Localization and identification of unknown target signal using oblique projection

This subsection presents the signal model which is considered for source localization and waveform identification. Assume that K far-field narrowband signals impinge upon an antenna array from distinct directions θ₀,θ₁,θ₂,⋯,θ_K−1. The antenna array consists of N sensors with arbitrary array geometry. The N×1 vector of received signals at time t can be expressed as [3]

where

and

where s _k (t) refers to the baseband signal waveform of the kth incident signal, and a(θ _k ) denotes the corresponding steering vector. n(t) represents the additive noise component with covariance \(\mathbf {R}_{n} = \mathbb {E}\left [\mathbf {n}(t)\mathbf {n}^{\mathrm {H}}(t)\right ]\). \(\mathbb {E}[\cdot ]\), (·)^T and (·)^H stand for statistical expectation, transposition and Hermitian transposition, respectively.

Both the received signals and the noise are assumed to be sampled from zero-mean and uncorrelated stationary random processes. Furthermore, the noise is temporally and spatially independent of the received signals, and R _n =σ²I, where σ² is referred to as the noise power and I stands for the identity matrix. Thus, the covariance matrix of the received data is given by

where \(\mathbf {R}_{S} = \mathbb {E}\left [\mathbf {S}(t)\mathbf {S}^{\mathrm {H}}(t)\right ]\).

This subsection formulates the localization and waveform identification problem. Without loss of generality, the signal s₀(t) is assumed to be the SOI and the remaining K−1 received signals are treated as interference signals. For the SOI and the interference signals, neither the source locations (i.e., θ₁,θ₂,⋯,θ_K−1) nor the signal waveforms (i.e., s₀(t),s₂(t),⋯,s_K−1(t)) are prior known.

The localization and waveform identification problem addressed in this paper is to obtain the DOA θ₀ and waveform s₀(t) from multiple snapshots \(\left \lbrace \mathbf {x}\left (t_{l}\right) \right \rbrace _{l=1}^{L}\), where L represents the number of snapshots.

In the next section, it will be shown that the waveform of the SOI can be identified as follows, i.e.,

where W₀ refers to the optimum weight, and

And, according to (6), the location of the SOI is given by the conventional method of single target localization.

To illustrate the principle of localization and waveform identification of the SOI, the DOAs of all incident signals are temporarily assumed to be known in this subsection. According to (1) and (6), it is natural that the SOI component in the received data can be given by

where

and

where the symbol ∖ signifies set difference.

To separate the SOI from the multiple incident signals by (7), we employ oblique projection to obtain the interference signals, i.e.,

where \(\phantom {\dot {i}\!}\mathbf {E}_{\mathbf {B}|\mathbf {a}\left (\theta _{0}\right)}\) denotes the oblique projection operator whose range space is \(\mathcal {R}\left \lbrace \mathbf {B}\right \rbrace \) and whose null space contains \(\mathcal {R}\left \lbrace \mathbf {a}\left (\theta _{0}\right)\right \rbrace \), and

where

and

where \(\mathcal {R}\left \lbrace \cdot \right \rbrace \), (·)^⊥, and (·) ^† denote range space, orthogonal complement, and Moore-Penrose pseudoinverse, respectively.

According to (7), only the SOI is contained in x₀(t _l ), and the interferences signals are removed. Thus, the SOI can be estimated by solving the following minimization problem, i.e.,

Solving (15), it is obtained that

and

where \(\mathbf {P}_{\mathbf {a}(\theta)}^{\bot } = \mathbf {I} - \mathbf {a}(\theta)\mathbf {a}^{\dagger }(\theta)\), and \(\mathbf {R}_{0} = \mathbb {E}\left [\mathbf {x}_{0}(t)\mathbf {x}^{\mathrm {H}}_{0}(t)\right ]\).

Comparing (5) and (16), it follows that the optimum weight W₀ to obtain the SOI is given by

where (·)⁻¹ denotes inverse.

