Passive Cascaded - Lattice Structures for Low - Sensitivity FIR Filter Design , with Applicati & s to Filter Banks t ’

A class of nonrecursive cascaded-lattice structures is derived. for the implementation of finite-impulse response (FIR) digital filters. The building blocks are lossless and the transfer function can be implemented as a sequence of planar rotations. The structures can be used for the synthesis of any scalar FIR transfer function H(z) with no restriction on the location of zeros; at the same time, all the lattice coefficients have magnitude bounded above by unity. The structures have excellent passband sensitivity because of inherent passivity, and are automatically internally scaled, in an L, sense. The ideas are also extended for the realization of a bank of M FIR transfer functions as a cascaded lattice. Applications of these structures in subband coding and in multirate signal processing are outlined. Numerical design examples are included. I. INTRODDCTI~N T HERE EXIST A number of methods [l]-[8] for the design of low-sensitivity infinite impulse response (IIR) digital filters. Notable among these are the wellknown wave-digital filters [l], [2], orthogonal digital filters 141, and certain types of lattice filters [6], [7]. Some of these methods are based on the notion of pseudopassivity, while certain others are based on orthogonality of internal computations. The lattice digital filters reported thus far can also be related to autoregressive/moving average modeling techniques [9]-[ll]. The relation between these families of filters has also been known for some time [14], [24], [25], WI. In principle, any stable IIR digital filter transfer function can be implemented as an orthogonal filter [4], [5], [14], with planar rotation building blocks. The purpose of this paper is to develop passive structures for arbitrary finite impulse response (FIR) transfer functions and FIR filter banks based on planar rotation building blocks. An important feature of many of the well-known IIR digital filter structures having favorable finite-wordlength properties is that some or all of the internal building blocks are passive in a certain sense, and this has been favorably exploited [7], [12], [13] to obtain low-sensitivity designs free of limit cycles. A normalized version of the passivitybased structures results in orthogonal implementations [6], Manuscript received July 2, 1985; revised June 6, 1986. This work was supported m part by the National Science Foundation under Grant ECS 84-04245 and in part by Caltech’s program in Advanced Technology sponsored by Aerojet General, General Motors, GTE, and TRW. The author is with the Department of Electrical Engineering, Cahfomia Institute of Technology, Pasadena, CA 91125. IEEE Log Number 8610176. [14] for IIR filters, which have the additional advantage that all internal signals are scaled in the 1, sense, thus rendering additional scaling effort unnecessary. When implementing FIR digital filters, there is clearly no possibility of limit cycles, as long as the structure is nonrecursive.’ However, it is still of interest to obtain structures that have low sensitivity. It is also desirable to have normalized structures, which in addition have low noise. Schuessler [15], [16] has studied quantization effects in FIR filters, and a variety of interesting structures based on polynomial-interpolation theory can be found in [16]. Chan and Rabiner [17], [lS] have done extensive research on cascade-form FIR filters, with particular emphasis on the ordering of the sections to attain optimal noise/dynamic range performance. The general conclusion seems to be that direct-form implementations of FIR filters have poor stopband sensitivity while cascade-form realizations have reasonably good stopband sensitivity, even though the passband behavior tends to deteriorate when the multiplier coefficients are quantized. Moreover, unless the cascading order is properly chosen, the roundoff noise gain can be excessively large. A class of FIR lattice digital filters has been studied by Makhoul and others [lo], [ll], [21] in the context *of linear-predictive coding of signals. A subset of these structures naturally arises while solving the “normal equations” via Levinson’s recursion. A main feature of these structures is that, as long as the multiplier coefficients k, (called the “reflection” coefficients) are bounded (in magnitude) by unity, a pair of FIR transfer functions G(z) and H(z) can be realized where G(z) has minimum phase and H(z) has maximum phase. However, with kf < 1, one cannot realize arbitrary transfer functions, for example, an equiripple FIR filter with some of its zeros on the unit circle of the z-plane. (It is, however, easy to force all zeros to be on the unit circle [22] simply by setting the “rightmost k,” to *l.) The FIR lattice filters arising out of linear-predictive coding of signals have an IIR counterpart, typically called the “synthesis filters” [ll]. The IIR or recursive counterpart, also known as the Gray-and-Markel filter structure ‘Recursive realizations of FIR filters, such as the frequency-sampling structures, can, theoretically support limit cycles. 