# Fast Tensor-Based Joint Estimation for Time Delay and Angle of Arrival in OFDM System

Authors: Jinzhi Du; Weijia Cui; Bin Ba; Chunxiao Jian; Haiyun Xu
Journal: International Journal of Antennas and Propagation
 (2022)

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## Abstract

Nowadays, the joint estimation of time delay (TD) and angle of arrival (AOA) using conventional vector structure suffers from the considerable complexity of multidimensional spectrum search. Therefore, a fast estimation method using orthogonal frequency division multiplexing (OFDM) technology and uniform planar array (UPA) is proposed in this paper, which adopts low-complexity tensor-based operations and spatial-frequency features to reconfigure the channel frequency response. To begin with, the array response is integrated with the OFDM signal characteristics to build an extended array in tensor form. Afterwards, we process the covariance matrix of the tensor structure by CANDECOMP/PARAFAC decomposition (CPD) to separate the respective signal subspaces of TD and AOA estimates. Finally, we conduct a one-dimensional (1-D) spectrum search to locate the TD estimates and a two-dimensional (2-D) spectrum search to locate the AOA estimates. The simulated performance demonstrates that the proposed algorithm offers precise estimates at low signal-to-noise ratios in a multipath environment and outperforms traditional vector-based algorithms with respect to computational complexity.


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## Body

## 1. Introduction

Orthogonal frequency division multiplexing (OFDM), a multicarrier digital modulation technique, uses multiple parallel subcarriers to achieve serial high-rate data communication. The subcarriers are orthogonal to each other, which can combat frequency-selective fading in an effective way. Moreover, the OFDM system can offer data traffic and positioning services to users, which are extensively applied in 5G mobile communication [1], IEEE 802.11 protocol [2], satellite communication [3], intelligence steer [4], and underwater acoustic communication [5]. Time delay (TD) and angle of arrival (AOA) are significant factors for positioning systems, as for indoor localization [6] and radar [7]. The frequency-domain AOA estimation algorithm [8] is studied with the same multiple signal classification (MUSIC) approach as the time-domain narrowband signal model. However, its capability is limited due to the size of the array aperture. The algorithm in [9] proposes a TD estimation method for OFDM signals, but this algorithm is not available in multipath environments. As researchers choose super-resolution approaches like MUSIC [10], estimating signal parameters via the rotational invariance techniques (ESPRIT) [11], the propagator method (PM) [12], and compressive sensing [13], it is difficult for them to enhance the estimation accuracy due to the restrictions of the signal bandwidth.The joint estimation of TD and AOA with spatio-time parameter coupling characteristics not only increase the accuracy but also decrease the amount of receiving nodes, thus reducing the system overhead and improving the efficiency of the positioning system. Hence, the joint estimation methods employed are important. In addition, a method presented in [14] solves TD under wideband signal conditions and then estimates AOA according to the triangular geometry with TD inequality, but its performance is not significantly improved. In further, a method for constructing an extended channel frequency response to obtain highly accurate joint estimation using an OFDM system is proposed in [15]. Nevertheless, the complexity of the method is extremely high due to the need for full-field-of-view search.Tensor as an efficient way to process multidimensional data have achieved extensive research and applications in fields such as machine learning [16], image processing [17], MIMO radar [18], and vehicle-to-everything communication [19]. Unlike traditional vector bases, tensor processing, with the help of data Kronecker structural properties, breaks up large-scale matrix operations into multiple small-scale matrix operations, avoiding additional overhead due to repeated calculations.Most of the above algorithms use uniform linear arrays (ULA) [20] for parameter estimation. Due to the one-dimensional (1-D) structure, only azimuthal angles can be estimated, and two-dimensional (2-D) angle search cannot be achieved. This paper studies the joint estimation for TD and AOA of OFDM techniques under uniform planar arrays (UPA) [21], which are commonly used in practical engineering to obtain stable 2-D angle estimates.Furthermore, we propose a joint estimation algorithm for multipath environments that functions to keep the estimation precision highly while reducing the considerable complexity associated with spectrum peak search methods. The channel frequency response of multiple subcarriers of the OFDM signal combined with the array response of the receiving antenna can be applied to construct an extended virtual array response with accurate estimation. We reconfigure the extended virtual array response by taking advantage of the structure of the tensor structure, which reduces the spectrum peak search dimension while maintaining the original estimation accuracy. Moreover, we can estimate TD and AOA separately, which greatly reduces the complexity.The rest of the paper is summarized as follows. At first, we introduce the signal model and the corresponding joint estimation algorithm applied to the vector-based algorithm in Section2. In Section 3, we describe the transformation of the signal model under a tensor structure and the corresponding joint estimation algorithm and further outline the algorithm procedure. We perform the complexity analyses in Section 4. For Section 5, we analyze the simulation performance. In the end, we summarize our efforts in Section 6. The notations used in this paper are described in Table 1.Table 1
Notations.
Identity matrixITranspose•TConjugate•∗Hermitian transpose•HKronecker product⊗Khatri-rao product⊙Tensor outer product°Orthogonalizationorth•Statistical expectationE•Tensor contraction along theqth dimension•,•q


