Cooperative DoA-only localization of primary users in cognitive radio networks
© Penna and Cabric; licensee Springer. 2013
Received: 2 July 2012
Accepted: 26 March 2013
Published: 19 April 2013
In this paper, we investigate localization techniques based on direction-of-arrival (DoA) measurements to estimate the position of primary users in cognitive radio networks. In the proposed approach, different multi-antenna sensors estimate the DoA of a primary signal by subspace-based techniques and the target position is estimated from the available DoA measurements using maximum-likelihood, least-squares, or Stansfield estimators. The resulting localization performance is evaluated numerically and compared to the Cramr-Rao bound derived under a considered problem setting. The impact of several system parameters (in particular, number of sensors and number of antennas per sensors) is thoroughly analyzed and discussed.
KeywordsCognitive radio Spectrum sensing Localization DoA estimation Cramr-Rao bound
After more than 10 years of research on cognitive radio (CR) systems [1, 2], a large variety of spectrum sensing techniques have been proposed (see, e.g., [3–5]). Even the most sophisticated sensing techniques, however, fail to provide full protection to primary users (PUs) due to unavoidable detection delays (depending on the number of samples needed before a decision is made) and vulnerability to adverse radio conditions (e.g., fading, physical obstacles). This conclusion has been recently acknowledged by the Federal Communications Commission, whose latest memorandum  replaced spectrum sensing by a database-oriented approach such that PU positions and times of activity are known in advance.
Knowledge of PU position, in particular, provides important information for planning and management of a CR network: it can be exploited to identify ‘spectrum holes in space’ , to implement power control mechanisms, to design location-aware routing protocols, and to handle mobility of secondary users (SUs). Also, performance and reliability of a CR network can be greatly improved if SUs are able to localize PU signal sources besides detecting only their presence, like in traditional spectrum sensing. PU localization capabilities are particularly beneficial for cognitive networks operating in spatially diverse radio environments without databases.
The problem of PU localization in CR networks has attracted some attention over the past few years. For example, in , a Bayesian approach based on range and phase measurements is proposed, while a semi-range method is introduced in . However, range-based approaches are difficult to apply in practice because the transmission power of PUs is typically unknown as well as the propagation channel between PUs and SUs. In addition, theoretical bounds and approximate maximum-likelihood algorithms based on received signal strength measurements were derived in , showing that the achievable estimation accuracy is relatively poor if all PU transmission parameters are unknown and SUs can rely only on energy measurements. Therefore, different approaches need to be explored for application in CR networks. Time-difference-of-arrival (TDoA) techniques could be adopted in principle, but they require perfect synchronization among different SU receivers, which is not always possible in CR systems. A weighted centroid approach was proposed in , which is suitable for dense CR networks with many sensors close to the PU position.
In this paper, we consider PU localization based on direction-of-arrival (DoA) measurements only. As an advantage compared to range-based localization, the DoA approach does not require knowledge of PU transmission power nor of the propagation channel. This localization technique, often referred to as bearings-only target location, was originally proposed and analyzed in the navigation literaturea (see, for example, [12–16]).
DoA estimates can be obtained in different ways, for example, by directional antennas or antenna arrays . This work concentrates on the latter case. Note that arrays may be multiple antennas of one receiver or multiple users cooperating with each other (‘virtual arrays’). In the following, we will use the general term sensor to denote a sensing unit (single device or virtual array) composed of a number m > 1 of antennas. Each sensor can estimate the DoA of the PU signal using classic array processing techniques  such as the multiple signal classification (MUSIC) algorithm .
The contributions of this paper are as follows:
Derivation of Cramér-Rao bound (CRB) of the achievable PU localization accuracy as a function of number of sensors in the network, number of antennas per sensor, distance of the sensors from the target, and array orientation with respect to the target.
Comparison of the above bound with practical algorithms. It is shown that a combination of MUSIC and Stansfield algorithm  can nearly reach the CRB in conditions of practical interest for CR networks.
