A direct approach to H/sub 2/ optimal deconvolution of ... · The optimal signal reconstruction...

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 50, NO. 7, JULY 2002 1685

A Direct Approach toH2 Optimal Deconvolutionof Periodic Digital Channels

Huan Zhou, Member, IEEE, Lihua Xie, Senior Member, IEEE, and Cishen Zhang, Member, IEEE

Abstract—This paper studies the 2 optimal deconvolutionproblem for periodic finite impulse response (FIR) and infiniteimpulse response (IIR) channels. It shows that the 2 norm ofa periodic filter can be directly quantified in terms of periodicsystem matrices and linear matrix inequalities (LMIs) without re-sorting to the commonly used lifting technique. The optimal signalreconstruction problem is then formulated as an optimizationproblem subject to a set of matrix inequality constraints. Underthis framework, the optimization of both the FIR and IIR periodicdeconvolution filters can be made convex, solved using the interiorpoint method, and computed by using the Matlab LMI Toolbox.The robust deconvolution problem for periodic FIR and IIRchannels with polytopic uncertainties are further formulated andsolved, also by convex optimization and the LMIs. Compared withthe lifting approach to the design of periodic filters, the proposedapproach is simpler yet more powerful in dealing with multiobjec-tive deconvolution problems and channel uncertainties, especiallyfor IIR deconvolution filter design. The obtained solutions areapplied to the design of an optimal filterbank yielding satisfactoryperformance.

Index Terms—Channel equalization, filterbanks, 2 optimiza-tion, linear matrix inequalities, periodic systems.

I. INTRODUCTION

T HE channel deconvolution problem is to inverse the con-volution process of the transmission channel and the signal

transmitted through the channel and remove the effect of noiseto reconstruct the signal transmitted. This is a fundamental andpractically important problem and has been widely explored inthe literature. It has found many applications in telecommuni-cations and signal processing. See, for example, [2]–[7].

For linear time-invariant (LTI) channels, the optimal decon-volution problem has been solved under the least squares andperformance specifications by the Wiener filtering techniquein the frequency domain; see [4], [5], [7], and [8], and by theKalman filtering technique in the time domain, see [6] and [9]. Avast number of results on the optimal deconvolution problem hasbeen reported based on the Wiener and Kalman filtering tech-niques for the past few decades.

Periodically operated devices and algorithms, such asmodulation/demodulation, multiplexing, multirate samplingand frequency multiplying, and dividing processes are widely

Manuscript received May 16, 2001; revised March 26, 2002. The associateeditor coordinating the review of this paper and approving it for publication wasDr. Xiang-Gen Xia.

H. Zhou and L. Xie are with the School of Electrical and Electronic En-gineering, Nanyang Technological University, Singapore (e-mail: [email protected]).

C. Zhang is with the Department of Electrical and Electronic Engineering,The University of Melbourne, Parkville, Australia.

Publisher Item Identifier S 1053-587X(02)05656-8.

used in telecommunication and signal processing applications.Channels and filters involving the use of periodic devices andalgorithms can be modeled by finite impulse response (FIR)or infinite impulse response (IIR) filters with periodicallytime-varying parameters. This class of linear periodicallytime-varying (LPTV) filters represents a class of practicallyimportant systems and has recently attracted a great deal ofattention from researchers.

There have been results on optimal periodic filtering sys-tems under the and least squares performance indices. Forexample, formulations and methods for the design of optimalmultirate filterbanks are presented in [10]–[12]. The optimal de-sign in [11] considered the data quantization noise. This, how-ever, was not considered in [10] and [12]. Further, the results of[10]–[12] did not take the system modeling uncertainties intoaccount, and they are only for the design of FIR filterbanks,which are a special class of periodic filters.

In the analysis and design of LPTV filters, a technique calledlifting is commonly adopted. The basic concept of lifting is totransform the LPTV filter into an equivalent LTI model withgreater input and output dimension. An advantage of the liftingis that the existing method for LTI filters can be applied to thelifted LTI model for the analysis and design of the LPTV filter.Various lifting approaches exist, and the lifting can be carriedout in the time domain [13], [14] and the frequency domain [15],[16].

