filter Structure Design:The delay-free loop problemCan

9、onic structure:the number of delays is equal to the order of the difference equation. Otherwise, it is a noncanonic structure.8.2 Equivalent StructuresBased on transpose operation, the equivalent structure can be obtained.Steps:(1)Reverse all paths,(2)Replace pick-off nodes by adder, and vice verse,

10、(3)Interchange the input and the output nodes.Equivalent StructuresExample 8.2:Original structureTransposed structureEquivalent StructuresRedrawn transposed structure is shown below:Equivalent StructuresAll other methods for developing equivalent structures are based on a specific algorithm for each

11、 structure.In practice, due to the finite wordlength limitations, a specific realization behaves differently from its other equivalent realization.8.3 Basic FIR Digital Filter StructuresThe characteristics of FIR filters:(1)h(n) is a length-N sequence.(2)H(z) is convergent in |z|0.(3)In structure ,t

12、here is non-recursion.If the h(n) of a FIR filter is a length-N sequence, 0nN-1, then its H(z) is: 8.3.1 Direct Forms(Transversal Forms, Convolution Forms)Its the convolution of LTI system, based on the equation, we can get the direct form:X(n)y(n)h(0)h(1)h(2)h(N-2)h(N-1)z -1z -1z -1We also call it

13、a tapped delay line.And this kind of structure is canonic.It is sensetive for quantization. 8.3.2 Cascade FormDecompose H(z) into a string of first-order or second-order transfer functions:)(ny.Discuss:if N is even, K=N/2 ; if N is odd, K=(N+1)/2 and2k=0.8.3.3 Polyphase RealizationDecompose H(z) int

14、o L-branch polyphase decomposition:The subfilters Em(z) are also FIR filters and can be realized using any of the methods.It is often used in multirate DSP application for computationally efficient realizations.Where:Application:DxnynPolyphase Realization (decimation)Application:DxnynDDApplication:I

15、xnynPolyphase Realization (interpolation)Application:IxnynI I8.3.4 Linear-phase FIR StructuresA linear-phase FIR filter of N-order with hn:The characterization can reduce the number of multipliers (almost a half) indirect form.Eg. The implementations of length 7 and 8 with symmetric impulse response

16、 : 8.3.5 FFT StructuresGiven:Length-N DFTLength-N DFTXLength-N IDFTx(n)h(n)y(n)X(k)H(k)Y(k) Then:The structure is:8.4 Basic IIR Digital Filter StructuresThe characteristics of IIR filters:(1)hn is an infinite length sequence.(2)The poles of H(z) is in finite Z-plane.(3)In structure ,there is recursi

17、on.If hn is the impulse response of a IIR filter, then its H(z) is:8.4.1 Direct Forms(1) Direct form I (noncanonic)Based H(z), we can get the input-output relation is:)(nx)(ny1/z1/z0b1b1/z1/z1/z.1/z1/z1/z1a2aNa2bMbIt is sensitive for quantization. (2) Direct form II (canonic)From direct form I , bas

18、ed on:In LTI system, if change the order of the cascade subsystems,the H(z) will not change. Direct Form II (canonic)1/z1/z1/z1/z1a2aNanx0b1b2bNbnyDirect FormsExample 8.3:Consider the third-order IIR transfer function:Then given the block form.8.4.2 Cascade RealizationsDecompose H(z) into a string o

19、f first-order or second-order transfer functions:(M=N) If the number of real zeros or poles is odd, the there is one2kor2k is zero.Cascade RealizationsExample 8.4:Consider the third-order IIR transfer function:Cascade RealizationsCascade formCascade RealizationsAnother cascade realization:8.4.3 Para

20、llel Realizations It is easy to modify poles with identical form.It is needed partial-fractional expansion. Parallel RealizationsExample 8.5:A partial-fraction expansion ofin z-1 yieldsParallel RealizationsThe corresponding parallel form I realization is shown right.Parallel RealizationsLikewise, a

21、partial-fraction expansion of H(z) in z yieldsThe corresponding parallel form II realization is shown on the rightFor example:例：用典范型和一阶级联型、并联型实现方程： 解：正准型、一阶级联和并联的系统函数表示：图示如下：8.5 Realization of Basic Structure Using MATLABCascade RealizationThe factorization of a polynomial can be carried out in MATL

22、AB using the function factor, which determines the second-order factors directly form the specified transfer function H(z).And the other function is zp2sos.Realization of Basic Structure Using MATLABParallel RealizationParallel forms I and II can be developed using the functions residuez and residue

23、, respectively.8.7 Computational Complexity of Digital Filter StructureTable 8.1 Computational complexity comparison of various realization of an FIR filter of order NstructureNo. of multipliersNo. of two-input addersDirect formCascade formPolyphaseCascade latticeLinear phaseN+1N+1N+12(N+1)INT(N+2)/

24、2NNN2N+1NComputational Complexity of Digital Filter StructureTable 8.2 Computational complexity comparison of various realization of an IIR filter of order NstructureNo. of multipliersNo. of two-input addersDirect form II and IItCascade formParallel form2N+12N+12N+12N2N2NHomework:Problems 8.4，8.9，8.

25、11，8.13，8.25，8.28M8.1，M8.2 Problem 8.5 Analyze the digital filter structure of Figure P8.4, and determine its transfer function H(z)=A(z)/X(z).(a) Is this a canonic structure?(b) What should be the value of the multiplier coefficient k so that H(z) has a unity gain at w=0?(c) What should be the valu

26、e of the multiplier coefficient k so that H(z) has a unity gain at w=0?(d) Is there a difference between these two values of K? If not, why not?Problem 8.5 Figure P8.4Z-1Z-1Z-1Z-1K-1-1X(z)Y(z)Problem 8.10Realize the FIR transfer functionH(z)=(1-0.6z-1)6=1-3.6z-1 + 5.4z-2 - 4.32z-3 + 1.944zz-4 - 0.4666z-5 + 0.0467z-6In the following forms: (a) two different direct forms, (b) cascade of six first-order sections, (c) cascade of three second-order sections, (d) cascade of two third-order sections, and (e) cascade o