The Digital All-Pass Filter A Versatile Signal

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The Digital All-Pass Filter: A Versatile SignalProcessing Building BlockPHILLIP A. REGALIA, STUDENT MEMBER, IEEE,AN D P. P. VAIDYANATHAN, MEMBER, IEEE

The digital all-pass filter is a computationally efficient signal pro-cessing building block which is quite useful in many signal pm-cessing applications. In this tutorial paper we review the propertiesof digital all-pass filters, and provide a broad overview of the diver-sity of applications in digital filtering. Starting with the definitionand basic properties of a scalar all-pass function, a variety of struc-tures satisfying the all-pass property are assembled, with emphasisplaced on the concept of structural losslessness. Applications arethen outlined in notch filtering, complementary filtering and filterbanks, multirate filtering, spectrum and group-delay equalization,Hilbert transformat ions, and so on. In all cases, the structural loss-lessness property induces very robust performance in the face ofmultiplier coefficient quantization. Finally, the state-space mani-festations of the all-pass property are explored, and it is shown thatmany all-pass filter structures are devoid of limit cycle behaviorand feature very low roundoff noise gain.I . INTRODUCTION

In many signal processingapplications, hedesigner mustdetermine the transfer function o f a digital filter subject toconstraints on the frequency selectivity andlor phaseresponse which are dictated by the application at hand.Once a suitable transfer function i s found, the designermust select a filter structure from the numerous choicesavailable. Ultimately, fin ite precision arithmetic i s used inany digital filter computation, and traditionally the round-off noise and coefficient sensitivity characteristics haveformed the basis of selecting one filter structure in favorof another.

In he quest for low coefficient sensitivityand low round-off noise, an elegant theory of losslessness and passivity inthe discrete-time domain has evolved[I], (21. Although thistheory has been motivated by the desire to obtain d igitalfilters with predictable behavior under finite word-lengthconditions, many useful by-products have emerged whichhave contributed to a better understanding of computa-

Manuscript received JulyIO,1987; evised October 16,1987. hiswork was supported by the National Science Foundation underGrants MIP 85-08017 nd MIP 8404245.P. A. Regalia and S.K. Mitra are wi th the Department of Electricaland Computer Engineering, Universi ty of Cal ifornia, Santa Bar-bara, CA 93106,USA.P. P. Vaidyanathan is with the Department of Electrical Engi-neering, Cal ifornia Institute of Technology, Pasadena, CA 91125,USA.I E E E Log Number 8718413.

SANJIT K. MITRA, FELLOW, IEEE,

tionally efficient filter structures, unable filters, filter bankanalysis/synthesis systems, multira te filtering, and stabili tyunder linear and nonlinear (i.e., quantized) environments.

This paper considersa basic scalar lossless building block,which i s a stable all-pass function. The interconnections ofsuch lossless building blocksform useful solutions to manypractical filter ing problems. The many results presentedhere can be derived by appealing to elegant theoreticalforms; however, in maintaining a tutorial tone it s our aimto expose the salient features using direct discrete-timeconcepts, in he hope that the references cited will furtheraid both hedesigner and researcher alike. Weshould poin tout that many of the results which are developed in termsof scalar all-pass functions in one dimension can be gen-eralized ovector or matrixall-pass functions [3]and to mul-tidimensional filtering [4 ] , hough for the presentwe restrictour attention to the one-dimensional scalar case.

We begin in Section I by defining a scalar all-pass func-tion and reviewing some basic properties. Section Il lassembles avariety of all-pass filter structures, wi th empha-si s placed on the concept of structurallosslessness. SectionIV outlines applications o notch iltering, SectionV to com-plementary filters and filter banks, Section V I to multiratesignal processing, SectionVII to tunablefilters, and SectionV l l l togroupdelay equalization. Finally, Section IX exploresstate-space representations of lossless transfer funct ions,and the implications of losslessness in obtainingvery robustperformance under fin ite word-length constraints.II. DEFINITIONSND PROPERTIES

The frequency responseA(ei'")of an all-pass filter exhibitsunit magnitude at all frequencies, i.e.,

JA(e'")(*= 1, for all U . (2.1)The transfer function of such a filter has all poles and zerosoccurring in conjugate reciprocal pairs, and takes the form

(2.2)For stability reasons we assume (Ykl < 1 for a ll k to placeall the poles nsidethe uni t circle. Now, ifA(z) sconstrainedto be a real function, we must have0 = 0 or 0 = T ,and any

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complex pole at z = Yk must be accompanied bya complexconjugate pole at z = 7 In this caseA(z) can be expressedin the form

(2.3)In effect, the numerator polynomial is obtained from thedenominator polynomial by reversing he order of the coef-ficients. For example,

a2 + alz-l + z - ~I + alz-' + a 2 2 - 2A(z) =isasecond-order all-pass functionoftheform of(2.3)above,

since the numerator coefficients appear in the reverse orderofthose in thedenominator. In hiscase,thenumeratoranddenominator polynomials are said to form a mirror-imagepair.

If we lif t the restr iction that A(z) be a real function, thenA(z) takes the more general form

z - M D*(l/z*)D(z)A(z) = (2.5)

The numerator and denominator polynomials now form aHermitian mirror-image pair. For example,

with0 = arg(a;/ao), isrecognizedasacomplexall-passfunc-tion due to the Hermitian mirror-image relation betweenthe numerator and denominator polynomials.Properties

A(z) = Y(z)/U(z) revealsFrom the definition of an all-pass function in (2.1), setting

1 y(eju)12= I u(eju)12, for all w. (2.7)Upon integrating both sides from w = -- a to ra nd applyingParseval's relation [I], we obtain

m mc I y(n)12 = ldn)l2. (2.8)It s convenient to interpret the two sides of (2.8)as the out-put energy and input energy of the digital filter, respec-tively [I], [5]. Thus an all-pass filter is lossless, since the out-put energy equals the input energy for all finite energyinputs. If the all-pass filter i s stable as well, it s termed Loss-less Bounded Real (LBR) [2], or more generally LosslessBounded Complex [6] if the coefficients are not all real.

Another useful property follows from (2.1) with the aidof the maximum modulus theorem. In particular, since astableall-pass functionhasall t s poles nsidethe unitcircle,all i t s zeros outside, and exhibits unit magnitude along the

"= -m

This relation i s useful in verifying stability in lattice real-izations of all-pass filters.

The last property of interest is the change in phase forreal all-pass filter over the frequency range w E [0 ,TI. estart with the group delay function T(W) of an all-pass filter,which i s usually defined as

ddw7 ( w ) = -- arg A(e/")]. (2.10)

Note that the phase function must be taken as continuousor "unwrapped" [5] if 7 ( w ) i s to be well-behaved. Since anall-pass unction i s devoid of zeros on the unit circle accord-ing to (2.1), the phase function arg A(ei") can always beunwrapped with no ambiguities. Now, the phase responseof a stable all-pass function i s a monotonically decreasingfunction of w, so that T(W) i s everywhere positive. An Mth-order real all-pass function, in fact, satisfies the propertylou(w ) dw = M r. (2.11)The interpretation of (2.11) i s that the change in phase ofthe all-phase function as w goes from 0 to r s -MT radians.1 1 1 . ALL-PASSILTER S T R U C T U R E S

The (Hermitian) mirror-image symmetry relation betweenthe numerator and denominator polynomials of an all-passtransfer function can be exploited to obtain a computa-tionally efficient filter realization with a minimum numberof multipliers. To see this, consider the second-order all-pass function of (2.4) which, upon expressing A(z) =Y(z)/U(z), corresponds o the second-orderdifference equa-tion

y(n) = -32[u(n)- y(n - 2)1+ al [u(n- 1)- y( n - I ) ] + u(n - 2) (3.1)

in which terms have been grouped in such a way that onlytwo multiplications are required. A similar strategy can beapplied to an arbitrary Mth-order all-pass filter, such thatonly M multiplications are required to compute each out-put sample. On the other hand, a direct-form filter real-ization would in general require 2M + 1 multiplications tocomputeeach output sample. I n his sense,an all-pass filterrepresents a computationally efficient structure.

Thedifferenceequation as expressed n (3.1) requiresfourdelay (or storage) elements to be realized as a filter struc-ture. Since the difference equation i s of second order, thisdoes not represent a canonic realization. However, mini -mum multiplier delay-canonic all-pass filter structures canbe developed using the multiplier extraction approach[q,[8]. For example, consider the digital two-pair network ofFig. 1,which has a constraining multiplier b, at the second"port": U2(z)= bl Y2(z).The transfer function as seen fromthe remaining port s constrained to be a first-order all-pass

unit circle, one can deduce 1

(2.9) Fig. 1. Digital two-pair network con strained with a multi.>I, or IzI < 1. plier of value b,.