However, there are disadvantages of using the lifting tech-nique in the design of LPTV deconvolution filters because ofthe following difficulties.

i) Structural constraint:The lifted model of the LPTV filtermust satisfy a spacial structural constraint to ensure thatthe corresponding LPTV filter is causal. Although thisconstraint is always satisfied when a causal LPTV filteris lifted into an LTI model, the available LTI methods arenot capable of designing the lifted LTI filter that meetsthe model structural constraint.

ii) Inverse of lifting:The design of the deconvolution filterusing the lifting technique results in a lifted LTI modelwith higher input and output dimensions. To convertthe designed lifted model to a LPTV model with lowerinput and output dimensions, which is required bythe computation and implementation procedures, willinvolve very complicated procedure and computation,especially when an IIR LPTV filter is to be sought. Itis a much more tedious and difficult procedure than therelatively straightforward lifting procedure.

iii) State dimension consistency:In the general case of LPTVfilters with time-varying state dimension, there is an in-

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1686 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 50, NO. 7, JULY 2002

consistency between the state dimension of the LPTVfilter and its lifted LTI model, i.e., the time-varying statedimension can be different from the constant state dimen-sion of the LTI lifted model from time to time [17]–[19].Thus, it is possible that the state dimension of the LPTVfilter is higher than that of the lifted model at some time.Because of this, the desirable state dimension of the de-signed LPTV filter may not be obtained by the indirectapproach based on lifting and by specification of the statedimension of the lifted model.

iv) Channel uncertainties:Parameters of the lifted modelare nonlinear combinations of that of the original LPTVchannel. Because of this, physically and numericallymeaningful properties represented by the parameters ofthe original LPTV channel are no longer preserved bythe lifted model. This will cause inaccurate and veryconservative quantification of the LPTV channel whenit has modeling uncertainties.

Although the causality constraint and channel uncertaintymay become trivial when FIR channel and deconvolution filtermodels are considered, they impose major difficulties in dealingwith IIR deconvolution filters or channels with uncertaintyusing the lifting technique. Note that the causality constraintis nonlinear in parameters of the corresponding periodic filter.Because of this, there has been no efficient numerical approachto the constrained optimal lifted filter and its causal periodicfilter, although the design of the lifted optimal deconvolutionfilter itself is a standard convex optimization problem.

In this paper, we will propose a direct approach to the optimaldeconvolution problem for periodic channels with both the FIRand IIR LPTV filters. The approach will be able to overcomethe aforementioned difficulties associated with the lifting tech-nique, especially when an IIR LPTV filter is to be considered.

To achieve the objective, we first derive a direct quantificationof the norm of the periodic filter in terms of its periodic pa-rameters. This allows us to cast the deconvolution probleminto a convex optimization problem. As a result, the solutions forthe optimal FIR and IIR periodic deconvolution filters arederived via the convex optimization subject to a set of linear ma-trix inequalities (LMIs). The direct optimization approach andthe LMI solution are then extended to solve the robust optimaldeconvolution problem for FIR and IIR periodic channels withpolytopic modeling uncertainties. It is worth noting that the LMIapproach has been widely applied in solving control problemsin the past few years [1], [20]. The extension of the approach tothe optimal deconvolution of periodically time-varying channelsoffers an efficient numerical technique for a variety of signalprocessing problems, including the multiobjective multirate fil-terbank design.

To demonstrate the performance and computation of theobtained solutions, we apply our solution for the periodicchannel deconvolution problem to the design of an optimalmutirate filterbank system. The simulation results clearlydemonstrate the superiority of our approach to the solution forperfect reconstruction (PR) in the presence of the channel noiseand uncertainties.

The rest of the paper is organized as follows. Section IIpresents the definition of the norm of periodic digital

filters and its quantification in terms of matrix inequalities.The periodic channel deconvolution problem is formulated inSection III. Section IV presents convex optimization solutionsto the FIR and IIR channel deconvolution problem. The robustchannel deconvolution problem with polytopic modelinguncertainties in the FIR and IIR channels is solved in Section V.Application examples of the obtained proposed solutions tothe design of an LPTV deconvolution filter and an optimalmultirate filterbank are demonstrated in Section VI.

II. PERIODIC DIGITAL FILTERS AND THE NORM

In this section, we present the definition of the norm of pe-riodic digital filters and present a matrix inequality formulationfor optimizing the norm of periodic filters. This is derivedfrom a periodic Lyapunov equation.

Consider a linear -periodic digital filter in thefollowing th-order state equation form.

(2.1)

(2.2)

where is the state vector, , are,respectively, the input and output of the filter, and ,

, , and are -periodicmatrices of the filter that satisfy

(2.3)

Let the transition matrix of the periodic digital filter (2.1) and(2.2) be

.(2.4)

It is known that the eigenvalues (also called characteristic mul-tipliers) of are independent of and that the periodicdigital filter (2.1) and (2.2) is stable if and only if all the eigen-values of are within the unit circle of the complex plane[21].

