Optimal Noise Reduction in Oversampled PR Filter ... - Semantic Scholar
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Optimal Noise Reduction in Oversampled PR Filter Banks Li Chai, Member, IEEE, Jingxin Zhang, Member, IEEE, Cishen Zhang, Member, IEEE, and Edoardo Mosca, Life Fellow, IEEE
Abstract—This paper studies the optimal noise reduction problem for oversampled filter banks (FBs) with perfect reconstruction (PR) constraint. Both the optimal design and worst case design are considered, where the former caters for the noise with known power spectral density (PSD) and the latter for the noise with unknown PSD. State-space based explicit formulae involving only algebraic Riccati equation and matrix manipulations are provided for the general (IIR or FIR) oversampled PR FBs, and the relations between different cases are analyzed and revealed. Extensive numerical examples are provided to illustrate the proposed design methods and to show their effectiveness. Index Terms—Dual frame, noise reduction, oversampled filter banks, perfect reconstruction, state-space approach.
I. INTRODUCTION
O
VERSAMPLED filter banks (FBs) with redundant signal expansions has been extensively studied in recent years. The noise reduction properties, extra design freedom and improved capacity for signal and information representation are their main advantages [1]–[6]. Oversampled FBs have found applications in broad real world problems, for example, the analysis and design of subband coders, precoders and equalizers in data compression and communications [7]–[9], and the analysis and design of imaging and reconstruction process in parallel magnetic resonance imaging (pMRI) [10]. In these applications, perfect reconstruction (PR), or equivalently zero forcing equalization in communications, is often desired. Because of the redundance in oversampled FBs, the PR synthesis FBs are not unique for a given analysis FB, which renders other design objectives in conjunction with PR in the design of synthesis FBs. One of these objectives is subband noise reduction. Manuscript received October 14, 2008; accepted April 26, 2009. First published June 02, 2009; current version published September 16, 2009. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Bogdan Dumitrescu. This work was partially supported by National Science Foundation of China under Grants 60672064 and 60874109, the program NCET-08-0674, the Australian Research Council’s Discovery funding scheme (project number DP0343057), and Singapore Ministry of Education Grants T208A1213 and RG8/06. L. Chai is with the Engineering Research Center of Metallurgical Automation and Measurement Technology, and the Key Lab of Metallurgical Equipment and Control, Wuhan University of Science and Technology, Wuhan, 430081, China (e-mail: [email protected]). J. Zhang is with the Department of Electrical and Computer Systems Engineering, Monash University, VIC3800, Australia (e-mail: jingxin.zhang@eng. monash.edu.au). C. Zhang is with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore (e-mail: [email protected]). E. Mosca is with the Dipartimento di Sistemi e Informatica, Universityá di Firenze, Firenze 50139, Italy (e-mail: [email protected]). Digital Object Identifier 10.1109/TSP.2009.2024280
Subband noises arise from subband quantization, subband channel uncertainties and subband contaminations, which lower the coding gain in subband coding [7], increase the bit error rate in data communication [8] and introduce image artifacts in pMRI [10]. Therefore, the optimal noise reduction of PR synthesis FBs is an issue of practical and theoretical importance. Using an elegant frame theoretic approach, [1], [2], and [7] have proven that an oversampled analysis FB with stable (not necessarily causal) inverse constitutes a frame in , and the PR synthesis FBs for this analysis FB are its dual frames [11]. Among all these dual frames, the canonical dual, given by the parapseudo inverse of the analysis FB, has the minimum norm and is optimal for subband noise reduction if the noises are independent and white. In [1], a parameterization of all dual frame synthesis FBs is given, which provides a general framework for the optimal design of PR synthesis FBs with respect to different criteria for noise reduction. This parameterization is further refined in [12] and [13] by revealing the true dimension of the free parameters. However, to our best knowledge, there has been no direct application of this parameterization to any optimal design of PR synthesis FBs. This paper studies the following problem: Given an oversampled analysis FB satisfying PR condition, find one synthesis FB from the entire set of all dual frame PR synthesis FBs, which is optimal with respect to different criteria for subband noise reduction, and compute efficiently and reliably this optimal synthesis FB. This problem is cast into a generic problem of finding an optimal parameter matrix to minimize the weighted norm of synthesis FB. The solution is derived using a novel state-space parameterization of all PR synthesis FBs. A key feature of the state-space parameterization and computation is that the state-space model can represent the infinite dimensional frames of the FBs in the finite dimensional statespace. Thus, it enables explicit and precise analysis and evaluation of the proposed noise reduction problem with algebraic operations of finite dimensional numerical matrices. This resolves the difficulties of the inevitable complicated symbolic operations and infinite dimensional computations in the polyphase representation and evaluation of FBs. Since the dual frame PR synthesis FBs are generally noncausal and IIR, their state-space parameterization is nontrivial and is deduced from the authors’ recent results [5] on the frame theory based analysis and design of oversampled FBs. This new parameterization, together with the results of [5], renders efficient and reliable optimization methods. These methods involve only algebraic Riccati equation and finite dimensional matrix operations without the approximation generally required in the current literature.
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The above generic solution is used to address two classes of subband noises under three different measures of noise reduction: the colored noises with known or unknown power spectral , generalized and norms of the density (PSD); the synthesis FB with nonsingular or singular weighting matrices. The optimal solution and state-space based efficient computation method are obtained for each case, and the relations between different cases are analyzed and revealed. It is important to pointed out that the noise reduction design of synthesis FBs has been considered previously by many authors, e.g., [9] and [14]–[18]. But the problems considered and results obtained in these works are different from those in this paper. The FBs considered in [14]–[18] are critically sampled, which have no redundance. Hence, the synthesis FBs deoptimal signed in [14]–[18] do not have PR property. The zero forcing equalizer (PR synthesis FB) designed in [9] is restricted to be causal, thus it is applicable only to the analysis FBs having causal stable inverse. Whereas the problem studied in this paper is frame theory based: the analysis FBs considered are FB frames with stable but not necessarily causal inverse, and the synthesis FBs designed are their dual frames which are not necessarily causal (however stability is guaranteed). Because this paper studies a different problem, its results differ significantly from those of [9] and [14]–[18], which are and filtering. The most startling rebased on standard sult of this paper is that of Theorems 4 and 6: for an analysis FB that constitutes a frame, the optimal dual frame PR synthesis FB for subband noise reduction is the same for all the two classes of noises and the three different norms discussed above. To our best knowledge, this result and the result of state-space computation of the optimal dual frame PR synthesis FBs have not been reported previously. The paper is organized as follows. The background knowledge of FB, frame theory, state-space method are presented in Section II. Section III presents an exact parameterization of all PR synthesis FBs. In Section IV, the optimal design problems of PR oversampled FB are formulated and solved systematically by the tool of matrix completion. The optimal design for noises with known PSD and worst-case design for noises with unknown PSD are studied in Sections V and VI, respectively, by using the theory developed in Section VI. The optimal design subject to singular weighting matrix is studied in Section VII based on the result of Section VI. Numerical examples are given in Section VIII to demonstrate the effectiveness of our algorithms. Finally, conclusions are drawn in Section IX. The notations of this paper is quite standard. For a matrix , its transpose, conjugate transpose, trace and eigenvalues are de, , and , respectively. Denote noted by if is full column rank and if is full row rank. II. PRELIMINARIES A. PR Oversampled FBs Consider the -channel oversampled FB with decimation shown in Fig. 1. In the figure, and are factor the input and reconstructed signals; , are the subband noises; and , , ,
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Fig. 1. Oversampled filter bank with
N M.
are the transfer functions of the analysis and synthesis filters in the form
where cients of and ters with
and
are impulse response coeffiand , respectively. Denote the polyphase matrices of the analysis filand the synthesis filters , respectively, and , for and . The FB achieves PR when . satisfy The synthesis FBs that achieve PR when and are called PR synthesis FBs. Their existence condition and characterization are given in the lemma below which is from [1] and [2]. with polyphase Lemma 1: Given an analysis FB , there exist PR synthesis FBs if and matrix for all . If this condionly if tion holds, then all the PR synthesis polyphase matrices are given by (1) where matrix
is a free parameter of any
stable transfer
(2) is (the pseudo inverse of ) the PR synthesis FB corre[1], [2], sponding to the canonical dual frame of and is the conjugate transpose of . Hereafter, and given in (1) and (2) will be called the dual frame and canonical dual frame synthesis FBs, respectively. B. State-Space Representations and Computation is called A transfer matrix causal if for all , and is called anti-causal if for all . For any rational causal transfer matrix , if the matrices , and are such that , then is called a state-space realization of
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and denoted as
. The realization
is minimal if
, where and are respectively and are the causal stable and anti-causal stable part of given by
has the minimal dimension among
all realizations of [19]. For any rational anti-causal transfer matrix , if the matrices and are such that , then is called an anti-causal state-space realization of and denoted as
(7)
(8)
. For any transfer matrix
, if it has a causal state-space , then
realization
.
