Fast Estimation Method of Space-Time Two-Dimensional Positioning Parameters Based on Hadamard Product

The estimation speed of positioning parameters determines the effectiveness of the positioning system. The time of arrival (TOA) and direction of arrival (DOA) parameters can be estimated by the space-time two-dimensional multiple signal classification (2D-MUSIC) algorithm for array antenna. However, this algorithm needs much time to complete the two-dimensional pseudo spectral peak search, which makes it difficult to apply in practice. Aiming at solving this problem, a fast estimation method of space-time two-dimensional positioning parameters based on Hadamard product is proposed in orthogonal frequency division multiplexing (OFDM) system, and the Cramer-Rao bound (CRB) is also presented. Firstly, according to the channel frequency domain response vector of each array, the channel frequency domain estimation vector is constructed using the Hadamard product form containing location information. Then, the autocorrelation matrix of the channel response vector for the extended array element in frequency domain and the noise subspace are calculated successively. Finally, by combining the closed-form solution and parameter pairing, the fast joint estimation for time delay and arrival direction is accomplished. The theoretical analysis and simulation results show that the proposed algorithm can significantly reduce the computational complexity and guarantee that the estimation accuracy is not only better than estimating signal parameters via rotational invariance techniques (ESPRIT) algorithm and 2D matrix pencil (MP) algorithm but also close to 2D-MUSIC algorithm. Moreover, the proposed algorithm also has certain adaptability to multipath environment and effectively improves the ability of fast acquisition of location parameters.


Introduction
Wireless target localization has found widespread application requirements in military and civilian fields.Parameter estimation is an important prerequisite for positioning algorithm, which determines the overall effectiveness of the positioning system.The location parameters include time of arrival (TOA), direction of arrival (DOA), time difference of arrival (TDOA), and received signal strength.The key technique of location finding based on TOA method is to estimate the propagation delay of the radio signal arriving from the direct line-of-sight (DLOS) propagation path accurately.The existing TOA methods mainly include cross-correlation method [1], multiple signal classification (MUSIC) algorithm [2], delay estimation algorithm based on Markov Monte Carlo [3], propagator algorithm [4], and matrix pencil algorithm [5].If the array antenna can be used to obtain the signal DOA for joint estimation [6,7], then it will effectively reduce the number of nodes and system overhead of the localization system.In theory, a reference node can lock the location of the target source.
OFDM technology is a multicarrier modulation and high-speed transmission technology, which has been widely used in underwater acoustic communication system, IEEE 802.11a/g/n/ac [8], OFDM new system radar [9], WiMAX system [10], 3GPP LTE/LTE-Advanced [11], 5G mobile communication systems, and so on.However, the demand for target location information of OFDM technology is also increasing, especially in indoor, underground places, and other environments.Therefore, using OFDM array antenna receiving system to achieve accurate positioning of the target signal has become a hot spot of current research.
Due to the limitation of bandwidth, sampling rate, and the attributes of narrowband signals, OFDM signals cannot achieve joint estimation of TOA and DOA employing the triangle geometry relation as in [12,13].Ni et al. [14] introduces a TOA estimation algorithm for OFDM wireless signal, which combines coarse estimation with fine estimation.The algorithm needs to use the technology-guided detection to suppress multipath propagation, but the positioning accuracy is limited.Three methods including peak detection, modified maximum peak leak rate detection, and channel frequency domain response reconstruction are proposed in [15].In the multipath environment of WINNER A1 LOS channel, the channel frequency response reconstruction method can provide the best performance, but this method is mainly based on the received preamble sequence.In the environment of indoor multipath, the twodimensional matrix pencil (2D MP) algorithm is presented in [16] to achieve the joint estimation of time delay and arrival angle, which has low computational complexity, but sacrifices the effective array element and bandwidth.Zhou et al. [17] provide a joint estimation algorithm of delay and DOA, which uses the frequency domain data of single antenna for MUSIC time delay estimation.This algorithm uses the array data for MUSIC DOA estimation, but it fails to give a matching method of time delay and arrival angle, and its computational complexity is high.Ba et al. [18] present a joint estimation of delay and angle using 2-MUSIC for two-dimensional searching, which solves the pairing problem, but its computational complexity is very high, which makes it hard to apply in practice.
In this paper, a fast estimation method of space-time twodimensional positioning parameters based on Hadamard product is presented, which can offer the closed-form solution of parameter estimation directly without pseudo spectral peak search.This method can implement parameter pairing and improves the efficiency significantly.The computational complexity analysis, CRB derivation, and simulation are also presented.The simulation results show that the proposed algorithm has higher precision of parameter estimation close to the 2D-MUSIC algorithm, but the computational complexity is greatly reduced, which effectively improves the ability of fast acquisition of the positioning parameters.
The rest of the paper is organized as follows: Section 2 introduces the data model.Section 3 presents the method of fast estimation and the computational complexity analysis in detail.The CRB for the method is derived in Section 4. Simulation and experimental results are presented in Section 5. Conclusions are drawn in Section 6.

