Dual H∞ Algorithms for Signal Processing— Application to Speech Enhancement

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 11, NOVEMBER 2007 5195

Dual H Algorithms for Signal Processing—Application to Speech Enhancement

David Labarre, Eric Grivel, Mohamed Najim, Fellow, IEEE, and Nicolai Christov

Abstract—This paper deals with the joint signal and parameterestimation for linear state–space models. An efficient solution tothis problem can be obtained by using a recursive instrumentalvariable technique based on two dual Kalman filters. In that case,the driving process and the observation noise in the state–spacerepresentation for each filter must be white with known variances.These conditions, however, are too strong to be always satisfied inreal cases. To relax them, we propose a new approach based on twodual filters. Once a new observation of the disturbed signal isavailable, the first algorithm uses the latest estimated parame-ters to estimate the signal, while the second algorithm uses theestimated signal to update the parameters. In addition, as thefilter behavior depends on the choice of various weights, we presenta way to recursively tune them. This approach is then studied inthe following cases: 1) consistent estimation of the AR parametersfrom noisy observations and 2) speech enhancement, where no apriori model of the additive noise is required for the proposed ap-proach. In each case, a comparative study with existing methods iscarried out to analyze the relevance of our solution.

Index Terms—Autoregressive (AR) modeling, filtering,Kalman filtering, speech enhancement.

I. INTRODUCTION

I N the framework of signal processing, parametric methodsbased on the autoregressive (AR) model are used in a wide

range of applications such as speech coding [4] and enhance-ment [13], [16], channel modeling in wireless communications[2], or biomedical signal spectral analysis [20]. In that case, thesignal is modeled by the following th order AR process:

(1)

where denote the AR parameters and the drivingprocess, assumed to be zero mean.

Manuscript received July 17, 2006; revised March 13, 2007. The associateeditor coordinating the review of this manuscript and approving it for publica-tion was Dr. Ilya Pollak.

D. Labarre was with the Equipe Signal et Image, LAPS, UMR 5218 IMS, Uni-versity of Bordeaux, 33402 Talence Cedex, France. He is now with the EquipeNoyaux Gris Centraux, Dynamiques Normales et Pathologiques, UMR 5227MAC, Université Victor Segalen Bordeaux 2, 33076 Bordeaux Cedex, France(e-mail: [email protected]).

E. Grivel is with the Equipe Signal et Image, LAPS, UMR 5218 IMS, Univer-sity of Bordeaux, 33402 Talence Cedex, France (e-mail: [email protected]).

M. Najim is with the Equipe Signal et Image, LAPS, UMR 5218 IMS, Univer-sity of Bordeaux, 33402 Talence Cedex, France (e-mail: [email protected]).

N. Christov is with the Laboratoire d’Automatique, Génie Informatique etSignal, Université des Sciences et Technologies de Lille, 59655 Villeneuved’Ascq, France (e-mail: [email protected]).

Digital Object Identifier 10.1109/TSP.2007.899587

To be carried out, a parametric approach usually requires apreliminary model parameter estimation step. Thus, whenis white, the minimum mean square error (MMSE) criterionis considered to estimate the AR parameters and leads to theso-called Yule–Walker (YW) equations. Furthermore, when

is assumed to be a Gaussian process with variance ,the MMSE estimators provide the maximum likelihood (ML)estimates of the AR parameters.

However, in real cases, two constraints have to be considered.First, few samples (e.g., a few hundreds) are available. Second,the sequence may be disturbed by an additive zero-meannoise , yielding the following observations:

(2)

It should be noted that, given (1) and (2), the observa-tion vector satisfies the following relation:

......

......

...

(3)

where and .In that case, since the process is not white, the MMSE

methods provide biased estimations of the AR parametervector . Therefore, various approaches have been proposed toovercome the bias problem when the driving process andthe additive noise are two zero-mean, white, and Gaussianprocesses, respectively, with variances and . The offlinemethods are based, for instance, on an iterative bias correc-tion scheme [45], a generalized eigenvalue decomposition[5], or a Newton–Raphson-algorithm-based technique [21].When dealing with online approaches, the least mean square(LMS) algorithm can be modified to counteract the effectsof the additive noise and various algorithms such as theso-called -LMS [38], -LMS [42], and -LMS [44] can beused. Although these methods provide significant results whenthousands of observations are used, they are no longer reliablewhen only a few hundreds of samples are available.

