 Review
 Open Access
An efficient central DOA tracking algorithm for multiple incoherently distributed sources
 Sonia Ben Hassen^{1}Email author and
 Abdelaziz Samet^{1, 2}
https://doi.org/10.1186/s1363401502760
© Hassen and Samet. 2015
 Received: 19 April 2015
 Accepted: 15 October 2015
 Published: 2 November 2015
Abstract
In this paper, we develop a new tracking method for the direction of arrival (DOA) parameters assuming multiple incoherently distributed (ID) sources. The new approach is based on a simple covariance fitting optimization technique exploiting the central and noncentral moments of the source angular power densities to estimate the central DOAs. The current estimates are treated as measurements provided to the Kalman filter that model the dynamic property of directional changes for the moving sources. Then, the covariancefittingbased algorithm and the Kalman filtering theory are combined to formulate an adaptive tracking algorithm. Our algorithm is compared to the fast approximated power iterationtotal least squareestimation of signal parameters via rotational invariance technique (FAPITLSESPRIT) algorithm using the TLSESPRIT method and the subspace updating via FAPIalgorithm. It will be shown that the proposed algorithm offers an excellent DOA tracking performance and outperforms the FAPITLSESPRIT method especially at low signaltonoise ratio (SNR) values. Moreover, the performances of the two methods increase as the SNR values increase. This increase is more prominent with the FAPITLSESPRIT method. However, their performances degrade when the number of sources increases. It will be also proved that our method depends on the form of the angular distribution function when tracking the central DOAs. Finally, it will be shown that the more the sources are spaced, the more the proposed method can exactly track the DOAs.
Keywords
 Tracking DOAs
 Incoherently distributed sources
 Covariance fitting technique
 Kalman filter
1 Review
1.1 Introduction
The most commonly considered system model in the direction of arrival (DOA)finding techniques is the point source model where the signals are assumed to be generated from farfield point sources [1–4]. However, in real surroundings, especially in modern wireless communication systems, local scattering in the source vicinity causes angular spreading. Therefore, the researchers considered a more realistic signal model called spatially distributed source model. Depending on the nature of scattering, distributed sources have been classified into two types: coherently and incoherently distributed (CD and ID) sources [5, 6]. For ID source model contrary to the CD case, the rank of the noisefree covariance matrix is different to the number of sources. Moreover, it increases with the angular spread. Therefore, traditional subspacebased methods become not applicable in this case. To deal with this problem, efforts have been directed to specifically design new techniques to estimate the angular parameters for ID sources. Some of them that are able to handle a single ID source were developed in [7–13]. Many other estimators were also developed to estimate the angular parameters of multiple ID sources. In fact, some subspace methods were proposed in [5, 14, 15] wherein the effective dimension of the signal subspace is defined as the number of the first eigenvalues of the noisefree covariance matrix where most of the signal energy is concentrated.
Despite their high accuracy, these methods suffer from a heavy computational load. To reduce the complexity, the socalled TLSESPRIT algorithm was derived in [16] which is based on the total least square (TLS) estimation of signal parameters via rotational invariance technique (ESPRIT).
Moreover, more computationally attractive but less efficient methods using the beamforming techniques were developed in [17, 18]. Later, Shahbazpanahi et al. proposed a new algorithm based on central and noncentral moments of the sources angular power densities [19]. In addition to its reduced complexity, this method is applicable to the multisource scenarios with different angular power densities. Recently, authors derived in [20] a new algorithm for the estimation of the angular parameters of multiple ID sources. It exploits the property of the inverse of the covariance matrix to estimate the angular parameters.