Data Model
Considering a node equipped with an equidistant line array composed of M antenna elements, the distance between the signal source and the antenna array is far enough, and θ i and τ i represent DOA and propagation delay of the multipath signal, respectively.s t is the OFDM symbol transmitted with N subcarriers.On the account of the wireless multipath propagation, the received signal at the mth antenna element can be formulated as where L P is the number of the multipath components and a i = a i e jφ i and τ i,m are the complex attenuation of the ith path and propagation delay of the ith path in mth antenna, respectively.The phase of the complex attenuation φ i is normally assumed random from one snapshot to another with a uniform probability density function U 0, 2π [19].n m t is additive Gauss white noise.The time delay in each antenna element depends not only on the propagation delay but also on the direction of arrival.In particular, for a uniform linear array, the propagation delay associated to be the ith arriving path in mth antenna is given by τ i,m = τ i + md sin θ i /c, 0 ≤ m ≤ M − 1, with d being the distance between adjacent array elements in the array and c the speed of light.After the Fourier transform of the received signal r m t , the sampled discrete frequency domain channel response of the kth subcarrier at the mth antenna element is given by where 0 ≤ k ≤ N − 1.In (2), it is clear that the effect of time delay md sin θ i /c coming from the antenna array is negligible compared with τ i , so this formula can be further simplified as where n m,k denotes additive Gauss white noise with mean zero and variance σ 2 at the kth subcarrier and the mth antenna element, f c , is the carrier frequency of OFDM signal, T = NT S is the net data length of OFDM symbol, and T S is the sampling interval.We can then write this formula (3) in a vector form.
where 2 International Journal of Antennas and Propagation and ⊕ denotes Hadamard product.Then, So, the frequency domain response vector of M extended array elements can be expressed as By eigenvalue decomposition of R XX = E XX H , we can obtain mutually orthogonal subspace U S and noise subspace U N based on Hadamard product-extended signal.Using space-time pseudo spectral search can estimate τ i and θ i , but this method is very time-consuming, which costs large amount of calculation, and is hard to practical application.Here, the fast estimation method of two-dimensional positioning parameters based on Hadamard product is discussed.
For the reference antenna element, the eigendecomposition of the covariance matrix with P snapshot measurement data can be expressed as where ÛS and ÛN are, respectively, the signal subspace and noise subspace and ΛS and ΛN are the signal eigenvalue diagonal matrix and noise eigenvalue diagonal matrix.By using the noise eigenvector based on Hadamard product to extract space parameter information, MUSIC-type algorithm function can be constructed as The zero value of the function gives the estimation of the time delay, which is equivalent to the value obtained by pseudo spectral peak search.Let z τ = e −j2πτ/T and s z τ = 1, z τ , … , z N−1 τ T be the power term of z * τ , and ( 11) is not the polynomials of z τ .Then, , the polynomials of Root-MUSIC for TOA, can be expressed as Then, by utilizing the sampled N frequency points at the reference element, the corresponding delay estimation can be obtained as where ẑτ 0 , ẑτ 1 , … , ẑτ L P −1 are L P roots which are the closest to the unit circle in (12).Similarly, the DOA estimation of the frequency response of the channel at the first frequency point of the M antenna elements can be obtained by the above algorithm.

