FIXED-INTERVAL smoothers (see [1] [10]) can outperform

Transcription

1 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL 54, NO 3, MARCH Optimal Robust Noncausal Filter Formulations Garry A Einicke, Senior Member, IEEE Abstract The paper describes an optimal minimum-variance noncausal filter or fixed-interval smoother The optimal solution involves a cascade of a Kalman predictor an adjoint Kalman predictor A robust smoother involving H predictors is also described Filter asymptotes are developed for output estimation input estimation problems which yield bounds on the spectrum of the estimation error These bounds lead to a priori estimates for the scalar in the H filter smoother design The results of simulation studies are presented, which demonstrate that optimal, robust, extended Kalman smoothers can provide performance benefits Index Terms H, Kalman filtering, noncausal filtering, robustness, smoothing I INTRODUCTION FIXED-INTERVAL smoothers (see [1] [10]) can outperform filters when processing blocks of noisy measurements Application of the optimal Wiener Kalman solutions requires knowledge of the plant statistics of the noise processes These optimal solutions minimize the 2-norm of the map from the input noises to the estimation error can perform poorly when there are errors or uncertainties in the problem assumptions This has motivated the development of robust estimation techniques (eg, see [8] [19]) In particular, the estimation techniques minimize the -norm of the map from the input noises to the estimation error can accommodate worst-case conditions The optimal noncausal filter or fixed-interval smoother was first described in the frequency domain by Wiener [1], [2] An early time-domain fixed-interval smoothing algorithm using the Kalman framework was developed by Rauch [3] Some further developments of [3] are reported by Mayne in [4] Rauch, Tung Striebel developed a maximum-likelihood smoother [5], which uses a Kalman filter, stores the corrected states, predicted covariances or corrected covariances, employs them within a backward recursion In the Fraser Potter smoother [6], Kalman filters are applied in forward backward directions The smoothed state estimate arises as a linear combination of forward backward state estimates weighted by the inverse of the underlying error covariance matrices Catlin subsequently showed that the same smoother can be derived without assuming that the forward backward errors are independent [7] A continuous-time robust fixed-interval smoother for the case when the noises the induced norm of the plant are Manuscript received October 13, 2004; revised May 9, 2005 The associate editor coordinating the review of this manuscript approving it for publication was Dr Dominic K C Ho The author is with the Division of Exploration Mining, Commonwealth Scientific Industrial Research Organization (CSIRO), Kenmore, QLD 4069, Australia ( garryeinicke@csiroau) Digital Object Identifier /TSP bounded but are otherwise arbitrary, is presented in [8] The smoother arises as a combination of forward states from an filter that involves an algebraic Riccati equation (ARE) adjoint states that evolve according to a Hamiltonian matrix A discrete-time case is described in [9], which employs an Riccati difference equation (RDE) within a filter to calculate forward states that are combined with adjoint states to obtain smoothed estimates The use of both a forward a backward ARE in the solution of a continuous-time robust smoothing problem is detailed in [10] An algorithm for discrete-time fixed-lag smoothing that relies on the solution of an RDE is described in [11] The work of [11] also establishes the tradeoff between the minimum smoothing lag achieving a prescribed performance level A Krein space approach which yields existence conditions for a continuous-time fixed-lag smoother is developed in [12] An alternative to the approach of [9] [10] for accommodating model uncertainties is presented in [13], where a quadratic Lyapunov function for all possible uncertainties is defined a convex optimization problem with linear matrix inequality constraints is solved Another technique for managing model uncertainties within robust designs is through the solution of a regularized weighted recursive least-squares problem, which is detailed in [14] This paper develops smoothing formulations for problems in which uncertainties may be present in the problem specifications In previous work [15], [16], robust techniques have been developed for nonlinear filtering problems Some robustness was achieved in [16] by assuming fixed solutions to the underlying Riccati equations In [15], an