Many classical direction of arrival (DOA) estimation algorithms suffer from sensitivity to array errors. A simple but efficient method is presented for direction finding in the presence of gain and phase errors as well as mutual coupling errors. By applying a group of auxiliary sensors, DOAs and gain and phase coefficients can be simultaneously estimated, and mutual coupling coefficients can also be estimated by utilizing a novel decoupling method. The proposed algorithm does not require iterative operation or any calibration sources or spectral peak searching. Simulation results demonstrate the effectiveness of the proposed method.
The methods of the direction of arrival (DOA) estimation have received considerable attention because of their importance in a great variety of applications such as radar, sonar, and communication [
In recent decades, many researchers attempt to estimate the signal DOAs in the presence of array errors. To calibrate errors of mutual coupling, a number of calibration algorithms [
Besides researching the problem of mutual coupling errors, sensor gain and phase errors calibration methods [
In practical situations, various sources of array errors may be present simultaneously, but the aforesaid calibration methods are capable of analyzing statistical performance of circumstances where only one array error model presents. Assuming that sensor locations are known, array calibration can be considered as estimating the DOAs in the presence of sensor gain and phase errors and mutual coupling errors. Generally, calibration methods can be classified as active calibration methods [
In this paper, we propose a direction finding method in the presence of sensor gain and phase errors as well as mutual coupling errors. By applying a group of auxiliary sensors, we can eliminate mutual coupling errors by the inherent mechanism of the proposed method, assuming that DOA estimation is independent of mutual coupling coefficients. Then DOAs as well as gain and phase coefficients can be jointly estimated in closed form based on extended leastsquares ESPRIT (LSESPRIT) algorithm. Finally, mutual coupling coefficients can also be estimated by utilizing a new decoupling method. The proposed method requires no spectral search or iteration, so the complexity is low, which makes it simpler but more efficient, and it can realize the estimation of DOAs, gain/phase error coefficients, and mutual coupling coefficients simultaneously.
The rest of the paper is organized as follows. Section
Consider a uniform linear array (ULA) consisting of
According to (
The eigendecomposition of
Considering mutual coupling errors and gain and phase uncertainties of the array, the corresponding array output vector is rewritten by
As we know, mutual coupling coefficients of sensors are inversely related to their distance. On the basis of ULA structure, MCM can be expressed as a banded symmetric Toeplitz matrix. Suppose there are
Actually, mutual coupling errors and gain and phase errors coexist. Assuming that the gain and phase of
Here, we are interested in the problem of how to estimate the DOAs
Auxiliary sensors with known gain and phase are utilized to estimate DOAs, gain and phase coefficients, and mutual coupling coefficients on the basis of original
Assuming that
Array geometry. Solid points represent uncertain sensors and circles represent auxiliary sensors.
Let
We define matrix
The MCM
Taking the unknown gain and phase uncertainties into account, their gain and phase vectors can be denoted by
Considering auxiliary sensors, the output of the middle array can be expressed as
Substituting
According to the above analysis, MCM in the steering vector of the middle array is converted to the scalar function related to mutual coupling coefficients, thus realizing decoupling. Therefore, the steering vectors of two subarrays of the middle array are given by
The relationship between
In the case that gain and phase errors and mutual coupling errors exist simultaneously, the received data of two subarrays are stacked into a column
As it is known,
Obviously, (
Thus
As it is known that
Considering the finite sampling data, the two signal subspace
Substituting formula (
In that way,
It should be noted that
Therefore, according to the definition of the inverse matrix, we can get
Based on (
Now, substituting (
Since
After obtaining DOAs of incident signals and gain and phase coefficients, we will discuss mutual coupling coefficients estimation. Here, the output of the whole array is used. For the whole array with all sensors
Since sample data is finite,
Thus mutual coupling coefficients have been estimated. This mutual coupling coefficients estimation method is applicable to any form of antenna array; just the transform matrix
The estimation method of DOAs, sensor gain and phase coefficients, and mutual coupling coefficients can be summarized as follows.
Assuming a ULA with
Perform eigendecomposition of
Based on the extending LSESPRIT algorithm of this paper and (
Gain and phase coefficients
Mutual coupling coefficients
In this paper, we focus on the simple solution to tackle sensor gain and phase errors and mutual coupling errors. Based on derivation in previous sections, it can be seen that the proposed method is computationally efficient because no spectral search and iteration are required. The contention WF method in [
With detailed analysis, there are some necessary conditions worthy of attention. First, in order to make the LSESPRIT algorithm work, we must use auxiliary sensors, and the number of
In this section, we present several simulation results to illustrate the performance of the proposed method. In the following examples, we consider a ULA of 14 sensors with half wavelength spacing, and 9 of them are auxiliary sensors, where
In the first example, we present numerical simulation results to evaluate the superior performance of the proposed method in DOA estimation, in comparison with the gain and phase calibration method in [
RMSE of two signals DOA estimation versus SNR.
RMSE of two signals DOA estimation versus snapshot number.
According to Figures
Next, performance comparison is made between the proposed method and the WF method which can calibrate these two types of errors at the same time through several experiments. The experiment results of MUSIC using the whole array with known uncertainties are also obtained, and CRB is displayed. Assuming three signals from −10°, 10°, and 20°, Figure
RMSE of three signals DOA estimation versus SNR.
RMSE of three signals DOA estimation versus snapshot number.
According to Figures
Experimental results above have proved good performance of the proposed method for DOA estimation. The proposed method also can be used to estimate gain and phase coefficients. Assuming three signals from −10°, 10°, and 20° the snapshot number is 200. In Figures
Mean and variance of gain and phase coefficients for all uncertain sensors at SNR = 0 dB and SNR = 20 dB. True values are
SNR  0 dB  20 dB  