An alternative consists in estimating both the noise-free ARprocess and its corresponding parameters. Instead of using anapproach dedicated to nonlinear state–space models, such asthe extended Kalman filter (EKF), the unscented Kalman filter(UKF) [14], and particle filtering [39], other solutions can bedeveloped.

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5196 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 11, NOVEMBER 2007

On the one hand, the expectation–maximization (EM) algo-rithm, initially proposed by Dempster in [6], aims at computing

by iterating over two steps [7]. In the E-step, the conditionalexpectation of the likelihood function is evaluated given the cur-rent values of the AR parameters and the variance of the additivenoise. This can be done by carrying out a Kalman smoothing.Then, the M-step provides the ML estimate of the AR param-eters given the noisy observations . However, the EM algo-rithm [6] guarantees the convergence to a local maximum. Itssuccess depends on the initial conditions.

On the other hand, recursive filters can be cross coupled tosolve the so-called dual estimation issue [1]. Each time a newobservation is available, the first filter uses the latest estimatedAR parameters to estimate the signal, while the second filteruses the estimated signal to update the AR parameters. Iter-ating between signal estimation and parameter estimation ona sample-by-sample basis often has the advantage of getting acomputational cost lower than a standard EM algorithm [6]. Itshould be noted that some authors view this solution as a sequen-tial version of the EM algorithm [14]. In [27], a variant basedon two interacting Kalman filters has been developed in whichthe variance of the innovation process in the first filter is used todefine the gain of the second filter. As this solution can be seenas a recursive instrumental variable (IV) technique,1 consistentestimates of the AR parameters are obtained.

In the aforementioned methods, the Kalman filter is optimaland gives the ML estimates of the state vector if and only ifthe driving process and the observation noise in the state–spacerepresentation are white and Gaussian, with exactly known vari-ances. However, in real cases, even if one attempts satisfyingthese assumptions, two main problems remain. First, sinceand are usually colored, they are also to be modeled, whichcontributes to modeling uncertainties. Second, the variancesand the model parameters have to be estimated from a reducednumber of observations, which introduces estimation errors.

To relax the previous assumptions, we propose an alternativeapproach based on estimation techniques. Indeed, al-gorithms are appealing because no statistical assumptions haveto be made on the driving process and the observation noise. Inaddition, these methods aim at minimizing the worst possibleeffects of the disturbances on the estimation error and are morerobust against the estimation errors and model uncertainties thanKalman filtering, according to [23] and [36].

Whereas the theory has been widely applied in the fieldof control, its use in signal processing is subject to more andmore attention. Thus, estimation techniques have beenstudied in the fields of active noise cancellation [32], [33],signal reconstruction by means of multirate filter banks [41],equalization [10], [24], [46], multiuser detector [40], and jointestimation of channel and symbols [25].

When dealing with AR modeling in a noisy environment,Shimizu et al. [37] and Shen et al. [35], [36] have both de-veloped -filter-based approaches for noise reduction. The

1The IV technique makes it possible to provide consistent estimate of theAR parameters from noisy observations. The principle is to design the so-calledinstrumental matrix Z , which satisfies

EfZ � g = 0; for any k and l

where the vector � is introduced in (3).

Fig. 1. Principle of the proposed scheme combining H smoothing and Hfiltering.

methods have been derived in the fields of orthogonal frequencydivision multiplexing (OFDM)-based wireless communications[3] and speech enhancement [35], [36] where the AR process,respectively, models the channel and the speech signal. Theydiffer in the way the AR parameters are estimated. Thus,Shimizu et al. [37] suggest using a total least square algorithmto estimate the AR parameters. Shen et al. carry out two con-secutive filters: The first one provides the AR parameterestimates; then, the second one uses the AR parameter esti-mates to retrieve the AR process from the noisy observations.Both filters are successively and directly applied on a blockof noisy observations. However, the resulting AR parameterestimates are biased [26]. Therefore, one could iterate betweensignal estimation and AR parameter estimation, as this is donein the simplified Kalman-filter-based EM algorithm2 [16].Nevertheless, when the number of noisy observations is large,this method is less attractive because its computational cost ishigh.3

In this paper, we propose a new approach that can be appliedin a wide range of applications, from biomedical signal analysisto speech enhancement. Our purpose is to find a better com-promise between computational cost and parameter estimationerror. Therefore, we propose an online approach for the joint es-timation of the signal and its AR parameters based on dualestimators. See Fig. 1. More particularly, an smoothing iscarried out to estimate the signal using the latest AR parame-ters, while the signal estimate feeds an filter to update theAR parameter estimates. In addition, as the filter behaviordepends on how some weights are tuned, we present a way torecursively adjust them.