extra parameter was introduced, which bounds a model uncertainty A naïve approach is adopted here, where some uncertainty is accommodated implicitly via an appropriate choice of the scalar within the default framework of [17], which evolved from the J-factorization approach of [18] A minimum-variance noncausal filter or fixed-interval smoother is described which derives from classical frequency domain (Wiener) filtering The noncausal filter involves a cascade of a forward Kalman predictor an adjoint Kalman predictor A robust fixed-interval smoother is constructed similarly using predictors However, the efficacy of an approach depends on finding an optimal scalar To this end, filter asymptotes are developed for time-invariant output estimation input estimation problems which yield bounds on the spectrum of the estimation error The derivation of these bounds is also motivated by the remarks within [19] lead to a priori estimates for in the filter smoother designs The remainder of the paper is organized as follows Some notation the realization of discrete-time adjoint systems is described in Section II-A The time-invariant time-varying optimal minimum-variance noncausal filter formulations are de X/$ IEEE

2 1070 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL 54, NO 3, MARCH 2006 veloped in Sections II-B II-C, respectively The optimal solutions rely on the assumption that the underlying processes are Gaussian that the covariances are known Section III introduces a robust smoother formulation that attempts to accommodate uncertain input processes Asymptotes are developed for single-input-single-output (SISO) output estimation input estimation examples It is shown that the maximum magnitudes of the robust solutions approach the asymptotes closer than the optimal minimum-variance solutions Linear nonlinear applications are described in Section IV, where it is demonstrated that optimal, robust extended Kalman smoothers can provide performance benefits Fig 1 General filtering problem Consider a linear system that maps which has the state-space realization II OPTIMAL NONCAUSAL FILTERING A Adjoint Systems In Section II-B, where a minimum-variance noncausal filter is derived, the discrete-time adjoint system is required Let be a linear operator between two Hilbert spaces Suppose that has the state space realization where,,,,,,,,, Let denote the inner product Then the adjoint of is the linear system that maps satisfies [18] Following the continuous-time approach [18] (1) Then it is straightforward to show that for an arbitrary B A Linear Time-Invariant Noncausal Filter Formulation Consider the general filtering problem depicted in Fig 1, where is given by (1) The reference system maps has the state space realization where, A linear system mapping is to be designed that produces estimates of the output of, denoted by, from the measurements (3) (4) (5) where,, is an independent process with Denote the estimation error as define (6) where the adjoint is the linear system The adjoint (2) has the same 2-norm the same -norm as the original system (1) Note that the adjoint system is anticausal, ie, it evolves backward in time Suppose, hypothetically, that is asymptotically stable it is desired to realize The cannot be realized directly because it is unstable By exploiting, the can be realized by operating on the transposed time-reversed input data then taking the transposed time reverse of the result (see [20, Ex 23]) (2) The is known as the map from the inputs to the estimation error By exping completing the squares, it can be shown that, where, (see [19]) Assuming that the inputs are independent, zeromean Gaussian processes, the optimal solution that minimizes the causal part of hence is the Kalman filter which is given by where one-step-ahead predictor gain, (7) is the Kalman

3 EINICKE: OPTIMAL AND ROBUST NONCAUSAL FILTER FORMULATIONS 1071 is the Kalman filter gain, is the solution of the ARE The form (7) follows from the results of [17] specialized for (1), (4), (5), (6) In the case of state estimation no feedthrough terms, the solution (7) reverts to the better known formulation (16) (111) of [21] In linear time-invariant estimation problems, the causal Wiener filter is equivalent to the minimum-variance Kalman filter This motivates the derivation of a minimum-variance noncausal filter Theorem: For the time-invariant filtering problem defined by (1), (4), (5), (6), suppose that the factor satisfies the existence conditions defined in [22], that the inverse exists Then the following applies (i) In the general case, where, the solution (8) (9) (10) difference matrix approach of [22] the Kalman