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Eigenstructure Algorithms for Multirate Adaptive Lossless FIR Filters Phillip A. Regalia, Fellow, IEEE, and Dong-Yan Huang, Senior Member, IEEE

Abstract—This paper addresses the problem of adaptively optimizing a two-channel lossless finite-impulse-response (FIR) filter bank, which finds application in subband coding and wavelet signal analysis. Instead of using a gradient decent procedure—with its inherent problem of becoming trapped in local minima of a nonquadratic cost function—two eigenstructure algorithms are proposed. Both algorithms feature a priori bounds on the output variance at any convergent point, which, based on simulations, lead to solutions that lie acceptably close to a global minimum point of an output variance objective function. Moreover, a sufficient condition for such stationary points based on fixed-point theory is shown. It is shown that the convergence rate of both algorithms increases as the separation of eigenvalues of the input covariance matrix increases. Simulations for synthetic and real data support the conclusions. Index Terms—Adaptive filter banks, a priori bounds, stationary points, wavelet analysis.

I. INTRODUCTION

T

HE widespread use of adaptive algorithms for various real-time applications, including digital communications, biomedical engineering, and financial engineering, has motivated adaptive algorithms for multirate lossless finite-impulse-response (FIR) filters. Their role is well recognized in wavelet signal processing and subband coding [1]. The basic two-channel maximally decimated filter bank is shown in rotation angles if the filter order is Fig. 1, comprising . A common problem in many applications is to design a filter bank in the form of Fig. 1, such that the variance of the output is “small,” possibly minimized with respect to some criterion. If the spectral density of the input signal prior to decimation is available, then the design problem is essentially deterministic, and solutions are easily approached [2]. Recently, much effort has been made to design such filter banks by finding the eigenvalues of the autocorrelation matrix of the input signal [2]–[8], although such methods may become trapped in local

Manuscript received September 1, 2004; revised May 31, 2005. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Henrique Malvar. P. A. Regalia is with the Electrical Engineering and Computer Science Department, Catholic University of America, Washington, DC 20064 USA, on leave from the Département Communication, Images et Traitement de l’Information, CNRS/SAMOVAR UMR 5157, Institut National des Télécommunications, 91011 Evry Cedex, France (e-mail: [email protected]). D.-Y. Huang is with the Media Processing—Multimedia Signal Processing Laboratory, Institute for Infocomm Research, Singapore 119316 (e-mail: [email protected]). Digital Object Identifier 10.1109/TSP.2006.870618

Fig. 1. Two-channel lossless filter bank.

minima, except the algorithm [9] which relates the cost funcas a convex optimization problem. In addition, tion the computational complexity of such schemes may prove prohibitive for real-time applications. In real-time applications, adaptive designs for the rotation may form an attractive alterangles native to costly offline optimization methods. The most immediate approach is to use a gradient descent procedure applied [10], [11]; adaptive algorithm deto the cost function computations sign parallels [12] closely, leading to order per time sample. Alternatively, a clever gradient scheme from [13] can reduce the complexity to order . An inherent drawis back of such an approach is that the cost function , nonquadratic in the rotation angles and local minima often result. Although the global minimum will certainly lead to the “best” solution, local minima can yield suboptimal performance. Avoiding local minima requires either global search methods, or adaptation algorithms that do not follow the negative gradient . The latter possibility is explored of the cost function complexity. in this paper, leading to two algorithms of order as the extremal eigenBoth algorithms seek to embed value of a covariance matrix; if successful, a priori bounds on after convergence may be developed, versus the filter order . After a review of the two-channel lossless FIR filter in Section II, we present the first eigenstructure algorithm in Section III and deduce its stationary points. A modified algorithm, which is observed in simulations to converge more reliably, is then derived in Section IV, which exploits a modulated version of the input sequence. Some results on the existence of stationary points are developed in Section V, with simulation examples for synthetic and real audio and speech subband coding presented in Section VI. Concluding remarks are synthesized in Section VII.