Despite its efficiency, this method requires that the multiple sources have an identical angular distribution. Moreover, it estimates the angular parameters (central DOAs and angular spreads) together with a 2D search.
All the aforementioned techniques assume the sources to maintain the same positions, and then their central DOAs are constant in time. This assumption is, however, too restrictive in wireless communication applications where mobile terminals often change their spatial positions. Therefore, several approaches have been proposed to deal with the problem of tracking the discrete DOAs at different time points. Most of them are based on subspace tracking techniques such as the projection approximate subspace tracking (PAST) of [21] and the orthonormal projection approximate subspace tracking (OPAST) of [22].
In this context, authors have proposed in [23] a simple DOA tracking scheme involving the PASTalgorithm and the Kalman filter to track the DOAs. Later, in [24], a fast implementation of the power iterations method for subspace tracking (FAPI) was derived. Recently, the researchers proposed in [25–27] new tracking algorithms based on some modifications of the Kalman filter and the existing subspace tracking techniques. All these tracking algorithms are limited to the point source model.
To deal with the problem of estimating the timevarying DOAs in scattering channels, a simple DOA tracking method based on the TLSESPRIT [16] and subspace updating via FAPI algorithm has been recently proposed in [28] for ID sources. Despite its relatively reduced computational cost, this method was shown to exhibit poor tracking success rate at low signaltonoise ratios (SNRs). More recently, a DOA tracking method based on a support vector regression approach was developed in [29] for coherently distributed source. Therefore, we aim in this paper to consider the problem of tracking DOAs at different time points in scattering channels. As the assumption of uncorrelated ID sources has been shown to be relevant in wireless communications environments with a high base station than the CD source case [30], we assume in this paper that the sources are ID.
We then derive a new method that outperforms the method derived in [28]. This method is based on a simple covariance fitting optimization technique as developed in [19] to estimate the central DOAs in each observed time interval. In fact, this technique can estimate the central DOAs regardless of the angular spreads, and it is applicable to the multisource scenarios with different angular power densities. The new tracking method also uses the famous Kalman filter to model the mobility of the sources and track the different DOAs during the tracking period. The Kalman filter (KF) can reduce estimation errors and avoid the data association problem when applied to angle tracking due to the nature of prediction correction.
This paper is organized as follows. In subsection 1.2, we introduce the system model that will be used throughout the article and we define the problem in terms of notation and assumptions. In subsection 1.3, we formulate the new algorithm. In subsection 1.4, the proposed algorithm is compared to the one of [28] through computer simulations.
Throughout this paper, matrices and vectors are represented by bold upper and lowercase characters, respectively. Vectors are, by default, in column orientation, while (.)^{∗}, (.)^{ T } and (.)^{ H } refer to conjugate, transpose, and conjugate transpose, respectively. Moreover, E{.} and tr(.) stand for the statistical expectation and trace operators, respectively. Furthermore, eig(.) and diag(.) represent the eigenvalues of a matrix and the diagonal matrix of a vector, respectively. Finally, ⊗ stand for the Kronecker operator.
1.2 System model
Finally, n _{ l }(t) represents the additive zeromean, Gaussian distributed, circular, spatially and temporally white noise (i.e., uncorrelated between the receiving antenna elements and between different snapshots).
In principle, the motion of each source is slowly changing within a time increment T. Hence, we suppose that the change of the angular spread of each source is negligible during the tracking period. Then, the tracking problem aims at estimating \(\bar {\boldsymbol {\theta }}(t)\), t=T,2T,… from N snapshots of array data measured within each time increment T while keeping the angular spread the same during the tracking period.
1.3 Derivation of the DOA tracking algorithm for incoherently distributed sources