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function(3.2)

This allows one t o solve for the internal parameters of thedigital two-pair. By following this and similar strategies,numerous first-order and second-order all-pass filter struc-tures have been catalogued [7-[10], many with roundoffnoise expressions as functions of the pole locations, andwith the minimum multiplier property. Such filter struc-tures have the property that, upon quantizing the multi-plier coefficients, t he numerator polynomial and denom-inator polynomial retain a mirror-image relationship, andthus the all-pass characteristic s independent of any mul-tiplier quantization. Filter structures with this property aretermed structurally lossless or structurallyLBR [2].

Another useful structure for realizing all-pass functionsis the Gray and Markel lattice filter [Ill.he synthesis pro-cedure uses the fo llowing recursion [12]:

m = M, M - 1, * ,1 (3.3a)where

k = A(=) (3.3b)Now, with AJz) a stable all-pass function, it can be verifiedusing(2.lO)that k < ,andthatA,-l(z) isastableall-passfunction of one order lower. The structural interpretationof (3.3) for the first step in the recursion appears as Fig. 2,

A&)AF - +)

Fig. 2. Two multip lier lattice nterpre tation of all-passfunc-tion order reduction process.where AMWl(z) is an (M - Vth-order all-pass function. Therecursion of (3.3) then continues on AM-l(z), and so on,which leads to the cascaded lattice realization of Fig. 3,,m-*,mAo

A&) A&) A,(dFig. 3. The cascaded lattice implemen tation of t he all-passfunction A,(z).where the constraining mult iplier A has uni t magnitude.Each lattice two-pair i s realized as in Fig. 2, although theoverall transfer function s unaltered i f each lattice two-pairi s implemented as per Fig. 4(a) or (b). Fig. 4(a) is the singlemultiplier form [Ill,hich requires the fewest number ofmultipliers. Fig. 4(b) i s the normalized form [13], wh ich hasthe advantage that all internal nodes are automaticallyscaled in the f2 sense [14]. Although more multip liers arerequired, the four multiplications can be performed simul-taneously using CORDlC processor techniques [15].With all the lattice structures above, stability of the filter

(a)/ I - Ik 1

(b)Fig. 4. (a) The single multiplier lattice two-p air. (b) he nor-malized lattice two-pair.i s equivalent to the condition that Ik,J < 1 or a l l m. Assuch, the all-pass lattice filters form a useful model forchecking stability of transfer functions; the connection ofthese lattice filters with system stability tests is explainedin [12]. In addition, the mirror-image relation between thenumerator and denominator polynomials holds in spite ofcoefficient quantization (except for the normalized form).Accordingly, the all-pass property for these lattice filters i sstructurally induced. These structures, in fact, have otherfavorable inite word-length properties which are exploredin greater detail i n Section IX .The above discussion has assumed that AM(z) s a real all-pass function, although the lattice filter i s easily general-ized for complex all-pass functions as well [6], 1161. In thiscase, the recursion of (3.3) becomes

rn = M, M - 1, * ,1 (3.4a)where now

k, = A:(=) (3.4b)The complex lattice two-pair now appears as Fig. 5. As inthe real coefficient case, stability of AM(z) s equivalent to

Fig. 5. Lattice filter interpretation of order reduction pro-cess for com plex all-pass functio ns. The signals at all nodesare assumed to b e complex-valued.

the condition I k < 1 for all rn . The complex general-izationsof thestructuresof Fig.4appearas Fig.6,with prop-erties similar to those of their real arithmetic counterparts.

An alternate method of realizing complex all-pass func-tions i s to cascade irst-order complex all-pass filters. Refer-ring to (2.21, each ter m in the product corresponds to the

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(a)

Fig. 6. Complex generalizations of the "single multiplier"lattice two-pair (a), and the no rmalized two-pair (b).The sig-nals at all nodes are in general complex valued.first-order difference equation

y(n) = Y k y h - 1) + Y;u(n) - u(n - 1) (3.5)which may be arranged as

y(n)= Re { y k } y(n - 1) + u(n)l+ j Im { y k } y(n - 1) - u(n)l- u(n - 1). (3.6)

plex all-pass ilter has a pole and zero on the imaginary axis.As shown in Section VI, such structures form useful build-ing blocks in certain multirate fil tering systems.

The structures presented here have emphasized he con-cept of losslessness n he discrete-timedomain. We shouldpoint out that the connection between discrete-time loss-lessness (and passivity) and continuous-time lossless (andpassive) networks i s well established I]. Thus many of theall-pass ilter s tructures above can be developed as the wavedigital counterpart of the appropriate lossless electricalnetworks. Indeed, a great body of literature has beendevoted to this subject; the interested reader i s referred tothe overview in [19].IV . DIGITALOTCH ILTERS

The first fil tering application of the all-pass filter we w illinvestigate s the design of a digital notch filter, which isuseful for removing a single-frequency component f rom asignal, such as an unmodulated carrier i n communicationsystems, or power supply hum from a sampled analog sig-nal, etc. This i s easily implemented using the circuit of Fig.8, where the transfer function realized i s

G(z) = ;[I + A(z)].

Fig. 7. Implementation of structurally lossless first-ordercomplex all-pass filter. The signals at all n odes are in gene ralcomplex-valued.multiplication) and offering again the structurally inducedall-pass property.

Finally, a complex all-pass function can be obtained froma real all-pass function using the frequency transformation[17l, [I81(3.7)- ' -+ eimz-' = (cos + + j sin 4)z-l.

(4.1)

The all-pass function i s chosen as second-order, so that thechange in phase of A(ei") as w goes from 0 to x i s -2x radi-ans. As such, from (4.1) it follows that

G(eiw) = o (4.2)where coo is the angular frequency at which the all-pass filterprovides a phase shift of A radians. The notch character-istics of (4.2) are structurally induced provided A(z) i s real-ized in a structurally lossless form. Design procedures anda catalogue of minimum-mul tiplier structures are detailedin [20].A particularly useful choice for the second-order all-pass filter i s the lattice filter, shown in minimum multiplierform for convenience in Fig. 9, where

k2 + kl(l + k2)z- ' + z- '1 + kl(l + k2)z-l + k2z-2'(z) = (4.3)

This transformation rotates the po le-zero plot of a real all-pass filter by + radians in the z-plane. In practice though,the multipliers which implement cos + and sin + cannot,in general, be quantized such that their squares sum tounity. As such, i f the transformation of (3.7) is applied to aminimum multiplier real all-pass filter, the structurallyinduced osslessroperty ma y not be preserved. one se-ful exception i s with + =

z - l - kjz-'. (3.8) Fig. 9. The second-order all-pass filter A ( z ) realized usinga cascade of single mu ltiplier lattice two-pairs.One can show that this choice of all-pass filte r allows inde-7r12, whence (3.7) becomespendent tuning of the notch frequency wo and the 3-dBattenuation bandwidth i according to

k l = -COS WO (4.4a)This transformation requires no explicit multipliers, andretains the structurally induced all-pass property. If (3.8) i sapplied to a first-order real all-pass filter, the resulting com-

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1 - an (6212)- I tan (6212)k - (4.4b)

Note from (4.3) that k2 = r 2 ,wherer is the pole radius. Somefrequency response examples are shown in Fig. O . Fig. 10(a)shows how the 3-dB attenuation bandwidth 62 can bechanged by adjusting only the mul tipl ier k,,while Fig. 10(b)

I " " I " " I " " I " " ]1.0

0.5

0.00.0 0. 1 0.2 0. 3 0. 4 0.5Normalized Frequency

(a)

k, = 0.875

0.0 0.1 0.2 0 .3 0.4 0.5Normalized Frequency(b)

Fig. 10. The "orthogonal" tuning of the notch filter: (a)illustrates how the rejection bandwidth can be adjusted byvarying a single coefficient k while (b) illustrates how thenotch frequency can be tuned by varying only k, .illustrates howtheno tch requencywocanbechangedwith-out affecting the 3-dB attenuation bandwidth by adjustingonly kl. n view of the "orthogonal" tun ing obtainedaccordingto(4.4),and thefactthat stability strivial oensureinaquantizedenvironment,thestructure sseen to bequiteamenable to adaptive notch filter applications.An interesting modification to the circuit,of Fig.8 resultsby replacing he delay variablez -' yz- N. The circuit thenprovidesNcomplex-conjugatezero pairs along the unit cir-cle, located at angles of

2 m & woN radians, n = 0,1, , N - 1, (4.5)

with coo given by (4.4a). Equation (4.4b) holds once 62 is takenas the total 3-dB attenuation bandwidth; the 3-dB atten-uation band about each notch frequency has width 62lN.

One very useful alignment results with kl= 0 in (4.4a),whichsets w0 = d 2 and places 2N zeros spaced equally along theunit circle. In this case, the frequency w = r12N and all itsodd harmonics are filtered out, which allows the removalof symmetric periodic signals with period 7 = 4N of oth-erwise arbitrary waveshape.