predictor equality of [24] Applying the matrix inversion lemma [25, Lemma 67] results in (8), the adjoint yields (9) (ii) By minmizing, it follows that the optimal smoother for the output estimation problem is defined by The optimal smoother for output estimation (8), (9), (12) involves a cascade of a evolving forward in time followed by a evolving backward in time It can be implemented via the following three-step procedure Step 1) Realize the system (8) operating on the measurements Step 2) In lieu of (9), which is unrealizable, exploit, that is, operate (8) on the time-reversed transpose of the then take the time-reversed transpose of the result Step 3) Apply (12) The Kalman filter (7) can be written as Therefore, the difference between the filter smoother is It can be seen that the benefit of smoothing diminishes when the measurement noise becomes negligible In the case of input estimation or equalization where, the optimal smoother solution is given by This equalizer can be realized similarly by evolving forward in time, followed by a backward evolution of subsequently C A Linear Time-Varying Noncausal Filter Formulation Consider the linear time-varying systems,,, having the state space realizations, the solu- satisfies (ii) In the output estimation case, where tion (8), (9), (11) (12) satisfies Proof: (i) By minimizing, it follows that the optimal smoother for the general problem is defined by It can be shown that comparing the ARE together with the identity (3) leads to A derivation is shown in [21] Alternative derivations follow from the continuous-time case in [23], the return respectively, where,,,, Suppose that exists, where Then is defined similarly where The Kalman filter for time-varying plants is given by (13) where,, is the solution of the RDE

4 1072 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL 54, NO 3, MARCH 2006 The Kalman filter is the optimal causal solution that minimizes, which is equivalent to minimizing the variance of the estimation error Next it is shown that a time-varying noncausal filter can be constructed which also minimizes Corollary: For the above time-varying filtering problem where, suppose that the factor its inverse Fig 2 Noncausal filter for output estimation exist Then the following applies (i) In the general case, where, the solution (14) matrix inversion lemma results in (14) its adjoint yields (15) (ii) The result follows by substituting within The structure of the noncausal filter for output estimation is shown in Fig 2 in which denotes the Kalman predictor Although the plants are time-varying, the factorization is constructed analogously to the time-invariant case It follows from the corollary that when the plant the input processes are locally stationary, ie, the time-varying parameters change sufficiently slowly, then The conditions for stationarity include the monotonicity results for time-varying Riccati equations that are reported in [26] Nonlinear smoothers can be constructed similarly using forward adjoint extended Kalman predictors an example is presented subsequently In some nonlinear applications, such as FM demodulation (see [16] [21]), the error covariance can converge, ie, so, which suggests that the false algebraic Riccati technique of [24], [27] [16] can be readily applied within nonlinear smoothers satisfies (15) (16) (17), the solu- (ii) In the output estimation case, where tion (14), (15) satisfies Proof: (i) The result follows by substituting within Comparing with the identity (3) leads to (18) Applying the III ROBUST NONCAUSAL FILTERING A A Robust Noncausal Filter Formulation Suppose that the inputs are not Gaussian A causal solution designed for worst-case inputs is the filter (see [8] [15], [17] [19]), which minimizes the -norm of the map from the input processes to the estimation error The filter achieves the performance for all, a given In the stationary case, this performance criterion is equivalent to the maximum magnitude of the estimation error power spectral density being bounded above by From the results of [17], the time-varying filter possesses the structure of (7) in which