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REGALIA AND HUANG: EIGENSTRUCTURE ALGORITHMS FOR MULTIRATE ADAPTIVE LOSSLESS FIR FILTERS

II. MULTIRATE LOSSLESS FIR FILTERS We begin with the two-channel lossless FIR filter of Fig. 1, rotation angles . This system using may be described in state-space form as

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are the polyphase components (here, the even-indexed and oddindexed terms) from the input spectral density function prior to subsampling with

(1)

where the input vector,

(8)

In the same way, the output power spectral density is

is the state vector, the output matrix is partitioned

vector, and the conformally as indicated. The input vector derives from a scalar process by way of subsampling by a factor of 2, i.e.,

(9) The lower-right entry of this matrix gives the spectral density of as

(2) (10) is a zero mean, wide-sense stationary We assume that process. The matrix in (1) is the interconnection of rotations, and therefore is always orthogonal. This implies that the overall , as in transfer matrix

Using Parseval’s theorem, the average value of the even-indexed component is (11)

(3) From this, the objective function

can be expressed as

is para-unitary [1], i.e., for all

(4)

. The objective irrespective of the rotation angles in the present context is to adjust the rotation angles according to be to the input signal so as to force the variance “small.” In order to study the objective function, let us introduce the input power spectral density matrix

(12) From this, we observe that the even-indexed correlation terms contribute nothing to the objective function , save which contributes a constant. Therefore, only for the term intervene in studying the odd-indexed correlation terms may vary according to which lossless is how implemented. As such, if vanishes, i.e., for odd, , so that then

(5) derives from a scalar process Since subsampling, one may show [14] that

by way of

(13) as well. This case gives, in particular (14)

(6) in which

(7)

we use. Accordingly, if irrespective of which para-unitary is negligible compared to , as can happen if the input is nearly symmetric about , the obspectrum versus is fairly flat, and reduces to a jective function vanishes. This simple fact constant function whenever will be exploited in Section IV.

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Fig. 2.

Error-performance surface of three rotation angles and the corresponding contour plot.

III. EIGENSTRUCTURE ALGORITHM

Consider now the covariance matrix

is non quadratic in the The objective function rotation angles , and a gradient descent algorithm (such as [10]–[13]) applied to this objective function may become trapped in a local minima. As an example, Fig. 2 plots the obversus rotation angles1 when jective function , , the input autocorrelation function is , , , . In this case, there are two local minima. As a gradient descent algorithm may become stuck in a local as minimum, our approach seeks instead to embed an extremal eigenvalue of a covariance matrix. With denoting the parameter vector of rotation angles in Fig. 1, the first algorithm we propose takes the form

If we partition the matrix

corresponding to , then

from (1) in the form

,

, and

(15) where

, with

This algorithm is reminiscent of an earlier one proposed in rational subspace estimation [15]. Before embarking on a convergence study, let us isolate the stationary points of this algorithm. Any such stationary point, , is characterized by those rotation angles which, if denoted held fixed, would result in the mean value of the update term from (15) vanishing, i.e.,

This vanishes if and only if the column vector is in the null space of . Since is orthogonal, the null space in question is spanned by ; the orthogonality to be colinear with constraint therefore constrains . A stationary point is thus characterized by the eigenequation

Since variance

has unit norm, the resulting eigenvalue is the output

The following conjecture claims that the eigenvalue at any con. vergent point should be the smallest eigenvalue of Conjecture 1: At any convergent point , the output varishould coincide with the smallest eigenvalue of ance 1The rotation angle  is uniquely determined as a function of the lowerindexed rotation angles, which allows us to plot the surface versus M 1 = 2 rotation angles here.

0

REGALIA AND HUANG: EIGENSTRUCTURE ALGORITHMS FOR MULTIRATE ADAPTIVE LOSSLESS FIR FILTERS

This property has been repeatedly verified in simulations; some supporting arguments are offered in the Appendix. Subject to this conjecture, we can bound the output variance at any convergent point. Property 1: Subject to conjecture 1, at any convergent point in mean of the eigenstructure algorithm from (15)

where

odd-indexed terms as we develop next.

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of the input autocorrelation function,

IV. MODULATION ALGORITHM To obtain an algorithm whose evolution is guided only by the odd-indexed terms of the input autocorrelation function, consider the covariance matrix obtained by modulating the scalar process by

is the input autocorrelation matrix of dimensions

.. .

..

.

..

.

..

.

.. .

.. .

.. .

We can then check that whose eigenvalues are set in decreasing order: . vary To verify, we note that although the eigenvalues of with the rotation angles , a uniform bound in terms of the eigenvalues of may be developed as follows. To begin, iterating the state equation into the past gives

However, since the filter is FIR of order therefore

, we have

are the matrices built from the even- and odd-indexed terms, respectively, of the input autocorrelation function. For example, , we would have with

; Property 2: If is an eigenvalue of , so is . . To verify, choose a vector arbitrarily, and set is equivalent Negating each even-indexed component of to negating each odd-indexed component of . Thus, if , then . Now choose as an eigenvector, so that

.. .

