Joint angle and delay estimation for underwater acoustic multicarrier CDMA systems using a vector sensor
2016; Institution of Engineering and Technology; Volume: 10; Issue: 4 Linguagem: Inglês
10.1049/iet-rsn.2015.0362
ISSN1751-8792
AutoresKun Wang, Jin He, Ting Shu, Zhong Liu,
Tópico(s)Indoor and Outdoor Localization Technologies
ResumoIET Radar, Sonar & NavigationVolume 10, Issue 4 p. 774-783 Research ArticleFree Access Joint angle and delay estimation for underwater acoustic multicarrier CDMA systems using a vector sensor Kun Wang, Kun Wang Department of Electronic Engineering, Nanjing University of Science and Technology, Nanjing, 210094 People's Republic of ChinaSearch for more papers by this authorJin He, Jin He Department of Electronic Engineering, Shanghai Key Laboratory of Intelligent Sensing and Recognition, Shanghai Jiaotong University, Shanghai, 200240 People's Republic of ChinaSearch for more papers by this authorTing Shu, Corresponding Author Ting Shu tingshu@sjtu.edu.cn Department of Electronic Engineering, Shanghai Key Laboratory of Intelligent Sensing and Recognition, Shanghai Jiaotong University, Shanghai, 200240 People's Republic of ChinaSearch for more papers by this authorZhong Liu, Zhong Liu Department of Electronic Engineering, Nanjing University of Science and Technology, Nanjing, 210094 People's Republic of ChinaSearch for more papers by this author Kun Wang, Kun Wang Department of Electronic Engineering, Nanjing University of Science and Technology, Nanjing, 210094 People's Republic of ChinaSearch for more papers by this authorJin He, Jin He Department of Electronic Engineering, Shanghai Key Laboratory of Intelligent Sensing and Recognition, Shanghai Jiaotong University, Shanghai, 200240 People's Republic of ChinaSearch for more papers by this authorTing Shu, Corresponding Author Ting Shu tingshu@sjtu.edu.cn Department of Electronic Engineering, Shanghai Key Laboratory of Intelligent Sensing and Recognition, Shanghai Jiaotong University, Shanghai, 200240 People's Republic of ChinaSearch for more papers by this authorZhong Liu, Zhong Liu Department of Electronic Engineering, Nanjing University of Science and Technology, Nanjing, 210094 People's Republic of ChinaSearch for more papers by this author First published: 01 April 2016 https://doi.org/10.1049/iet-rsn.2015.0362Citations: 2AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract In this study, the problem of joint angle and delay estimation for underwater acoustic multicarrier CDMA systems using a vector sensor is addressed. By jointly utilising the temporal, pressure and velocity information inherent in the received signal, the temporal-vector-sensor manifold vector is constructed. Then, a MUSIC-based algorithm that contains several one-dimensional searches for jointly estimating azimuth angle, elevation angle and time-delay parameters of a 'desired source' operating in the multi-source and multipath environment is proposed. In addition, a closed-form ESPRIT-based algorithm is also presented for the scenario where the multipath propagation signals are emanated from a single source. Simulation results are finally presented to verify the efficacy of the proposed algorithms. 1 Introduction In underwater acoustic communication, the signal emitted by a source normally suffers multiple reflections and scattering along the transmission path, hence arriving the receiver via different angles, path delays and fading [1]. Among these three propagation parameters, angle and delay are of great importance, and many joint angle and delay estimation (JADE) algorithms have been presented [2-10] in the past decade. These algorithms are developed with the use of arrays of pressure (scalar) sensors, which cannot measure the particle-velocity information embedded in underwater acoustic wavefield. Recently, the authors in [11] have proposed the use of velocity vector sensors for underwater communications, and have shown that an underwater wireless communication channel with a receive array of velocity vector sensors in conjunction with the conventional pressure sensors is able to provide a performance better than that of a channel with pressure sensors only. An acoustic vector sensor consists of three identical, collocated, but orthogonally oriented velocity sensors plus a pressure sensor. This acoustic vector sensor thus distinctly measures each Cartesian component of the particle velocity vector plus the pressure scalar, all at the same point in