Using (5), (7), (10), (17), and (18), both the location θ₀ and the waveform s₀(t) of the SOI can be computed. Besides, the localization and waveform identification processes for the SOI can also be extended, i.e., the localization and waveform identification process for the interference signals are feasible if each of the interference signals is regarded as the SOI.

Recall that the localization and waveform identification process in Section 3.2 requires known DOAs of all the incident signals, which is infeasible in practical applications. In this paper, it is considered that the DOA of the SOI is prior unavailable, and the prior DOAs of the interference signals are also unknown. Toward the purpose, the presented localization and waveform identification process is implemented iteratively. The proposed OPLI firstly estimate the parameters of the SOI, so that a rough information of the SOI is available. And then, the parameters of each interference signals are updated one after another. Through iteration, not only the parameters of the SOI are estimated, but also the parameters of the interference signals are gradually achieving high precision.

At the ith iteration, the estimated DOAs \(\hat {\theta }_{{0}}^{(i)}\), \(\hat {\theta }_{1}^{(i)}\), ⋯, \(\hat {\theta }_{k-1}^{(i)}\), \(\hat {\theta }_{k}^{(i-1)}\), \(\hat {\theta }_{k+1}^{(i-1)}\), ⋯, \(\hat {\theta }_{{{K-1}}}^{(i-1)}\) are utilized as initial values to compute \(\hat {\theta }_{k}^{(i)}\) and \(\hat {s}_{k}^{(i)}(t)\), where i≥1 and k=0,1,2,⋯,K−1. \(\hat {\theta }_{k}^{(i)}\) and \(\hat {s}_{k}^{(i)}(t)\) denote the estimation of θ _k and s _k (t) at the ith iteration (similarly hereinafter), respectively.

If the kth incident signal is regarded as the SOI, using (7) ∼ (10) and (17), it follows that

where

and

where

and

Besides, using (5) and (18), it follows that

where

During the proposed iterative process, the relative change of the following cost function, which is derived from maximum likelihood approximation [53], between two consecutive iterations is utilized to indicate the convergence, i.e.,

Referring to the well-known relaxed iterative approach in [50, 51], the localization and waveform identification procedure of the proposed OPLI is summarized in Table 1. It is shown that the proposed OPLI starts with the initialization \({\mathbf {a}\left ({{{\hat \theta }_{0}^{(-1)}}}\right)} = {\mathbf {0}}\), \({\mathbf {a}\left ({{{\hat \theta }_{1}^{(-1)}}}\right)} = {\mathbf {0}}\), ⋯, \({\mathbf {a}\left ({{{\hat \theta }_{K-1}^{(-1)}}}\right)} = {\mathbf {0}}\). At the ith iteration, \(\hat {\theta }_{j}^{(i)}\) is determined within the range scope \(\left [\hat {\theta }_{j}^{(i-1)}-\epsilon _{j}, \hat {\theta }_{j}^{(i-1)}+\epsilon _{j}\right ]\), where K−1≥j≥0, and ε _j is a user-selected parameter which indicates the range of the jth signal in the spatial domain.

In this subsection, the computational complexity of the proposed OPLI is analyzed.

Using L number of snapshots, the computation of matrix \(\hat {\mathbf {R}}_{k}^{(i)}\) in (20) can be given as \(\sum _{l = 1}^{L}\hat {\mathbf {x}}_{k}^{(i)}\left (t_{l}\right) \left (\hat {\mathbf {x}}_{k}^{(i)}\left (t_{l}\right)\right)^{\mathrm {H}}/L\), which takes about O(LN²+N²) flops. Herein, a flop is defined as a complex floating-point addition or multiplication operation. The number of flops roughly required to compute (21) is O(N(K−1)L+NL) flops. The calculation of (22) requires approximately O((K−1)³+2N(K−1)²+N²(K−1)+(K−1)NL) flops, where the calculation of \(\mathbf {E}_{\hat {\mathbf {B}}_{k}^{(i)}|\mathbf {a}\left (\hat {\theta }_{k}^{(i-1)}\right)}\) additionally takes about O(3N²(K−1)+2N(K−1)²+(K−1)³+2N²+N) flops. Thus, the computational complexity of (20) ∼ (22) is roughly O(N²(L+4K−1)+N(4(K−1)²+(2K−1)L+1)+2(K−1)³) flops in total.