## 2. Vector-Based Algorism

### 2.1. Signal Model

The array sensors used in this paper consist ofM×M array sensors as illustrated in Figure 1. The function of this model is to estimate TD and AOA without needing to set the signal source position. Scholars typically model multipath wireless propagation channels typically as complex low-pass equivalent impulse responses. Following channel estimation, the channel impact response of the array sensor at location x,y in the sth snapshot is denoted as follows:where K indicates the quantity of multipaths, αksejβkS is the complex attenuation in the kth path, αks is its amplitude, βks is its phase, which matches the uniform distribution with density function U0,2π.Figure 1
Signal arrives in UPA.(1)hx,yst=∑k=1Kαksejβksδt−τk−ξk,x,y,We assume that the reference array sensor is the array sensor at the origin of the coordinate axis. In the multipath environment, the propagation delay of the signal source reaching the reference array sensor via thekth path is τk. And, the relative delay of the array sensor at x,y is ξk,x,y, denoted as follows:(2)ξk,x,y=λsinθkxcosφk+ysinφk2c,In which λ is the impinging signal wavelength; φk,θk is the direction of the incident signal.We assume that the quantity of OFDM subcarriers isL. After the Fourier transform of (1), the channel frequency domain response for the lth subcarrier at the x,y array sensor can be obtained as follows:(3)Hl,x,ys=∑k=1Kαksejβkse−j2πfc+Δflτk+ξk,x,y+nl,x,ys,where Δf is the subcarrier spacing, fc is the carrier frequency, and nl,x,ys is additive white Gaussian noise at power σ2. From (3), the channel frequency response for lth subcarrier is obtained as follows:(4)Hls=Hl,0,0sHl,0,1s⋮Hl,M−1,M−1s=Alτ,ξρs+nls,where(5)τ=τ1τ2⋯τKT,ξ=ξ1ξ2⋯ξK,ξk=ξk,0,0ξk,0,1⋯ξk,M−1,M−1T,ρks=αksejβks,ρs=α1sejβ1sα2sejβ2s⋯αksejβksT,Alτ,ξ=alτ1,ξ1⋯alτK,ξK,alτk,ξk=alτk,ξk,0,0⋯alτk,ξk,M−1,M−1T.In which alτk,ξk,x,y=e−j2πfc+Δflτk+ξk,x,y, and(6)nls=nl,0,0snl,1,0s⋯nl,M−1,M−1sT.In case there are a total ofS snapshots, (4) is denoted by the following equation:(7)Hl=Alτ,ξρ+nl,where(8)ρ=ρ1ρ2⋯ρS,nl=nl1nl2⋯nlS.According to the space-time equivalence, OFDM signal subcarriers can be analogous to the array sensors. Based on the space signal processing algorithm, an extended channel frequency response (EX-Response) matrix H∈ℂM2L×S with coupled spatial-frequency information can be constructed whose expression is(9)H=H0H1⋮HL−1=A0τ,ξA1τ,ξ⋮AL−1τ,ξρ+N=Aτ,ξρ+N,where(10)N=n0Tn1T⋯nL−1TT.The model converts the time domain information to the frequency domain for processing and has the same structure as the traditional time domain narrowband model. In addition, building an EX-Response has two obvious merits. On the one hand, the virtual bandwidth is enlarged toM2 multiples of the actual bandwidth. On the other hand, the virtual aperture is enlarged to L multiples of the actual aperture. From the perspective of parameter estimation, the bandwidth and aperture expansion will improve the performance of estimating TD and AOA.


### 2.2. The Joint TD and AOA Estimation

Combining (9), the covariance matrix can be defined for;(11)RH=1SHHH=Aτ,ξRρAτ,ξH+σ2IM2L,where Rρ is a complex attenuation covariance matrix. When the complex attenuations are independent, rankRρ=K. Due to rankAτ,ξ=K and rankAτ,ξRρAτ,ξH=K, the situation satisfies the requirements for using the MUSIC method. Furthermore, the eigenvalue decomposition of RH can be expressed as follows:(12)RH=USΣSUSH+UNΣNUNH,where ΣN=σ2IM2−K is a diagonal matrix of M2−K eigenvalues, and UN is the noise subspace. Therefore, the joint estimated spatial spatial expression is expressed as follows:(13)Pτ,θ,φ=UNHAτ,ξ2−2.Althoughτ^,θ^,φ^ is obtained by the three-dimensional (3-D) MUSIC approach with excellent estimation accuracy, the 3-D spectral peak search complexity is unacceptably high.


