# Fast Signal Recovery in the Presence of Mutual Coupling Based on New 2-D Direct Data Domain Approach

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## Abstract

The performance of adaptive algorithms, including direct data domain least square, can be significantly degraded in the presence of mutual coupling among array elements. In this paper, a new adaptive algorithm was proposed for the fast recovery of the signal with one snapshot of receiving signals in the presence of mutual coupling, based on the two-dimensional direct data domain least squares (2-D D^{3}LS) for uniform rectangular array (URA). In this method, inverse mutual coupling matrix was not computed. Thus, the computation was reduced and the signal recovery was very fast. Taking mutual coupling into account, a method was derived for estimation of the coupling coefficient which can accurately estimate the coupling coefficient without any auxiliary sensors. Numerical simulations show that recovery of the desired signal is accurate in the presence of mutual coupling.

### Keywords

Coupling Coefficient Adaptive Algorithm Antenna Element Mutual Coupling Uniform Linear Array## 1. Introduction

Adaptive antenna arrays are strongly affected by the existence of mutual coupling (MC) effect between antenna elements; thus, if the effects of MC are ignored, the system performance will not be accurate [1, 2]. Research into compensation for the MC has been mainly based on the idea of using open circuit voltages, firstly proposed by Gupta and Ksienski [2]. While this method has calculated the mutual impedance, the presence of other antenna elements has been ignored and a very simplified current distribution has been assumed for each antenna elements. Many efforts have been made to compensate for the MC effect for uniform linear array (ULA) and uniform circular array (UCA) [2, 3, 4, 5, 6, 7, 8, 9]. In [3], an adaptive algorithm was used to compensate for the MC effect in a ULA. In [7], the authors introduced a minimum norm technique MC compensation method, which is based on the technique in [2] for general arrays with arbitrary elements and more accurate. In [9], a new method was proposed to compensate for the MC effect which relied on the calculation of a new definition of mutual impedance. however, the authors did not deal with 2-D DOA estimation problem.

On the other hand, many algorithms of the 1-D DOA estimation have been extended to solve the 2-D cases [10, 11]; however, a few have considered the effect of mutual coupling or any other array errors [12]. Besides, most of these proposed adaptive algorithms are based on the covariance matrix of the interference. However, these statistical algorithms suffer from two major drawbacks. First, they require independent identically-distributed secondary data in order to estimate the covariance matrix of the interference. Unfortunately, the statistics of the interference may fluctuate rapidly over a short distance, limiting the availability of homogeneous secondary data. The resulting errors in the covariance matrix reduce the ability to suppress the interference. The second drawback is that the estimation of the covariance matrix requires the storage and processing of the secondary data. This is computationally intensive, requiring many calculations in real-time. Recently, direct data domain algorithms have been proposed to overcome these drawbacks of statistical techniques [13, 14, 15, 16]. The approach is to adaptively minimize the interference power while maintaining the array gain in the direction of the signal. The sample support problem is eliminated by avoiding the estimation of a covariance matrix which leads to enormous savings in the required real-time computations. The performance of this algorithm is affected by the MC effect, too [17] and must be compensated.

Unfortunately, the MC matrix tends to change with time due to environmental factors, so full elimination of its effect and prediction of its variability are impossible. Therefore, calibration procedures based upon signal processing algorithms are needed to estimate and compensate for the effect of the MC. The most likely way is to carry out some measurements for calibration. However, this procedure has the drawbacks of being time-consuming and very expensive [18]. Some other researches suggested self-calibration adaptive algorithms for damping the MC effect [19, 20, 21].

In this paper, a new adaptive algorithm was proposed for the fast recovery of the signal with one snapshot of receiving signals in the presence of mutual coupling, based on 2-D D^{3}LS algorithm for URA. Then, utilizing the 2-D D^{3}LS algorithm properties, a novel technique for the coupling coefficients estimation, without using any auxiliary sensors is presented.

This paper is organized as follows. Section 2, conventional 2-D D^{3}LS algorithm is reviewed. In Section 3, a fast adaptive algorithm of direct data domain including mutual coupling effect is presented. In Section 4, a new technique is presented for compensation of the MC effect. In Section 5, numerical simulations illustrate these proposed techniques which can accurately recover the desired signal in the presence of MC.

## 2. 2-D Direct Data Domain Algorithm

## 3. 2-D Fast Signal Recovery Algorithm in the Presence of Mutual Coupling

where Open image in new window denotes, Open image in new window rows from the vector. Open image in new window is computationally intensive and requires many calculations in the real-time because evaluation of the inverse requires an Open image in new window process (here Open image in new window denotes "on the order of"). Therefore, (14) can be replaced with (16) and the number of processes would be an Open image in new window .

## 4. Mutual Coupling Compensation

^{3}LS algorithm. If the mutual coupling effect is ignored, the term Open image in new window , for Open image in new window and Open image in new window will have no signal components. However, in the presence of MC, for the edge elements in the URA, the above term can be written as the following:

where Open image in new window is the estimation of Open image in new window and Open image in new window is Open image in new window with replacement of Open image in new window , Open image in new window , Open image in new window with Open image in new window , Open image in new window , Open image in new window .

## 5. Numerical Examples

**.**

Parameters for the desired signal and interferer.

Magnitude | Phase | |||
---|---|---|---|---|

Signal | 1–10 V/m | 0 | 75° | 45° |

Jammer1 | 1000 V/m | 0 | 43° | −77° |

Later on, the performance of the proposed method is illustrated by the various simulations. The amplitude of the desired signal accuracy is measured by the root mean-squared error (RMSE), and Open image in new window is the number of Monte Carlo runs.

## 6. Conclusion

In this paper, the problems of 2-D D^{3}LS algorithms were studied for recovering of the signal in the presence of mutual coupling and driving a new formulation to recover the signal in the presence of MC. Without using the moment of method and impedance matrix calculation, coupling coefficients can be automatically estimated and without computing the inverse matrix, the desired signal can be recovered. Because we did not use the inverse MC matrix, the amount of computation would be reduced. Moreover, simulation results were confirmed when SNR was high and the RMSE of the method was very close to the ideal D^{3}LS in the absence of MC.

## Appendix

(a) Absence of the Mutual Coupling

If the one row from each column is multiplied by Open image in new window and subtracted from the next row and then the result of each column is multiplied by Open image in new window and subtracted from the next column, in the absence of mutual coupling, this will cancel out all the signals and only noise and interferer will be left

(b) Presence of the Mutual Coupling

## Notes

### Acknowledgment

The authors want to acknowledge the Iran Telecommunication Research Centre (ITRC) for their kindly supports.

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