# AOA, Delay, and Complex Propagation Factor Estimation for the Monostatic MIMO Radar System

Authors: Saleh O. Al-Jazzar; Sami Aldalahmeh
Journal: International Journal of Antennas and Propagation
 (2018)

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## Abstract

In this paper, we propose a solution to find the angle of arrival (AOA), delay, and the complex propagation factor for the monostatic multiple-input multiple-output (MIMO) radar system. In contrast to conventional iterative computationally demanding estimation schemes, we propose a closed form solution for most of the previous parameters. The solution is based on forming an approximate correlation matrix of the received signals at the MIMO radar receiver end. Then, an eigenvalue decomposition (EVD) is performed on the formed approximate correlation matrix. The AOAs of the received signals are deduced from the corresponding eigenvectors. Then, the delays are estimated from the received signal matrix properties. This is followed by forming structured matrices which will be used to find the complex propagation factors. These estimates can be used as initializations for other MIMO radar methods, such as the maximum likelihood algorithm. Simulation results show significantly low root mean square error (RMSE) for AOAs and complex propagation factors. On the other hand, our proposed method achieves zero RMSE in estimating the delays for relatively low signal-to-noise ratios (SNRs).


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## Body

## 1. Introduction

Multiple-input multiple-output (MIMO) radar is becoming increasingly popular due to its ability to overcome the fluctuation in the received power caused by varying radar cross section [1]. Thus, it is proposed in the literature as a tool to estimate the location of different targets [2–8]. Radar signals are transmitted from multiple transmitter antennas; then, they are reflected off targets and received back at the radar, hence providing the required receiver diversity. However, this operation requires estimating the angle of arrivals (AOAs) and delays of the received signals, which is usually done via a maximum likelihood (ML) method [9]. Nonetheless, the ML is computationally expensive because of its iterative nature. Worse yet, it might converge to a local minima leading to erroneous estimated parameters, providing inappropriate initial values fed to the ML algorithm. As a result, a need rises to provide a closed form solution to the estimation problem, which might also serve as an appropriate initial guess for the ML.Several articles in the literature proposed direction finding methods for the bistatic MIMO radar system where the transmitter and receiver are located in different positions such as in [5]. On the other hand, many MIMO radar models were proposed in the literature such as in [6], which utilize the monostatic MIMO radar model where the transmitter and receiver are colocated. In this paper, we utilize the monostatic MIMO radar model in [6] with some extensions to fit the proposed solution. The proposed method starts by computing an approximate correlation matrix upon which we apply eigenvalue decomposition (EVD) to find the AOAs from the corresponding eigenvectors. However, it is noticed that when the received signal is stacked in a matrix form, a specific number of columns at the beginning of the matrix should be ideally zeros. The number of these columns is related to the delay of the received signal reflected off the nearest target. This property is used to independently estimate the delays. Then, the estimated AOAs and delays are used to form structured matrices which are utilized to find the complex propagation factors for each target. These estimates can be used as initializations for other adaptive MIMO radar estimating methods as in [6].The rest of the paper is organized as follows: Section2 presents the problem formulation part, Section 3 contains the proposed solution, and Section 4 shows simulation results. Finally, we conclude in Section 5.