## 2.1. Signal Model

The array sensors used in this paper consist ofM×M array sensors as illustrated in Figure 1. The function of this model is to estimate TD and AOA without needing to set the signal source position. Scholars typically model multipath wireless propagation channels typically as complex low-pass equivalent impulse responses. Following channel estimation, the channel impact response of the array sensor at location x,y in the sth snapshot is denoted as follows:where K indicates the quantity of multipaths, αksejβkS is the complex attenuation in the kth path, αks is its amplitude, βks is its phase, which matches the uniform distribution with density function U0,2π.Figure 1
Signal arrives in UPA.(1)hx,yst=∑k=1Kαksejβksδt−τk−ξk,x,y,We assume that the reference array sensor is the array sensor at the origin of the coordinate axis. In the multipath environment, the propagation delay of the signal source reaching the reference array sensor via thekth path is τk. And, the relative delay of the array sensor at x,y is ξk,x,y, denoted as follows:(2)ξk,x,y=λsinθkxcosφk+ysinφk2c,In which λ is the impinging signal wavelength; φk,θk is the direction of the incident signal.We assume that the quantity of OFDM subcarriers isL. After the Fourier transform of (1), the channel frequency domain response for the lth subcarrier at the x,y array sensor can be obtained as follows:(3)Hl,x,ys=∑k=1Kαksejβkse−j2πfc+Δflτk+ξk,x,y+nl,x,ys,where Δf is the subcarrier spacing, fc is the carrier frequency, and nl,x,ys is additive white Gaussian noise at power σ2. From (3), the channel frequency response for lth subcarrier is obtained as follows:(4)Hls=Hl,0,0sHl,0,1s⋮Hl,M−1,M−1s=Alτ,ξρs+nls,where(5)τ=τ1τ2⋯τKT,ξ=ξ1ξ2⋯ξK,ξk=ξk,0,0ξk,0,1⋯ξk,M−1,M−1T,ρks=αksejβks,ρs=α1sejβ1sα2sejβ2s⋯αksejβksT,Alτ,ξ=alτ1,ξ1⋯alτK,ξK,alτk,ξk=alτk,ξk,0,0⋯alτk,ξk,M−1,M−1T.In which alτk,ξk,x,y=e−j2πfc+Δflτk+ξk,x,y, and(6)nls=nl,0,0snl,1,0s⋯nl,M−1,M−1sT.In case there are a total ofS snapshots, (4) is denoted by the following equation:(7)Hl=Alτ,ξρ+nl,where(8)ρ=ρ1ρ2⋯ρS,nl=nl1nl2⋯nlS.According to the space-time equivalence, OFDM signal subcarriers can be analogous to the array sensors. Based on the space signal processing algorithm, an extended channel frequency response (EX-Response) matrix H∈ℂM2L×S with coupled spatial-frequency information can be constructed whose expression is(9)H=H0H1⋮HL−1=A0τ,ξA1τ,ξ⋮AL−1τ,ξρ+N=Aτ,ξρ+N,where(10)N=n0Tn1T⋯nL−1TT.The model converts the time domain information to the frequency domain for processing and has the same structure as the traditional time domain narrowband model. In addition, building an EX-Response has two obvious merits. On the one hand, the virtual bandwidth is enlarged toM2 multiples of the actual bandwidth. On the other hand, the virtual aperture is enlarged to L multiples of the actual aperture. From the perspective of parameter estimation, the bandwidth and aperture expansion will improve the performance of estimating TD and AOA.


## 2.2. The Joint TD and AOA Estimation

Combining (9), the covariance matrix can be defined for;(11)RH=1SHHH=Aτ,ξRρAτ,ξH+σ2IM2L,where Rρ is a complex attenuation covariance matrix. When the complex attenuations are independent, rankRρ=K. Due to rankAτ,ξ=K and rankAτ,ξRρAτ,ξH=K, the situation satisfies the requirements for using the MUSIC method. Furthermore, the eigenvalue decomposition of RH can be expressed as follows:(12)RH=USΣSUSH+UNΣNUNH,where ΣN=σ2IM2−K is a diagonal matrix of M2−K eigenvalues, and UN is the noise subspace. Therefore, the joint estimated spatial spatial expression is expressed as follows:(13)Pτ,θ,φ=UNHAτ,ξ2−2.Althoughτ^,θ^,φ^ is obtained by the three-dimensional (3-D) MUSIC approach with excellent estimation accuracy, the 3-D spectral peak search complexity is unacceptably high.


