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

Authors: Haiwen Li; Nae Zheng; Xiyu Song; Yinghua Tian
Journal: International Journal of Antennas and Propagation
 (2018)

---

## Abstract

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.


---

## Body

## 1. 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 two-dimensional 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 two-dimensional 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: Section2 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.


## 2. Data Model

Considering a node equipped with an equidistant line array composed ofM antenna elements, the distance between the signal source and the antenna array is far enough, and θi and τi represent DOA and propagation delay of the multipath signal, respectively. st 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
(1)rmt=∑i=0LP−1aist−τi,m+nmt,where LP is the number of the multipath components and ai=aiejφi and τi,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 φi is normally assumed random from one snapshot to another with a uniform probability density function U0,2π [19]. nmt 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 τi,m=τi+mdsinθi/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 rmt, the sampled discrete frequency domain channel response of the kth subcarrier at the mth antenna element is given by
(2)xm,k=Hm,k+nm,k=∑i=0LP−1αie−j2πfcτie−j2πk/Tτi+mdsinθi/ce−j2πfcmdsinθi/c+nm,k,where 0≤k≤N−1. In (2), it is clear that the effect of time delay mdsinθi/c comingfrom the antenna array is negligible compared withτi, so this formula can be further simplified as
(3)xm,k=Hm,k+nm,k=∑i=0LP−1αie−j2πfc+k/Tτie−j2πmdsinθi/λ+nm,k,where nm,k denotes additive Gauss white noise with mean zero and variance σ2 at the kth subcarrier and the mth antenna element, fc, is the carrier frequency of OFDM signal, T=NTS is the net data length of OFDM symbol, and TS is the sampling interval. We can then write this formula (3) in a vector form.
(4)Xm=Hm+nm=Dmτ,θα+nm,where
(5)Dmτ,θ=dm,0τ0,θ0dm,0τ1,θ1⋯dm,0τLP−1,θLP−1dm,1τ0,θ0dm,1τ1,θ1⋯dm,1τLP−1,θLP−1⋮⋮⋱⋮dm,N−1τ0,θ0dm,N−1τ1,θ1⋯dm,N−1τLP−1,θLP−1N×LP.Moreover,dm,kτi,θi=e−j2πfc+k/Tτie−j2πmdsinθi/λ, set dmτi,θi=sτi⊕amθi, and ⊕ denotes Hadamard product. Then,
(6)sτi=e−j2πfc+0/Tτie−j2πfc+1/Tτi⋯e−j2πfc+K‐1/TτiΤ,amθi=e−j2πmdsinθi/λe−j2πmdsinθi/λ⋯e−j2πmdsinθi/λΤ.So, the frequency domain response vector ofM extended array elements can be expressed as
(7)X=H0H1⋮HM−1+n=D0τ,θD1τ,θ⋮DM−1τ,θα+n=Dτ,θα+n,where
(8)Dτ,θ=dτ0,θ0dτ1,θ1⋯dτLP−1,θLP−1MN×LP,dτi,θi=d0Ττi,θid1Ττi,θi⋯dM−1Ττi,θiΤ.By eigenvalue decomposition ofRXX=EXXH, we can obtain mutually orthogonal subspace US and noise subspace UN based on Hadamard product-extended signal. Using space-time pseudo spectral search can estimate τi and θi, 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.