space. The use of vector sensor has the following potential advantages in underwater communications when compared with an array of pressure hydrophones. First, a single vector sensor can measure both elevation and azimuth angles while occupying very little space [12]. Second, since a single vector sensor contains no time-delay phase factor among its four components, it can process wideband signals in the same way as narrowband signals [13]. Note that the underwater acoustic signals are wideband in nature due to the small ratio of the carrier frequency to the signal bandwidth [14, 15]. Third, the array response of a vector sensor, unlike that of a spatially displaced array, is independent of the frequencies of the source signals [12]. This implies that different subcarriers of a signal would have the same vector sensor array response. This fact is pivotal to the JADE algorithm to be proposed. Therefore, it would be worthwhile to develop JADE algorithms for underwater acoustic communications, where acoustic vector sensors are employed. In fact, since Nehorai and Paldi [13] first introduced the acoustic vector sensor measurement model to the signal processing community, a variety of studies regarding angle estimation using an array of vector sensors have been extensively investigated [12, 16-26]. However, little work [27, 28] has been done to JADE using acoustic vector sensor arrays, especially in a multiple signal environment. Hence, in this paper a novel JADE algorithm is proposed for multiple underwater acoustic multicarrier code-division multiple access (MC-CDMA) systems using a single vector sensor. We first incorporate the temporal dimension to the vector sensor response vector to construct the temporal-vector-sensor (TEVES) response vector for the MC-CDMA signals. Then, the subcarrier information is used to decorrelate the signal coherency and a MUSIC-based algorithm is proposed for JADE of a 'desired source [In multisource environment, a 'desired source' means the source of interest, whose signature waveform is known.]' operating in the multipath environment. The proposed algorithm is in a tree structure, containing two τ-MUSIC steps for estimation of delays, two θ-MUSIC steps for estimation of elevation angles and one ϕ-MUSIC step for estimation of azimuth angles. In addition, a closed-form ESPRIT-based algorithm is presented for the scenario where the multipath propagation signals are emanated from a single source. The Cramér–Rao bound (CRB) for the problem under consideration is also provided. Throughout the paper, scalar quantities are denoted by lowercase letters. Bold lowercase letters are used for vectors and uppercase ones for matrices. Superscripts T, H, * and † represent the transpose, conjugate transpose, complex conjugate and pseudo inverse, respectively, while ⊗ and symbolise the Kronecker-product operator and the Khatri-Rao (column-wise Kronecker) matrix product, respectively. 2 System model 2.1 Transmitted signal Consider an underwater acoustic system with K MC-CDMA transmitted signals and a four-component receive vector sensor, which is composed of three velocity sensors and a pressure sensor, co-located in space. The velocity sensor measures one Cartesian component of the incident wavefield's velocity-vector, and the pressure sensor measures the acoustic wavefield's pressure. The binary data bit bk(n) of the kth source signal is spread using an M-chip direct sequence (DS) spreading waveform ck(t). To simplify the analysis, we assume that the bit duration of the data stream is Tb and the chip duration is Tc = Tb/M. The DS-spread signals are then modulated using P subcarriers, with the subcarrier separation being set to Δf. Therefore, the transmitted signal of the kth source signal in the nth data bit can be expressed as [29] (1) where βk is the amplitude of the kth transmitted signal, denotes the signature waveform of the kth signal, with ck(m) taking values of +1 or −1 with equal probability, and ψ(t) being a normalised chip waveform of duration Tc, fp = fc + pΔf is the pth subcarrier frequency, fc being the carrier frequency, and ϕp,k represents the phase angle introduced in the carrier modulation process of the signal k in the pth subcarrier and is uniformly distributed over [0, 2π). For simplicity of analysis, a sufficiently long cyclic prefix is employed to avoid the delay-spread-induced inter-symbol interference. 