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Nonuniform Filter Bank Design with Noises Yu-Min Cheng and Bor-Sen Chen, Senior Member, IEEE

Abstract—This work solves the signal reconstruction problem involving nonuniform filter bank systems with rational decimation factors and noises. Three main nonuniform filter bank systems, i.e., filter-block decimator (FBD) structure, upsamplerfilter-downsampler (UFD) structure, and tree structure, are included in this study. According to different operating conditions, two different signal reconstruction problems for nonuniform filter bank systems with noises under the unknown but identifiable input signal model and the unknown input signal model are discussed, respectively. At the first stage, a unified block state space model for different nonuniform filter bank systems with noises is developed. Then, by incorporating the identified input signal model with this unified state space model and appropriate choice of the augmented state vector, the signal reconstruction problem is reduced to an equivalent state estimation problem for resulting augmented systems if the input signal is identifiable. If the input signal is lacking in modeling, the signal reconstruction is discussed from the minimax estimation point of view. Two state estimation techniques involving robust Kalman filtering filtering are employed, respectively, to treat the sigand nal reconstruction problem of nonuniform filter bank systems according to different a priori knowledge of the input signal. Finally, several numerical examples are presented to illustrate the proposed algorithms and exhibit the performances.

H1

I. INTRODUCTION

M

ULTIRATE analysis/synthesis filter bank systems are conventionally used to decompose signals into separate frequency bands [1]–[3], [13], [14]. The most frequent studied cases of filter bank systems are the ones with integer sampling factors. Such decompositions, e.g., uniform and tree-structured systems based on two-band decompositions, have conformed to the one with integer sampling factors. However, integer band decompositions often do not satisfy many application requirements. For example [5], in the subband coding of speech and audio signals, the ideal decomposition should match the critical bands of the ear. These bands are neither uniformly spaced nor easily represented by tree structures based on uniform two-band filter bank systems. Moreover, the desired passband locations and bandwidths usually violate the integer band sampling constraint [5]. If we want to analyze the signal into unequal subbands, rational sampling factors have to be allowed. However, relatively few investigations have studied the design of nonuniform filter bank systems. Previous literature has concentrated primarily on perfect reconstruction analysis/synthesis filter bank design based on uniform filter bank systems [1]–[3], [13], [14]. Manuscript received February 12, 1996; revised March 9, 1998. This work was supported by National Science Council, R.O.C., under Contract NSC 862213-E-007-035. The associate editor coordinating the review of this paper and approving it for publication was Dr. Truong Q. Nguyen. The authors are with the Department of Electrical Engineering, National Tsing Hua University, Hsinchu, Taiwan, R.O.C. Publisher Item Identifier S 1053-587X(98)05944-3.

Recently, the task of perfectly reconstructing nonuniform filter bank systems has received considerable attention. A previous work [8] aimed at alias cancellation. Smith et al. treated this problem through tree splitting [4]. In [5], a time domain approach was proposed for nonuniform filter bank systems through numerical solutions. In [6], the perfect reconstruction filter bank theory with rational sampling factors was developed on the basis of two transforms. The multidimensional filter bank design with rational sampling factors was also discussed in [7]. Notably, all of the above studies involving either uniform or nonuniform filter bank systems did not consider transmission noise. However, in practical applications, the transmission signals are often corrupted by noises, which may be due to quantization noises in coding processes or external noises from environment. Under this circumstance, the above filter bank design methods fail to treat this problem. In the work of [21], the influence of quantization errors in filter bank systems has been studied. However, external noises are still not discussed. Recently, Chen et al. [11] proposed an effective optimal design method to solve the signal reconstruction problem of uniform filter bank systems. Obviously, the performance of signal reconstruction under noises markedly improved the conventional perfect reconstruction design case. However, this optimal design method cannot be applied to nonuniform filter bank systems directly. Moreover, an exact model of the input signal is needed and incorporated in this design methodology. This requirement restricts the areas of applications because an exact model of the input signal is usually difficultly obtained in most practical applications. This work is motivated by solving the signal reconstruction problem for nonuniform filter bank systems with noises. Three main nonuniform filter bank systems involving FBD structure, UFD structure, and tree structure are discussed. We first develop a unified state space description for the three nonuniform filter bank systems with additive noises. According to different a priori knowledge of the input signal, two different signal reconstruction problems are investigated in this study. Unlike the previous work, the work in [11] requires an exact model of the input signal; an approximate model of the input signal is used in this study. Such an approximate model can be obtained by some system identification techniques [24]. An augmented system then can be obtained by combining this approximate model and the state space model of nonuniform filter bank systems. By appropriate choice of augmented state vector, the input signal is embedded in the state of the augmented system. The signal reconstruction problem then becomes an equivalent state estimation problem of the augmented system. Since an approximate model of the input signal is used, the effect of the unmodeling errors must be

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CHENG AND CHEN: NONUNIFORM FILTER BANK DESIGN WITH NOISES

taken into consideration. Instead of using the Kalman filtering technique in [11], the robust Kalman filtering technique [18], [19] is employed in this study to treat the equivalent state estimation problem. Notably, this design method is signal independent, owing to the fact that the approximate model of the input signal can be identified on-line. However, a lot of computing time is necessary because an additional identification is introduced. To overcome the drawback of the previous methods, the filtering technique [17], [20] is also used in this study. The filtering problem has received much attention in recent years. The problem lies in finding a filtering method to minimize (or in the suboptimal sense) the worst-case estimation error energy over all possible exogenous inputs (include input signals and filtering design external noises). In other words, this achieves the minimax state estimation over all possible input filtering design does not depend on signals. Since the the exogenous inputs, it is very suitable for treating our signal reconstruction problem to obtain a signal-independent synthesis filter bank design with less computational time. In developing a signal-independent design to treat the signal reconstruction problem of nonuniform filter bank systems, we first specify an appropriate weighting function that reflects the importance or contribution of the block input signal in the power spectrum. By incorporating this weighting function into the unified state space model of nonuniform filter bank systems and appropriate choice of the state vector, a new augmented system is obtained. In particular, the input signal is embedded in this augmented state vector. Consequently, the signal reconstruction problem is reduced to an equivalent state estimation problem without the a priori knowledge of the filtering technique, input signal. Then, by employing the the signal reconstruction can be solved from the minimax estimation point of view. The remainder of this paper is organized as follows. In Section II, we describe three different nonuniform filter bank systems and formulate different signal reconstruction problems. A unified state space description of the three nonuniform filter bank systems with noises is developed in Section III. In Section IV, we discuss the synthesis filter bank design where the robust Kalman filtering technique filtering-based synthesis filter bank design is used. The is discussed in Section V. To illustrate the proposed design methods and exhibit the performances, several numerical examples are presented in Section VI. In Section VII, some conclusions are made. II. SYSTEM DESCRIPTIONS AND PROBLEM STATEMENTS In this section, the signal reconstruction problem of nonuniform filter bank systems with rational sampling factors and noises is investigated. First, nonuniform filter bank systems are briefly described. Next, the problem studied in this work is formulated. A. System Descriptions Nonuniform filter bank systems can have a number of different structures. According to the classification of nonuniform

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Fig. 1. FBD structure nonuniform filter bank systems.

Fig. 2.

UFD structure nonuniform filter bank systems.

filter bank systems in [5], two typical structures are discussed. In addition, the tree-structured filter bank system [2]–[4] is also included. 1) Filter-Block Decimator (FBD) Structure: Fig. 1 illus-channel FBD nonuniform filter bank system. trates an Within this FBD structure, the input signal is directly sent analysis filters . Notably, to nonuniformly divide the spectrum subbands. The filtered signals then pass through into block decimators with block decimation factors for . In the th channel, samples of block of the filtered signal are retained, and the rest are size discarded in the block decimator. For perfect reconstruction with maximum decimation, the block decimation rates for must satisfy the fmaximum decimation relation [5] (1) transmitted signals are corrupted In transmission channels, additive noises, which may be a result of quantization by noisy signals are noises or external noises. Then, the received by the synthesis filter bank to restore the original input signal. 2) Upsampler-Filter-Downsampler (UFD) Structure: Fig. 2 illustrates an -channel UFD nonuniform filter bank system. Unlike the nonuniform filter bank system with FBD structure, the input signal passes through upsamplers with at first. Then, the resulting upsampling rates upsampled signals pass through analysis filters , which nonuniformly divide the spectrum subbands. The filtered signals then pass through into downsamplers with downsampling rates to transmitted signals. Similar to FBD nonuniform generate filter bank systems, the maximum decimation relation (1) must be satisfied for perfect reconstruction with maximum

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Fig. 3.