For the -periodic filter with inputs and outputs, letbe the th column vector of the identity matrix, and

let be the Dirac delta function. Then, is theimpulse applied to theth input of at , and is thecorresponding output of under the zero initial condition. Foreach , it is simple to verify that

(2.5)

If is a stable filter, the norm of the output vector sequencefor each exists, i.e., we have (2.6),

shown at the bottom of the next page. Recall that the standarddefinition of the norm of linear time-invariant filters is the

ZHOU et al.: DIRECT APPROACH TO OPTIMAL DECONVOLUTION 1687

norm of the filter impulse response, which is also equiva-lent to the averaged energy of the output due to a white noiseinput with unity power spectrum. A natural extension of thenorm of linear time-invariant filters to -periodic filters is totake the averaged norm of the filter impulse response overthe period . This leads to the definition of the norm of theperiodic filter as (2.7), shown at the bottom of the page. Wenow present three technical results in the following for quantifi-cation of the norm of periodic digital filters. These resultsare important to the solution of the optimal deconvolutionproblem.

Lemma 2.1:The -periodic digital filter in the form (2.1)and (2.2) is stable if and only if there exists an-periodic sym-metric matrix with and

for such that the following periodicLyapunov equation is satisfied:

(2.8)

Proof: Consecutively using the Lyapunov (2.8), we obtain

(2.9)

If (2.1) and (2.2) is a stable filter, the state matrixsatisfies

This, together with (2.9), yields

(2.10)

Apparently, and are satisfied. Itfurther follows from the stability of (2.1) and (2.2) that , as in

(2.9), exists and is a periodic solution of the Lyapunov equation(2.8).

On the other hand, the existence of in (2.8)implies that the limit

exists as approaches infinity. Then, the stability of (2.1) and(2.2) follows.

Lemma 2.2: If the -periodic digital filter in the form(2.1) and (2.2) is stable, then its norm satisfies

trace (2.11)

where is the periodic solution of the Lyapunov (2.8).Proof: It follows immediately from (2.6) and (2.10) that

trace (2.12)

Substituting this into (2.7) yields (2.11).The norm of the periodic filter can also be characterized

in terms of LMIs.Lemma 2.3: If the -periodic digital filter in the form

(2.1) and (2.2) is stable, then its norm satisfies

trace (2.13)

subject to

(2.14)

(2.15)

simultaneously for , wherewith .

trace (2.6)

trace (2.7)


Fig. 1. Error system model.

Proof: In conjunction with (2.11) and (2.8), the result fol-lows from the monotonicity of Lyapunov equation and the well-known Schur complements.

III. DECONVOLUTION PROBLEM

Consider a transmission channel modeled by a stable linear-periodic digital filter . Let and be the measure-

ment output and the measurement noise of, respectively, i.e.,

Consider that the measurement signalof the transmissionchannel is the input to a receiver channel modeled by a linear

-periodic digital filter . Let be the delayoperator such that . In signal processing andcommunication applications, it is often required that the trans-mitted signal be reconstructed with an admissible time delay.Let be the error signal between the receiverchannel output and the delayed signal . A block diagramof the error system is shown in Fig. 1. The channel deconvo-lution problem is to find a linear -periodic filter for thereceiver channel such that the error signal is minimized undersome performance specification. In this paper, we considerthe channel deconvolution problem under theperformanceindex.

For the brevity of presentation, in the sequel, we will con-fine our attention to the channel deconvolution problem forsingle-input single-output (SISO) channels. It is noted that itwill be straightforward to extend the solution to the multi-inputmulti-output (MIMO) channel case.

Consider the transmission channel filterin the followingth-order state equation form

(3.1)

(3.2)

where is the state vector, is the input,is the measurement output of the filter, and ,

, , and are -periodic matrices ofthe filter and satisfy (2.3). Throughout this paper, it is assumedthat the transmission channel filter (3.1) and (3.2) is stable.

Let the -periodic digital receiver filter be in the followingth-order state equation form:

(3.3)

(3.4)

where is the state vector, and ,, , and are -periodic matrices of

the filter and satisfy

(3.5)

Let

......

...

We use these to write a state equation for the-delay systemas

(3.6)

(3.7)

where .

Introduce an augmented state vector . Then, we

can use the state equations (2.1), (2.2), (3.3), (3.4), (3.6), and(3.7) to write a state equation for the error signalas

(3.8)

(3.9)

where is the input to the error state equation,and

(3.10)

(3.11)

(3.12)

(3.13)

Let denote the -periodic error system (3.8) and(3.9). Then, the optimal deconvolution problem is stated asfollows: Find periodic matrices , , , and for thereceiver filter in the form (3.3) and (3.4) such that (3.3) and (3.4)is stable and the norm of the error system is minimized.

