版《数字信号处理(英)》课件ch8-Digital-Filter-Structures.ppt
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- 数字信号处理英 数字信号 处理 课件 ch8 Digital Filter Structures
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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
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