-stage tree structure nonuniform filter bank systems.

decimation. In transmission channels, additive noises are introduced due to quantization noises or external noises. noisy signals are received by the synthesis Finally, the filter bank to reconstruct the original input signal. Remark 1: In UFD structure nonuniform filter bank systems, the band coverage of analysis filters is more complicated than other filter bank systems. An efficient method to assign the band coverage for every analysis filter can be found in [5]. 3) Tree Structure: In general, tree structure filter bank systems have many different structures for different elementary modules and different connections [2]–[4]. For simplicity, only the octave tree-structured nonuniform filter bank system [2], [3] is discussed in this study. Fig. 3 illustrates a -stage treestructured nonuniform filter bank system. Within this structure, a two-channel uniform filter bank with a lowpass filter and a complementary highpass filter is specified as an elementary module. By successively sending the lowfrequency part of the output signal of the elementary module to the further elementary module, a tree-structured nonuniform filter bank system is constructed. Note that the tree-structured nonuniform filter bank system is also a maximally decimated filter bank system. In transmission channels, the signals are corrupted by additive noises. Then, the noisy signals are received by the synthesis filter bank to reconstruct the original input signal. Remark 2: According to the classification in [5], a filter bank system can have uniform, integer nonuniform, FBD nonuniform, or UFD nonuniform structures. However, it is clear that uniform and integer nonuniform filter bank systems can be seen as a special case of FBD or UFD nonuniform , both nonuniform filter bank systems. In the case of filter bank systems with FBD structure and UFD structure are reduced to the integer nonuniform decimation filter bank system. Moreover, uniform filter bank systems are also obtained and for , . if

operating conditions, two different signal reconstruction problems are formulated as follows. • If the model of input signal is unknown but can be identified, how do we find a synthesis filter bank design to restore the input signal from received noisy signals? • If the input signal is lacking in modeling, how do we find a synthesis filter bank design that will reconstruct the input signal from received noisy signals? Remark 3: In the previous work [11], an optimal synthesis filter bank design was proposed by using the Kalman filtering technique. In such a formulation, the model of the input signal must be known exactly. However, an exact model for a given signal is usually unknown in most practical applications. Hence, the above problems are addressed to meet the practical requirements.

III. BLOCK STATE SPACE MODEL OF NONUNIFORM FILTER BANK SYSTEMS In this section, the block state space model of nonuniform filter bank systems with additive noises is established. A. FBD Structure Consider the FBD structure nonuniform filter bank systems as shown in Fig. 1. Let the analysis filter of the th channel be

(2)

and are some positive constants specified by the where . Let denote the maximum designers for and . The corresponding state space representation of of in (2) is analysis filter

B. Problem Statements In this study, the signal reconstruction problem for the above three nonuniform filter bank systems with rational sampling factors and noises are investigated. According to different

(3)

CHENG AND CHEN: NONUNIFORM FILTER BANK DESIGN WITH NOISES

where we have the expression shown at the bottom of the page, for and . Note is a matrix with dimension for that . block state space models of (5), By combining the the resulting integrated block state space model of FBD nonuniform filter bank systems is obtained as

where .. .

.. .

.. .

..

.. .

.

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(6) for , and [12]. Note that if , and if . Let us define as the ; then, the block least common multiple of state space model of the th channel from to with block is obtained as length

where ..

(4)

.. .

.

..

.

.. .

where

for noises, we have

.. .

. After corrupted by additive

(7) where .. .

.. .

..

.

.. .

and we have the expression shown at the bottom of the next , and . page, for for , and . After the operation of the block decimator with decimation rate , only the signal points from the th element to the th element of the block signal are preserved, and the others are discarded, for , and . Hence, the can be derived as block state space model from to

(5)

.. .

.. .

..

.

.. .

B. UFD Structure For UFD nonuniform filter bank systems, it is relatively difficult to derive the block state space model directly, owing to its structural complexity. However, by employing the concept of identical input–output representation as shown in Fig. 4 [2], [3], an equivalent filter bank system is obtained denotes the least common mulas in Fig. 5. Note that upsampling rates . Comparing tiple of the this equivalent structure to the FBD nonuniform filter bank system clearly reveals that the equivalent filter bank system

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Fig. 4.

Input–output identities.

Fig. 5. Equivalent system of the UFD structure nonuniform filter bank systems.

consists of a preupsampler with upsampling rate and an intermediate FBD nonuniform filter bank system with block for . Based on decimation factor this equivalent structure, the block state space model of UFD nonuniform filter bank systems can be obtained. Consider the equivalent system in Fig. 5. Let the analysis filter of the th channel be the same as in (2) or (3); then the can be obtained as state space model of

then, the corresponding block state space model from with block length is of the form

(9) where

(8) where

.. . .. .

..

.

.. .

.. . denotes an identity matrix with the same dimension as , and , and denote the zero matrices matrix , , and , for with the same dimensions as , and . Let the block length be the least ; common multiple of

to

for

, and

.. .

..

.

.

.. .

CHENG AND CHEN: NONUNIFORM FILTER BANK DESIGN WITH NOISES

After the operation of the downsampler with downsampling , only the signal points at the th element rate are preserved, and the others are of the block signal , and discarded for . Hence, we can get the block state space model

(10) where we have the first expression shown at the bottom of the and . Note that page for is a matrix for . By combining the -block state-space models of (10), we get the integrated block state space representation

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After corruption with additive noises, we have (12) where

and we also have the second expression at the bottom of the , and . page for with block Now, consider the block input signal length . The relation (13) is easily derived, where

(11) where ..

.. .

.. .

.

..

and where .

.. .

Then, the noisy integrated block state space model of UFD nonuniform filter bank systems becomes (14)

for

.

where

.. .

.. .

..

.

.. .

for

.

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Fig. 6. Equivalent system of the -stage tree structure nonuniform filter bank systems for N

where

C. Tree Structure Consider a -stage octave tree-structured nonuniform filter and be bank system as shown in Fig. 3. Let the lowpass filter and highpass filter of elementary modules. By employing the input–output identity as in Fig. 4 [2], [3], an equivalent -channel nonuniform filter bank system can . Obviously, be obtained as in Fig. 6, where this equivalent system is a special case of FBD nonuniform filter bank systems with , and

.. .

.. .

.. .

..

.

.. .

(15)

.. .

for , , and . After the , operation of the downsampler with downsampling rate , only for th element of the block signal the signal points at the are preserved, and the others are discarded, for , , and . Hence, the block state space model from to can be derived as

Rewriting (15) into state space representation yields (16) , and for block state space representation from can be obtained as

= + 1.

. The corresponding to with block length (18) (17)

.. .

.. .

..

.

where we have the expression shown at the bottom of the page, and . for

.. .

CHENG AND CHEN: NONUNIFORM FILTER BANK DESIGN WITH NOISES

is a matrix with dimension for . By combining the above block state space models, the resulting integrated block state space model is obtained as

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IV. SYNTHESIS FILTER BANK DESIGN VIA ROBUST KALMAN FILTERING

Note that

(19) where

and

are the same as in (6), ..

,

, and

are

.

.. .

for we have

. After corrupted by additive noises,

Similar to the treatment in uniform filter bank design [11], the Kalman filtering technique can be used to treat the signal reconstruction problem of nonuniform filter bank systems with noises if the input signal model is known exactly. If the signal model is unknown or uncertain, then the Kalman filtering technique is no longer useful. However, it is usually difficult to derive an exact model for a given input signal in most practical applications. If some uncertainties are permitted, then we can easily obtain an approximate model of the input signal via some system identification techniques [24]. Because an approximate model of the input signal is used, the effect of unmodeling errors has to be taken into account in the design of synthesis filter banks. The robust Kalman filtering strategy is thus proposed to treat this problem in this section. Without loss of generality, the input signal model is assumed to be described by the state space form

(20) where

(22) is a zero-mean, Gaussian, white-noise with variance and are the known nominal matrices that can be obtained via some system identification are the methodologies [24]; and . The corresponding unmodeling quantities for block state space model with block length is derived as where

and

(23)

for

, and

,

where

.

D. Summary Based on the previous analysis, the above three nonuniform filter bank systems with additive noises can be described by the unified state space representation (21)

.. .

where , , ,

for FBD structure for UFD structure for Tree structure

(4), for FBD structure (13), for UFD structure (17), for Tree structure for

.

.. .

.. .

..

.

.. .

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corresponding correlations of derived as

for FBD and tree structures, and

and

..

can be

.

.. . .. .. .

.. .

..

.

.. .

To treat the robust state estimation problem in a more general and manner, we also assume that the correlations are unknown but bounded and contained in some nonempty convex and compact sets as

and

and the uncertainty where the matrices corresponding unmodeling parts for Let us define

and

and

.

is contained in a compact set as

are the .

(24)

Then, by incorporating the block state space model of the input signal in (23) with the block state space description of nonuniform filter bank in (21), we get the augmented system

where and are the known nominal parts of the actual and , respectively, , , and are unknown values is the square root some known positive constants, and for a matrix . Then, of the maximum eigenvalue of according to the result in [18], we describe the robust Kalman filtering algorithm as follows. Robust Kalman Filtering Algorithm • One Stage Prediction Algorithm: (26)

(25)

• Observation Algorithm:

where

(27) • Filtering Algorithm:

(28) • A Priori Variance Algorithm:

for . By observing the above augmented system, the input signal is embedded in the augmented state vector. Therefore, the signal reconstruction problem of the nonuniform filter bank system with noises is reduced to the state estimation problem . In for the augmented system (25) with uncertainties the following, the robust Kalman filtering technique [18], [19] is employed to solve this state estimation problem. For simplicity, let us assume that the additive noises are all zero-mean, Gaussian, white-noises with for . Moreover, we variance are mutually independent. The also assume

(29) • Robust Kalman Gain Algorithm:

(30)

CHENG AND CHEN: NONUNIFORM FILTER BANK DESIGN WITH NOISES

• A Posteriori Variance Algorithm: (31)

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Remark 5: If the nominal input signal model is described by a time-varying model

• Initial Setting:

(32) Note that gain matrix,

for , the corresponding nominal block state space model can be derived as

is the recursive block robust Kalman , and are defined as

where we have the equations shown at the bottom of the page. The nominal part of the augmented state equation of (25) becomes respectively, for some constant and . After the estimate of the augmented state is obtained, the reconstruction signal can be derived by

where

(33) is an identity matrix with compatible dimension of where , and and are zero matrices with the vector , and , respectively. compatible dimension of the vector Remark 4: In case where all information on the input signal and are known exactly, model and exogenous inputs , , , and are all zero matrices, , . Consequently, we have , and and . The value of can be chosen to be unity. In this situation, the above robust multirate Kalman filtering algorithm is reduced to the multirate Kalman filtering algorithm. It can be seen as the extension of the previous work [11] to the nonuniform filter bank case.

