Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 969042, 11 pages

http://dx.doi.org/10.1155/2015/969042

## CC-MUSIC: An Optimization Estimator for Mutual Coupling Correction of L-Shaped Nonuniform Array with Single Snapshot

^{1}School of Electronics and Information Engineering, Harbin Institute of Technology, 92 West Dazhi Street, Nangang, Harbin 150001, China^{2}Institute of Engineering Mechanics, China Earthquake Administration, No. 29 Xuefu Road, Nangang, Harbin 150080, China

Received 2 September 2014; Revised 6 February 2015; Accepted 6 February 2015

Academic Editor: Massimo Scalia

Copyright © 2015 Yuguan Hou et al. 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

For the case of the single snapshot, the integrated SNR gain could not be obtained without the multiple snapshots, which degrades the mutual coupling correction performance under the lower SNR case. In this paper, a Convex Chain MUSIC (CC-MUSIC) algorithm is proposed for the mutual coupling correction of the L-shaped nonuniform array with single snapshot. It is an online self-calibration algorithm and does not require the prior knowledge of the correction matrix initialization and the calibration source with the known position. An optimization for the approximation between the no mutual coupling covariance matrix without the interpolated transformation and the covariance matrix with the mutual coupling and the interpolated transformation is derived. A global optimization problem is formed for the mutual coupling correction and the spatial spectrum estimation. Furthermore, the nonconvex optimization problem of this global optimization is transformed as a chain of the convex optimization, which is basically an alternating optimization routine. The simulation results demonstrate the effectiveness of the proposed method, which improve the resolution ability and the estimation accuracy of the multisources with the single snapshot.

#### 1. Introduction

The L-shaped linear array generally consists of two mutually perpendicular configuration uniform linear arrays, which could be horizontal or vertical. It can be used to estimate the target signal’s two-dimensional information of the elevation angle and the azimuth angle. In addition, its structure is simpler than the planar linear array. In practical applications, the antenna performance of the L-shaped linear array is influenced by the mutual coupling between two array elements [1]. Ignoring the impact of the mutual coupling, the direction of arrival (DOA) estimation algorithm makes a serious degradation [2]. Reducing the impact of the mutual coupling of an antenna array, therefore, becomes an important task. The mutual coupling correction approach is a research hotspot in recent years. In the past two decades, many array calibration algorithms have been proposed for the mutual coupling problem. In [2], the authors pointed out that the mutual coupling reduced the eigenstructure decomposition algorithm performance, and a method of compensation and correction was introduced. In [3], a maximum likelihood algorithm for compensating the mutual coupling of the array element gain and phase has been proposed. This scheme considered all the obstacles without the calibration correction; however, a set of calibration sources in known locations were required. In [4], Lin and Yang considered the uniform circular arrays and presented a blind calibration method for the mutual coupling between elements. In [5], a method that links the DOA estimation and correction of the uniform linear array was presented. This method employed the Toeplitz structure of the array covariance matrix to compensate the gain and phase of the array elements. In [6], the self-correction of the array’s mutual coupling was achieved by setting the amount of the auxiliary array element. Furthermore, the problems caused by iteration were also avoided. In [7], an online mutual coupling compensation algorithm for the uniform and linear arrays was presented. It could simultaneously compensate for the mutual coupling and estimate the direction-of-arrivals of signals. An alternating minimization procedure based on the closed-form solutions was performed to estimate the mutual coupling matrix in the field of complex symmetric Toeplitz matrices. In [8], a method for the estimation of the direction of arrival in the presence of multipath propagation and the mutual coupling for a frequency hopping system was proposed. An iterative alternating minimization algorithm for finding the mutual coupling and the DOA parameters in an alternate manner was formulated by using the pilot symbols and assuming perfect time-frequency synchronization for a linear array. In [9], a new mutual coupling compensation method based on the minimum norm solution to an underdetermined system of equations was introduced. The formulation was proved to be independent of the type of the antenna element and provide good results in situations where signal strengths vary considerably. The analysis of mutual coupling was applied in the context of a code division multiple access communication system. In [10], a computationally efficient algorithm for the direction of arrival estimation of uncorrelated sources and for self-calibration of mutual coupling between the sensors consisting of several uniformly spaced subarrays was proposed. In [11], a simple method was presented for the estimation and the compensation of mutual coupling in antenna systems of arbitrary geometries, including antenna arrays in the vicinity of scatters. The method included both theoretical and experimental schemes, whereas it did not resort to assumptions that often encountered in the previous mutual coupling estimation approaches. In [12], a decoupled method for 2D direction of arrival estimation in the presence of the elevation-dependent mutual coupling was proposed for the compact uniform circular arrays (UCAs) based on the rank reduction theory. In [13], a very simple but effective MUSIC DOA estimation algorithm was introduced by decoupling the antenna mutual coupling in the coupled noise component with the assumption that the uncoupled noise power can be determined a priori and can be removed from the array received power. In [14], the mutual-coupling problems in transmitting and receiving antenna arrays were revisited. The differences between the mutual coupling and mutual impedances for transmitting and receiving antenna arrays are explained. In [15], provided the angularly-independent mutual coupling was treated as angularly-dependent complex array gains, the middle subarray was found to have the same complex array gains. Consequently, a way for parameterizing the steering vector was proposed and the corresponding method for joint estimation of DOAs and mutual coupling matrix using the whole array data was derived based on subspace principle. In [16], the system identification method was applied to the wideband mutual coupling compensation of the receiving arrays. Using the receiving mutual impedances of an antenna array which was calculated at different frequencies, a multiport compensation network was identified for the wideband mutual coupling compensation.