Lemma 2 and Theorem 1 below provide a computationally effective method to obtain the canonical dual frame synthesis FB given in (2). Their proofs are given in [21] and [5], respectively. Lemma 2: For with a minimal causal real, assume the following. has full column rank (the rank equals
A1.
is the solution of the following Lyapunov equation
has an anti-causal
state-space realization
ization
where
Remark 1: For filter banks with rational filters (causal or non-causal), it follows from Weierstrass’s Theorem [20] that always has a causal realization (maybe unstable) by adding some delays. Or in other words, one can make the with large enough. transfer matrix proper by multiplying C. The Norms of Signals and Systems Let be a wide sense stationary (WSS) vector signal with finite power (variance) and PSD function . The power is defined as (semi) norm of
. the number of columns) for all can be factorized in the form Then (3) and are rational transfer matrices, and where is causal stable, satisfying . Moreover, and can be obtained from the state-space realization
and the peak norm of is defined as . and be the output and input of a system with Let . Assume that is WSS with power transfer matrix and that is stable (but not necessarily causal). norm Then and are related by
(4) where
and
are given by
and and are both bounded. The following three norms of can be defined as follows. (or ) norm of , denoted as , is given by The (5)
and equation
is the unique solution to the algebraic Riccati
(9) The
(6) which guarantees that all the eigenvalues of the unit circle. Theorem 1: Let
(or Chebyshev) norm [9], [21] of , is given by
, denoted as
are within
be a polyphase ma-
trix of an analysis FB satisfying the condition A1. Let , and be as defined in (5) and (6). The canonical dual frame synthesis FB is given by
(10) where
is the largest eigenvalue.
CHAI et al.: OPTIMAL NOISE REDUCTION IN OVERSAMPLED PR FILTER BANKS
The generalized given by
norm of
, denoted as
, is
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Theorem 2: Let
be a polyphase matrix
of an analysis FB satisfying the condition A1. The set of all dual frame PR synthesis FBs are given by (14)
(11)
is given in Theorem 1, is given by (the reciprocal conjugate transpose of) is any stable transfer matrix. (13), and Proof: Using (2) and (3) and Lemma 3, it is easy to show that where
As seen from the above, measures the power gain of for white noise inputs, measures the maximum meapower gain for all power bounded inputs, while sures the maximum peak of the output for all power bounded inputs. For more details about these norms of signals and systems, see [21], [23], and [25]. III. STATE-SPACE PARAMETERIZATION OF PR SYNTHESIS FBS As seen from Lemma 1, for oversampled FB, the PR synthesis FBs (if exist) are not unique and all these FBs can be . parameterized by Therefore, one can use the design freedom embedded in the to achieve other design objectives in adfree parameter dition to PR. This can be done by first computing the canonwith Theorem 1 and then ical dual frame synthesis FB searching the free parameter matrix . In [5], effective comin the putation algorithms have been provided to obtain state-space form (7)–(8) and to obtain the state-space represen. Using these results, the state-space tation of based searching algorithms of may be developed. Howwith the transfer matrix ever, parameterizing does not reflect the true freedom available for designing . In [12], [13] it has been shown that the freedom embedded in is actually only . Theorem 2 below will extend the results of [5], [12], [13] to derive a state-space pa. To present the rameterization for the optimal design of theorem, the following technical results are introduced. For an analysis FB with polyphase representation and ated with
, let
as given in (5) and (6). Let and
and
be associ,
be such that (12)
Hence, . It . then follows from (1) that The proof is completed by setting for the free parameter in (1) and by noting that . Remark 2: i) Theorem 2 coincides with the result of [12] and being [13] on the true dimension of the free parameter . But different from [1], [12], and [13], the parameterization (14) is in state-space which involves only finite dimensional parameter matrices [confer (7)-(8) and (12)-(13)]. Thus, its manipulation can be easily performed without approximation. Since direct manipulation of transfer matrices is in general difficult, this state-space parameterization is essential for . ii) Together with Theorem 1, Thecomputing optimal orem 2 provides a simpler state-space parameterization of all dual frame PR synthesis FBs than that of [5]. This is because the in (14) is dimension of the free parameter transfer matrix , instead of , and the searching space of is reduced by . iii) It provides a technical tool for the analysis of optimal design of PR synthesis FB. See the next section for details. and to comTo use Theorem 2 in design, one needs . Finding and that satisfy (12) has been conpute sidered in [24], but the algorithm provided there is not easy to use since it involves singular value decomposition. A simpler algorithm based on QR factorization is provided in Appendix A.1, which is a trivial extension of the result in [24].
(13) IV. WEIGHTED OPTIMAL DESIGN OF PR SYNTHESIS FBS Then the following lemma can be easily proved by direct system manipulations [21], [24]. is unitary, i.e., Lemma 3:
With the above lemma, the following theorem can be established.
Consider the polyphase equivalent of the oversampled FB , the recongiven in Fig. 2. For PR FBs satisfying struction error is given by . As will be exemplified in later sections, most of the noise reduction design problems of PR synthesis FBs can be formulated as , find a synthesis FB follows: Given an analysis FB that minimizes , the norm of , subject to , where is a weighting transfer , matrix. The norms to be considered in this paper are and , which measure the gain from the weighted
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Fig. 2. Polyphase equivalent of oversampled filter bank with subband noises.
in relation to the subband noises presented in [7] carries over without any change. For details of this analysis, the to reader is referred to [7]. ii) As seen from Theorems 1 and 4, and hence are IIR in general. They can be FIR is unimodular [2]. if and only if V. OPTIMAL DESIGN FOR KNOWN NOISES
subband noises to the reconstruction error under different noise assumptions and error metrics as defined in Section II. Using Theorem 2, the above constrained optimization problem can be converted to an unconstrained one:
Suppose that the blocked noise in Fig. 2 is WSS with PSD . Then the PSD of the reconstruction error is given by , and its power (semi)norm and peak norm are given respectively by
(15) where and are as given in Theorem 2. This section , will present a unified and simple solution for the cases of and . , , and be defined as Theorem 3: Let above. Assume that is of full column rank on the unit circle. Then
(18) (19) Note that is positive real and can always be factorized as , where is a causal, stable and minimum phase transfer matrix. Hence, by Theorem 2 and the norm and generalized norm, and definitions of can be respectively written as
(16) and , is the unique optimal solution of (15) for . Therefore, it is an and one of the optimal solutions for optimal solution of (15) for , , . The proof of this theorem is given in Appendix A.2. Theorem 3 reveals an important property of the solution to (15) and , , . The theprovides a unified optimal solution for orem below presents an alternative formula for direct compuprovided that is nonsingular for all tation of . The proof of this theorem is given in Appendix A.3. and a weighting Theorem 4: Given an analysis FB transfer matrix , the optimal PR synthesis FB that , for , is given by minimizes
It follows from Theorem 4 that given and (hence ), the optimal minimizing and is given , where . by is also shown in [7]. However, This result for minimizing even for the simplest case of white noise with , no [1]. numerically efficient algorithms exist for the general This section will combine the results of Theorems 1 and 4 to provide the state-space computational formulae for the design . of optimal PR synthesis FB and in the state-space form Write , and
(17) where
and
is defined by
. Then the cas-
caded state-space realization [22] of by
and
. Remark 3: Theorems 3 and 4 show that the optimal solution for the three different norms defined above is the same , where is the pseudo-inand is given by , which can be verse of the augmented polyphase matrix optimal result is easily computed by Theorem 1. While the already provided in [1], [7], Theorem 4 reveals that the optimal solution is also an optimal and generalized solution. This is a striking property of noncausal PR synthesis FBs, which is in general not true for the filtering and control problems widely studied in the literature [14], [15], [18], [21] where the filters and controllers are confined to the causal set. Remark 4: i) Using the same argument as that of [7], it is satisfies the condition A1, then trivial to show that if given in (17) is a dual frame of . Thus, the frame-theoretical analysis of the reconstruction error
Define of the following Riccati equation
, where
is given
(20) is the solution
(21) and equation
. Let
be the solution of the Lyapunov
(22) Then the following theorem can be established. Theorem 5: Suppose that the state-space model is given by (20). Let , and be defined as
CHAI et al.: OPTIMAL NOISE REDUCTION IN OVERSAMPLED PR FILTER BANKS
above. Then the optimal PR synthesis FB minimizing , where given by
is
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to minimize the worst case power amplification given by (the system norm induced by the signal power norm)
Therefore, the optimization problem is to find an from the set of all PR synthesis filter banks (causal and/or noncausal) is minimized. such that with a minimal realizaTheorem 6: Suppose that Proof: The result follows immediately from applying Theorem 1 to the generalized system given in (20). is with If the PSD matrix of the noise being a positive definite constant matrix, then a more compact and simpler solution can be obtained. Let be the solution of the following Riccati equation
(23)
tion
satisfies the rank assumption A1 given in
Lemma 2. Then the PR synthesis FB that minimizes is given by the canonical dual frame synthesis FB in , where Theorem 1, namely, and are given by (7) and (8), respectively. in (14) of Proof: Recall to (16) in Theorem 3, we Theorem 2. Substituting know that an optimal minimizing is given by
and define
(24) Further, let
be the solution of the Lyapunov equation (25)
Then the following corollary can be established. is Corollary 1: Assume that the PSD matrix of the noise with being a positive definite constant matrix. , the PR synthesis FB minimizing Then for a given is given by , where and are given by
where ,
and
and are defined by (23), (24), and (25), respectively.
