Joint Error Calibration-Based Angle Estimation Method for Multi-Channel WAS-GMTI

The utilization of multiple-channel echo signals provides increased system degrees of freedom, resulting in a superior performance compared to previous single-channel radar systems. However, the inevitable channel errors between the multiple channels can significantly impact the estimation of angles for moving targets, leading to an inability to accurately determine the ground position of the moving targets. To address this issue, this paper proposes a joint error calibration angle measurement method. Firstly, according to the array element arrangement and channel division information, the effective phase center of multi-channel radar is recalculated. Then, the beam pointing is calibrated using the estimated Doppler center to constrain the search range of the maximum likelihood (ML) angle estimation method. Next, the weight vector of the steering vector is re-estimated from the covariance matrix of the sampled data to compensate for phase errors caused by channel spacing. Finally, joint error calibration is applied to the measured data from three channels. The results demonstrate that the proposed method effectively suppresses the ambiguity in the ML angle measurement caused by channel errors, resulting in a higher accuracy in angle measurement.


INTRODUCTION
The airborne wide-area surveillance ground moving target indication (WAS-GMTI) mode is an important operational mode of airborne battlefield surveillance radar.It utilizes an airborne longrange battlefield surveillance radar system to achieve real-time monitoring of moving targets in a wide sector on one side of the aircraft through azimuth scanning or azimuth-elevation two-dimensional scanning.Due to its advantages of long range, wide surveillance area, strong real-time capabilities, high revisit rates, and ability to monitor moving targets, as well as its immunity to weather conditions and day-night data collection capabilities, it holds important application value [1].
Research has shown that multi-channel radar systems provide more degrees of freedom.For multi-channel airborne WAS-GMTI radar systems, the spatial freedom of the system can be fully utilized by employing space time adaptive processing (STAP) to suppress clutter and improve the detection capability of low-altitude, slowspeed, and small-sized moving targets within the main lobe clutter.Super-resolution Direction of Arrival (DOA) estimation methods are used to estimate target angles and further enhance the accuracy of target angular measurement and localization [2,3].
The ML method is theoretically an optimal estimation method and is one of the super-resolution DOA estimation algorithms.By utilizing the model of the received signal vector (the statistical characteristics of the signal and noise), it can obtain the most accurate DOA estimation results by maximizing the likelihood function.
It performs well even in low signal-to-noise ratio environments and can be applied in multi-channel receiver arrays.Unlike the subspace-based methods, the ML method can maintain good performance even when the original signals are correlated.It effectively overcomes the effects of signal propagation in multipath and interference environments.When the noise is stationary and the number of samples tends to infinity, the error convergence of the deterministic ML method approaches zero [4].
However, the ML estimation method also faces several problems.It cannot guarantee convergence to the global minimum due to the influence of the initial estimate of the arrival direction.Secondly, accurate prior knowledge of the statistical characteristics of the signal and noise is required, but there may be errors between theory and actual measurement, making it more susceptible to errors in the steering vector, and so on [5].
Therefore, this article combines the overall data processing flow of WAS-GMTI with the problem of ML angle ambiguity to propose a joint error calibration angle measurement method.Calibration is performed on the effective phase center, beam pointing, channel spacing, and steering vector of the multi-channel radar.Then, the ML-DOA estimation method is used to locate and measure the angle of the target.The experimental results demonstrate that the proposed method effectively suppresses the ambiguity in the ML angle measurement caused by channel errors, resulting in reliable positioning results.