The remainder of this paper is organized as follows. InSection II, we recall the filtering approach. In Section III,we present the proposed method. In Section IV, we point outthe relevance of our approach for the AR parameter estimation

2It should be noted that such a procedure is not clearly mentioned in [3] and[36].

3The computational cost approximately corresponds to the one of two H

filters, multiplied by the number of iterations (e.g., 2 to 3).

LABARRE et al.: DUAL ALGORITHMS FOR SIGNAL PROCESSING—APPLICATION TO SPEECH ENHANCEMENT 5197

Fig. 2. Transfer operator from disturbances to estimation error considered forH -norm-based estimation.

from noisy observations. In Section V, we analyze the rele-vance of the proposed parametric approach in the framework ofspeech enhancement.

II. FILTERING FOR SIGNAL PROCESSING

Before introducing the filtering, we recall the normof a transfer operator that maps the discrete-time input tothe discrete-time output

(4)

where denotes the norm of the causal sequence de-fined as follows:

(5)

Therefore, the norm can be viewed as the maximum en-ergy gain from the input to the output .

Let us define the following state–space representation of thesystem:

(6)

where denotes the state vector at time , the transitionmatrix, the input vector, the observation vector, and avector which relates the state vector to the quantity to be es-timated. Furthermore, and are, respectively, the so-calleddriving sequence and the observation noise.

filtering provides the estimation of based on theobservations by minimizing the worst possible ef-fects of the unknown sequences and on the estimationerror. See Fig. 2. The estimation is obtained by minimizingthe following criterion:

(7)

where stands for the norm of the systems (1) and (2)and denotes the number of the noisy observations available.In addition, and are two weighting positive scalars to betuned, in order to adjust the filter behavior. More particularly,Shen et al. underline in [36] that and , respectively, play

the same role as the variances of the excitation process and theobservation noise when using the Kalman filter.

Thus, as mentioned by Grimble et al. in [18], the optimalestimation problem consists in designing an estimator that min-imizes the peak error power in the frequency domain, whereasKalman filtering aims at minimizing the average error power.This hence prevents small disturbances such as additive noisefrom leading to large estimation errors [22].

However, a closed-form solution of the previous filteringissue does not always exist [23]. Therefore, one commonly con-siders the following “suboptimal” criterion:

(8)

where is a prescribed noise attenuation level.It should be noted that the estimation problem can be

formulated from a game theory point of view [43]. Indeed, aplayer, say the statistician, aims at minimizing the estimationerror while his adversary, say nature, disturbs the system to max-imize it. This hence leads to the so-called minimax problem

(9)where the scalar weights in that case the contributions of thedriving sequence and the observation noise.

Many methods have been proposed to obtain a filter whichsatisfies condition (8). More particularly, in addition to fre-quency-domain solutions such as the polynomial approaches[18], a great deal of attention has been paid to the state–spaceapproaches. They are based either on the resolution of a convexoptimization problem under linear matrix inequality (LMI)constraints [15] or on an algebraic Riccati equation (ARE) [34].

In this paper, we focus our attention on the ARE-based so-lution because they lead to a simple recursive implementationof the filter. Moreover, it allows the differences between theKalman filter and the filter to be highlighted.

In that case, the estimation fulfilling the condition (8) ex-ists if and only if

(10)

where the matrix satisfies a Riccati recursion, which canbe split into the two following relations:

(11a)

(11b)

with

(12)

According to Yaesh et al. [43], the matrix corre-sponds to an upper bound of the error covariance matrix whenusing Kalman filter.


When the condition (10) is satisfied, the a posteriori estima-tions of both the state vector and are updated as follows:

(13a)

(13b)

(13c)

where denotes the estimation of the state vector given

observations, is thefilter gain and is the so-called innovationsequence.

Equations (11)–(13) raise the two following comments.1) When considering the same state–space representation (6),

the computational cost of the filter (11)–(13) is some-what higher than the Kalman filter due to relations (11b)and (12).