1  2  3  4  5  1  2  3  4  5 

0.8087  1.2690  1.5651  0.7702  1.3699  0.7998  1.2504  1.5308  0.7504  1.3600 












0.6316  −1.0510  −0.6257  0.7878  −0.3139  0.6284  −1.0470  −0.6282  0.7854  −0.3137 











RMSE of gain coefficient estimation versus SNR.
RMSE of phase coefficient estimation versus SNR.
The experimental results show that the proposed method has a better performance in estimation of gain and phase coefficients as SNR increases. When SNR is higher than 5 dB, RMSEs of gain and phase can be less than 10%, which proves that when SNR is high enough, the coefficients could be estimated, with higher accuracy and stability.
In the following experiments, the statistical efficiency of the proposed method in mutual coupling coefficient estimation is studied. Assuming three signals from −10°, 10°, and 20° the snapshot number is 200. Figure
Mean and variance of the real and imaginary part of all mutual coupling coefficients at different SNRs. True values are
SNR  −5 dB  5 dB  15 dB  20 dB  


1  2  1  2  1  2  1  2 

0.1387  0.2266  0.4189  0.2599  0.4459  0.2603  0.4497  0.2598 










0.3669  −0.1058  0.5185  −0.1432  0.5344  −0.1492  0.5359  −0.1500 









RMSE of mutual coupling coefficients versus SNR.
Figure
In summary, simulation results have proved the effectiveness and excellent performance of the proposed method. As discussed in Section
In this paper, a simple and efficient method is proposed for direction finding in the presence of gain and phase errors as well as mutual coupling errors. By applying a group of auxiliary sensors, DOAs as well as gain and phase coefficients can be simultaneously estimated, and mutual coupling coefficients can also be estimated by utilizing the proposed decoupling method. Iterative operation, calibration sources, and spectral peak searching are not required for this method. However, auxiliary sensors are necessary to implement this algorithm, whose number affects algorithm performance. Different from other methods, the proposed method can achieve good performance with only a few auxiliary sensors. As shown in simulation results, the effectiveness and excellent performance of the proposed method are compared with other popular methods. DOA estimation can achieve good performance even at low SNRs and few snapshots. Moreover, gain and phase coefficients and mutual coupling coefficients can be accurately estimated when SNR reaches a certain threshold.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors thank the reviewers and editors for their corresponding contributions in making the paper more presentable. This work was supported in part by the National Science Foundation of China under Grant 61201410 and in part by the Fundamental Research Funds for the Central Universities under Grant no. HEUCF130804.