Study of the impact of number of sensors and number of antennas on localization accuracy and numerical evaluation of the trade-off between these two factors.
We remark that results on the CRB of multi-sensor localization were already derived in the navigation literature (e.g., [13–16]) and later applied to wireless sensor networks (see, e.g., the survey paper ). However, our work is, to the best of our knowledge, the first to consider the joint effect of the number of sensors and the number of antennas per sensor in a multi-sensor, multi-antenna target localization system. The results of this paper show the impact of several parameters (number of sensor elements, choice of DoA fusion algorithm, array orientation, signal-to-noise ratio) on the overall localization performance. Applied to cognitive radio networks, these results provide important design guidelines for implementing practical PU localization systems. In this paper, as common in the localization literature, we adopt the position root mean square error (RMSE) as performance metric to evaluate the proposed algorithms. The RMSE has a theoretical justification, in that it allows for direct comparison with the CRB. In practical CR applications, an error in the estimation of PU position (expressed by the RMSE) leads to an erroneous identification of the boundaries of a spectrum hole in space and, ultimately, to a reduction of the throughput for the secondary network or to an increase of interference for the primary network. While the relation between PU localization accuracy and PU/SU communication performance is strongly application-dependent, minimizing the PU position RMSE translates in general into maximizing the efficiency of dynamic spectrum usage.
The paper is organized as follows: the problem model is introduced in section 2. In section 3, we briefly state preliminary results on the DoA estimation variance achieved by the MUSIC algorithm and its corresponding CRB. In section 4, we address the problem of localization by combining DoA estimates of multiple sensors: we first derive the CRB of the achievable localization error, highlighting its dependence on distances and angles between sensors and target, and then we introduce practical algorithms: maximum-likelihood (ML), least-squares (LS), and Stansfield estimators. In section 5, we evaluate by numerical simulations the performance of the proposed methods for DoA estimation and localization; we study the impact of various system parameters and investigate the trade-off between number of sensors and number of antennas per sensor. In section 6, we discuss implementation issues of the proposed method. Section 7 concludes the paper and summarizes the key results.
2 Mathematical model
is the array response matrix. If primary signals are uncorrelated, . We then define the signal-to-noise ratio (SNR) of signal i as .
Remark 1. By convention, the DoA support is limited to one semi-plane, e.g., θi,j∈ [− π /2, π /2] in Equation 2. This choice is derived from the π -ambiguity of MUSIC and other subspace-based methods using uniform linear arrays with isotropic antennas. Such ambiguity can be resolved if some prior information about PU location is available (e.g., a coarse, initial estimate obtained by directional antennas) so that the reference semi-plane is known in advance.
3 DoA estimation error
3.1 Cramér-Rao bound
and . Note that A is a short-hand notation for .
The factor can be interpreted as the effect of the orientation of the ULA with respect to the line between target signal and sensor. The DoA estimation CRB is minimized when the ULA is perpendicular to such line (), whereas it tends to ∞ when the ULA is aligned to the same direction of the received signal ().
3.2 Practical algorithms
which converges to the corresponding CRB (Equation 5) as the SNR ρ i increases. The symbol [X] i j represents the element (i,j) of a matrix X.
4 Localization error
4.1 Cramér-Rao bound
where, recalling that the MUSIC estimator is asymptotically unbiased and Gaussian distributed (see section 3.2), error variables wi,j are modeled as zero-mean Gaussian random variables: .
These expressions show that the estimation error variance scales linearly with the DoA estimation variance and quadratically with the target distance.
for k = 0,1,2. This result is consistent with , where it was shown from geometric considerations that non-observability occurs if the sensors are placed on a straight line passing through the target. For M=2, this condition corresponds exactly to θi,1 − θi,2 = π.
A physical interpretation is that the ambiguity of π in DoA estimation makes it impossible to distinguish between angles whose difference is π.
If we increase the number of sensors (M ≥ 3), configurations such that the denominator of Equation 26 is 0 (i.e., all sensors lying on the same straight line) become events with zero probability. We conclude that the minimum number of sensors should be M = 3.
