Low-complexity DOA estimation from short data snapshots for ULA systems using the annihilating filter technique
- Faouzi Bellili^{1}Email authorView ORCID ID profile,
- Souheib Ben Amor^{1},
- Sofiène Affes^{1} and
- Ali Ghrayeb^{2}
DOI: 10.1186/s13634-017-0480-1
© The Author(s) 2017
Received: 2 November 2016
Accepted: 23 May 2017
Published: 30 June 2017
Abstract
This paper addresses the problem of DOA estimation using uniform linear array (ULA) antenna configurations. We propose a new low-cost method of multiple DOA estimation from very short data snapshots. The new estimator is based on the annihilating filter (AF) technique. It is non-data-aided (NDA) and does not impinge therefore on the whole throughput of the system. The noise components are assumed temporally and spatially white across the receiving antenna elements. The transmitted signals are also temporally and spatially white across the transmitting sources. The new method is compared in performance to the Cramér-Rao lower bound (CRLB), the root-MUSIC algorithm, the deterministic maximum likelihood estimator and another Bayesian method developed precisely for the single snapshot case. Simulations show that the new estimator performs well over a wide SNR range. Prominently, the main advantage of the new AF-based method is that it succeeds in accurately estimating the DOAs from short data snapshots and even from a single snapshot outperforming by far the state-of-the-art techniques both in DOA estimation accuracy and computational cost.
Keywords
DOA estimation Root-MUSIC Annihilating filter Array signal processing NDA estimation1 Introduction
In recent years, there has been a surge of interest in array signal processing applications in both military and civil domains [1, 2]. The concept of direction of arrival (DOA) estimation find its use in applications related to radar or sonar systems. In addition, in modern mobile communication systems, for example, based only on the data received at the antenna array, estimating the DOAs of the desired users and those of the interference signals allows their extraction and cancellation, respectively, by beamforming technologies [3, 4] in order to improve the wireless systems’ performance.
Roughly speaking, depending on the a priori knowledge of the transmitted signals, DOA estimators can be categorized as data-aided (DA) or non-data-aided (NDA). In plain English, DA approaches base the estimation process on a priori perfectly known symbols. Unfortunately, although being simple and accurate, these approaches may suffer from the major drawback of limiting the whole throughput of the system by periodically sending a reference (known) signal [5]. It should be mentioned here that superimposed pilots do not affect the throughput but increase the complexity of the channel estimation process. Hence, the ever increasing demand for channel bandwidth spurred the more practically oriented minds to develop new estimation techniques that rely on the received data samples only and which are therefore commonly known as NDA techniques. NDA estimators themselves are referred to as deterministic or stochastic if the unknown transmitted signal is assumed deterministic or completely random, respectively. So far, from maximum likelihood-based to subspace-based methods, many NDA DOA estimators have been proposed and extensively studied in the literature [6–8]. The NDA maximum likelihood approaches are undoubtedly the most accurate, but unfortunately, they are often computationally very expensive. To circumvent this challenging problem, covariance-based estimators are often a trend—in NDA estimation schemes—to alleviate this burden of computational cost. Fortunately, usually, they also provide sufficiently accurate DOA estimates, especially in the presence of sufficiently large number of received samples. But in situations of short data snapshots, they may not be reliable and one would be obliged to trade low complexity for more accurate estimation by simply applying the maximum likelihood approaches. Yet, the maximum likelihood estimators are analytically intractable in the NDA case especially in the presence of random transmitted symbols/signals. Therefore, they are often tackled numerically via multidimensional grid search approaches. Their accuracy/resolution is therefore dictated by the discretization step of the grid. A very dense discretization (small step) is able to provide very accurate estimates even at low operational SNRs, but the complexity of the underlying ML algorithm would be extremely high and even prohibitive since its complexity grows exponentially with the number of the parameters to be estimated. Another alternative is to solve the ML criterion using pilot/reference symbols/signals only where a closed-form solution may be feasible. Unfortunately, this approach is not able to provide in-service estimates as the receiver is compelled to wait for the next pilot signals in order to update the estimates.
Motivated by these facts, we develop in this paper a new covariance-based DOA estimation method for ULA configurations which succeeds in estimating the DOA from very short data records. It is based on the annihilating filter technique: finding the roots of an annihilating filter (AF) which are directly related to the unknown DOAs. It should be noted that the AF technique has been well known for a very long time in the mature field of spectral estimation. About a decade ago, it was also used to successfully develop the so-called finite-rate-of-innovation (FRI) sampling method [9] where it led to signal sampling and reconstruction paradigms at the minimal possible rate (far below the traditional Nyquist rate). In this contribution, we apply for the first time the AF approach to DOA estimation for ULA configurations and, therefore, we will henceforth refer to our new technique as the AF-based method. The coefficients of the corresponding AF are calculated by the singular value decomposition (SVD) of a matrix whose elements are built from second-order cross moments across the receiving antenna elements of the received samples. Interestingly, this matrix is of reduced dimensions thereby yielding a very low computational load of the SVD decomposition.