2.2 Received signal We assume that the kth transmitted signal arrives at the receiver via Ik multipaths. Let θk,i, ϕk,i and τk,i, respectively, be the elevation angle, azimuth angle and delay of the ith path of the kth signal. Furthermore, the subcarrier signals are assumed to be orthogonal to one another. Hence, we can arrange the received signal in a subcarrier-by-subcarrier format [30]. With a total of K signals, the baseband signal vector associated with the pth subcarrier can be expressed as (2) where is the 4 × Ik vector sensor array response matrix associated with the kth signal and the pth subcarrier, represents the complex path coefficient for the ith path of the kth signal at pth subcarrier. The magnitude and phase of are typically modelled by independent and identically distributed (i.i.d.) Rayleigh random variable and i.i.d. uniform random variable on the interval of [0, 2π), respectively [31], is the Ik × 1 signal vector for the kth signal, , and is the 4 × 1 additive noise vector. Since the array response of the vector sensor is independent of the frequency spectra of the source signals, we have , where , with (3) being the vector sensor's 4 × 1 array response vector with regard to the ith path signal of the kth source [13, 16]. In (3), uk,i = sin θk,i cos ϕk,i, vk,i = sin θk,i sin ϕk,i and wk,i = cos θk,i, respectively, signify the direction cosines along the x-, y- and z-axes. 3 JADE: multiple sources case With the system model given in Section 2, the problem of interest is to determine the angle and delay parameters of the K signals. 3.1 Preprocessing The vector in (2) can be rewritten as (4) where , and . denotes the operator which produces a diagonal matrix by placing the vector in (·) on the main diagonal, and Note that in each data bit, ck(t) can be written in a 1 × M vector form as (5) Thus, we can construct a 4 × M matrix from (4) and (5) as (6) where , with For simplicity of analysis, the delay τk,i is assumed to be an integer multiple of Tb/M, so that τk,i can be modelled as , with . Further, the chip duration is normalised to Tc = Tb/M = 1. In this case, and can be expressed as , where (7) To obtain the angle and delay estimation, we need to convert the 4 × M matrix into a 4M × 1 vector. Let vec(X) be the operator that enumerates the entries of X in a row-wise order. Then, we have (8) where , and ηp, k(n) = γp, kbk(n). It is noteworthy that the constructed data model in (8) coincides with the classical array signal model [32]. The only difference is that the columns of the manifold matrix depend on three parameters: elevation angle, azimuth angle and propagation delay. In this form, the columns of matrix can be considered as TEVES response vectors, which are formed by incorporating the temporal dimension to the vector sensor response vector. Obviously, the second-order statistics of the vector would become rank deficient with the dimension of the signal covariance matrix being reduced to unity. This is due to the fact that the signals ηp, k(n) associated with the different multipaths are all some complex multiples of a common signal bk(n). Consequently, the subspace-based algorithms, such as MUSIC [32] and ESPRIT [33], cannot be applied to due to the rank deficiency. To restore the rank of the signal covariance matrix, we will exploit the multicarrier property of the signals. Referring to (8), the data vector associated with each subcarrier provides a different linear combination of the vectors , as would be the case for uncorrelated signals. Therefore, the information acquired by the P different subcarriers would help in obtaining a signal covariance matrix with full rank. For all p = 1, …, P, it is shown in [10] that under the condition P ≥ max{I1, …, IK}, the matrix S in subcarrier smoothed matrix R is of full rank, where (9) with , , with , and . 3.2 MUSIC-based algorithm for JADE Let . To efficiently proceed with the subspace-based algorithms, we assume and . Furthermore, the columns of B are assumed to be linearly independent. For simplicity of analysis, we will take the kth source as the 'desired source' and perform JADE to obtain the angle and delay parameter estimates of the Ik path signals of this source. Further consider the scenario where the path signals impinging on the vector sensor may be either the same in angle or in delay, but not both. Let us eigen-decompose R to construct the signal-subspace matrix , whose columns correspond to the 4M × 1 eigenvectors associated with the largest eigenvalues of R. Then, we form the following MUSIC pseudo-spectrum