Frequency Diversity Array for DOA Estimation

The localization of targets has been presented in this article. DOA (Direction of Arrival) is an important parameter to be determined by radar. The MLE (Maximum Likelihood Estimator) has been widely used to accurately and efficiently estimate the DOAs of multiple targets. The targets at different ranges result in a variation in amplitude of the received signals, so an MLE estimator has to operate at all ranges. For accurate results of DOA, the complex amplitudes of multiple targets should not be much different and also the prior information of Doppler and number of targets is required. In this paper, an approach is proposed which uses the classical 2D algorithm to estimate range, Doppler and number of targets and then FDA (Frequency Diversity Array) is used to focus power in a particular range. As a result, the MLE can get data from a particular range cell where all targets have almost same amplitude and thus MLE can accurately estimate the DOAs of multiple targets. Simulations and results have confirmed the effectiveness of proposed approach.


INTRODUCTION
MLE for DOA estimation by antenna scanning is presented in [20] which accurately and efficiently estimate the DOAs of multiple targets. The targets at different ranges result in power variation of signals received after reflection from the targets. DOA estimationthrough MLE requires data from targets in same range cell (approximately same amplitude). So the MLE estimator has to operate on all ranges as the complex amplitude of multiple targets should not be varied. So for targets at different ranges, it gives errors. Also the prior information of doppler and number of targets is needed [20]. As [20], the accuracy of DOA estimation through the MLE of amplitude highly depends on power of signals reflected D OA estimation is an important area in radar, sonar and communication signal processing.
In the last two decades, many techniques have been proposed for DOA estimation [1][2][3][4][5]. High resolution techniques have also been studied by radar researchers [6][7][8][9][10]. The monopulse technique has been used in many practical radar systems for DOA estimation [11][12][13][14]. The problem with the monopulse system is that when multiple targets are present in an azimuth cell, it gives only one direction for all the targets [15][16]. MLE methods have been presented for radar array signal processing [17][18].
Multiple targets can be correctly resolved in an azimuth cell by the using the MLE technique [19].
from targets in a cell. So if targets have large variation in power due to range difference, the DOA estimation has errors. When the range separation of targets is less, their DOAs are accurately estimated, but when the range separation increases, error is produced because now the amplitude of reflected signal from targets largely varies.
In this paper, an approach is presented which uses the 2D (Two Dimensional) Range/Doppler algorithm [21] to estimate range, Doppler and number of targets and then FDA [22] is used to focus the power in a particular range.
We have proposed a methodology in which first the range of multiple targets is estimated then DOA estimation is performed by focusing the main beam in a particular range through FDA [23][24] so that power of the received signals in a batch remains almost same. FDA is used to concentrate beam on a particular range [22]. Thus FDA helps in reducing the DOA estimation error. Doppler and number of targets' information are used by the MLE to find the DOA of targets.
The rest of the paper is organized as follows. Section 2 describes the signal modeling and array system. Section 3 develops the proposed methodology. Section 4 shows the simulation results of the DOA estimator. Section 5 presents conclusion of this paper.

SIGNAL MODELING
Consider a ULA (Uniform Linear Array) with 2L sensors with l=-L,…,-1,1,…,+L. The targets are assumed to be in far-field so that plane waves are incident on the sensors.
The distance between 2 sensors is d. Incident wave makes angle θ with the normal which is alternate angle to horizontal axis i.e. both are same. Fig. 1 shows the array configuration.
If H o is the hypothesis that no target is present (noise only) and H 1 is the hypothesis that a target is present, we can write mathematically data from l th sensor as: The distance between sensors is d=λ/2. It is assumed that the rotation is slow across different pulses, but fast within a pulse so that these assumptions can be made: Consider the radar antenna rotates mechanically with angular velocity ω R rad/s, the number of pulses (M) for

PROPOSED ALGORITHM
Estimation of DOA requires three steps. The proposed algorithm is presented in the coming sub-sections.