## 3. Fast Estimation Method of Two-Dimensional Parameters

### 3.1. Algorithm Description

According to (7), the frequency domain estimation vector of the reference antenna element is obtained as
(9)X0,:=H0,:+n0,:=Sτa′+n0,:,where X0,:=x0,:,x0,:,…,x0,:Τ and H0,:=H0,:,H0,:,…,H0,:Τ are the estimation vectors and response vectors of the channel in frequency domain, respectively, Sτ=sτ0,sτ1,…,sτLP−1 is Vandermonde matrix of the N frequency points at the reference antenna element, and sτi=1,e−j2πτi/T,…,e−j2πN−1τi/TΤ, α′=α0′,α1′,…,αLP−1′Τ, αi′=αie−j2πfcτi, and n0,:=n0,:,n0,:,…,n0,:Τ are the additive Gauss white noise vector.For the reference antenna element, the eigendecomposition of the covariance matrix withP snapshot measurement data can be expressed as
(10)R^X0,:X0,:=1P∑j=1PX0,:jHX0,:jH=U^SΛ^SU^SH+U^NΛ^NU^NH,where U^S and U^N are, respectively, the signal subspace and noise subspace and Λ^S and Λ^N 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
(11)PMUSIC′τ=sHτU^NU^NHsτ.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. Letzτ=e−j2πτ/T and szτ=1,zτ,…,zτN−1T be the power term of zτ∗, and (11) is not the polynomials of zτ. Then, sHzτ=bΤz−1=1,zτ−1,…,zτ−N−1, the polynomials of Root-MUSIC for TOA, can be expressed as
(12)f^zτ=zτN−1bHzτU^NU^NHbzτ.Then, by utilizing the sampledN frequency points at the reference element, the corresponding delay estimation can be obtained as
(13)τ^i=−anglez^τiT2π,i=0,1,…,LP−1,where z^τ0,z^τ1,…,z^τLP−1 are LP 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 theM antenna elements can be obtained by the above algorithm.
(14)θ^i=arcsin−anglez^θiλ2πd,i=0,1,…,LP−1.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 matrixY is constructed as
(15)Y=X0,:X1,:=D0τ,θD1τ,θα+n0,:n1,:=D′τ,θα+n′,where D′=d′τ0,θ0,d′τ1,θ1,…,d′τLP−1,θLP−1,
(16)d′τi,θi=1,e−j2πτi/T,…,e−j2πN−1τi/T,e−j2πdsinθi/λ,…,e−j2πdsinθi/λe−j2πN−1τi/TT.The covariance matrix ofY can be an eigenvalue decomposed as R^YY=1/P∑j=1PYYH=V^SΦ^SV^SH+V^NΦ^NV^NH, where V^S∈C2MN×Lp and V^N∈C2MN×2MN−Lp are, respectively, the signal subspace and the noise subspace; obviously, the column vector d′τ,θ and noise subspace V^N are orthogonal, which make d′τ,θ·V^N=0. In order to achieve the pairing of delay and arrival angle, the cost function can be obtained by
(17)hτ,θ=d′Hτ,θV^NV^HNd′τ,θ.So the pairing of the estimated parameters is to minimize the cost function(18)τ^i,θ^j=mini,jhτ^i,θ^j,i,j=0,1,…,LP−1.Firstly, select a fixedτ^i from τ^ii=0LP−1, and let θ^i traverse θ^ii=0LP−1, then the τ^i,θ^j corresponding to the minimum value of the cost function is the correct pairing value.


### 3.3. Computational Complexity Analysis

The computational complexity of the algorithm in this paper consists of the following four aspects: the estimation of the covariance matrixR^X:,0X:,0 and R^X0,:X0,:, the computational complexity is OM2P and ON2P, respectively; the eigenvalue decomposition of the covariance matrix R^X:,0X:,0 and R^X0,:X0,:, the computational complexity is OM3 and ON3; the complexity of constructed polynomial is OM2−MLp and ON2−NLp; the computational complexity of parameter pairing is O2N2P+2N3+2N2Lp!+4NLp!−2NLpLp!. So the total complexity of the algorithm is OM3+9N3+M2P+1+N25P+1+4Lp!−MLp−NLp+4NLp!−2NLpLp!.The total complexity of using 2D-MUSIC algorithm isOM3N3+M2N2P+MNMN−LpWτWθ+MN−LpWτWθ, and Wτ and Wθ 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 O2N3+2M3+4P+1N2+Lp2N. Table 1 compares the computational complexity of the three algorithms.Table 1
Algorithm complexity comparison.
Algorithm type
Computational complexity
2D-MUSIC
OM3N3+M2N2P+MNMN−LpWτWθ+MN−LpWτWθ
ESPRIT
O2N3+2M3+4P+1N2+Lp2N
Proposed algorithm
OM3+9N3+M2P+1+N25P+1+4Lp!−MLp−NLp+4NLp!−2NLpLp!AssumingM=4, P=100, Wτ=1000, Wθ=1000, and Lp=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.Figure 1
Computational complexity of different algorithms.