4.2 Practical algorithms
In this section, we introduce different DoA-only localization methods (ML, LS, and Stansfield estimators) and discuss their application in CR networks. Before continuing, the following preliminary observations need to be made:
For simplicity of presentation, we consider M DoA measurements to be collected by the fusion center at the same time. In practice, such measurements may also be obtained during different time slots. In this case, the considered estimators are implemented through iterative numerical methods (which, in fact, can be used in the case of simultaneous measurements as well). However, multiple measurements per sensor are beneficial only if sensors move; otherwise, they may result in matrix singularities which ultimately degrade the estimation quality.
We do not consider the extended Kalman filter because this method provides poor performance when applied to DoA-only localization, as proven in .
While CRB results presented in section 4.1 hold for an arbitrary number of coexisting signals (as long as n<m), in practice, data association of multiple DoAs observed by multiple sensors is a non-trivial problem (precisely, it is NP-hard. See, e.g.,  and references therein). For this reason, in this paper, we assume ideal identification of PUs by SUs, i.e., either there is a single PU observed by multiple sensors or there are multiple PUs, but each of them can be unambiguously identified by SUs by exploiting other information, e.g., detection of carrier frequency, bandwidth, modulation type. We thus remove index i in the following (e.g., θ j ≡ θi,j).
4.2.1 Maximum-likelihood estimator
The minimization of Equation 29 is a non-linear least-squares problem and can be performed by iterative methods. The most popular of these methods is the Gauss-Newton algorithm [15, 16], but it is very sensitive to the initial value. More robust iterative methods are the Levenberg-Marquardt algorithm  or the trust-region-reflective (TRR) algorithm .
As shown in , the ML algorithm, asymptotically in the number of measurements M, is unbiased and achieves the CRB. However, we will show in section 5.2 that the theoretical performance of the ML estimator is degraded when adopting iterative implementations as cases of non-convergence often occur.
4.2.2 Stansfield estimator
and . Note that the Stansfield estimator involves the distances R j between sensors and target; however, as discussed in  and confirmed by simulation results (see section 5.2), the dependency of the estimator on R is quite weak. For this reason, if distances are unknown (as it is the case in CR networks), R can be roughly estimated from the received signal strength or simply replaced by an identity matrix. In the rest of the paper, we will consider the latter case (worst-case scenario).
The Stansfield estimator was shown  to provide excellent performance in terms of mean square error (which can be even smaller than that of the ML estimator when M is small), but it was also shown to be biased (hence, not directly comparable to the CRB).
4.2.3 Least-squares estimator
The linear LS estimator is not asymptotically unbiased: as shown in simulation results, its estimation accuracy tends to saturate as M grows.
5 Numerical results and analysis
In this section, we investigate the performance of the methods presented in the previous sections as a function of several systems parameters. We first consider DoA estimation (section 3): we compare the performance of MUSIC against the CRB and show how DoA estimation accuracy scales with the number of antennas m. We then consider the DoA-based localization algorithms presented in section 4: we compare the position error achieved by LS, ML, and Stansfield estimators and analyze the impact of the number of sensors M as well as the number of antennas per sensor m. We finally investigate the trade-off between the above parameters, in case of ideal conditions (uniform distance of all sensors from target, optimal orientation of sensor arrays) or realistic conditions (random distance and orientation).
5.1 DoA estimation
The DoA estimation variance achieved by the MUSIC algorithm using a ULA is given by Equation 14, and the corresponding CRB is given by Equation 12. In order to compare MUSIC performance against the CRB, we evaluate the DoA RMSE, i.e., the square root of the average (Δ θ)2 over several simulations, with Δ θ defined as in Equation 30. Without loss of generality, we assume ideal orientation of the array, such that in Equations 12 and 14. Different array orientations would result in a constant (in logarithmic scale) penalty term.