5 EINICKE: OPTIMAL AND ROBUST NONCAUSAL FILTER FORMULATIONS 1073,, where (19) smoothers can be constructed using the structure of (14) (18), in which,, are substituted with,, respectively Some examples that demonstrate how performance benefits can arise are presented in Sections III-C, III-D, IV B Some General Properties of Filters This section remarks on three properties of filters First, the filter is designed to revert to the Kalman filter at Second, the cost of an solution is an increased upper bound for the state error covariance Third, the solution of the RDE monotonically approaches the solution of the Kalman RDE It follows from the monotonicity property that a designer is required to search iteratively for a that yields a performance benefit; to this end, some a priori guidance is presented in Sections III-C III-D The increased error covariance property indicates that filters can exhibit degraded performance when the problem assumptions are, in fact, correct Conversely, when uncertainty in the problem assumptions is accommodated successfully, robust solutions can be advantageous, which is illustrated in Section IV An equivalent form of the quadratic term of the RDE (19) can be obtained via (241) of [17] as, where (20), shown at the bottom of the page, in which Lemma 1: at Lemma 2: The cost of an solution is an increased upper bound for the state error covariance Lemma 3: As, the solution of the RDE (19) or equivalently (20) monotonically approaches the solution of the Kalman RDE The proofs are described in [28] techniques can also accommodate model uncertainty explicitly as is shown in [15] However, the efficacy of estimation depends on finding an optimal, which is discussed in the subsequent sections C A Stationary Output Estimation Case In the problem (1), (4), (5) (6), consider a time-invariant, SISO, autoregressive (AR) plant having the canonical form, Since the plant is time-invariant, the transfer function exists is denoted by Consider the output estimation case where in (4) Some notation is defined prior to stating some observations for this problem Suppose that an filter has been constructed for the above plant Let the ARE solution, predictor gain, filter gain, predictor, filter smoother transfer function matrices be denoted by,,,,, respectively The filter transfer function matrix may be written as, (20)

6 1074 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL 54, NO 3, MARCH 2006 where The transfer function matrix of the map from the inputs to the filter output estimation error is (21) The smoother transfer function matrix can be written as Similarly, let,,, denote the minimum-variance ARE solution, predictor gain, filter gain, filter smoother transfer function matrices, respectively Proposition 1: In the above output estimation problem: (i) (ii) (iii) (22) (23) achieves the performance, when (from (24)), an a priori design estimate is D A Stationary Input Estimation Case Consider a time-invariant, SISO, AR plant having the transfer function where, are defined in Section III-C, together with Consider an input estimation (or equalization) problem with, within (1), (4), (5) (6) The transfer function matrix of the causal solution that estimates the input of the plant is The transfer function matrix of the map from the inputs to the input estimation error is (27) The noncausal transfer function matrix of the input estimator can be written as Proposition 2: For the above input estimation problem: (i) (28) (iv) (v) (24) (25) (ii) (iii) (29) (26) Outline of Proof: (i) Let denote the (1,1) component of The low measurement noise observation (22) follows from (ii) Observation (23) follows from, which results in (iii) Observation (24) follows immediately from the application of (22) in (21) (iv) Observation (25) follows from (v) Observation (26) follows from which results in An interpretation of (22) (25) is that when, the maximum magnitudes of the filters smoothers asymptotically approach a short circuit (or zero impedance) From (23) (26), as, the maximum magnitudes of the solutions approach the short circuit asymptote closer than the optimal minimum-variance solutions That is, for low measurement noise, the robust solutions accommodate some uncertainty by giving greater weighting to the data Since the filter (iv) (v) (30) (31) (32) Outline of Proof: (i) (iv) The high measurement noise observations (28) (31) follow from (ii) (v) The observations (29) (32) follow from, which results in (iii) The observation (30) follows immediately from the application of (28) in (27) An interpretation of (28) (31) is that when, the maximum magnitudes of the equalizers asymptotically approach an open circuit (or infinite impedance) From (29)