## 2. Problem Formulation

In this section, we present the system model for the monostatic MIMO radar which is similar to the one used in [6]. Assume that the radar consists of colocated NTx and NRx transmitting and receiving uniform linear array (ULA) antennas, respectively. Let the transmitting antennas transmit a pulsed signal at that is modulated by the sequences mit (for i=1⋯NTx), having a period of Tb and consisting of Nc chips. These signals are uncorrelated over time, and antenna elements, i.e.,
(1)Emitmjt−τ=0,ifi≠jorτ≠0,1,otherwise,where E· is the expectation operator and τ is some delay.ForK targets, the manifolds of the transmitted and received signals are, respectively,
(2)sTxθk=expjrx,ry,rzTxkθkNTx×1,sRxθk=exp−jrx,ry,rzRxkθkNRx×1,kθk=2πfcccosθk,sinθk,0T,where ·T is the transpose operator, rx,ry,rzTx are the coordinates of transmitting antennas, rx,ry,rzRx are the coordinates of receiving antennas, fc is the carrier frequency used, c is the speed of light, and θk is the AOA of the kth target (it is worth mentioning that we are considering the azimuth angle only in this paper).Neglecting the clutter effect, the radar signal is reflected off the targets, hence experiencing different propagation factors and delays. Therefore, the received signal vectorxt at time t is represented as
(3)xt=∑k=1KβksRxθksTxHθkmt−τk+ntNRx×1,where ·H is the Hermetian operator and
(4)mt=m1t,m2t,⋯,mNTxtTNTx×1 with βk is the complex propagation factor, which includes the path loss and reflection coefficient factor off the kth target. Also, τk is the delay for the kth target, nt is a NRx×1 complex Gaussian noise vector of zero mean, and σ2INRx×NRx covariance matrix, with INRx×NRx is NRx×NRx identity matrix. The radar system is shown in Figure 1.Figure 1
Monostatic radar system model.Recall that the transmitted pulse has a total duration ofTb and a chip duration of Tc, where Nc=Tb/Tc chips per pulse. Then, the quantized delay is
(5)lk=τkTcmodNc,where τk/TcmodNc is the ceil of Nc modulus of τk/Tc.Now, the received signal can be represented in a compact matrix form by stacking the discretized vectors as(6)X=∑k=1KβksRxθksTxθkHMJlk+NNRx×L,where M is the matrix of stacked discretized mt vectors having a size of NTx×L, where L=Nc+Q is the number of pulses and Q is the number of zero bins appended to mt which is the maximum tolerable delay. Whereas J is the discrete shifting matrix constructed as
(7)J=0L−1×1,IL−1×L−10,01×L−1L×L,where 0m×n is an all-zero m-by-n matrix and N is an appropriate discrete NRx×L complex Gaussian noise matrix.Equation (6) can be further simplified as follows:
(8)X=ΓRxβΓTxHΥΩ+N,where ΓRx is a NRx×K matrix with its kth column contains sRxθk, ΓTx is NTx×K matrix constructed in a similar manner from sTxθk, β is a K×K diagonal matrix with its elements containing βk, and
(9)Υ=M,⋯,MNTx×LK,Ω=Jl1,⋯,JlKTLK×L,where Υ is a matrix with M is repeated K times and Ω is the universal shift matrix.


## 3. The Proposed Method

Our proposed method localizes the targets without using computationally demanding methods such as the ML. This is done by estimating the parameters of the first target, then subtracting its formed signal from the received signal in (6), and subsequently estimating the parameters of the second target and so on. The proposed method starts estimating the AOAs (θk’s), then quantized time delays (lk’s) and finally the complex propagation factors (βk’s).


