The Digital All-Pass Filter: A Versatile Signal Processing Building Block PHILLIP A. REGALIA, STUDENT MEMBER, IEEE, SANJIT K. MITRA, FELLOW, IEEE, AND P. P. VAIDYANATHAN, MEMBER, IEEE The digital all-pass filter is a computationally efficient signal pro- tionally efficient filter structures, tunable filters, filter bank cessing building block which is quite useful in many signal pm- analysis/synthesis systems, multirate filtering, and stability cessing applications. In this tutorial paper we review the properties of digital all-pass filters, and provide a broad overview of the diver- under linear and nonlinear (i.e., quantized) environments. sity of applications in digital filtering. Starting with the definition This paper considersa basic scalar lossless building block, and basic properties of a scalar all-pass function, a variety of struc- which is a stable all-pass function. The interconnections of tures satisfying the all-pass property are assembled, with emphasis such lossless building blocksform useful solutions to many placed on the concept of structural losslessness. Applications are practical filtering problems. The many results presented then outlined in notch filtering, complementary filtering and filter banks, multirate filtering, spectrum and group-delay equalization, here can be derived by appealing to elegant theoretical Hilbert transformations, and so on. In all cases, the structural loss- forms; however, in maintaining a tutorial tone it is our aim lessness property induces very robust performance in the face of to expose the salient features using direct discrete-time multiplier coefficient quantization. Finally, the state-space mani- concepts, in the hope that the references cited will further festations of the all-pass property are explored, and it is shown that many all-pass filter structures are devoid of limit cycle behavior aid both thedesigner and researcher alike. Weshould point and feature very low roundoff noise gain. out that many of the results which are developed in terms of scalar all-pass functions in one dimension can be gen- I. INTRODUCTION eralized tovector or matrixall-pass functions [3]and to mul- tidimensional filtering [4],though for the presentwe restrict In many signal processingapplications, thedesigner must our attention to the one-dimensional scalar case. determine the transfer function of a digital filter subject to We begin in Section II by defining a scalar all-pass func- constraints on the frequency selectivity andlor phase tion and reviewing some basic properties. Section Ill response which are dictated by the application at hand. assembles avariety of all-pass filter structures, with empha- Once a suitable transfer function is found, the designer sis placed on the concept of structurallosslessness. Section must select a filter structure from the numerous choices IV outlines applications to notch filtering, Section V to com- available. Ultimately, finite precision arithmetic is used in plementary filters and filter banks, Section VI to multirate any digital filter computation, and traditionally the round- signal processing, SectionVII to tunablefilters, and Section off noise and coefficient sensitivity characteristics have Vlll togroupdelay equalization. Finally, Section IX explores formed the basis of selecting one filter structure in favor state-space representations of lossless transfer functions, of another. and the implications of losslessness in obtainingvery robust In the quest for low coefficient sensitivityand low round- performance under finite word-length constraints. off noise, an elegant theory of losslessness and passivity in the discrete-time domain has evolved [I], (21. Although this II. DEFINITIONSAND PROPERTIES theory has been motivated by the desire to obtain digital filters with predictable behavior under finite word-length The frequency responseA(ei'")of an all-pass filter exhibits conditions, many useful by-products have emerged which unit magnitude at all frequencies, i.e., have contributed to a better understanding of computa- JA(e'")(* = 1, for all U. (2.1) Manuscript received July IO, 1987; revised October 16,1987.This The transfer function of such a filter has all poles and zeros work was supported by the National Science Foundation under occurring in conjugate reciprocal pairs, and takes the form Grants MIP 85-08017and MIP 8404245. P. A. Regalia and S. K. Mitra are with the Department of Electrical and Computer Engineering, University of California, Santa Bar- (2.2) bara, CA 93106, USA. P. P. Vaidyanathan is with the Department of Electrical Engi- For stability reasons we assume (Ykl 1 for all k to place neering, California Institute of Technology, Pasadena, CA 91125, < USA. all the poles insidethe unit circle. Now, ifA(z) isconstrained IEEE Log Number 8718413. to be