A covariancefittingbased DOA estimator that exploits the central and noncentral moments of the source angular power densities in order to estimate the central DOAs over [n T,(n+1)T] and

A Kalman filter that tracks these DOAs during the tracking period.
In fact, to estimate the central DOAs of the uncorrelated ID sources over [n T,(n+1)T], we propose to exploit the technique developed in [19] for constant DOAs. This method consists of approximating the covariance matrix using central and noncentral moments of the source angular power densities. Based on this approximation, these moments are estimated using a simple covariance fitting optimization technique. Finally, the central DOAs are obtained from the moment estimates. The advantage of this method is that it has a reduced computational cost and it addresses multiple sources with different angular power densities.
Once the estimated central DOAs are obtained, we then propose a Kalmanfilteringbased tracking algorithm to model the dynamic property of directional changes for the sources. The KF ensures the association between the estimates made at different time points thanks to its predictability characteristic. Indeed, at each stage during the tracking process, the central DOAs predicted by the Kalman filter are used to smooth the central DOAs estimated via the covariancefittingbased algorithm.
1.3.1 Estimation of the central angles via the covariancefittingbased algorithm
In this subsection, we will recall briefly the covariance fitting optimization technique developed in [19] to estimate the central DOAs from N samples received over the time interval [n T,(n+1)T] over which the central DOAs are assumed to be invariant. Therefore, we will use \(\bar {\theta }_{k}\) and ψ _{ k } instead of \(\bar {\theta }_{k}(t)\) and ψ _{ k }(t), respectively.
In (16), \(\tilde {\bar {\theta }}_{k}\) represents a coarse initialization of the true central DOA \(\bar {\theta }_{k}\) of the kth source. Note here that if \(\tilde {\bar {\theta }}_{k}=\bar {\theta }_{k}\), then \(M_{\textit {nk}}(\tilde {\bar {\theta }}_{k})\) becomes the nth central moment, \(M_{\textit {nk}}(\bar {\theta }_{k})\), of the kth source angular power density.
Therefore, we conclude from (17) that in order to estimate the central DOA of the kth source, one should calculate an estimate for the first noncentral moment of its angular power density. To do so, one use a simple covariance fitting optimization technique.
1.3.1.1 Covariance fitting optimization technique
with i=(l−1)R+m+1,j=(k−1)R+r+1(1≤l,k≤K,0≤m,r<R).
Note here that if the matrix \(\boldsymbol {Q}(\tilde {\bar {\boldsymbol {\theta }}})\) is singular, we can replace its inverse by pseudoinverse.
where \(\widehat {M}_{1k}(\tilde {\bar {\theta }}_{k})\) is the first estimated noncentral moment of the angular density of the kth source obtained from \(\widehat {\boldsymbol {m}}(\tilde {\bar {\boldsymbol {\theta }}})\). Moreover, \(\tilde {\bar {\theta }}_{k}\) is an arbitrary DOA that should be chosen sufficiently close to \(\bar {\theta }_{k}\) to reduce the estimation errors.
To obtain a more accurate value of the estimated DOA, we replace \(\tilde {\bar {\theta }}_{k}\) by \(\widehat {\bar {\theta }}_{k}\) already calculated, and we solve again (27) to obtain the estimates of the central moments \(\widehat {\boldsymbol {m}}(\widehat {\bar {\boldsymbol {\theta }}})\). Finally, we obtain a new value of the estimated central angle from the first estimated central moment \(\widehat {M}_{1k}(\widehat {\bar {\theta }}_{k})\). This operation is repeated a few times.
1.3.1.2 Algorithm for the estimation of the central DOAs
 1.
Compute the sample covariance matrix \(\widehat {\boldsymbol {R}}_{\textit {xx}}\) and specify the initial values of \(\tilde {\bar {\theta }}_{k}\), k=1,2,…,K;
 2.
Compute \(\widehat {\boldsymbol {m}}(\tilde {\bar {\boldsymbol {\theta }}})\) from (34) and deduce \(\widehat {M}_{1m}(\tilde {\bar {\theta }}_{k})\) using (23) and (24);
 3.
Compute \(\widehat {\bar {\theta }}_{k}=\widehat {M}_{1k}(\tilde {\bar {\theta }}_{k})+\tilde {\bar {\theta }}_{k}\) and update \(\tilde {\bar {\theta }}_{k}=\widehat {\bar {\theta }}_{k}\); and
 4.
Repeat steps 2 and 3 few times to obtain good estimates of the central DOAs.
These estimates will be treated as measurements and provided to the celebrated Kalman filter to track the DOAs.
1.3.2 The KF tracking algorithm
In this paper, similar to [23, 31], we suppose that \({\sigma ^{2}_{w}}\) is constant during the tracking period and small. Besides, v _{ k }(t) is the measurement noise which is supposed to be zeromean with variance \(\sigma ^{2}_{\textit {vk}}(t)\) and uncorrelated with w _{ k }(t).
As in the proposed KF tracking scheme, instead of using the array output directly as the measurement process, we use the most current data to form the DOA estimates via the covariancefittingbased algorithm, the measurement noise is then due to the estimation inaccuracy of the covariancefittingbased method.
Otherwise, the snapshots x(t)_{ t=1,…,N } are random and mutually independent. Then, using the classical central limit theorem [32], we have for a large number of snapshots N that \(\widehat {\boldsymbol {R}}_{\textit {xx}}\) follows a normal distribution. We can then conclude from [33] that \(\widehat {M}_{1k}(\tilde {\bar {\theta }}_{k})=\text {alg}(\widehat {\boldsymbol {R}}_{\textit {xx}})\) is asymptotically Gaussian distributed. Consequently, the measurement noise is Gaussian. We will also verify in subsection 1.4 that the estimation error of the covariancefittingbased algorithm is Gaussian distributed.
Then, the variances of the measurement noise \(\{\sigma _{\textit {vk}}^{2}(t)\}_{k=1}^{K}\) can be approximated by the diagonal entries of \(\textrm {CRB}(\bar {\boldsymbol {\theta }})\) at each time interval [ n T,(n+1)T] for large number of samples.