The robust Kalman filtering algorithm in (26)–(32) is then still with . useful by replacing Note that a forgetting factor that puts more emphasis on recent data than older data is usually introduced while a timevarying system is considered. Since the identified model of the input signal plays an important role to guide the signal reconstruction in the proposed design method, the use of the forgetting factor can be incorporated into the mechanism of system identifications for the time-varying input signal. The detailed discussion can be found in [24].

.. .

.. . for FBD and Tree structures for UFD structure.

.. .

..

.

..

.

.. .

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Remark 6: For many practical applications, a reasonable delay is tolerated. Consequently, a smoothing algorithm is preferable to a filtering algorithm. According to the result in [22], we briefly discuss the smoothing algorithm in the is tolerated. Note following. Suppose a reasonable delay must be a multiple of the block length for the that . Based requirement of signal reconstruction, i.e., on the conduction of the smoothing algorithm in [22], a new augmented system for (25) is obtained as

where

.. .

..

.

.. .

.. .

and

denotes the th lag version of for . By applying the robust Kalman filtering algorithm (26)–(32) to this new augmented system, the estimation of is obtained. Then, the corresponding smoothed state of (25) can be derived as

for . Consequently, the smoothed block signal can be derived by

Remark 7: The choice of the value in (29) and (30) is very important because it relates to the convergence (or stability) of the proposed robust Kalman filtering algorithm (26)–(32). Let us define

V. SYNTHESIS FILTER BANK DESIGN VIA

FILTERING

In using the previous design methods, the model of the input signal can be identified via some system identification techniques. If the input signal cannot be modeled by an ARMA process, then the previous results are no longer useful. In filtering scheme is introduced to treat this section, an the signal reconstruction problem of nonuniform filter bank filtering scheme has been introduced to systems. The solve the state estimation problem without the knowledge of input signals and noises [17], [20] (and the reference therein). Because a priori knowledge of the statistical properties of the exogenous signals is unnecessary, it is very suitable in treating the signal reconstruction problem of filter bank systems. Consider the nonuniform filter bank systems as shown in Figs. 1–3. The corresponding state space representation can be be a prescribed stable weighting described by (21). Let function that would reflect the importance or contribution of in the power spectrum with the block input signal

for , where is an unknown exogenous signal. Remark 8: Proper choice of the weighting system for the block input signal is a very important work in filtering-based design scheme. It is heavily related the to the reconstruction performance. A feasible selection of the is described as follows. weighting function denote a prescribed weighting function for the Let input signal that reflects the importance or contribution of in the power spectrum. The state space realization is denoted . Then, the state space realization by of can be obtained as

.. . and an auxiliary stable system

Then, can be chosen as the maximum eigenvalue of the . To guarantee the convergence of the robust matrix Kalman filtering algorithm (26)–(32), the condition [18]

must be satisfied, where

and for

is a positive constant satisfying

is the transition matrix of the auxiliary stable system .

.. .

.. .

..

.

.. .

for FBD and Tree structures for UFD structure. Note that the role of the prescribed weighting function is very similar to the exact model of the input signal in [11] or the identified model of the input signal in the previous section. However, it is more flexible in practical applications, especially in the case where the input signal cannot be modeled by an ARMA process because it can be specified arbitrarily by the designer.

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• Filtering Algorithm:

Let us define (34) By rearranging the unified state space description (21), the augmented system

(41) • Error Variance Algorithm:

(35) is obtained, where (42) where for . Note that the input signal is embedded in the state vector of the augmented system (35). Hence, the signal reconstruction problem of nonuniform filter bank systems is reduced to the state estimation problem of the augmented system (35). Let the filtered error be

•

Gain Algorithm:

(36) and define the

performance criterion as

(43) • Initial Setting: (44)

(37)

Note that matrix, and

is the recursive block is defined as

gain

where terminal time; positive definite weighting matrix which reflects a is to the priori knowledge as to how close ; initial guess optimal performance measure. Physical interpretation of the performance in (37) finds an such that the smallest estimation optimal design method error energy over all possible exogenous signals is achieved. optimization problem is very difficult to In general, the solve. An alternative treatment is developed from a suboptimal viewpoint to find a strategy satisfying

respectively, for . After the estimate of the augmented state is obtained, the reconstruction signal can be derived by (45) is an identity matrix with compatible dimension of where and are zero matrices with compatible the vector , respectively. The derivadimension of the vectors and smoothing algorithm is similar to the tion of the use of discussion in Remark 6. filtering strategy Remark 9: From [17] and [20], the satisfying (37) exists if and only if for

(38) , where and are all in for some given . According to the result in [17] and [20], we summarize filtering algorithms in the following. the a posteriori Filtering Algorithm • One Stage Prediction Algorithm: (39) • Observation Algorithm: (40)

where . Hence, to guarantee the satisfactority of (38), the above condition must be checked for each iteration. Remark 10: According to the discussion in [17] and [20], filtering algorithm has the same structure as the the . This means Kalman filtering algorithm, whereas that an alternative version of the multirate Kalman filtering design for nonuniform filter bank systems is obtained in case of all a priori knowledge of the input signal and exogenous noises being known exactly. Notably, the role of the weighting of the input signal in Remark 8 is the same function as the exact model of the input signal in this situation.

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Fig. 7. SNRr -SNRc curves for two-band FBD case.

VI. NUMERICAL SIMULATIONS This section provides several simulation examples to illustrate the effectiveness of the proposed design methods for different nonuniform filter bank systems with additive noises. Because the tree-structured nonuniform filter bank system can be seen as a special case of FBD nonuniform filter bank systems, only FBD and UFD structures are tested for simplicity. To evaluate the performance of the proposed design methods, two definitions of signal-to-noise ratio SNR

for

SNR are used to characterize the corruption of additive noises and the reconstruction error. For convenience, all SNR are chosen to be the same value as SNR . We divide the simulations into the following two parts. A. Theoretical Demonstrations In the following, a two-band and a five-band nonuniform filter bank system for both FBD and UFD structures with band and , and are used as the illussplits trative examples. All the analysis filters are designed by MATLAB command “fir1,” which implements the classical method of windowed linear-phase FIR filter design with order 16. , and in Seven cases with SNR is assumed to be decibels are tested. The input signal generated by an AR model that is driven by a zero-mean, Gaussian, white noise with signal length 3000. An eight-pole

AR model with pole locations at and is chosen as the input AR model. To use the robust Kalman filtering algorithm, the identified signal model is assumed to be an AR model with pole locations . In at -based design method, the weighting function the proposed is chosen according to the discussion in Remark 8 for is obtained by spectral estimation of the input which with order 4. By employing the proposed design signal methods to solve the signal reconstruction problem of the three filter bank systems, the performance is presented by SNR -SNR curves as shown in Figs. 7, 9, and 11. In the case where only quantization errors are considered, the performance can be presented in terms of rate-distortion curves (SNR versus bitrate). In this case, the linear quantizers with bit lengths 2, 4, 6, 8, 10, 12, 14, and 16 are adopted in the three nonuniform filter bank systems. The simulation results are shown in Figs. 8, 10, and 12. To compare the reconstruction performance of the proposed design method with the perfect reconstruction (PR) design, the simulation results using the PR design method [5] are also shown in Figs. 7–12. According to the discussions in Remarks 4 and 10, the reconstruction performance using the Kalman filtering design method is also shown in Figs. 7–12 to reflect the limit of the performance improvement of the proposed design methods. From the results in Figs. 7, 9, and 11, it is found that the proposed design methods provide a more marked performance improvement than the PR design, especially in the low SNR case. A similar conclusion is obtained in Figs. 8, 10, and 12 in the low bitrate case. However, less performance improvement in the proposed design methods is observed in the high bitrate case in Figs. 8,

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Fig. 8.

SNRr -bitrate curves for two-band FBD case.

Fig. 9.

SNRr -SNRc curves for two-band UFD case.

10, and 12. This is because the quantization noise is very small in the high bitrate case. Hence, the reconstruction performance is dominated by the effect of unmodeling errors in robust Kalman filtering design or the choice of the weighting function filtering design. From Figs. 7–12, it is also found that a in much more marked performance improvement of the Kalman filtering design to that of the PR design is obtained, even

though it is the high bitrate case. Since the reconstruction performance of the Kalman filtering design can be seen as the limit of the performance improvement of the proposed design methods, we conclude that a better performance in the proposed design methods will be obtained if a more precise model of the input signal or a more suitable weighting function is chosen.