## 3. Tensor-Based Algorism

### 3.1. Signal Model

In the joint direction matrixAτ,ξ of the vector basis, the propagation delay phase and the relative delay phase are stacked on the matrix columns by the Kronecker product, which are M2L×K dimensional matrices. The covariance matrix RH generated by Aτ,ξ contains a large number of redundant terms, which imposes a serious computational burden on the 3-D spectrum peak search. In tensor operations, the large-scale matrix computation is decomposed into small-scale matrix computation, thus considerably reducing the computational load. In order to preserve the spatial-frequency characteristics of the EX-Response, the S snapshots are connected along the time dimension to form a 3-D tensor. Thus the tensor equivalent ℋ∈ℂM2×L×S of (9) is(14)ℋ=Aτ,ξ°ρT+N=V°T°ρT+N,where.(15)Tk=e−j2πfcτke−j2πfc+Δfτk⋮​e−j2πfc+Δfl−1τk​​⊙​e−j2πΔf0ξk,0,0e−j2πΔf1ξk,0,1⋮​e−j2πΔfl−1ξk,M−1,M−1​,T=T1T2⋯TK.Considering the practical situation, the relative time is considerably less compared to the propagation delay, and the subcarrier relative delay is even smaller, which means that it can be neglected. Thus, the equation can be established as follows:(16)Tk=e−j2πfcτke−j2πfc+Δfτk⋯e−j2πfc+Δfl−1τkT≈Tk,T=T1T2⋯TK≈T.Therefore, in order to separate the propagation time delay phase from the relative time delay phase, (3) can be approximated as follows:(17)ℋ≈V°T°ρT+N,where(18)Vk=e−j2πfcξk,0,0e−j2πfcξk,0,1⋯e−j2πfcξk,M−1,M−1T,V=V1V2⋯VK,In which N is additive white Gaussian noise in the same dimension as ℋ with power of σ2.


### 3.2. The Joint TD and AOA Estimation

In (17), the tensor covariance matrix ℛℋ∈ℂM2×L×M2×L can be obtained as follows:(19)ℛℋ=Eℋ,ℋ∗3=∑k=1Kβk2Vk°Tk°Vk∗°Tk∗+Nℛ,where βk2=EρkTρkT∗ is the complex decay of the kth path and ρk=ρk1ρk2⋯ρkS; Nℛ=EN,N∗3 is the noise term. In practice, the tensor covariance matrix ℛℋ can be estimated as follows:(20)ℛ^ℋ=1Sℋ,ℋ∗3.The CANDECOMP/PARAFAC decomposition (CPD) [22] is a common method for splitting a high-dimensional tensor. We perform CPD on the four-dimensional tensor ℛ^ℋ to obtain a sum ℛCP∈ℂM2×K,L×K,M2×K,L×K of component rank-one tensors as follows:(21)ℛCP=VCP,TCP,VCP∗,TCP∗.While the complex attenuations are independent, then there is(22)rankVCP+rankTCP+rankVCP∗+rankTCP∗≥2K+N−1.Furthermore, the uniqueness of the CPD is satisfied [22], where N is the number of matrices obtained by the CPD or the dimension of the decomposed matrix. We extract the signal subspace TCP from ℛCP for TD estimation and the signal subspace VCP for AOA estimation.


#### 3.2.1. TD Estimation

We obtain the noise subspaceUntUntH for TD estimation from the orthogonal complementary subspace of the signal subspace TCP as follows:(23)UntUntH=I−orthTCPorthTCPH.Then, the spatial spectrum expression of the TD estimate can be obtained as follows:(24)Pτ=1TτHUntUntHTτ.WhenK<L, we adopt the 1-D MUSIC method to obtain the τ^ value, which has a higher estimation accuracy and greatly reduces the computational complexity compared to the 3-D algorithm.


#### 3.2.2. AOA Estimation

We obtain the noise subspaceUnaUnaH for AOA estimation from the orthogonal complementary subspace of the signal subspace VCP as follows:(25)UnaUnaH=I−orthVCPorthVCPH.Then, the spatial spectrum expression of the AOA estimate can be obtained as follows:(26)Pθ,φ=1Vθ,φHUnaUnaHVθ,φ.WhenK<M, we use the 2-D MUSIC method to obtain the θ^,φ^ value, which has a higher estimation accuracy and greatly reduces the computational complexity compared to the 3-D algorithm.


#### 3.2.3. Algorithm Steps

Table2 lists the main processes of the proposed algorithm.Table 2
Algorithm steps.
step1:The tensor EX-Response matrixℋ is constructed according to (26).step2:The tensor covariance matrixℛℋ⌢ is constructed based on (30).step3:Perform a CPD ofℛ^ℋ, which solves the signal subspaces TCP and VCP, and then the corresponding noise subspaces and are obtained by (33) and (35), respectively. UntUntH and UnaUnaH.step4:Proceeded to conduct a 1-D spectrum peak search forUntUntH to solve the τ^ by (34), and a 2-D spectrum peak search for UnaUnaH to solve the θ^,φ^ by (36).