Squaring this up and taking expectations gives (16) is orthogonal for all , it follows that for all (e.g., [1]). Since the rows of are orthonormal, the Poincaré separation theorem [16] allows in terms of those of as us to bound the eigenvalues of Now, since

We then observe that is just , to give the bound of Property 1. One may easily find inputs for which the algorithm of (15) cannot converge. For example, if the polyphase component of the input spectral density is negligible compared to , then for all . But may, according to the even-indexed input autocorrelation terms, be considerably smaller than . Property 1 then cannot apply. We remark that, in such cases, the error surface is rather flat, such that little coding gain is available anyway. This consideration motivates nonetheless an alternate algorithm that uses only the

for some eigenvalue . Then , which is also an eigenvalue. shows that Consider now applying the modulated input sequence to the same lossless filter bank, as in Fig. 3. With hatted accents absorbed throughout, the state-plus-input covariance matrix becomes, analogously to (16)

This allows us to introduce

(17) in which only the odd-indexed terms of the input autocorrelation intervene.

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Fig. 3.

0

Filter bank whose driving sequence is modulated by ( 1) .

An adaptive algorithm that seeks an extremal eigenvalue of takes the form

(18) For a fixed , the mean value of the correction term (minus the diagonal scaling from ) becomes

to and that the relation (17) mapping , the Poincaré separation theorem [16] again gives

From Property 2, the eigenvalues are nonnegative, while the eigenvalues are nonpositive, with . The bound of Property 3 then follows from the observation that . V. EXISTENCE OF STATIONARY POINTS

A stationary point for which this vanishes must render colinear with , giving the eigenequation

At any such stationary point, the eigenvalue takes the form

Since the eigenstructure algorithms do not seek the minimum point of some cost function, the existence of stationary points does not follow immediately. We develop in this section how the Brouwer fixed-point theorem can be applied to deduce conditions under which a stationary point must exist. We begin with the Brouwer fixed-point theorem itself [17, p. 45], [18, p. 84]. Theorem 1: Let be a closed, bounded, and convex set in , and let be any continuous function which maps into . Then admits a fixed point in . and let For our application, set be restricted to its principal angle range (19)

We note next that depends only on the even-indexed terms of the input autocorrelation function; it may thus be reduced to an integral of the form (14), then gives giving the value . Solving for

The following property claims an a priori bound on this variance. of the modulation Property 3: At any convergent point algorithm (18)

This clearly gives a closed, bounded and convex set . denote a second set of rotation angles, Let likewise in , and consider now the parametrization of the unitas norm vector

.. .

(20)

Return now to the covariance matrix from Section III, and, for a fixed , let be an extremal eigenvector: . Then choose such that (that such a always exists may be deduced from, e.g., [19]). Doing this for each in , we can construct an equation in the form That the extremal eigenvalue of consequence of conjecture 1, once

should intervene is a replaces . Considering

(21)

REGALIA AND HUANG: EIGENSTRUCTURE ALGORITHMS FOR MULTIRATE ADAPTIVE LOSSLESS FIR FILTERS

Fig. 4.

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(a) Pole-zero plot of the input spectral shaping filter; (b) frequency responses of two-channel filter bank at global minimum.

which allows us to consider as a function of A fixed point of this map [of the form eigenequation of the form

:

. ] yields an

(22) corresponding to a stationary point of the eigenstructure algorithm. By the Brouwer fixed point theorem, if can be written as a continuous function of without leaving , then a fixed point will exist. be a simple (as opA sufficient condition is that posed to multiple) eigenvalue for all , since this ensures that its corresponding eigenvector varies continuously without encounbecomes a multiple tering a branch point [20]. If, instead, eigenvalue for a particular value of , say , then the minimum eigenvector is no longer unique at ; rather, it lies in a subspace of dimension equal to the multiplicity of . In can vary with the direction that apthis case, proaches , which can break continuity at . Note finally that by to reach the same conclusion we may replace is simple for all for the modulation algorithm: if , then the algorithm admits a stationary point. Translating this sufficient condition in terms of the input spectral density, on the other hand, is less immediate.