2) Although ARE (11) reduces to the conventionalKalman ARE when , a solution to the problem (8)is not always guaranteed since the ARE has indefinitequadratic terms. This hence leads to condition (10).

In the next section, we present the proposed approach.

III. PROPOSED METHOD

A. Fixed-Lag Smoothing to Estimate the Signal

When the observations are available, anfixed-lag smoothing of the signal consists in estimating thequantity , where denotes the lag. In that case,the following state vector is chosen:

(14)

and the vector is defined by

(15)

Given the systems (1) and (2) and the state vector (14), thematrices of the state–space representation (6) are defined asfollows:

. . .. . .

.... . .

. . ....

. . .

.... . .

.... . .

. . ....

(16)

and

(17)

At time , the a posteriori filtered version of the signal isgiven by

(18)

while its smoothed estimate at time satisfies

(19)

Since the estimations of the AR parameters are requiredin (16), a second algorithm, say a filter, is introduced toestimate from .

B. Second Filter to Estimate the AR Parameters

First, let the signal be assumed quasi stationary. In that case,the AR parameters are constant

(20)

Second, let us express as a function of the AR parame-ters at time . When observations are available, combining(13b), (13c), and (19) leads to

(21)

where denotes the component of the gainat time .

Furthermore, for varying from 1 to , when the noisyobservations are available, one has

(22)

Thus, by combining (21) and (22), one can express asfollows:

(23)

In addition, according to (13b), (13c), and (18), the estimationof the AR process at time satisfies

(24)

where corresponds to the first components of.

Therefore, by substituting (24) into (23), one obtains

(25)

with

(26)

and

(27)


The set of (20)–(25) defines the state–space representationused to estimate the AR parameters by means of an filter.It should be noted that, due to (20), this second estimator onlyrequires the additional weight .

Thus, given the two following noise attenuation levels and, both estimators can be carried out. In Section III-C,

selecting the noise attenuation levels and and tuning theweights , , and are addressed.

C. Tuning the Noise Attenuation Level and the Weights

In the framework of signal processing, few papers explicitlydeal with the selection made on the noise attenuation level. Asan alternative to an a priori fixed value, an adaptive tuning isconsidered in some specific cases. Indeed, in [36] and [46], theauthors propose to take advantage of condition (10). Given (10)and since the matrix is the identity matrix when dealingwith the AR parameter estimation, it can be easily shown thatthe noise attenuation level must satisfy the following:

(28)

where denotes the maximum eigenvalue of thematrix .

At that stage, Shen et al. [36] or Zhuang et al. [46] proposeto update the noise attenuation level as follows:

(29)

where is a scalar larger than one.Instead of tuning the weights , , and manually, an au-

tomatic and heuristic approach is here presented. Due to the in-terpretation of and given by Shen et al. [36], we proposeto adjust the weights by taking into account a similar argumentas the one we propose in [27] for the Kalman filter case.

First, when assuming that the characteristics of the additivemeasurement noise vary slowly with time, can be tuned orupdated during nonsignal periods.

Second, we exploit the Kalman-filtering-based results in [27]to define . Indeed, using the Riccati recursion (11a) and (11b),

can be tuned by extracting the information relative to the in-novation sequence at each step of the algorithm, leading tothe following recursion [27]:

(30)

where and.

Third, according to the Kalman filter theory, the in-novation is a white random process with variance

[1]. Thus, given (27), the weightcan be tuned as follows:

(31)In Section IV, we illustrate the performances of the proposed

approach for AR parameter estimation. The equations of thealgorithm are summarized in Fig. 3.

Fig. 3. Equations of the proposed algorithm.

IV. COMPARATIVE STUDIES ON AR PARAMETER ESTIMATION

BASED ON TRUE AR DATA

A. Estimation of AR Parameters From Data Disturbed by anAdditive White Noise

A comparative study between existing techniques is carriedout on an AR process disturbed by an additive zero-mean whiteGaussian noise with variance . The seven following methodsare tested.

1) Standard Kalman filter directly applied to the noisy data.2) Standard filter directly applied to the noisy data, as

suggested by Shen et al. in [36] with either the adaptiveselection (29) or an a priori fixed value of the noise atten-uation level .