### 3.1. AOA Estimation

Estimating the AOAs starts by forming the following approximate correlation matrix:(10)R=XXHNRx×NRx.From the definition ofX in (8), it can be shown that
(11)R=ΓRxΨΓRxH+σ2INRx×NRx,(12)Ψ=βΓTxHΥΩβΓTxHΥΩH=βΓTxHΥΩΩHΥHΓTxβH.Next, let us denoteG=ΥΩΩHΥH which is a NTx×NTx matrix, then (12) becomes
(13)Ψ=βΓTxHGΓTxβH.Now, from the assumption in (1) and assuming that Nc is large enough, it can be shown that the matrix G is a diagonal matrix and its diagonal elements are g0,g1,⋯,gNTx−1. These diagonal elements are positive-real and close in the value to each other. Similarly, denote the K×K matrix H=ΓTxHGΓTx. In this case and for a ULA antenna element system, it can be shown that the diagonal elements of H are
(14)hu,u=∑p=0NTx−1gp,where hu,u is the u,u element of the H matrix. Also, it can be shown that the off diagonal elements of H are given by
(15)hu,v=∑p=0NTx−1gpe−jpwΔxcosθu−cosθv+Δysinθu−sinθvforu≠v,where hu,v is the u,v element of the H matrix and w=2πfc/c. Also, Δx and Δy are the distances between successive antenna elements on the x and y axes, respectively. Without loss of generality, the above can be simplified by setting Δy to zero yielding
(16)hu,v=∑p=0NTx−1gpe−jpwΔxcosθu−cosθvforu≠v.It is clear thathu,v is the sum of vectors of almost equal magnitudes and different phases. Thus, these off diagonal elements of H will sum up to values which are very small when compared to its diagonal elements (shown in (14)), i.e.,
(17)hu,v≪hu,uforu≠v.We will call matrices satisfying the condition in (17) a quasidiagonal matrix. Quasidiagonal matrices retain the essential properties of diagonal matrices as it is shown in the simulation section (Section 4). So, H is a quasidiagonal matrix and so is Ψ, since β is a diagonal matrix.The next step will be taking the EVD ofR as follows:
(18)R=UDUH,where U is NRx×NRx size and its columns containing the eigenvectors of R and D is a NRx×NRx diagonal matrix containing the corresponding eigenvalues.By comparing (11) and (18) and neglecting the noise term σ2INRx×NRx for the meanwhile, then,
(19)ΓRxΨΓRxH=UDUH.So, it can be deduced that(20)ΓRx=UT,where T is the appropriate matrix to change the basis vectors of size NRx×K. Furthermore,
(21)D=TΨTH.It is already proven in [10] that since D is diagonal and Ψ is quasidiagonal (which retain the properties of a diagonal matrix) and considering the first K eigenvectors of U, which correspond to the K largest eigenvalues of D, then we can deduce the AOAs of the K targets θk implicitly as follows:
(22)Δxcosθk+Δysinθk=λζk2π, where λ is the wavelength of the transmitted signal. Now, by setting Δy=0 (without loss of generality) to simplify the equation, we have
(23)Δxcosθk=λζk2π, where
(24)ζk=−∑μ=1NRx−1∠uμ+1,k/uμ,kNRx−1 and uμ,k is the μth element of the kth eigenvector. More elaboration on how to find θk from (22) will be discussed in Section 4.Algorithm1 summarizes the phase estimation process for the targets.Algorithm 1: Estimating θk’s.
1:Construct R via (10).
2:Find U via the EVD of R.
3:Initialize target counter k=1.
4:for k≤K do
5:Find ς^k via (24).
6:Find θ^k via (23).
7:end forNotice thatf^ represents the estimate of some parameter f.


### 3.2. Time Delay and Complex Propagation Factor Estimation

In order to estimate the delays (lk’s), we note that the received signal matrix X in (6) should have l1 all-zero columns at the beginning of X in the noiseless case, which is the delay required for the signal reflected off the nearest target to reach the receiver. Hence, l1 can be estimated by identifying the all-zero columns from X. However, because of the noise presence, these values will not equal to zero exactly but will have relatively small magnitudes. So, to detect these values, we propose direct comparison with some threshold, which is chosen to be proportional to the magnitude average value of all elements of X. The estimation process is summarized in Algorithm 2.Algorithm 2: Estimating lk.
1:Initialize counter: m=1.
2:Set threshold: γ.
3:Compute matrix average: x¯=1/L2∑m=1L∑n=1LXn,m.
4:Compute column average: x¯m=1/L∑n=1LXn,m.
5:while x¯m<γx¯ do
6:Increment counter m by 1.
7:Compute column average: x¯m=1/L∑n=1LXn,m.
8:end while
9:Set estimate: l^k=m.Note thatx¯m is the magnitude average of the mth column elements of X and x¯ is the magnitude average of all elements of X. Also, γ is an appropriately chosen threshold.Next, we can computeβk for the kth target (as proven in the Appendix and similar to what is shown in [8]), as follows:
(25)β^k=sRxθ^kHXJl^kHMHsTxθ^kNRxsTxθ^kHMJl^kJl^kHMHsTxθ^k.


### 3.3. Localization Algorithm

Now, that we have the means to find all the required parametersθ^k,l^k, and β^k, we can reformulate the received signal for a specific target as
(26)X^k=β^ksRxθ^ksTxθ^kHMJl^k.As stated earlier, the estimated signals are subtracted fromX in (6) in order to improve the k+1th estimation in a successive manner. The complete localization algorithm is presented in Algorithm 3 (note that the outer loop in the algorithm refines the parameter estimation process).Algorithm 3: Localization algorithm.
1:Estimate AOAs, θk, from X via Algorithm1.
2:Initialize target counter b=1.
3:Initialize X^k=0.
4:for b≤K do
5:Initialize target counter k=1.
6:for k≤K do
7:X~k,b=X−∑n=1n≠kKX^n
8:Estimate propagation delays, lk, from X~k,b (instead of X) via Algorithm2
9:Estimate propagation factors, βk, from X~k,b (instead of X) via (25).
10:Compute X^k via (26).
11:end for
12:end for