## 3.1. Algorithm Description

According to (7), the frequency domain estimation vector of the reference antenna element is obtained as
(9)X0,:=H0,:+n0,:=Sτa′+n0,:,where X0,:=x0,:,x0,:,…,x0,:Τ and H0,:=H0,:,H0,:,…,H0,:Τ are the estimation vectors and response vectors of the channel in frequency domain, respectively, Sτ=sτ0,sτ1,…,sτLP−1 is Vandermonde matrix of the N frequency points at the reference antenna element, and sτi=1,e−j2πτi/T,…,e−j2πN−1τi/TΤ, α′=α0′,α1′,…,αLP−1′Τ, αi′=αie−j2πfcτi, and n0,:=n0,:,n0,:,…,n0,:Τ are the additive Gauss white noise vector.For the reference antenna element, the eigendecomposition of the covariance matrix withP snapshot measurement data can be expressed as
(10)R^X0,:X0,:=1P∑j=1PX0,:jHX0,:jH=U^SΛ^SU^SH+U^NΛ^NU^NH,where U^S and U^N are, respectively, the signal subspace and noise subspace and Λ^S and Λ^N 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
(11)PMUSIC′τ=sHτU^NU^NHsτ.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. Letzτ=e−j2πτ/T and szτ=1,zτ,…,zτN−1T be the power term of zτ∗, and (11) is not the polynomials of zτ. Then, sHzτ=bΤz−1=1,zτ−1,…,zτ−N−1, the polynomials of Root-MUSIC for TOA, can be expressed as
(12)f^zτ=zτN−1bHzτU^NU^NHbzτ.Then, by utilizing the sampledN frequency points at the reference element, the corresponding delay estimation can be obtained as
(13)τ^i=−anglez^τiT2π,i=0,1,…,LP−1,where z^τ0,z^τ1,…,z^τLP−1 are LP 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 theM antenna elements can be obtained by the above algorithm.
(14)θ^i=arcsin−anglez^θiλ2πd,i=0,1,…,LP−1.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 matrixY is constructed as
(15)Y=X0,:X1,:=D0τ,θD1τ,θα+n0,:n1,:=D′τ,θα+n′,where D′=d′τ0,θ0,d′τ1,θ1,…,d′τLP−1,θLP−1,
(16)d′τi,θi=1,e−j2πτi/T,…,e−j2πN−1τi/T,e−j2πdsinθi/λ,…,e−j2πdsinθi/λe−j2πN−1τi/TT.The covariance matrix ofY can be an eigenvalue decomposed as R^YY=1/P∑j=1PYYH=V^SΦ^SV^SH+V^NΦ^NV^NH, where V^S∈C2MN×Lp and V^N∈C2MN×2MN−Lp are, respectively, the signal subspace and the noise subspace; obviously, the column vector d′τ,θ and noise subspace V^N are orthogonal, which make d′τ,θ·V^N=0. In order to achieve the pairing of delay and arrival angle, the cost function can be obtained by
(17)hτ,θ=d′Hτ,θV^NV^HNd′τ,θ.So the pairing of the estimated parameters is to minimize the cost function(18)τ^i,θ^j=mini,jhτ^i,θ^j,i,j=0,1,…,LP−1.Firstly, select a fixedτ^i from τ^ii=0LP−1, and let θ^i traverse θ^ii=0LP−1, then the τ^i,θ^j corresponding to the minimum value of the cost function is the correct pairing value.


## 3.3. Computational Complexity Analysis

The computational complexity of the algorithm in this paper consists of the following four aspects: the estimation of the covariance matrixR^X:,0X:,0 and R^X0,:X0,:, the computational complexity is OM2P and ON2P, respectively; the eigenvalue decomposition of the covariance matrix R^X:,0X:,0 and R^X0,:X0,:, the computational complexity is OM3 and ON3; the complexity of constructed polynomial is OM2−MLp and ON2−NLp; the computational complexity of parameter pairing is O2N2P+2N3+2N2Lp!+4NLp!−2NLpLp!. So the total complexity of the algorithm is OM3+9N3+M2P+1+N25P+1+4Lp!−MLp−NLp+4NLp!−2NLpLp!.The total complexity of using 2D-MUSIC algorithm isOM3N3+M2N2P+MNMN−LpWτWθ+MN−LpWτWθ, and Wτ and Wθ 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 O2N3+2M3+4P+1N2+Lp2N. Table 1 compares the computational complexity of the three algorithms.Table 1
Algorithm complexity comparison.
Algorithm type
Computational complexity
2D-MUSIC
OM3N3+M2N2P+MNMN−LpWτWθ+MN−LpWτWθ
ESPRIT
O2N3+2M3+4P+1N2+Lp2N
Proposed algorithm
OM3+9N3+M2P+1+N25P+1+4Lp!−MLp−NLp+4NLp!−2NLpLp!AssumingM=4, P=100, Wτ=1000, Wθ=1000, and Lp=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.Figure 1
Computational complexity of different algorithms.