VI. WORST-CASE DESIGN FOR UNKNOWN NOISES In some applications, it is often unrealistic to obtain the PSD matrix of the noises. The method in the previous section is therefore infeasible. In such situations, the worst-case criterion is appropriate for the design of oversampled PR FBs, which has been widely used in signal processing and control [14], [16]. In Section III, it is shown that the solution to the worst case design of the PR synthesis FB coincides with the canonical dual frame synthesis FBs given in Theorem 1. Hence, it is simpler than the filtering problem where causality is required. standard depends on , As seen from (18), the PSD of . When no information on is available except its power norm being bounded, the best one can do is
Hence, the resulting optimal synthesis filter bank is , which was given in Theorem 1. Theorem 6 reveals that the canonical dual frame synthesis FB in Theorem 1 is also optimal in the sense of worst case power amplification (the system norm induced by the signal power norm). The result is simpler than the well-known filtering design [14]–[16]. This is a striking property of the dual frame synthesis FB which allows non-causality. Note that the is not unique and is synthesis FB that minimizes just one of the optimal solutions. VII. OPTIMAL DESIGN FOR SINGULAR WEIGHTING MATRIX may be singular or almost singular The weighting with very large condition number. This happens, for example, when there are no noises in some channels a priori, or data from some subband channels are totally lost. In these cases, the state-space methods based on the generalized transfer matrix in Sections V and VI are not applicable. The optimal design now has to be done differently as described in the following corollary. Corollary 2: Denote the canonical dual of . Then one of , that minimizes , the optimal PR synthesis FBs, , and , is given by for (26) can be solved with Theorem 1 provided is of full column rank. given by (16) into in (14) Proof: Substituting gives where
This completes the proof.
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It follows from Corollary 2 that the pseudo-inverses of two and , have to be computed using transfer matrices, Theorem 1. One has to consider the numerical complexity. is usually However, the computation of is simpler since the size of , which is usually small.
VIII. NUMERICAL EXAMPLES 1) Example 1: Consider the oversampled FB with channel number and decimation number , where are low-pass, bandpass and high-pass Butterworth filters generated respectively by MATLAB , command and . The frequency responses , and are shown in Fig. 3. Using of and the algorithm in Appendix A.1, are computed and is consequently obtained, by (13), as
Fig. 3. Example 1: Frequency responses of the analysis filters. ( ): dotted; ( ): dashed.
H z
H z
H (z): solid;
To see how the noise spectral information in Sections V and VI affects the optimal synthesis FB, consider four different values of as follows:
It follows from Theorem 1 that [see the first equation shown at the bottom of the page]. Thus, by Theorem 2, the set of all PR synthesis FBs is given by (27) is any stable 2 1 transfer matrix. As discussed where before, the parameterization (27) is quite useful for practical design of synthesis FBs, since the only freedom is to choose to satisfy the design criteria.
The optimal synthesis filters for (the white noises), , and are given below, and their frequency responses are shown in Fig. 4(a)–(d), respectively. : See equation (28) at the bottom of the page. : See the first equation at the bottom of the next page.
(28)
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: See the second equation at the bottom of the page. : See the last equation at the bottom of the page. It follows from Theorem 6 that the synthesis filters by the worst case design are given by (28), the same as that of . As seen from Fig. 4(a), the frequency response of the three filters are ‘balanced.’ Roughly speaking, the noises with are much larger in the first channel than in others. To fully reject the noise from the first channel, the optimal solution is , but the synthesis filters have to satisfy the PR cannot be zero. Fig. 4(b) shows that constraint, hence the norm of is still much smaller than others although are much it cannot be zero. Similarly, the noises with larger in the second channel than in others. Fig. 4(c) also shows is much smaller than others although it that the norm of cannot be zero. Example 2: Consider the oversampled FB with channels and decimation number , where
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and . The coeffiare known as binomial QMF filter coefficients cients of , [27, Table 4.2]. The frequency response of are shown in Fig. 5. Note that and are the same. It can be readily calculated that for this example , , By Theorem 1
It then follows from Theorem 2 that the set of all PR synthesis FBs is given by
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Fig. 4. Example 2: Frequency responses of the synthesis filters. F (z ): solid; F (z ): dotted; F (z ): dashed.
where and is any stable 2 1 transfer matrix. For this example, we first consider a non-constant weighting transfer matrix. Let
Then, using Theorem 5, the following optimal synthesis filters are obtained for the above . See the equation at the bottom . of the page, and Next, we consider the optimal synthesis filters for the following different ’s:
Fig. 5. Example 2: Frequency responses of the analysis filters. H (z ): solid; H (z ): dotted; H (z ): dashed.
CHAI et al.: OPTIMAL NOISE REDUCTION IN OVERSAMPLED PR FILTER BANKS
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Fig. 6. Example 2: Frequency responses of the synthesis filters. F (z ): solid; F (z ): dotted; F (z ): dashed.
TABLE I OPTIMAL SYNTHESIS FB COEFFICIENTS FOR S =
S
TABLE II OPTIMAL SYNTHESIS FB COEFFICIENTS FOR S =
S
TABLE III OPTIMAL SYNTHESIS FB COEFFICIENTS FOR S =
S
TABLE IV OPTIMAL SYNTHESIS FB COEFFICIENTS FOR S =
S
The frequency responses of the optimal synthesis filters for (white noises), , the noises with and are shown in Fig. 6(a)–(d), and their coefficients are given in Tables I–IV, respectively. Because is unimodular, these filters are FIR. and are the As seen from Table I and Fig. 6(a), does not change much same. Fig. 6(b) and (c) show that
when the weighing of the first channel is increased. This is difmay be ‘small’ when the ferent from Example 1 where is designed to satnoise are ‘large’. The reason is that isfy the PR constraint first. For Example 2, there is no redun. On the other hand, if is chosen, corredancy for sponding to a ‘large’ noise in the third channel, Fig. 6(d) shows is very ‘small’, which could be viewed as reducing that the noise.
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Fig. 7. Example 4: The input and reconstructed signals and the reconstruction errors of FBS4 (left) and FBS1 (right).
Example 3: Consider again the oversampled FB given in Example 2 but with the following singular weighting matrices: and
. By di-
rect computation, it is easy to show that
Thus
It then follows from Corollary 2 that the optimal and is given by weighting matrices
for the
where is given by Example 2. The corresponding syn, , 0, where thesis filters are then given by and are shown in Table I. For this example, is zero since there is no noise in the third channel while the second cannot be zero since it has channel is perfect. However, to satisfy PR constraint. Example 4: This example presents simulation results of the synthesis FBs designed above. Two FBs with subbband noises are implemented in SIMULINK. The analysis filters in both , , given in Example 2, and the FBs are the , , given in Tables I and synthesis FBs are the
IV, respectively. Since these FBs are designed for the noise and , respectively, they are referred PSDs to as FBS1 and FBS4 here. The FBS1 and FBS4 are subject and to the same subband noises with PSD the same sinusoidal input signal. The corresponding input , reconstructed signal and reconstruction signal are plotted in Fig. 7. As seen from errors the plots, FBS1 performs poorly because it is designed with , i.e., no noise weighting in optimization; whereas FBS4 gives much better reconstruction since it is designed with . The calculation of the optimal noise weighting SNR shows that SNR (12.8558 dB) and SNR (69.4361 dB). The optimal design boosts the SNR by over 16 times. IX. CONCLUSION Optimal noise reduction in general (IIR or FIR) PR oversampled FBs has been studied in this paper. Using a new state-space parameterization of all dual frame PR synthesis FBs, the generic formulation and general solution of the optimal PR synthesis FBs for noise reduction have been presented. Two classes of subband noises under three different measures of noise reduction have been addressed using the general solution. The optimal solution and state-space based efficient design method are obtained for each case, and the relations between different cases are analyzed and revealed. Extensive examples are presented to demonstrate the effectiveness of the obtained design methods. The results of this paper have provided deeper insight into the optimal subband noise reduction in oversampled FBs, which can be generalized to the study of noise reduction in generic frames. The technical tools of this paper can also be extended to other optimal design problems such as the optimal noise reduction in oversampled FBs. These results will be presented elsewhere.
CHAI et al.: OPTIMAL NOISE REDUCTION IN OVERSAMPLED PR FILTER BANKS
APPENDIX A1. Algorithm for Computing
and
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Therefore in (12)–(13)
, , Recall that and . Step 1. Apply QR factorization to obtain
where
,
Since
where triangular,
is an isometry, we have
is lower and is co-isometry, that is,
. Step 2. Take (29) or (30) Lemma 4: and generated by the above algorithm satisfy (12). Proof: The first equation of (12) follows from
Obviously and are positive semi-defand inite on the unit circle. Therefore, are also positive semi-definite. It is well-known that for two positive semi-definite matrices and , and , if and only if [26]. moreover, Recalling the definition of three different norms, we have
The second equation of (12) comes from the fact that
By taking well-defined since , we have
This completes the proof.
which is is of full-row rank for any
A.2 Proof of Theorem 3 Proof: Since unit circle, we define
is of full column rank on the Hence (31)
where and let , i.e.,
, . It is easy to check that is an isometry, and that
with
the
optimal
. Moreover, .