Greater flexibility in the notch characteristic can begained by using a first-order complex all-pass filter, withtransfer function

y* - - l1 - yz-1A ( z ) = e'' ~

where y = a + jb . We begin by wr iting (4.6) in the formA ( z ) = e'[GI(z) + jH l ( z ) l (4.7)

wheref ( z )D ( z )

a - 1 + a 2 - b2)z- '+ a z - 21 - 2az-' + (a 2 + b 2 ) z - 2G1(z)=- (4.8a)

(4.8b)( z ) b - 2abz-' +D ( z )H , ( z ) =- I 2az-1 + (a2 + b2)z-2'From (4.8) the following observations are noted. First, bothGl(z) and Hl(z)are second-order eal ransfer functions witha complex-conjugate pole pair at z = a f b, and second,both pEz) and Q ( z ) are symmetric polynomials, with thezeros of these polynomials determined by the po le loca-tions. Symmetry of f ( z )and Q ( z ) mplies the zeros of eitherpolynomial occur as a complex-conjugate pair on the uni tcircleorasareciprocal pair on the realaxis. Now, byabsorb-ing the factor e'' in (4.7) we obtain

(4.9)(z ) = G z ( z )+ ~Hz(z)where now

(4.10b)By varying 8, the zeros of G2(2) nd H 2 ( z ) an be moved alongthe unit circle (or along the real axis) without affecting thepole locations.As such, low-pass ribtch and high-pass notchcharacteristicscan be realized. In particular, given a pair ofpoles at z = a & jb (which then determines D ( z ) , ( z ) ,andQ ( z ) ) ,he zeros of G 2 ( z ) an be placed at z = e lw oprovidedthat

(4.11)Since both P(z)and Q(z)arf symmetric polynomials, the left-hand side of (4.11) i s real or anywo,which implies a solutionfor B always exists. The filtering function G2( z )s realized byfeeding a real input sequence to the complex all-pass filterA(z) and retaining the real component of the outputsequence. The application of such filters in low-sensitivitycascade realizations of elliptic transfer functions can befound in [21].V. DOUBLYOMPLEMENTARYILTERS

Two filters are said to be complementary f the passbandsof one match the stopbands of the other. Complementaryfilters f ind applications in various signal processing sys-

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tems where different frequency bands are to be processedseparately to measure signal strengths in each band or t oachieve, for example, compression or noise reduction. Inthis section we consider two stable transfer functions whichare all-pass complementary

IG(e/") + H(e/")J= 1, for all w (5.1)as well as power complementary

I C(e/")I2 + 1 H(e/w)12= I, for all w . (5.2)Transfer function pairs which satisfy both complementaryproperties in (5.1) and (5.2) are termed doubly complemen-tary[22]. We shall see that all-pass filters play a natural ro lein doubly complementary filters.

First, by combining (5.1) and (5.2) we obtainI G(e/") + H(e'")l2 = 1 G(e/")I2 + I H(elW)l2 I ,

for all w. (5.3)Now, by in terpreting G(e/") and H(e/") as phasors in thecomplex plane, application of t he law of cosines to the leftequality of (5.3) reveals that G(e/") and /+(e/") must exhibitphase quadrature with respect to each other for all w . Assuch, the phasors [G(e/") + H(e /" ) ] nd [G(e/")- H(e/")]mustexhibit the same magnitude, namely unity, for all W . By ana-lytic continuation we thus obtain

G(z) + H(z) = Al(z)G(z) - H(z) = A ~ ( z ) (5.4)

where Al(z) and A2(z) are stable all-pass funct ions. Solvingfor G(z) and H(z ) results in

H(z) = ;[Ai(z) - A~(z ) ] . (5.5)A structural implementation of (5.5) appears as Fig. 11 , inthe form of the sum and difference of two all-pass filters.Referring to Fig. 11, it i s seen that, as G(z) is the sum of twoall-pass functions, the passband (respectively, stopband) ofG(e/") occurs for frequencies for which the tw o all-passfunctions are in phase (respectively, out of phase) wi th eachother. Since H(z ) i s the difference of the tw o all-pass filters,H(el") has a stopband where G(e/") as a passband, and viceversa, in accordance with (5.2).

Thestructureof Fig. 11was first recognized i n digital filterdesign in terms of wave digital lattice filters [23], wherebybilinearly transformed versions of G(z) and H(z ) are inter-preted as the reflection and transmission coefficients,respectively, of a real symmetric lossless two-port networkwith equal resistiveterminations. Byexploitingce rtainanal-ogies with classical network theory, Gazsi [24] has devel-oped explicit formulas for the multip lier coefficients of theall-pass filters t o mplement odd-ordered classical low-passfilter approximations.

The structure of Fig. 11can also be examined in terms ofdirect z-domain design techniques [IO], [22], [25]. For exam-ple, Liu and Ansari [IO] have shown optimized bandpassfilter designs which cannot be derived from classical low-pass filte r approximations. We brief ly review some designconsiderations in what follows.

LetAl(z) be an Mlth real order all-pass funct ion. LetA2(z)be an M2th real order all-pass funct ion, and in phasor

Fig. 11. Implementation of the doubly complementary f i l -ter pair as the sum and differenceof all-pass functions.

notation letA, (e / " ) = e/@?(")A2 (e / " )= e/@2(w) (5.6)

where +l(w) and @2(w)are monotone decreasing functionsof U ,with + , ( O ) = 42(0)= 0 ,+,(a) = -M1a, and +2(a) -M2a.Note that this implies for the dc values of the transfer func-tions G(e/') = 1 and H(e1') = 0. Then from (5.5) we obtain

I G(e/")l = 1 e/[6l(w)-+2(~)111

In particular, at w = 71

I H(e/")l= 1 e/[M1-M21*- 1 (5.8)which shows that a low-pass-high-pass complementary pairrequires

(5.9)But if m i s nonzero, this leads to mult iple passbands, andso for a low-pass-high-pass pair

Mi M2 = +I. (5.10)

M1- M2= 2m + 1, m = integer.

Similarly, fo r a bandpass-bandstop pairMi M2 = +2. (5 .11)

In general, the conditionM1 - M2 = + L (5.12)

leads to a total of at least L + 1 passbands (counting hoseof G (e 9 plus those of Me'")). From (5.5) it i s clear that G(z)will have the poles of both Al(z) and A2(z), and thus it s orderwill be M1+ M2. n view of (5.10) then, it is clear that M1+M2must be an odd integer if a low-pass-high-pass com-plementary pair is desired.By lifting the restriction that the all-pass filters be real,even-ordered low-pass-high-pass complementary filterscan be realized as well [16]. Thus let Al(z) be a complex all-pass func tion such that complex poles appear withou t theirconjugates, and let A2(2) be obtained from A,(z) by replac-ing each coefficient with i t s complex conjugate. Accord-ingly, we have

A2(z) = A;(z*). (5.13)Two real transfer functions C(z ) and H ( z ) can then bedefined according to

G(z) = $[A(z) + A*(z * ) ]jH(z ) = i [A(z ) - A*(z * ) ] (5.14)

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where the numbered subscript has been dropped from thecomplex all-pass unction. A(z) contains one pole rom eachcomplex conjugate pai r of G(z), and so its order i s half thatof G(z). The terms G(z) andjH(z) in (5.14) above can be inter-preted as the conjugate symmetric and conjugate antisym-metric parts, respectively, of the complex function A(z) [5].As such, only one complex all-pass function need be imple-mented; the real and imaginary parts of the complex outputsequence are identified as the outputs of the filters G ( z ) ndH(z), respectively, (Fig. 12). In effect, when processinga real

i n p u t j T p (z)(real) I imag'-+ H(z)

Fig. 12. Separating the real and imaginary outputs of acomplex all-pass filter to realize an even-ordered com ple-mentary filter pair.

inpu t sequence, the outputs of the filters A(z) and A*(z*) arecomplex conjugates of each other, and thus implement ingboth complex all-pass ilters in (5.14) is redundant. By invert-ing (5.14) we obtain

G(z) + jH(z) = A(z)G(z) - H(2) = A*(z*) (5.15)

which shows thecomplex all-pass complementary propertyof G(z) and jH(z). With the power complementary relationof (5.2) lef t intact, C(z) and jH(z)are seen to form a doublycomplementary pair.

The realization scheme of Fig. 12 has important practicalsignificance, as one can show that all even order Butter-worth, Chebyshev, and elliptic low-pass (and high-pass)transfer functions can be decomposed as per (5.14) [26].