## 3.1. Signal Model

In the joint direction matrixAτ,ξ of the vector basis, the propagation delay phase and the relative delay phase are stacked on the matrix columns by the Kronecker product, which are M2L×K dimensional matrices. The covariance matrix RH generated by Aτ,ξ contains a large number of redundant terms, which imposes a serious computational burden on the 3-D spectrum peak search. In tensor operations, the large-scale matrix computation is decomposed into small-scale matrix computation, thus considerably reducing the computational load. In order to preserve the spatial-frequency characteristics of the EX-Response, the S snapshots are connected along the time dimension to form a 3-D tensor. Thus the tensor equivalent ℋ∈ℂM2×L×S of (9) is(14)ℋ=Aτ,ξ°ρT+N=V°T°ρT+N,where.(15)Tk=e−j2πfcτke−j2πfc+Δfτk⋮​e−j2πfc+Δfl−1τk​​⊙​e−j2πΔf0ξk,0,0e−j2πΔf1ξk,0,1⋮​e−j2πΔfl−1ξk,M−1,M−1​,T=T1T2⋯TK.Considering the practical situation, the relative time is considerably less compared to the propagation delay, and the subcarrier relative delay is even smaller, which means that it can be neglected. Thus, the equation can be established as follows:(16)Tk=e−j2πfcτke−j2πfc+Δfτk⋯e−j2πfc+Δfl−1τkT≈Tk,T=T1T2⋯TK≈T.Therefore, in order to separate the propagation time delay phase from the relative time delay phase, (3) can be approximated as follows:(17)ℋ≈V°T°ρT+N,where(18)Vk=e−j2πfcξk,0,0e−j2πfcξk,0,1⋯e−j2πfcξk,M−1,M−1T,V=V1V2⋯VK,In which N is additive white Gaussian noise in the same dimension as ℋ with power of σ2.


## 3.2. The Joint TD and AOA Estimation

In (17), the tensor covariance matrix ℛℋ∈ℂM2×L×M2×L can be obtained as follows:(19)ℛℋ=Eℋ,ℋ∗3=∑k=1Kβk2Vk°Tk°Vk∗°Tk∗+Nℛ,where βk2=EρkTρkT∗ is the complex decay of the kth path and ρk=ρk1ρk2⋯ρkS; Nℛ=EN,N∗3 is the noise term. In practice, the tensor covariance matrix ℛℋ can be estimated as follows:(20)ℛ^ℋ=1Sℋ,ℋ∗3.The CANDECOMP/PARAFAC decomposition (CPD) [22] is a common method for splitting a high-dimensional tensor. We perform CPD on the four-dimensional tensor ℛ^ℋ to obtain a sum ℛCP∈ℂM2×K,L×K,M2×K,L×K of component rank-one tensors as follows:(21)ℛCP=VCP,TCP,VCP∗,TCP∗.While the complex attenuations are independent, then there is(22)rankVCP+rankTCP+rankVCP∗+rankTCP∗≥2K+N−1.Furthermore, the uniqueness of the CPD is satisfied [22], where N is the number of matrices obtained by the CPD or the dimension of the decomposed matrix. We extract the signal subspace TCP from ℛCP for TD estimation and the signal subspace VCP for AOA estimation.


### 3.2.1. TD Estimation

We obtain the noise subspaceUntUntH for TD estimation from the orthogonal complementary subspace of the signal subspace TCP as follows:(23)UntUntH=I−orthTCPorthTCPH.Then, the spatial spectrum expression of the TD estimate can be obtained as follows:(24)Pτ=1TτHUntUntHTτ.WhenK<L, we adopt the 1-D MUSIC method to obtain the τ^ value, which has a higher estimation accuracy and greatly reduces the computational complexity compared to the 3-D algorithm.


### 3.2.2. AOA Estimation

We obtain the noise subspaceUnaUnaH for AOA estimation from the orthogonal complementary subspace of the signal subspace VCP as follows:(25)UnaUnaH=I−orthVCPorthVCPH.Then, the spatial spectrum expression of the AOA estimate can be obtained as follows:(26)Pθ,φ=1Vθ,φHUnaUnaHVθ,φ.WhenK<M, we use the 2-D MUSIC method to obtain the θ^,φ^ value, which has a higher estimation accuracy and greatly reduces the computational complexity compared to the 3-D algorithm.


### 3.2.3. Algorithm Steps

Table2 lists the main processes of the proposed algorithm.Table 2
Algorithm steps.
step1:The tensor EX-Response matrixℋ is constructed according to (26).step2:The tensor covariance matrixℛℋ⌢ is constructed based on (30).step3:Perform a CPD ofℛ^ℋ, which solves the signal subspaces TCP and VCP, and then the corresponding noise subspaces and are obtained by (33) and (35), respectively. UntUntH and UnaUnaH.step4:Proceeded to conduct a 1-D spectrum peak search forUntUntH to solve the τ^ by (34), and a 2-D spectrum peak search for UnaUnaH to solve the θ^,φ^ by (36).


