- Open Access
Single-snapshot DOA estimation by using Compressed Sensing
© Fortunati et al.; licensee Springer. 2014
- Received: 25 February 2014
- Accepted: 1 July 2014
- Published: 30 July 2014
This paper deals with the problem of estimating the directions of arrival (DOA) of multiple source signals from a single observation vector of an array data. In particular, four estimation algorithms based on the theory of compressed sensing (CS), i.e., the classical ℓ1 minimization (or Least Absolute Shrinkage and Selection Operator, LASSO), the fast smooth ℓ0 minimization, and the Sparse Iterative Covariance-Based Estimator, SPICE and the Iterative Adaptive Approach for Amplitude and Phase Estimation, IAA-APES algorithms, are analyzed, and their statistical properties are investigated and compared with the classical Fourier beamformer (FB) in different simulated scenarios. We show that unlike the classical FB, a CS-based beamformer (CSB) has some desirable properties typical of the adaptive algorithms (e.g., Capon and MUSIC) even in the single snapshot case. Particular attention is devoted to the super-resolution property. Theoretical arguments and simulation analysis provide evidence that a CS-based beamformer can achieve resolution beyond the classical Rayleigh limit. Finally, the theoretical findings are validated by processing a real sonar dataset.
- Compressive sensing
- Angular sparsity
- DOA estimation
- Fourier beamformer
- LASSO algorithm
- SPICE algorithm
- IAA-APES algorithm
The problem of estimating the directions of arrival (DOA) of a certain number of sources has been an active research area for decades [1, 2], with applications to monostatic and multi-static radar systems [3–7] and remote sensing [8, 9]. The first approach to carrying out space processing, i.e., DOA estimation, from data sampled by an array of sensors was the well-known Fourier beamformer (FB). However, the main drawbacks of the FB are the high level of secondary lobes and poor angular resolution . In fact, the FB suffers from the Rayleigh resolution limit, which is independent of the signal-to-noise ratio (SNR). In order to overcome these limitations, adaptive beamformers, such as Capon  and MUSIC , have been proposed, and their performance is widely investigated, also in the presence of multiplicative noise [8, 9] and array errors . However, most of these adaptive algorithms rely on asymptotic assumptions, e.g., high SNR level and large number of snapshots. In many practical applications, for example, in sonar processing, due to physical constraints, e.g., sound speed, only a very small number of snapshots or, in the worst case, a single snapshot is available for DOA estimation [13, 14]. Another application in which the number of available snapshots is a critical parameter is the DOA estimation in automotive radar systems (see, e.g., ). In the single-snapshot scenario, the adaptive algorithms that require calculating the inverse of the estimated noise covariance matrix, e.g., the Sample Covariance Matrix (SCM), cannot be used since the estimate is rank deficient. Recently, new algorithms, based on the emerging field of the Compressed Sensing (CS) theory have been proposed in the array processing literature (see e.g., [16–18]).
The aim of this paper is to investigate the statistical properties of CS-based beamformers. The analysis is carried out in the single-snapshot scenario, which is of practical relevance in sonar and in automotive radar applications. The multi-snapshot scenario is left to future works. The focus here is on three statistical properties: (i) the estimation performance, i.e., the efficiency in the DOA estimation; (ii) the detection performance, i.e. evaluation of the receiver operating characteristic (ROC) curves; and (iii) the resolution capability. In particular, we show that a CS-based DOA estimator is able to guarantee the super-resolution property, typical of the adaptive estimation algorithms.
The remainder of this paper is organized as follows. Section 2 describes the single-snapshot DOA estimation problem. In particular, a brief description of the classical FB and of the four considered CSBs is provided. In Section 3, the estimation and detection performance of the four CSBs are evaluated and compared with that of the FB for two different noise models. The super-resolution property of the CSBs is investigated in Section 4, whereas some results on real sonar data are presented in Section 5. Finally, our conclusions are summarized in Section 6.
2.1 The measurement model
where K is the number of sources and are their K spatial frequencies, relative to the K DOAs , which are the parameters to be estimated. In this paper, when the random signal model is adopted to characterize the multi-source scenario, the sources are assumed to be independent.