given by , where , is the unique solution for
and and
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A.3 Proof of Theorem 4
REFERENCES
Proof: Note that
Since ties of
and are invertible by assumption and the properand , it follows that
Using our notations, it can be rewritten in a compact formula as follows:
This implies that
[1] H. Bölcskei, F. Hlawatsch, and H. G. Feichtinger, “Frame-theoretic analysis of oversampled filter banks,” IEEE Trans. Signal Process., vol. 46, no. 12, pp. 3256–3268, Dec. 1998. [2] Z. Cvelkovic´ and M. Vetterli, “Oversampled filter banks,” IEEE Trans. Signal Process., vol. 46, no. 5, pp. 1245–1255, May 1998. [3] L. Gan and K.-K. Ma, “Oversampled linear-phase perfect reconstruction filterbanks: Theory, lattice structure and parameterization,” IEEE Trans. Signal Process., vol. 51, no. 3, pp. 744–759, Mar. 2003. [4] T. Strohmer, “Finite and infinite-dimensional models for oversampled filter banks,” in Modern Sampling Theory: Mathematics and Applications, J. J. Benedetto and P. J. S. G. Ferreira, Eds. Cambridge, MA: Birkhäser, 2001. [5] L. Chai, J. Zhang, C. Zhang, and E. Mosca, “Frame theory based analysis and design of oversampled filter banks: Direct computational method,” IEEE Trans. Signal Process., vol. 55, no. 2, pp. 507–519, Feb. 2007. [6] L. Chai, J. Zhang, C. Zhang, and E. Mosca, “Efficient computation of frame bounds using LMI-based optimization,” IEEE Trans. Signal Process., vol. 56, no. 7, pp. 3024–3033, Jul. 2008. [7] H. Bölcskei and F. Hlawatsch, “Noise reduction in oversampled filter banks using predictive quantization,” IEEE Trans. Inf. Theory, vol. 47, pp. 155–172, Jan. 2001. [8] A. Scaglione, G. B. Giannakis, and S. Barbarossa, “Redundant filterbank precoders and equalizers Part I: Unification and optimal designs,” IEEE Trans. Signal Process., vol. 47, no. 7, pp. 1988–2006, Jul. 1999. [9] G. Gu and L. Li, “Worst-case design for optimal channel equalization in filterbank transceivers,” IEEE Trans. Signal Process., vol. 51, no. 9, pp. 2424–2435, Sep. 2003. [10] Z. Chen and J. Zhang, “FB analysis of PMRI and its application to SENSE reconstruction,” in Proc. 14th IEEE Int. Conf. Image Processing (ICIP), San Antonio, TX, Sep. 2007, pp. III-129–132. [11] O. Christensen, Frames and Bases An Introductory Course. Cambridge, MA: Birkhäuser, 2008. [12] F. Labeau, “Synthesis filters design for coding gain in oversampled filter banks,” IEEE Signal Process. Lett., vol. 12, pp. 697–700, Oct. 2005. [13] F. Labeau, R. Chiang, M. Kieffer, P. Duhamel, and L. Vandendorpe, “Oversampled filter banks as error correcting codes: Theory and impulse noise correction,” IEEE Trans. Signal Process., vol. 53, no. 12, pp. 4619–4630, Dec. 2005. [14] T. Chen and B. A. Francis, “Design of multirate filter banks by optimization,” IEEE Trans. Signal Process., vol. 43, no. 12, pp. 2822–2830, Dec. 1995. [15] B.-S. Chen, C.-L. Tsai, and Y.-F. Chen, “Mixed filtering design in multirate transmultiplexor systems: LMI approach,” IEEE Trans. Signal Process., vol. 49, no. 11, pp. 2693–2701, Nov. 2001. [16] H. Vikalo, B. Hassibi, A. T. Erdogan, and T. Kailath, “On robust signal reconstruction in noisy filter banks,” Signal Process., vol. 85, pp. 1–14, 2005. [17] H. Zhou, L. Xie, and C. Zhang, “A direct approach to H2 optimal deconvolution of periodic digital channels,” IEEE Trans. Signal Process., vol. 50, no. 7, pp. 1685–1698, Jul. 2002. [18] J. Zhang, R. Yang, C. Zhang, and E. Mosca, “Design of IIR multirate filter banks subject to subband noises,” in Proc. 5th Asian Control Conf., Melbourne, Australia, 2004, pp. 1333–1338. [19] T. Kailath, Linear Systems. Englewood Cliffs, NJ: Prentice-Hall, 1980. [20] L. V. Ahlfors, Complex Analysis. New York: McGraw-Hill, 1979. [21] K. Zhou, J. C. Doyle, and K. Glover, Robust Optimal Control. Englewood Cliffs, NJ: Prentice-Hall, 1996. [22] H. Shu and T. Chen, “On causality and anticausality of cascaded discrete-time systems,” IEEE Trans. Circuits Syst. I, vol. 43, pp. 240–242, Mar. 1996. [23] D. A. Wilson, M. Neekoui, and G. D. Halikias, “An LQR wieghted control,” Int. J. Control, selection approach to discrete generalized vol. 71, no. 1, pp. 93–101, 1998. [24] D. W. Gu, M. C. Tsai, S. D. O’Young, and I. Postlethwaite, “Statespace formulae for discrete-time optimization,” Int. J. Control, vol. 49, no. 5, pp. 1683–1723, 1989. [25] W. A. Gardner, Statistical Spectral Analysis: A Nonprobability Theory. Englewood Cliffs, NJ: Prentice-Hall, 1988. [26] R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge, U.K.: Cambridge Univ. Press, 1985. [27] A. N. Akansu and R. A. Haddad, Multiresolution signal decomposition: transforms, subbands, and wavelets. New York: Academic, 1992.
H
H
(32) On the other hand, direct computation using (16) and gives
It then follows from (32) that
h =h
H
h
CHAI et al.: OPTIMAL NOISE REDUCTION IN OVERSAMPLED PR FILTER BANKS
Li Chai (S’00–M’03) received the B.S. degree in applied mathematics and the M. S. degree in control science and engineering, both from Zhejiang University, China, in 1994 and 1997, respectively, and the Ph.D. degree in electrical engineering from Hong Kong University of Science and Technology in 2002. In September 2002, he joined Hangzhou Dianzi University, China. He worked as a Postdoctoral Research Fellow at the Monash University, Clayton, Australia, from May 2004 to June 2006. In December 2007, he joined Wuhan University of Science and Technology, where he is currently a Xiang-Tao Professor. His research interests include multirate signal processing, wavelets and control with communication constraints.
Jingxin Zhang (M’03) received the M.Eng. and Ph.D. degrees in electrical engineering from Northeastern University, China. Between 1988 and 1992, he was an Associate Professor with Northeastern University, China. Since 1989, he has held research positions in the University of Florence, Italy; the University of Melbourne, Australia; and the Cooperative Research Centre for Sensor Signal and Information Processing, Australia; and a Senior Lecturer position in the University of South Australia and Deakin University, Australia. He is currently with the Department of Electrical and Computer Systems Engineering, Monash University, Australia. He is the author and coauthor of over 140 research papers in diverse areas such as adaptive and predictive control, time varying systems, robust filtering, multirate signal processing, and medical imaging. He is the first inventor of a provisional international patent on parallel magnetic resonance image reconstruction. His current research interests are in control and signal processing and their applications to biomedical and industrial systems. Dr. Zhang is the Editorial Board Member of The Open Automation and Control Systems Journal. He is the recipient of 1989 Fok Ying Tong Educational Foundation (Hong Kong) for the outstanding Young Faculty Members in China and 1992 China National Education Committee Award for the Advancement of Science and Technology.
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Cishen Zhang (M’89) received the B.Eng. degree from Tsinghua University, China, in 1982 and Ph.D. degree in electrical engineering from Newcastle University, Australia, in 1990. Between 1971 and 1978, he was an Electrician with Changxindian (February Seven) Locomotive Manufactory, Beijing, China. He carried out research work on control systems at Delft University of Technology, The Netherlands, from 1983 to 1985. After his Ph.D. study from 1986 to 1989 at Newcastle University, he was with the Department of Electrical and Electronic Engineering at the University of Melbourne, Australia, as a Lecturer, Senior Lecturer, and Associate Professor and Reader until October 2002. He is currently with the School Electrical and Electronic Engineering and School of Chemical and Biomedical Engineering at Nanyang Technological University, Singapore. His research interests include signal processing, medical imaging and control.
Edoardo Mosca (F’97–LF’06) received the Dr.Eng. degree in electronics engineering from the University of Rome ”La Sapienza,” Italy, in 1963. He then spent four years in the aerospace industry, where he worked on the research and development of advanced radar systems, particularly, optimal signal synthesis and processing, and phased-array radar systems. Thereafter, from 1968 to 1972, he held academic positions at the University of Michigan, Ann Arbor, and McMaster University, ON, Canada. Since 1972, he has been with the Engineering Faculty, University of Florence, Italy: from 1972 to 1974 as an Associate Professor, and since 1975 as a full Professor of Control Engineering. In the latter capacity, he founded in 1981 the Department of Systems and Computer Science and Engineering, of which he was the first chair until 1987, and which he conceived and set up as a department, including all the academic staff and the related teaching and research activities in computer and control of the university. He has been a Visiting Professor in universities and research centers in many different countries. From 1995 to 1998 he has been the President of the Italian Association of Researchers in Automatic Control, and from 1983 to present the coordinator of several national research projects in the field of automatic control. He is the author of many research papers spanning various diversified fields such as radar signal synthesis and processing, radio communications, system identification, adaptive, predictive, switching supervisory control, and detection of performance degradation in feedback-control systems. He is the author of a book Optimal, Predictive, and Adaptive Control (Prentice-Hall, 1995). Dr. Mosca is an editor of the following journals: the European Journal of Control; the International Journal of Adaptive Control and Signal Processing (Wiley); and the IEE Proceedings—Control Theory and Applications. He is the Italian NMO representative in IFAC (International Federation of Automatic Control). He has been a Council member of EUCA (European Union Control Association) until 1998, and from 1996 to 2002 a Council member of IFAC . In 2001, he was awarded ”honoris causa” the Doctor in Information Engineering degree from the Universidade Tecnica de Lisboa, Lisbon, Portugal, and in 1997 elected to the grade of Fellow of the IEEE for his contributions to adaptive and predictive control.