International Journal of Antennas and Propagation
Since the delay and DOA are estimated separately, the corresponding parameter pairing is required.
3.2.Parameter Pairing.By using the frequency domain channel responses at the reference array element and the next element, the matrix Y is constructed as where The covariance matrix of Y can be an eigenvalue decomposed as RYY = 1/P∑ P j=1 YY H = VS ΦS VH S + VN ΦN VH N , where VS ∈ C 2MN×L p and VN ∈ C 2MN× 2MN−L p are, respectively, the signal subspace and the noise subspace; obviously, the column vector d′ τ, θ and noise subspace VN are orthogonal, which make d′ τ, θ • VN = 0.In order to achieve the pairing of delay and arrival angle, the cost function can be obtained by So the pairing of the estimated parameters is to minimize the cost function τi , θj = min i,j h τi , θj , i, j = 0, 1, … , L P − 1 18 Firstly, select a fixed τi from τi , and let θi traverse θi L P −1 i=0 , then the τi , θj corresponding to the minimum value of the cost function is the correct pairing value.

Computational Complexity Analysis.
The computational complexity of the algorithm in this paper consists of the following four aspects: the estimation of the covariance matrix RX :,0 X :,0 and RX 0,: X 0,: , the computational complexity is O M 2 P and O N 2 P , respectively; the eigenvalue decomposition of the covariance matrix RX :,0 X :,0 and RX 0,: X 0,: , the computational complexity is O M 3 and O N 3 ; the complexity of constructed polynomial is O M 2 − ML p and O N 2 − NL p ; the computational complexity of parameter pairing is O 2N 2 P + 2N 3 + 2N 2 L p + 4NL p − 2NL p L p .So the total complexity of the algorithm is O M 3 The total complexity of using 2D-MUSIC algorithm and W τ and W θ are, respectively, the number of grid of delay and angle of arrival search, that is, the range of measurement search is divided by the search step.The computational complexity of ESPRIT algorithm is O 2N 3 + 2M 3 + 4P + 1 N 2 + L p 2 N .Table 1 compares the computational complexity of the three algorithms.
Assuming M = 4, P = 100, W τ = 1000, W θ = 1000, and L p = 3, the computational complexity curves of the subcarrier number N in the three algorithms are plotted.It can be seen from Figure 1 that the computational complexity of this paper is slightly higher than that of the ESPRIT algorithm, but much lower than that of the 2D-MUSIC algorithm, which greatly reduces the amount of computation.

The Cramer-Rao Bound
The CRB is the lower bound of minimum variance which an unbiased estimation can achieve.The following is to derive the CRB of the model.Let the parameter vector be defined as σ 2 η Τ Τ , where η = τ Τ θ Τ Τ . According to (7) and the relevant initial conditions, the joint probability density function of Thus, the log-likelihood function of (19), omitting the constant term, is To calculate the Fisher information matrix (FIM), the derivatives of (20) with respect to σ 2 , the real part Re α j of α j , the imaginary part Im α j of α j , and η can be obtained as Since η is a two-dimensional vector, the derivatives of (20) with respect to η are different from the ones in the angle-spread only model.

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and similarly, 4 International Journal of Antennas and Propagation where g θ i is the derivative with respect to θ i of the ith column of D. Similarly, g τ i is the derivative with respect to τ i of the ithcolumn of D. Due to d m τ i , θ i = s τ i ⊕ a m θ i , then Further, written more compactly where Re diag α j H G H n j 26 By using the results proven in [20], the following expressions can be attained Finally, the FIM for the parameters is obtained.Using the results in [20], the CRB of the parameters can be shown as

Algorithm type
Computational complexity which completes the proof.