VI. SIMULATION RESULTS We present here simulation results that verify the utility of the proposed algorithms.

A. A Priori Bounds generated from the Consider an input sequence output of a filter whose poles and zeros are shown in Fig. 4(a), . when driven by white noise, normalized to yield Choosing , the global minimum of was found at

giving

The resulting frequency responses of low-pass filter and highpass filter are shown in Fig. 4(b). The a priori bound for the eigenstructure algorithm (15) is, according to Property 1

The algorithm exhibited a unique convergent point at

For the modulation algorithm (18), the a priori bound from Property 3 is far more conservative, as follows:

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TABLE I SUMMARY OF PARAMETER VALUES OF THE SECOND-ORDER AR MODELING PROBLEM

Fig. 5. (a) Optimal value E [y (n)] of the eigenstructure algorithm and bound for (R) = 3; (b) optimal value E [y (n)] of the modulation algorithm and bound for (R) = 3.

The convergent point is nonetheless quite satisfactory, as follows:

giving

Both algorithms have, for this example, found an acceptable neighborhood of the global minimum [21]. B. Convergence Behavior We present the simulation results for examining the transient behavior of these two algorithms applied to an autoregressive (AR) process which is described by the second-order difference equation

where the sample is drawn from a white-noise process of zero mean and variance . The transient behavior of the two algorithms is evaluated in following settings: • comparing the steady-state value of ; • varying the eigenvalue separation of input with a fixed step size ; • varying the step-size parameter with a fixed eigenvalue separation of input. In the first experiment, we study the influence of the input and step size on the steady-state value .

We take four sets of AR parameter values of the second-order AR modeling problem in Table I [22]. We choose these sets of parameters in order to show that the performance and the transient behavior of our algorithms are quite different from a . standard gradient descent algorithm applied to In this experiment, we tune the step-size to make the both . The algorithms converge to their optimal values of results are shown in the Table I. We observe that the eigenstructure algorithms converge rather slowly when the eigenvalues of covariance matrix are poorly separated, while the modulation algorithm converges to its minimum value, which is better than that of the steepest descent algorithm (which can be trapped in a local minimum). Both algorithms are observed or , to converge to the minimum eigenvalue of and the steady-state residual values of eigenstructure algorithm are generally smaller than for the modulation algorithm. The steady-state values of both algorithms for case 2 and 3 set parameters are examples shown in Figs. 5 and 6. In the second experiment, the step size is fixed at 0.3, and the evaluations are made for the two sets of AR parameters in cases 2 and 3, shown in Figs. 7 and 8 for the eigenstructure , and modulation algorithms for the cases 10, respectively. We observe that the simulations confirm the fact that good eigenvalue separation leads to rapid convergence, while poorly separated eigenvalues lead to slow convergence; the modulation algorithm is observed to still converge more rapidly than the eigenstructure one. In order to better judge the convergence behavior, the reand the convergence sults about the optimal values

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Fig. 6. (a) Optimal value E [y (n)] of the eigenstructure algorithm and bound (R) = 10; (b) optimal value E [y (n)] of the modulation algorithm and bound for (R) = 10.

Fig. 7.

 (R )

(a) Trajectory of rotation angles of the eigenstructure algorithm for (R) = 3,  = 0:3; (b) rotation angle trajectories for the modulation algorithm for  = 0 :3 .

= 3,

behavior for ten rotation angles for lossless FIR filters with ten , are shown rotation angles for the cases in Figs. 9–12: C. Application: Audio and Speech Signal Coding We outline here the application of the algorithms to a perceptual audio coding for which the basic block diagram is shown in Fig. 13. • In order to exploit the spectral masking properties of the human auditory system, the perceptual audio coding schemes employ an analysis–synthesis filter bank pair in a tree structure using the elementary two-channel lossless FIR filter bank, while the two-channel analysis filter bank is designed using adaptive eigenstructure algorithms (15) and (18). The tree structure forms wavelet packet filter

banks, which decompose the audio signals into spectral subbands. • The subband signals are quantized using vector quantization and coding. In order to achieve a perceptually transparent coding at low bit rates, the quantization must be sufficiently coarse in order not to surpass the target bit rate with the bit allocation procedure, while the quantization noise should be shaped to lie below the estimated masked threshold by MPEG psychoacoustic model II if possible. • The quantized values with side information are multiplexed into one bitstream. • The decoder decodes the bitstream and the entropy coded subband signals. It then calculates the reconstructed signals using adaptive lossless FIR filter bank algorithms (15) and (18). Therefore, the synthesis filter bank has the same complexity as the analysis filter bank.