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Fig. 10.

Fig. 11.

SNRr -bitrate curves for two-band UFD case.

SNRr -SNRc curves for five-band FBD (or UFD) case.

B. Practical Applications To demonstrate, the proposed design methods can be used in practical applications. In this subsection, an 8-kHz speech signal “hai-lauo-hir” with signal length 8192 and an audio signal “Danger Zone” in the movie Top Gun with signal length 199 680 is employed as the input signal. For simplicity, only

the case of the two-band FBD nonuniform filter bank system, which is the same as the demonstrative example, with additive dB, is tested. To use the robust Kalman noise SNR filtering algorithm to treat the signal reconstruction problem of the nonuniform filter bank system with additive noises, the identified AR model with order 60 is derived by the linear predictive coding method [23] with frame length 128. The

CHENG AND CHEN: NONUNIFORM FILTER BANK DESIGN WITH NOISES

Fig. 12.

Fig. 13.

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SNRr -bitrate curves curves for five-band FBD (or UFD) case.

Waveform of the partial reconstructed signal for speech signal in the noise case SNRc = 10 dB.

reconstruction performances with SNR dB and dB are obtained for speech and audio cases, SNR filtering-based respectively. By employing the proposed method to treat the problem, the weighting function is chosen according to the result in Remark 8 that is given by spectral estimation of the input signal with order 60. dB The reconstruction performances with SNR dB are derived for speech and audio and SNR

cases, respectively. The reconstruction performances that use the PR design are SNR dB (speech case) and dB (audio case). The waveforms of partially SNR reconstructed signals, which are obtained by the proposed design methods, and the different original input signals are shown in Figs. 13 and 14. It is shown that the proposed design methods can be applied to practical applications with a good reconstruction performance.

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Fig. 14.

Waveform of the partial reconstructed signal for audio signal in the noise case SNRc = 10 dB.

Fig. 15.

Curve of SNRr versus order of identified AR model for speech case.

Remark 11: In general, it is very difficult to discuss theoretically how good the input signal has to be approximated by the ARMA model in the proposed robust Kalman filteringbased design method to obtain an improved performance over PR design. To reveal this information from simulation, several additional numerical experiments for both speech and audio dB are presented. cases under the additive noise SNR is derived by a The identified AR model with order

linear predictive coding method [23] with frame length 128. The reconstruction performances with respect to the variation of order for the AR model are shown in Figs. 15 and 16. It is shown that the approximate AR model of the speech signal with order of at least 16 will guarantee a performance improvement over the PR design. The worst-case requirement of the order for the approximate AR model of the audio signal to guarantee improvement over the PR design is 18. Obviously,

CHENG AND CHEN: NONUNIFORM FILTER BANK DESIGN WITH NOISES

Fig. 16.

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Curve of SNRr versus order of identified AR model for audio case.

the required order of the speech signal is less than that of the audio signal. The reconstruction performance of the speech signal is better than that of audio signal in Figs. 15 and 16. This is because an audio signal is inherently more complex than a speech signal and, therefore, cannot be modeled very well by an AR model.

ACKNOWLEDGMENT The authors would like to thank the reviewers and the associate editor for their constructive comments and suggestions, which have greatly improved the quality of this manuscript.

REFERENCES VII. CONCLUSION This study has investigated the signal reconstruction problem of nonuniform filter bank systems in FBD, UFD, and tree structures with rational decimation rates and noises. The main contributions are the following. • We have developed a unified state space model to describe the different nonuniform filter bank systems. • We have proposed a robust Kalman filtering-based method to treat the signal reconstruction problem for nonuniform filter bank systems if the input signal is unknown but identifiable. filtering-based method to solve • We have proposed an the signal reconstruction problem for nonuniform filter bank systems without any a priori knowledge of the input signal and noises. Based on the proposed design methods, orthogonal band splits are not necessary in analysis filter designs. In addition, it can be performed recursively. Consequently, the proposed design methods are very suitable for filter bank system designs and can be applied to comprehensive areas such as speech, audio, or image subband coding systems with some waveform coding algorithms. Simulation results have also confirmed the effectiveness of the proposed methods.

[1] R. E. Crochiere and L. R. Rabiner, Multirate Digital Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1983. [2] P. P. Vaidyanathan, Multirate System and Filter Banks. Englewood Cliffs, NJ: Prentice Hall, 1993. [3] N. J. Fliege, Multirate Digital Signal Processing. New York: Wiley, 1994. [4] M. J. T. Smith and T. P. Barnwell, III, “Exact reconstruction techniques for tree structured subband coders,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-34, pp. 434–441, June 1986. [5] K. Nayebi, T. P. Barnwell, III, and M. J. T. Smith, “Nonuniform filter banks: A reconstruction and design theory,” IEEE Trans. Signal Processing, vol. 41, pp. 1114–1127, Mar. 1993. [6] J. Kovaˇcevi´c and M. Vetterli, “Perfect reconstruction filter banks with rational sampling factors,” IEEE Trans. Signal Processing, vol. 41, pp. 2047–2066, June 1993. [7] R. A. Gopinath and C. S. Burrus, “On upsampling, downsampling, and rational sampling rate filter banks,” IEEE Trans. Signal Processing, vol. 42, pp. 812–824, Apr. 1994. [8] P. Q. Hoang and P. P. Vaidyanathan, “Nonuniform multirate filter banks: Theory and design,” in Proc. IEEE Int. Symp. Circuits Syst., Portland, OR, 1989, pp. 371–374. [9] S. Wada, “Design of nonuniform division multirate FIR filter banks,” IEEE Trans. Circuits Syst. II, vol. 42, pp. 115–121, Feb. 1995. [10] M. R. Azimi-Sadjadi, T. Lu, and E. M. Nebot, “Parallel and sequential block Kalman filtering and their implementations using systolic arrays,” IEEE Trans. Signal Processing, vol. 39, pp. 137–147, Jan. 1991. [11] B. S. Chen, C. W. Lin, and Y. L. Chen, “Optimal signal reconstruction in noisy filter bank systems: Multirate Kalman synthesis filtering approach,” IEEE Trans. Signal Processing, vol. 43, pp. 2496–2505, Nov. 1995. [12] M. Gopal, Modern Control System Theory. New Delhi, India: Wiley, 1987.

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[13] K. Nayebi, T. P. Barnwell, III, and M. J. T. Smith, “Time domain filter bank analysis: A new design theory,” IEEE Trans. Signal Processing, vol. 40, pp. 1412–1429, June 1992. [14] R. V. Cox, “The design of uniformly and nonuniformly spaced pseudo quadrature mirror filters,” IEEE Trans. Acoust., Speech, Signal Processing, vol. 34, pp. 1090–1096, May 1986. [15] J. M. Mendel, Lessons in Digital Estimation Theory. Englewood Cliffs, NJ: Prentice-Hall, 1987. [16] V. P. Sathe and P. P. Vaidyanathan, “Effects of multirate systems on the statistical properties of random signals,” IEEE Trans. Signal Processing, vol. 41, pp. 131–146, Jan. 1993. optimality of the [17] B. Hassibi, A. H. Sayed, and T. Kailath, “ LMS algorithm,” IEEE Trans. Signal Processing, vol. 44, pp. 267–280, 1996. [18] B. S. Chen and S. C. Peng, “Robust stability analysis of Kalman filter under parametric and noise uncertainties,” in Approximate Kalman Filtering, G. Chen, Ed. Singapore: World Scientific, 1993. [19] Y. Theoder and U. Shaked, “Robust discrete time minimum variance filtering,” IEEE Trans. Signal Processing, vol. 44, pp. 181–189, 1996. [20] B. Hassibi, A. H. Sayed, and T. Kailath, “Linear estimation in Krein spaces—Part I: Theory, Part II: Applications,” IEEE Trans. Automat. Contr., vol. 41, pp. 18–49, 1996. [21] J. Kovaˇcevi´c, “Subband coding systems incorporating quantizer models,” IEEE Trans. Image Processing, vol. 4, pp. 543–553, 1995. [22] B. D. O. Anderson and J. B. Moore, Optimal Filtering. Englewood Cliffs, NJ: Prentice-Hall, 1979. [23] S. Furui, Digital Speech Processing, Synthesis, and Recognition. New York: Marcel Dekker, 1989. [24] T. S¨oderstr¨om and P. Stoica, System Identification. Englewood Cliffs, NJ: Prentice-Hall, 1989.

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Yu-Min Cheng was born in Taoyuan, Taiwan, R.O.C. He received the B.S. degree in nuclear engineering and the M.S. degree in electrical engineering from the National Tsing Hua University, Hsinchu, Taiwan, in 1990 and 1992, respectively. He is currently working toward the Ph.D. degree in electrical engineering at National Tsing Hua University. His research interests include robust control, fuzzy control, adaptive control, and signal processing.

Bor-Sen Chen (M’82–SM’89) received the B.S. degree from Tatung Institute of Technology, Taiwan, R.O.C., in 1970, the M.S. degree from National Central University, Taiwan, in 1973, and the Ph.D. degree from the University of Southern California, Los Angeles, in 1982. He was a Lecturer, Associate Professor, and Professor at Tatung Institute of Technology from 1973 to 1987. He is now a Professor at National Tsing Hua University, Hsinchu, Taiwan. His current research interests include control and signal processing. Dr. Chen has received the Distinguished Research Award from National Science Council of Taiwan four times.