7 EINICKE: OPTIMAL AND ROBUST NONCAUSAL FILTER FORMULATIONS 1075 (32), as, the maximum magnitude of the solutions approaches the open circuit asymptote closer than that of the optimum minimum-variance solutions That is, at high measurement noise, the robust solutions accommodate some uncertainty by giving less weighting to the data Since the solution achieves the performance, when (from (30)), an a priori design estimate is Proposition 1 follows intuitively Indeed, the short circuit asymptote is sometimes referred to as the singular filter Proposition 2 may appear counter-intuitive warrants a further explanation When the plant is minimum phase the measurement noise is negligible, the equalizer inverts the plant Conversely, when the equalization problem is dominated by measurement noise, the solution is a low gain filter; that is, the estimation error is minimized by giving less weighting to the data (a) IV APPLICATIONS A Speech Enhancement In high-measurement noise applications, sampled speech is often modeled as an AR1 process (33) (34) where,, the speech message measurement noise are uncorrelated, zero-mean, Gaussian processes of variances, respectively Suppose that a filter designed for (33) (34) has produced state estimates from measurements Using the approach of [29] assuming that, it is straightforward to show that the maximum-likelihood estimates (MLEs) for the unknown parameters are, A voiced speech utterance a eiou was sampled at 8 khz for the purpose of comparing smoother performance Simulations were conducted with the zero-mean, unity-variance speech sample interpolated to a 16-kHz sample rate, to which 200 realizations of Gaussian noise were added the signal-to-noise ratio (SNR) was varied from 5 to 5 db The were calculated at 20-dB SNR using Kalman filter state estimates within an expectation maximization algorithm The results of simulations that compare the performance of the minimum-variance smoother with the Fraser Potter smoother [6] the maximum likelihood smoother [5] are presented in Fig 3(a) It can be seen that the Fraser Potter smoother exhibits the worst MSE is suboptimal In the case of this example, indistinguishable performance is exhibited by the maximum-likelihood smoother the minimum-variance smoother (14), (15) (18) which achieves when designs require an optimum value for the scalar within (19) If is too small the of (19) is not positive definite, the filter will lack stability Conversely, if is too large, the filter design will be too conservative The a priori initial design (from Proposition 1) permits a one-sided search (b) Fig 3 (a) Fixed-interval smoother performance comparison: (i) minimumvariance smoother (dotted line); (ii) Fraser Potter smoother (dashed line); (iii) maximum-likelihood smoother (crosses) (b) Speech estimate performance comparison: (i) data (crosses); (ii) Kalman filter (dotted line); (iii) H filter (dashed line); (iv) minimum-variance smoother (dotted dashed line); (v) robust smoother (solid line) instead of a two-sided search for an optimal Here an approximate MLE for was calculated online an initial value of was incremented by 10% until the solution of (19) was positive definite for each noise realization The results of simulations are shown in Fig 3(b) It can be seen for this example that the filter outperforms the Kalman filter By appealing to Proposition 1, it is argued that the performance improvement arises as a consequence of giving greater weighting to the data The time-varying minimum-variance smoothers were applied to the above-mentioned noisy speech data the simulation results are indicated by the dotted dashed line the solid line in Fig 3(b), respectively It can be seen that smoothers outperform causal filters The figure indicates that the robust smoother exhibits about 4-dB reduction in MSE compared to the Kalman filter at 0-dB SNR The improvement was observed to be manifested by the estimate trajectories exhibiting fewer errors in the vicinity of the speech message peaks, which is consistent with the stated interpretation of Proposition 1 Lemma 2 states that the cost of the robust filter is a degradation in MSE when the plant is known However, Fig 3(b) demonstrates that combining robust forward backward state predictors within (8), (9), (12) yielded an MSE improvement Therefore, the AR1 plant (33) together with the MLEs

8 1076 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL 54, NO 3, MARCH 2006 of do not adequately model voiced speech Indeed, higher order models are used in speech coding For example, the full-rate GSM speech coder employs an eighth order linear predictive coding analysis [30] Nevertheless, this first order example does demonstrate that trading off optimality is appropriate in the presence of uncertainty B FM Demodulation Consider a nonlinear plant of the form,, in which the nonlinearities, are assumed to be smooth, differentiable functions of appropriate dimensions (see [21]) An extended Kalman smoother for the corresponding output estimation problem can be implemented using the linearizations akin to the extended Kalman filter (EKF) via the following three-step procedure Step 