2.2 Classical beamforming
Due to the fundamental importance of the DOA estimation problem in a multitude of practical applications, many estimation algorithms have been proposed in the literature. Without claiming to be complete, the estimation methods associated with Equations 1 and 2 can be categorized in two large classes: the non-parametric (spectral-based) algorithms and the parametric algorithms . The non-parametric algorithms (e.g., Fourier, Capon, and MUSIC beamformers) exploit some spectrum-based function of the parameters to be estimated, e.g., the DOAs. More precisely, the DOA's estimation problem is solved by finding the locations of the highest peaks of a spectrum-based function. The parametric techniques, e.g., Deterministic  and Stochastic  Maximum Likelihood (DML and SML) algorithms, on the other hand, fully exploit the statistical characterization of the measurements and, in general, require a simultaneous search over all the unknown parameters to be estimated. The latter approach often guarantees higher estimation performance than the spectral-based algorithm, albeit at the expense of an increased computational complexity.
However, almost all these algorithms (both spectral-based and parametric methods) have to work in the so-called asymptotic region, i.e., they need high SNR values and a large enough number of snapshots in order to provide reliable estimates. However, in some applications, e.g., sonar applications, due to physical constraints, only a very small number of snapshots or, in the worst case, a single snapshot is available for DOA estimation. In the single-snapshot scenario, adaptive algorithms (such as e.g., Capon, MUSIC, DML, and SML) that rely on an estimate of the noise covariance matrix C cannot be applied. In fact, if the standard Sample Covariance Matrix (SCM) estimator is used, the resulting estimate of C would be rank deficient (see, e.g., ). In the single snapshot case then, the only feasible algorithm is the FB.
The noise vector n is a complex zero-mean Gaussian-distributed random vector with covariance matrix C = σ n 2I.
The number of sources K in the scenario is equal to 1.
ρ is a deterministic unknown complex factor.
Again, it must be stressed that the FB suffers of two main drawbacks: the high level of secondary lobes and the Rayleigh resolution limit, which is a problem when K > 1.
2.3 A CS approach to single-snapshot DOA estimation
Some consideration on the linear model in Equation 6 should now be done. It is well known from basic CS theory that in order to reconstruct the sparse signal x using the ℓ1 minimization problem given in Equation 8, the matrix A in Equation 5 must satisfy the restricted isometry property (RIP). It is easy to shown that the matrix A does not satisfy the RIP, since a submatrix composed of a very small number of contiguous columns is already very close to singular . However, in a recent paper , the problem of reconstructing a sparse signal from incomplete frequency samples is discussed and analyzed. In particular, consider a discrete time signal x ∈ ℂ G and a randomly chosen set of frequencies Ω. It has been shown in  that it is still possible to exactly reconstruct x from the partial knowledge of its Fourier coefficients on the set Ω. We return to this result later on, when the super-resolution property is discussed. As it is obvious from the previous discussion, also in the CS-based approach, the spatial frequency ν is assumed to be a discrete variable. It must be noted that recent works deal with the more challenging case of continuous parameter space (see e.g., [23, 25]). However, these recent results fall beyond the scope of this paper.
2.4 CS-based beamformers
In this work, four different algorithms are used to find a feasible solution for the constrained optimization problem in Equation 8, i.e., the classical ℓ1 minimization (L1) algorithm (or least absolute shrinkage and selection operator, LASSO), the fast smoothed ℓ0 minimization (SL0) algorithm, the sparse iterative covariance-based estimator (SPICE) algorithm and the iterative adaptive approach for amplitude and phase estimation (IAA-APES) algorithm. Even if these four algorithms have been derived starting under different hypotheses, we will show that they are strictly related. In the following, a brief description of the main advantages and drawback of each algorithm is provided.
2.4.1 The ℓ 1 minimization (L1)algorithm
In its most general form, the problem in Equation 8 belongs to the well-known class of constrained optimization problem that can be solved using a LASSO solver (see e.g., ). One big advantage of the LASSO algorithm is that it promotes sparse solutions irrespective of the particular noise distribution. On the other hand, the LASSO solver requires the setting of some additional parameters, which have to be chosen heuristically by the user. A wrong choice of these parameters could compromise the convergence of the minimization algorithm. A LASSO-based algorithm is used in  to solve the DOA estimation problem. An example of a critical parameter is the threshold value δ in the constraint of Equation 8. Clearly, δ is a function of the noise covariance matrix C that is, in general, unknown, but there are few theoretical studies on this point and the analytical relation between δ and C has not been explicitly derived so far. Moreover, an estimator of δ from the data snapshot y is not yet available in the literature. For the numerical simulation, the NESTA  algorithm is used to evaluate the LASSO solution of the minimization problem in Equation 8.