The computation of \(\hat {\theta }_{k}^{(i)}\) in (19) requires roughly \(O\left (\left (N^{3} + 2N^{2} + 2N\right)\tilde {N}_{\hat {\theta }_{k}^{(i)}}\right)\) flops, where \(\tilde {N}_{\hat {\theta }_{k}^{(i)}}\) denotes the number of potential source locations in the region scope \(\left [\hat {\theta }_{k}^{(i-1)}-\epsilon _{k}, \hat {\theta }_{k}^{(i-1)}+\epsilon _{k}\right ]\). The computation of \(\hat {s}_{k}^{(i)}(t)\) in (25) requires roughly O(NL) flops, where the calculation of \(\hat {\mathbf {W}}_{k}^{(i)} \) additionally takes about O(2N+1) flops. The computation of \(\mathcal {L}^{(i)}\) in (27) requires roughly O((K+2)NL) flops.

Therefore, according to Table 1, the computational complexity of the proposed OPLI is roughly \(O\left ((K + 2)NL\right) + O\left (\sum _{k=0}^{K-1}\left (\left (N^{3} + 2N^{2} + 2N\right)\tilde {N}_{\hat {\theta }_{k}^{(0)}} + N^{2}(L+4K-1) +\right.\right. N\left (4(K-1)^{2} +(2K-1)L+1\right) + 2(K-1)^{3} + NL + 2N + 1 +\sum _{i=1}^{N_{r_{k}}}\left (\sum _{j=0}^{k}\left ((N^{3} \,+\, 2N^{2} \,+\, 2N)\tilde {N}_{\hat {\theta }_{j}^{(i)}} \,+\, N^{2}(L\,+\,4K-1) +\right.\right. N(4(K-1)^{2} +(2K-1)L+1) + 2(K-1)^{3} + NL + 2N \!\left.\left.\left.\left.\!\!\!\!{\vphantom {\left (\sum _{k=0}^{K-1}\left (\left (N^{3} + 2N^{2} + 2N\right)\tilde {N}_{\hat {\theta }_{k}^{(0)}} + N^{2}(L+4K-1) + N\left (4(K-1)^{2} +(2K-1)L+1\right) + 2(K-1)^{3} + NL + 2N + 1\right.\right.}} + 1 \right)+ (K + 2)NL\right)\right)\right)\)flops in total, where \(N_{r_{k}}\) denotes the number of iterations employed in the kth outer loop. Particularly, the total complexity is approximately \(O\left (\sum _{k=0}^{K-1}\left (N^{3}\tilde {N}_{\hat {\theta }_{k}^{(0)}} + \sum _{i=1}^{N_{r_{k}}} \sum _{j=0}^{k}N^{3}\tilde {N}_{\hat {\theta }_{j}^{(i)}} \right)\right)\) flops, when L≫N>K, \(\tilde {N}_{\hat {\theta }_{k}^{(i)}} \gg L/N\), which occurs often in practical applications.

This subsection evaluates the waveform identification performance of the proposed OPLI. The minimum variance distortionless response (MVDR) beamformer [36], the diagonally loaded sample matrix inversion (LSMI) beamformer [38], the general linear combination (GLC)-based beamformer [39], and the optimal beamformer which is based on the maximum output SINR principle [46] are employed for performance comparison. The diagonal loading factor of the LSMI beamformer is set to be equal to the noise power.