0098-4094/86/1100-1045$01.00 01986 IEEE 1046 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. CAS-33, NO. 11, NOVEMBER 1986 [6], [7], has several equivalent forms, one of them being the “normalized” lattice form [6]. These IIR structures are known to have passivity properties [7], and in addition, the normalized lattice is based on a cascade of orthogonal (planar rotation) building blocks. The IIR structures have excellent “robustness” properties under quantized conditions. Even though this IIR version has passivity properties, the FIR counterpart does not (this will be elaborated on in Section V). The FIR lattice structures we introduce in this paper are not related to linear-predictive coding in the sense that the “corresponding” IIR versions do not necessarily represent a maximum-entropy spectral model [ll] of any appropriate time series. The new FIR structures are characterized by a set of coefficients k, such that k,? < 1, yet any arbitrary FIR transfer function H(z) with no restriction on the zero locations can be realized. (Accordingly, the “corresponding” IIR version does not necessarily represent a stable filter.) These FIR structures are based on simple interconnections of orthogonal (planar rotation) building blocks, and are “passive” in the same sense (to be elaborated on) as the well-known orthogonal and wave-digital IIR filters. They are automatically internally scaled and have low passband sensitivity. In spite of the fact that they are in the form of a nonrecursive cascade, we do not encounter serious roundoff noise problems. Indeed, the noise-variance gain is bounded above by the filter order, exactly as in the direct-form structure [19], [20]. Finally, these structures seem to be a natural choice for certain types of filter banks and multirate subband coding applications involving”quadrature mirror filters” [23]. In Section II, we briefly review the structural-passivity concept and its relation to low sensitivity. Section III introduces the new FIR lossless lattice structures. A synthesis procedure is presented in this section so that an arbitrary FIR transfer function (not necessarily linear phase) can be implemented in this form. The result of synthesis is a one-input two-output structure, with transfer functions G(z) and H(z), where G(z) is the desired FIR function and H(t) is an auxiliary function such that (G(ej”)(2 + ]H(ej”)12 =l. We thus have a bank of two filters. Section IV is a study of certain important properties of this class of structures. Next, in Section V, we review the relation between the new structures and certain well-known lattice structures for digital filtering. In Section VI, we extend the idea of Section III to the case of a filter bank of M FIR transfer functions G,,(z), G,(z),. .> G,-,(z) realized as a cascaded lossless FIR lattice structure. The bank is such that ]Go(eiW) (* + . . . ]G,,(e@)]’ = 1. Section VII presents a general roundoff-noise/dynamic range analysis, which applies to all structures presented in the paper. In Section VIII, applications in analysis/synthesis filter banks are presented. Finally, in Section IX, we present an application of the structures in multirate digital filtering, where sampling-rate changes are involved in addition to linear time-invariant filtering. Notations Used in the Sequel: In this paper, superscript t stands for matrix transposition, whereas superscript dagger (t) stands for transposition followed by complex conjugation. Bold-faced letters indicate vectors and matrices. The tilde accent stands for transposition followed by reciprocation of functional argument; for example, E?(z) = H’( z-l). The notation A < B (where A and B are square matrices of equal dimensions) is abbreviation for “B A is positive semidefinite.” Similarly, A < B means “B A is positive definite.” 1, (with subscript possibly omitted) denotes the identity matrix of dimension m X m. For a (real symmetric) positive definite matrix P, we define its square root P1l2 according to the factorization P = P’/2P’/2, where P’12 is the transpose of P112. II. STRUCTURAL BOUNDEDNESS AND Low SENSITIVITY A stable digital filter transfer function H(z) with real coefficients is said to be bounded real (BR) if (ff(e+)( Gl, for all w. 0) If equality holds in (1) for all w, then H(z) is said to be lossless bounded real (LBR). An implementation is said to be structurally passive or structurally bounded [24] if, regardless of the multiplier values (as long as they are in a well-defined range such as 1~ mk < l), the transfer function satisfies (1). If the multiplier values in a structurally passive implementation are such that ]H(ejw)] attains the upper bound of unity in the passband( then the structure has low passband sensitivity [14], [24]. It is useful to extend these definitions to the matrix case. A stable transfer matrix H(z) with real coefficients is BR if Ht(ej”)H(ej”) Q I for all w, and LBR (or “allpass”) if this inequality becomes an equality for all o. Given an FIR BR transfer function H(z) of order N satisfying (1) we can always find another FIR BR function G(z) such that lG(ejw)12 + JH(ej”)J2 =l, for all w. (2) The vector G,(z) = [H(z) G(z)]’ of order N is, hence, FIR LBR (or “allpass”), and we show how this FIR vector’ can be realized in terms of FIR lossless building blocks. Such a realization exhibits low sensitivity in the passbands of both G(z) and H(z). This idea of embedding H(z) into a vector GN(z) is easily generalized. Thus, given M 1 FIR BR transfer functions G,(z), G,(z), . . . , G,-,(z) such that IGo(ej”)12flGl(e~w)12+ --’ +]G,-,(ej”)]*<l (3) for all w, we can always find an FIR BR function G,-,(z) such that the vector G,(z) = [G,(z),G,(z); . .,G,_i(z)]’ is FIR LBR (or “allpass”). III. LA~ICE STRUCTURES FOR TWO-COMPONENT FIR ALLPASS VECTORS Let us assume that we have an FIR allpass vector ‘%-I(Z) = k-l(z) QN-dd1’ VAIDYANATHAN: PASSIVE CASCADED-LATTICE STRUCTURES 1047 Fig. 1. Realization of C,+,(~J.