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Fig. 8. (a) Rotation angle trajectories of the eigenstructure algorithm for (R) = 10,  = 0:3; (b) rotation angle trajectories of the modulation algorithm for (R) = 10,  = 0:3.

Fig. 9. (a) Optimal value E [y (n)] of the eigenstructure algorithm and bound for (R) = 3; (b) optimal value E [y (n)] of the modulation algorithm and bound for (R) = 3.

The efficiency of wavelet packet audio codec depends on various properties in the filter bank, including perfect reconstruction and the appropriate filter length to guarantee no appearance of pre-echo problems while retaining low complexity and energy compaction. In the wavelet packet coding, the selection of two-channel wavelet basis plays a critical role in the performance of the codec. In order to compare the adaptive filter bank design algorithms with fixed parameter filter banks and the gradientbased algorithm, objective quality measures are applied. The segmental signal-to-mask ratio is used to replace the measurement of segmental signal-to-noise (SNR) for perceptual audio coding design. Here we compare the energy compaction and the delay of analysis/synthesis filter bank. The energy compaction

can be measured by coding gain, which is defined for a perfect reconstruction filter bank as [23]

where is the variance of the th subband signal. indicates the factor by which the mean-square reconstruction error is reduced when applying an optimal separate quantizer to each transform component, as compared to quantizing the signal samples directly (as in pulse code modulation).

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Fig. 10. (a) Optimal value E [y (n)] of the eigenstructure algorithm and bound (R) = 10; (b) optimal value E [y (n)] of the modulation algorithm and bound for (R) = 10.

Fig. 11. (a) Trajectory of ten rotation angles of the eigenstructure algorithm for (R) = 3;  = 0:21; (b) rotation angle trajectories for the modulation algorithm for (R) = 3,  = 0:21.

In the simulations from [24], the Daubechies filters were slightly preferable among 25 candidate filters, and so are compared with the eigenstructure and modulation wavelets in what follows. In [24], the authors pointed out that the choice of the Daubechies filters together with its length affects the separation of the subband signals. Considering the trade-off between complexity and the separation of the subband signals, the length of Daubechies filter is chosen to be ten, which is the length we use for the proposed algorithms as well. Table II shows the values of the coding gain in decibels calculated for different wavelets [25] for MPEG standard and some private-collection audio clips. From Table II, we notice that the adaptive algorithms offer improved performance in this coding application compared to standard subband oriented filter banks. In particular,

the modulation algorithm is preferable to others in terms of high coding gain and speed. This stems from the fact that the adaptive algorithms can follow the signal characteristics, unlike a fixed-parameter solution. We next compare the delay of adaptive analysis filter banks for eigenstructre, modulation and gradient algorithms as a function of coding gain for the audio signal Svega shown in the Table III. From above simulations, the following conclusions emerge. • Low-order filters (three rotation angles for most audio clips) are enough to achieve low bit rate with satisfactory coding gain. • The modulation algorithm consistently gives better coding gain for our tested audio clips than the fixed-parameter

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Fig. 12.

 (R )

(a) Rotation angle trajectories of the eigenstructure algorithm for (R) = 10,  = 0:21; (b) rotation angle trajectories of the modulation algorithm for  = 0:21.

= 10,

Fig. 13.

Structure of adaptive wavelet packet perceptual audio codec.

filter bank, the eigenstructure algorithm, and gradient algorithm due to its fast convergence. • With no quantization noise, the adaptive two-channel lossless analysis/synthesis filter bank can be implemented as a time-varying perfect-reconstruction scheme without side information. In effect, the coefficients of the synthesis filter bank can be recovered using the initial values of the rotations, the decomposed signals and the adaptive algorithms (15) and (18). The time-varying coefficients of the analysis filter bank need no longer be quantized and transmitted, offering an attractive bit rate savings. This is in contrast to conventional wavelet packet methods, for which the coefficients must be transmitted as side information. This implementation method has been used successfully in a lossless audio coding system using adaptive linear predictor [26] and extends to the adaptive twochannel lossless filter bank studied here. • The adaptive filter bank tracks sufficiently well to offer improved coding gain for the nonstationary audio signals, compared with half-band fixed-parameter approximations. • As the wavelet packet is designed using adaptive eigenstructure algorithms and performs an instance-based anal-