3) Davila’s algorithm [5]: This method is based on the reso-lution of the so-called noise-compensated YW equations,which are nonlinear in and in . It consists in viewingthe estimation issue as a quadratic eigenvalue problem.


TABLE IAVERAGE AR PARAMETER ESTIMATION FOR EXPERIMENT 1, N = 2000

4) The -LMS filter [42]: This method consists in usingthe LMS filter on enhanced observations, where the ARparameter vector depends on the previous AR parametervector multiplied by a value defined from the varianceof the noise and the LMS gain.

5) EM algorithm based on Kalman smoothing [6], [7].6) Dual-Kalman-filtering-based IV method [27], denoted

dual Kalman.7) Proposed method, denoted dual : For the sake of sim-

plicity, a priori fixed values for the noise attenuation levelsand are used.

A sixth-order AR process is first generated according to thepoles4 , , and

and with variance . Then, 100 realizationsof a zero-mean white Gaussian noise are used to disturb thesamples of the signal. The signal-to-noise ratio (SNR), definedby SNR , is equal to 10 dB.Two experiments are considered.

Experiment 1: To analyze the consistency of the AR param-eter estimates, samples of the noisy AR process areassumed to be available.

Experiment 2: Only samples are available.First, let us have a look on the use of Kalman and filters

directly applied to noisy observations. In both experiments, theylead to biased estimates of the AR parameters. The simulationresults also confirm that used in the filtering plays thesame role as in the Kalman filtering. In addition, when anadaptive tuning of the noise attenuation level is considered,two drawbacks can be pointed out as follows.

1) Due to the eigenvalue decomposition carried out at eachtime in (29), the computational cost is significantlyhigher than an a priori fixed value for the noise attenua-tion level .

2) The choice of in (29) must satisfy a compromise be-tween the estimation error variance and the estimation errorbias. Indeed, let us have a look on the Itakura distance de-fined as follows:

(32)

where denotes the Toeplitz autocorrelation matrixof the clean signal .

4The transfer function H(z) corresponding to the filter associated to themodel given in (1) is defined as follows: H(z) = 1=(1 + a z ) =1=( (1� p z )) where fp g are the so-called poles.

Fig. 4. Experiment 1. (a) Itakura distance averaged over 100 realizations and(b) its standard deviation as a function of the parameter � , N = 2000, andSNR = 10 dB.

When carrying out Experiment 1, the direct use of thefilter may produce estimated poles outside the unit circle in the

-plane if is too low (equal to 2, for instance). A minimumvalue of the Itakura distance mean is obtained when is ap-proximately equal to 6. However, the corresponding standarddeviation is too high. Then, when selecting a higher value for

, the Itakura distance mean increases and converges towardsthe value obtained with the Kalman filter (see Fig. 4).

When the number of samples available is large (e.g., a fewthousands), methods 3–7 provide significant results. The simu-lation results also confirm that the way is defined in (31) isrelevant. See Table I.

When the number of observations is lower (e.g., a few hun-dreds), the offline methods may lead to estimated poles outsidethe unit circle. Davila’s method even provides a negative vari-ance for the driving process in 20 realizations among 100. Itshould be noted that these realizations are not taken into ac-count in the results presented in Fig. 5 and Table II. Moreover,the -LMS algorithm would require at least a few thousands ofobservations to converge. The EM algorithm provides estimated


Fig. 5. Experiment 2: pole localization in the z-plane and the corresponding spectra for various approaches. The 100 realizations are plotted for each method:N = 300 and SNR = 10 dB.

AR poles close to the true ones, providing several iterations(at least 5) are completed. This, however, leads to the highestcomputational cost. The dual Kalman filtering and the proposedmethod make it possible to determine the main frequency reso-nances in the signal spectrum.

B. Estimation of AR Parameters From Data Disturbed bya Colored Noise

In this section, our purpose is to analyze the limitationsof -filtering-based approaches when the AR process is


Fig. 6. Experiments 3: pole and zero localization in the z-plane and the corresponding spectra for various approaches. The 100 realizations are plotted for eachmethod: N = 512 and SNR = 5 dB.


Fig. 7. Temporal evolutions and spectrograms of the speech signal “Le tribunal doit bientôt rendre son jugement” for (a) clean speech, (b) noisy speech, and(c) enhanced speech with the proposed method—SNR = 10 dB, white noise case.

disturbed by additive colored noise. Thus, we carry out a com-parative study between the four following methods.