scalar function for the kth source (10) where . Using the orthogonality property between the signal and noise subspaces, the Ik values of (θ, ϕ, τ) that maximise V(θ, ϕ, τ) would constitute the estimates of the elevation angle, azimuth angle and time delay of the path signals of the kth source. However, this estimator has a high computational cost in that it obtains the angle and delay estimates by performing a three-dimensional (3D) search to find Ik maxima of the above highly non-linear cost function. In the following, we will propose a computationally efficient MUSIC-based algorithm, which requires only several one-dimensional (1D) searches, to estimate the angle and delay parameters. Observing the 4M × Ik matrix , the first M rows of , which are associated with the pressure sensor, depend only on the time delays but not on the angles. The second and third M rows of , which are, respectively, associated with the velocity sensors aligning along the x- and y-axes, depend on both elevation angles, azimuth angles and time delays. The last M rows of , which are associated with the velocity sensor aligning along the z-axis, depend on the elevation angles and time delays but not on the azimuth angles. These observations indicate that the submatrix , which denotes the first M rows of , is dependent only on the time delays but not on the angles. The submatrices and , which denote the second and third M rows of , are dependent on both elevation angles, azimuth angles and time delays. The submatrix , which denotes the last M rows of , is dependent on the elevation angles and time delays but not on the azimuth angles. These observations also suggest a 1D MUSIC-based algorithm, which is described as follows. Step 1) τ-MUSIC step: We apply the 1D MUSIC to the data measured by the pressure sensor to yield a set of reference delay estimates. This can be accomplished by performing the following 1D search (11) It is noteworthy that if some of the path signals have the same delay, we can only obtain Q(Q ≤ Ik) different delay estimates. In other words, Q groups of delay estimates are obtained. These group delay estimates can help the partition of the path signals into Q groups, with each group containing paths with the same delay but different angles. We assume that each group has Kq, q = 1, …, Q delay estimates. The resulting delay estimates from (11) are written as . Step 2) θ-MUSIC step: With a set of reference delay estimates obtained in the Step 1), the following 1D MUSIC search is performed to obtain a set of reference elevation angle estimates for the path signals in group q(12) where (13) Note that in group q, if some of the path signals have the same elevation angle, we can only obtain Lq(Lq ≤ Kq) subgroups of elevation angle estimates. We assume that each subgroup has , elevation angle estimates. The estimated reference elevation angles for path signals in group q are denoted as . Step 3) ϕ-MUSIC step: Based on the estimated delay and elevation angle , we can obtain the azimuth angle estimates of the path signals, which are associated with the delay estimate and the elevation angle estimate , by finding the peaks of the following pseudo-spectrum function (14) Then, for q = 1, …, Q and , the azimuth angle estimates of all the Ik path signals can be obtained. These estimates are denoted as . Step 4) θ-MUSIC step: Suppose the estimated is associated with the reference delay estimate . The elevation angle estimate of the ith path signal, which is denoted as , can be determined by performing the following pseudo-spectrum (15) Step 5) τ-MUSIC step: With the precisely estimated elevation and azimuth angles above, we can isolate each path signal since the angles of the path signals inside each group are well separated as assumed earlier. Therefore, we can finally invoke τ-MUSIC step again to obtain a more precise estimate of the delay for each isolated path signal. The resulting delay estimate is denoted as , and can be found by the 1D search given by (16) Thus far, the proposed MUSIC-based algorithm realises the estimation of elevation angles, azimuth angles and time delays of the Ik path signals with several 1D searches, which include two τ-MUSIC steps for estimation of delays, two θ-MUSIC steps for estimation of elevation angles and one ϕ-MUSIC step for estimation of azimuth angles. In summary, the overall structure of the proposed algorithm is shown in Fig. 1. Fig. 1Open in figure viewerPowerPoint Flowchart of the proposed 1D