Simulation Results
In this section, the performance of fast estimation of twodimensional positioning parameters of TOA and DOA is studied.Monte Carlo simulation is utilized to verify the practicability and robustness of the proposed algorithm.
The relevant parameters of narrowband array antenna in OFDM system are set as follows: the system bandwidth is B = 20 MHz, the FFT cycle is T = 3 2 μs, the number of subcarriers is N = 64, the carrier frequency is f c = 2 3 GH z, the number of elements is M = 12, the array spacing is d = λ/2 meters, and so on.

Simulation 1.
In the case of low SNR at 0 dB and 5 dB, assuming that the multipath number of the received signals is L P = 3, the arrival time is τ 1 = 100 ns, τ 2 = 140 ns, and τ 3 = 180 ns, respectively; the direction of arrival is θ 1 = 30 °, θ 2 = 40 °, and θ 3 = 60 °, respectively.Simulation time of the algorithm is S = 100, the scatter diagrams of the joint estimation of TOA and DOA combined with three multipath components are obtained, Figures 2 and 3 show that the algorithm has a good performance for the joint estimation at low SNR conditions.

Simulation 2.
Under different SNRs, conditions are the same as simulation 1, and the scatter plots of TOA and DOA can be obtained.It can be seen from Figure 4 and Figure 5 that with the increase of the SNR, the dispersion of TOA and DOA gradually tends to a point, and the estimation precision is higher.At low SNR, the estimation accuracy amplitudes of TOA and DOA are all within a unit.as , where xn is the parameter estimated value obtained by the nth simulation and x is the true value of the corresponding parameter.The TOA and DOA estimation performances of the respective first path are shown in Figures 8 and 9, respectively.The number of multipath is L p = 2, L p = 3, and L p = 4, respectively.Since the 1D MP algorithm is based on a single snapshot data, little information is available, and the 1D MP algorithm is the worst in low SNRs.However, the 2D MP algorithm extends the information, and the availability of the information is relatively increased so that the performance of 2D MP algorithm is better than that of 1D MP algorithm.In essence, the proposed algorithm is another form of MUSIC algorithm and has a similar estimation performance as 2D-MUSIC algorithm, of which the complex two-dimensional pseudo spectral peak search is avoided in

Conclusion
The fast estimation of location parameter is a necessary condition for the quality of wireless positioning system.As the fact that 2D-MUSIC algorithm needs a long time twodimensional pseudo spectral peak search, which makes it difficult to be applied in practice, a fast estimation method of two-dimensional positioning based on Hadamard product in OFDM system is proposed.The construction of the model, the theoretical analysis, and the process of CRB derivation are presented.The computation complexity of the proposed method by solving the closed-form solution and parameter pairing is much lower than that of 2D-MUSIC algorithm.The simulation results show that the parameter estimation performance of the proposed algorithm is not only better than that of ESPRIT algorithm and 2D matrix pencil algorithm, but also close to that of 2D-MUSIC algorithm, and it has certain adaptability to the multipath environment and certain practical value.

5. 3 . 5 . 4 . Simulation 4 .
Simulation 3. At low SNR of SNR = 5 dB, other conditions are the same as simulation 1, and the estimation error of TOA and DOA is defined as Δτ S,i = τS,i − τ i and Δθ S,i = θS,i − θ i , respectively.Then, the scatter diagram of Δτ S,1 , Δτ S,2 , and Δτ S,3 corresponding to three multipath components is shown in Figure6, and the corresponding Δθ S,1 , Δθ S,2 , and Δθ S,3 distribution are shown in Figure7.It can be seen from Figures6 and 7that the algorithm at low SNR can almost achieve the unbiased estimation of TOA and DOA.Define the root mean square error (RMSE)

Figure 9 :Figure 10 :
Figure 9: DOA estimation performance of the first path under different multipath numbers.

Figure 11 :
Figure 11: Performance comparison of different algorithms for DOA estimation.