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Robust FIR Filter Design with Envelope Constraints and Channel Uncertainty
Ching-Min Lee and I-Kong Fong
Abstract—In this note, a finite impulse response (FIR) filter design
problem is considered.The signals to be filtered are assumed to be corrupted by the channel noise.In addition, the channel characteristics are assumed to contain uncertainties.The linear matrix inequalities approach is adopted to provide two optimization procedures for designing optimal filters and robust filters subject to filter output envelope constraints.A numerical example is presented to illustrate the proposed filter design methods.
Index Terms—Bounded stability, integral quadratic constraints, linear
matrix inequality, robust FIR filter, time-domain envelope constraint.
I. INTRODUCTION
In the field of signal processing, many filter design problems can be cast as constrained optimization problems. The constraints are usually defined by the specifications of the desired filters, and these specifica-tions arise either from the standards set by certain regulatory bodies or from practical considerations. The time-domain envelope-constrained filter designis one example of these problems, which ofteninvolve re-quirements on the transient responses, such as the pulse-shape require-ments in digital data transmission systems. In particular, these kinds of filters may be seeninapplications like the pulse compressionfor many communication and radar systems, the TV waveform equalization with respect to the K-mask, and the data channel equalization or deconvo-lution, [1]–[3].
As to the optimizationpart, theH1optimizationtheory has been widely used in robust control and signal processing problems [4]–[8].
Manuscript received Janary 4, 2003; revised July 21, 2003. This work was supported by the National Science Council of the Republic of China under Grant NSC 90-2213-E-002-084. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Yuan-Pei Lin.
The authors are with the Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan 10617, R.O.C. (e-mail: d8921011@ee.ntu.edu.tw; ikfong@cc.ee.ntu.edu.tw).
Digital Object Identifier 10.1109/TSP.2004.827142
Fig. 1. Deconvolution filtering system.
Take the deconvolution filter design, for example. The objective of the problem may be set to the minimization of theH1norm of the filtering error transfer function. In this approach, the system formulation allows the inclusion of the transmission channel and/or signal models. In ad-dition, by utilizing methods developed in [7] and [9], the time-domain envelope constraints may be accommodated simultaneously. However, there are still other important factors to take care of, such as the system uncertainties. In the literature [10], [11], there are some discussions about input uncertainty of the filter, but consideration of input uncer-tainty bounds only [11] does not fully use information about system un-certainties that may be available. A more direct and complete approach for handling the transmission channel model uncertainties is desirable. In this correspondence, the H1 optimal finite impulse response (FIR) filtering problem with envelope constraints and channel un-certainties is studied. The linear matrix inequality (LMI) framework is adopted, and the uncertainties in the channel are formulated as satisfying the integral quadratic constraints (IQCs) [12], [13]. The design method is also enhanced to ensure that the output of the filter is not too close to the constraining envelopes [8], [10], [14]. Compared with some existing method [14], which deals with the output envelope constraint problem by optimization procedures depending on the
quasi-Newtonmethod and goldensectionmethod, the LMI-based
method handling a convex optimization problem is numerically more attractive. To illustrate the effectiveness of the proposed design method, a numerical example is presented.