MUSIC-based JADE algorithm 3.3 Remarks The algorithm presented in above subsection obtains the angle and delay estimates of the Ik path signals of a 'desired source' that propagates in a multipath environment. Note that the 3D parameter estimation has been extracted by using just one solitary vector sensor and no planar arrays. The elevation angle estimates, azimuth angle estimates and the time-delay estimates are all automatically paired without any extra processing. Note also that the number of identifiable path signals is no longer limited by the number of vector sensor components. In particular, the manifold surface associated with all sources is embedded in an M-dimensional complex space and so long as the total number of identifiable path signals is less than M, the proposed algorithm will operate properly. Moreover, the subcarrier separation Δf can be completely arbitrary so long as a set of i.i.d. complex path coefficients for different subcarriers are preserved. This would guarantee the signal subspace of R in (9) to be of full rank. The efficient implementation of the proposed algorithm also exploits the point-like structure of the employed four-component vector sensor, whose array response remains the same for the signals with different subcarrier frequencies. If the vector sensor is replaced by an array of spatially displaced sensors, the present algorithm would not work. This is because the array response vector of a spatially displaced sensor array varies at each of the P subcarriers, which destroys the model expression in (4) for the implementation of the algorithm. This point will be examined in simulations. Lastly, the authors in [34] have found that the maximum number of sources uniquely identifiable by one vector sensor is two. This finding does not contradict the earlier assertion in this paper that the number of identifiable path signals is not limited by the number of vector sensor components. There is no contradiction because the present algorithm makes the additional restriction of multicarrier CDMA signals and the further construction of the temporal vector sensor response vectors. In other words, this algorithm has presumed a certain observable space-time structure in the data set, which has not been made in [34]. 4 JADE: single source case In this section, we propose an ESPRIT-based algorithm for JADE of a single source propagating in the multipath environment. This ESPRIT-based algorithm has a much reduced complexity than the MUSIC-based algorithm in Section 3 in that it does not require the computationally expensive search processes. In the single source case, . Then, using the basic idea of ESPRIT [33], we have (17) where T is an unknown but non-singular I1 × I1 coupling matrix, , , , and , with (18) (19) (20) Then, it follows from (17)–(20) that (21) (22) (23) Thus, the parameters u1,k, v1,k and w1,k can be found from the kth ESPRIT's eigenvalue, i.e., the eigenvalue of , and , respectively. It is seen from (21)–(23) that the matrices , and have the same set of eigenvectors. Eigenvalues of these matrices corresponding to the same eigenvector therefore give the direction cosines of the same path signal. However, the estimated , and from the finite data would have different sets of eigenvectors. Let , and , respectively, be the estimation of T from the , and estimates. Then, the matching set of eigenvectors corresponding to each path signal can be obtained by matching the columns of , and . Consider the matrix products and , and let ji, ℓi denote respectively the column indices of the matrix elements with the largest absolute values in the ith rows of and . It is inferred that the eigenvectors constituting the ith column of , the jith column of and the ℓith column of , correspond to the same path signal. Hence, for each i ∈ {1, 2, …, I1}, the eigenvalues corresponding to the ith column of , the jith column of and the ℓith column of are the estimates of the direction cosines of the ith path signal. Now, using (17)–(20) again, the matrix may be estimated as (24) Let be the ith column of . Next, consider the fact that the estimated would suffer the unknown scaling ambiguities of the columns, which may further result in the ±1 ambiguities in the estimation of . In other words, the estimate may be equal to either or . Unlike the case of angle estimation, there would be ambiguity in estimating the time delays. Indeed, the following two sets of delay estimates can be obtained (25) where denotes