We propose two different versions of the new AF-based solution^{1} depending on the SNR threshold. The first one, referred to as “version I”, is more advantageous at high SNR levels. It exploits each consecutive 2K+1 correlation coefficients along the columns and rows of the covariance matrix (K being the number of sources). The second one, referred to as “version II”, exploits the Toepltiz structure of the covariance matrix in order to enhance the estimation performance at low SNR levels. In both versions, the obtained DOA estimates are then used to find the unknown sources’ powers along with the noise variance.
In the multiple snapshot case, both versions of the proposed AF-based technique are compared in accuracy performance to the Cramér-Rao lower bound (CRLB) [10] and to the root-MUSIC algorithm—a popular and powerful technique of DOA estimation for ULA systems—which is also based on polynomial rooting [11]. In the single-snapshot scenario, however, it is compared to another Bayesian method that was designed precisely for the challenging single-snapshot case [12] as well as the deterministic ML (DML) estimator. We mention here that a more recent iterative technique that handles the single-snapshot case has also been proposed in [13]. Unfortunately, in its NDA version, it relies on the prior availability of an initial guess about all the unknown DOAs whose accuracy affects the overall performance of the method. Therefore, for the sake of fairness, this technique is not considered since none of the considered techniques (including our AF-based estimator itself) requires an initial guess about the DOAs. Even more, it has been recently recognized in a comparative study of various DOA estimators [14] that DML is indeed the most attractive one if the DOAs are to be estimated from a single snapshot. It will be shown by Monte-Carlo simulations that the new AF-based method is able to accurately estimate the DOAs from short data snapshots and even from a single-shot measurement. Furthermore, it outperforms the classical Bayesian and DML estimators over a wide SNR range with a slight performance advantage for the latter in the low SNR region but at the cost of an extremely high computational load.
We organize the rest of this paper as follows. In Section 2, we introduce the system model that will be used throughout this article. Then in Section 3, we develop our new AF-based DOA estimation technique. In Section 4, we exploit these new AF-based DOA estimates to develop new estimates for the channel powers. In Section 5, we assess the performance of the new estimators. Finally, we draw out some concluding remarks in Section 6.
We mention beforehand that some of the common notations will be used throughout this paper. Vectors and matrices are represented by lower- and upper-case bold fonts, respectively. Moreover, {.}^{ H } and {.}^{ T } denote the Hermitian (i.e. transpose conjugate) and transpose of any vector or matrix, respectively. The operators {.}^{∗} and |.| return the conjugate and amplitude of any complex number, respectively, and j is the pure complex number that verifies j ^{2}=−1. Moreover, N _{ a } refers to the number of antenna elements in a uniform linear array (ULA). The statistical expectation is denoted as E{.}, and the notation \(\triangleq \) is used for definitions.
2 System model
where at time index n, a _{ k }(n)^{2} is the signal (or symbol) transmitted by the k ^{ th } source and w _{ i }(n) is the noise component on the i ^{ th } antenna branch that is modelled by a zero-mean complex Gaussian random variable with independent real and imaginary parts, each of variance σ ^{2}. The complex channel coefficients corresponding to the K sources are assumed to be unknown, and they are denoted by \(\{h_{k}=|h_{k}|e^{j\phi _{k}}\}_{k=1}^{K}\) where ϕ _{ k } stands for any possible channel distortion phase. Moreover, \(\{\theta _{k}\}_{k=1}^{K}\) are the unknown DOAs (to be estimated) of the planar waves impinging from the K sources.Note here that the receiving antenna elements are supposed to be spaced by half the wavelength, i.e. d=λ/2 where d is the distance between two consecutive antenna branches and λ is the carrier wavelength of the signal. Note also that although the vector/matrix representation of the received signals is more compact and widely adopted in the open literature, we settle here on the scalar form of the received signals (i.e. the elementary received signals on each antenna element). We believe that this representation allows for an easy grasp of the theoretical foundations of the new estimator since it is—as will be seen later—based on the explicit expression for each cross-covariance between the elementary received signals.
3 Formulation of the new AF-based DOA estimator
and this sample average does not coincide with the statistical average given in (5) unless the observation window size, N, is very large. Yet, we will see in the simulations section that the new AF-based estimator performs very well with very short-data records and even from a single snapshot.
3.1 Robustness to the presence of short data records
We observe from (29) that the second-order moments estimated with short data records (or even a single snapshot) exhibit the interesting property of a “weighted sum of sinusoids” and therefore the DOAs can still be accurately estimated from the roots of their annihilating filter.