II. PROBLEMFORMULATION
Consider the deconvolution filtering system shown in Fig. 1. In the system, the source signals(k) 2 R is assumed to be generated by the signal model
6S: xs(k) = Cs(k + 1) = Asxs(k) + Bsw(k)
sxs(k) + Dsw(k) (1)
wherexs(k) 2 Rn is the model state vector,w(k) 2 l2[0; 1) is the driving signal of the model, andAs,Bs,Cs, an dDsare known con-stant matrices of appropriate dimensions. The source signal is trans-mitted through a channel with an uncertain characteristic modeled by
6C: xc(k + 1) = Acxc(k) + Bcs(k) + p
i=1Hc1ici(k) zc(k) = Ccxc(k) + Dcs(k) + pi=1Hc2ici(k) (2) wherexc(k) 2 Rn is the channel state vector,ci(k) 2 Rn,i = 1; 2; . . . ; p is the ith uncertain vector satisfying the IQC [12]
k=0 kci(k)k2 k=0 kE1ixc(k) + E2is(k) + E3ic(k)k2 (3) as ! 1, cT(k) = [Tc1(k) 1 1 1 cpT(k)], an d Ac, Bc, Cc, Dc, Hc1i, Hc2i, E1i, E2i, an d E3i are known constant ma-trices with appropriate dimensions. For subsequent usage, we define Hc1 = [Hc11 1 1 1 Hc1p], Hc2 = [Hc21 1 1 1 Hc2p], ET 1 = [ET11 1 1 1 E1pT], E2T = [E21T 1 1 1 E2pT], an d E3T = [ET 31 1 1 1 E3pT]. 1053-587X/04$20.00 © 2004 IEEE
At the receiving end, the measured signaly(k) is equal to zc(k) + v(k), where v(k) is the energy-bounded channel noise. To optimally recover the source signals(k), the signal y(k) is deconvoluted by a filter
6F : x^s(k) = Cf(k + 1) = Afxf(k) + Bfy(k)
fxf(k) + Dfy(k) (4)
wherexf(k) 2 Rn is the filter state vector, andAf,Bf,Cf, an dDf are constant matrices to be designed. Let the filtering error be defined ase(k) = s(k) 0 ^s(k). Then, it satisfies
6e: e(k) = Cxe(k + 1) = Aexe(k) + Bewe(k) + H1ec(k) exe(k) + Dewe(k) + H2ec(k) (5) wherexTe(k) = [xTs(k) xTc(k) xTf(k)], weT(k) = [wT(k) vT(k)], an d Ae= As 0 0 BcCs Ac 0 BfDcCs BfCc Af Be= Bs 0 BcDs 0 BfDcDs Bf ; H1e= 0 Hc1 BfHc2 Ce= [Cs0 DfDcCs 0 DfCc 0 Cf]; De= [Ds0 DfDcDs 0 Df]; H2e= 0DfHc2: (6) The purpose of this paper is to designanFIR filter6F with three desired properties, of which the first one is given by the following def-inition.
Definition 1: [12] The filtering error dynamics6eis called bounded stable if there exists a constant 0 such that kxe(k)k for all k 0, no matter what the initial condition xe(0) and input we(k) 2 l2[0; 1) are.
The second desired property of the filter is that for a given filter input signaly(k) of finite duration, the output signal ^s(k) has to be bounded above and below by
ls(k) ^s(k) us(k) (7)
fork = 0; 1; . . . ; n, where ls(k) and us(k) are the lower and upper bounds of the time-domain mask, respectively, andn+1 is the duration of^s(k). Finally, it is desired that when xe(0) = 0, the filter has the H1 performance 01 k=0 ke(k)k2 k=0 kwe(k)k2 (8)
for some scalar > 0 and all we6= 0, as ! 1. III. FIR FILTERDESIGN
To start, theS-procedure [15] is applied to convert two of the above design objectives into a quadratic condition, and a theorem is derived.
Theorem 1: Under all admissible channel uncertainties satisfying
the IQC (3), the error dynamics6eis bounded stable and satisfies (8) whenxe(0) = 0 if there exists a symmetric positive definite matrix X 2 Rn +n +n and positive scaling scalars
1; 2; . . . ; p, satis-fying the following inequality:
(Aexe+ Bewe+ H1ec)TX(Aexe+ Bewe+ H1ec) 0 xT eXxe+ p i=1 i kE1ixc+ E2is + E3ick20 kck2 + 01kCexe+ Dewe+ H2eck20 kwek2< 0 (9) for all[xTe; wTe; cT]T 6= 0.