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 10, OCTOBER 2009
Optimal Noise Reduction in Oversampled PR Filter Banks Li Chai, Member, IEEE, Jingxin Zhang, Member, IEEE, Cishen Zhang, Member, IEEE, and Edoardo Mosca, Life Fellow, IEEE
Abstract—This paper studies the optimal noise reduction problem for oversampled filter banks (FBs) with perfect reconstruction (PR) constraint. Both the optimal design and worst case design are considered, where the former caters for the noise with known power spectral density (PSD) and the latter for the noise with unknown PSD. State-space based explicit formulae involving only algebraic Riccati equation and matrix manipulations are provided for the general (IIR or FIR) oversampled PR FBs, and the relations between different cases are analyzed and revealed. Extensive numerical examples are provided to illustrate the proposed design methods and to show their effectiveness. Index Terms—Dual frame, noise reduction, oversampled filter banks, perfect reconstruction, state-space approach.
I. INTRODUCTION
O
VERSAMPLED filter banks (FBs) with redundant signal expansions has been extensively studied in recent years. The noise reduction properties, extra design freedom and improved capacity for signal and information representation are their main advantages [1]–[6]. Oversampled FBs have found applications in broad real world problems, for example, the analysis and design of subband coders, precoders and equalizers in data compression and communications [7]–[9], and the analysis and design of imaging and reconstruction process in parallel magnetic resonance imaging (pMRI) [10]. In these applications, perfect reconstruction (PR), or equivalently zero forcing equalization in communications, is often desired. Because of the redundance in oversampled FBs, the PR synthesis FBs are not unique for a given analysis FB, which renders other design objectives in conjunction with PR in the design of synthesis FBs. One of these objectives is subband noise reduction. Manuscript received October 14, 2008; accepted April 26, 2009. First published June 02, 2009; current version published September 16, 2009. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Bogdan Dumitrescu. This work was partially supported by National Science Foundation of China under Grants 60672064 and 60874109, the program NCET-08-0674, the Australian Research Council’s Discovery funding scheme (project number DP0343057), and Singapore Ministry of Education Grants T208A1213 and RG8/06. L. Chai is with the Engineering Research Center of Metallurgical Automation and Measurement Technology, and the Key Lab of Metallurgical Equipment and Control, Wuhan University of Science and Technology, Wuhan, 430081, China (e-mail: [email protected]). J. Zhang is with the Department of Electrical and Computer Systems Engineering, Monash University, VIC3800, Australia (e-mail: jingxin.zhang@eng. monash.edu.au). C. Zhang is with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore (e-mail: [email protected]). E. Mosca is with the Dipartimento di Sistemi e Informatica, Universityá di Firenze, Firenze 50139, Italy (e-mail: [email protected]). Digital Object Identifier 10.1109/TSP.2009.2024280
Subband noises arise from subband quantization, subband channel uncertainties and subband contaminations, which lower the coding gain in subband coding [7], increase the bit error rate in data communication [8] and introduce image artifacts in pMRI [10]. Therefore, the optimal noise reduction of PR synthesis FBs is an issue of practical and theoretical importance. Using an elegant frame theoretic approach, [1], [2], and [7] have proven that an oversampled analysis FB with stable (not necessarily causal) inverse constitutes a frame in , and the PR synthesis FBs for this analysis FB are its dual frames [11]. Among all these dual frames, the canonical dual, given by the parapseudo inverse of the analysis FB, has the minimum norm and is optimal for subband noise reduction if the noises are independent and white. In [1], a parameterization of all dual frame synthesis FBs is given, which provides a general framework for the optimal design of PR synthesis FBs with respect to different criteria for noise reduction. This parameterization is further refined in [12] and [13] by revealing the true dimension of the free parameters. However, to our best knowledge, there has been no direct application of this parameterization to any optimal design of PR synthesis FBs. This paper studies the following problem: Given an oversampled analysis FB satisfying PR condition, find one synthesis FB from the entire set of all dual frame PR synthesis FBs, which is optimal with respect to different criteria for subband noise reduction, and compute efficiently and reliably this optimal synthesis FB. This problem is cast into a generic problem of finding an optimal parameter matrix to minimize the weighted norm of synthesis FB. The solution is derived using a novel state-space parameterization of all PR synthesis FBs. A key feature of the state-space parameterization and computation is that the state-space model can represent the infinite dimensional frames of the FBs in the finite dimensional statespace. Thus, it enables explicit and precise analysis and evaluation of the proposed noise reduction problem with algebraic operations of finite dimensional numerical matrices. This resolves the difficulties of the inevitable complicated symbolic operations and infinite dimensional computations in the polyphase representation and evaluation of FBs. Since the dual frame PR synthesis FBs are generally noncausal and IIR, their state-space parameterization is nontrivial and is deduced from the authors’ recent results [5] on the frame theory based analysis and design of oversampled FBs. This new parameterization, together with the results of [5], renders efficient and reliable optimization methods. These methods involve only algebraic Riccati equation and finite dimensional matrix operations without the approximation generally required in the current literature.
1053-587X/$26.00 © 2009 IEEE
CHAI et al.: OPTIMAL NOISE REDUCTION IN OVERSAMPLED PR FILTER BANKS
The above generic solution is used to address two classes of subband noises under three different measures of noise reduction: the colored noises with known or unknown power spectral , generalized and norms of the density (PSD); the synthesis FB with nonsingular or singular weighting matrices. The optimal solution and state-space based efficient computation method are obtained for each case, and the relations between different cases are analyzed and revealed. It is important to pointed out that the noise reduction design of synthesis FBs has been considered previously by many authors, e.g., [9] and [14]–[18]. But the problems considered and results obtained in these works are different from those in this paper. The FBs considered in [14]–[18] are critically sampled, which have no redundance. Hence, the synthesis FBs deoptimal signed in [14]–[18] do not have PR property. The zero forcing equalizer (PR synthesis FB) designed in [9] is restricted to be causal, thus it is applicable only to the analysis FBs having causal stable inverse. Whereas the problem studied in this paper is frame theory based: the analysis FBs considered are FB frames with stable but not necessarily causal inverse, and the synthesis FBs designed are their dual frames which are not necessarily causal (however stability is guaranteed). Because this paper studies a different problem, its results differ significantly from those of [9] and [14]–[18], which are and filtering. The most startling rebased on standard sult of this paper is that of Theorems 4 and 6: for an analysis FB that constitutes a frame, the optimal dual frame PR synthesis FB for subband noise reduction is the same for all the two classes of noises and the three different norms discussed above. To our best knowledge, this result and the result of state-space computation of the optimal dual frame PR synthesis FBs have not been reported previously. The paper is organized as follows. The background knowledge of FB, frame theory, state-space method are presented in Section II. Section III presents an exact parameterization of all PR synthesis FBs. In Section IV, the optimal design problems of PR oversampled FB are formulated and solved systematically by the tool of matrix completion. The optimal design for noises with known PSD and worst-case design for noises with unknown PSD are studied in Sections V and VI, respectively, by using the theory developed in Section VI. The optimal design subject to singular weighting matrix is studied in Section VII based on the result of Section VI. Numerical examples are given in Section VIII to demonstrate the effectiveness of our algorithms. Finally, conclusions are drawn in Section IX. The notations of this paper is quite standard. For a matrix , its transpose, conjugate transpose, trace and eigenvalues are de, , and , respectively. Denote noted by if is full column rank and if is full row rank. II. PRELIMINARIES A. PR Oversampled FBs Consider the -channel oversampled FB with decimation shown in Fig. 1. In the figure, and are factor the input and reconstructed signals; , are the subband noises; and , , ,
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Fig. 1. Oversampled filter bank with
N M.
are the transfer functions of the analysis and synthesis filters in the form
where cients of and ters with
and
are impulse response coeffiand , respectively. Denote the polyphase matrices of the analysis filand the synthesis filters , respectively, and , for and . The FB achieves PR when . satisfy The synthesis FBs that achieve PR when and are called PR synthesis FBs. Their existence condition and characterization are given in the lemma below which is from [1] and [2]. with polyphase Lemma 1: Given an analysis FB , there exist PR synthesis FBs if and matrix for all . If this condionly if tion holds, then all the PR synthesis polyphase matrices are given by (1) where matrix
is a free parameter of any
stable transfer
(2) is (the pseudo inverse of ) the PR synthesis FB corre[1], [2], sponding to the canonical dual frame of and is the conjugate transpose of . Hereafter, and given in (1) and (2) will be called the dual frame and canonical dual frame synthesis FBs, respectively. B. State-Space Representations and Computation is called A transfer matrix causal if for all , and is called anti-causal if for all . For any rational causal transfer matrix , if the matrices , and are such that , then is called a state-space realization of
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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 10, OCTOBER 2009
and denoted as
. The realization
is minimal if
, where and are respectively and are the causal stable and anti-causal stable part of given by
has the minimal dimension among
all realizations of [19]. For any rational anti-causal transfer matrix , if the matrices and are such that , then is called an anti-causal state-space realization of and denoted as
(7)
(8)
. For any transfer matrix
, if it has a causal state-space , then
realization
.