Applying Lemma 2.3 to the error system (3.8) and (3.9), weobtain the following matrix inequality formulation for mini-mizing the norm of the error system.

trace (3.14)


subject to

(3.15)

(3.16)

for , simultaneously, wherewith .

Remark 3.1:The above formulation of the optimal de-convolution of periodic channels is a direct approach in thesense it uses directly the periodic parameters of the channel fil-ters. In contrast to this, an indirect deconvolution of FIRchannels was presented in [22], where the periodic channel fil-ters are first transformed into linear time-invariant models by alifting technique. Then, the optimal filter design is applied tothe lifted linear time-invariant models. The advantages of thedirect approach over the indirect approach via lifting is thatthe direct approach is more powerful in dealing with IIR filtersand modeling uncertainties. Moreover, it does not encounter thedifficulty of structural constraint on the causality of the liftedmodels.

IV. OPTIMAL DECONVOLUTION OF PERIODIC

FIR AND IIR CHANNELS

In this section, we present convex optimization solutions interms of LMIs to the design of optimal FIR and IIR deconvolu-tion filters under the performance. Because of some essen-tial difference between the design of the periodic FIR andIIR deconvolution filters, as will be pointed out in the sequel,we deal with the two problems separately.

A. Design of FIR Periodic Deconvolution Filters

For a given transmission channel modeled by a known peri-odic FIR or IIR filter, consider an FIR deconvolution filter ofthe form

(4.1)

where is the length of the FIR deconvolution filter, ,are periodic coefficients to be determined.

The FIR deconvolution filter (4.1) has a state-space represen-tation in the form (3.3) and (3.4) with

... (4.2)

(4.3)

In this situation, it can be observed from (3.10) that andare known matrices of the error

system, and hence, the matrix inequalities (3.15) and (3.16) arelinear in , and the FIR filter parameters as contained in

and . It follows that the optimization in (3.14) subjectto (3.15) and (3.16) is convex, and the optimaldeconvolutionfilter can be obtained easily by using the convex optimization,as given in (3.14)–(3.16). Moreover, there is no constraint onthe choice of the order of the FIR deconvolution filter in theabove LMI solution.

B. Design of IIR Periodic Deconvolution Filters

When an IIR periodic filter is to be designed, the matrixinequalities (3.15) and (3.16) are no longer linear in the filterparameters , , , and . Therefore, they cannot bedealt with directly by the LMI optimization [23]. To tackle thisproblem, we will introduce variable transformations to recoverthe convexity of the problem. We assume that the IIR decon-volution filter to be designed is of full order, i.e., .In fact, the optimization problem for a reduced-order IIRdeconvolution filter with is an essentially nonconvexproblem, which is beyond the scope of this paper. In addition,the order constraint is an essential difference between the FIRand IIR deconvolution filters.

To derive the optimal solution and under the condition, we first decompose the symmetric matrix

, and its inverse of (3.15) and (3.16)as

(4.4)

where , , , , , , and , , , areapparently symmetric matrices. In addition, denote

(4.5)Next, we introduce the change of variables

(4.6)

(4.7)

(4.8)

It is easy to verify that the matrices of (4.4)–(4.8) satisfy thefollowing matrix equations:

(4.9)

(4.10)

(4.11)

(4.12)

(4.13)

Pre- and post-multiplying (3.15) by diag and

diag , respectively, and using (4.9)–(4.13), wecan transform (3.15) into the equivalent matrix inequality in(4.14), shown at the bottom of the next page. Similarly, bypre- and post-multiplying (3.16) by diag

and diag , respectively, (3.16) is equivalent to


(4.15), shown at the bottom of the page. Observe that the matrixinequalities (4.14) and (4.15) are linear in the-periodicmatrices , , , , , , and . This leads to thefollowing result.

Theorem 4.1:The optimal decovolution problem for thestable -periodic transmission channel filter (3.1) and (3.2) canbe solved by optimizing

trace (4.16)

subject to the periodic LMIs (4.14) and (4.15) with.

Remark 4.1:Since the transmission channel is assumed tobe stable, the optimal FIR or IIR deconvolution problem isalways solvable. Hence, the solution matrices, , , ,

, , and of the optimization in Theorem 4.1 exist. Whena solution of the optimization in Theorem 4.1 is obtained, then

(4.17)

is satisfied. Since , it is easy to verify that

and that . Hence, the matrix is nonsin-gular such that nonsingular solutions and can always befound for (4.17). Consequently, the rest of the optimal periodicreceiver filter parameters, i.e., , , , ,can be obtained from (4.6)–(4.8).