## 3.1. AOA Estimation

Estimating the AOAs starts by forming the following approximate correlation matrix:(10)R=XXHNRx×NRx.From the definition ofX in (8), it can be shown that
(11)R=ΓRxΨΓRxH+σ2INRx×NRx,(12)Ψ=βΓTxHΥΩβΓTxHΥΩH=βΓTxHΥΩΩHΥHΓTxβH.Next, let us denoteG=ΥΩΩHΥH which is a NTx×NTx matrix, then (12) becomes
(13)Ψ=βΓTxHGΓTxβH.Now, from the assumption in (1) and assuming that Nc is large enough, it can be shown that the matrix G is a diagonal matrix and its diagonal elements are g0,g1,⋯,gNTx−1. These diagonal elements are positive-real and close in the value to each other. Similarly, denote the K×K matrix H=ΓTxHGΓTx. In this case and for a ULA antenna element system, it can be shown that the diagonal elements of H are
(14)hu,u=∑p=0NTx−1gp,where hu,u is the u,u element of the H matrix. Also, it can be shown that the off diagonal elements of H are given by
(15)hu,v=∑p=0NTx−1gpe−jpwΔxcosθu−cosθv+Δysinθu−sinθvforu≠v,where hu,v is the u,v element of the H matrix and w=2πfc/c. Also, Δx and Δy are the distances between successive antenna elements on the x and y axes, respectively. Without loss of generality, the above can be simplified by setting Δy to zero yielding
(16)hu,v=∑p=0NTx−1gpe−jpwΔxcosθu−cosθvforu≠v.It is clear thathu,v is the sum of vectors of almost equal magnitudes and different phases. Thus, these off diagonal elements of H will sum up to values which are very small when compared to its diagonal elements (shown in (14)), i.e.,
(17)hu,v≪hu,uforu≠v.We will call matrices satisfying the condition in (17) a quasidiagonal matrix. Quasidiagonal matrices retain the essential properties of diagonal matrices as it is shown in the simulation section (Section 4). So, H is a quasidiagonal matrix and so is Ψ, since β is a diagonal matrix.The next step will be taking the EVD ofR as follows:
(18)R=UDUH,where U is NRx×NRx size and its columns containing the eigenvectors of R and D is a NRx×NRx diagonal matrix containing the corresponding eigenvalues.By comparing (11) and (18) and neglecting the noise term σ2INRx×NRx for the meanwhile, then,
(19)ΓRxΨΓRxH=UDUH.So, it can be deduced that(20)ΓRx=UT,where T is the appropriate matrix to change the basis vectors of size NRx×K. Furthermore,
(21)D=TΨTH.It is already proven in [10] that since D is diagonal and Ψ is quasidiagonal (which retain the properties of a diagonal matrix) and considering the first K eigenvectors of U, which correspond to the K largest eigenvalues of D, then we can deduce the AOAs of the K targets θk implicitly as follows:
(22)Δxcosθk+Δysinθk=λζk2π, where λ is the wavelength of the transmitted signal. Now, by setting Δy=0 (without loss of generality) to simplify the equation, we have
(23)Δxcosθk=λζk2π, where
(24)ζk=−∑μ=1NRx−1∠uμ+1,k/uμ,kNRx−1 and uμ,k is the μth element of the kth eigenvector. More elaboration on how to find θk from (22) will be discussed in Section 4.Algorithm1 summarizes the phase estimation process for the targets.Algorithm 1: Estimating θk’s.
1:Construct R via (10).
2:Find U via the EVD of R.
3:Initialize target counter k=1.
4:for k≤K do
5:Find ς^k via (24).
6:Find θ^k via (23).
7:end forNotice thatf^ represents the estimate of some parameter f.