Asan illustration, supposewedesireafilterwith less that0.025-dB attenuation for frequencies less than 0.14 rad/s(normalized requency), and greater than 45-dB attenuationforfrequenciesgreaterthan 0.2 rad1s. Itturnsoutthatasixth-order e lliptic transfer function wil l satisfy these specifica-tions, w ith the poles of the transfer function located at

0.468823 f 10.221266z = 0.475711 -+ j0.575375i .501533 f 0.780218and three zero pairs on the unit circle. Following design

procedures n [16], the desired transfer function G ( z ) an bedecomposed into complex all-pass functions as per (5.14),where A(z) i s in the form of (2.2) with M = 3 and

7, = 0.468823 + 10.2212667 2 = 0.475711 - 0.5753757 3 = 0.501533 + j0.780218

e = 0.510542.A realization ofA(z) can be obtained by cascading irst ordercomplex all-pass filters as in Fig. 7. A frequency responseplot for G(z )appears as Fig. 13. Inc luded on the p lot i s thefrequency response of the transfer function H(z)obtainedfrom the imaginary part of the complex output, which i spower complementary to that of G(z), as expected.

0 I F , , 8 , .I I , , , I 1 1 , I , , ,Ii

-20hBv

0)4 -40r!ild-60

-800.0 0.1 0.2 0.3 0.4 0.5Normalized Frequency (o/2n)

Fig. 13. Frequency responses of the sixth-order cornple- kmentary filter pair.

Low Sensit ivity PropertiesLet usexaminethefiltersof Figs.11and 12 ingreaterdetail.

With respect to Fig. 11, recall that with A,(z) and A2(z) anMjth - and M2th-order all-pass function, respectively, G ( z )has an order N = M, + M2. Now, with the all-pass filtersrealized in a minimum mu ltiplier form, the overall filterrequires only N multiplications to compute each outputsample, and so i s canonic with respect to the number ofmultiplications. Recall that a direct- form realization wouldrequire typically 2N + 1 multiplications to compute eachoutput sample. But the structureof Fig. 11in fact makes twocomplementary transfer functions available, and so tep-resents an optimum i n terms of computational efficiency.

The structure of Fig. 12 has similar Computationa l econ-omies. With G( z ) an Nth-order transfer function, A(z)becomes an N12th-order complex all-pass function. Thisallows a filter realization using no more that N12 + 1com-plex multiplications to compute each (complex) outputsample. If the complex all-pass filter processes a real inputsequence, this can be configured as 2N + 2 real multipli-cationsfor each complexoutput sample, but requiring onlyN + 2 distinct (real) coefficient values. The number of mul-tiplications is comparable to that of a direct-form realiza-tion, and slightly better than that df a cascade realizationif the filter order i s larger than four. But, as above, a com-plementary transfer function i s made available at no extracost.Letus now turn to the sensitivity properties. Referring o(5.5) or (5.14), if the all-pass filters are realized n structurallylossless orm, it i s clear that IG(e/")l can never exceed uni tyfor any value of W. That is, the magnitude function 1 G(e9 )i s structurally bounded above by one. Suppose now thatG( z ) s designed such that at specific frequencies, t o bedenoted W k , the passband amplitude achieves the upperbound of unity, i.e., I G(e"')l = 1.Regardless of the sign ofany multiplier perturbation (due to quantization) the mag-nitude of the transfer function at w = W k can only decrease.We can thus apply Orchard's argument [27J at these fre-quencies (the so-called points of maximum power transfer)to establish the low-passband sensitivity behavior.

Returning to the design example above, Fig. 14(a) showsthe passband response obtained for G(z) using an &bitbinary representation for each multiplier coefficient of the



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and R2 i s two-poin t DFT matrix. This formulation naturallysuggests the following generalization [29]:

(5.18a)1N(z) = - R,a(z)

g(z) = [Gdz) GAZ) . . . GN(Z)ITwhere now

a(z) = [A&) A~(z) . . AN(z)IT (5.18b)and RN is an N-point DFT matrix. The doubly complemen-tary defin itions of (5.1) and (5.2) can now be extended to Ntransfer functions. I n particular, the application of Parse-Val's relation [5] to (5.18) reveals2

I G,(e/")12 = gel") g(e/") = 1 i(e/w)a(el")= - c IA,(el")12= 1

I = ' N(5.19)

which shows the power complementary property of the Ntransfer functions. Also, by explo iting he properties of DFTmatrices, it i s easily verified that

I NN ='

-0.200.00 0 05 0.10 0.15Normalized Frequency

(a)

Complex Allpass-80

I-100 " " "" ' " I " " " "0.0 0.1 0.2 0.3 0. 4 0. 5

Normalized Frequency(b)

Fig. 14. Effectsof coefficient quantization on C(e'")( usinga complex all-pass-based realization and a cascade real-ization . (a) Passband details and (b) stopband performance.complex all-pass filter.' Shown for comparison purposes i sa conventional cascade realization of G(z) using again8 bitsper multip lier coefficient. The complex all-pass scheme isseen to exhibit much better passband performance. Thestopband behavior i s superior f or the cascade realizationthough (Fig. 14(b)), as it i s known hat the cascade structurehas very good stopband sensitivity characteristics (281.Filter Bank Extensions

Let us rewrite the doubly complementary filter pair of(5.5) as

In matrix form this becomes

where

'The quantization is such that the mantissa of a floating-pointrepresentation is rounded to the specified number of bits, so thatthe relat ivecoefficient accuracy does no t depend on ts magnitude.

(5.20)which shows the all-pass complementary property. More-over, the low-passband sensitivity argument above remainsintact.

One attractive design strategy i s to obtain the all-passfunct ion elements of a(z) from a polyphase decompositionof an Nth-band ow-pass filter [30]-[32]. In his case,a(z) akesthe form

a(z) = [Al(zN) z-'A2(zN) z- ~ A, ( z~ ). . z-~+'AN(z~)] ' .(5.21)

The transfer functions elements of g(z) in (5.18a) can beshown to form a unifo rm filter bank [33] (i.e., the frequencyresponse for each band i s a frequency shifted version ofthat for the adjacent band). Such filter banks are compu-tationally efficient in applications requiring the output sig-nals to be decimated [32]. I n addition, computer-aideddesign procedures for choosing the all-pass functions in(5.21) have been reported [31], [32].

Although in (5.16) we have interpreted R2 as a DFT matrix,other unitary matrix families reduce to the same R2 in the2 x 2 case [34], [35]. As such, using the N-band extensionof (5.18), doubly complementary filter banks can be devel-oped for RN chosen as virtually any N x N unitary matrix.VI . MULTIRATEPPLICATIONS

In he coding and transmission of signals, it is often con-venient to split a signal into i t s various subband compo-nents so that perceptual and statistical properties whichdiffer in each frequency band can be exploited separately.Thistechnique istermed subband coding[33],[36].The basictwo-band system s depicted i n Fig. 15, where the filters G(z)and H(z)are chosen typically as half-band low-pass and high-pass filters, respectively. Since the signal at each analysisfilter bank output occupies only half the available digital

'The tild e notation indicates the conjugate transpose operation.

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________-______.analysis codjng and synthesisfdterbank traosmission fdter bankFig. 15 . Basic two-channel maximally decimated subb andcodin g system.bandwidth, every other sample may bediscarded at the out-put, thereby minimizing the system bit rate. The signals arethen coded, transmitted to the receiver, and decoded. Inthe synthesis filte r bank stage, the signals are up-sampled,filtered, and summed into a composite output.

In a practical system, the nonperfec t frequency separa-tion of the analysis bank filter pair results in aliasing dis-tor tion upon discarding every other sample. This aliasingdistortion then propagates through the remainder of thesystem, and i s considered a rather objectionable by-prod-uct of the decimation operation. In he absence of codingor transmission errors, straightforward analysis reveals thecomposite output may be expressed as

+ ~ U ( - Z ) [ G ( Z )( - Z ) - H ( z ) H ( - z ) ] . (6.1)The first term on the right-hand side of (6.1) i s a filtered ver-sion of the desired signal, while the second term representsthe aliasing distortion. Consider now the following choiceof filtering functions:

G(z) = + z-lA2(z2)1H ( z ) = - z- 'A2(z2 ) ] (6.2)

where A , ( - )and A 2 ( . )are real all-pass functions. Note that(6.2) implies H ( z ) = C ( - z ) , so that t he frequency responsesof the filter pair form a "mirror image" about w = d2 . This,combined with the phasequadrature condition pointed outin Section V, eads to the name Quadrature M irro r Filters(QMF) [36], [37l. Butterworth and symmetric elliptic half-band filter pairs, for example, can be decomposed as per(6.2), and thus be long to the class of quadrature mirr or fil-ters.