Lemma 2 and Theorem 1 below provide a computationally effective method to obtain the canonical dual frame synthesis FB given in (2). Their proofs are given in [21] and [5], respectively. Lemma 2: For with a minimal causal real, assume the following. has full column rank (the rank equals
A1.
is the solution of the following Lyapunov equation
has an anti-causal
state-space realization
ization
where
Remark 1: For filter banks with rational filters (causal or non-causal), it follows from Weierstrass’s Theorem [20] that always has a causal realization (maybe unstable) by adding some delays. Or in other words, one can make the with large enough. transfer matrix proper by multiplying C. The Norms of Signals and Systems Let be a wide sense stationary (WSS) vector signal with finite power (variance) and PSD function . The power is defined as (semi) norm of
. the number of columns) for all can be factorized in the form Then (3) and are rational transfer matrices, and where is causal stable, satisfying . Moreover, and can be obtained from the state-space realization
and the peak norm of is defined as . and be the output and input of a system with Let . Assume that is WSS with power transfer matrix and that is stable (but not necessarily causal). norm Then and are related by
(4) where
and
are given by
and and are both bounded. The following three norms of can be defined as follows. (or ) norm of , denoted as , is given by The (5)
and equation
is the unique solution to the algebraic Riccati
(9) The
(6) which guarantees that all the eigenvalues of the unit circle. Theorem 1: Let
(or Chebyshev) norm [9], [21] of , is given by
, denoted as
are within
be a polyphase ma-
trix of an analysis FB satisfying the condition A1. Let , and be as defined in (5) and (6). The canonical dual frame synthesis FB is given by
(10) where
is the largest eigenvalue.
CHAI et al.: OPTIMAL NOISE REDUCTION IN OVERSAMPLED PR FILTER BANKS
The generalized given by
norm of
, denoted as
, is
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Theorem 2: Let
be a polyphase matrix
of an analysis FB satisfying the condition A1. The set of all dual frame PR synthesis FBs are given by (14)
(11)
is given in Theorem 1, is given by (the reciprocal conjugate transpose of) is any stable transfer matrix. (13), and Proof: Using (2) and (3) and Lemma 3, it is easy to show that where
As seen from the above, measures the power gain of for white noise inputs, measures the maximum meapower gain for all power bounded inputs, while sures the maximum peak of the output for all power bounded inputs. For more details about these norms of signals and systems, see [21], [23], and [25]. III. STATE-SPACE PARAMETERIZATION OF PR SYNTHESIS FBS As seen from Lemma 1, for oversampled FB, the PR synthesis FBs (if exist) are not unique and all these FBs can be . parameterized by Therefore, one can use the design freedom embedded in the to achieve other design objectives in adfree parameter dition to PR. This can be done by first computing the canonwith Theorem 1 and then ical dual frame synthesis FB searching the free parameter matrix . In [5], effective comin the putation algorithms have been provided to obtain state-space form (7)–(8) and to obtain the state-space represen. Using these results, the state-space tation of based searching algorithms of may be developed. Howwith the transfer matrix ever, parameterizing does not reflect the true freedom available for designing . In [12], [13] it has been shown that the freedom embedded in is actually only . Theorem 2 below will extend the results of [5], [12], [13] to derive a state-space pa. To present the rameterization for the optimal design of theorem, the following technical results are introduced. For an analysis FB with polyphase representation and ated with
, let
as given in (5) and (6). Let and
and
be associ,
be such that (12)
Hence, . It . then follows from (1) that The proof is completed by setting for the free parameter in (1) and by noting that . Remark 2: i) Theorem 2 coincides with the result of [12] and being [13] on the true dimension of the free parameter . But different from [1], [12], and [13], the parameterization (14) is in state-space which involves only finite dimensional parameter matrices [confer (7)-(8) and (12)-(13)]. Thus, its manipulation can be easily performed without approximation. Since direct manipulation of transfer matrices is in general difficult, this state-space parameterization is essential for . ii) Together with Theorem 1, Thecomputing optimal orem 2 provides a simpler state-space parameterization of all dual frame PR synthesis FBs than that of [5]. This is because the in (14) is dimension of the free parameter transfer matrix , instead of , and the searching space of is reduced by . iii) It provides a technical tool for the analysis of optimal design of PR synthesis FB. See the next section for details. and to comTo use Theorem 2 in design, one needs . Finding and that satisfy (12) has been conpute sidered in [24], but the algorithm provided there is not easy to use since it involves singular value decomposition. A simpler algorithm based on QR factorization is provided in Appendix A.1, which is a trivial extension of the result in [24].
(13) IV. WEIGHTED OPTIMAL DESIGN OF PR SYNTHESIS FBS Then the following lemma can be easily proved by direct system manipulations [21], [24]. is unitary, i.e., Lemma 3:
With the above lemma, the following theorem can be established.
Consider the polyphase equivalent of the oversampled FB , the recongiven in Fig. 2. For PR FBs satisfying struction error is given by . As will be exemplified in later sections, most of the noise reduction design problems of PR synthesis FBs can be formulated as , find a synthesis FB follows: Given an analysis FB that minimizes , the norm of , subject to , where is a weighting transfer , matrix. The norms to be considered in this paper are and , which measure the gain from the weighted
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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 10, OCTOBER 2009
Fig. 2. Polyphase equivalent of oversampled filter bank with subband noises.
in relation to the subband noises presented in [7] carries over without any change. For details of this analysis, the to reader is referred to [7]. ii) As seen from Theorems 1 and 4, and hence are IIR in general. They can be FIR is unimodular [2]. if and only if V. OPTIMAL DESIGN FOR KNOWN NOISES
subband noises to the reconstruction error under different noise assumptions and error metrics as defined in Section II. Using Theorem 2, the above constrained optimization problem can be converted to an unconstrained one:
Suppose that the blocked noise in Fig. 2 is WSS with PSD . Then the PSD of the reconstruction error is given by , and its power (semi)norm and peak norm are given respectively by
(15) where and are as given in Theorem 2. This section , will present a unified and simple solution for the cases of and . , , and be defined as Theorem 3: Let above. Assume that is of full column rank on the unit circle. Then
(18) (19) Note that is positive real and can always be factorized as , where is a causal, stable and minimum phase transfer matrix. Hence, by Theorem 2 and the norm and generalized norm, and definitions of can be respectively written as
(16) and , is the unique optimal solution of (15) for . Therefore, it is an and one of the optimal solutions for optimal solution of (15) for , , . The proof of this theorem is given in Appendix A.2. Theorem 3 reveals an important property of the solution to (15) and , , . The theprovides a unified optimal solution for orem below presents an alternative formula for direct compuprovided that is nonsingular for all tation of . The proof of this theorem is given in Appendix A.3. and a weighting Theorem 4: Given an analysis FB transfer matrix , the optimal PR synthesis FB that , for , is given by minimizes
It follows from Theorem 4 that given and (hence ), the optimal minimizing and is given , where . by is also shown in [7]. However, This result for minimizing even for the simplest case of white noise with , no [1]. numerically efficient algorithms exist for the general This section will combine the results of Theorems 1 and 4 to provide the state-space computational formulae for the design . of optimal PR synthesis FB and in the state-space form Write , and
(17) where
and
is defined by
. Then the cas-
caded state-space realization [22] of by
and
. Remark 3: Theorems 3 and 4 show that the optimal solution for the three different norms defined above is the same , where is the pseudo-inand is given by , which can be verse of the augmented polyphase matrix optimal result is easily computed by Theorem 1. While the already provided in [1], [7], Theorem 4 reveals that the optimal solution is also an optimal and generalized solution. This is a striking property of noncausal PR synthesis FBs, which is in general not true for the filtering and control problems widely studied in the literature [14], [15], [18], [21] where the filters and controllers are confined to the causal set. Remark 4: i) Using the same argument as that of [7], it is satisfies the condition A1, then trivial to show that if given in (17) is a dual frame of . Thus, the frame-theoretical analysis of the reconstruction error
Define of the following Riccati equation
, where
is given
(20) is the solution
(21) and equation
. Let
be the solution of the Lyapunov
(22) Then the following theorem can be established. Theorem 5: Suppose that the state-space model is given by (20). Let , and be defined as
CHAI et al.: OPTIMAL NOISE REDUCTION IN OVERSAMPLED PR FILTER BANKS
above. Then the optimal PR synthesis FB minimizing , where given by
is
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to minimize the worst case power amplification given by (the system norm induced by the signal power norm)
Therefore, the optimization problem is to find an from the set of all PR synthesis filter banks (causal and/or noncausal) is minimized. such that with a minimal realizaTheorem 6: Suppose that Proof: The result follows immediately from applying Theorem 1 to the generalized system given in (20). is with If the PSD matrix of the noise being a positive definite constant matrix, then a more compact and simpler solution can be obtained. Let be the solution of the following Riccati equation
(23)
tion
satisfies the rank assumption A1 given in
Lemma 2. Then the PR synthesis FB that minimizes is given by the canonical dual frame synthesis FB in , where Theorem 1, namely, and are given by (7) and (8), respectively. in (14) of Proof: Recall to (16) in Theorem 3, we Theorem 2. Substituting know that an optimal minimizing is given by
and define
(24) Further, let
be the solution of the Lyapunov equation (25)
Then the following corollary can be established. is Corollary 1: Assume that the PSD matrix of the noise with being a positive definite constant matrix. , the PR synthesis FB minimizing Then for a given is given by , where and are given by
where ,
and
and are defined by (23), (24), and (25), respectively.
