ChapterChapter EightEightDigital Filter Structures Main Contents in this chapter1. Block Diagram Representation of the LTI Digital Filter2. Basic FIR Filter Structure3. Basic IIR Filter Structure Introduction The actual implementation of a LTI digital filter could be either in software or hardware form, depending on applications.But ,how can we realize those filters?For a LTI FIR system we knowFor a LTI IIR systemThe convolution sum description of an LTI discrete-time system can, in principle, be used to implement the systemFor an IIR finite-dimensional system this approach is not practical as here the impulse response is of infinite lengthHowever, a direct implementation of the FIR finite-dimensional system is practicalDigital filter StructuresIntroduction The above representations are not clear enough for us to implement those filters(such as in hardware form or improving our algorithms). The structural representation provides the relations between some pertinent internal variables with the input and the output that in turn provide the keys to the implementation.8.1 Block Diagram Representation1. The basic building blocks and signal-flowing diagram: 8.1 Block Diagram Representation8.1 Block Diagram RepresentationExample Its block diagram is as below:8.1 Block Diagram Representation The advantages of representation in block diagram form:(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.8.1 Block Diagram Representation2. The Delay-Free Loop ProblemIt is physically impossible ! We must change it,otherwise ,it cannot be realized physically! (Fig8.3)8.1 Block Diagram Representation3.canonic structure:the number of delays is equal to the order of the difference equation. Otherwise, it is a noncanonic structure.8.2 Equivalent StructureEquivalent Structure:They have the same transfer function.We can get the equivalent structure via the transpose operation:(1) Reverse all paths.(2) Replace pick-off nodes by adders,and vice versa.(3) Interchange the input and the output nodes.8.3 Basic FIR Filter Structure8.3.1 Direct Forms It is called a tapped delay line or a transversal filter.A causal FIR filter of order N8.3 Basic FIR Filter StructureFor Its direct form structure is as belowBoth direct forms are canonic structures.Its Equivalent Structure is as below8.3 Basic FIR Filter Structure8.3.2 Cascade Form Cascade form is also canonic structure. Where H(z) is divided into first-order( equal zero) or second-order transfer functions.8.3 Basic FIR Filter Structure8.3.3 Polyphase realizationSee Fig8.7cSee Fig8.7bSee Fig8.7a8.3 Basic FIR Filter StructureThe above structures are noncanonic.For Where If the delay(z-L) in all subfilters are shared,the structure is canonic. See Fig8.8 The polyphase structures are often used in multirate digital signal processing applications for computation efficient realizations.8.3 Basic FIR Filter Structure8.3.4 Linear-Phase FIR Structures Linear-Phase FIR filter : hn is symmetric, or antisymmetric: The characterization can reduce the number of multipliers(almost a half) indirect form. Example The implementations of length 7 and 8 with symmetric impulse response : 8.3 Basic FIR Filter Structure It is noted that a similar savings occurs in the case of an FIR filter with an antisymmetric impulse response. 8.3 Basic FIR Filter Structure8.3 Basic FIR Filter StructureExample: Consider an FIR filter as belowIts cascade form structure is wanted.Solution:8.3 Basic FIR Filter Structure8.3 Basic FIR Filter StructureIts cascade form structure is as below:8.4 Basic IIR Structure8.4.1 Direct FormFor We can use the possible realization schemeWhere 8.4 Basic IIR StructureFor H1(z),we can get its structure as belowFor H2(z),we can get its structure as belowFig8.128.4 Basic IIR StructureWhile H(z)=H1(z)H2(z),so we can get the direct structure of H(z) as belowIts Equivalent Structure is as below8.4 Basic IIR StructureBecause H(z)=H1(z)H2(z)=H2(z)H1(z),so by exchange H1(z)and H2(z) we can get other two direct structures.8.4 Basic IIR Structure All direct forms above are noncanonic structures. But from the last two structures we can easily get its canonic structures as below.8.4 Basic IIR Structure For direct form structure, It is difficult to modify zeros and poles. 8.4 Basic IIR Structure8.4.2 Cascade Forms It is easy to modify zeros and poles.But the forms of realization are no identical. 8.4 Basic IIR Structure8.4.3 Parallel Form 8.4 Basic IIR StructureIt is needed partial-fractional expansion. It is easy to modify poles with identical form.Example For a system as below Its transfer function 8.4 Basic IIR Structureny1/z1/z4/38/1-3/1nxIts direct form is as rightThe cascade form with first-order ny1/z4/13/12/11/znx8.4 Basic IIR Structure1/z1/z3/10nx3/7-ny4/12/1The paralle form with first-order 8.4 Basic IIR StructureIts direct form is as belowExample For a system as below 8.4 Basic IIR StructureIts cascade form with second-order 8.4 Basic IIR StructureIts paralle form with second-order 8.5 Realization Using MATLABThe cascade form requires the factorization of the transfer function which can be developed using the M-file zp2sosThe statement sos = zp2sos(z,p,k) generates a matrix sos containing the coefficients of each 2nd-order section of the equivalent transfer function H(z) determined from its pole-zero form HomeworkProblems 8.5, 8.10, 8.24, 8.26 M8.1, M8.2, M8.3