This choice of filter pair ha s the following significance.First, using H ( z ) = G ( - z ) it follows that

G(z) G ( - z ) - H ( z ) H ( - z )= G(z) G(-z) - G ( - z ) G ( z ) = 0. (6.3)

As such, the aliased signal term in (6.1) vanishes. This resultis important because, despite critica l subsampling of theanalysis filter bank outputs, the synthesis bank output sig-nal is completely free of aliasing effects. Note also that thisresult makes no assumption as to the quality of frequencyselectivity provided by the analysis bank filte r pair.3

Next, with the aliasing effects removed, the overall sys-tem becomes linear and shift-invariant with transfer

31n practice though, any reduction in the transmitted bit rate viathe coding stage is very dependent on adequate frequency selec-tivity of the analysis bank filter pair.

function

1212

= - G(z) + H(z)l[G(z) - H(z)l(6.4)

The last substitution in (6.4) is easilyverified using (6.2). Theoverall transfer function of the multirate system is thus anall-pass function (save for the scale factor error of i) ,whichindicates perfect magnitude reconstruction of the originalsignal from i ts critically sampled subbands. Moreover, wit hthe alLpassfilters realized n a structurally lossless orm, thealiasing cancellation and perfect magnitude reconstructionproperties become structurally induced.

The resul ting system may be redrawn as Fig. 16(a). Notethat the terms A1(z2)and A2(z2) re even functions of z . Assuch, multirate identities [33] allow the decimators andinterpolators to be passed h rough he all-pass ilters, whi lereplacingz2 by z (Fig. 16(b)), which leads finally to the poly-phase arrangement of Fig. 16(c). The polyphase structure

= - z - ' A , ( z 2 )A2(z2) .

Fig. 16. (a) Two-band maximally decimated quadrature-mirror filter system u sing all-pass filters. (b) Equivalent sys-tem obtained by passing the decimators and interpolatorsthrough the all-pass functions. (c) The polyphase imple-mentation.i s attractive because it allows the all-pass filters to operateat the reduced sampling rate. Further design considera-tions may be found in [381.

Recall from SectionV hat low-pass-high-pass filter pairsmust be of odd order to be realizedas per (5.5)(or (6.2)). Forthe sake of completeness, it i s of interest to develop a cor-responding result for even ordered low-pass-high-passpairs. To this end, consider the modified twochannel mul-tirate scheme of Fig. 17 [39], where the analysis filter bankoutput signals are now subsampled alternately. For this sys-- .Fig. 17. Mo difie d two-band m ultirate system using even-ordered filters G(z) an d H(z ) .



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Fig. 21 . Implementation of acomplex half-band ilter usingreal all-pass filters. If the inp ut signal i s real-valued, he sys-tem reduces o a parallel all-pass network Hilber t transformsystem.

system becomes

(6.12). -1= T /Z A,(-z) A ~ ( - z )which is an all-pass function as expected.sible, as (6.11) reveals

If the input signal is real, further simplifications are pos-

H(z) = C*(z*). (6.13)As such, the two channels of Fig. 15 process signals which,at each stage, share a complex conjugate relat ion, wi th theoutput formed from the difference of these two channels.This allows one channel of the complex multirate systemto bediscarded, with the imaginary output from the remain-ing channel retained as the desired (real) output signal.Using Fig. 21 as the model for c(z) , he real-equivalent sig-nal flowgraph of Fig. 22(a) is obtained. The analysis bankoutput signals may be interpreted as the real and imaginaryparts of a complex analytic sequence wit h only positivefrequency components. A polyphase arrangement s read-ily obtained as Fig. 22(b). This example thus reaff irms thefeasibility of downsampling complex analytic signals [41],

-

-1421.

+@+Re G(n)

(b)Fig. 22 . (a) Multirate Hilbert transform system offering noaliasingdistort ion and perfect magnitude reconstruction ofthe inpu t spectrum. (b) Equivalent polyphase system.

Although we have derived a Hilbert transform systemfrom a real half-band ilter pair, it should be pointed out thatthe all-pass functions in the filter pair of (6.11) can also beobtained using the design formulas in [43]. Several otherauthors have also discussed design techniques for obtain-ing Hilbert transform systems using all-pass sections [44]-[46]. As explained n 42], many of these designs can be den-tically obtained by applying the transformation z- -+- j z - to known real half-band filter designs. This i s an

extension of a result by Jackson [47] showing that theapproximation problem for Hilbert transformers is closelyrelated to that for half-band low-pass filters. This has beenexemplified in (6.10) for half-band filters obtained as thesum and differenceof real all-passfunctions, although anal-ogous results can be developed for those obtained fromcomplex all-pass functions as per (6.6), as well as half-banddesigns obtained from approximately inear phase ilters (tobe introduced in Section VIII). The resu lting designs havepassbands which are symmetric about o = d4, and mul-tirate results n each case can be developed [42]. Finally, wepoint out that Hilbert transform systems with passbandsthat are not symmetric about o = 1~/4an be obtained fromthe above-mentioneddesigns by using standard requencytransformations[a].o such systems, however, multirateresults are not applicable.VII. TUNABLEILTERS AND GENERALIZED COMPLEMENTARYFILTERS

Let us return to the doubly complementary filter of Fig.11, and consider the filter pair which results rom choosing

A&) = 1(7.1)k2+ kl(l + k2)z- + z- = 1 + kl(l + k2) - + k2z-

The transfer function G(z) s recognized as the dig ital notchfilter o f Section IV. In view of the power complementaryrelation of (5.2), H(z)must be a bandpass filter with centerfrequency oo = cos- (-kl). Consider now fhe transferfunction formed wit h the following linear combination:

F(z) = G(z)+ KH(z). (7.2)A realization s sketched in Fig. 23. Due to the phase quad-rature relationship between G(ej) and H(ej9, it i s easily

1+TJ?&+-2

Fig. 23. Tunable frequency response equalization filter.shown that

In view of the filter type realized for G(z) ndH(z),7.3) showsthat the frequency response F(ej9 modifies the gain aboutthe center frequency of /-/(e) while leaving he remainderof the spectrum unaffected. Such a filter i s ideally suitedas a magnitude equalization filter [49], and finds extensiveapplication in digita l audio systems [50]-[52] to compensatefor frequency response deficiencies n an acoustic playbackenvironment.

Design equations can be developed as an extension ofthose presented n (4.4). Thus let coo be the frequency aboutwhich boost or cut i n the magnitude response is desired,and let 0 be the 3-dB notch bandwidth obtained for K = 0.The design equations become

kl = -COS WO

REGALIA et a l . : THE DIGITAL ALL-PASS FILTER 29


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1 - tan (012)- I tan ( ~ 1 2 )k -

K = F(e/"O). (7.4)It can also be shown that w ith K > 0, the circuit providesminimum phase equalization [49]. Some frequencyresponse examples are shown in Fig. 24, which demon-

h%?val5B2

hmUv

K = 0.8125

k, = -0.3125. l O L , , , , , , , I , , , I , , , , , , , j0.0 0.1 0.2 0.3 0.4 0.5Normalized Frequency

(a)10 - -

k, = 0.625 -

K = 3.0, k, = -0.3125

0.0 0.1 0.2 0.3 0.4 0.5Normalized Frequency

(b)kl = -0.3125

K = 3.0, k, = 0.75I , 1 1 1 1 1 1 1 1 1 1 1 , 1 , 1 I I

0.0 0.1 0.2 0.3 0.4 0.5Normalized Frequency(C)

Fig. 24. Illustrating the parametric adjustment of the fre-quency response. (a) Variable gain at the ce nter frequency,(b) adjusting the m odification band width, and (c) tuning thecenter frequency.

strate the true parametric tuning ability of the circuit. Bycascading a few such circuits, a complete parametricallyadjustable digital frequency response equalizer may berealized. By comparison, the designs in [50]-[52] requireprecomputing the multiplier coefficient values for alldesired equalizer settings. This does not represent a tun-able design, and as such has the drawback of requiringexcessive coefficient storage.

More general tunable filters can be realized by using aless trivial choice than (7.1) for the all-pass functions. Sucha tunable filter realization i s attractive since it allows boththe poles and zeros of an Nth-order transfer function G(z)to be tuned by varying only N coefficients in the all-passfilters. I f the all-pass filters are realized in attice form, sta-bility i s trivial to ensure simply by constraining the latticeparameters to have magnitudes less than unity. Moreover,the frequency response type (i.e., low-pass, high-pass,bandpass, etc.) i s related to the orders of the all-pass filtersin view of the discussion surrounding (5.10)-(5.12). If, forexample, Al(z) and A2(z) have orders which differ by one,then we are guaranteed in some sense a low-pass char-acteristic (though not always optimal) or any choice of theall-pass filter parameters, assuming of course that stabilityi s not impaired.

