International Journal of Antennas and Propagation
Volume 2018 (2018), Article ID 2054271, 7 pages
https://doi.org/10.1155/2018/2054271

Research Article

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

Saleh O. Al-Jazzar and Sami Aldalahmeh

Al-Zaytoonah University of Jordan, Amman, Jordan

Correspondence should be addressed to Saleh O. Al-Jazzar; saleh.g@zuj.edu.jo

Received 19 June 2018; Revised 29 August 2018; Accepted 12 September 2018; Published 22 November 2018

Academic Editor: Giorgio Montisci

Copyright © 2018 Saleh O. Al-Jazzar and Sami Aldalahmeh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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).

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: Section 2 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 and transmitting and receiving uniform linear array (ULA) antennas, respectively. Let the transmitting antennas transmit a pulsed signal that is modulated by the sequences (for ), having a period of and consisting of chips. These signals are uncorrelated over time, and antenna elements, i.e.,

where is the expectation operator and is some delay.

For targets, the manifolds of the transmitted and received signals are, respectively,

where is the transpose operator, are the coordinates of transmitting antennas, are the coordinates of receiving antennas, is the carrier frequency used, is the speed of light, and is the AOA of the th 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 vector at time is represented as

where is the Hermetian operator and

with is the complex propagation factor, which includes the path loss and reflection coefficient factor off the th target. Also, is the delay for the th target, is a complex Gaussian noise vector of zero mean, and covariance matrix, with is 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 of and a chip duration of , where chips per pulse. Then, the quantized delay is

where is the ceil of modulus of .

Now, the received signal can be represented in a compact matrix form by stacking the discretized vectors as

where is the matrix of stacked discretized vectors having a size of , where is the number of pulses and is the number of zero bins appended to which is the maximum tolerable delay. Whereas is the discrete shifting matrix constructed as

where is an all-zero -by- matrix and is an appropriate discrete complex Gaussian noise matrix.

Equation (6) can be further simplified as follows:

where is a matrix with its th column contains , is matrix constructed in a similar manner from , is a diagonal matrix with its elements containing , and

where is a matrix with is repeated times and 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 (’s), then quantized time delays (’s) and finally the complex propagation factors (’s).

3.1. AOA Estimation

Estimating the AOAs starts by forming the following approximate correlation matrix:

From the definition of in (8), it can be shown that

Next, let us denote which is a matrix, then (12) becomes

Now, from the assumption in (1) and assuming that is large enough, it can be shown that the matrix is a diagonal matrix and its diagonal elements are . These diagonal elements are positive-real and close in the value to each other. Similarly, denote the matrix . In this case and for a ULA antenna element system, it can be shown that the diagonal elements of are

where is the element of the matrix. Also, it can be shown that the off diagonal elements of are given by

where is the element of the matrix and . Also, and are the distances between successive antenna elements on the and axes, respectively. Without loss of generality, the above can be simplified by setting to zero yielding

It is clear that is the sum of vectors of almost equal magnitudes and different phases. Thus, these off diagonal elements of will sum up to values which are very small when compared to its diagonal elements (shown in (14)), i.e.,

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, is a quasidiagonal matrix and so is , since is a diagonal matrix.

The next step will be taking the EVD of as follows:

where is size and its columns containing the eigenvectors of and is a diagonal matrix containing the corresponding eigenvalues.

By comparing (11) and (18) and neglecting the noise term for the meanwhile, then,

So, it can be deduced that

where is the appropriate matrix to change the basis vectors of size . Furthermore,

It is already proven in [10] that since is diagonal and is quasidiagonal (which retain the properties of a diagonal matrix) and considering the first eigenvectors of , which correspond to the largest eigenvalues of , then we can deduce the AOAs of the targets implicitly as follows:

where λ is the wavelength of the transmitted signal. Now, by setting (without loss of generality) to simplify the equation, we have

where

and is the th element of the th eigenvector. More elaboration on how to find from (22) will be discussed in Section 4.

Algorithm 1 summarizes the phase estimation process for the targets.

1: Construct R via (10).

2: Find U via the EVD of R.

3: Initialize target counter

4: for do

5:        Find via (24).

6:        Find via (23).

7: end for

Algorithm 1: Estimating ’s.

Notice that represents the estimate of some parameter .

3.2. Time Delay and Complex Propagation Factor Estimation

In order to estimate the delays (’s), we note that the received signal matrix in (6) should have all-zero columns at the beginning of in the noiseless case, which is the delay required for the signal reflected off the nearest target to reach the receiver. Hence, can be estimated by identifying the all-zero columns from . 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 . The estimation process is summarized in Algorithm 2.

1: Initialize counter:

2: Set threshold:

3: Compute matrix average:

4: Compute column average:

5: while do

6:        Increment counter by 1.

7:        Compute column average:

8: end while

9: Set estimate:

Algorithm 2: Estimating .

Note that is the magnitude average of the th column elements of and is the magnitude average of all elements of . Also, is an appropriately chosen threshold.

Next, we can compute for the th target (as proven in the Appendix and similar to what is shown in [8]), as follows:

3.3. Localization Algorithm

Now, that we have the means to find all the required parameters , and , we can reformulate the received signal for a specific target as

As stated earlier, the estimated signals are subtracted from in (6) in order to improve the th 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).

1: Estimate AOAs, from X via Algorithm¹.

2: Initialize target counter

3: Initialize

4: for do

5:        Initialize target counter

6:        for do

7:                

8:                Estimate propagation delays, from (instead of X) via Algorithm²

9:                Estimate propagation factors, , from (instead of X) via (25).

10:            Compute via (26).

11:    end for

12: end for

Algorithm 3: Localization algorithm.

4. Simulation

Simulations are performed to assess the performance of the proposed method. Two targets are assumed, i.e., with  = 30° and  = 70°. Also, the complex propagation factors are assumed to be and . The delays are assumed chips and chips with and . The nonzero part of the message signal matrix is formed using Gold sequences. The results are averaged over ensembles. The number of antenna elements at the transmitter and receiver sides are and , respectively. The threshold is assumed to be . Also, and . So, (22) is simplified to

The number of iterations of the outer loop in Algorithm 3 is two.

Figures 2, 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 for .

Figure 3: Root mean square error for the delay in chip time vs. SNR in dB for .

Figure 4: Root mean square error for the complex propagation factor magnitude vs. SNR in dB for .

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.

Appendix

In this section, we will prove the equation in (25). To do so, let us minimize the mean squared error (MSE) of the received signal in (6) with respect to , i.e.,

which can be written in the form

Now, take the derivative of (A.2) with respect to and equate it to zero as follows:

Now, dividing both sides by and multiplying the left side and right side of (A.3) by and , respectively, lead to

But, . So,

Dividing both sides of (A.5) by , then,

Next, rearranging (A.6) leads to

So, (25) is proven.

Data Availability

The data is available through simulation of the derived equations in the paper.

Conflicts of Interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

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