Note that both OPLI and RELAX implement waveform identification without prior DOA information. Whereas the MVDR, LSMI, and GLC beamformers require prior DOA of the SOI to calculate beamforming weights. Unless otherwise stated, in the subsection, it is assumed that the prior DOA is considered without error.

Figure 1 presents the output SINR of the proposed OPLI, where the input SNR ranges from −20 to 30 dB. K=3 signals are considered, and L=200 snapshots are employed. The SOI impinges on the array from 5.3°, and the interference signals are impinging from − 5.2°and 45.5°. The simulation results show that the output SINR of the proposed OPLI coincides well with the optimal output SINR when the input SNR within moderate to high region. Herein, RELAX shows a similar performance to the proposed OPLI. OPLI exhibits better performance than the counterpart MVDR, LSMI and GLC beamformers, and this is mainly due to the fact that OPLI employs oblique projection, which has excellent performance for suppressing interference signals, to identify the waveform of the SOI. The proposed OPLI tends to have a performance drop when the input SNR is lower than 0 dB, this is because the employed oblique projection increases the noise variance (cf. [26, 30]). The simulation result verifies the theoretical analysis given in Section 4.

Figure 2 presents the output SINR of the proposed OPLI, where the number of snapshots L ranges from 2 to 1000. The input SNR is fixed at 10 dB. Other simulation conditions remain the same as previous experiment. It is seen that the output SINR gradually approximate to the optimal output SINR when the number of snapshots increases. Compared to RELAX, OPLI exhibits the same performance. While comparing to the MVDR, LSMI, and GLC beamformers, the proposed OPLI has a better performance for waveform identification even if the number of snapshots is much limited.

Figure 3 presents the output SINR of the proposed OPLI, where the DOA error of the SOI ranges from − 5°to 5°. In this simulation, L=200 snapshots are considered, and the input SNR is fixed at 10 dB. Other simulation conditions remain the same as previous experiment, except that the prior DOA of the SOI is considered with different error for the MVDR, LSMI, and GLC beamformers. The prior DOA of the SOI is set to 5.3°+δ, where δ is varied from − 5° to 5°. The simulation results show that the MVDR, LSMI, and GLC beamformers are sensitive to the prior DOA uncertainties of the SOI. The larger the DOA error goes, the quicker the output SINR drops. Comparing to the MVDR, LSMI, and GLC beamformers, the proposed OPLI as well as RELAX remains the same output SINR. This is due to the fact that both RELAX and OPLI implement waveform identification without prior DOA information.

This subsection evaluates the source localization performance of the proposed OPLI. The well-known RELAX [51], multiple signal classification (MUSIC) [55], and perturbed SBL (PSBL) [56] are included for performance comparison.

Figure 4 presents the RMSE of the DOA estimates for the proposed OPLI, where the input SNR ranges from −20 to 20 dB. K=3 signals impinging from − 5.2°, 5.3°, and 45.5° are considered, and L=200 snapshots are employed. It is seen that the RMSE of the proposed OPLI coincides well with the CRB within a moderate to high SNR region. Compared to PSBL, the proposed OPLI has a much better performance when the input SNR is high. Compared to MUSIC and RELAX, the proposed PSBL exhibits a better performance when the input SNR is low.

Figure 5 verifies the influence of the number of snapshots on DOA estimation, where the number of snapshots L is varied from 1 to 400. K=3 signals impinging from − 5.2°, 5.3°, and 45.5° are considered, and the input SNR is fixed at 15 dB. It can be seen that the proposed OPLI as well as RELAX is computationally efficient for DOA estimation even if the number of snapshots is small, whereas the conventional MUSIC fails in this case. Compared to PSBL, the DOA estimation performance of the proposed OPLI is more attractive. When the number of snapshots increases, the RMSE of the proposed OPLI decreases. The RMSE of OPLI, MUSIC, and RELAX can coincide well with the CRB when a large number of snapshots is given.