It is possible that the matrix is numerically close

to being singular if the matrix is saturated at the op-

timum (satisfying ). This problem can be resolvedby introducing an additional LMI constraint with a parameterin the optimization as follows:

(4.18)

Then, the procedure maximizes the minimal eigenvalueof and yields the desirable well-conditioned matrix

(see [24] for details).

Remark 4.2:The solution of the optimization in Theorem 4.1subject to the matrix inequality (3.16) implies that the resultanterror system satisfies the Lyapunv inequality

It follows that the error system and, hence, the designed periodicreceiver filter (3.3) and (3.4) is stable.

V. DECONVOLUTION OFUNCERTAIN PERIODIC CHANNELS

As pointed out earlier, the direct approach to the optimal pe-riodic channel deconvolution has an advantage in dealing withmodeling uncertainties. In applications such as wireless com-munications, there inevitably exist channel modeling uncertain-ties due to the time-varying nature of channels and/or modelingand quantization errors. In this section, we consider periodicchannels with modeling uncertainties in a polytope. That is, pe-riodic matrices , , , and of the filter in (3.1) and(3.2) reside in the following uncertainty polytope:

(5.1)

where , , , and are the matrices at thethvertex of the polytope. It is, again, assumed that the periodictransmission channel filter (3.1) and (3.2) with the polytopicuncertainty is stable.

It is noted that a polytopic representation describes thechannel modeling uncertainties with a convex combination ofseveral sets of matrices for the state equation. This is a generalrepresentation of the channel uncertainties and covers theinterval parameter uncertainty as a special case. Because thepolytopic uncertainty deals directly with the parameter values,it is a more realistic and, hence, less conservative representation

(4.14)

(4.15)


than the norm-bounded modeling uncertainty representation[25].

A. Deconvolution of FIR Uncertain Periodic Channels

In this subsection, we consider periodic transmission chan-nels modeled by the FIR filter

This can be represented in the state equation:

...

(5.2)

Assume that there exist modeling uncertainties in the FIR filtercoefficients , . In addition, uncertainties aredescribed by a polytope for the coefficient matricesandas

(5.3)

where the matrices , are -periodic, with respect to,and are the matrices at theth vertex of the polytope.

Our objective here is to find a periodic FIR deconvolutionfilter that will minimize an upper bound of the norm of theerror system (3.8) and (3.9) for all uncertainties residing in thepolytope (5.3). The result is stated in the following theorem.

Theorem 5.1:The optimal FIR filter that minimizes anupper bound of the norm of the deconvolution error system(3.8) and (3.9) over the family of transmission channels charac-terized by the polytope (5.3) can be obtained by the followingoptimization:

trace (5.4)

subject to

(5.5)

(5.6)

for and , simultaneously,where

(5.7)

...

(5.8)

and and are as in (3.10) and .Proof: By substituting (3.10) into (3.15) and (3.16) at each

vertex of (5.3), we have two matrix inequalities

(5.9)

(5.10)

where denotes the entry that can be deduced from the sym-metry of the matrices and

Note that the above characterization is not convex. However, inview of the fact that the matrices , , , and are knownand have the forms of (4.2) and (5.2), a simple rectification canbe made by introducing

(5.11)

and rearranging the matrices and as

(5.12)

where and are as in (5.8). That is, we confine the lastcolumn and last row of the matrix to be vertex-indepen-dent.

Hence, (5.5) and (5.6) follow. Obviously, (5.5) and (5.6) areconvex, and the optimization of (5.4) subject to (5.5) and (5.6)leads to an optimized upper bound of the norm of the de-convolution error system (3.8) and (3.9).

Remark 5.1: In Theorem 5.1, we have exploited the fact thatthe matrices , , , and are vertex-independent and al-lowed the first sub-block


of , i.e., to be vertex dependent. Although it is some-what conservative due to the constraint that and arefixed over all the vertices, the result will greatly reduce the con-servativeness of the design as is now allowed to be vertexdependent. Note that the optimization in Theorem 5.1 can becarried out easily by using the Matlab LMI Toolbox [20].

B. Deconvolution of IIR Uncertain Periodic Channels

In this section, we investigate the design of more general IIRfilter to achieve optimal channel deconvolution under thechannel modeling uncertainty.

Recall that the optimization in (3.14) subject to (3.15) and(3.16) is not convex for the IIR filter case. For the periodic filterwith the polytopic uncertainty in the form (5.1), a common prac-tice by setting the matrix to be vertex-independent will leadto a very conservative design. To reduce the conservativeness ofthe design, we introduce some additional free parameters thatare similar to the method described in [26] to obtain an alterna-tive quantification of the norm to avoid the product of ma-trices and or and in (3.15) and (3.16). Thisresult is given in the following lemma, where is used todenote a matrix at theth vertex of the poly-tope .