The next prob lem i s to determine the tuning algorithmsfor the all-pass filter coefficients to achieve the desired tun-ability of G(z). Suppose that Al(z) and A2(z)n (5.2) are deter-mined so that their sum G(z) s a satisfactory low-pass filterwith cutoff frequency wl. A new low-pass filter with a dif-ferent cutoff frequencycan beobtained from G(z) using thefrequency transformations of Constantinides [48]

z -1 -+ P(z) (7.5)where P(z) s a stable all-pass funct ion, so that the unit circlein hez-plane maps to the uni t circle in he P(z)-plane. Thusby wri ting @(el")= e- o("), the transformation of (7.5) may beunderstood as the frequency mapping

-+ +). (7.6)Procedures for selecting P(z) can be fou nd in many texts ondigital fiIters[5],[28].The important poin t for ourdiscussioni s that th e transformation of (7.5) maps an all-pass functionto an all-pass function. Hence if G(z) s the sum of tw o all-pass functions, G(p(z-')) i s also the sum of two all-passfunctions. Thus the problem of t unability i s "solved" byimplementing the all-pass functions Al(P(z -I)) andAz(/3(z-')) in (5.5). I n practice though, t he direct imple-mentation of (7.5) may lead to delay-free loops if P(z) doesnot contain a pure delay factor. For example, a low-pass-to-low-pass transformation results for [48]

z - 1 - f,1 - ff1z-Iz -1 + P(z) =

with(7.7a)

(7.7b)where w2 i s the new desired cutoff frequency. Substituting(7.7a) direct lyfor each delay element inAl(z) andA,(z) wouldintroduce delay-free loops. One remedy i s to express thecoefficients of A1(P(z-')) and A2(p(z-l ))as functions of thevariableal. The resultingcoefficientscaneach beexpanded

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as a Taylor series in al.f a, s small (corresponding to asmall shift i n the cutoff frequency), each series expansionmay be truncated after the inear term toobta in a simplifiedexpression for the coefficients [53]. For example, with

(7.8)we find

By expanding the coefficient (a o+ al)/(l+ alao) n a Taylorseries in a, nd truncating after the linear term, we obtainthe approximation

which can be realized using any all-pass implementationscheme by adding to the m ultiplie r branch of value a. theparallel uning branch of value ba,, where b = 1- i . Like-wise, if A2(z) s a second-order all-pass function

(7.11)a similar procedure results inAz(B(z

[a2 + a,al(l - a,)] - al + a1(2 + 2a2 - a:)lz-' + z- 'I - a, + a1(2 + 2a2- a:)] z - l + [a2 + a,al(l - a,)]z-"

(7.12)Noteagain hat thetransfer func tion coefficients haveatun-ing branch ofthetypebial ,i= 1,2, in parallelwith hen om-inal multiplier values a, and a,. Higher order all-pass func-tions can be factored into first-order and second-orderfactors, with the results of (7.10) and (7.12) immediatelyapplicable. Although the derivation of (7.10) and (7.12)assumes a small tun ing variation for al,uning ranges ofseveral octaves have been reported for narrow-band low-pass filters [53].

Another useful choice for P(z) s the fo llow ing low-pass-to-band-pass transformation:

I

with CY, = -cos (a3), here w3 is the desired center fre-quency of the band-pass filter. Note that (7.13) can bedirectlysubstituted for each delay element inA,(z) andA2(z)without introducingdelay-free loops. This results n a band-pass filter G(B(z-')) whose center frequency may be tunedby adjusting a single parameter a2 .Generalized ComplementaryFilters

Let us remove any constraints rom the all-pass unct ions,and consider now the transfer function pair which resultsfrom

F~(z) KlG(z) + K,H(z)F~(z)= K~C(Z) K,H(z). (7.14)

By exploiting the phase quadrature relationship betweenG(e/") and /-/(e/") we obta in

I Fl(e/")I2= K : 1 G(e/")(' + K $ l H(e"91'I F,(e/"))' = K : I G(e/")I2+ K : I (7.15)

Using the fact that C(e/") and H(e'") are power comple-mentary, we find

IFl(e'")12 + I F2(e/")1' = K : + K $ . (7.16)Such filter pairs may be termed generalized complemen-tary filters. Now, substituting (5.5) into (7.14) results in

where

e = tan-' -(2 J 3.(7.17)

Note that wi th 0 = 1r/4 and r = Il&, the doubly comple-mentary filters of Section V result , save for a minus sign inthe second transfer function. Using (7.15) he following i seasily inferred:

(7.18)Equation (7.18) holds true upon replacing "max" every-where with "min." As such, if neither K1nor K, iszero, thenthe transfer function pair provides no stopband per se.However, such filters find application in multi-level pass-band filters [54], [55], and generalized magnitude equal-ization [52].

max, I Fl(e/")l = max, I Fz(el")l = max (1 K1(, 1 K 2 ( ) .

V I I. GROUP-DELAYQUALIZATIONThus far we have overlooked the obvious implica tion of

(2.1):An all-pass filter can be cascaded with another filterto alter the phase response while leaving the magnituderesponse unaffected. In this section, we provide an over-view of applications to group delay equalization, and callattention t o a class of II R filters which has intrinsically goodphase response properties.

Recall from (2.13) that the group-delay response T ~ ( w ) fa given filter F(z) s defined as the negative of the derivativeof it s phase response

With the transfer funct ion F(z) expressed in rational form(8.2)

one can show [56] that the group-delay response becomes

where the prime denotes differentiation w ith respect to z.



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Applying (8.3) to an all-pass function

we obtain the result

Our goal then i s to choose A(z) such that, when cascadedwi th F(z), the resulting group-delay func tion approximatessome desired response (U) ver the passband interval($of F(e/"). Recalling hat the group-delay response i s additivein a cascade connection, our problem i s equivalent to min-imizing the magnitude of the error function

(8.6)over the passband interval($ of F(el") with proper choice of7A(w). Typically, (U) is chosen as a constant over the pass-band interval(s) of F(e/").

If a minimax approximation to a constant group delay i sdesired, then one s tempted to appeal tot he powerful alter-nation theorem for rationa l functions [ 5 7 . Roughly stated,this theorem asserts that the unique best optimum 7A(w)approximating (U) - ~ ~ ( w )n a Chebyshev (or minimax)sense is found when the error function E(W) in (8.6) alter-nates in sign from extremum to extremum wi th equal mag-nitude. However, as pointed out by Deczky[58], group-delayfunctions do not satisfy the conditions of the alternationtheorem, and as such an equiripple error funct ion does notensure a minimax solution.

Nonetheless, an equ iripp le approximation to a constantgroup delay is attractive in applications where waveformdistortion s to be avoided. Unfortunately, the op timizationof 7A(w) imposes nonlinear constraints on the parametersof the all-pass func tion A(z), and closed-form solutions forthe all-pass fil ter parameters are not generally available. Assuch, one must resort to iterative computer approximationmethods. An early report of all-pass filter design for group-delay equalization using he Fletcher-Powell algorithm wasgiven by Deczky [59]. An improvement in speed using amodified Remez-type exchange algorithm was subse-quently reported in [58].

Design procedures for all-pole filters offering an equi-ripple group-delay response have also been reported in heliterature [60]-[62]. These methods can be used for group-delay equalization follow ing a minor reformulation of theproblem. Consider the all-pole counterpart t o the all-passfunct ion of (8.4)

= [(a) 7F(w) + 7A(w)]

(8.7)The group-delay response 7&) corresponding to B(e/") canbe found w ith the aid of (8.3)

Thus let the error func tion of (8.6) be rewritten as;(U) = ;[(U) 7F(w) - M ] - 7B(w). (8.9)

We can thus take ; [ (U ) - F ( W ) - MI as the "ideal" group-delay response, and search for the all-pole filter B(z) =

I/D(z)wi th the best group-delay approximation, using themethods in [60]-[62]. The all-pass function A(z) obtainedfrom (8.4) then achieves the desired group-delay equal-ization of F(z).

Consider now one filter from the doubly complementarypair of (5.5)

G(Z) ;[A~(z)+ A2(~)1. (8.10)if we let 7 , (w)and ~ ~ ( w )enote, respectively, the group-delayresponses of A,(e/") and A,(e'"), then using (8.3) reveals, fol -lowing some algebra, that the group-delay response 7c(w)for G(e/") becomes

(8.11)Recall that the passband(s) or G(e/") occur whereA,(el") andA2(e'") are in-phase. Hence, by equalizing the passbandgroup delay of G(e/"), we approximate lyequalize the group-delay functions 7 , (w) and ~ ~ ( w )ver this same frequencyrange. This suggests that good magnitude and group-delayresponses can be simultaneously achieved by choosingA,(z)such that its group-delay response i s favorable over thedesired passband region, and then choosingA,(z) such thatit phase response i s in-phase and out-of-phase wi th respectto A,(z) over the passband and stopband regions, respec-tively.

