Digital Filter Structures Te convolution sum description of an LTI discrete-time system can, in principle, be used to implement te system For an IIR finite-dimensional system tis approac is not practical as ere te impulse response is of infinite lengt However, a direct implementation of te IIR finite-dimensional system is practical
Digital Filter Structures Here te input-output relation involves a finite sum of products: N M y n = = d y n k k= 0 pk x n On te oter and, an FIR system can be implemented using te convolution sum wic is a finite sum of products: N y n = = 0 k x n k k k k k 2
Digital Filter Structures Te actual implementation of an LTI digital filter can be eiter in software or ardware form, depending on applications In eiter case, te signal variables and te filter coefficients cannot be represented wit finite precision 3
Digital Filter Structures 4 However, a direct implementation of a digital filter based on eiter te difference equation or te finite convolution sum may not provide satisfactory performance due to te finite precision aritmetic It is tus of practical interest to develop alternate realiations and coose te structure tat provides satisfactory performance under finite precision aritmetic
Digital Filter Structures 5 A structural representation using interconnected basic building blocks is te first step in te ardware or software implementation of an LTI digital filter Te structural representation provides te key relations between some pertinent internal variables wit te input and output tat in turn provides te key to te implementation
Block Diagram Representation In te time domain, te input-output relations of an LTI digital filter is given by te convolution sum y n = k= k x n k or, by te linear constant coefficient difference equation N M k= k pk x n k y n = d y n k k= 0 6
Block Diagram Representation For te implementation of an LTI digital filter, te input-output relationsip must be described by a valid computational algoritm To illustrate wat we mean by a computational algoritm, consider te causal first-order LTI digital filter sown below 7
Block Diagram Representation Te filter is described by te difference equation y n = d y n p x n p x n 0 Using te above equation we can compute yn for n 0 knowing te initial condition y and te input xn for n : 8
Block Diagram Representation y 0 = dy p0x0 px y = dy0. p0x px0 y 2 =. dy p0x2 px We can continue tis calculation for any value of te time index n we desire 9
Block Diagram Representation Eac step of te calculation requires a knowledge of te previously calculated value of te output sample delayed value of te output, te present value of te input sample, and te previous value of te input sample delayed value of te input As a result, te first-order difference equation can be interpreted as a valid computational algoritm 0
Basic Building Blocks Te computational algoritm of an LTI digital filter can be conveniently represented in block diagram form using te basic building blocks sown below A yn xn xn wn Adder Unit delay yn xn xn Multiplier xn Pick-off node yn xn
Basic Building Blocks Advantages of block diagram representation Easy to write down te computational algoritm by inspection 2 Easy to analye te block diagram to determine te explicit relation between te output and input 2
Basic Building Blocks 3 3 Easy to manipulate a block diagram to derive oter equivalent block diagrams yielding different computational algoritms 4 Easy to determine te ardware requirements 5 Easier to develop block diagram representations from te transfer function directly
Analysis of Block Diagrams 4 Carried out by writing down te expressions for te output signals of eac adder as a sum of its input signals, and developing a set of equations relating te filter input and output signals in terms of all internal signals Eliminating te unwanted internal variables ten results in te expression for te output signal as a function of te input signal and te filter parameters tat are te multiplier coefficients
Analysis of Block Diagrams Example - Consider te single-loop feedback structure sown below 5 Te output E of te adder is E = X G2 Y But from te figure Y = G E
6 Analysis of Block Diagrams Analysis of Block Diagrams Eliminating E from te previous two equations we arrive at wic leads to 2 X G Y G G = 2 G G G X Y H = =