The above-mentioned methods for the mutual coupling correction are based on the case of the multiple snapshots. For the case of the single snapshot, the integrated SNR gain could not be obtained without the multiple snapshots, and the performance of those methods degrades significantly. Therefore, it is a challengeable work to calibrate the mutual coupling in the case of the single snapshot with the lower SNR level. Moreover, for the correlated signals and the nonuniform array geometry cases, an additional optimization for the interpolated matrix is required. The main contribution of the paper is a joint optimization method for the online calibration of the mutual coupling in the cases of the single snapshot, the correlated signals, and the L-shaped nonuniform array geometry. A Convex Chain MUSIC (CC-MUSIC) for the L-shaped nonuniform array with single snapshot is proposed. It uses a novel iterative approach to improve the self-calibration algorithm for the mutual coupling correction. Neither the prior knowledge of the correction matrix initialization nor the calibration source with the known position is required for the proposed algorithm. The virtual array interpolation method is employed to transform the L-shaped linear nonuniform array to the virtual L-shaped linear uniform array for the spatial smoothing technique, which achieves a good DOA resolution and estimation for the coherent source signals. Subsequently an optimum solution for the approximation between the no mutual coupling covariance matrix without the interpolated transformation and the covariance matrix with the coupling and the interpolated transformation is derived. Moreover, a global optimization problem is formed for the mutual coupling correction and the spatial spectrum estimation. To avoid solving the nonconvex optimization problem, a chain of the convex optimization is adopted.

#### 2. Signal Model of Horizontal L-Shaped Nonuniform Linear Array for Coherent Sources

Without loss of generality, we have the following assumptions to simplify the problem: the number of sources is smaller than that of the array elements, thus assuming that the number of the rows of the array bearing matrix is greater than that of the columns. The target signals are the far-field narrowband signals. For the horizontal L-shaped linear array [1], we assume that the reference element is at the origin of the antenna array coordinate axis as shown in Figure 1. The horizontal L-shaped nonuniform linear array is formed based on the horizontal L-shaped uniform linear array and some elements of which are failed or do not exist due to the restriction of the location place.