Proof: Summing up the left-hand side of the inequality in (9)
along any trajectory of the error dynamics6e, one gets xTe( + 1)Xxe( + 1) 0 xTe(0)Xxe(0) + p i=1 i k=0 kE1ixc(k) + E2is(k) + E3ic(k)k2 0 k=0 kci(k)k2 + 01 k=0 ke(k)k20 k=0 kwe(k)k2 < 0 (10)
for allwe 6= 0. From (3), it follows that
xTe( + 1)Xxe( + 1) < xTe(0)Xxe(0) + k=0
kwe(k)k2 as ! 1. Thus, the bounded stability of the filtering error dynamics is implied. As to theH1performance, whenxe(0) = 0, it follows from (3) and (10) that (8) holds.
A. H1Optimal FIR Filter Design
To develop a set of easy-to-use conditions for designing the FIR fil-ters, the parameter matrices of the filter6F are assumed to be of the form [5] Af = 0 1 0 1 1 1 0 0 0 1 1 1 1 0 .. . ... ... . .. ... 0 0 0 1 1 1 1 0 0 0 1 1 1 0 n 2n ; Bf = 0 0 .. . 0 1 n 21 Cf = [f(nf) f(nf01) 1 1 1 f(1)] ; Df = f(0) wheref(0); f(1); 1 1 1 ; f(nf) are the parameters to be determined. Ac-cordingly, the transfer function of the filter is
Cf(zI 0 Af)01Bf+ Df
= f(0) + f(1)z01+ f(2)z02+ 1 1 1 + f(n
f)z0n : (11) At this point, it is appropriate to consider the second desired property of the filter. Suppose that a signalfy(0); y(1); y(2); . . . ; y(m); 0; 0; . . .g is given, and it is required that (7) be satisfied. Let [7]
y = y(0) y(1) .. . y(m) ; f = f(0) f(1) .. . f(nf) ; lb= ls(0) ls(1) .. . ls(n) ; ub= us(0) us(1) .. . us(n) (12) and Y = y(0) 0 1 1 1 0 y(1) y(0) 1 1 1 0 .. . y(1) 1 1 1 ... y(m) ... y(0) 0 y(m) ... y(1) .. . ... ... 0 0 1 1 1 y(m) (13)
whereY is an n 2 (nf+ 1) matrix, and n = m + nf+ 1. Therefore, the time-domain constraints in (7) are equivalent to
where“diag(1)”denotesadiagonalmatrixformedbyputtingtheelements of the argument vector on the diagonal positions of the matrix. Based on the inequality (14) and Theorem 1, the following theorem can be estab-lished to provide a convex optimization problem with the LMI formula-tion, which may be easily solved for finding the filter parameters.
Theorem 2: AnH1optimal FIR filter of the form (11) satisfying the three desired properties formulated inSectionII exists if the fol-lowing optimization problem has solutions:
min
X;f;J (15)
subject to (16), shownat bottom of the page, and
diag(lb) diag(Y f) (17) diag(Y f) diag(ub) (18) X > 0; J > 0; > 0 (19) where J = diag( 1In ; 1 1 1 ; pIn ), Ce = [Cs 0 0] 0 Df [DcCs Cc 0] 0 Cf[0 0 I], De = [Ds 0] 0 Df[DcDs I], Ee = [E2Cs E1 0], an d Ew= [E2Ds 0].
Proof: Note that (9) canbe rewrittenas
(Aexe+Bewe+H1ec)TX(Aexe+Bewe+H1ec)0xTeXxe + (Eexe+Ewwe+E3c)TJ(Eexe+Ewwe+E3c) 0 TcJc+ 01(Cexe+Dewe+H2ec)T 2 (Cexe+Dewe+H2ec) 0 wTewe = xT e weT Tc M xTe weT Tc T < 0:
B. RobustH1FIR Filter Design
WhenanH1optimal FIR filter is designed using Theorem 2 in the above subsection, the time-domain constraints in (7) will be satisfied. However, insome cases, the output response of the filter may be very close to or even touch the constraint envelopes. In certain applications, this is regarded as undesired. To avoid this kind of situation, the enve-lope constraints and objective function considered in Theorem 2 can be modified as ms(k) 0 1 es(k) ^s(k) ms(k) + 1 es(k) (20) fork = 0; 1; . . . ; n, an d min X;f;J(1 + 2) (21) respectively, where ms(k) = 12(us(k) + ls(k)) (22) es(k) = 12(us(k) 0 ls(k)) (23)
1and2 are tunable weighting parameters, and0 < < 1 is a new scaling parameter to be determined. Note that (20) is equivalent to the LMIs
diag(mb) 0 1 diag(eb) diag(Y f) (24)
and
diag(Y f) diag(mb) + 1 diag(eb) (25)
where mb= ms(0) ms(1) .. . ms(n) ; eb= es(0) es(1) .. . es(n) andf and Y are defined in (12) and (13).