VI. WORST-CASE DESIGN FOR UNKNOWN NOISES In some applications, it is often unrealistic to obtain the PSD matrix of the noises. The method in the previous section is therefore infeasible. In such situations, the worst-case criterion is appropriate for the design of oversampled PR FBs, which has been widely used in signal processing and control [14], [16]. In Section III, it is shown that the solution to the worst case design of the PR synthesis FB coincides with the canonical dual frame synthesis FBs given in Theorem 1. Hence, it is simpler than the filtering problem where causality is required. standard depends on , As seen from (18), the PSD of . When no information on is available except its power norm being bounded, the best one can do is
Hence, the resulting optimal synthesis filter bank is , which was given in Theorem 1. Theorem 6 reveals that the canonical dual frame synthesis FB in Theorem 1 is also optimal in the sense of worst case power amplification (the system norm induced by the signal power norm). The result is simpler than the well-known filtering design [14]–[16]. This is a striking property of the dual frame synthesis FB which allows non-causality. Note that the is not unique and is synthesis FB that minimizes just one of the optimal solutions. VII. OPTIMAL DESIGN FOR SINGULAR WEIGHTING MATRIX may be singular or almost singular The weighting with very large condition number. This happens, for example, when there are no noises in some channels a priori, or data from some subband channels are totally lost. In these cases, the state-space methods based on the generalized transfer matrix in Sections V and VI are not applicable. The optimal design now has to be done differently as described in the following corollary. Corollary 2: Denote the canonical dual of . Then one of , that minimizes , the optimal PR synthesis FBs, , and , is given by for (26) can be solved with Theorem 1 provided is of full column rank. given by (16) into in (14) Proof: Substituting gives where
This completes the proof.
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It follows from Corollary 2 that the pseudo-inverses of two and , have to be computed using transfer matrices, Theorem 1. One has to consider the numerical complexity. is usually However, the computation of is simpler since the size of , which is usually small.
VIII. NUMERICAL EXAMPLES 1) Example 1: Consider the oversampled FB with channel number and decimation number , where are low-pass, bandpass and high-pass Butterworth filters generated respectively by MATLAB , command and . The frequency responses , and are shown in Fig. 3. Using of and the algorithm in Appendix A.1, are computed and is consequently obtained, by (13), as
Fig. 3. Example 1: Frequency responses of the analysis filters. ( ): dotted; ( ): dashed.
H z
H z
H (z): solid;
To see how the noise spectral information in Sections V and VI affects the optimal synthesis FB, consider four different values of as follows:
It follows from Theorem 1 that [see the first equation shown at the bottom of the page]. Thus, by Theorem 2, the set of all PR synthesis FBs is given by (27) is any stable 2 1 transfer matrix. As discussed where before, the parameterization (27) is quite useful for practical design of synthesis FBs, since the only freedom is to choose to satisfy the design criteria.
The optimal synthesis filters for (the white noises), , and are given below, and their frequency responses are shown in Fig. 4(a)–(d), respectively. : See equation (28) at the bottom of the page. : See the first equation at the bottom of the next page.
(28)
CHAI et al.: OPTIMAL NOISE REDUCTION IN OVERSAMPLED PR FILTER BANKS
: See the second equation at the bottom of the page. : See the last equation at the bottom of the page. It follows from Theorem 6 that the synthesis filters by the worst case design are given by (28), the same as that of . As seen from Fig. 4(a), the frequency response of the three filters are ‘balanced.’ Roughly speaking, the noises with are much larger in the first channel than in others. To fully reject the noise from the first channel, the optimal solution is , but the synthesis filters have to satisfy the PR cannot be zero. Fig. 4(b) shows that constraint, hence the norm of is still much smaller than others although are much it cannot be zero. Similarly, the noises with larger in the second channel than in others. Fig. 4(c) also shows is much smaller than others although it that the norm of cannot be zero. Example 2: Consider the oversampled FB with channels and decimation number , where
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and . The coeffiare known as binomial QMF filter coefficients cients of , [27, Table 4.2]. The frequency response of are shown in Fig. 5. Note that and are the same. It can be readily calculated that for this example , , By Theorem 1
It then follows from Theorem 2 that the set of all PR synthesis FBs is given by
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Fig. 4. Example 2: Frequency responses of the synthesis filters. F (z ): solid; F (z ): dotted; F (z ): dashed.
where and is any stable 2 1 transfer matrix. For this example, we first consider a non-constant weighting transfer matrix. Let
Then, using Theorem 5, the following optimal synthesis filters are obtained for the above . See the equation at the bottom . of the page, and Next, we consider the optimal synthesis filters for the following different ’s:
Fig. 5. Example 2: Frequency responses of the analysis filters. H (z ): solid; H (z ): dotted; H (z ): dashed.
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Fig. 6. Example 2: Frequency responses of the synthesis filters. F (z ): solid; F (z ): dotted; F (z ): dashed.
TABLE I OPTIMAL SYNTHESIS FB COEFFICIENTS FOR S =
S
TABLE II OPTIMAL SYNTHESIS FB COEFFICIENTS FOR S =
S
TABLE III OPTIMAL SYNTHESIS FB COEFFICIENTS FOR S =
S
TABLE IV OPTIMAL SYNTHESIS FB COEFFICIENTS FOR S =
S
The frequency responses of the optimal synthesis filters for (white noises), , the noises with and are shown in Fig. 6(a)–(d), and their coefficients are given in Tables I–IV, respectively. Because is unimodular, these filters are FIR. and are the As seen from Table I and Fig. 6(a), does not change much same. Fig. 6(b) and (c) show that
when the weighing of the first channel is increased. This is difmay be ‘small’ when the ferent from Example 1 where is designed to satnoise are ‘large’. The reason is that isfy the PR constraint first. For Example 2, there is no redun. On the other hand, if is chosen, corredancy for sponding to a ‘large’ noise in the third channel, Fig. 6(d) shows is very ‘small’, which could be viewed as reducing that the noise.
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Fig. 7. Example 4: The input and reconstructed signals and the reconstruction errors of FBS4 (left) and FBS1 (right).
Example 3: Consider again the oversampled FB given in Example 2 but with the following singular weighting matrices: and
. By di-
rect computation, it is easy to show that
Thus
It then follows from Corollary 2 that the optimal and is given by weighting matrices
for the
where is given by Example 2. The corresponding syn, , 0, where thesis filters are then given by and are shown in Table I. For this example, is zero since there is no noise in the third channel while the second cannot be zero since it has channel is perfect. However, to satisfy PR constraint. Example 4: This example presents simulation results of the synthesis FBs designed above. Two FBs with subbband noises are implemented in SIMULINK. The analysis filters in both , , given in Example 2, and the FBs are the , , given in Tables I and synthesis FBs are the
IV, respectively. Since these FBs are designed for the noise and , respectively, they are referred PSDs to as FBS1 and FBS4 here. The FBS1 and FBS4 are subject and to the same subband noises with PSD the same sinusoidal input signal. The corresponding input , reconstructed signal and reconstruction signal are plotted in Fig. 7. As seen from errors the plots, FBS1 performs poorly because it is designed with , i.e., no noise weighting in optimization; whereas FBS4 gives much better reconstruction since it is designed with . The calculation of the optimal noise weighting SNR shows that SNR (12.8558 dB) and SNR (69.4361 dB). The optimal design boosts the SNR by over 16 times. IX. CONCLUSION Optimal noise reduction in general (IIR or FIR) PR oversampled FBs has been studied in this paper. Using a new state-space parameterization of all dual frame PR synthesis FBs, the generic formulation and general solution of the optimal PR synthesis FBs for noise reduction have been presented. Two classes of subband noises under three different measures of noise reduction have been addressed using the general solution. The optimal solution and state-space based efficient design method are obtained for each case, and the relations between different cases are analyzed and revealed. Extensive examples are presented to demonstrate the effectiveness of the obtained design methods. The results of this paper have provided deeper insight into the optimal subband noise reduction in oversampled FBs, which can be generalized to the study of noise reduction in generic frames. The technical tools of this paper can also be extended to other optimal design problems such as the optimal noise reduction in oversampled FBs. These results will be presented elsewhere.
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APPENDIX A1. Algorithm for Computing
and
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Therefore in (12)–(13)
, , Recall that and . Step 1. Apply QR factorization to obtain
where
,
Since
where triangular,
is an isometry, we have
is lower and is co-isometry, that is,
. Step 2. Take (29) or (30) Lemma 4: and generated by the above algorithm satisfy (12). Proof: The first equation of (12) follows from
Obviously and are positive semi-defand inite on the unit circle. Therefore, are also positive semi-definite. It is well-known that for two positive semi-definite matrices and , and , if and only if [26]. moreover, Recalling the definition of three different norms, we have
The second equation of (12) comes from the fact that
By taking well-defined since , we have
This completes the proof.
which is is of full-row rank for any
A.2 Proof of Theorem 3 Proof: Since unit circle, we define
is of full column rank on the Hence (31)
where and let , i.e.,
, . It is easy to check that is an isometry, and that
with
the
optimal
. Moreover, .