In Figure 2, the DoA RMSE, averaged over 104 iterations, is plotted as a function of the number of antennas m for different values of SNR (ρ = 0 dB and ρ = −5 dB). The number of received signal samples is set to N = 100. The following observations can be made:
Validation of MUSIC performance expressions. The formula of MUSIC estimation variance Equation 14 matches with the simulation results.
MUSIC vs. CRB. The gap between MUSIC estimation error and CRB is small and tends to reduce as the SNR increases (see section 3.2).
Impact of number of antennas. The DoA estimation accuracy (for both MUSIC and CRB) monotonically improves with m, following an almost linear trend on a logarithmic scale. No saturation is observed for practical values of m (i.e., m ≤ 10), which suggests that the more antennas are used by each sensor, the more accurate DoA estimates (and, therefore, PU localization) would be obtained.
We now analyze the performance of the ML, LS, and Stansfield localization methods introduced in section 4.
5.2.1 Error distributions
computed over 1,000 simulations. The true PU position is set, without loss of generality, to x P = [0,0] T .
At every simulation, M sensor positions are selected randomly on a unit-radius circle centered in x P so that all sensors are at equal distance (R j = 1 unit) from the target. The DoA estimation variance of each sensor () is random, uniformly distributed between 10° and 30°. For each of the considered estimators, we test two versions: weighted (W) estimators (W-ML, W-Stansfield, W-LS) that assume knowledge of the values , i.e., of the matrix W in Equations 30, 32, and 36, respectively, and non-weighted estimators (ML, Stansfield, LS) that replace W by an identity matrix, thus assigning equal weight to all sensors.
The ML method is implemented iteratively, starting from an initial guess selected randomly at each Monte Carlo simulation, with σ0=1.5 distance units. The TRR implementation is adopted as it proves to be more robust than the Gauss-Newton algorithm (which does not converge if the initial guess is far from the true value).
Results, shown for three cases M = 3, M = 5, and M = 8, reveal the following:
The Stansfield method provides the best performance among the three considered estimators. It reaches convergence in 100% of cases, while the ML CDF saturates around 90/95% due to cases of non-convergence (which become more frequent as M increases). Notice that suboptimality of the ML estimator is due exclusively to iterative implementation, which is sensitive to the initial estimate and may converge to local minima. The Gauss-Newton implementation provides even worse results, i.e., it fails to converge in 20%/25% of cases in the considered setting. The LS solution is close to the Stansfield solution for M = 3, but it becomes much more suboptimal as M grows.
Weighted estimators provide in general a modest performance improvement over non-weighted ones, for the considered DoA variance range (). The gap increases for large M when more different DoA measurements (hence more weights) are available.
5.2.2 Position RMSE and CRB vs. number of sensors
We now compare the performance of ML, LS, and Stansfield estimators against the CRB derived in section 4.1, and we study their evolution with the sensor number M. To this purpose, we consider the position RMSE computed as the square root of the average square error over multiple iterations and random sensor positions. The lower bound of the position RMSE for unbiased estimators is given by the square root of the average of (see section 4.1) over the same random sensor positions.
In our simulation, we assume sensors to be equidistant from the target PU (R j = 1 distance unit ∀ j) and, at every new random configuration, we generate M sensor positions as with ϕ j ∼U [0,2π). Angles ϕ j are defined as the angles of sensors j with respect to target, i.e., ϕ j = (θ j − π) mod 2 π; hence, angles θ j are also random with the same uniform distribution as ϕ j . We generate 100 random sensor positions (i.e., 100 random vectors [ϕ1,…,ϕ M ]), where we compute the CRB and test the performance of LS, ML, and Stansfield methods by running 1,000 Monte Carlo simulations per configuration.
The simulation is repeated for values of M from 2 to 10. Note that LS and Stansfield estimators are not directly comparable to the CRB because they are not unbiased (so their RMSE could be, in principle, lower than the CRB); however, the CRB still represents an important benchmark to evaluate the quality of the position RMSE achieved by practical algorithms. For M = 2, when generating [ϕ1,ϕ2], we impose the constraints 5° <|ϕ1⊖ϕ2|<175° so as to avoid configurations close to non-observability conditions (see section 4.1) that would result in RMSE and CRB → ∞. In this case, the DoA estimation variance is constant () so that weighted and non-weighted algorithms are identical.