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 46, NO. 9, SEPTEMBER 1998

Nonuniform Filter Bank Design with Noises Yu-Min Cheng and Bor-Sen Chen, Senior Member, IEEE

Abstract—This work solves the signal reconstruction problem involving nonuniform filter bank systems with rational decimation factors and noises. Three main nonuniform filter bank systems, i.e., filter-block decimator (FBD) structure, upsamplerfilter-downsampler (UFD) structure, and tree structure, are included in this study. According to different operating conditions, two different signal reconstruction problems for nonuniform filter bank systems with noises under the unknown but identifiable input signal model and the unknown input signal model are discussed, respectively. At the first stage, a unified block state space model for different nonuniform filter bank systems with noises is developed. Then, by incorporating the identified input signal model with this unified state space model and appropriate choice of the augmented state vector, the signal reconstruction problem is reduced to an equivalent state estimation problem for resulting augmented systems if the input signal is identifiable. If the input signal is lacking in modeling, the signal reconstruction is discussed from the minimax estimation point of view. Two state estimation techniques involving robust Kalman filtering filtering are employed, respectively, to treat the sigand nal reconstruction problem of nonuniform filter bank systems according to different a priori knowledge of the input signal. Finally, several numerical examples are presented to illustrate the proposed algorithms and exhibit the performances.

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I. INTRODUCTION

M

ULTIRATE analysis/synthesis filter bank systems are conventionally used to decompose signals into separate frequency bands [1]–[3], [13], [14]. The most frequent studied cases of filter bank systems are the ones with integer sampling factors. Such decompositions, e.g., uniform and tree-structured systems based on two-band decompositions, have conformed to the one with integer sampling factors. However, integer band decompositions often do not satisfy many application requirements. For example [5], in the subband coding of speech and audio signals, the ideal decomposition should match the critical bands of the ear. These bands are neither uniformly spaced nor easily represented by tree structures based on uniform two-band filter bank systems. Moreover, the desired passband locations and bandwidths usually violate the integer band sampling constraint [5]. If we want to analyze the signal into unequal subbands, rational sampling factors have to be allowed. However, relatively few investigations have studied the design of nonuniform filter bank systems. Previous literature has concentrated primarily on perfect reconstruction analysis/synthesis filter bank design based on uniform filter bank systems [1]–[3], [13], [14]. Manuscript received February 12, 1996; revised March 9, 1998. This work was supported by National Science Council, R.O.C., under Contract NSC 862213-E-007-035. The associate editor coordinating the review of this paper and approving it for publication was Dr. Truong Q. Nguyen. The authors are with the Department of Electrical Engineering, National Tsing Hua University, Hsinchu, Taiwan, R.O.C. Publisher Item Identifier S 1053-587X(98)05944-3.

Recently, the task of perfectly reconstructing nonuniform filter bank systems has received considerable attention. A previous work [8] aimed at alias cancellation. Smith et al. treated this problem through tree splitting [4]. In [5], a time domain approach was proposed for nonuniform filter bank systems through numerical solutions. In [6], the perfect reconstruction filter bank theory with rational sampling factors was developed on the basis of two transforms. The multidimensional filter bank design with rational sampling factors was also discussed in [7]. Notably, all of the above studies involving either uniform or nonuniform filter bank systems did not consider transmission noise. However, in practical applications, the transmission signals are often corrupted by noises, which may be due to quantization noises in coding processes or external noises from environment. Under this circumstance, the above filter bank design methods fail to treat this problem. In the work of [21], the influence of quantization errors in filter bank systems has been studied. However, external noises are still not discussed. Recently, Chen et al. [11] proposed an effective optimal design method to solve the signal reconstruction problem of uniform filter bank systems. Obviously, the performance of signal reconstruction under noises markedly improved the conventional perfect reconstruction design case. However, this optimal design method cannot be applied to nonuniform filter bank systems directly. Moreover, an exact model of the input signal is needed and incorporated in this design methodology. This requirement restricts the areas of applications because an exact model of the input signal is usually difficultly obtained in most practical applications. This work is motivated by solving the signal reconstruction problem for nonuniform filter bank systems with noises. Three main nonuniform filter bank systems involving FBD structure, UFD structure, and tree structure are discussed. We first develop a unified state space description for the three nonuniform filter bank systems with additive noises. According to different a priori knowledge of the input signal, two different signal reconstruction problems are investigated in this study. Unlike the previous work, the work in [11] requires an exact model of the input signal; an approximate model of the input signal is used in this study. Such an approximate model can be obtained by some system identification techniques [24]. An augmented system then can be obtained by combining this approximate model and the state space model of nonuniform filter bank systems. By appropriate choice of augmented state vector, the input signal is embedded in the state of the augmented system. The signal reconstruction problem then becomes an equivalent state estimation problem of the augmented system. Since an approximate model of the input signal is used, the effect of the unmodeling errors must be

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CHENG AND CHEN: NONUNIFORM FILTER BANK DESIGN WITH NOISES

taken into consideration. Instead of using the Kalman filtering technique in [11], the robust Kalman filtering technique [18], [19] is employed in this study to treat the equivalent state estimation problem. Notably, this design method is signal independent, owing to the fact that the approximate model of the input signal can be identified on-line. However, a lot of computing time is necessary because an additional identification is introduced. To overcome the drawback of the previous methods, the filtering technique [17], [20] is also used in this study. The filtering problem has received much attention in recent years. The problem lies in finding a filtering method to minimize (or in the suboptimal sense) the worst-case estimation error energy over all possible exogenous inputs (include input signals and filtering design external noises). In other words, this achieves the minimax state estimation over all possible input filtering design does not depend on signals. Since the the exogenous inputs, it is very suitable for treating our signal reconstruction problem to obtain a signal-independent synthesis filter bank design with less computational time. In developing a signal-independent design to treat the signal reconstruction problem of nonuniform filter bank systems, we first specify an appropriate weighting function that reflects the importance or contribution of the block input signal in the power spectrum. By incorporating this weighting function into the unified state space model of nonuniform filter bank systems and appropriate choice of the state vector, a new augmented system is obtained. In particular, the input signal is embedded in this augmented state vector. Consequently, the signal reconstruction problem is reduced to an equivalent state estimation problem without the a priori knowledge of the filtering technique, input signal. Then, by employing the the signal reconstruction can be solved from the minimax estimation point of view. The remainder of this paper is organized as follows. In Section II, we describe three different nonuniform filter bank systems and formulate different signal reconstruction problems. A unified state space description of the three nonuniform filter bank systems with noises is developed in Section III. In Section IV, we discuss the synthesis filter bank design where the robust Kalman filtering technique filtering-based synthesis filter bank design is used. The is discussed in Section V. To illustrate the proposed design methods and exhibit the performances, several numerical examples are presented in Section VI. In Section VII, some conclusions are made. II. SYSTEM DESCRIPTIONS AND PROBLEM STATEMENTS In this section, the signal reconstruction problem of nonuniform filter bank systems with rational sampling factors and noises is investigated. First, nonuniform filter bank systems are briefly described. Next, the problem studied in this work is formulated. A. System Descriptions Nonuniform filter bank systems can have a number of different structures. According to the classification of nonuniform

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Fig. 1. FBD structure nonuniform filter bank systems.

Fig. 2.

UFD structure nonuniform filter bank systems.

filter bank systems in [5], two typical structures are discussed. In addition, the tree-structured filter bank system [2]–[4] is also included. 1) Filter-Block Decimator (FBD) Structure: Fig. 1 illus-channel FBD nonuniform filter bank system. trates an Within this FBD structure, the input signal is directly sent analysis filters . Notably, to nonuniformly divide the spectrum subbands. The filtered signals then pass through into block decimators with block decimation factors for . In the th channel, samples of block of the filtered signal are retained, and the rest are size discarded in the block decimator. For perfect reconstruction with maximum decimation, the block decimation rates for must satisfy the fmaximum decimation relation [5] (1) transmitted signals are corrupted In transmission channels, additive noises, which may be a result of quantization by noisy signals are noises or external noises. Then, the received by the synthesis filter bank to restore the original input signal. 2) Upsampler-Filter-Downsampler (UFD) Structure: Fig. 2 illustrates an -channel UFD nonuniform filter bank system. Unlike the nonuniform filter bank system with FBD structure, the input signal passes through upsamplers with at first. Then, the resulting upsampling rates upsampled signals pass through analysis filters , which nonuniformly divide the spectrum subbands. The filtered signals then pass through into downsamplers with downsampling rates to transmitted signals. Similar to FBD nonuniform generate filter bank systems, the maximum decimation relation (1) must be satisfied for perfect reconstruction with maximum

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Fig. 3.