Lemma 5.1:The optimal IIR filter that minimizes anupper bound of the norm of the deconvolution error systemover the family of transmission channels with the polytopic un-certainty can be obtained by the following optimization:

trace (5.13)

subject to

(5.14)

(5.15)

for and , simultaneously,where the periodic matrices and are symmetric, and

is a full matrix that does not present any struc-tural constraint such as symmetry.

Proof: First, observe from (5.1) and (3.10) that with the in-volvement of the uncertain transmission channel filter, the errorsystem matrices have a similar convex form of(5.1). Then, it follows from (5.14) and (5.15) that

(5.16)

(5.17)

where

We now show that the above two LMIs hold if and only if theLMIs (3.15) and (3.16) hold. The sufficiency is straightforwardto establish since (3.15) and (3.16) withlead to (5.14) and (5.15). To prove the necessity, we notice from(5.14) and (5.15) the fact that

It follows that is nonsingular and

are satisfied. This enables us to obtain

(5.18)

(5.19)

Multiplying (5.18) and (5.19) on the right bydiag and on the left by , we recovermatrix inequalities (3.15) and (3.16). Therefore, (5.13) pro-vides an optimized upper bound of the norm of thedeconvolution error system.

Remark 5.2: It is important to note that due to the separationbetween the Lyapunov matrices and the system dynamics orinput matrices, the robust optimal deconvolution filter can bedesigned with parameter-dependent Lyapunov functions.This can significantly reduce the conservativeness as comparedwith the approach of using the constrained Lyapunov matrix.

Observe that the matrix inequalities (5.14) and (5.15)are still not linear in the deconvolution filter parameters

, as contained in .To resolve this nonconvexity problem, similar to Section IV-B,we introduce variable transformations. To this end, let

(5.20)

where denotes insignificant blocks in the matrices. By ap-plying some congruence transformations and noting the abovechange of variables, the matrix inequalities (5.14) and (5.15) canbe converted into new LMIs, as given in the following lemma.

Lemma 5.2:An optimized upper bound of the norm ofsignal reconstruction error system over the family of transmis-


sion channels with polytopic uncertainty can be obtained by theoptimizations in (5.21)–(5.23), shown at the bottom of the page,for , simultaneously.

With the aid of the above lemma, the periodic IIR filter thatminimizes an upper bound of the norm of deconvolutionerror system for all admissible uncertainties in the polytope canbe obtained as in the following theorem.

Theorem 5.2:The robust optimal IIR filter deconvolu-tion problem for the error system (3.8) and (3.9) with the poly-topic uncertainty (5.1) can be solved by the convex optimiza-tion in (5.24)–(5.26), shown at the bottom of the page, for

and , simultaneously, where. In addition, the parameters

of the IIR deconvolution filter can be computed from

(5.27)

(5.28)

(5.29)

Proof: From Lemma 5.2, we define and

(5.30)

Multiply (5.21) on the left by diag and onthe right by diag , and multiply (5.23) onthe left by diag and on the right bydiag . We obtain (5.31) and (5.32), shownat the bottom of the next page, where

The LMIs (5.25) and (5.26) are then derived from (5.31) and(5.32) by defining

(5.33)

(5.34)

(5.35)

trace (5.21)

(5.22)

(5.23)

trace (5.24)

(5.25)

(5.26)


The above solutions to the LMIs provide the computation of thedeconvolution filter parameters in (5.27)–(5.29).

The above solution of the robust periodic IIR deconvolutionfilter satisfies the original matrix inequality (3.16), which isequivalent to a Lyapunov inequality and, hence, guarantees thestability of the deconvolution filter.

VI. EXAMPLES

This section demonstrates two examples of the proposedoptimal design of periodic deconvolution filters. The first ex-ample serves the purpose of comparing the direct approach ofthis paper with that of [22] by using the indirect approach vialifting. The second example presents an application to the de-sign of an optimal multirate filterbank in comparison with theexisting PR filterbank.

A. Example 1

We consider the numerical example, as investigated in [22],where a lifting technique is applied to solve the deconvolutionproblem. The third-order LPTV FIR filter of period 2 is de-scribed by

with

for

for

We first design a third-order LPTV FIR inverse filtersuch thatthe norm of the error signal against the input and measure-ment noises is minimized. By the direct approach described inSection IV, we obtain the optimal periodic FIR inverse filter

of the form (4.2) and (4.3), with

for

for

The norm of the error system is 0.1527, which is about thesame as 0.1528, which is given by the lifting technique [22].