5.2.3 Position RMSE and CRB vs. DoA (for M = 2)
Values of θ Δ from 0 to 2π are considered, with a step of π/20. It can be observed that, as predicted by the theory, θ Δ = 0 and θ Δ = π are non-observability conditions; therefore, both the CRB and the Stansfield error tend to diverge as θ Δ approaches such critical values. The divergence of the CRB is sharp, while the Stansfield method exhibits a smoother behavior. The Stansfield RMSE nearly attains the CRB for values of θ Δ far from 0 and π, and it becomes suboptimal in the regions close to non-observability.
5.2.4 Position RMSE vs. number of antennas
We observe that, both from the point of view of the CRB and of practical algorithms, the position RMSE is reduced by increasing the number of antennas. Thus, the localization accuracy as a function of the number of antennas is not affected by saturation phenomena that occur when increasing the number of sensors (see Figure 4). We also note that the RMSE of the Stansfield algorithm tends to achieve the CRB as m→∞.
5.3 Trade-off between number of sensors and number of antennas
So far, we have analyzed the position accuracy as a function of different system parameters, in particular, the number of sensors M and the number of antennas m. A natural question may be asked: which one of these two factors is more important with regard to PU localization accuracy? To address this question, let us consider a CR network with a total of K antennas; assume that such antennas can be arbitrarily organized into arrays (for simplicity, ULAs) to localize a PU transmitter; then, if every array has m elements, the number of arrays (hence, of DoA measurements) is M = ⌊K/m⌋.
We then formulate the problem as follows: given a fixed number of antennas K, we want to determine the optimal trade-off [m,M] between number of antennas per sensor and number of sensors, with m×M≤K. It has been shown in Figure 6 that increasing m constantly improves the localization accuracy; on the contrary, increasing the number of sensors besides a certain value (around M = 4) does not provide significant advantages, as shown in Figure 4. For this reason, we expect that the best trade-off will be obtained for m>M.
In ideal conditions (Figure 7), the optimal trade-off is located, as expected, in the region m > M for all curves. In particular, according to the CRB and to the Stansfield method, the optimal combination is [m = 9,M = 4], while using the LS method, the optimum is at [m = 12, M = 3]. Such discrepancy is explained by the results of Figure 4: for a given DoA estimation variance, the LS localization RMSE does not significantly improve from M = 3 to M = 4; therefore, it is more advantageous to keep M = 3 and increase the number of antennas from 9 to 12 so as to improve . We note, then, that the gap between the two CRBs is very moderate (i.e., MUSIC by itself provides excellent DoA estimation performance, as already observed in Figure 2), and that the Stansfield RMSE equals CRB() except for M = 3 (cf. Figure 4).
When introducing variable distances (Figure 8), a similar behavior is observed. The only difference is that the Stansfield estimator becomes slightly suboptimal compared to the CRB because we assume distances to be unknown, so R is still considered an identity matrix in Equation 32. The conclusion is that different distances of sensors from the target only have a minor impact on the resulting localization accuracy,
Very different results are obtained when considering non-ideal array orientation (Figure 9). Note that the assumption of random uniformly distributed may correspond either to a scenario where each sensor has a different, random orientation or to a scenario where all sensors have equal orientation (in this case, + a constant offset; therefore, is random like θ j ). We consider here two variants of the LS and Stansfield estimators: one where the orientation is known and coefficients are incorporated in the weight matrix W, and one where the orientation is unknown, so W = I. Figure 9 shows that ‘weighted’ estimators, W-LS and W-Stansfield, achieve good performance (close to the CRB for W-Stansfield), while standard estimators are strongly suboptimal and provide nearly constant RMSE for all values of m and M. The best trade-off is located at values of M lower than in the previous scenarios: [m = 6, M = 6] for Stansfield and CRB, and [m = 9,M = 4] for LS, i.e., one step less than in the other cases (Figures 7 and 8). This behavior can be explained by observing that, when each sensor has a random orientation, more ‘sensor diversity’ is necessary to achieve good performance. In terms of absolute values, the best RMSE achievable with random orientation is approximately 0.02, while it is approximately 0.008 in the case of ideal orientation. The gap is approximately 4 dB.