2.4.2 The fast smoothed ℓ 0 minimization(SL0) algorithm
where F is some continuous function that approximates the ℓ0 norm. Of course, the SL0 is a suboptimal algorithm for the DOA estimate. In fact, as it can be seen from Equation 9, the SL0 algorithm does not take into account the measurement noise. In , the authors claim that the SL0 is robust with respect to the noise, but there is no theoretical guarantee for this. However, the SL0 algorithm has two advantages with respect to the classical LASSO algorithm: (i) the numerical minimization algorithm (a gradient-based algorithm) is very fast, and (ii) the SL0 algorithm requires the choice of a very small number of critical parameters.
2.4.3 The SPICE algorithm
The SPICE algorithm is an iterative algorithm that, as the previous two algorithms, provides an estimate of a spectrum-like function pSPICE(Ω) of the data snapshot on an assigned set Ω of possible spatial frequencies. The SPICE algorithm was derived for the single snapshot case in  and then generalized to the multi-snapshot case in . The SPICE algorithm has a different and stronger statistical foundation with respect to the LASSO algorithm. Moreover, it does not require any difficult and heuristic selection of parameters, since they are jointly estimated within the iterations. In the following, a brief description of the fundamental concepts behind the SPICE algorithm is provided.
where || · || F is the Frobenius norm. The minimization problem in Equation 13 has an iterative closed form solution [29, 30]. Interestingly, even if they have been derived from two completely different perspectives, the SPICE and the LASSO algorithms are strictly related. This connection is based on the Elfving theorem  and it has been extensively discussed in  and .
2.4.4 The IAA-APES algorithm
The IAA-APES algorithm  is an iterative and non-parametric algorithm that provides an estimate of a spectrum-like function pIAA ‒ APES(Ω) of the data snapshot on an assigned set Ω of possible spatial frequencies. As for the SPICE algorithm, it does not require any selection of parameters and can deal with the single snapshot case. In the following, a brief description of the basic principles of the IAA-SPICE algorithm is provided.
Since to estimate the spectrum-like function , the IAA-APES algorithm requires the matrix T, which itself depends on the unknown signal power, it has to be implemented as an iterative algorithm . Remarkably, as shown in , the IAA-APES algorithm is a close approximation of the ML estimator in the multi-source scenario.
where Q(ξ) has the Toeplitz structure typical of an AR (1) process, i.e., [Q(ξ)]i,j = (ξ|i-j|)* where the asterisk defines the conjugate operator.
3.1 RMSE and CRLB on DOA estimation
In order to evaluate the RMSE, the measurement model in Equation 1 has been adopted with the following parameters:
n is a white, zero-mean, complex Gaussian vector with covariance matrix C = σ n 2I with σ n 2 = 1.
|ρ| is a complex unknown scalar factor with , where SNR is the signal-to-noise ratio.
The number of independent Monte Carlo trials is 104.
The number N of array sensors is 32.
The nominal value of the target spatial frequency is chosen uniformly at random between -0.5 and 0.5, i.e., .
Since the number of grid points is chosen to be equal to 29 for all the beamformers (FB and CSBs), then |ϒ| = |Ω| = G = 29.
where Σ = I - C- 1/2v(v H C- 1v)- 1v H C- 1/2, . For notation simplicity, we omitted the dependence of the steering vector v on the actual spatial frequency . As before, the measurement model in Equation 1 has been adopted with the following parameters:
n is a white, zero-mean, complex Gaussian vector with covariance matrix σ n 2I + σ c 2Q(ξ), where σ n 2 = 1, [Q(ξ)] ij = (ξ|i-j|)*, ξ = 0.98e jϑ , ϑ is uniformly distributed in [0, 2π), and σ c 2 is chosen accordingly to the given clutter-to-noise ratio (CNR) value, . In this simulation, CNR = 15 dB.
|ρ| is a complex unknown scalar factor with , where SINR is the signal-to-interference-plus-noise ratio.
Since the number of grid points is chosen to be equal to 210 for all the beamformers (FB and CSBs), then |ϒ| = |Ω| = G = 210.
All the other parameters are equal to the ones used in the white noise case. As we can see from Figure 2, also in this case, the RMSEs of the five beamformers are very close to each other. However, for high SINR values, i.e., greater than 25 dB, all the CSBs slightly outperform the FB, that is, no more ML estimator in this scenario .
3.2 The ROC curves
where p(·) is one of the ‘periodograms’ described in Section 2.4. For both the FB and the CSBs, the size of the CUT is chosen to be equal to the Rayleigh resolution limit (see Section 4).