SIGNAL MODEL AND CALIBRATION METHODS 2.1 Signal model and DOA estimation method
As shown in figure 1, the data are acquired using an antenna array with receiving (Rx) channels arranged along the azimuth or flight direction The multichannel signal model in expressed as: where is the complex reflectivity of the scatterer, is the radar carrier wavelength, is the number of Rx channels, ( ) is the slant range between the target and the array center, ( ( )) and respectively represent the gain error and phase error of the th receiving channel, ( − 1) • is the position of the antenna phase center in azimuth direction with respect to the array origin, the channel spacing of the Rx channels will be discussed in detail in the next section when introducing the effective phase center.The ( ( )) is the beamforming vector and is the directional cosine with respect to the antenna array axis and is the azimuth time [6,7].
The beamforming vector is generally used for estimating the target's directional cosine, The ML-DOA estimation method is as follows: The DOA angle of the target is then computed as: When calculating the measured data, we found that under high signal-to-noise ratio conditions, it is possible to ignore ˆ −1 to reduce the bias in performance evaluation caused by clutter suppression.In addition, we can find that the search range of and the beamforming vector have a significant impact on the ML-DOA estimation, and targeted analysis and improvement are needed.

Ective phase center calibration
For a multi-channel phased array radar system arranged along the track direction, it is usually operated in a one transmitter (Tx) to multiple receivers mode, where the positions of the transmission channel do not align with the receiving channels.The inter-channel spacing cannot be directly determined by the center distance between the receiving channels.Instead, it needs to be equivalently transformed into a virtual system that overlaps between the transmission and receiving channels, in order to calculate the effective phase center position [8,9].Figure 2 illustrates an example of a one-transmit, three-receive multi-channel phased array radar system, where the array elements are distributed in equidistant intervals along the azimuth and elevation directions with distances of and .Assuming the number of azimuth array elements is 18, divided into three receiving channels in the azimuth direction, the typical radar system operates with all elements transmitting and each channel receiving.The Figure 2: The spatial sampling model of a radar system with three azimuth receive channels direct calculation of the azimuth center distance between channels should be equal to 18 /3 = 6 .However, it is effectively converted into a phase center that overlaps between the transmitting and receiving channels, located at the center position between Tx and Rx1 in Figure 2. The effective spacing between channels is 3 .

Calibration of pointing error
Due to the effects of flight attitude, atmospheric turbulence, platform vibrations, and other factors at high altitudes, the beam pointing direction of the radar platform may differ from the intended direction, which in turn can affect the search range for ML angle estimation.There exists a fixed relationship between beam pointing and Doppler frequency.The true beam pointing can be derived by reverse calculation based on the Doppler center frequency value.Because it is difficult to obtain a more accurate Doppler center frequency solely based on certain attitude angle information from the inertial navigation data through calibration, a method of jointly calculating and estimating the Doppler center frequency using a combination of formulas and measured data can be employed.
Firstly, based on the measured data, the ambiguous Doppler center frequency ˆ 0 can be estimated using the correlation function method.Then, a rough estimation of the Doppler center frequency is derived from the inertial navigation data.Finally, based on this frequency, the estimated values can be de-ambiguated.The final formula for estimating the Doppler center frequency is [10,11]: where is the pulse repetition frequency, is the preliminary rough estimation of the Doppler center frequency based on geometric relationships, and (•) represents the rounding function.The true beam pointing angle can be obtained from the following formula: where is the beam pitch angle, as shown in figure 1.

Calibration of antenna spacing and steering vector
Although the inter-channel spacing can be calculated as an effective phase center based on the spacing between elements and the number of elements in each channel division, in practical applications, errors in the spacing may exist due to factors such as installation and environmental changes.In actual engineering, the inter-channel spacing can be estimated through the covariance matrix of the training samples from the sampled data between channels, and then used to calibrate the errors of the guiding vector.Considering three-channel received data, for the given th distance bin and Doppler bin , we have [12]: where is the number of pulse accumulations for a resident beam and Δ is the distance the radar traveled within a pulse repetition interval.represents the equivalent effective inter-channel spacing after phase center calibration, where the spacing and gain are both referenced to the first receiving channel.
For each Doppler bin, the covariance matrix for the receiving channels is: where is the steering vector, can be estimated from the eigendecomposition of the covariance matrix ( ), which can be further decomposed from the sample covariance in the STAP clutter suppression process to reduce computational complexity.The a( )a ( ) is a 3 × 3 Rank-1 matrix.Since the majority of targets in the scene are stationary targets, the eigenvectors corresponding to the large eigenvalues after decomposition serve as the steering vectors for clutter in the a( )a ( ) matrix.Specifically, the column eigenvector corresponding to the maximum eigenvalue in the a( )a ( ) matrix represents the calibration coefficients for the proposed steering vector calibration method, which are used in the construction of the steering vectors.These coefficients can be normalized using the value of the first item in the eigenvectors.