1) Dual-Kalman-filtering-based IV method [27], denoteddual Kalman. A model is necessary for the additive noise.

2) Standard filtering directly applied on the noisy data.No a priori model is used for the additive noise.

3) EM algorithm based on Kalman smoothing proposed byGannot et al. [13]. A model is used for the additive noise.

4) Proposed method, denoted dual . In that case, no apriori model is used for the additive noise.

The following experiment is considered.

Experiment 3: A narrowband second-order AR process of512 samples is first generated according to the poles

and with variance . Then, 100 realizationsof a fourth-order MA process whose zeros areand are used to disturb the signal. The SNR isequal to 5 dB. It should be noted that for the Kalman-filter-basedmethods, a fourth MA process is used to model the additivecolored noise.

For both experiments, using a standard filter directly onthe noisy observations leads to biased AR parameter estimates.According to the results shown in Table III and Fig. 6, the


TABLE IIAVERAGE AR PARAMETER ESTIMATION FOR EXPERIMENT 2, N = 300

TABLE IIIAVERAGE AR PARAMETER ESTIMATION FOR EXPERIMENT 3

dual Kalman method underestimates the variance of the drivingprocess and provides resonances in the signal spectrum that aresharper than expected. When using the EM algorithm, we havenoticed that the results mostly depend on the initialization step,and more particularly, the way the additive noise parametersare initialized. Furthermore, this approach requires once againseveral iterations to converge. The dual method providesthe more accurate results. It is still attractive because it corre-sponds to a good compromise between performances and com-putational cost. Furthermore, it has the advantage of avoidingthe use of any a priori model for the additive noise. Given theresults of this comparative study, we propose to derive the pro-posed method in the framework of speech enhancement.

V. APPLICATION TO SPEECH ENHANCEMENT

A. State of the Art

In the framework of a single-microphone-based speech en-hancement, parametric approaches based on an a priori modelfor the speech can be developed in addition to short-time spec-tral attenuation techniques [9]. Thus, when the signal is mod-eled as a sum of complex exponentials, subspace methods en-able the speech to be estimated from the noisy observationsin the MMSE or minimum variance sense. This approach isusually based on the eigen (respectively, singular) value de-composition of the observation autocorrelation matrix (respec-tively, the observation Hankel matrix). See [8], for an exhaustivepresentation.

When an AR process is used to model the speech, Kalman fil-tering is usually carried out. In Paliwal et al.’s pioneering work[31], the AR parameters and the variances and are, re-spectively, estimated from the clean speech and the noise se-quence, both assumed to be available. However, this approachcannot be applied in practice. Then, various methods have beenproposed. Among them, Gibson et al. [16] suggest filtering the

noisy speech iteratively with a Kalman filter. At each itera-tion, the AR parameter estimates are updated from the enhancedspeech. In addition, in this method, a voice activity detector[4] is required to detect the silent frames for the noise varianceestimation.

To avoid it, Gannot et al. [13] propose an EM approach toestimate the signal, the AR parameters, and the noise variance.In [19], subspace methods for identification makes it possible toview speech enhancement as a realization issue, i.e., to estimatethe system matrices in the state–space representation, includingthe estimation of the additive noise variance. An alternative so-lution consists in avoiding the need to explicitly know both theadditive noise and the driving process variances [12] by takingadvantage of Mehra’s work [30].

It should be noted that even if the existing Kalman-filter-based methods provide significant SNR improvements in av-erage, the authors in [19] and [31] carry out a Kalman smoothingto weaken the residual noise.

However, the additive observation noise is often colored andhas to be modeled either by an AR process [13], [16], [19] or bya moving average (MA) process [26]. This implies the estima-tion of the parameters of the noise during the silent frames andleads to a significantly higher computational cost.

Therefore, we propose to consider the speech enhancementissue in an setting. In Section V-B, we carry out a compara-tive study between Kalman-filter- and -filter-based methodswhen the additive noise is white or colored.

B. Simulation Protocol

The performances of the four following methods arecompared:

1) Shen’s method [36];2) Gannot’s algorithm [13];3) dual-Kalman-filter-based approach [27];4) method proposed in this paper.