given by , where , is the unique solution for
and and
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A.3 Proof of Theorem 4
REFERENCES
Proof: Note that
Since ties of
and are invertible by assumption and the properand , it follows that
Using our notations, it can be rewritten in a compact formula as follows:
This implies that
[1] H. Bölcskei, F. Hlawatsch, and H. G. Feichtinger, “Frame-theoretic analysis of oversampled filter banks,” IEEE Trans. Signal Process., vol. 46, no. 12, pp. 3256–3268, Dec. 1998. [2] Z. Cvelkovic´ and M. Vetterli, “Oversampled filter banks,” IEEE Trans. Signal Process., vol. 46, no. 5, pp. 1245–1255, May 1998. [3] L. Gan and K.-K. Ma, “Oversampled linear-phase perfect reconstruction filterbanks: Theory, lattice structure and parameterization,” IEEE Trans. Signal Process., vol. 51, no. 3, pp. 744–759, Mar. 2003. [4] T. Strohmer, “Finite and infinite-dimensional models for oversampled filter banks,” in Modern Sampling Theory: Mathematics and Applications, J. J. Benedetto and P. J. S. G. Ferreira, Eds. Cambridge, MA: Birkhäser, 2001. [5] L. Chai, J. Zhang, C. Zhang, and E. Mosca, “Frame theory based analysis and design of oversampled filter banks: Direct computational method,” IEEE Trans. Signal Process., vol. 55, no. 2, pp. 507–519, Feb. 2007. [6] L. Chai, J. Zhang, C. Zhang, and E. Mosca, “Efficient computation of frame bounds using LMI-based optimization,” IEEE Trans. Signal Process., vol. 56, no. 7, pp. 3024–3033, Jul. 2008. [7] H. Bölcskei and F. Hlawatsch, “Noise reduction in oversampled filter banks using predictive quantization,” IEEE Trans. Inf. Theory, vol. 47, pp. 155–172, Jan. 2001. [8] A. Scaglione, G. B. Giannakis, and S. Barbarossa, “Redundant filterbank precoders and equalizers Part I: Unification and optimal designs,” IEEE Trans. Signal Process., vol. 47, no. 7, pp. 1988–2006, Jul. 1999. [9] G. Gu and L. Li, “Worst-case design for optimal channel equalization in filterbank transceivers,” IEEE Trans. Signal Process., vol. 51, no. 9, pp. 2424–2435, Sep. 2003. [10] Z. Chen and J. Zhang, “FB analysis of PMRI and its application to SENSE reconstruction,” in Proc. 14th IEEE Int. Conf. Image Processing (ICIP), San Antonio, TX, Sep. 2007, pp. III-129–132. [11] O. Christensen, Frames and Bases An Introductory Course. Cambridge, MA: Birkhäuser, 2008. [12] F. Labeau, “Synthesis filters design for coding gain in oversampled filter banks,” IEEE Signal Process. Lett., vol. 12, pp. 697–700, Oct. 2005. [13] F. Labeau, R. Chiang, M. Kieffer, P. Duhamel, and L. Vandendorpe, “Oversampled filter banks as error correcting codes: Theory and impulse noise correction,” IEEE Trans. Signal Process., vol. 53, no. 12, pp. 4619–4630, Dec. 2005. [14] T. Chen and B. A. Francis, “Design of multirate filter banks by optimization,” IEEE Trans. Signal Process., vol. 43, no. 12, pp. 2822–2830, Dec. 1995. [15] B.-S. Chen, C.-L. Tsai, and Y.-F. Chen, “Mixed filtering design in multirate transmultiplexor systems: LMI approach,” IEEE Trans. Signal Process., vol. 49, no. 11, pp. 2693–2701, Nov. 2001. [16] H. Vikalo, B. Hassibi, A. T. Erdogan, and T. Kailath, “On robust signal reconstruction in noisy filter banks,” Signal Process., vol. 85, pp. 1–14, 2005. [17] H. Zhou, L. Xie, and C. Zhang, “A direct approach to H2 optimal deconvolution of periodic digital channels,” IEEE Trans. Signal Process., vol. 50, no. 7, pp. 1685–1698, Jul. 2002. [18] J. Zhang, R. Yang, C. Zhang, and E. Mosca, “Design of IIR multirate filter banks subject to subband noises,” in Proc. 5th Asian Control Conf., Melbourne, Australia, 2004, pp. 1333–1338. [19] T. Kailath, Linear Systems. Englewood Cliffs, NJ: Prentice-Hall, 1980. [20] L. V. Ahlfors, Complex Analysis. New York: McGraw-Hill, 1979. [21] K. Zhou, J. C. Doyle, and K. Glover, Robust Optimal Control. Englewood Cliffs, NJ: Prentice-Hall, 1996. [22] H. Shu and T. Chen, “On causality and anticausality of cascaded discrete-time systems,” IEEE Trans. Circuits Syst. I, vol. 43, pp. 240–242, Mar. 1996. [23] D. A. Wilson, M. Neekoui, and G. D. Halikias, “An LQR wieghted control,” Int. J. Control, selection approach to discrete generalized vol. 71, no. 1, pp. 93–101, 1998. [24] D. W. Gu, M. C. Tsai, S. D. O’Young, and I. Postlethwaite, “Statespace formulae for discrete-time optimization,” Int. J. Control, vol. 49, no. 5, pp. 1683–1723, 1989. [25] W. A. Gardner, Statistical Spectral Analysis: A Nonprobability Theory. Englewood Cliffs, NJ: Prentice-Hall, 1988. [26] R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge, U.K.: Cambridge Univ. Press, 1985. [27] A. N. Akansu and R. A. Haddad, Multiresolution signal decomposition: transforms, subbands, and wavelets. New York: Academic, 1992.
H
H
(32) On the other hand, direct computation using (16) and gives
It then follows from (32) that
h =h
H
h
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Li Chai (S’00–M’03) received the B.S. degree in applied mathematics and the M. S. degree in control science and engineering, both from Zhejiang University, China, in 1994 and 1997, respectively, and the Ph.D. degree in electrical engineering from Hong Kong University of Science and Technology in 2002. In September 2002, he joined Hangzhou Dianzi University, China. He worked as a Postdoctoral Research Fellow at the Monash University, Clayton, Australia, from May 2004 to June 2006. In December 2007, he joined Wuhan University of Science and Technology, where he is currently a Xiang-Tao Professor. His research interests include multirate signal processing, wavelets and control with communication constraints.
Jingxin Zhang (M’03) received the M.Eng. and Ph.D. degrees in electrical engineering from Northeastern University, China. Between 1988 and 1992, he was an Associate Professor with Northeastern University, China. Since 1989, he has held research positions in the University of Florence, Italy; the University of Melbourne, Australia; and the Cooperative Research Centre for Sensor Signal and Information Processing, Australia; and a Senior Lecturer position in the University of South Australia and Deakin University, Australia. He is currently with the Department of Electrical and Computer Systems Engineering, Monash University, Australia. He is the author and coauthor of over 140 research papers in diverse areas such as adaptive and predictive control, time varying systems, robust filtering, multirate signal processing, and medical imaging. He is the first inventor of a provisional international patent on parallel magnetic resonance image reconstruction. His current research interests are in control and signal processing and their applications to biomedical and industrial systems. Dr. Zhang is the Editorial Board Member of The Open Automation and Control Systems Journal. He is the recipient of 1989 Fok Ying Tong Educational Foundation (Hong Kong) for the outstanding Young Faculty Members in China and 1992 China National Education Committee Award for the Advancement of Science and Technology.
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Cishen Zhang (M’89) received the B.Eng. degree from Tsinghua University, China, in 1982 and Ph.D. degree in electrical engineering from Newcastle University, Australia, in 1990. Between 1971 and 1978, he was an Electrician with Changxindian (February Seven) Locomotive Manufactory, Beijing, China. He carried out research work on control systems at Delft University of Technology, The Netherlands, from 1983 to 1985. After his Ph.D. study from 1986 to 1989 at Newcastle University, he was with the Department of Electrical and Electronic Engineering at the University of Melbourne, Australia, as a Lecturer, Senior Lecturer, and Associate Professor and Reader until October 2002. He is currently with the School Electrical and Electronic Engineering and School of Chemical and Biomedical Engineering at Nanyang Technological University, Singapore. His research interests include signal processing, medical imaging and control.
Edoardo Mosca (F’97–LF’06) received the Dr.Eng. degree in electronics engineering from the University of Rome ”La Sapienza,” Italy, in 1963. He then spent four years in the aerospace industry, where he worked on the research and development of advanced radar systems, particularly, optimal signal synthesis and processing, and phased-array radar systems. Thereafter, from 1968 to 1972, he held academic positions at the University of Michigan, Ann Arbor, and McMaster University, ON, Canada. Since 1972, he has been with the Engineering Faculty, University of Florence, Italy: from 1972 to 1974 as an Associate Professor, and since 1975 as a full Professor of Control Engineering. In the latter capacity, he founded in 1981 the Department of Systems and Computer Science and Engineering, of which he was the first chair until 1987, and which he conceived and set up as a department, including all the academic staff and the related teaching and research activities in computer and control of the university. He has been a Visiting Professor in universities and research centers in many different countries. From 1995 to 1998 he has been the President of the Italian Association of Researchers in Automatic Control, and from 1983 to present the coordinator of several national research projects in the field of automatic control. He is the author of many research papers spanning various diversified fields such as radar signal synthesis and processing, radio communications, system identification, adaptive, predictive, switching supervisory control, and detection of performance degradation in feedback-control systems. He is the author of a book Optimal, Predictive, and Adaptive Control (Prentice-Hall, 1995). Dr. Mosca is an editor of the following journals: the European Journal of Control; the International Journal of Adaptive Control and Signal Processing (Wiley); and the IEE Proceedings—Control Theory and Applications. He is the Italian NMO representative in IFAC (International Federation of Automatic Control). He has been a Council member of EUCA (European Union Control Association) until 1998, and from 1996 to 2002 a Council member of IFAC . In 2001, he was awarded ”honoris causa” the Doctor in Information Engineering degree from the Universidade Tecnica de Lisboa, Lisbon, Portugal, and in 1997 elected to the grade of Fellow of the IEEE for his contributions to adaptive and predictive control.