## 3.2.1. TD Estimation

We obtain the noise subspaceUntUntH for TD estimation from the orthogonal complementary subspace of the signal subspace TCP as follows:(23)UntUntH=I−orthTCPorthTCPH.Then, the spatial spectrum expression of the TD estimate can be obtained as follows:(24)Pτ=1TτHUntUntHTτ.WhenK<L, we adopt the 1-D MUSIC method to obtain the τ^ value, which has a higher estimation accuracy and greatly reduces the computational complexity compared to the 3-D algorithm.


## 3.2.2. AOA Estimation

We obtain the noise subspaceUnaUnaH for AOA estimation from the orthogonal complementary subspace of the signal subspace VCP as follows:(25)UnaUnaH=I−orthVCPorthVCPH.Then, the spatial spectrum expression of the AOA estimate can be obtained as follows:(26)Pθ,φ=1Vθ,φHUnaUnaHVθ,φ.WhenK<M, we use the 2-D MUSIC method to obtain the θ^,φ^ value, which has a higher estimation accuracy and greatly reduces the computational complexity compared to the 3-D algorithm.


## 3.2.3. Algorithm Steps

Table2 lists the main processes of the proposed algorithm.Table 2
Algorithm steps.
step1:The tensor EX-Response matrixℋ is constructed according to (26).step2:The tensor covariance matrixℛℋ⌢ is constructed based on (30).step3:Perform a CPD ofℛ^ℋ, which solves the signal subspaces TCP and VCP, and then the corresponding noise subspaces and are obtained by (33) and (35), respectively. UntUntH and UnaUnaH.step4:Proceeded to conduct a 1-D spectrum peak search forUntUntH to solve the τ^ by (34), and a 2-D spectrum peak search for UnaUnaH to solve the θ^,φ^ by (36).


## 4. Algorithm Complexity Analysis

This section analyzes the complexity of the proposed tensor-based algorithm (EX-Proposed) under the EX-Response model and compares it with the corresponding vector-based algorithm (EX-Vector-Based). For the simulation comparison in V, the complexity of the tensor-based algorithm (Proposed) and the corresponding vector-based algorithm (Vector-Based) employing a single-frequency model [8] is also compared.The complexity of the algorithms can be split into vector-based covariance matrix computation, tensor-based covariance matrix computation, eigenvalue decomposition, CPD, and 1-D spectrum peak search, which areOSM4L2, OSM4L, OM6L3, O2NKM4L2+NK3, and OM2LM2L−KG, separately, in which G represents the number of spectrum points in the 1-D search. Therefore, the complexity of the EX-Proposed is OM4LS+2NKL+NK3+M2LM2L−KGφGθ+Gτ. The cost of the EX-Vector-Based is OS+M2LM4L2+M2LM2L−KGφGθGτ, in which Gφ, Gθ and Gτ indicate the quantity of searches for azimuth, elevation, and propagation delay, separately. Furthermore, the complexity of the single-frequency model algorithms, i.e., Proposed and Vector-Based are OM4SL+2NK+NK3+M2M2−KGφGθ+Gτ and OSL+M2M4+M2M2−KGφGθGτ, separately. In order to compare clearly, Table 3 summarizes the complexity of all methods. Furthermore, the complexity of the algorithms are compared in terms of snapshots S, the number of sensors M , and the searching step (Δτ and Δφ=Δθ, where Gφ=360/Δφ, Gθ=90/Δθ) in Figure 2(a)–2(d), respectively. For correspondence with subsequent simulations, we set L=64, K=3, and N=4. The other parameters are described in Figure 2.Table 3
Computational complexity comparison.
AlgorithmComplexityProposedOM4SL+2NK+NK3+M2M2−KGφGθ+GτVector-basedOSL+M2M4+M2M2−KGφGθGτEX-proposedOM4LS+2NKL+NK3+M2LM2L−KGφGθ+GτEX-vector-basedOS+M2LM4L2+M2LM2L−KGφGθGτFigure 2
Complexity comparison(a) versusS where M=4, Δθ=0.05°, and Δτ=0.001ns(b)versus M where S=500, Δθ=0.05°, and Δτ=0.001ns(c)versus Δθ where M=4, S=500, and Δτ=0.001ns(d)versus Δτ where M=4, S=500, and Δθ=0.05°.
(a)(b)(c)(d)From Figure2, the complexity of EX-Vector-based and Vector-based are extremely high due to the huge number of spectrum points, particularly in the condition of small spectrum steps. In contrast, tensor-based Proposed and EX-Proposed use CPD to handle the covariance matrix for reducing the dimensionality of the spectrum peak search. Thus, both methods significantly decrease the complexity. In contrast to the Proposed method, EX-Proposed adopts an EX-Response model, which has higher complexity and better estimation accuracy.