6 Implementation issues
Availability or multi-antenna SU devices (possibly with rotation capabilities),
Existence of a wired or wireless link for sending DoA measurements from distributed sensors to a fusion center, and
Sufficient processing power and energy supply to support DoA estimation at the sensor nodes and DoA fusion at the fusion center.
Regarding the first point, multi-antenna cognitive radios are nowadays a common assumption in the spectrum sensing literature (see, for example,  and references therein) and can be therefore exploited to combine DoA estimation with traditional signal detection with little additional complexity. Furthermore, the number of antennas we considered in our numerical examples is within practical limits of current base stations and Wi-Fi access points. We also remark that, if antenna arrays are not available, directional antennas or electronically steerable antennas  can be used as well. A more challenging requirement is the availability of rotating arrays, which may be feasible only in certain scenarios (e.g., vehicle terminals, dedicated spectrum monitors, base stations). However, the availability of such type of devices is not a prerequisite of the algorithm but only a possible way to optimize performance (see Figure 10). In the absence of rotating arrays, localization performance can be still improved by increasing the number of sensors or using proper weighting matrices (see Figure 9).
Coming to the second point, let us first consider the case of physical antenna arrays (SUs equipped with multiple antennas). The proposed algorithm involves exchange of DoA measurements from M−1 distributed sensors to a fusion center (it is assumed that the fusion center itself is one of the M cooperating sensors). In addition, if the array orientation of sensor nodes is unknown to the fusion center, M−1 additional angle measurements should be sent from the distributed sensors in order to establish a common coordinate system. Hence, if a scalar angle measurement is encoded by one ‘information unit’ (e.g., 1 byte), the signaling overhead of the algorithm amounts to 2(M − 1) = O(M) information units, i.e., it scales linearly with the number of sensors. Given the moderate bandwidth requirement, the algorithm lends itself well to both wired and wireless implementations. In the latter case, the sensors can communicate to the fusion center via a dedicated channel (‘common control channel’ [26, 27]), which may be either predefined or selected dynamically after a preliminary spectrum sensing phase. In the case of virtual arrays (m single-antenna users performing MUSIC cooperatively), there is an additional signaling overhead within each sensor array. Assuming that one user acts as a local fusion center, each of the remaining M − 1 users must transmit N received signal samples over the wireless channel, hence the signaling overhead per sensor is N(m − 1) complex numbers. In this case, the number of samples (N) should not be too large, and synchronization between the users within a virtual array is crucial.
Finally, numerical complexity (and, consequently, energy consumption) is dominated by the eigenvalue or singular value decomposition (EVD/SVD) necessary to implement MUSIC at each sensor node. The number of floating-point operations required for EVD (applied to the m × m covariance matrix) is ≈ 10m3, while for SVD (applied to the m × N data matrix) it is ≈k1N m2+k2m3, where k1,k2 are constants [28, 29]. Therefore, in general, the complexity of EVD/SVD scales as O(m3). Clearly, this task may be critical in terms of processing power and battery consumption for sensor nodes. We note, however, that (1) according to our simulation results, accurate localization can be achieved with a limited number of antennas per sensor, resulting in an affordable complexity for typical state-of-the-art embedded processors (for a practical DSP implementation of MUSIC, we refer to ); (2) although we have considered here the classic MUSIC algorithm, low-complexity variants exist in the literature (see, for example, [31–33]). The DoA fusion step (using LS or Stansfield estimation) then involves solving a system of linear equations (Equations 32 and 34). This task can be performed by LU factorization and Gaussian elimination  with complexity O(m3). Since it is reasonable to assume for the fusion center higher computation power than the other sensor nodes and wired connection to the network backbone, the computational burden of DoA fusion is generally less critical than the one involving distributed sensors.