TABLE II CODING GAIN IN DECIBELS FOR THE TEST CASES

LENGTH

OF

TABLE III ADAPTIVE FILTER BANKS VIA THE CODING GAIN DECIBELS FOR AUDIO SIGNAL SVEGA

IN

ysis, the wavelet packet causes an additional delay identical to the filter bank length. Therefore, the delay of the adaptive wavelet packet filter bank can be kept low to reach the minimum bit-rate with satisfaction coding gain, as required of real-time two-way communications.

REGALIA AND HUANG: EIGENSTRUCTURE ALGORITHMS FOR MULTIRATE ADAPTIVE LOSSLESS FIR FILTERS

VII. CONCLUSION In this paper, we have presented an eigenstructure approach for optimizing multirate adaptive lossless FIR filters, leading to as two algorithms seeking to embed the cost function the extremal eigenvalue of a covariance matrix. Conditions ensuring the existence of stationary points have been established based on fixed-point theory. Both algorithms feature a priori bounds on the output error variance at any convergent point, which is important since a standard gradient descent procedure may become trapped in a local minimum, for which such a priori bounds need not apply. The convergence rate of both algorithms increases as the spread of eigenvalues of the input covariance matrix increases, as indicative of signals offering improved coding gain; the modulation algorithm tends to converge more rapidly than the eigenstructure algorithm due to its better eigenvalue separation of its input covariance matrix. Simulations for synthetic and real data validate the interest of an adaptive filter bank over its fixed coefficient counterpart. APPENDIX Here we provide supporting arguments for conjecture 1, relating to the eigenstructure algorithm, since a parallel develop. ment applies to the modulation algorithm. Let from (1) can be partitioned row-wise in Now, the matrix terms of derivatives of its final row

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The stationary point is locally stable for small step-size choices if and only if the derivative matrix

evaluated at a stationary point , has all its eigenvalues in the right-half plane (e.g., [27], [28]). can now be calculated as The th column of .. .

.. .

.. . To handle the first term, we note that , which leaves us evaluating terms of the form

Now, since

for all

, we have

As such, the first two terms in the expansion of bine, leaving us with

com-

, and therefore

.. . .. .

with and . A verification is esinto planar tablished by exploiting the factorization of rotations (akin to [19]) and is omitted. As such, with , we may write

.. . For a fixed , the mean value of the update term from the algorithm (15) therefore appears as the vector

.. .

A stationary point occurs where this vanishes (i.e., ), corresponding to , with .

.. . This gives

in the form .. .

.. . We observe that is singular, containing in its null , whose columns are orspace. The matrix thogonal to , then effects a congruence transformation to obtain the first term on the right-hand side, the inertia of which , by must agree with that of the remaining eigenvalues of Sylvester’s theorem. These remaining eigenvalues are all positive iff i) is the smallest eigenvalue of and ii) is simple. The second term may be shown strictly upper triangular, upon is unaffected by rotation angle for . noting that state In particular, the trace (or sum of diagonal elements) of the