IV. NUMERICALEXAMPLE
Inthis section, anexample is worked out to illustrate the proposed designalgorithms. Suppose that the system showninFig. 1 has the signal model6Swith the following system matrices:
ATs = 02:3060 1 0 0 0 0 02:9625 0 1 0 0 0 02:2590 0 0 1 0 0 01:0922 0 0 0 1 0 00:3009 0 0 0 0 1 00:0325 0 0 0 0 0 ; Bs= 1 0 0 0 0 0 Cs= [0 0 0 0 0:0062 0:2170]; Ds= 0
and the channel model6Cwith the following system matrices: Ac= 01:90 01:18 00:24 1 0 0 0 1 0 ; Bc= 1 0 0 Cc= [00:360 0 0:153 0 0:027]; Dc= 0:45:
To simplify the discussion, assume that there is only one uncer-tain variable in the channel. The related matrix parameters are HT
c1 = Hc11T = [0:02 0:03 0:01], Hc2 = Hc21 = 0:024,
E1 = E11 = [0:62 0:51 0:55], E2 = E21 = 0:73, an d
E3 = E31 = 00:082. Inaddition, assume, to start, that there is no channel noise (i.e.,v 0). In Fig. 2, the filter output constraint envelopes corresponding to a particular channel output y(k) are shown. Thisy(k) is generated by the unit-impulse response of the signal model when the uncertain variable of the channel model equals zero. Then, theH1optimal filter is designed using the LMI Toolbox of Matlab [16] with nf selected to be 15. The resultant optimal = 3:5330, and the filter output with respect to the above y(k) is also displayed inFig. 2. Clearly, the output is quite close to the lower envelope. Thus, the robustH1filter is designed with1 = 1 and2 = 1:5. The resultant optimal and are 3.8891 and 0.6243, respectively. InFig. 3, the new filter output is displayed, which is very close to the envelope center line, as desired.
To see the robustness of the two filters, the filter outputs are com-puted againwith respect to the same signals(k), but this time, a nor-mally distributed white channel noisev(k) with zero meanand vari-ance equal to1 2 1004is added. In addition, the channel uncertain variablec1(k) is set to be 03:356k=(k2+ k + 1) for k 0. The re-sults are showninFigs. 4 and 5. It is seenthat most of the time, the filter
M = AT eXAe0 X + EeTJEe ATeXBe+ EeTJEw ATeXH1e+ EeTJE3 CeT BT eXAe+ EwTJEe BTeXBe+ EwTXEw0 I BeTXH1e+ EwTJE3 DTe HT 1eXAe+ E3TJEe H1eTXBe+ E3TJEw H1eTXH1e+ E3TJE30 J H2eT Ce De H2e 0 I < 0 (16)
Fig. 2. Filter output ( ) (dotted line), the corresponding filter output constraint envelopes (dashed lines), and output of the optimal filter (solid line).
Fig. 3. Center line of the mask (dash-dotted line) and output of the robust filter (solid line).
Fig. 4. Filter output of the optimal filter (solid line) with channel noise and uncertainty.
Fig. 5. Filter output of the robust filter (solid line) with channel noise and uncertainty.
output of the robustH1filter is kept withinthe envelopes, whereas the situationfor the optimalH1filter is relatively worse.