In Figure 3, the ROC curves relative to the white-noise-only case are shown. In this case, the FB slightly outperforms the CSBs. However, this behavior is somehow expected since, as shown in Section 3.1, the FB is the ML estimator, so it is, at least asymptotically, the most efficient. Nevertheless, the loss in terms of PD for a given PFA of the CSBs with respect to the FB is small. In particular, we observe that the FB and the SL0-CSB algorithms have almost the same performance.In Figure 4, the ROC curves for the scenario characterized by a spatially correlated clutter model are reported. We note that in this case, all the three CSBs outperform the classical FB. In particular, the SPICE and the L1 algorithms have the best detection performance.
It is well known that the FB suffers from the Rayleigh resolution limit, which is independent of the SNR. Some adaptive methods, e.g., MUSIC and Capon, are able to resolve two sources within a Rayleigh cell. However, as discussed before, to achieve super-resolution, they need a sufficiently high SNR level and a suitable number of temporal snapshots (to estimate the disturbance covariance matrix). In this section, we investigate the super-resolution property of the four proposed CSBs. The results show that unlike Capon and MUSIC estimators, a CSB can achieve the super-resolution with only one temporal snapshot, without the need to estimate the disturbance covariance matrix.
Then, if two sources are spaced by less than the Rayleigh resolution limit Δv, they cannot be resolved by a classical non-adaptive FB. Instead, providing a sufficient level of SNR and a suitable number of grid points, a CSB is able to resolve sources that are in the same Rayleigh resolution cell. The ability of a CSB to achieve super-resolution has been also discussed in recent works. For example, in , the authors investigate the super-resolution property for a CSB for the single and the multi-snapshot scenarios. However, only a qualitative proof of this property is provided, neither a strong theoretical justification nor a statistical characterization of the CS super-resolution capability is reported in . In this paper, a fundamental result of the CS theory is exploited to provide theoretical justification and a rigorous definition of the CS super-resolution capability. Moreover, this property of the CSB is also statistically characterized.
The ability of a CSB to resolve two sources below the Rayleigh limit, even using a single snapshot, is strictly related to the fundamental Theorem 1.3 in . Roughly speaking, this theorem claims that under the sparsity assumption, it is possible to exactly reconstruct (with overwhelming probability) a complex signal x from a very low number of its Fourier coefficients (or anti-coefficient). This theorem is clearly related to the CS beamforming by the Equation 6. In fact, as discussed previously, by assuming that the true source spatial frequencies belong to the set Ω Equation 6 is equivalent to the measurement model of Equation 2, then it is clear from the particular structure of the matrix A(Ω) that the entries of the measurement vector y represent N Fourier (anti-) coefficients of the complex vector x ∈ ℂ G (with N ≪ G = |Ω|) corrupted by noise. Theorem 1.3 can be recast in the following form, more suitable in the array processing framework.
then, with probability at least η = 1 - O(G-d), the solution of the problem in Equation 24 is unique and is equal to x.
It can be seen that while the Rayleigh resolution limit in Equation 23 decreases as N-1, the CS super-resolution limit in Equation 26 decreases as exp ( - C d N/K). In particular, given a fixed number of array elements N, the minimum possible spatial frequency separation between two sources is at least Δν = 0.886N- 1 if FB is used, while it is at least if CSB is used. Roughly speaking, the Rayleigh resolution limit decreases linearly with the number N of sensors, while the CS super-resolution limit decreases exponentially with N.
Finally, some comment needs to be made regarding the constant C d . As stated in , the asymptotic value of C d provided by Theorem 1.3 is a very conservative value. In other words, the same value of Pex can be guaranteed by a smaller value of C d than the one provided in Theorem 1.3 and therefore, a finer resolution can be achieved in practice. Additional theoretical investigations are necessary to refine the value of C d , and this is an active research area within the CS community. A first attempt to refine this constant can be found in .
Theorem 1.3 refers to the noise-free case. Moreover, the true spatial frequency is assumed to belong to the set Ω. In array processing applications, however, since a certain amount of noise is always present, then the optimization problem in Equation 24 should be replaced with the problem in Equation 8. Of course, in the noisy case and in the presence of the off-grid events, one cannot expect exact recovery  and Theorem 1.3 is no longer valid. In the following, the robustness of the previous results on CS super-resolution is verified against the measurement noise and the off-grid effects as a function of the SNR.
In the following, Pres is evaluated as a function of the SNR and of the frequency separation between the two sources. The measurement noise generated in the simulation is white Gaussian.
|ϒ| = |Ω| = G = 29.
ν m = 0.3, while ν1 and ν2 are sampled, in the same Rayleigh resolution cell (Δν ≃ 0.0277), from two uniform and independent probability density functions, such that where μ1 = 0.2922 and μ2 = 0.3078. This allows us to model the grid-off effects.