-stage tree structure nonuniform filter bank systems.

decimation. In transmission channels, additive noises are introduced due to quantization noises or external noises. noisy signals are received by the synthesis Finally, the filter bank to reconstruct the original input signal. Remark 1: In UFD structure nonuniform filter bank systems, the band coverage of analysis filters is more complicated than other filter bank systems. An efficient method to assign the band coverage for every analysis filter can be found in [5]. 3) Tree Structure: In general, tree structure filter bank systems have many different structures for different elementary modules and different connections [2]–[4]. For simplicity, only the octave tree-structured nonuniform filter bank system [2], [3] is discussed in this study. Fig. 3 illustrates a -stage treestructured nonuniform filter bank system. Within this structure, a two-channel uniform filter bank with a lowpass filter and a complementary highpass filter is specified as an elementary module. By successively sending the lowfrequency part of the output signal of the elementary module to the further elementary module, a tree-structured nonuniform filter bank system is constructed. Note that the tree-structured nonuniform filter bank system is also a maximally decimated filter bank system. In transmission channels, the signals are corrupted by additive noises. Then, the noisy signals are received by the synthesis filter bank to reconstruct the original input signal. Remark 2: According to the classification in [5], a filter bank system can have uniform, integer nonuniform, FBD nonuniform, or UFD nonuniform structures. However, it is clear that uniform and integer nonuniform filter bank systems can be seen as a special case of FBD or UFD nonuniform , both nonuniform filter bank systems. In the case of filter bank systems with FBD structure and UFD structure are reduced to the integer nonuniform decimation filter bank system. Moreover, uniform filter bank systems are also obtained and for , . if

operating conditions, two different signal reconstruction problems are formulated as follows. • If the model of input signal is unknown but can be identified, how do we find a synthesis filter bank design to restore the input signal from received noisy signals? • If the input signal is lacking in modeling, how do we find a synthesis filter bank design that will reconstruct the input signal from received noisy signals? Remark 3: In the previous work [11], an optimal synthesis filter bank design was proposed by using the Kalman filtering technique. In such a formulation, the model of the input signal must be known exactly. However, an exact model for a given signal is usually unknown in most practical applications. Hence, the above problems are addressed to meet the practical requirements.

III. BLOCK STATE SPACE MODEL OF NONUNIFORM FILTER BANK SYSTEMS In this section, the block state space model of nonuniform filter bank systems with additive noises is established. A. FBD Structure Consider the FBD structure nonuniform filter bank systems as shown in Fig. 1. Let the analysis filter of the th channel be

(2)

and are some positive constants specified by the where . Let denote the maximum designers for and . The corresponding state space representation of of in (2) is analysis filter

B. Problem Statements In this study, the signal reconstruction problem for the above three nonuniform filter bank systems with rational sampling factors and noises are investigated. According to different

(3)

CHENG AND CHEN: NONUNIFORM FILTER BANK DESIGN WITH NOISES

where we have the expression shown at the bottom of the page, for and . Note is a matrix with dimension for that . block state space models of (5), By combining the the resulting integrated block state space model of FBD nonuniform filter bank systems is obtained as

where .. .

.. .

.. .

..

.. .

.

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(6) for , and [12]. Note that if , and if . Let us define as the ; then, the block least common multiple of state space model of the th channel from to with block is obtained as length

where ..

(4)

.. .

.

..

.

.. .

where

for noises, we have

.. .

. After corrupted by additive

(7) where .. .

.. .

..

.

.. .

and we have the expression shown at the bottom of the next , and . page, for for , and . After the operation of the block decimator with decimation rate , only the signal points from the th element to the th element of the block signal are preserved, and the others are discarded, for , and . Hence, the can be derived as block state space model from to

(5)

.. .

.. .

..

.

.. .

B. UFD Structure For UFD nonuniform filter bank systems, it is relatively difficult to derive the block state space model directly, owing to its structural complexity. However, by employing the concept of identical input–output representation as shown in Fig. 4 [2], [3], an equivalent filter bank system is obtained denotes the least common mulas in Fig. 5. Note that upsampling rates . Comparing tiple of the this equivalent structure to the FBD nonuniform filter bank system clearly reveals that the equivalent filter bank system

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Fig. 4.

Input–output identities.

Fig. 5. Equivalent system of the UFD structure nonuniform filter bank systems.

consists of a preupsampler with upsampling rate and an intermediate FBD nonuniform filter bank system with block for . Based on decimation factor this equivalent structure, the block state space model of UFD nonuniform filter bank systems can be obtained. Consider the equivalent system in Fig. 5. Let the analysis filter of the th channel be the same as in (2) or (3); then the can be obtained as state space model of

then, the corresponding block state space model from with block length is of the form

(9) where

(8) where

.. . .. .

..

.

.. .

.. . denotes an identity matrix with the same dimension as , and , and denote the zero matrices matrix , , and , for with the same dimensions as , and . Let the block length be the least ; common multiple of

to

for

, and

.. .

..

.

.

.. .

CHENG AND CHEN: NONUNIFORM FILTER BANK DESIGN WITH NOISES

After the operation of the downsampler with downsampling , only the signal points at the th element rate are preserved, and the others are of the block signal , and discarded for . Hence, we can get the block state space model

(10) where we have the first expression shown at the bottom of the and . Note that page for is a matrix for . By combining the -block state-space models of (10), we get the integrated block state space representation

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After corruption with additive noises, we have (12) where

and we also have the second expression at the bottom of the , and . page for with block Now, consider the block input signal length . The relation (13) is easily derived, where

(11) where ..

.. .

.. .

.

..

and where .

.. .

Then, the noisy integrated block state space model of UFD nonuniform filter bank systems becomes (14)

for

.

where

.. .

.. .

..

.

.. .

for

.

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Fig. 6. Equivalent system of the -stage tree structure nonuniform filter bank systems for N

where

C. Tree Structure Consider a -stage octave tree-structured nonuniform filter and be bank system as shown in Fig. 3. Let the lowpass filter and highpass filter of elementary modules. By employing the input–output identity as in Fig. 4 [2], [3], an equivalent -channel nonuniform filter bank system can . Obviously, be obtained as in Fig. 6, where this equivalent system is a special case of FBD nonuniform filter bank systems with , and

.. .

.. .

.. .

..

.

.. .

(15)

.. .

for , , and . After the , operation of the downsampler with downsampling rate , only for th element of the block signal the signal points at the are preserved, and the others are discarded, for , , and . Hence, the block state space model from to can be derived as

Rewriting (15) into state space representation yields (16) , and for block state space representation from can be obtained as

= + 1.

. The corresponding to with block length (18) (17)

.. .

.. .

..

.

where we have the expression shown at the bottom of the page, and . for

.. .

CHENG AND CHEN: NONUNIFORM FILTER BANK DESIGN WITH NOISES

is a matrix with dimension for . By combining the above block state space models, the resulting integrated block state space model is obtained as

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IV. SYNTHESIS FILTER BANK DESIGN VIA ROBUST KALMAN FILTERING

Note that

(19) where

and

are the same as in (6), ..

,

, and

are

.

.. .

for we have

. After corrupted by additive noises,

Similar to the treatment in uniform filter bank design [11], the Kalman filtering technique can be used to treat the signal reconstruction problem of nonuniform filter bank systems with noises if the input signal model is known exactly. If the signal model is unknown or uncertain, then the Kalman filtering technique is no longer useful. However, it is usually difficult to derive an exact model for a given input signal in most practical applications. If some uncertainties are permitted, then we can easily obtain an approximate model of the input signal via some system identification techniques [24]. Because an approximate model of the input signal is used, the effect of unmodeling errors has to be taken into account in the design of synthesis filter banks. The robust Kalman filtering strategy is thus proposed to treat this problem in this section. Without loss of generality, the input signal model is assumed to be described by the state space form

(20) where

(22) is a zero-mean, Gaussian, white-noise with variance and are the known nominal matrices that can be obtained via some system identification are the methodologies [24]; and . The corresponding unmodeling quantities for block state space model with block length is derived as where

and

(23)

for

, and

,

where

.

D. Summary Based on the previous analysis, the above three nonuniform filter bank systems with additive noises can be described by the unified state space representation (21)

.. .

where , , ,

for FBD structure for UFD structure for Tree structure

(4), for FBD structure (13), for UFD structure (17), for Tree structure for

.

.. .

.. .

..

.

.. .

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corresponding correlations of derived as

for FBD and tree structures, and

and

..

can be

.

.. . .. .. .

.. .

..

.

.. .

To treat the robust state estimation problem in a more general and manner, we also assume that the correlations are unknown but bounded and contained in some nonempty convex and compact sets as

and

and the uncertainty where the matrices corresponding unmodeling parts for Let us define

and

and

.

is contained in a compact set as

are the .

(24)

Then, by incorporating the block state space model of the input signal in (23) with the block state space description of nonuniform filter bank in (21), we get the augmented system

where and are the known nominal parts of the actual and , respectively, , , and are unknown values is the square root some known positive constants, and for a matrix . Then, of the maximum eigenvalue of according to the result in [18], we describe the robust Kalman filtering algorithm as follows. Robust Kalman Filtering Algorithm • One Stage Prediction Algorithm: (26)

(25)

• Observation Algorithm:

where

(27) • Filtering Algorithm:

(28) • A Priori Variance Algorithm:

for . By observing the above augmented system, the input signal is embedded in the augmented state vector. Therefore, the signal reconstruction problem of the nonuniform filter bank system with noises is reduced to the state estimation problem . In for the augmented system (25) with uncertainties the following, the robust Kalman filtering technique [18], [19] is employed to solve this state estimation problem. For simplicity, let us assume that the additive noises are all zero-mean, Gaussian, white-noises with for . Moreover, we variance are mutually independent. The also assume

(29) • Robust Kalman Gain Algorithm:

(30)

CHENG AND CHEN: NONUNIFORM FILTER BANK DESIGN WITH NOISES

• A Posteriori Variance Algorithm: (31)

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Remark 5: If the nominal input signal model is described by a time-varying model

• Initial Setting:

(32) Note that gain matrix,

for , the corresponding nominal block state space model can be derived as

is the recursive block robust Kalman , and are defined as

where we have the equations shown at the bottom of the page. The nominal part of the augmented state equation of (25) becomes respectively, for some constant and . After the estimate of the augmented state is obtained, the reconstruction signal can be derived by

where

(33) is an identity matrix with compatible dimension of where , and and are zero matrices with the vector , and , respectively. compatible dimension of the vector Remark 4: In case where all information on the input signal and are known exactly, model and exogenous inputs , , , and are all zero matrices, , . Consequently, we have , and and . The value of can be chosen to be unity. In this situation, the above robust multirate Kalman filtering algorithm is reduced to the multirate Kalman filtering algorithm. It can be seen as the extension of the previous work [11] to the nonuniform filter bank case.