Signaling overhead and numerical complexity
Signaling overhead (virtual array case)
We finally remark that the proposed method is robust to possible failures of sensor nodes (due to low battery or other reasons), as long as at least three sensors remain operational.
In this paper, we have investigated PU localization in CR networks by DoA-based techniques, where DoA estimates are obtained by multi-antenna sensors. The key advantage of the proposed approach is that it does not rely on range measurements (difficult to obtain in a CR network where PUs do not collaborate with SUs) nor on precise synchronization of different sensors (needed for TDoA techniques).
The following conclusions can be drawn from the results presented in the paper:
Very accurate PU localization can be obtained using the MUSIC algorithm for DoA estimation at each sensor and the Stansfield method to combine DoA measurements together. In this way, the PU position RMSE nearly achieves the CRB.
Other DoA fusion methods, namely the linear LS estimator and the ML estimator implemented iteratively, are inferior to the Stansfield estimator.
In order to maximize the localization accuracy, the effect of the number of antennas per sensor prevails over the effect of the number of sensors.
If sensor arrays have random orientation, the performance is affected by a major penalty factor, which should be compensated by properly weighting DoA measurements from different sensors or rotating sensors (when possible) so as to optimize their orientation.
aIn the navigation context, it typically assumed that multiple observations are obtained by a unique sensor at different positions (i.e., a maneuvering ship), whereas in CR networks we assume that observations are taken by multiple sensors at the same time, and then gathered by one node acting as a fusion center.
bIf subspace methods (e.g., MUSIC) are used, the maximum number of identifiable primary signals is M − 1. Thus, we assume n<m.
cThe adopted formulation ensures that the difference between two arbitrary angles is always computed correctly (e.g., |1° ⊖351° | = 10°. On the contrary, |1° − 351 ° | = 1 ° ⊖351° = 350° is formally correct too, but leads to overestimate an error that is in fact small, which is a problem for iterative implementation of the ML method).
dTRR is implemented using the Optimization Toolbox of Matlab v7.8.
eAn exact ML solution would give at least the same performance of the Stansfield estimator, which is in fact an approximation of the ML estimator for Δ θ →0. However, an ML implementation through exhaustive search (e.g., on a grid of points) would be prohibitive in terms of complexity and still suboptimal due to discretization of the search space.
fThe main source of complexity of DoA estimation is given by the eigenvalue decomposition of the covariance matrix, which is also required for most of the existing multi-antenna detection techniques.
gThe sensors are assumed to be aware of their own position as well as that of the fusion center so that they are able to estimate the relative angle. This step is not necessary if the positions of sensor nodes are fixed and/or known in advance at the fusion center.
FP received his M.S. degree (with Hons.) and Ph.D. degree in communications engineering from the Politecnico di Torino, Italy in 2008 and 2012, respectively. He is currently a research associate with the Fraunhofer Heinrich Hertz Institute in Berlin, Germany. His research interests include cognitive radio networks, distributed signal processing, multi-sensor signal detection, and cooperative localization. DC received her Dipl. Ing. degree from the University of Belgrade, Serbia in 1998 and her M.Sc. degree in electrical engineering from the University of California, Los Angeles (UCLA) in 2001. She received her Ph.D. degree in electrical engineering from the University of California, Berkeley in 2007. She was a member of the Berkeley Wireless Research Center, University of California, Berkeley. In 2008, she joined the faculty of Electrical Engineering, UCLA, as an assistant professor. Her key contributions involve the novel radio architecture, signal processing, and networking techniques to implement spectrum sensing functionality in cognitive radios. DC was awarded a Samueli Fellowship in 2008 and an Okawa Foundation research grant in 2009.
This research was supported by the National Science Foundation under CNS grant 1149981 and by the German Federal Ministry of Education and Research under grant 01BU1224. Some preliminary results of this paper were presented at the IEEE Global Communications Conference (GLOBECOM), Houston, TX, USA, December 2011 .
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