## 4. 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 asσ2ηΤΤ, where η=τΤθΤΤ. According to (7) and the relevant initial conditions, the joint probability density function of X1,X2,…,XK is
(19)fXη,α=∏j=1K12πMNσ2/2MNexp−1σ2Xj‐Dαj22.Thus, the log-likelihood function of (19), omitting the constant term, is
(20)lnf=−MNKlnσ2−1σ2∑j=1KXj‐DαjHXj‐Dαj.To calculate the Fisher information matrix (FIM), the derivatives of (20) with respect to σ2, the real part Reαj of αj, the imaginary part Imαj of αj, and η can be obtained as
(21)∂lnf∂σ2=−MNKσ2+1σ4∑j=1KnHjnj,∂lnf∂Reαj=2σ2ReDHnj,∂lnf∂Imαj=2σ2ImDHnj.Sinceη is a two-dimensional vector, the derivatives of (20) with respect to η are different from the ones in the angle-spread only model.
(22)∂lnf∂θi=1σ2∑j=1K∂∂θi2ReαHjDHXj−αHjDHDαj=2σ2∑j=1KReαijgθiHnj,i=0,…,LP−1,and similarly,
(23)∂lnf∂τi=2σ2∑j=1NReαijgτiHnj,i=0,…,LP−1,where gθi is the derivative with respect to θi of the ith column of D. Similarly, gτi is the derivative with respect to τi of the ithcolumn of D. Due to dmτi,θi=sτi⊕amθi, then
(24)gθi=sτi⊕∂a0θi∂θisτi⊕∂a1θi∂θi⋮sτi⊕∂aM−1θi∂θi,gτi=∂sτi∂τi⊕a0θi∂sτi∂τi⊕a1θi⋮∂sτi∂τi⊕aM−1θi.Further, written more compactly(25)∂lnf∂θ=2σ2∑j=1KRediagαjHGθHnj,∂lnf∂τ=2σ2∑j=1KRediagαjHGτHnj,where Gθ=gθ0,…,gθLP−1 and Gτ=gτ0,…,gτLP−1. Let G=GτGθMN×2LP and Bj=I2⊗diagαj, then we can obtain
(26)∂lnf∂η=2σ2∑j=1KRediagαjHGHnj.By using the results proven in [20], the following expressions can be attained
(27)E∂lnf∂σ22=MNKσ4,E∂lnf∂Reαj∂lnf∂ReαpΤ=2σ2ReDHDδj,p,E∂lnf∂Reαj∂lnf∂ImαpΤ=−2σ2ImDHDδj,p,E∂lnf∂Imαj∂lnf∂ImαpΤ=2σ2ReDHDδj,p,E∂lnf∂Reαj∂lnf∂ηΤ=2σ2ReDHGBj,E∂lnf∂Imαj∂lnf∂ηΤ=2σ2ImDHGBj,E∂lnf∂η∂lnf∂ηΤ=2σ2∑j=1KReBHrGHGBj.Finally, the FIM for the parameters is obtained. Using the results in [20], the CRB of the parameters can be shown as
(28)CRBη=σ22∑j=1KReBHjGHI−DDHD−1DHGBj−1,which completes the proof.


## 5. Simulation Results

In this section, the performance of fast estimation of two-dimensional 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 isB=20MHz, the FFT cycle is T=3.2μs, the number of subcarriers is N=64, the carrier frequency is fc=2.3GHz, the number of elements is M=12, the array spacing is d=λ/2 meters, and so on.


### 5.1. Simulation 1

In the case of low SNR at 0 dB and 5 dB, assuming that the multipath number of the received signals isLP=3, the arrival time is τ1=100ns, τ2=140ns, and τ3=180ns, respectively; the direction of arrival is θ1=30°, θ2=40°, and θ3=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.Figure 2
The scatter plot of joint TOA and DOA estimation, SNR = 0 dB.Figure 3
The scatter plot of joint TOA and DOA estimation, SNR = 5 dB.


### 5.2. Simulation 2

Under different SNRs, other conditions are the same as simulation 1, and the scatter plots of TOA and DOA can be obtained. It can be seen from Figure4 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.Figure 4
The scatter plot of TOA estimation under different SNRs.Figure 5
The scatter plot of DOA estimation under different SNRs.


### 5.3. 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 asΔτS,i=τ^S,i−τi and ΔθS,i=θ^S,i−θi, respectively. Then, the scatter diagram of ΔτS,1, ΔτS,2, and ΔτS,3 corresponding to three multipath components is shown in Figure 6, and the corresponding ΔθS,1, ΔθS,2, and ΔθS,3 distribution are shown in Figure 7. It can be seen from Figures 6 and 7 that the algorithm at low SNR can almost achieve the unbiased estimation of TOA and DOA.Figure 6
The distribution of TOA estimation error, SNR = 5 dB.Figure 7
The distribution of DOA estimation error, SNR = 5 dB.