the Frobenius norm. For the above two sets of estimates , we may determine as to which one is the true one by first constructing the vectors and , and then taking if or otherwise. To avoid the above ±1 ambiguity problem, we present another method to estimate the time delays. This method is based on the application of Fourier transformation to map the delays to certain phase progressions [6]. Let us define the following preprocessor (26) where F is an M × M discrete Fourier transform (DFT) matrix, and denotes the operator which produces a diagonal matrix by placing the vector in ( · ) on the main diagonal. By applying to (6), is transformed to (27) where , with , and . Obviously, previous developments in Sections 3 and 4 and all equations from (8)–(24) still hold, with replacing , and by , and , respectively. Therefore, the ESPRIT estimator in (24) would give the estimate , whose ith column equals to with an known complex scaling. That is, , where denotes the ith column of , and ci represents a complex constant. Unlike the scaling ambiguity in , the scaling ambiguity in can be easily resolved by take the first element of as a reference and normalising with respect to it. After removing the complex constant ci, the time-delay estimate of the ith path signal can be computed as (28) where is the mth element of . 5 Cramér–Rao bound The CRB provides a lower bound on the variance of any unbiased estimator. The CRB for vector sensor angle estimation without delay spread has been thoroughly analysed in [12, 13]. In this section, we adapt the results in [13] to the JADE problem (8). For simplicity of expression, we only provide the result for the single source and single data bit case. This result can be readily extended to the multiple sources and multiple data bits scenario. For the case K = 1 and N = 1, where N represents the number of data bits, (8) can be rewritten as (29) where , , , η(p) = ηp,1(1) and . Let , and . Then, we have the following theorem. Theorem.Assume η(p) to be unknown deterministic quantities and n(p) to be complex, stationary, zero-mean Gaussian random processes that are uncorrelated from subcarrier to subcarrier. The CRB for θ, ϕ and τ can be expressed as (30) where (31) (32) (33) (34) (35) (36) (37) (38) We omit the proof of this result since it is obtained in a way similar to that in [13]. 6 Simulation results In this section, we provide simulation results to illustrate the performance of the proposed algorithms. The shallow water channel modelled by employing a direct path and two paths reflected at surface and bottom boundaries is considered, as shown in Fig. 2. The water is 100 m depth. A vector sensor is deployed at 60 m depth. The constant sound speed at 1500 m/s is assumed. The bottom considered is rigid without loss. A source of interest is located at 20 m depth and 117 m range, unless otherwise stated. Letting the vector sensor be located at the origin, the location of source is assumed to be (100, 60, 40), with the angle and delay parameters of the direct path are θ1,1 = 71°, ϕ1,1 = 31° and τ1,1 = 10Tc (sample starting at 50 m and Tc = 5 ms). The image method is used to generate the angle and delay parameters of the two reflected paths, which are calculated as: θ1,2 = 55.6°, ϕ1,2 = 31°, τ1,2 = 12Tc, θ1,3 = 134.2°, ϕ1,3 = 31° and τ1,3 = 16Tc. The source centre frequency is set as f0 = 10 kHz, with bandwidth 4 kHz [35]. Fig. 2Open in figure viewerPowerPoint Illustration of shallow water multipath channel considered in simulations First, we consider the scenario where the above mentioned source operates in the presence of two interferers with locations (120, 40, 30) and (90, 40, 50). Each source is being assigned a unique signature waveform with M = 32 chips. The collected number of data bits is assumed to be equal to N = 128. The number of subcarriers is set as P = 24. Fig. 3 shows the implementation of the proposed MUSIC-based algorithm for estimating the angle and delay parameters of the desired source. In the first step, τ-MUSIC is conducted to obtain the reference delay , and estimates, see plot (a). In the second step, θ-MUSIC is conducted thrice to obtain the reference elevation angle estimates , and , which are associated with , and , respectively, see plot (b). In the third step, ϕ-MUSIC is performed thrice to obtain the azimuth angle estimates ϕ1,1, ϕ1,2 and ϕ1,3, which are associated with and , respectively, see plot (c). In the fourth step, θ-MUSIC is performed again to obtain