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second term vanishes, so that the trace of equals that of the first term of the right-hand side. Thus, although this second term influences the eigenvalues of , it affects the real parts by compensating amounts. If this term could be bounded versus the first term, then it may become possible to show that the eigenvalue loci of could be restricted to the right-half plane when . Lacking such a bound, however, we can only emit conjecture 1, which has been verified extensively in simulations. REFERENCES [1] P. P. Vaidyanathan, Multirate Systems and Filter Banks. Englewood Cliffs, NJ: Prentice Hall, 1993. [2] M. K. Tsatsanis and G. B. Giannakis, “Principal component filter banks for optimal multiresolution analysis,” IEEE Trans. Signal Process., vol. 43, no. 8, pp. 1766–1777, Aug. 1995. [3] H. Caglar, Y. Liu, and A. N. Akansu, “Statistically optimized PR-QMF design,” in Proc. SPIE, Visual Communications Image Processing, vol. 1605, Nov. 1991, pp. 86–94. [4] O. Rioul and P. Duhamel, “A Remez exchange algorithm for orthonormal wavelets,” IEEE Trans. Circuits Systems II, Analog. Digit. Signal Process., vol. 41, no. 8, pp. 550–560, Aug. 1994. [5] A. Mertins, “Statistical optimization of PR-QMF filter banks and wavelets,” in Proc. Int. Conf. Acoustics, Speech, Signal Processing, vol. 3, Apr. 1994, pp. 113–117. [6] T. Q. Nguyen, “A quadratic-constrained least-squares approach to the design of digital filter banks,” in Proc. Int. Symp. Circuits Systems, vol. 3, 1992, pp. 1344–1347. [7] P. P. Vaidyanathan, T. Q. Nguyen, Z. Doganata, and T. Saramäki, “Improved technique for design of perfect reconstruction FIR QMF banks with lossless polyphase matrices,” IEEE Trans. Acoust., Speech, Signal Process., vol. 37, no. 7, pp. 1042–1056, Jul. 1989. [8] M. Unser, “On the optimality of ideal filters for pyramid and wavelet signal approximation,” IEEE Trans. Signal Process., vol. 41, no. 12, pp. 3591–3596, Dec. 1993. [9] J.-C. Pesquet and P. L. Combettes, “Wavelet synthesis by alternating projections,” IEEE Trans. Signal Process., vol. 44, no. 3, pp. 728–732, Mar. 1991. [10] P. Delsarte, B. Macq, and D. T. M. Slock, “Signal adapted multiresolution transform for image coding,” IEEE Trans. Inf. Theory, vol. 38, no. 2, pp. 897–904, Mar. 1992. [11] B. Macq and J. Y. Mertes, “Optimization of linear multiresolution transforms for scene adaptive coding,” IEEE Trans. Signal Process., vol. 41, no. 12, pp. 3568–3572, Dec. 1993. [12] M. Tummala and S. R. Parker, “A new efficient adaptive cascade lattice structure,” IEEE Trans. Circuits Syst., vol. 34, pp. 707–711, 1987. [13] O. S. Jahromi, M. A. Masnadi-Shirazi, and M. Fu, “A fast ( ) algorithm for adaptive filter bank design,” in Proc. Int. Conf. Acoust., Speech, Signal Processing (ICASSP), 1988, pp. 1325–1328. [14] V. P. Sathe and P. P. Vaidyanathan, “Effects of multirate systems on the statistical properties of random signals,” IEEE Trans. Signal Process., vol. 41, pp. 131–146, Jan. 1993. [15] P. A. Regalia and P. Loubaton, “Rational subspace estimation using adaptive lossless filters,” IEEE Trans. Signal Process., vol. 40, no. 10, pp. 2392–2405, Oct. 1992. [16] R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge, U.K.: Cambridge Univ. Press, 1985. [17] T. L. Saaty and J. Bram, Nonlinear Mathematics. New York: McGrawHill, 1964. [18] J.-P. Aubin, L’Analyze Non Linéaire et ses Motivations Economiques. Paris, France: Masson, 1984.

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Phillip A. Regalia (S’86–M’88–SM’96–F’00) was born in Walnut Creek, CA. He received the B.Sc. (highest hons.), M.Sc., and Ph.D. degrees in electrical and computer engineering from the University of California at Santa Barbara in 1985, 1987, and 1988, respectively, and the Habilitation à Diriget des Recherches degree from the University of Paris-Orsay, France, in 1994. Currently, he is with the Department of Electrical Engineering and Computer Science of the Catholic University of America, Washington, DC, and an Adjunct Professor with the Institut National des Télécommunications, Evry, France. His research interests include adaptive filtering and wireless communications.

Dong-Yan Huang (M’97–SM’04) was born in Xi’an, Shaanxi, China, in 1963. She received the B.Sc. degree in control and information engineering and the M.Sc. degree in electrical engineering from Xi’an Jiaotong University, Xi’an, China, in 1985 and 1988, respectively, and the Ph.D. degree in signal processing from Conservatoire National des Arts et Métiers (CNAM), Paris, France. In December 1996, she began postdoctoral research work on low-delay high-quality audio and speech codec design at UFR Mathématiques et Informatique, Université René Descartes, Paris V, Paris, France. From December 1997 to December 2002, she was Senior Research Engineer with the Institute of Microelectronics, Singapore. Currently, she is a Research Scientist with the Institute for Infocomm Research, Singapore. Her research interests include lossy/lossless audio and image compression, adaptive filtering. and robust wireless multimedia transmission.