A slight variation to his strategy results if we consider thephase response, rather than the group-delay response, tobe important. Thus let us choose one all-pass function asa pure delay

A,(z) = z - ~ (8.12)and choose A2(z) to be in-phase and out-of-phase w ithrespect to the delay over the passband and stopbandregions, respectively. The passband magnitude and phasecharacteris tics of G(e/") are simultaneously opt imized byforc ing A2(e/") o approximate a linear phase characteristicover the passband region of G(e/"). Such filters are termedapproximately linearphase [63], [64]. Although the order ofthese filters i s typically higher than that of an elliptic filter(wi th nonlinear phasecharacteristics) meeting he samefre-quency-selective specifications [64], the signal delay i s typ-ically less than that obtained using an FI R filter. These filtersare thus attractive in applications requiring low waveformdistortion with small delay.Ix. STATE-SPACE MANIFESTATIONSF THE ALL-PASS P R O P E R T Y

The state-space description provides a powerful frame-work whereby many concepts of system theory can beaddressed in a unified manner. The losslessness propertysatisifed by all-pass functions induces some elegant prop-erties on a state-space description, which we summarizebelow in he discrete-time lossless bounded real lemma [65].Some applications of this lemma to finite word-lengtheffects in digital filters are outlined.

We consider a single-input/single-output (SISO) state-space description

x( n + 1) = Ax(n) + bu (n )y(n) = c'x(n) + du(n) (9.1)

where A is N x N, b and car e N x 1, d is 1 x 1, and x(n)= [x,(n) . . . xN (n ) ] ' s the state vector. The transfer funct ion

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Y(z)IU(z) is given by the orthogonality of R i s in turn equivalent toy'(n) y(n) + x'(n + I)(n + I)

- x'(n) x(n) = u'(n) u(n). (9.10)Equation (9.10) states that, at time n, the instantaneousout-put energy y'(n)y(n) plus the instantaneous rfcrease n stateenergy [x'(n + 1) x(n + 1) - x(n)'x(n)] i s precisely equal tothe instantaneous nput energy u'(n) u(n). Hence (9.10) s anenergybalance relation. Note that th is statement s strongerthan that of (2.8); anystructuresatisfyingtheenergybalancerelation certainly satisfies the (external) osslessness con-dition of (2.8),but he converse snotnecessarily rue. How-ever, given any lossless structure satisfying 2.8), there existsa similarity transformation which renders the structure inan energy balanced form. (Such a transformation can, infact, be constructed from the observabilitygrammian of thesystem[66].)rom this observation, we may restate the sca-lar discrete-time LBR lemma as follows:

Y(z)/U(z) s a n all-pass function i f a n d only i f t admits arealization whose state-space description satisfies theenergy balance relation of (9. IO).

Consider, for example, cascaded lattice structure of Fig.3. Bychoosingtheoutputsof thedelayvar iablesasthestatesx&), and implementing each two-pair in normalized form(cf. Fig. 4(b)), the structure is known [14], [65] to satisfy theenergy balance realization of (9.10). Sincean arbitrary stableall-pass function can be realized using the normalized cas-caded lattice, we have identif ied an energy balanced struc-ture. In fact, many more such structures can be identifiedas, e.g., orthogonal digital filters [ 67 , 68], properly scaledwave digital filters [191, and LBR digital filters [2] .Applications to Roundoff Noise Gain

The state-spacedescription of (9.1) i s depicted in Fig. 25(a).In a practical implementation, quantizers must be intro-duced into he feedback loop to prevent an unlimitedaccu-mulation of the number of bits required to represent thesignals. The quantization error i s typically modeled byintroducing an error vector e(n) in the feedback loop, asshown in Fig. 25(b). Consider again the cascaded latticestructure of Fig. 3. If one quantizer i s inserted after eachlattice section ust pri or to each delay, the model of Fig. 25(b)holds. Since the model does not permit any quantizersinside the lattice sections, a bi t accumulation occurs pro-

- - - d + c'(z1 - A ) - %U(Z) (9.2)We assume also that the realizat ion s minimal in the num-ber of states, so that the order of Y(z)lU(z) s N (i.e., thereare no pole-zero cancellations). Now, given any nonsin-gular N x N matrix T, we can invoke a similarity transfor-mation by replacing the set { A , b, c} with

A I = T - ' A T b, = T - ' b c i = c'T. (9.3)This transformation has no effect on the external transferfunction Y(z)IU(z), and wi th various choices of T, we canderivean unlimited number of structures to mplement thegiven transfer function. (In fact, a ll minimal realizations ofa given transfer function are related through a similaritytransformation.) The internal properties of the systemthough, such as scaling at internal nodes, overflow char-acteristics, roundoff noise, and coefficient sensitivit ies,canvary markedly among different representations.If Y(z)/U(z) s an all-pass function, the state-space param-eters {A,b,c,d } in (9.1) satisfysomeveryelegant propertieswhich have implications in roundoff noise gain, scalingproperties, and limit cycle behavior. We begin by quotinga scalar (SISO)version of the discrete-time ossless boundedreal (LBR) lemma:

lemma [65]: Given the state-space system o f (9.1), thetransfer function Y(z)/U(z) in (9.2) is a stable all-pass func-tion i fand only i f here exists an N x Nsymmetric positive-definite matrix P such that

A'PA + CC ' = P (9.4a)b'P b + d'd = 1 (9.4b)

I A ' Pb + cd = 0. (9.4c)To better understand his lemma, wecan derive from (9.4)

an important "energy balance" result. SincePi s symmetricand positive-definite, there exists a nonsingular matrix Tsuch that P = T'T. Consider the new set of state-spaceparameters which result from the (inverse)similarity trans-formation

A2 = T A T - ' b2 = T b C: = c 'T - ' . (9.5)Wri ting (9.4) in erms of the parameters {A2 ,b2 ,c2,d } resultsin

(9.6a)(9.6b)(9.6~)

A g 2 + c2c: = Iba2+ d'd = 1A 8 2 + ~ 2 d 0 .

By considering the (N + 1) x (N + 1)matrix R formed as(9.7)

the constraints of (9.6) become equivalent t o orthogonalityof R

R'R = 1 (9.8)Now, upon recognizing that (9.1) may be writ ten as

(9.9)

(a)

(b)Fig. 25. (a) State-space filter description. (b) Error vectormodel for quantization noise.



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gressing from left to right along the upper portion of thelattice, although this accumulation i s finite (proportional toN ) . Such accumulation can be avoided by inserting twoquantizers (rather han one) between successive attice sec-tions. We shall proceed with the model of Fig. 25(b), though,to keep the d iscussion manageable.

Our goal is to minimize the error component in the out-put sequence introduced by the quantization error vectore (n ) ,subject to the constraint that the transfer function toeach state variable be proper ly scaled to minimize the prob-ability of overflow. Let us introduce two vectors

andg(n) = [gl(n) * * gN(n)l'

such that f&) i s the response at the kth state variable to aunit pulse input, and &(n) i s the response at the outpu t toa unit pulseatthekth statevariable.Thesevectorsareeasilyfound as

f ( n ) = An b g'(n) = C'A". (9.11)It s convenient to introduce the controllability and observ-ability Grammian matrices K and W associated with thesetwo vectors:

m mK = f (n ) '(n) = Anb(Anb)' (9.12a)n = O n =O

m mW = g(n)g' (n) = (c'A")'(C'A"). (9.12b)n = O n = O

These matrices are positive-definite provided there are nopole-zero cance llations in the f ilter [69], and can be iden-tified as solutions to the equations

K = A K A' + b b' W = A'W A + c c'. (9.13)Moreover, if A has a ll its eigenvalues inside the uni t circle,the so lutions Kand Win (9.13) are unique [70]. In particular,the diagonal elements of either matrix are the squares ofthe P2 norms of the e lements of f and g

cm

K k k = 11 fk1I2 = ngofk(n)I2w k k = llgk1I2 = Igk(n)I2* (9.14b)

Assume now that the components of the noise vector e(n)areuncorrelatedand thateach iswhitewithvariancea:, i.e.,the covariance matrix of e(n) s a:/. Under these assump-tions, the variance of the error at the filter output causedby the quantizers i s equal to

(9.14a)m

n = O

N= a2 Wkk. (9.15)

Now, the diagonal elements K k k of the matrix K representthesquareoftheP,normsofthe impulse reponsesequencesfk(n) o the state variable nodes. For a structure scaled in ant2sense, these quantities should equal unity. This i s easilyaccomplished by using adiagonal similarity transformationmatrix T , such that

r = diag { 1 / 1 1 f k k l l } . (9.16)

k = l

In effect, each term fk(n) s replaced by fk(n)/ll k l l , and eachterm & ( n) is replaced by 11 fk I I & ( n ) to compensate. Hence,the roundoff noisevariance or a scaled realization n ermsof the unscaled parameters i s

N Nk = l k = l= 0: I)fk11211gk112= of K k k Wkk. (9.17)

For a given transfer function, the termNe K k k W k kk = l

depends on the state-space description, and can be con-sidered a"noise gain" fo rthe realization in question. It hasbeen shown [69] that this noise gain i s minimized when{ A , b , c} are such that K and W satisfy the fol lowing twoproperties:

1) K = A W A or some diagonal matrixA of positive ele-ments.2) K k k w k k = constant independent of k. (9.18)

In fact, a minimum-noise description satisfying hese prop-erties always exists [69].