TABLE IVSPEECH ENHANCEMENT COMPARATIVE STUDY: SNR IMPROVEMENTS IN DECIBELS FOR AN INPUT SNR OF 15 dB

TABLE VSPEECH ENHANCEMENT COMPARATIVE STUDY: SNR IMPROVEMENTS IN DECIBELS FOR AN INPUT SNR OF 10 dB

We use a set of six sentences sampled at 8 kHz,5 composed ofevery French language phoneme. The order of the AR modelis set to 10. Every method is based on a 256 sample frame-by-frame approach and a 50% add–overlap procedure to reconstructthe speech signal. Two cases are investigated. First, the speechsignal is disturbed by a white Gaussian noise (denoted WGN);second, corresponds to a real noise recorded in a car (denotedREAL). In that case, no parametric model is used for the additivenoise when dealing with -filter-based approaches, whereasthe Kalman-filter-based approaches are based an a priori sixth-order MA model for the noise. In addition, the MA parametersare estimated during the silent frames.

5“Le tribunal doit bientôt rendre son jugement,” which means “The court willsoon send its verdict.”

“Mes yeux s’accoutument lentement à la pénombre,” which means “My eyesslowly get accustomed to the darkness.”

“Quatorze éditeurs lui renvoyèrent son manuscrit,” which means “14 editorssent back his manuscript.”

“Le musée ouvrira ses portes en décembre,” which means “The museum’sgrand opening is in December.”

“Marion s’est accroupie près de moi,” which means “Marion sat close to me.”“Il n’aimait pas qu’on le regarde travailler,” which means “He did not like to

be seen when he worked.”

C. Results and Comments

First, the proposed approach provides similar results to thedual Kalman filtering, but its computational cost is lower sinceno model for the additive noise is required.

Second, in the colored noise case, Gannot’s method requiresa parametric model for the additive noise. Initializing the noiseparameters is a key step. Moreover, at least five iterations arerequired for the EM algorithm.

Although Gannot’s method provides the most significant re-sults, its computational cost is the highest among the testedapproaches.

Third, when using Shen’s method [36], a broadband residualnoise disturbs the enhanced speech in addition to a slight mu-sical noise.6 Thanks to our method, the broadband residual noiseis weakened. See Fig. 7. Such a phenomenon is confirmed by in-formal subjective tests (IST). This is mainly due to the fact thatthe AR parameters are no longer estimated from the noisy obser-vations, but from a smoothed version of the observations.

6The musical noise corresponds to short sinusoids or tones randomly dis-tributed over time and frequency.


TABLE VISPEECH ENHANCEMENT COMPARATIVE STUDY: SNR IMPROVEMENTS IN DECIBELS FOR AN INPUT SNR OF 5 dB

More particularly, when looking at the enhanced speechobtained with our method, the quality in terms of SNR improve-ment and subjective listening test increases with the lag . InTables IV–VI, a tradeoff between computational cost and SNRimprovement is achieved by considering .

Remark 1: We have also investigated the relevance of variousperceptual postfiltering based on masking properties of audi-tory systems [28], [29], initially developed in the frameworkof Kalman-filtering-based speech enhancement. This kind ofapproach operates as follows. First, the frequency maskingthreshold is computed from the signal . Second, whenthe components of the power spectral density of arehigher (respectively, lower) than the masking threshold, thecorresponding spectrum components are weighted by a factorwhich depends on the threshold (respectively, a constant usuallyequal to 0.8). According to our listening tests, the techniquepresented in [29] can slightly improve the sound quality,whereas the algorithm proposed in [28] introduces too manydistortions.

Remark 2: As suggested by Goh et al. in [17], the drivingprocess could be modeled by the so-called long-delay predictor[4], to capture the harmonic details in the spectrum of thespeech, especially when processing a “voiced” sound such asthe vowels “a” and “ e,” which are quasi periodic. However, avoiced/unvoiced (U/UV) decision should be made and the pitchshould be estimated from noisy observations. The resultingapproach would have, therefore, a much heavier computationalcost.

VI. CONCLUSION

To solve the joint signal and parameter estimation for linearstate–space models, a sample-by-sample dual scheme using

estimators is presented. Moreover, we propose to recur-sively tune the weights and .

This method has the following properties.1) Whereas Kalman filtering minimizes the mean square error

of the state vector estimation, techniques aim at mini-mizing the worst possible effects of the disturbances, suchas the additive noise, on the estimation error. This has the

advantage of relaxing the assumptions on the noise se-quences. No a priori modeling for the additive noise isrequired.