a real function, we must have 0 = 0 or 0 = T,and any 00189219/88/0100-0019601.000 1988 IEEE PROCEEDINGS OF THE IEEE, VOL. 76, NO. 1, JANUARY 1988 19 complex pole at z = Yk must be accompanied by a complex This relation is useful in verifying stability in lattice real- conjugate pole atz = 7:. In this caseA(z) can be expressed izations of all-pass filters. in the form The last property of interest is the change in phase for real all-pass filter over the frequency range w E [0, TI.We (2.3) start with the group delay function T(W) of an all-pass filter, which is usually defined as In effect, the numerator polynomial is obtained from the d denominator polynomial by reversingthe order of the coef- 7(w) = - - [arg A(e/")]. (2.10) ficients. For example, dw a2 + alz-l + z-~ Note that the phase function must be taken as continuous A(z) = I + alz-' + a22-2 or "unwrapped" [5] if 7(w) is to be well-behaved. Since an all-pass function is devoid of zeros on the unit circle accord- isasecond-order all-pass functionoftheform of(2.3)above, ing to (2.1), the phase function arg A(ei") can always be since the numerator coefficients appear in the reverse order unwrapped with no ambiguities. Now, the phase response ofthose in thedenominator. In thiscase,thenumeratorand of a stable all-pass function is a monotonically decreasing denominator polynomials are said to form a mirror-image function of w, so that T(W) is everywhere positive. An Mth- pair. order real all-pass function, in fact, satisfies the property If we lift the restriction that A(z) be a real function, then A(z) takes the more general form lou7(w) dw = Mr. (2.11) z -M D*(l/z*) A(z) = (2.5) D(z) The interpretation of (2.11) is that the change in phase of The numerator and denominator polynomials now form a the all-phase function as w goes from 0 to r is -MT radians. Hermitian mirror-image pair. For example, 111. ALL-PASSFILTER STRUCTURES The (Hermitian) mirror-image symmetry relation between the numerator and denominator polynomials of an all-pass transfer function can be exploited to obtain a computa- tionally efficient filter realization with a minimum number of multipliers. To see this, consider the second-order all- with0 = arg(a;/ao), isrecognizedasacomplexall-passfunc- pass function of (2.4) which, upon expressing A(z) = tion due to the Hermitian mirror-image relation between Y(z)/U(z), corresponds to the second-orderdifference equa- the numerator and denominator polynomials. tion Properties y(n) = -32[u(n) - y(n - 2)1 From the definition of an all-pass function in (2.1), setting + al[u(n- 1) - y(n - I)] + u(n - 2) (3.1) A(z) = Y(z)/U(z) reveals in which terms have been grouped in such a way that only 1 y(eju)12 = I u(eju)12, for all w. (2.7) two multiplications are required. A similar strategy can be Upon integrating both sides from w = --a to rand applying applied to an arbitrary Mth-order all-pass filter, such that Parseval's relation [I], we obtain only M multiplications are required to compute each out- put sample. On the other hand, a direct-form filter real- m m ization would in general require 2M + 1 multiplications to c I y(n)12 = ldn)l2. (2.8) " = -m computeeach output sample. In this sense,an all-pass filter represents a computationally efficient structure. It is convenient to interpret the two sides of (2.8) as the out- Thedifferenceequation as expressed in (3.1) requiresfour put energy and input energy of the digital filter, respec- delay (or storage) elements to be realized as a filter struc- tively [I], [5]. Thus an all-pass filter is lossless, since the out- ture. Since the difference equation is of second order, this put energy equals the input energy for all finite energy does not represent a canonic realization. However, mini- inputs. If the all-pass filter is stable as well, it is termed Loss- mum multiplier delay-canonic all-pass filter structures can less Bounded Real (LBR) [2], or more generally Lossless be developed using the multiplier extraction approach [q, Bounded Complex [6] if the coefficients are not all real. [8]. For example, consider the digital two-pair network of Another useful property follows from (2.1) with the aid Fig. 1, which has a constraining multiplier b, at the second of the maximum modulus theorem. In particular, since a "port": U2(z)= bl Y2(z).The transfer function as seen from hasall its poles insidethe unitcircle, stableall-pass function the remaining port is constrained to be a first-order all-pass all its zeros outside, and exhibits unit magnitude along the unit circle, one can deduce <I, for Iz( > 1 (2.9) Fig. 1. Digital two-pair network constrained with a multi. >I, for IzI < 1. plier of value b,. 20 PROCEEDINGS OF THE IEEE, VOL. 76, NO. 1, JANUARY 1988 function (3.2) This allows one to solve for the internal parameters of the digital two-pair.