V. DISCUSSION ANDCONCLUSION
In this correspondence, an LMI-based optimization approach is proposed to designFIR filters that satisfy prespecified time-domain envelope constraints at the output. In addition to the widely treated channel noise, the less-considered channel uncertainties are accom-modated using the versatile IQC formulation. Both theH1optimal filter and the robustH1filter are offered so that the tradeoff between theH1performance and constraint robustness may be carried out.
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[2] B. Vo, A. Cantoni, and K. L. Teo, “Envelope constrained filter with linear interpolator,” IEEE Trans. Signal Processing, vol. 45, pp. 1405–1414, June 1997.
[3] , Filter Design With Time Domain Mask Constraints: Theory and Applications. Dordrecht, The Netherlands: Kluwer, 2001.
[4] A. Sideris and H. Rotstein, “Single-input-single-output control with time domain constraints,” Automatica, vol. 29, pp. 969–983, July 1993.
[5] H. Rotsteinand A. Sideris, “ optimizationwith time domaincon-straints,” IEEE Trans. Automat. Contr., vol. 39, pp. 762–779, Apr. 1994. [6] M. J. Grimble, “ inferential filtering, prediction and smoothing
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design,” IEEE Trans. Circuits Syst. II, vol. 47, pp. 79–82, Jan. 2000. [8] Z. Zang, A. Cantoni, and K. L. Teo, “Envelope-constrained IIR filter
designvia optimizationmethods,” IEEE Trans. Circuits Syst., vol. 46, pp. 649–653, June 1999.
[9] Z. Tan, Y. C. Soh, and L. Xie, “Envelope-constrained filter design: anLMI optimizationapproach,” IEEE Trans. Signal Processing, vol. 48, pp. 2960–2963, Oct. 2000.
[10] B. Vo and A. Cantoni, “Continuous-time envelope constrained filter design with input uncertainty,” Trans. Circuits Syst. I, vol. 47, pp. 1445–1454, Oct. 2000.
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[12] H. Li and M. Fu, “A linear matrix inequality approach to robust filter,” IEEE Trans. Signal Processing, vol. 45, pp. 2338–2350, Sept. 1997.
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[16] P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali, LMI Control Toolbox—for Use With MATLAB. Natick, MA: The Math Works Inc., 1995.
Eigenstructure Approach for Complete Characterization of Linear-Phase FIR Perfect Reconstruction Analysis Length
Filterbanks
Anamitra Makur, Arigovindan Muthuvel, and P. Viswanadha Reddy
Abstract—The eigenstructure based characterization of -channel fi-nite impulse response perfect reconstruction (FIR PR) filterbanks in a pre-vious paper by the authors is extended here to the linear-phase case.Some results relating to linear-phase filterbanks is derived by finding appropriate restrictions on the eigenstructure of the analysis polyphase matrix.Conse-quently, a complete and minimal characterization for such filterbanks with all analysis length2 and any synthesis length is developed.Parameteri-zation and design examples are also presented.
Index Terms—Eigenstructure, FIR filterbank, linear phase.
I. INTRODUCTION
In[1], we used the eigenstructure representation of the polyphase
matrix to propose complete characterizations of finite impulse response perfect reconstruction (FIR PR)M-channel filterbanks with first-order
analysis polyphase matrix. Linear-phase FIR perfect reconstruction fil-terbanks (LPFBs) find application in many signal and image processing fields. In this correspondence, we extend the eigenstructure represen-tationto obtaina complete characterizationof linear-phase FIR perfect reconstructionM-channel filterbanks with all analysis filter lengths being2M (hence first-order analysis polyphase matrix), which are re-ferred to henceforth as FOLPFBs. The synthesis filter length in this characterizationis not restricted to2M as is conventionally done but may take a value up toM2.