## 5. Simulation Results

This section performs a simulation experiment analysis in which the proposed tensor-based algorithm is compared with the vector-based algorithm. Furthermore, the improved algorithm using the EX-Response model is also compared with the corresponding vector-based algorithm. The Cramer–Rao bound (CRB) [23] is a threshold for the unbiased estimation variance of the proposed model and can be used as a performance reference benchmark.Firstly, we assume that the parameter estimates are all performed individually withS snapshots, so the joint probability density function is expressed as follows:(27)fH1,…,HS=12πM2LSσ2/2M2LSe​−1/σ2∑s=1SHs−AρsHHs−Aρs.Then, taking the log-likelihood function of (25), we obtain the following equation:(28)LoH1,…,HS=−M2LSln2π−M2LSlnσ2/2−1σ2∑s=1SHs−AρsHHs−Aρs.Define η=τT,θT,φTT. In addition, ρ˜s and ρ¯s are the imaginary part and real part of ρs, respectively, which are indicated as ρ¯s=Reρs and ρ˜s=Imρs. The Fisher information matrix is Ω=EψψT, where(29)ψT​=∂Lo∂σ2​​ρ¯T1​​ρ˜T1​​⋯​​ρ¯TS​​ρ˜TS​​ηT.For the Fisher information matrix, the CRB of η conforms,(30)CRBη=σ22∑s=1SReFHsBHPA⊥BFs−1,where(31)B=BθBφBτ,Bθ=bθ1bθ2⋯bθK,Bφ=bφ1bφ2⋯bφK,Bτ=bτ1bτ2⋯bτK,bθk=∂AkT∂θkT,bφk=∂AkT∂φkT,bτk=∂AkT∂τkT,A=A1A2⋯AK,PA⊥=I−PA=I−AAHA−1AH,=I−AAHA−1AH,Fs=I⊗diagρs.We perform simulations using OFDM signals withL=64 subcarriers, fast Fourier transform period TFFT=32us, carrier frequency fc=2.4GHz , and bandwidth B=20MHz. Used UPA contains 4×4 array sensors. We set the spectrum steps of Δτ=0.001ns and Δθ=0.05°. For evaluating the precision of the methods, we calculate the root mean square error (RMSE) by the following equation:(32)RMSE=1QL∑q=1Qλ−λ^i2,where Q, λ^i, and λi are the amount of Monte Carlo simulations, the estimated values, and the ith true values, separately.


### 5.1. Performance at Low Signal-to-Noise Ratio (SNR)

Assume that the quantity of multipath is three, the associated delays are3.5ns, 13.5ns, and 23.5ns, related azimuth angles are −20°, 0° and 30°, and related elevation angle are 20°, 30°, and 45°, separately. Furthermore, Q=200 is chosen and the distribution of TD and AOA under S=500 and SNR=−5dB is determined, as illustrated in Figure 3.Figure 3
Estimated distribution atSNR=−5dB.Figure3 indicates that the proposed algorithm (EX-Proposed) is successful in solving the parameters and the estimated values are concentrated surrounding the true values. Furthermore, the proposed algorithm performs effectively in low SNR.


### 5.2. Performance versus SNR

This part analyzes the capabilities of EX-Proposed, EX-Vector-Based, Proposed, Vector-Based, and CRB under multipath component conditions. Suppose the multipath components are three with the same parameters as those simulated in A. We chooseQ=200, S=500 and the spectrum steps of Δτ=0.001ns and Δφ=Δθ=0.05°. In addition, the RMSE performance versus SNR, with ranges of −15dB to 20dB in 5dB intervals, as illustrated in Figure 4.Figure 4
Performance comparison versus SNR: (a) azimuth; (b) elevation; (c) TD.
(a)(b)(c)Besides, Figure4 indicates that the RMSEs of both EX-Proposed and EX-Vector-Based are much higher than those of Proposed and Vector-Based using the single-frequency model. The reason is that both algorithms extend the channel frequency response by combining space-time characteristics from the vector basis and tensor basis, separately. Therefore, the algorithms based on the EX-Response model have higher estimation accuracy and are closer to the CRB than the corresponding single-frequency algorithms.However, the RMSEs of the tensor-based algorithms (EX-Proposed and Proposed) are both slightly higher than those of the corresponding vector-based algorithms (EX-Vector-Based and Vector-Based). This is due to the CPD process (31) of the tensor algorithm is an approximate operation and the decomposition process has a partial loss of virtual aperture and virtual bandwidth, but its estimation is still accurate. When comparing the complexities, the complexities of EX-Proposed and EX-Vector-Based are aboutO1.39×1013 and O3.39×1017, respectively. Therefore, the proposed algorithm (EX-Proposed) not only obtains a value estimated with high accuracy but also decreases the complexity considerably.