## 3.2. Time Delay and Complex Propagation Factor Estimation

In order to estimate the delays (lk’s), we note that the received signal matrix X in (6) should have l1 all-zero columns at the beginning of X in the noiseless case, which is the delay required for the signal reflected off the nearest target to reach the receiver. Hence, l1 can be estimated by identifying the all-zero columns from X. However, because of the noise presence, these values will not equal to zero exactly but will have relatively small magnitudes. So, to detect these values, we propose direct comparison with some threshold, which is chosen to be proportional to the magnitude average value of all elements of X. The estimation process is summarized in Algorithm 2.Algorithm 2: Estimating lk.
1:Initialize counter: m=1.
2:Set threshold: γ.
3:Compute matrix average: x¯=1/L2∑m=1L∑n=1LXn,m.
4:Compute column average: x¯m=1/L∑n=1LXn,m.
5:while x¯m<γx¯ do
6:Increment counter m by 1.
7:Compute column average: x¯m=1/L∑n=1LXn,m.
8:end while
9:Set estimate: l^k=m.Note thatx¯m is the magnitude average of the mth column elements of X and x¯ is the magnitude average of all elements of X. Also, γ is an appropriately chosen threshold.Next, we can computeβk for the kth target (as proven in the Appendix and similar to what is shown in [8]), as follows:
(25)β^k=sRxθ^kHXJl^kHMHsTxθ^kNRxsTxθ^kHMJl^kJl^kHMHsTxθ^k.


## 3.3. Localization Algorithm

Now, that we have the means to find all the required parametersθ^k,l^k, and β^k, we can reformulate the received signal for a specific target as
(26)X^k=β^ksRxθ^ksTxθ^kHMJl^k.As stated earlier, the estimated signals are subtracted fromX in (6) in order to improve the k+1th estimation in a successive manner. The complete localization algorithm is presented in Algorithm 3 (note that the outer loop in the algorithm refines the parameter estimation process).Algorithm 3: Localization algorithm.
1:Estimate AOAs, θk, from X via Algorithm1.
2:Initialize target counter b=1.
3:Initialize X^k=0.
4:for b≤K do
5:Initialize target counter k=1.
6:for k≤K do
7:X~k,b=X−∑n=1n≠kKX^n
8:Estimate propagation delays, lk, from X~k,b (instead of X) via Algorithm2
9:Estimate propagation factors, βk, from X~k,b (instead of X) via (25).
10:Compute X^k via (26).
11:end for
12:end for


## 4. Simulation

Simulations are performed to assess the performance of the proposed method. Two targets are assumed, i.e.,k=2 with θ1 = 30° and θ2 = 70°. Also, the complex propagation factors are assumed to be β1=0.7+0.7j and β2=0.5−0.25j. The delays are assumed l1=4 chips and l2=9 chips with NTc=127 and Q=25. The nonzero part of the message signal matrix M is formed using Gold sequences. The results are averaged over 1000 ensembles. The number of antenna elements at the transmitter and receiver sides are NTx=7 and NRx=9, respectively. The threshold γ is assumed to be 0.4. Also, Δx=λ/2 and Δy=0. So, (22) is simplified to
(27)θk=cos−1ζkπ.The number of iterations of the outer loop in Algorithm3 is two.Figures2, 3, and 4 show the root mean squared error (RMSE) of the AOAs, delays, and complex propagation factor magnitude estimation versus different signal-to-noise ratios (SNRs). It is clear from Figure 2 that the performance of AOA estimation for both targets is enhanced as the SNR increases up to some value where the performance plateaus due to the cross-correlation between different target signals (remember that Gold sequences have some small cross-correlation values). Interestingly, Figure 3 shows that the delay RMSE reaches zero at relatively low SNRs. Whereas Figure 4 clearly shows that the RMSE of the complex propagation factor magnitude decreases for both targets as the SNR increases, as expected.Figure 2
Root mean square error for the AOA in degrees vs. SNR in dB forγ=0.4.Figure 3
Root mean square error for the delay in chip timeTc vs. SNR in dB for γ=0.4.Figure 4
Root mean square error for the complex propagation factor magnitude vs. SNR in dB forγ=0.4.


## 5. Conclusion

In this paper, we propose a computationally inexpensive algorithm to find the AOAs, delays, and complex propagation factors for monostatic MIMO radar systems. Most of the parameters are found in a closed-form manner. The algorithm starts by computing an approximate correlation matrix, then applying EVD to find the AOAs from the corresponding eigenvectors. The delay is estimated from the zero submatrix in the received signal stack. Then, we utilize the structured matrices from the estimated AOAs and delays to find the complex propagation factors for each target. These estimates can be used as initializations for other iterative MIMO radar methods. Simulation results, on one hand, show relatively low RMSE for the AOAs and complex propagation factors. And on the other hand, simulation results show virtually zero RMSE for the delay estimation.



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*Source: 2054271-2018-11-22.xml*