### 5.4. Simulation 4

Define the root mean square error (RMSE) asΩRMSE=1/N∑n=0N−1x^n−x2, where x^n 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 Lp=2, Lp=3, and Lp=4, respectively. The results are based on 100 Monte Carlo simulations. As expected, the TOA and DOA estimation performances of the first path decrease with the increase of the number of multipath. At low SNR of SNR = 5 dB, the estimation accuracy under multipath conditions can still meet the positioning requirements, so the algorithm has certain adaptability to multipath environment.Figure 8
TOA estimation performance of the first path under different multipath numbers.Figure 9
DOA estimation performance of the first path under different multipath numbers.


### 5.5. Simulation 5

Under different SNRs, other conditions are the same as simulation 1. The proposed algorithm is compared with 2D-MUSIC algorithm, ESPRIT algorithm, 2D MP algorithm [16], and 1D MP algorithm [13]. The TOA and DOA RMSE curves of each algorithm’s first path are shown in Figures 10 and 11, respectively, and compared with the CRB of this model. As shown in Figures 10 and 11, the performance of the proposed algorithm is obviously superior to that of ESPRIT algorithm and 2D MP algorithm, and very close to 2D-MUSIC algorithm, and the accuracy of parameter estimation completely meets the positioning requirements.Figure 10
Performance comparison of different algorithms for TOA estimation.Figure 11
Performance comparison of different algorithms for DOA estimation.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 the proposed algorithm. Therefore, it can realize the fast estimation of two-dimensional location parameters.


## 5.1. Simulation 1

In the case of low SNR at 0 dB and 5 dB, assuming that the multipath number of the received signals isLP=3, the arrival time is τ1=100ns, τ2=140ns, and τ3=180ns, respectively; the direction of arrival is θ1=30°, θ2=40°, and θ3=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.Figure 2
The scatter plot of joint TOA and DOA estimation, SNR = 0 dB.Figure 3
The scatter plot of joint TOA and DOA estimation, SNR = 5 dB.


## 5.2. Simulation 2

Under different SNRs, other conditions are the same as simulation 1, and the scatter plots of TOA and DOA can be obtained. It can be seen from Figure4 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.Figure 4
The scatter plot of TOA estimation under different SNRs.Figure 5
The scatter plot of DOA estimation under different SNRs.


## 5.3. 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 asΔτS,i=τ^S,i−τi and ΔθS,i=θ^S,i−θi, respectively. Then, the scatter diagram of ΔτS,1, ΔτS,2, and ΔτS,3 corresponding to three multipath components is shown in Figure 6, and the corresponding ΔθS,1, ΔθS,2, and ΔθS,3 distribution are shown in Figure 7. It can be seen from Figures 6 and 7 that the algorithm at low SNR can almost achieve the unbiased estimation of TOA and DOA.Figure 6
The distribution of TOA estimation error, SNR = 5 dB.Figure 7
The distribution of DOA estimation error, SNR = 5 dB.


## 5.4. Simulation 4

Define the root mean square error (RMSE) asΩRMSE=1/N∑n=0N−1x^n−x2, where x^n 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 Lp=2, Lp=3, and Lp=4, respectively. The results are based on 100 Monte Carlo simulations. As expected, the TOA and DOA estimation performances of the first path decrease with the increase of the number of multipath. At low SNR of SNR = 5 dB, the estimation accuracy under multipath conditions can still meet the positioning requirements, so the algorithm has certain adaptability to multipath environment.Figure 8
TOA estimation performance of the first path under different multipath numbers.Figure 9
DOA estimation performance of the first path under different multipath numbers.


## 5.5. Simulation 5

Under different SNRs, other conditions are the same as simulation 1. The proposed algorithm is compared with 2D-MUSIC algorithm, ESPRIT algorithm, 2D MP algorithm [16], and 1D MP algorithm [13]. The TOA and DOA RMSE curves of each algorithm’s first path are shown in Figures 10 and 11, respectively, and compared with the CRB of this model. As shown in Figures 10 and 11, the performance of the proposed algorithm is obviously superior to that of ESPRIT algorithm and 2D MP algorithm, and very close to 2D-MUSIC algorithm, and the accuracy of parameter estimation completely meets the positioning requirements.Figure 10
Performance comparison of different algorithms for TOA estimation.Figure 11
Performance comparison of different algorithms for DOA estimation.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 the proposed algorithm. Therefore, it can realize the fast estimation of two-dimensional location parameters.


## 6. 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 two-dimensional 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.



---

*Source: 7306902-2018-03-26.xml*