Assume now that Y(z ) /U( z )s an all-pass function realizedin an energy balanced structure, so that the matrix R in (9.8)i s orthogonal. The orthogonality of R also implies or thog-onality of R', so that R R' = I ields three equations anal-ogous to (9.6). The first of these is

A A ' + b b ' = I. (9.19)The comparison of (9.19) with (9.13) reveals K = I, hichindicates an energy balanced filter is inherently scaled inan t2sense. Likewise, comparing (9.6a) wi th (9.13) revealsW= I,o that the roundoff noise i s minimized according tothe conditions (9.18) above. In other words, for any energy-balanced structure, the constraints of internal scaling andminimum roundoff noise are automatically satisfied. Forsuch structures, the output noise variance is found from(9.17) as

NU: = U2 K k k W k k = NU: (9.20)

sothat the noisegain isfound knowingonlythefilterorder.We poin t out that this result holds independent of the po lelocations of the filter.

Referring again to the normalized lattice structure, inpractice it may be convenient to introduce quantizers intoboth upper and lower branches connecting adjacent sec-tions to preventword lengthsfrom accumulating n he suc-cessive lattice stage computations. Upon effecting thismodification, the roundoff noise variance appearing at theoutput can be found as

k = l

ai = 2Na:.The state-space descriptions corresponding to the other

Gray-Markel lattice structures can be obtained from thatof the normalized form via a diagonal similarity transfor-mation matrix T. Accordingly, the matrices Kl and W l forsuch structures are given by

K~= r;IK r;' = ry2wl = r! ,w rl = r: .

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These matrices s t i l l satisfy conditions (9.18), which indicatesthat scaledversionsof these structures are minimum noisestructures. Of course, upon scaling these structures in an4 ense, the result ing statespacedescription coincideswiththat of the normalized form.Limi t Cycle Behavior

The quantizers in he feedback oop of Fig. 25(b) are non-linear elements, and can cause nonlinear oscillations knownas limit cycles, resulting in a periodic output even after theinput signal has been removed. We assume that the quan-tizers use magnitude truncation in the absence of an over-flow, and two's-complement overflow followed bymagnitude truncation otherwise. In circuits using suchquantizers, two types of limit cycles are commonlyobserved, namely granular oscillations and overflow oscil-lations [71]. A sufficient condition [72] for the absence ofzero-input limit cycles s the existence of a diagonal matrixDoof positive elements such that4

Do- A'DoA 2 0. (9.21)Inview of condition (9.6a), we see that (9.21) i s satisifiedwithDo= 1 That is, the energy balance cond ition ensures theabsence of lim it cycles. Moreover, if A satisfies condition(9.21),sodoes T - ' A Tfor any diagonal ransformation matrixT, simply by replacing Dowith T DOT.Hence scaling doesnot sacrifice the freedom from limit cycle property, andaccordingly, all the Gray-Markel lattice structures aredevoid of limit cycles 1141, 1721.Applicat ions to Complex All-Pass Functions

Inquoting he discretetime LBR lemmawe have assumedreal coefficient filtering. The many results above easilygen-eralize o complex filters by replacing, for example, matrixtransposition operations with conjugate transpositionoperations. As such, one can show that complex filterswhich satisfy the energy balance constraint are minimumnoise structures and are free from lim it cycles. Thus thecomplex lattice filters, for example, share the attractiveproperties of their real arithmetic counterparts.Complex all-pass ilters derived from other methods havedesirable finite word-length properties as well. For exam-ple, the complex transformation of (3.7)may be nterpretedas a diagonal similari ty transformation T, such that

T = diag {e'4k}. (9.22)This transformation matrix is unitary, and as such has noeffect on the energy balance properties (or lack thereof) ofthe structure to which it i s applied.

Finally, the (nonminimal) first-order complex all-passstructure of Fig. 7 admits the state-space description

a - b+ [, ] u(n)4The nequality in (9.21) ndicates that the (symmetric) matrix onthe left i s positive semidefinite.

where a + jb = y is the pole of the filter. Upon setting theinput u(n) o zero, t he feedback port ion reduces to

x,(n + 1) = yx,(n). (9.23)Now, stability of the filter implies

1 - y12 > 0 (9.24)which, in view of (9.21) (withDo= 1andA = y), ensures theabsence of zero-input lim it cycles.X. CONCLUDINGEMARKS

In this paper we have outlined the use of all-pass filtersin a variety of signal processing applications, includingcomplementary filter ing and filter banks, mult irate filter-ing, frequency response equalization, etc. Fundamental omany of these results s the lossless property exhibited byan all-pass function; provided this property i s structurallyinduced, the desirable features in each application exhibitvery robust performance in the face of coefficient quan-tization. Furthermore, by using filter structures which sat-isfy the energy balance relation of Section IX, lim it cyclesare avoided and the roundoff noise of the filter is mini-mized.ACKNOWLEDGMENT

The authors would like to thank Dr. M. Bellanger for use-ful criticisms.REFERENCES

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Phillip A. Regalia (Student Member, I E E E )was born in Walnut Creek, CA, in 1962. Hereceived the B.Sc. degree (highest honors)in electrical engineering from th e Univer-sity of California at Santa Barbara in 1985.Presently, he i s a Ph.D. candidate atUCSB, where he work s as a research assis-tant in he Signal Processing Laboratory. Hisresearch interests include analog circuittheory and digital signal processing.

Sanjit K. Mit ra (Fellow, IEEE) received theB.Sc. (Hons.) degree in physics in 1953fromUtkal University, Cuttack, India; the M.Sc.(Tech.) degree in radio physics and elec-tronics in 1956 from Calcutta University,Calcutta, India; and the M.S. and Ph.D.degrees in electrical engineering from theUniversity of California, Berkeley, in 1960and 1962, respectively. In May 1987, he wasawarded an Honorary Doctorate of Tech-nology degree by the Tampere Universityof Technology, Tampere, Finland.

Hewasa memberof thefacult yoftheCo rnell University, Ithaca,NY, from 1962 to 1965, and a member of theTechnic al Staff of theBell Laboratories from 1965 to 1967. He joined the faculty of theUniversity of California, Davis, in 1967, and transferred to he SantaBarbara campus in 1977 as a Professor of Electrical and Com puterEngineering, where he served as Chairman of the Department fr omJuly 1979 to June 1982. He has held vi siting appointme nts at uni-versities in Australia, Brazil, Finland, India, West Germany, andYugoslavia. He i s the re cipient of the 1973F.E. Terman Award andthe 1985 AT&T Foundation Award o f th e American Society of Engi-neering Education, a Visiting Professorship fr om t he JapanSocietyfor Promotio n of Science in 1972, and the Di stinguished F ulbrightLecturer Award for Brazil i n 1984 and Yugoslavia i n 1986.

Dr. Mitra i s a Fellow of the AAAS, and a member o f the ASEE,EURASIP, Sigma Xi, and Eta Kappa Nu. He s a member of the Advi-sory Council of the George R. Brown School of Engineering of th eRice University, Houston, TX and an Honorary Professor of theNort hern Jiaotong University, Beijing, China.

P. P. Vaidyanathan Member, IEEE) was bo rnin Calcutta, India, o n October 16,1954. Hereceived the B.Sc. (Hons.) degree i n phys-ics, and the B.Tech. and M.Tech. degrees inradiophysics and electronics from the Uni-versityof Calcutta, India, in 1974,1977, and1979, respectively, and the Ph.D. degree inelectrical and computer engineering fromthe University o f California, Santa Barbara,in 1982.He was a Postdoctoral Fellow at the Uni -versity of California, Santa Barbara, from September 1982 to Feb-ruary 1983. Since March 1983 he has been wit h the Californi a Ins ti-tu te of Technology, Pasadena, as an Assistant Professor of ElectricalEngineering. His m ain research interests are in digit al signal pro-cessing, linear systems, and f ilte r design. He was the recip ient ofthe Award for Excellence in Teaching at the Cali fornia Insti tute ofTechnology for 1983-1984, and a recipient of NSFs PresidentialYoung Investigator Award, starting from the year 1986.Dr. Vaidyanathan served as the Vice Chairman of the TechnicalProgram Committee f or the 1983 EE E International Symposium o nCircuits and Systems. He currently serves as an Associate Editorof the I E E E TRANSACTIONSN CIRCUITSND SYSTEMS.


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The Digital All-Pass Filter A Versatile Signal