3.2 Exploiting the Toeplitz structure of the covariance matrix
from which we construct a single vector, \(\widehat {\bar {\boldsymbol {r}}}_{\boldsymbol {\theta }}=[\widehat {\bar {r}}_{\boldsymbol {\theta }}(1), \widehat {\bar {r}}_{\boldsymbol {\theta }}(2),\cdots, \widehat {\bar {r}}_{\boldsymbol {\theta }}(N_{a}-1)]^{T}\). Then, the same procedure that was previously applied for all the eligible columns is now applied to the single vector \(\widehat {\bar {\boldsymbol {r}}}_{\boldsymbol {\theta }}\) since it also inherits the interesting property of weighted sum of exponentials. For ease of notation, we simply refer to this procedure as version II of the new AF-based estimator and we refer to the procedure described previously (column-wise and row-wise) as version I.This operation of averaging over the secondary diagonals is not only useful to combat the effect of the noise at low SNRs but also expected to improve the DOA estimation even for moderate SNR values whenever the number of sources to be localized is large. In fact, when K is high, the number N _{ a }−2K−1 of eligible columns in version I can be limited. For instance, for N _{ a }=8 and K=3, only the first column is eligible since N _{ a }−2K−1=1. Consequently, a large part of the covariance matrix is simply ignored although it carries a lot of information about the unknown DOAs. Yet, by averaging over the secondary diagonals, all the entries of the covariance matrix are incorporated in the estimation process and the whole information is being exploited. Therefore, as long as the SNR decreases or the number of sources increases (for a fixed number of receiving antenna elements), it is expected that the second version of the new estimator outperforms its first version. However, for sufficiently high SNR values, the estimated elementary cross-covariances (without averaging) are already quite accurate and can hence be reliably used to obtain more accurate^{7} DOA estimates with version I. The latter is even more recommended if the number of sources is also small since the number of eligible columns (and consequently the number of exploited cross-covariances) would be sufficiently high.
3.3 Complexity analysis
Complexity of the three single-shot techniques with N _{ a }=16 receiving antenna branches
K=2 | K=4 | |||
---|---|---|---|---|
R=100 | R=500 | R=100 | R=500 | |
Bayesian method | 35.32×10^{6} | 8.83×10^{8} | 7.9200×10^{11} | 4.95×10^{14} |
DML | 94.42×10^{6} | 2.3605×10^{9} | 1.1244×10^{12} | 7.0275 ×10^{14} |
AF-based | 2484 | 2484 | 7464 | 7464 |
4 Per-source channel power estimation
5 Simulation results
The NRMSE for the SNR estimator is defined likewise. DOA estimation will be basically organized in three subsections: (i) the case of multiple snapshots (including short-data records), (ii) the case of a single-shot measurement, and (iii) the case of time varying DOAs. Channel powers and SNR estimation will then follow.
5.1 Multiple and short-data records: comparison against root-MUSIC
We see that the two versions of the new estimator provide sufficiently accurate DOA estimates over the entire SNR range. In such comfortable situation where a very large number of measurements can be used in the estimation process, the classical root-MUSIC technique outperforms the two AF-based versions. It is also seen that as N _{ a } increases, version I of the AF-based estimator exhibits a performance gain against its version II at low SNR values. Actually, this is only true when the window size is large enough (e.g. N=1000 as considered in this figure) so that the elementary cross-covariances are quite accurate and therefore the elementary estimates \(\hat {\theta }^{(l,k)}\) are also sufficiently accurate. Indeed, since the number of these elementary estimates (N _{ a }−2K−1 eligible columns and rows) also increases with N _{ a }, this leads to a more accurate final averaged estimate than the single estimate obtained by applying version II. The same observation holds for sufficiently high SNR values even if N _{ a } is small (N _{ a }=8).