the elevation angle estimates θ1,1, θ1,2 and θ1,3, which are respectively, associated with ϕ1,1, ϕ1,2 and ϕ1,3, see plot (d). In the last step, τ-MUSIC is invoked again to obtain the delay estimates τ1,1, τ1,2 and τ1,3, which are respectively, associated with ϕ1,1, ϕ1,2 and ϕ1,3, see plot (e). It is seen from Fig. 3 that all the three multipaths are identified successfully using the proposed MUSIC-based algorithm with only 1D searches. Fig. 3Open in figure viewerPowerPoint MUSIC-based algorithm for estimating angles and delays of the desired source. θ1,1 = 71°, ϕ1,1 = 31° and τ1,1 = 10Tc, θ1,2 = 55.6°, ϕ1,2 = 31°, τ1,2 = 12Tc, θ1,3 = 134.2°, ϕ1,3 = 31° and τ1,3 = 16Tc a τ-MUSIC obtaining reference delay estimates b θ-MUSIC obtaining reference θ estimates c ϕ-MUSIC obtaining ϕ estimates d θ-MUSIC obtaining θ estimates e τ-MUSIC obtaining τ estimates Next, we examine the performance of the proposed method in the presence of closely spaced paths (with close but not the same angle and delay parameters). We assume that the source of interest is at 2 m depth, with location (100, 60, 58). The source signature waveform chips considered is M = 128. The angle and delay parameters of the three paths are θ1,1 = 63.56°, ϕ1,1 = 31° and τ1,1 = 53Tc, θ1,2 = 62°, ϕ1,2 = 31°, τ1,2 = 55Tc, θ1,3 = 130.2°, ϕ1,3 = 31° and τ1,3 = 87Tc, with Tc = 1 ms. The remaining conditions are the same as in the first simulation. Fig. 4 shows the result of the proposed MUSIC-based algorithm for estimating the angle and delay parameters of the desired source, in the presence of two closely spaced paths (direct path and surface reflected path). We can see from Fig. 4 that all the three multipaths, as well as two closely-spaced paths, are identified successfully using the proposed MUSIC-based algorithm. Incidentally, the proposed method would fail if some paths have exactly the same delay. Fig. 4Open in figure viewerPowerPoint MUSIC-based algorithm for estimating angles and delays of the desired source. θ1,1 = 63.56°, ϕ1,1 = 31° and τ1,1 = 53Tc, θ1,2 = 62°, ϕ1,2 = 31°, τ1,2 = 55Tc, θ1,3 = 130.2°, ϕ1,3 = 31° and τ1,3 = 87Tc a τ-MUSIC obtaining reference delay estimates b θ-MUSIC obtaining reference θ estimates c ϕ-MUSIC obtaining ϕ estimates d θ-MUSIC obtaining θ estimates e τ-MUSIC obtaining τ estimates Finally, we compare the performance of the four-component vector sensor with that of a 2 × 2 scalar sensor array, for the single source scenario (without interference sources). The scalar sensor array is of element spacing d = 0.5λ0 in both dimensions, where λ0 is wavelength at centre frequency. The waveform chips, number of data bits, and subcarrier numbers, are set as M = 64, N = 1 and P = 24, respectively. The geometry of the channel is the same as that used in the first simulation. The root mean squared errors (RMSEs) of the direct path's angle and delay parameters as a function of the signal-to-noise ratio (SNR), varying from 0 to 40 dB in steps of 5 dB, are shown in Fig. 5, where the CRBs are also plotted for comparison. We see from the figure that the algorithm with vector sensor has a performance significantly superior to that using scalar sensor array. Also note that the performance of the scalar sensor array remain almost unchanged when SNR ≥ 5 dB. This phenomenon is due to the fact that the array response vector of scalar sensor array varies at each of the P subcarriers, thus destroying the model expression in (4) for the implementation of the algorithm, as stated in Section 3.3. Fig. 5Open in figure viewerPowerPoint RMSEs of angle and delay estimates against SNR. θ1,1 = 71°, ϕ1,1 = 31° and τ1,1 = 10Tc, θ1,2 = 55.6°, ϕ1,2 = 31°, τ1,2 = 12Tc, θ1,3 = 134.2°, ϕ1,3 = 31° and τ1,3 = 16Tc. Five hundred independent experiments are conducted 7 Conclusions We have proposed the use of a single acoustic vector for joint space–time parameter estimation for underwater acoustic MC-CDMA systems. We have constructed the TEVES response vector by incorporating the temporal dimension to the vector sensor response vector. On the basis of this structure, a MUSIC-based algorithm that contains only several 1D searches has been presented for joint parameter estimation of a 'desired source' operating in the multi-source and multipath environment. Furthermore, a closed-form ESPRIT-based algorithm has also been presented for JADE for the scenario where the multipath propagation signals are emanated from a single source. 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