Characterizationof a subclass of LPFBs, such as orthogonal, M = 2, or M = 3, has beenreported onseveral occasions. The de-sign of FOLPFBs with a multistage structure using the discrete cosine transform (DCT) such that a fast implementation exists is reported in [2], where it is called the lapped biorthogonal transform. Designof anLPFB of any order is reported in[3], whenK filters are givensuch that the part polyphase matrix has rankK for all z01exceptz01= 0,
Manuscript received July 9, 2002; revised July 7, 2003. This work was done at the Electrical Communication Engineering Department, Indian Institute of Science, Bangalore, India.
A. Makur is with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore.
A. Muthuvel is with Swiss Federal Institute of Technology, Lausanne, Switzerland.
P. V. Reddy is with General Electric, Bangalore, India. Digital Object Identifier 10.1109/TSP.2004.827201
and the remaining filters are designed. In [4], symbolic computation is used to characterize an LPFB of any order. However, none of the above characterizations are complete. In [5], a lattice structure is used to characterize the LPFB so that the analysis polyphase matrix
of the FOLPFB becomes (1=2p2) (1 + z
01)U (1 0 z01)U
(1 0 z01)V (1 + z01)V
1 VU0 U0JM=2
0JM=2 0V0 , whereU, V, U0, an dV0are nonsingular M=2 2 M=2 matrices, and Jkis ak 2 k counter identity matrix. It is shown to be complete for FOLPFBs with synthesis length2M [6].
We briefly describe below the characterizationof [1]. Replacingz01 by, the lth-order analysis polyphase matrix E(z) is seenas a matrix polynomialEl(). Any matrix polynomial may be characterized by the Jordan pair (or decomposable pair or spectral data)(Y; T()) with Y = [ XF XR] and T() = diag(I00JF; JR0IMl00), where diag() represents a block diagonal matrix with the arguments as the blocks insequence,Ikis thek 2 k identity matrix, and 0 is the degree ofjEl()j. XF is theM 2 0 canonical set of Jordan chains, an d JFis the0 2 0 Jordan form of El() (finite Jordan pair or finite spectrum). JF is block diagonal with Jordan blocks of sizeb0; . . . ; bnsuch that biare nonincreasing positive integers summing up to0. Each Jordan block’s diagonal elements are eigenvalues ofEl(), upper off-diagonal elements are 1, and the remaining elements are 0.1 XRandJRare the
correspondingM 2 (Ml 0 0) and (Ml 0 0) 2 (Ml 0 0) matrices
of the reversed matrix polynomiallEl(01) for the zero eigenvalue (infinite Jordan pair or infinite spectrum). It follows that [7]
El() = A(IMl0 P)T()S01l01Q() (1) where Sk= XF XRJRk XFJF XRJRk01 .. . ... XFJFk XR ; Q() = IM IM .. . l01I M : (2)
A is an M 2 Ml matrix such that [ ST
l02 AT]T is nonsingular, and P = diag(I0; JR)S01l01[ IMl0M 0 ]TSl02.
For the FIR inverse to exist,El() has to be a matrix polynomial with a monomial determinant. This is equivalent to all eigenvalues of El() being zero. Thus, JF(as alsoJR) should have a zero diagonal. Further, for the first-order(l = 1) case, (1) simplifies to the block
diagonal characterization[1]
E1() = AT()Y01 (3)
where A and Y are any M 2 M nonsingular matrices. Note
that JF, JR are nilpotent matrices with indices of nilpotency
nF and nR (nF = b0, size of the largest Jordanblock of
JF, etc.). Then, the synthesis polyphase becomes E011 () = Ydiag(I001 + ni=2JFi010i; 0IM00 0 n 01i=1 iJRi)A01. It follows that the maximum length of the synthesis filters is (nF+ nR)M , and reconstruction delay is (nF+ 1)M 0 1. The char-acterization allows unconstrained parameter optimization and provides control over the length of the synthesis filters and reconstruction delay. 1For example, for0 = 4, = 3 1 and zero eigenvalue = 0 1 0 0
0 0 1 0 0 0 0 0 0 0 0 0 1053-587X/04$20.00 © 2004 IEEE