Initially, the covariancefittingbased algorithm is used twice to obtain two angle estimates \(\hat {\bar {\boldsymbol {\theta }}}(1)\) and \(\hat {\bar {\boldsymbol {\theta }}}(0)\) at t=0. Then, the initial state vector is \(\hat {\boldsymbol {y}}(00)=\left [\hat {\bar {\theta }}_{1}(0), (\hat {\bar {\theta }}_{1}(0)\hat {\bar {\theta }}_{1}(1))/T,0,\ldots,\hat {\bar {\theta }}_{K}(0),\right.\) \(\left. (\hat {\bar {\theta }}_{K}(0)\hat {\bar {\theta }}_{K}(1))/T,0\right ]^{T} \).

Calculate the initial variances of the measurement noise \(\{\sigma _{\textit {vk}}^{2}(0)\}_{k=1}^{K}\) which represent the variances of the estimation error of the covariancefittingbased algorithm at t=0. These variances can also be obtained from the diagonal entries of (54) for a large number of samples.

Calculate the covariance matrix of the initial state vector \(\hat {\boldsymbol {y}}(00)\) as follows:$$\begin{array}{@{}rcl@{}} {}\boldsymbol{P}(00)&=&\text{diag}\left(\left[\sigma^{2}_{v1}(0),\ldots,\sigma^{2}_{vK}(0)\right]\right)\otimes\left(\begin{array}{ccccc} 1&\frac{1}{T}&0\\ \frac{1}{T}&\frac{2}{T^{2}}&0\\ 0&0&0 \end{array}\right),\\ &=&\boldsymbol{R}(0)\otimes\left(\begin{array}{ccccc} 1&\frac{1}{T}&0\\ \frac{1}{T}&\frac{2}{T^{2}}&0\\ 0&0&0 \end{array}\right). \end{array} $$(61)

For t=1,2,…, we do the following steps

Prediction of DOA angles: obtain the predicted estimates \(\hat {\boldsymbol {y}}(tt1)\) of the state vector y(t) from the existing estimates \(\hat {\boldsymbol {y}}(t1t1)\) available at time t−1 and its covariance matrix P(tt−1) by the equations:$$\begin{array}{@{}rcl@{}} &{}\hat{\boldsymbol{y}}(tt1)=\bar{\boldsymbol{F}}\hat{\boldsymbol{y}}(t1t1), \end{array} $$(62)$$\begin{array}{@{}rcl@{}} &{}\qquad\quad\boldsymbol{P}(tt1)=\bar{\boldsymbol{F}}\boldsymbol{P}(t1t1)\bar{\boldsymbol{F}}^{T}+\bar{\boldsymbol{Q}}. \end{array} $$(63)Then, we obtain the predicted central DOAs vector \(\hat {\bar {\boldsymbol {\theta }}}(tt1)\) from \(\hat {\boldsymbol {y}}(tt1)\) as follows:$$\begin{array}{@{}rcl@{}} \hat{\boldsymbol{\theta}}(tt1)&=&\bar{\boldsymbol{H}}\hat{\boldsymbol{y}}(tt1). \end{array} $$(64)

Central DOA estimation via the covariancefittingbased method: we estimate the central angles \(\hat {\bar {\boldsymbol {\theta }}}(t)\) via the covariancefittingbased algorithm as described in the previous subsection.