Analysis of Block Diagrams Example - Analye te cascaded lattice structure sown below were te - dependence of signal variables are not sown for brevity 7
Analysis of Block Diagrams 8 Te output signals of te four adders are given by W = X α S 2 W2 = W δs W3 = S εw2 Y = βw γ S 2 From te figure we observe S 2 = W3 S = W2
Analysis of Block Diagrams 9 Substituting te last two relations in te first four equations we get W = X α W3 W 2 = W δ W2 W 3 = W2 εw2 Y = βw γ W3 From te second equation we get W / 2 = W δ and from te tird equation we get W = ε W 3 2
Analysis of Block Diagrams Combining te last two equations we get W ε 3 = W δ Substituting te above equation in W = X α W3, Y = βw γ W3 we finally arrive at Y β βδ γε γ2 H = = X δ αε α2 20
Te Delay-Free Loop Problem For pysical realiability of te digital filter structure, it is necessary tat te block diagram representation contains no delayfree loops To illustrate te delay-free loop problem consider te structure below 2
Te Delay-Free Loop Problem Analysis of tis structure yields u n = w n y n y n = B v n Au n wic wen combined results in y n = B v n A w n y n Te determination of te current value of yn requires te knowledge of te same value 22
Te Delay-Free Loop Problem However, tis is pysically impossible to acieve due to te finite time required to carry out all aritmetic operations on a digital macine Metod exists to detect te presence of delay-free loops in an arbitrary structure, along wit metods to locate and remove tese loops witout te overall input-output relation 23
Te Delay-Free Loop Problem Removal acieved by replacing te portion of te overall structure containing te delayfree loops by an equivalent realiation wit no delay-free loops Figure below sows suc a realiation of te example structure described earlier 24
Canonic and Noncanonic Structures A digital filter structure is said to be canonic if te number of delays in te block diagram representation is equal to te order of te transfer function Oterwise, it is a noncanonic structure 25
Canonic and Noncanonic Structures Te structure sown below is noncanonic as it employs two delays to realie a first-order difference equation y n = dy n p0x n p x n 26
Equivalent Structures 27 Two digital filter structures are defined to be equivalent if tey ave te same transfer function We describe next a number of metods for te generation of equivalent structures However, a fairly simple way to generate an equivalent structure from a given realiation is via te transpose operation
Equivalent Structures 28 Transpose Operation Reverse all pats 2 Replace pick-off nodes by adders, and vice versa 3 Intercange te input and output nodes All oter metods for developing equivalent structures are based on a specific algoritm for eac structure
Equivalent Structures 29 Tere are literally an infinite number of equivalent structures realiing te same transfer function It is tus impossible to develop all equivalent realiations In tis course we restrict our attention to a discussion of some commonly used structures
Equivalent Structures Under infinite precision aritmetic any given realiation of a digital filter beaves identically to any oter equivalent structure However, in practice, due to te finite wordlengt limitations, a specific realiation beaves totally differently from its oter equivalent realiations 30
Equivalent Structures Hence, it is important to coose a structure tat as te least quantiation effects wen implemented using finite precision aritmetic One way to arrive at suc a structure is to determine a large number of equivalent structures, analye te finite wordlengt effects in eac case, and select te one sowing te least effects 3
Equivalent Structures 32 In certain cases, it is possible to develop a structure tat by construction as te least quantiation effects We defer te review of tese structures after a discussion of te analysis of quantiation effects Here, we review some simple realiations tat in many applications are quite adequate
Basic FIR Digital Filter Structures A causal FIR filter of order N is caracteried by a transfer function H given by N = n H n = 0 n wic is a polynomial in In te time-domain te input-output relation of te above FIR filter is given by N k y n = = 0 k x n k 33