1) In view of (14), calculate,,, where,,, in which Step 2) In lieu of (15), calculate the by carrying out Step 1) operating on the transposed time-reversed data, then taking the transposed time-reverse of the result Step 3) Calculate Consider the problem of demodulating a unity amplitude frequency modulated (FM) signal Let where,, denote the instantaneous frequency, instantaneous phase, complex observations measurement noise respectively Simulations were conducted in which an FM signal was generated using the speech message described in Section IV-A The MLE parameter estimates of Section IV-A were used within an extended Kalman Filter demodulator An FM discriminator [31], ie,, serves as a benchmark as an auxiliary frequency measurement for an extended Kalman smoother The innovations within Steps 1) 2) are given, respectively, by, Fig 4 FM demodulation performance comparison: (i) FM discriminator (crosses); (ii) EKF (dotted line); (iii) Robust EKF (dashed line); (iv) extended Kalman smoother robust extended Kalman smoother (solid line) The SNR was varied in 15-dB steps from 3 to 15 db The MSEs were calculated over 200 realizations of Gaussian measurement noise are shown in Fig 4 It can be seen from the figure, that at 75 db SNR, the EKF improves on the FM discriminator MSE by about 12 db The improvement arises because the EKF demodulator exploits the signal model whereas the FM discriminator does not The figure shows that the extended Kalman smoother further reduces the MSE by about 2 db, which illustrates the advantage of exploiting all the data in the time interval Since the filter achieves the performance, an a priori design estimate is, where is obtained from an EKF demodulator implementation A robust EKF a robust extended Kalman smoother were designed in which an initial value of 05 times the above estimate for the was incremented by 10% until the solution of (19) was positive definite for each noise realization The resulting MSE performance is shown in Fig 4 It can be seen at 75-dB SNR that the robust EKF provides about a 1-dB performance improvement compared to the EKF, whereas the extended Kalman smoother the robust extended Kalman smoother performance are indistinguishable This nonlinear example illustrates once again that smoothers can outperform filters Since a first-order speech model is used the output mapping is linearized, some model uncertainty is present, so the robust design demonstrates a marginal improvement over the EKF A procedure that explicitly accommodates linearization errors within a robust EKF is described in [15] V CONCLUSION The paper introduces an optimum minimum-variance noncausal filter or fixed-interval smoother for discrete-time systems The smoother involves a cascade of a Kalman predictor an adjoint Kalman predictor It is shown that the solution is optimal in the sense that it minimizes the 2-norm of the map from the input processes to the estimation error

9 EINICKE: OPTIMAL AND ROBUST NONCAUSAL FILTER FORMULATIONS 1077 The advantage of a state space formulation is that timevarying, uncertain nonlinear applications can be described In particular, a robust smoother is developed which employs predictors designs require an optimal to be found Therefore, solution asymptotes are developed for stationary output estimation input estimation problems, which yield bounds on the spectrum of the estimation error These bounds lead to a priori estimates for the scalar in the robust filter smoother designs The solution asymptotes also provide an insight into how robust designs achieve performance benefits That is, when uncertainties exist, robust filters pay more attention to the data, whereas robust equalizers pay less attention to the data A speech enhancement example is discussed that demonstrates that combining forward adjoint predictors within a robust smoother can be beneficial An extended Kalman smoother is presented for nonlinear problems an a priori estimate for the within robust designs is described A FM demodulation example is presented which illustrates that optimal robust smoothers can provide performance benefits ACKNOWLEDGMENT The author would like to thank Prof M-A Poubelle for providing helpful advice over the Pacific Dr D Hainsworth for checking the manuscript REFERENCES [1] J S Meditch, A survey of data smoothing for linear nonlinear dynamic systems, Automatica, vol 9, pp , 1973 [2] T Kailath, A view of three decades of linear filtering theory, IEEE Trans Inf Theory, vol 20, no 2, pp , Mar 1974 [3] H E Rauch, Solutions to the linear smoothing problem, IEEE Trans Autom Control, vol 8, no 4, pp , Oct 1963 [4] D Q Mayne, A solution of the smoothing