EXPERIMENTAL DESIGN
In this section, the performance of the proposed joint error calibration method for angle estimation in multi-channel WAS-GMTI processing is verified using actual measurement data.The entire radar experiment was conducted using a self-developed three-channel radar system from the Institute of Space Information of the Chinese Academy of Sciences.As shown in figure 3, the radar was installed below the airborne platform and was equipped with a specially designed antenna cover to minimize the impact of external environment.The radar system operated in the X-band with a transmit igure 3: The flight experiment platform and radar system bandwidth of 50MHz.The antenna array was a uniform planar array (UPA) consisting of 72 elements in each row and 4 elements in each column.The original echo data received from these elements were directly synthesized into three-channel outputs.The main parameters of the radar system are shown in Table 1.The WAS-GMTI flight experiment took place in February 2022 in Chengde, Hebei Province, China, as the observed area.Figure 4 shows the flight route of the experiment, which begins in (117°4'10.16"E,40°55'50.98"N)and ends in (117°25'31.43"E,40°54'55.63"N).During the flight experiment, the radar operated in the side-view mode, illuminating the right-side area of the aircraft platform, with a scanning pattern from tail to head.The actual measurement data of the 6th cycle was selected for the experiment, and the enlarged image within figure 5 represents the conceptual monitoring area.

ANALYSIS OF EXPERIMENTAL RESULTS
The following figure shows the comparison results of the electronic map positioning of the moving target after using the proposed joint error calibration and before the uncalibration.Both images are within the monitoring area and use the same processing flow and ML-DOA estimation method, except for calibration.The yellow dots represent the results before calibration, and the red dots represent the results after calibration.
As shown in figure 5, the yellow and red boxes represent the comparison between the same moving target points before and   5, it can be observed that, except for a few false alarms, the red moving target points, obtained by applying the proposed calibration method followed by ML angle estimation, are accurately located on the main road.In contrast, the yellow target points, obtained without calibration, are erroneously positioned on the mountains or farmland, which does not match the actual situation.
In figure 6, the white solid line represents the actual S304 provincial highway.By comparing the results, it can be observed that the proposed multi-channel angle estimation method based on joint error calibration, presented in this paper, closely follows the distribution of the road in terms of moving target localization.In contrast, the yellow localization results without calibration are distributed on the mountain terrain.Clearly, the proposed joint error calibration method effectively mitigates the issue of inaccurate maximum likelihood angle estimation caused by channel errors in multi-channel WAS-GMTI angle estimation, resulting in more reliable localization results.

CONCLUSION
The channel error between multiple channels will seriously affect the angle estimation of moving targets.Therefore, this paper studies and proposes a multi-channel WAS-GMTI angle estimation method based on joint error calibration.The method starts from the echo model, data processing flow, and ML-DOA estimation method.It first equivalently calculates the real effective phase center interval between channels and calibrates the beam pointing direction by estimating the Doppler center value of the beam from the data.Then, it decomposes and calculates the compensation coefficient for the steering vector error from the echo data covariance matrix through joint error processing.The experimental results show that the method proposed in this paper effectively suppresses the problem of inaccurate maximum likelihood angle estimation caused by channel errors to a certain extent, and the localization results are more reliable.

Figure 1 :
Figure 1: The geometric model for data acquisition of Multichannel airborne radar with channels

2 2 (
is the energy accumulation along the range direction of first receiving channel, and a ( ) = 1

Figure 4 :Figure 5 :
Figure 4: Schematic diagram of flight routes and monitoring areas

Figure 6 :
Figure 6: Results of WAS-GMTI monitoring area II for locating moving targets

Table 1 :
The main parameters of the radar system