5.2 Single-shot case: comparison against the DML and Bayesian methods
The three existing estimators were simulated using a discretization step s=180/100 (in the remainder of this paper, we will characterize the grid step, s, by the integer number R where s=180/R). We observe from this figure that both versions of the newly developed AF-based estimator are still able to estimate the DOAs over a wide SNR range. We see also from Fig. 4 that for sufficiently high SNR values the MSE of version II saturates, contrarily to version I. This is because in this SNR region the signals are almost noise-free and therefore the elementary cross-covariances’ estimates are already noiseless. They can be thus exploited as they are (as done in version I) to provide a large number of sufficiently accurate estimates \(\hat {\theta }_{k}^{(l,p)}\) without prior averaging (as done in version II). In fact, averaging along the secondary diagonals would simply provide a number of statistics that are as accurate as the elementary cross-covariances themselves, and hence, the performance in terms of DOA estimation does not improve (saturation).On the other hand, the existing single-shot techniques (Bayesian, DML estimators and ST-BCS) exhibit a slight advantage at low SNR levels, but their computational load is extremely much higher. In fact, in light of the complexity analysis presented in Table 1 at the end of Section 3.3, the complexities of the DML and Bayesian algorithms are, respectively, in the order of \(N_{oper}^{\text {Bayesian}}= 35.32\times 10^{6}\) and \(N_{oper}^{\textrm {DML}}= 94.42\times 10^{6}\) operations against only \(N_{oper}^{\text {AF}}=2484\) operations for the proposed estimator. This amounts to complexity ratios in the order of \(\frac {N_{oper}^{\text {Bayesian}}}{N_{oper}^{\text {AF}}}\approx \frac {N_{oper}^{\textrm {DML}}}{N_{oper}^{\text {AF}}} \approx 10^{4}\). Yet, even at these extremely high computational loads, the traditional single-snapshot algorithms are not able to outperform the new estimator for medium to high levels. Of course, as stated previously, for extremely large values of R (very dense grid search), these two estimators would ultimately outperform our new method over the entire SNR range, but unfortunately their complexities become even more prohibitive^{9}. For example, under the same simulation setup of Fig. 4 (in particular N _{ a }=16 and K=2), these two estimators will outperform the AF-based technique, over the entire SNR range, by setting R=500 (i.e. estimating the DOAs at a grid resolution of 0.36°). However, the complexity ratios become in the order of \(\frac {N_{oper}^{\text {Bayesian}}}{N_{oper}^{\text {AF}}}\approx \frac {N_{oper}^{\textrm {DML}}}{N_{oper}^{\text {AF}}} \approx 10^{6}\).
The new method is therefore very useful (in terms of accuracy/complexity trade-offs) in applications where a single snapshot is to be used. This is encountered in many situations where a very high estimation update speed is required. These applications can be indeed enhanced by providing a DOA estimate once a single sample is acquired instead of waiting for a larger number of measurements. Furthermore, in many other practical situations, the DOAs may change appreciably from one snapshot to another due to the fast motion of the sources. For all these systems, our new AF-based estimator offers the best accuracy/complexity trade-offs.
5.3 Time-varying DOAs
5.4 Performance of the channel powers and SNR estimators
6 Conclusions
In this paper, we derived a new DOA estimation method for multiple planar waves impinging on a ULA antenna array. The transmitted sources and the noise components are assumed to be spatially and temporally white. The new method is based on the annihilating filter technique. It was seen that the new method exhibits accurate statistical performance while having a low computational cost. Its major advantage is its capability of accurately resolving DOAs as close as 10°from short data snapshots and even from a single snapshot. This capability makes this new estimator well geared toward applications that require DOA estimation of fast moving sources or require up-to-date estimates for the DOAs over very short observation windows. The estimated DOAs were then used to easily estimate the channel powers and SNRs for each source (or user).
7 Endnotes
^{1} Extensions of the proposed AF-based technique to the problem of joint angle and delay estimation (JADE) [21] falls beyond the scope of this paper.
^{2} The signal a _{ k }(n) can be complex symbols taken from any constellation such as QPSK, M-PSK and M-QAM or simply complex Gaussian.
^{3} This is because all the cross-covariances that belong to any given secondary diagonal of the covariance matrix have the same expression.
^{4} One could decide to consider the upper-triangular matrix, i.e. i<l. But this does not change the estimator, as seen from (5), since this will only introduce a negative sign in the exponential argument.
^{5} Note that the vector \({\boldsymbol {r}}_{{\boldsymbol {\theta }}}^{(l)}\) contains all the N _{ a }−l elements of the l ^{ th } column that are lying under the main diagonal of the covariance matrix.
^{6} We mention here that \(\boldsymbol {r}^{\prime (l)}_{\boldsymbol {\theta }}\) plays the role of \(\boldsymbol {r}^{(l)}_{\boldsymbol {\theta }}\) that was previously used when the estimation process was performed column-wise.
^{7} This is because this version provides a larger number of estimates for each DOA, which can be averaged to obtain a more refined final estimate.
^{8} Please note that the root-MUSIC techniques has almost the same complexity of the our AF-based estimator since it involves similar operations of SVD decomposition (but with different matrices sizes) and polynomial rooting. Also note that we evaluate and refer to the complexity of version I of the new AF-based estimator since it is more computationally expensive than version II.
^{9} Their complexities also increase exponentially with the number of unknown DOAs, K, contrarily to the proposed estimator whose complexity increases only polynomially with K (see Table 1 for K=4).
Declarations
Acknowledgments
This work was made possible by NPRP grant NPRP 5-250-2-087 from the Qatar National Research Fund (a member of Qatar Foundation). The statements made herein are solely the responsibility of the authors. Work published in part in [22].
Competing interests
The authors declare that they have no competing interests.
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