We further design an optimal third-order IIR periodic decon-volution filter using the result of Section IV. By using Matlabfor the computation, the parameters of the designed IIR peri-odic filter are obtained as follows:

for

(6.1)

(6.2)

for

(6.3)

(6.4)

The norm of the error system is 0.1169, which provides23.5% improvement over the norm of the third-order op-timal FIR filter.

We carried out statistical tests on the filtering error of thedesigned FIR and IIR filters based on 100 samples. The inputsignal and measurement noise are white noises with zero meanand unit variance. The mean and covariance of the two filtersare compared in Fig. 2. It is shown that both the designed FIRand IIR deconvolution filters provide satisfactory performance.

B. Example 2

Multirate filterbanks have been an active research area in re-cent years and have a wide range of applications in telecom-munications, speech processing, digital audio, and image com-pression. Basically, a multirate filterbank is a multichannel LTIsystem that consists of an analysis filterbank and a synthesisfilterbank. The analysis filterbank decomposes the input signalinto a number of subband signals, whereas the synthesis filter-bank performs reconstruction of the original input from the sub-band signals. Multirate filterbank systems are periodic systemsdue to the periodic multirate sampling operation.

(5.31)

(5.32)


Fig. 2. Comparison between FIR and IIR filters.

In its early stage of development, the study of multirate filter-banks was focused on the design of a synthesis filterbank thatrecovers the input signal perfectly, which is called PR. That is,the output signals are the scaled and delayed versions of the in-puts. However, due to channel noises and quantization errors,PR is practically unfeasible. This motivates the study of optimalfilterbank design [27], [28].

Multirate filterbanks can be represented in a number ofdifferent forms. For example, Fig. 3 shows a general real-ization for an -channel multirate filterbank, whereis the input signal, is the reconstructed signal, ,

, are the subband signals, anddenote -fold decimator and interpolator, respectively, and

and , represent LTI SISO filtersof the analysis filterbank and synthesis filterbank, respec-tively. Note that one property of this general realization isthat the -fold decimators and interpolators in each channeloperate simultaneously. In [29], an FIR LPTV multirate fil-tering scheme for PR is presented where the decimation andinterpolation processes operate sequentially but not simul-taneously. Further, the dynamics of the analysis and syn-thesis filters only require an SISO filter with periodicallytime-varying coefficients, and the multichannel decimationand interpolation process can be implemented with a simpleswitch-summer scheme. The structure of the LPTV filter andthe realization of decimation/interpolation scheme is shownin Fig. 4.

Fig. 3. Simultaneously operated multirate filterbank.

To perform the -fold decimation, the multiplexingswitch operates periodically and, at time , switchesto the th channel to provide the decimated subband signals

. On the other hand, to perform the-foldinterpolation, the demultiplexing summer simply sums upthe subband signals to give . At the time instant

, satisfies because at this time, allthe subband signals are zero for .

As pointed out in [29], the latter offers considerable improve-ments in computation and implementation over the former. Be-cause multichannel parallel processing is no longer required, thenumber of delay operators for implementation of the filter isminimal, and the memories for the storage of the filter coeffi-cients and the amount of multiplication and addition operationsare also considerably reduced.

We consider the numerical example given in [29] in which ananalysis filterbank with length 5 has three channels yielding thesubband signals for . The three FIR analysisfilters are given by

(6.5)

It is shown in [29] that these LTI filters are used to evaluate inputsignal of the bandwidths , , and

, respectively. Note that the analysis filterbankcan be converted to the periodic filter in Fig. 4, where

can be modeled as (3.2) with the equations at the bottomof the page. The allowed delay for signal reconstruction is set to

.


Fig. 4. Structure scheme of the LPTV filter and switch summer.

TABLE IPERIODIC SYNTHESIS FILTER DESIGN RESULTS

be . We assume that the input signal and all the threechannel noises are uncorrelated white noises with zero meanand variances 1 and 0.05, respectively, which is based on thesignal-to-noise ratio (SNR) of 20 dB. By applying the resultsof Section IV-B, we obtain the optimal fifth-order periodicFIR synthesis filter, as shown in Table I.

The tracking performance with some additional noise isshown in Fig. 5, where the solid line stands for reconstructeddelayed input signal, and the-marked line stands for theoriginal input signal. We also carry out statistical tests onthe error signal . Fig. 6 shows the mean and covariance ofdeconvolution error based on 100 samples. It is observed thatthe mean is around 0.0003 and that the covariance rangesfrom 0 to 0.0409.

It is clear that the designed synthesis filterbank gives a goodperformance on signal reconstruction in the presence of channelnoises.