problem for linear dnamic systems, Automatica, vol 4, pp 73 92, 1966 [5] H E Rauch, F Tung, C T Striebel, Maximum likelihood estimates of linear dynamic systems, AIAA J, vol 3, no 8, pp , Aug 1965 [6] D C Fraser J E Potter, The optimum linear smoother as a combination of two optimum linear filters, IEEE Trans Autom Control, vol AC-14, no 4, pp , Aug 1969 [7] D C Catlin, The independence of forward backward estimation errors in the two-filter form of the fixed interval Kalman smoother, IEEE Trans Automat Control, vol 25, no 6, pp , Dec 1980 [8] K M Nagpal P P Khargonekar, Filtering smoothing in an H setting, IEEE Trans Autom Control, vol 36, no 2, pp , Feb 1991 [9] Y Theordor, U Shaked, C E de Souza, A game theory approach to robust discrete-time H estimation, IEEE Trans Signal Process, vol 32, no 6, pp , Jun 1996 [10] S O R Moheimani, A V Savkin, I R Petersen, Robust filtering, prediction, smoothing observability of uncertain systems, IEEE Trans Circuits Syst, vol 45, no 4, pp , Apr 1998 [11] P Bolzern, P Colaneri, G De Nicolao, On discrete-time H fixed-lag smoothing, IEEE Trans Signal Process, vol 52, no 1, pp , Jan 2004 [12] H Zhang, L Xie, W Wang, X Lu, An innovation approach to H fixed-lag smoothing for continuous time-varying systems, IEEE Trans Autom Control, vol 49, no 12, pp , Dec 2004 [13] F Wang V Balakrishnan, Robust-state filtering for systems with deterministic stochastic uncertainties, IEEE Trans Signal Process, vol 51, no 10, pp , Oct 2003 [14] A Subramanian A H Sayed, Regularized robust filters for timevarying uncertain discrete-time systems, IEEE Trans Autom Control, vol 49, no 6, pp , Jun 2004 [15] G A Einicke L B White, Robust extended Kalman filtering, IEEE Trans Signal Process, vol 47, no 9, pp , Sep 1999 [16] G A Einicke, L B White, R R Bitmead, The use of fake Algebraic Riccati equations for co-channel demodulation, IEEE Trans Signal Process, vol 51, no 9, pp , Sep 2003 [17] D J N Limebeer, M Green, D Walker, Discrete-time H control, in Proc 28th Conf Decision Control, Tampa, FL, Dec 1989, pp [18] D J N Limebeer, B D O Anderson, P Khargonekar, M Green, A game theoretic approach to H control for time-varying systems, SIAM J Control Optim, vol 30, no 2, pp , 1992 [19] U Shaked, H - minimum error state estimation of linear stationary processes, IEEE Trans Autom Contr, vol AC-35, no 5, pp , May 1990 [20] C S Burrus, J H McClellan, A V Oppenheim, T W Parks, R W Schafer, H W Schuessler, Computer-Based Exercises for Signal Processing Using Matlab Englewood Cliffs, NJ: Prentice-Hall, 1994, pp [21] B D O Anderson J B Moore, Optimal Filtering Englewood Cliffs, NJ: Prentice-Hall, 1979 [22] U Shaked, A transfer function approach to the linear discrete stationary filtering the steady-state discrete optimal control problems, Int J Control, vol 29, pp , 1979 [23] A P Sage J L Melsa, Estimation Theory With Applications to Communications Control New York: McGraw-Hill, 1971, p 319 [24] R R Bitmead, M Gevers, V Wertz, Adaptive Optimal Control Sydney, Australia: Prentice-Hall, 1990, pp [25] T Söderström, Discrete-Time Stochastic Systems London, UK: Springer-Verlag, 2002, p 165 [26] G Freiling V Ionescu, Time-varying discrete Riccati equation: Some monotonicity results, Linear Algebra Its Appl, no 286, pp , 1999 [27] M-A Poubelle, I R Petersen, M R Gevers, R R Bitmead, A miscellany of results on an equation of Count J F Riccati, IEEE Trans Autom Control, vol 31, no 7, pp , Jul 1986 [28] G A Einicke, Robust filtering, PhD thesis, Dept of Electrical Electronic Engineering, Univ of Adelaide, Adelaide, Australia, Jun 1995 [29] S M Kay, Fundamentals of Statistical Signal Processing: Estimation Theory Englewood Cliffs, NJ: Prentice-Hall, 1993, ch 7, pp [30] L Besacier, S Grassi, A Dufaux, M Ansorge, F Pellini, GSM speech coding speaker recognition, in Proc Int Conf Acoustics, Speech, Signal Processing, vol 2, Jun 2000, pp [31] J Aisbett, Automatic modulation recognition using time domain parameters, Signal Process, vol 13, pp , 1987 Garry A Einicke (S 90 M 93 SM 05) received the Bachelor of Engineering, Master s of Engineering, PhD degrees from the University of Adelaide, South Australia, in 1979, 1991, 1996, respectively From 1980 to 1996, he was a Senior Research Scientist in the Signal Analysis Discipline with the Defence Science Technology Organization Since 1997, he has been with the Commonwealth Scientific Industrial Research Organization (CSIRO), Kenmore, Queensl, Australia, where he is a Principle Research Scientist leads mine communication research He also has been an Adjunct Associate Professor in the School of Information Technology Electrical Engineering at The University of Queensl