Figure 6 examines the influence of angular separation on DOA estimation, where K=2 signals impinging from 5.3°−Δ and 5.3° are considered. Δ ranges from 2° to 16°, the input SNR is fixed at 10 dB, and L=200 snapshots are employed. The simulation results indicate that the RMSE of OPLI coincides well with the CRB when the angular separation is large. Furthermore, the proposed OPLI exhibits a better DOA estimation performance than RELAX, MUSIC and PSBL when the angular separation is small. OPLI exhibits a better performance for source localization of spatially adjacent sources than the counterpart algorithms.

This subsection compares the comprehensive localization and waveform identification performances of the proposed OPLI with that of the combinational algorithms. Here, the combinational algorithms employ MUSIC and PSBL to estimate DOA of the SOI, and utilize MVDR, LSMI, and GLC to estimate the waveform of the SOI. For comprehensive performance evaluation, the average root mean square error (AveRMSE) is utilized, and it is defined as

Besides, the computational complexity of the proposed OPLI, which is evaluated by the running time of algorithm, is also compared with those of the combinational algorithms.

Figure 7 illustrates the AveRMSE of the proposed OPLI, where the input SNR ranges from −20 to 20 dB. K=3 signals are considered, and L=200 snapshots are employed. The SOI is incident from 5.3°, and the interference signals is incident from − 5.2° and 45.5°. From the simulation results, it is seen that the proposed OPLI and the well-known RELAX exhibit better performance than the counterpart algorithms, when the input SNR within moderate to high region. With a low input SNR, the performance of the proposed OPLI is not attractive, and this performance verifies the result of Fig. 1.

Figure 8 illustrates the AveRMSE of the proposed OPLI, where the number of snapshots ranges from 1 to 400. K=3 signals are considered, where the SOI is incident from 5.3° and the interference signals is incident from − 5.2° and 45.5°. The input SNR is fixed at 15 dB. It follows from the simulation results that the proposed OPLI and the well-known RELAX exhibit a better comprehensive performance than the other counterpart combinational algorithms, when the number of snasphots is much limited. With a few number of snasphots, the performance of the proposed OPLI is much attractive.

Figure 9 illustrates the AveRMSE of the proposed OPLI, where different angular separation is considered. K=2 signals are considered, the input SNR is fixed at 10 dB, and L=200 snapshots are employed. The SOI impinges from 5.3°, and the interference signals impinge from 5.3°−Δ, where Δ ranges from 2° to 16°. It can be seen that the proposed OPLI is superior to the well-known RELAX when the angular separation is small. With a small angular separation, comparing to the counterpart combinational algorithms, the proposed OPLI also shows a better performance, and this is mainly because the DOA estimation accuracy of the proposed OPLI is much higher than those of the counterpart algorithms (see also Fig. 5).

Figure 10 shows the running time of the proposed OPLI at each input SNR. Figures 11 and 12 illustrate the running time of the proposed OPLI when different number of snapshots and different angular separations are correspondingly considered. Parameters settings of Figs. 10, 11, and 12 remain the same as Figs. 7, 8, and 9, respectively. All the simulation experiments are performed using MATLAB 2013b running on a computer with a 2.3 GHz Intel Quad-Core processor and 8GB RAM, under Windows 8.1. It can be seen that the computational complexity of the proposed OPLI is lower than that of the RELAX, but is higher than that of the MUSIC-based combinational algorithms. This is mainly because the localization and waveform identification processes of the proposed OPLI is implemented iteratively, whereas the MUSIC-based combinational algorithms are not. Compared with the PSBL-based combinational algorithms, the proposed OPLI illustrates a relatively stable running time when the input SNR and the number of snapshots within moderate to high SNR region. However, when the SNR, the number of snapshots and the angular separation increase, the running time of the PSBL-based combinational algorithms fluctuates. This is due to the fact that the iteration number of PSBL is not stable and keeps varying with these parameters (cf. [56]).