Direct Form FIR Digital Filter Structures An FIR filter of order N is caracteried by N coefficients and, in general, require N multipliers and N two-input adders Structures in wic te multiplier coefficients are precisely te coefficients of te transfer function are called direct form structures 34
Direct Form FIR Digital Filter Structures A direct form realiation of an FIR filter can be readily developed from te convolution sum description as indicated below for N = 4 35
Direct Form FIR Digital Filter Structures 36 An analysis of tis structure yields y n = 0 x n x n 2 x n 2 3 x n 3 4 x n 4 wic is precisely of te form of te convolution sum description Te direct form structure sown on te previous slide is also known as a tapped delay line or a transversal filter
Direct Form FIR Digital Filter Structures Te transpose of te direct form structure sown earlier is indicated below Bot direct form structures are canonic wit respect to delays 37
Cascade Form FIR Digital Filter Structures A iger-order FIR transfer function can also be realied as a cascade of secondorder FIR sections and possibly a first-order section To tis end we express H as K 2 H = 0 k = βk β2k were K = N if N is even, and K = N if N 2 2 is odd, wit β2 K = 0 38
Cascade Form FIR Digital Filter Structures A cascade realiation for N = 6 is sown below 39 Eac second-order section in te above structure can also be realied in te transposed direct form
40 Polypase Polypase FIR Structures FIR Structures Te polypase decomposition of H leads to a parallel form structure To illustrate tis approac, consider a causal FIR transfer function H wit N = 8: 4 3 2 4 3 2 0 = H 8 7 6 5 8 7 6 5
4 Polypase Polypase FIR Structures FIR Structures H can be expressed as a sum of two terms, wit one term containing te evenindexed coefficients and te oter containing te odd-indexed coefficients: 8 6 4 2 8 6 4 2 0 = H 7 5 3 7 5 3 8 6 4 2 8 6 4 2 0 = 6 4 2 7 5 3
42 Polypase Polypase FIR Structures FIR Structures By using te notation we can express H as 3 2 7 5 3 = E 4 3 2 0 8 6 4 2 0 = E 2 2 0 E E H =
Polypase FIR Structures 43 In a similar manner, by grouping te terms in te original expression for H, we can reexpress it in te form 3 3 2 3 H = E0 E E2 were now 2 E 0 = 0 3 6 2 E = 4 7 2 E = 2 5 2 8
44 Polypase Polypase FIR Structures FIR Structures Te decomposition of H in te form or is more commonly known as te polypase decomposition 3 2 2 3 3 0 E E E H = 2 2 0 E E H =
Polypase FIR Structures 45 In te general case, an L-branc polypase decomposition of an FIR transfer function of order N is of te form were E L m H = = m = 0 n= 0 wit n=0 for n > N N / L m E m Ln L m m
Polypase FIR Structures Figures below sow te 4-branc, 3-branc, and 2-branc polypase realiation of a transfer function H 46 Note: Te expression for te polypase components are different in eac case E m
Polypase FIR Structures Te subfilters E L m in te polypase realiation of an FIR transfer function are also FIR filters and can be realied using any metods described so far However, to obtain a canonic realiation of te overall structure, te delays in all subfilters must be sared 47
Polypase FIR Structures Figure below sows a canonic realiation of a lengt-9 FIR transfer function obtained using delay saring 48
Linear-Pase FIR Structures 49 Te symmetry or antisymmetry property of a linear-pase FIR filter can be exploited to reduce te number of multipliers into almost alf of tat in te direct form implementations Consider a lengt-7 Type FIR transfer function wit a symmetric impulse response: H 2 = 0 2 3 4 5 2 0 6 3
Linear-Pase FIR Structures Rewriting H in te form 6 H = 0 2 4 5 2 3 we obtain te realiation sown below 3 50
Linear-Pase FIR Structures 5 A similar decomposition can be applied to a Type 2 FIR transfer function For example, a lengt-8 Type 2 FIR transfer function can be expressed as 7 6 H = 0 2 5 3 4 2 3 Te corresponding realiation is sown on te next slide
Linear-Pase FIR Structures 52 Note: Te Type linear-pase structure for a lengt-7 FIR filter requires 4 multipliers, wereas a direct form realiation requires 7 multipliers
Linear-Pase FIR Structures Note: Te Type 2 linear-pase structure for a lengt-8 FIR filter requires 4 multipliers, wereas a direct form realiation requires 8 multipliers Similar savings occurs in te realiation of Type 3 and Type 4 linear-pase FIR filters wit antisymmetric impulse responses 53