Updating the covariance matrix of the measurement noise vector R(t): we calculate the variances of the measurement noise \(\left \{\sigma _{\textit {vk}}^{2}(t)\right \}_{k=1}^{K}\), and we then deduce the covariance matrix \(\boldsymbol {R}(t)=\text {diag}\left (\left [\sigma ^{2}_{v1}(t),\ldots,\sigma ^{2}_{\textit {vK}}(t)\right ]\right)\).

Updating the estimated DOA angles: in this last step, we aim to find the estimate \(\hat {\boldsymbol {y}}(tt)\) of the state vector y(t). Therefore, we should first estimate the innovation errors \(\delta \bar {\boldsymbol {\theta }}\) defined as:$$\begin{array}{@{}rcl@{}} \delta\bar{\boldsymbol{\theta}}(t)&=&\hat{\bar{\boldsymbol{\theta}}}(t)\hat{\bar{\boldsymbol{\theta}}}(tt1). \end{array} $$(65)Because the motion of each source is mutually independent, the innovation errors vector δ θ(t) is zero mean with covariance \(\bar {\boldsymbol {H}}\boldsymbol {P}(tt1)\bar {\boldsymbol {H}}^{T}+\boldsymbol {R}(t)\). After the determination of \(\delta \bar {\boldsymbol {\theta }}(t)\), we update the state vector by the following equation:$$\begin{array}{@{}rcl@{}} \hat{\boldsymbol{y}}(tt)&=&\hat{\boldsymbol{y}}(tt1)+\boldsymbol{G}(t)\delta\bar{\boldsymbol{\theta}}(t), \end{array} $$(66)where G(t) is the Kalman gains vector at time t given by:$${\kern15pt} \begin{aligned} \boldsymbol{G} (t)&=\boldsymbol{P}(tt1)\bar{\boldsymbol{H}}^{T}\left[\bar{\boldsymbol{H}}\boldsymbol{P}(tt1)\bar{\boldsymbol{H}}^{T}+\boldsymbol{R}(t)\right]^{1}.\\ \end{aligned} $$The covariance matrix of \(\hat {\boldsymbol {y}}(tt)\) is obtained as:$$\begin{array}{@{}rcl@{}} \boldsymbol{P}(tt)&=&(I\boldsymbol{G}(t)\bar{\boldsymbol{H}})\boldsymbol{P}(tt1). \end{array} $$(67)Finally, we obtain the updated central DOA vector \(\hat {\bar {\boldsymbol {\theta }}}(tt)\) from \(\hat {\boldsymbol {y}}(tt)\) as follows:$$\begin{array}{@{}rcl@{}} \hat{\bar{\boldsymbol{\theta}}}(tt)&=&\bar{\boldsymbol{H}}\hat{\boldsymbol{y}}(tt). \end{array} $$(68)