The robust Kalman filtering algorithm in (26)–(32) is then still with . useful by replacing Note that a forgetting factor that puts more emphasis on recent data than older data is usually introduced while a timevarying system is considered. Since the identified model of the input signal plays an important role to guide the signal reconstruction in the proposed design method, the use of the forgetting factor can be incorporated into the mechanism of system identifications for the time-varying input signal. The detailed discussion can be found in [24].

.. .

.. . for FBD and Tree structures for UFD structure.

.. .

..

.

..

.

.. .

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Remark 6: For many practical applications, a reasonable delay is tolerated. Consequently, a smoothing algorithm is preferable to a filtering algorithm. According to the result in [22], we briefly discuss the smoothing algorithm in the is tolerated. Note following. Suppose a reasonable delay must be a multiple of the block length for the that . Based requirement of signal reconstruction, i.e., on the conduction of the smoothing algorithm in [22], a new augmented system for (25) is obtained as

where

.. .

..

.

.. .

.. .

and

denotes the th lag version of for . By applying the robust Kalman filtering algorithm (26)–(32) to this new augmented system, the estimation of is obtained. Then, the corresponding smoothed state of (25) can be derived as

for . Consequently, the smoothed block signal can be derived by

Remark 7: The choice of the value in (29) and (30) is very important because it relates to the convergence (or stability) of the proposed robust Kalman filtering algorithm (26)–(32). Let us define

V. SYNTHESIS FILTER BANK DESIGN VIA

FILTERING

In using the previous design methods, the model of the input signal can be identified via some system identification techniques. If the input signal cannot be modeled by an ARMA process, then the previous results are no longer useful. In filtering scheme is introduced to treat this section, an the signal reconstruction problem of nonuniform filter bank filtering scheme has been introduced to systems. The solve the state estimation problem without the knowledge of input signals and noises [17], [20] (and the reference therein). Because a priori knowledge of the statistical properties of the exogenous signals is unnecessary, it is very suitable in treating the signal reconstruction problem of filter bank systems. Consider the nonuniform filter bank systems as shown in Figs. 1–3. The corresponding state space representation can be be a prescribed stable weighting described by (21). Let function that would reflect the importance or contribution of in the power spectrum with the block input signal

for , where is an unknown exogenous signal. Remark 8: Proper choice of the weighting system for the block input signal is a very important work in filtering-based design scheme. It is heavily related the to the reconstruction performance. A feasible selection of the is described as follows. weighting function denote a prescribed weighting function for the Let input signal that reflects the importance or contribution of in the power spectrum. The state space realization is denoted . Then, the state space realization by of can be obtained as

.. . and an auxiliary stable system

Then, can be chosen as the maximum eigenvalue of the . To guarantee the convergence of the robust matrix Kalman filtering algorithm (26)–(32), the condition [18]

must be satisfied, where

and for

is a positive constant satisfying

is the transition matrix of the auxiliary stable system .

.. .

.. .

..

.

.. .

for FBD and Tree structures for UFD structure. Note that the role of the prescribed weighting function is very similar to the exact model of the input signal in [11] or the identified model of the input signal in the previous section. However, it is more flexible in practical applications, especially in the case where the input signal cannot be modeled by an ARMA process because it can be specified arbitrarily by the designer.

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• Filtering Algorithm:

Let us define (34) By rearranging the unified state space description (21), the augmented system

(41) • Error Variance Algorithm:

(35) is obtained, where (42) where for . Note that the input signal is embedded in the state vector of the augmented system (35). Hence, the signal reconstruction problem of nonuniform filter bank systems is reduced to the state estimation problem of the augmented system (35). Let the filtered error be

•

Gain Algorithm:

(36) and define the

performance criterion as

(43) • Initial Setting: (44)

(37)

Note that matrix, and

is the recursive block is defined as

gain

where terminal time; positive definite weighting matrix which reflects a is to the priori knowledge as to how close ; initial guess optimal performance measure. Physical interpretation of the performance in (37) finds an such that the smallest estimation optimal design method error energy over all possible exogenous signals is achieved. optimization problem is very difficult to In general, the solve. An alternative treatment is developed from a suboptimal viewpoint to find a strategy satisfying

respectively, for . After the estimate of the augmented state is obtained, the reconstruction signal can be derived by (45) is an identity matrix with compatible dimension of where and are zero matrices with compatible the vector , respectively. The derivadimension of the vectors and smoothing algorithm is similar to the tion of the use of discussion in Remark 6. filtering strategy Remark 9: From [17] and [20], the satisfying (37) exists if and only if for

(38) , where and are all in for some given . According to the result in [17] and [20], we summarize filtering algorithms in the following. the a posteriori Filtering Algorithm • One Stage Prediction Algorithm: (39) • Observation Algorithm: (40)

where . Hence, to guarantee the satisfactority of (38), the above condition must be checked for each iteration. Remark 10: According to the discussion in [17] and [20], filtering algorithm has the same structure as the the . This means Kalman filtering algorithm, whereas that an alternative version of the multirate Kalman filtering design for nonuniform filter bank systems is obtained in case of all a priori knowledge of the input signal and exogenous noises being known exactly. Notably, the role of the weighting of the input signal in Remark 8 is the same function as the exact model of the input signal in this situation.

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Fig. 7. SNRr -SNRc curves for two-band FBD case.

VI. NUMERICAL SIMULATIONS This section provides several simulation examples to illustrate the effectiveness of the proposed design methods for different nonuniform filter bank systems with additive noises. Because the tree-structured nonuniform filter bank system can be seen as a special case of FBD nonuniform filter bank systems, only FBD and UFD structures are tested for simplicity. To evaluate the performance of the proposed design methods, two definitions of signal-to-noise ratio SNR

for

SNR are used to characterize the corruption of additive noises and the reconstruction error. For convenience, all SNR are chosen to be the same value as SNR . We divide the simulations into the following two parts. A. Theoretical Demonstrations In the following, a two-band and a five-band nonuniform filter bank system for both FBD and UFD structures with band and , and are used as the illussplits trative examples. All the analysis filters are designed by MATLAB command “fir1,” which implements the classical method of windowed linear-phase FIR filter design with order 16. , and in Seven cases with SNR is assumed to be decibels are tested. The input signal generated by an AR model that is driven by a zero-mean, Gaussian, white noise with signal length 3000. An eight-pole

AR model with pole locations at and is chosen as the input AR model. To use the robust Kalman filtering algorithm, the identified signal model is assumed to be an AR model with pole locations . In at -based design method, the weighting function the proposed is chosen according to the discussion in Remark 8 for is obtained by spectral estimation of the input which with order 4. By employing the proposed design signal methods to solve the signal reconstruction problem of the three filter bank systems, the performance is presented by SNR -SNR curves as shown in Figs. 7, 9, and 11. In the case where only quantization errors are considered, the performance can be presented in terms of rate-distortion curves (SNR versus bitrate). In this case, the linear quantizers with bit lengths 2, 4, 6, 8, 10, 12, 14, and 16 are adopted in the three nonuniform filter bank systems. The simulation results are shown in Figs. 8, 10, and 12. To compare the reconstruction performance of the proposed design method with the perfect reconstruction (PR) design, the simulation results using the PR design method [5] are also shown in Figs. 7–12. According to the discussions in Remarks 4 and 10, the reconstruction performance using the Kalman filtering design method is also shown in Figs. 7–12 to reflect the limit of the performance improvement of the proposed design methods. From the results in Figs. 7, 9, and 11, it is found that the proposed design methods provide a more marked performance improvement than the PR design, especially in the low SNR case. A similar conclusion is obtained in Figs. 8, 10, and 12 in the low bitrate case. However, less performance improvement in the proposed design methods is observed in the high bitrate case in Figs. 8,

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Fig. 8.

SNRr -bitrate curves for two-band FBD case.

Fig. 9.

SNRr -SNRc curves for two-band UFD case.

10, and 12. This is because the quantization noise is very small in the high bitrate case. Hence, the reconstruction performance is dominated by the effect of unmodeling errors in robust Kalman filtering design or the choice of the weighting function filtering design. From Figs. 7–12, it is also found that a in much more marked performance improvement of the Kalman filtering design to that of the PR design is obtained, even

though it is the high bitrate case. Since the reconstruction performance of the Kalman filtering design can be seen as the limit of the performance improvement of the proposed design methods, we conclude that a better performance in the proposed design methods will be obtained if a more precise model of the input signal or a more suitable weighting function is chosen.