To evaluate the performance of the designed filterbankagainst the PR filter under different noise levels, we introducethe reconstruction SNR

SNR reconstruction SNR

Obviously, the larger the SNR, the better the signalreconstruction performance. Since the reconstruction SNRissignificantly influenced by the additive channel noise, a similarSNR to characterize different channel noise cases is defined as

SNR channel SNR

It is obvious that the signal reconstruction SNRis better inhigher SNR, which can also be observed from Table II. Theresult indicates that the optimal filterbank performs muchbetter than the PR filterbank in the presence of channel noises.

We further examined the performance of the proposedpe-riodic deconvolution filter when applied to the multirate filter-bank under modeling uncertainties. Consider, in the preceding

Fig. 5. Tracking performance of filterbank using LPTV filters(SNR =25 dB).

Fig. 6. Mean and variances of error signal(SNR = 20 dB).

filterbank example, that an uncertainty exists in the second anal-ysis FIR filter such that

, where . Usingthe result in Section V, we designed robust optimalFIR andIIR filterbanks. In addition, their performance in comparison


TABLE IISIGNAL RECONSTRUCTIONSNRS UNDER DIFFERENTCHANNEL NOISES

TABLE IIISIGNAL RECONSTRUCTION SNRS COMPARISON

FOR UNCERTAIN ANALYSIS FILTERBANK

with the PR synthesis filterbank is shown in Table III. It is shownthat in the table that although the PR filter has better trackingperformance when no uncertainty exists, its performance doesdegrade greatly, even without additional channel noise. On thecontrary, the robust FIR or IIR filterbank gives considerablybetter results across the SNRs from 0 to 25 dB.

VII. CONCLUSION

In this paper, we have presented a new direct approach to thedesign of optimal FIR and IIR periodic deconvolution filterssubject to the channel noise and uncertainties. Compared withthe indirect lifting approach to LPTV systems, our direct ap-proach is simpler and more powerful in tackling the multiobjec-tive filter design problem because it does not have difficulties ofthe causality constraint, inverse lifting, state dimension consis-tence, and quantization of channel uncertainties involved in thelifting approach. The optimal periodic solutions of this paper arederived as a convex optimization problem in the form of LMIs,which is an extension of the LMI applications to optimal LTIsystems. The LMI solution is numerically efficient and can bereadily computed by using the standard Matlab LMI Toobox.The obtained results have been demonstrated through an appli-cation example of multirate filterbank design. Because periodicdevices and algorithms are widely used in telecommunicationand signal processing applications, our optimization approachto periodic systems will have potential in solving other prac-tical problems.

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Huan Zhou (S’94–M’99) was born in Shanxi,China. She received the B.Eng. and M.Eng. degreesin automatic control from the Northeast University,Qinhuangdao, China, in 1994 and 1999, respectively.She is currently pursuing the Ph.D. degree atNanyang Technological University, Singapore.

Her research interests include robust control, mul-tirate filterbanks, and digital filter design.


Lihua Xie (SM’97) received the B.E. and M.E. de-grees in control engineering from Nanjing Universityof Science and Technology, Nanjing, China, in 1983and 1986, respectively, and the Ph.D. degree in elec-trical engineering from the University of Newcastle,Callaghan, Australia, in 1992.

From April 1986 to January 1989, he was aTeaching Assistant and then a Lecturer in the De-partment of Automatic Control, Nanjing Universityof Science and Technology. He joined the NanyangTechnological University, Singapore, in 1992, where

he is currently an Associate Professor with the School of Electrical andElectronic Engineering. From November 1998 to July 1999, he was a VisitingResearch Fellow with the Department of Electrical and Electronic Engineering,University of Melbourne, Parkville, Australia. He has authored and coauthoredone book and written more than 90 refereed international journal papers andmore than 80 conference papers. His current research interests include optimaland robust estimation, signal processing, two-dimensional systems, robustcontrol, and active noise control.

Dr. Xie is an Associate Editor of the Conference Editorial Board of the IEEEControl Systems Society.

Cishen Zhang (M’91) received the B.Eng. degreefrom Tsinghua University, Beijing, China, in 1982and the Ph.D. degree in electrical engineering fromNewcastle University, Australia, in 1990.

Between 1971 and 1978, he was an electrician withChangxindian (February Seven) Locomotive Manu-factory, Beijing. He carried out research work on con-trol systems at Delft University of Technology, Delft,The Netherlands, from 1983 to 1985. He has beenwith the Department of Electrical and Electronic En-gineering, University of Melbourne, Australia, since

1989 and is currently an Associate Professor and Reader. In 2002, he is vis-iting the School Electrical and Electronic Engineering, Nanyang TechnologicalUniversity, Singapore, for one year. His research interests include time-varyingsystems and adaptive systems for signal processing and control.

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