Summary of Algorithm
The overall algorithm flow is shown in Fig. 3.
To make the amplitude of received signals same, FDA concentrates beam on those targets which are at the same range and MLE can estimate the DOAs accurately. Fig.4 shows how the batch is processed.

Targets, Doppler, Range and Required Batch for DOA Estimation)
The array of sensors given in Fig. 1 is rotated about its axis and during the scan through azimuth cell (equal to beamwidth), all sensors' data is added.
The Range/Doppler Algorithm [21] is used to estimate range, Doppler and number of targets. Fig. 5 shows the steps where details are given in [21]. The range/doppler algorithm gives information about the number of targets, their range and doppler. Fig. 6 shows these results for a 10 element array and four targets  Table 1.
Target 1 and 2 are considerably closer; similarly, target 3 and 4 are close to each other. Now as 4 targets are present in different ranges so if DOA is estimated through MLE [20], error will occur due to amplitude variation. To avoid this, FDA is used to concentrate energy at specific range cell.

FDA for Focusing Beam on Particular Ranges
After the exploration of FDA [22], it is being used in many existing algorithms to improve their performance [25][26].
The FDA procedure given in [22] shows that increment in frequency gives another degree of freedom. Consider f is the fundamental frequency and Δf is the increment in frequency from one sensor to the next. So frequency of the l th sensor is given by f l =f+lΔf where l is the sensor number. Fig. 7 shows frequency of different sensors.
If r (reference) is the range between reference and target, then the range from l th sensor is given by r l = r -ld/2sindθ.  π π π π π π π π π   So the radiation also depends on range. The frequency increment and range are related by R=c/Δf [22]. So by using Δf we can scan at different ranges. FDA gives improved localization in range [25]. As seen in Fig. 6, 2 targets were found in the 320-360 km range cell. To collect pulses from that range, now the FDA is applied to scan range cell of 320-360 km as shown in Fig. 10. Now a frequency increment of 923 Hz is applied to get mainlobe focused on the desired 320-360 km range.

DOA Estimation through MLE of Amplitude
Now the data is collected from range-azimuth cells which contain multiple targets and MLE for DOA [20] is now used to get the DOA information as shown in Fig. 11.
The sum of data from all sensors can be written as: Where m is pulse number, c is cell number, M is total pulses, T is PRI. The beampattern is Gaussian with pattern shown in Fig.12.  2πfdi k) . The complex Gaussian PDF is given by [1]: where M is the noise covariance matrix and σ 2 is variance of noise. Methods of finding M can be found in [27]. For thermal noise, it is identity M=I [20]. Maximization of Equation is equivalent to minimization of the quadratic form: Minimization of the quadratic form is given now [28]: The gradient w.r.t b is: The DOA estimator is given by [16], Equation (9): Equation (5) gives the DOA (θ) of target(s).

SIMULATION RESULTS OF DOA ESTIMATOR
In this section we present simulation results of the proposed methodology. As MLE [20] produces errors in DOA estimation when there is large range separation between the targets. Now using the modified approach, first the range and Doppler information is obtained. For simulation, we are using L = 10 sensors. ULA with distance between sensors, d=λ/2. According to Table 1 and 60 km as shown in Fig. 21(a). The DOAs are estimated using Equation (5) as shown in Fig. 21(b). The DOAs are estimated correctly as the peak for target 1 lies at 45 o and for target 2 lies at 48 o . FDA is then used to scan targets present at 140 and 150 km as shown in Fig. 22(a). The main lobe is at the required range and the targets present at 58 and 50 o are estimated accurately using Equation (5) as shown in Fig. 22(b).

CONCLUSIONS
In this work, we have developed a complete methodology to estimate DOA via MLE using FDA. The problems faced by MLE are highlighted and solved via FDA. Also, the parameters such as range, Doppler and number of targets are estimated. It has been shown that multiple targets at different ranges can be accurately localized using the presented method. Currently we are also working on the technique through which Elevation angle can be estimated by scanning the beam electronically through elevation angles an each azimuth. Simulation results verify the effectiveness of the approach. The frequency shift in FDA can cause phase error if the number of elements is too large or the frequency shift is very much high.