1.4 Simulation results
In this subsection, we will present some figures to illustrate the effectiveness of our proposed algorithm in different scenarios and compare it to the method derived in [28] (referred to as FAPITLSESPRIT method). It will be seen that our new method outperforms the FAPITLSESPRIT method.
Throughout this section, we consider a zeromean, Gaussian distributed, spatially and temporally white noise. We also consider a uniform linear array of 11 sensors separated by a halfwavelength λ/2. The sources are tracked over an interval of 40 s with T=1 s. During each 1s interval, N=500 snapshots of sensor data are generated and used to estimate the central DOAs via the covariancefittingbased algorithm with R=3. In addition, three iterations of steps 2 and 3 of this algorithm are used in all our examples. Moreover, 100 independent simulation runs have been performed to obtain each simulated central DOA vector. To track the DOAs, we assume that the variance of the process noise in the KF is constant and equal to \({\sigma ^{2}_{w}}=0.0001\) while the variances of the measurement noise \(\left \{\sigma _{\textit {vk}}^{2}(t)\right \}_{k=1}^{K}\) are calculated at each time interval [n T,(n+1)T] as explained in the previous section.
To simulate the FAPITLSESPRIT method in a proper way, we consider two identical subarrays of 11 sensors with a halfwavelength interelement spacing. The intersubarray displacement is assumed to be δ=λ/10. Moreover, to estimate the signal subspace via the FAPI algorithm, we choose the forgetting factor β=0.985.
These figures illustrate the result proved in the first example. The proposed algorithm offers an excellent DOA tracking performance and outperforms the FAPITLSESPRIT method at low SNR values. Moreover, when we compare the estimation error of the first central DOA made by our proposed method in Fig. 3 to that in Fig. 8, we note that this error in the first case is lower than the corresponding error in the second case. The same holds for the variance of the first estimated central DOA. In fact, we see clearly from Fig. 4 that, in the first case, this variance is too low. This can be explained by the fact that in the first case, the sources are more spaced than in the second case. Indeed, in Figs. 3 and 4, the central DOAs are \(\bar {\theta }_{1}=10^{\circ}\) and \(\bar {\theta }_{2}=30^{\circ}\) while in Figs. 8 and 9 the central DOAs are \(\bar {\theta }_{1}=10^{\circ}\) and \(\bar {\theta }_{2}=30^{\circ}\). Therefore, we can conclude that the more the sources are spaced, the more the proposed method can exactly track the central DOAs.
We see from Fig. 10 that the two methods can exactly track the central DOAs at high SNR values. Moreover, we note from Figs. 11 and 12 that despite the random variation of the estimation errors and the variances of the central DOA estimates by the FAPITLSESPRIT algorithm, their values are close to those obtained with our proposed method. Furthermore, comparing Figs. 11 and 12 to Figs. 8 and 9 of the second example, we see clearly that the performances of the FAPITLSESPRIT method increase in particular at high SNR values. Therefore, we can conclude that the improvement made by our method with regard to the FAPITLSESPRIT method is more prominent for low SNR values.
Next, we reconsider the low SNR assumption (SNR=0 dB) but we consider two other ID sources. The first one is assumed to be uniform distributed (UID) with central DOA \(\bar {\theta }_{1}=10^{\circ}\) and angular spread σ _{1}=2.5° and the second one GID with central DOA \(\bar {\theta }_{2}=30^{\circ}\) and angular spread σ _{2}=1.5°.
It can be seen from these figures that our proposed method outperforms the FAPITLSESPRIT method even if the sources are closely spaced. However, comparing these figures to Figs. 2, 3, and 4 of the first example and Figs. 7, 8, and 9 of the second example where the sources are widely spaced, we note that the proposed method cannot properly track the DOAs when the sources are closely spaced. This confirms the conclusion previously proved: the more the sources are spaced, the more the proposed method can exactly track the DOAs.
2 Conclusions
In this paper, we developed a new method for tracking the central DOAs assuming multiple incoherently distributed (ID) sources. This method is based on a simple covariance fitting optimization technique to estimate the central DOAs in each observed time interval. It also uses the Kalman filter to model the mobility of the sources and track the different DOAs during the tracking period. Our method was compared to the FAPITLSESPRIT algorithm using the TLSESPRIT method and the subspace updating via FAPIalgorithm in different scenarios. We proved that this new method outperforms the FAPITLSESPRIT method. The improvement made by our method with regard to the FAPITLSESPRIT method is more prominent for low SNR values. We also showed that the proposed method can better track the central DOA when the source is GID or LID than when it is UID. Therefore, our method depends on the form of the angular distribution function when tracking the central DOAs. Moreover, we proved that the more the sources are spaced, the more the proposed method can exactly track the central DOAs. Finally, when the number of sources increases, the performances of our proposed algorithm decrease.
3 Endnote
^{1} We assume that the delay spread caused by the multipath propagation is small compared to the inverse bandwidth of the transmitted signals. This means that the narrowband assumption is valid in the presence of scattering.
Declarations
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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