The number of independent Monte Carlo trials is 103.
In accordance with the previous result on the estimation accuracy (see Figure 1), for SNR lower than 0 dB, all the estimators are in the non-asymptotic region: they do not provide reliable DOA estimates. Beyond 0 dB, the Pres of the four CSBs is much better than that of the FB. In particular, the Pres of the FB is much lower than that of the CSBs and independent of the SNR. Regarding the CSBs, the best estimator, at least in terms of Pres, is the SPICE algorithm: its Pres, as well as the one of the IAA-APES, tends to 1 as the SNR increases.
|ϒ| = |Ω| = G = 210,
ν m = 0.3, while, in order to take into account the grid-off effects, ν1 and ν2 are sampled, in the same Rayleigh resolution cell (Δν ≃ 0.0277), from two uniform and independent probability density function, such that where μ1 = 0.2930 and μ2 = 0.3066.
The number of independent Monte Carlo trials is 105.
As expected, the FB presents a high RMSE and a bias in the estimate of ν1 and ν2, since it is not able to resolve the two sources. The CSBs and the RELAX beamformers that have the super-resolution property provide better estimation performance.
The Cooperative Littoral ASW Behaviour (COLLAB) 2013 experiment has been conducted by CMRE in La Spezia waters, Italy, from 29 June to 7 July 2013. The purpose of the experiment was to test environmentally adaptive, collaborative area search algorithms and behaviors for Autonomous Underwater Vehicles (AUVs) which act as the receiving nodes in the Generic Littoral Intelligent Network Technology (GLINT) Autonomous Sensor Networks (AuSN) antisubmarine warfare (ASW) demonstration system [43, 44].
In this paper, some CS-based beamformers, i.e., the classical ℓ1 minimization (or LASSO), the fast smooth ℓ0 minimization, the SPICE, and the IAA-APES algorithms, have been analyzed and compared with the classical FB for target DOA estimation in a single-snapshot scenario. We analyze the estimation accuracy, the detection performance, and the resolution capability. The performance of the CSBs has been investigated, both in the presence of white Gaussian noise and in the presence of spatially correlated Gaussian noise. Regarding the estimation performance, the RMSE of the FB and of the four CSBs has been compared with the CRLB in the white noise scenario (that is the case when the FB is the ML estimator) and in the spatially correlated noise scenario. As concerning the estimation accuracy and the detection performance, we found that the FB slightly outperforms the CSBs in the white noise scenario, whereas the four CSBs outperform the classical FB in the spatially correlated noise scenario. In particular, the SPICE and the L1 algorithms have the best detection performance, especially at low SNR values. Concerning the resolution capability, we verified that the CSBs can achieve super-resolution beyond the Rayleigh limit even with a single pulse, while classical super-resolution algorithms like MUSIC need multiple snapshots. Theoretical arguments have been proposed here to link the super-resolution property of the CSBs to the CS theory, and a new rigorous definition of CS super-resolution limit has been provided. Moreover, a robustness analysis of the CS super-resolution property has been carried out exploiting a classical method already used in the array processing literature to statistically characterize the MUSIC super-resolution capability in terms of probability of resolution. The simulations have shown that the SPICE algorithm has the best super-resolution capability. Finally, the performance of the four CS-based beamformers has been tested on real sonar data. In particular, the range-spatial frequency maps at the output of the four CS-based beamformers have been evaluated and compared with the map at the output of the Fourier beamformer. From this comparison, the ability of the CS-based algorithms to reduce the secondary lobes and then, to reduce the probability of false alarm, becomes clear. Using the same range-spatial frequency maps, the super-resolution capability of the CS-based beamformers has been verified as well. As concerning the processing time of the four CSBs, both the simulated and the real data analyses have shown that the SL0 algorithm is at least one order of magnitude faster than SPICE, IAA-APES, and L1 algorithms. Since, in many practical applications, a low processing time is a stringent requirement, the SL0 algorithm could represent a good tradeoff between the statistical optimality and the practical implementation.
Future research efforts will be devoted to the multi-snapshot case. Moreover, a deeper comprehension of the statistical properties of the CSBs in different noise and clutter distributions, e.g., the widely known compound-Gaussian distributions, has to be developed.
aThe straddling losses arise because a source is not always precisely centered in a range-Doppler gate, so the acquired sample is not located at the maximum of the matched filter output.
bThe tests were performed on an Intel® Xeon® E5620 2.40 GHz multicore processor (Intel Corporation, Sta Clara, CA, USA) using a Matlab® (MathWorks Inc., Natick, MA, USA) implementation of the three CSBs.
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