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Fig. 10.

Fig. 11.

SNRr -bitrate curves for two-band UFD case.

SNRr -SNRc curves for five-band FBD (or UFD) case.

B. Practical Applications To demonstrate, the proposed design methods can be used in practical applications. In this subsection, an 8-kHz speech signal “hai-lauo-hir” with signal length 8192 and an audio signal “Danger Zone” in the movie Top Gun with signal length 199 680 is employed as the input signal. For simplicity, only

the case of the two-band FBD nonuniform filter bank system, which is the same as the demonstrative example, with additive dB, is tested. To use the robust Kalman noise SNR filtering algorithm to treat the signal reconstruction problem of the nonuniform filter bank system with additive noises, the identified AR model with order 60 is derived by the linear predictive coding method [23] with frame length 128. The

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Fig. 12.

Fig. 13.

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SNRr -bitrate curves curves for five-band FBD (or UFD) case.

Waveform of the partial reconstructed signal for speech signal in the noise case SNRc = 10 dB.

reconstruction performances with SNR dB and dB are obtained for speech and audio cases, SNR filtering-based respectively. By employing the proposed method to treat the problem, the weighting function is chosen according to the result in Remark 8 that is given by spectral estimation of the input signal with order 60. dB The reconstruction performances with SNR dB are derived for speech and audio and SNR

cases, respectively. The reconstruction performances that use the PR design are SNR dB (speech case) and dB (audio case). The waveforms of partially SNR reconstructed signals, which are obtained by the proposed design methods, and the different original input signals are shown in Figs. 13 and 14. It is shown that the proposed design methods can be applied to practical applications with a good reconstruction performance.

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Fig. 14.

Waveform of the partial reconstructed signal for audio signal in the noise case SNRc = 10 dB.

Fig. 15.

Curve of SNRr versus order of identified AR model for speech case.

Remark 11: In general, it is very difficult to discuss theoretically how good the input signal has to be approximated by the ARMA model in the proposed robust Kalman filteringbased design method to obtain an improved performance over PR design. To reveal this information from simulation, several additional numerical experiments for both speech and audio dB are presented. cases under the additive noise SNR is derived by a The identified AR model with order

linear predictive coding method [23] with frame length 128. The reconstruction performances with respect to the variation of order for the AR model are shown in Figs. 15 and 16. It is shown that the approximate AR model of the speech signal with order of at least 16 will guarantee a performance improvement over the PR design. The worst-case requirement of the order for the approximate AR model of the audio signal to guarantee improvement over the PR design is 18. Obviously,

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Fig. 16.

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Curve of SNRr versus order of identified AR model for audio case.

the required order of the speech signal is less than that of the audio signal. The reconstruction performance of the speech signal is better than that of audio signal in Figs. 15 and 16. This is because an audio signal is inherently more complex than a speech signal and, therefore, cannot be modeled very well by an AR model.

ACKNOWLEDGMENT The authors would like to thank the reviewers and the associate editor for their constructive comments and suggestions, which have greatly improved the quality of this manuscript.

REFERENCES VII. CONCLUSION This study has investigated the signal reconstruction problem of nonuniform filter bank systems in FBD, UFD, and tree structures with rational decimation rates and noises. The main contributions are the following. • We have developed a unified state space model to describe the different nonuniform filter bank systems. • We have proposed a robust Kalman filtering-based method to treat the signal reconstruction problem for nonuniform filter bank systems if the input signal is unknown but identifiable. filtering-based method to solve • We have proposed an the signal reconstruction problem for nonuniform filter bank systems without any a priori knowledge of the input signal and noises. Based on the proposed design methods, orthogonal band splits are not necessary in analysis filter designs. In addition, it can be performed recursively. Consequently, the proposed design methods are very suitable for filter bank system designs and can be applied to comprehensive areas such as speech, audio, or image subband coding systems with some waveform coding algorithms. Simulation results have also confirmed the effectiveness of the proposed methods.

[1] R. E. Crochiere and L. R. Rabiner, Multirate Digital Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1983. [2] P. P. Vaidyanathan, Multirate System and Filter Banks. Englewood Cliffs, NJ: Prentice Hall, 1993. [3] N. J. Fliege, Multirate Digital Signal Processing. New York: Wiley, 1994. [4] M. J. T. Smith and T. P. Barnwell, III, “Exact reconstruction techniques for tree structured subband coders,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-34, pp. 434–441, June 1986. [5] K. Nayebi, T. P. Barnwell, III, and M. J. T. Smith, “Nonuniform filter banks: A reconstruction and design theory,” IEEE Trans. Signal Processing, vol. 41, pp. 1114–1127, Mar. 1993. [6] J. Kovaˇcevi´c and M. Vetterli, “Perfect reconstruction filter banks with rational sampling factors,” IEEE Trans. Signal Processing, vol. 41, pp. 2047–2066, June 1993. [7] R. A. Gopinath and C. S. Burrus, “On upsampling, downsampling, and rational sampling rate filter banks,” IEEE Trans. Signal Processing, vol. 42, pp. 812–824, Apr. 1994. [8] P. Q. Hoang and P. P. Vaidyanathan, “Nonuniform multirate filter banks: Theory and design,” in Proc. IEEE Int. Symp. Circuits Syst., Portland, OR, 1989, pp. 371–374. [9] S. Wada, “Design of nonuniform division multirate FIR filter banks,” IEEE Trans. Circuits Syst. II, vol. 42, pp. 115–121, Feb. 1995. [10] M. R. Azimi-Sadjadi, T. Lu, and E. M. Nebot, “Parallel and sequential block Kalman filtering and their implementations using systolic arrays,” IEEE Trans. Signal Processing, vol. 39, pp. 137–147, Jan. 1991. [11] B. S. Chen, C. W. Lin, and Y. L. Chen, “Optimal signal reconstruction in noisy filter bank systems: Multirate Kalman synthesis filtering approach,” IEEE Trans. Signal Processing, vol. 43, pp. 2496–2505, Nov. 1995. [12] M. Gopal, Modern Control System Theory. New Delhi, India: Wiley, 1987.

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[13] K. Nayebi, T. P. Barnwell, III, and M. J. T. Smith, “Time domain filter bank analysis: A new design theory,” IEEE Trans. Signal Processing, vol. 40, pp. 1412–1429, June 1992. [14] R. V. Cox, “The design of uniformly and nonuniformly spaced pseudo quadrature mirror filters,” IEEE Trans. Acoust., Speech, Signal Processing, vol. 34, pp. 1090–1096, May 1986. [15] J. M. Mendel, Lessons in Digital Estimation Theory. Englewood Cliffs, NJ: Prentice-Hall, 1987. [16] V. P. Sathe and P. P. Vaidyanathan, “Effects of multirate systems on the statistical properties of random signals,” IEEE Trans. Signal Processing, vol. 41, pp. 131–146, Jan. 1993. optimality of the [17] B. Hassibi, A. H. Sayed, and T. Kailath, “ LMS algorithm,” IEEE Trans. Signal Processing, vol. 44, pp. 267–280, 1996. [18] B. S. Chen and S. C. Peng, “Robust stability analysis of Kalman filter under parametric and noise uncertainties,” in Approximate Kalman Filtering, G. Chen, Ed. Singapore: World Scientific, 1993. [19] Y. Theoder and U. Shaked, “Robust discrete time minimum variance filtering,” IEEE Trans. Signal Processing, vol. 44, pp. 181–189, 1996. [20] B. Hassibi, A. H. Sayed, and T. Kailath, “Linear estimation in Krein spaces—Part I: Theory, Part II: Applications,” IEEE Trans. Automat. Contr., vol. 41, pp. 18–49, 1996. [21] J. Kovaˇcevi´c, “Subband coding systems incorporating quantizer models,” IEEE Trans. Image Processing, vol. 4, pp. 543–553, 1995. [22] B. D. O. Anderson and J. B. Moore, Optimal Filtering. Englewood Cliffs, NJ: Prentice-Hall, 1979. [23] S. Furui, Digital Speech Processing, Synthesis, and Recognition. New York: Marcel Dekker, 1989. [24] T. S¨oderstr¨om and P. Stoica, System Identification. Englewood Cliffs, NJ: Prentice-Hall, 1989.

H1

Yu-Min Cheng was born in Taoyuan, Taiwan, R.O.C. He received the B.S. degree in nuclear engineering and the M.S. degree in electrical engineering from the National Tsing Hua University, Hsinchu, Taiwan, in 1990 and 1992, respectively. He is currently working toward the Ph.D. degree in electrical engineering at National Tsing Hua University. His research interests include robust control, fuzzy control, adaptive control, and signal processing.

Bor-Sen Chen (M’82–SM’89) received the B.S. degree from Tatung Institute of Technology, Taiwan, R.O.C., in 1970, the M.S. degree from National Central University, Taiwan, in 1973, and the Ph.D. degree from the University of Southern California, Los Angeles, in 1982. He was a Lecturer, Associate Professor, and Professor at Tatung Institute of Technology from 1973 to 1987. He is now a Professor at National Tsing Hua University, Hsinchu, Taiwan. His current research interests include control and signal processing. Dr. Chen has received the Distinguished Research Award from National Science Council of Taiwan four times.