### 5.3. Performance versus Snapshots

To highlight the impact of snapshots on RMSE, we setSNR=15dB, the quantity of snapshots varies within the range of S=20,50,100,200,500,1000,2000,5000, and the rest of the simulation parameters are identical to those simulated in B. As illustrated in Figure 5, the RMSE also decreases along with the increases in the number of snapshots, but the declines are gradually plateauing. The rest of the results are the identical to corresponding simulated in B. The RMSE of the proposed tensor-based algorithm (EX-Proposed) is higher than that of the single-frequency model algorithms (Proposed and Vector-Based), close to that of the vector-based algorithm using the EX-Response model (EX-Vector-Based), while the complexity is much lower.Figure 5
Performance comparison versus snapshots: (a) azimuth; (b) elevation; (c) TD.
(a)(b)(c)


## 5.1. Performance at Low Signal-to-Noise Ratio (SNR)

Assume that the quantity of multipath is three, the associated delays are3.5ns, 13.5ns, and 23.5ns, related azimuth angles are −20°, 0° and 30°, and related elevation angle are 20°, 30°, and 45°, separately. Furthermore, Q=200 is chosen and the distribution of TD and AOA under S=500 and SNR=−5dB is determined, as illustrated in Figure 3.Figure 3
Estimated distribution atSNR=−5dB.Figure3 indicates that the proposed algorithm (EX-Proposed) is successful in solving the parameters and the estimated values are concentrated surrounding the true values. Furthermore, the proposed algorithm performs effectively in low SNR.


## 5.2. Performance versus SNR

This part analyzes the capabilities of EX-Proposed, EX-Vector-Based, Proposed, Vector-Based, and CRB under multipath component conditions. Suppose the multipath components are three with the same parameters as those simulated in A. We chooseQ=200, S=500 and the spectrum steps of Δτ=0.001ns and Δφ=Δθ=0.05°. In addition, the RMSE performance versus SNR, with ranges of −15dB to 20dB in 5dB intervals, as illustrated in Figure 4.Figure 4
Performance comparison versus SNR: (a) azimuth; (b) elevation; (c) TD.
(a)(b)(c)Besides, Figure4 indicates that the RMSEs of both EX-Proposed and EX-Vector-Based are much higher than those of Proposed and Vector-Based using the single-frequency model. The reason is that both algorithms extend the channel frequency response by combining space-time characteristics from the vector basis and tensor basis, separately. Therefore, the algorithms based on the EX-Response model have higher estimation accuracy and are closer to the CRB than the corresponding single-frequency algorithms.However, the RMSEs of the tensor-based algorithms (EX-Proposed and Proposed) are both slightly higher than those of the corresponding vector-based algorithms (EX-Vector-Based and Vector-Based). This is due to the CPD process (31) of the tensor algorithm is an approximate operation and the decomposition process has a partial loss of virtual aperture and virtual bandwidth, but its estimation is still accurate. When comparing the complexities, the complexities of EX-Proposed and EX-Vector-Based are aboutO1.39×1013 and O3.39×1017, respectively. Therefore, the proposed algorithm (EX-Proposed) not only obtains a value estimated with high accuracy but also decreases the complexity considerably.


## 5.3. Performance versus Snapshots

To highlight the impact of snapshots on RMSE, we setSNR=15dB, the quantity of snapshots varies within the range of S=20,50,100,200,500,1000,2000,5000, and the rest of the simulation parameters are identical to those simulated in B. As illustrated in Figure 5, the RMSE also decreases along with the increases in the number of snapshots, but the declines are gradually plateauing. The rest of the results are the identical to corresponding simulated in B. The RMSE of the proposed tensor-based algorithm (EX-Proposed) is higher than that of the single-frequency model algorithms (Proposed and Vector-Based), close to that of the vector-based algorithm using the EX-Response model (EX-Vector-Based), while the complexity is much lower.Figure 5
Performance comparison versus snapshots: (a) azimuth; (b) elevation; (c) TD.
(a)(b)(c)


## 6. Conclusions

For joint estimation of TD and AOA at UPA and solving the problem of high computational complexity, we propose a fast joint estimation algorithm using tensor structures and OFDM techniques. Furthermore, we combine the receive antenna array response and the channel frequency response of OFDM subcarriers through a tensor structure to obtain more useable information with less complexity. In summary, the signal model and the algorithmic process under vector and tensor structures are first explained, from which we interpret the correspondence between the two structures. The related algorithms are then analyzed for their complexity, which demonstrates the relative advantage of the proposed method. Lastly, the simulation results illustrate that the proposed algorithm is considerably less complex than the conventional vector-based algorithm while maintaining a higher estimation accuracy.In the future, we will further exploit the dimensionality reduction advantage of the proposed algorithm to explore more algorithms that can match the model, such as spatial smoothing methods [24], which can increase the robustness under coherent multipath conditions. Moreover, the applicability of the tensor structure on the proposed model fully illustrates its structural advantages and greatly extends the possibilities of the joint estimation algorithm.



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*Source: 6856050-2022-09-27.xml*