2) Iterating between signal estimation and parameter estima-tion on a sample-by-sample basis makes it possible to finda good compromise between computational cost and per-formances. This is particularly of interest when the numberof noisy observations is large.

Our comparative study confirms the efficiency of our ap-proach to estimate AR parameters from noisy observations,even when a few hundred observations are available.

In the framework of speech enhancement, the method wepresent makes it possible to improve the existing -filter-based methods in terms of SNR improvement and subjectivelistening tests.

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David Labarre received the diploma of engineer inelectronics and the Ph.D. degree in signal processingfrom the University of Bordeaux, Talence, France, in2001 and 2005, respectively.

His research interest deals with the designof model-based statistical approaches for signalprocessing with main applications in biomedicalengineering and speech processing.

Eric Grivel received the diploma of engineerin electronics in 1996 and the Ph.D. degree insignal processing from the University of Bordeaux,Talence, France, in 1996 and 2000, respectively.

Currently, he is an Associate Professor at theTelecommunications Department, ENSEIRB, agraduate national engineering school in Bordeaux,France, and belongs to the signal and image researchgroup. His research interest includes the design ofparametric approaches for signal processing withapplications in mobile communication and speech

processing.


Mohamed Najim (M’74–SM’83–F’89) received theDr.Sci. degree (Doctorat d’Etat) from the Universityof Toulouse, Toulouse, France, in 1972.

In 1972, he joined the University of Rabat, Mo-rocco, as an Associate Professor, where he becamea Professor in 1974. Since 1988, he has been theProfessor with the ENSEIRB, University of Bor-deaux, Talence, France, where he founded the Signaland Image Processing Laboratory. He has managedvarious projects with industrial partners (includingTexas Instruments, Digital Equipment, ST Micro-

electronics). Currently, he is the Head of the LASIS a joint TOTAL/CNRSlaboratory. He supervised more than 50 Ph.D. theses and published over 220scientific papers and coauthored several books, including Parametric modelingin Image Processing (Paris, France: Masson, 1994). He is the author of thebook Modeling and Identification in Signal Processing (Paris, France: Masson,1988). He has worked in various fields, including microwaves, modelingand identification, adaptive filtering including H , and control. His currentresearch interests are modeling and identification in multidimensional signaland image processing with applications in speech, seismic, biomedical, radarsignal processing, textures, and image enhancement. He set up the “CADSoftware Library” group within the GDR-CNRS TdSI, which is a Frenchresearch program on signal and image processing.

Dr. Najim organized 15 international conferences on control, signal pro-cessing, and friendly exchange through the Internet. He co-organized withProf. Kailath (Stanford University, Stanford, VA) and Prof. Dewilde (DelftUniversity, Delft, The Netherlands) two workshops, respectively, dedicatedto math and systems (Saint Emilion, France, 1997) and structured algorithms(Cadzand, The Netherlands, 2002). He was the Co-Chairman of the 2005 IEEEStatistical Signal Processing Conference organized in Bordeaux, France. Hewas the Editor of numerous symposia proceedings. From 1981 to 1990, hewas a member of the IFAC Technical Board. Currently, he a member of theTechnical Committee on Digital Signal Processing of the IEEE Circuits andSystems Society. Since 1999, he has been an elected Associate Member of theThird World Academy of Sciences.

Nicolai Christov received the Engineer degree inelectrical engineering from the Technical Universityof Sofia (TUS), Sofia, Bulgaria, in 1971, the Engi-neer degree and Diplôme d’Etudes Approfondiesin control engineering from the Grenoble Instituteof Technology and Joseph Fourier University,Grenoble, France, in 1973, and the Ph.D. degree inmathematics from TUS in 1978.

He was a Research Scientist from 1978 to 1990 andan Associate Professor from 1990 to 2004 at the De-partment of Automatics, TUS. From 1997 to 2004,

he was the Co-Director of the TUS UNESCO Chair of Engineering for Devel-opment. Since 2004, he has been a Professor with the Department of ComputerScience and Electrical Engineering, Lille University of Science and Technology,Lille, France. His research interests include linear systems theory, optimal androbust control and estimation, and sensitivity analysis and numerical methodsfor linear multivariable systems.

Dual H∞ Algorithms for Signal Processing— Application to Speech Enhancement

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