Mathematical Problems in Engineering

Volume 2017, Article ID 7904356, 7 pages

https://doi.org/10.1155/2017/7904356

## MVDR Algorithm Based on Estimated Diagonal Loading for Beamforming

^{1}China University of Mining and Technology, Xuzhou 221116, China^{2}University of Chinese Academy of Sciences, Beijing 101408, China

Correspondence should be addressed to Hongsheng Yin; moc.anis@shyuohzux

Received 1 June 2017; Revised 24 August 2017; Accepted 11 September 2017; Published 12 October 2017

Academic Editor: Thomas Schuster

Copyright © 2017 Yuteng Xiao 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

Beamforming algorithm is widely used in many signal processing fields. At present, the typical beamforming algorithm is MVDR (Minimum Variance Distortionless Response). However, the performance of MVDR algorithm relies on the accurate covariance matrix. The MVDR algorithm declines dramatically with the inaccurate covariance matrix. To solve the problem, studying the beamforming array signal model and beamforming MVDR algorithm, we improve MVDR algorithm based on estimated diagonal loading for beamforming. MVDR optimization model based on diagonal loading compensation is established and the interval of the diagonal loading compensation value is deduced on the basis of the matrix theory. The optimal diagonal loading value in the interval is also determined through the experimental method. The experimental results show that the algorithm compared with existing algorithms is practical and effective.

#### 1. Introduction

The beamformer is used for the purpose of detecting the desired signals and suppressing the interference signals. It filters the interference signals and outputs the desired signals whose mainlobe points to desired direction. Beamformer, which is the key technology in many fields, is widely used in radar, communications, sonar, GPS, and other fields. For example, weather radar system needs the algorithm to get the desired signals [1] from the meteorological information. In MIMO communication system, the beamformer is used to achieve large array gains [2].

MVDR (Minimum Variance Distortionless Response) is a typical beamforming (beamformer) algorithm, which was proposed by Capon in 1967 [3]. It makes the outputting power with minimum interference and noise in the desired direction through adjusting a weight factor. However, the performance of the beamforming algorithm will decline dramatically with the inaccuracy in the steering vector and the covariance matrix [4]. To reduce the dispersive degree of the eigenvalues of the covariance matrix, Carlson proposed diagonal loading by pulsing a constant diagonal loading compensation value to the diagonal elements of the covariance matrix [5]. On this basis of constant diagonal loading compensation, the literature [6] selects the loading value which is higher than the background noise. However, when the snapshots are small, there is a large inaccuracy in the sample covariance matrix, and the constant diagonal loading compensation value cannot correct the inaccuracy. Literatures [7–9] reconstruct the covariance matrix which is also one of the diagonal loadings. In literature [10], an optimal algorithm employing variable diagonal loading of the sample covariance matrix eigenvalues is proposed. In literature [11], an algorithm is used to compute the diagonal loading level automatically from the given data with a shrinkage method of enhancing covariance matrix. Literature [12] uses beam-to-reference ratio which is estimated as a weighting factor for variable diagonal loading. These algorithms have solved the problem that the diagonal loading value is not easy to determine and cannot be adjusted automatically with the sample changing. However, these algorithms are more complex and less efficient in the calculation.

In this paper, we will adopt the diagonal loading method to improve MVDR algorithm. It uses the adaptive diagonal loading compensation method, adjusting the diagonal elements in sample covariance matrix. The sharp decline, which is caused by the inaccuracy in the covariance matrix, is improved in performance.

The rest of this paper is organized as follows. In Section 2, the beamforming array signal model is introduced. In Section 3, the model with beamforming MVDR algorithm is proposed and the model is solved by Lagrange multiplier. Then, the model with MVDR optimization based on diagonal loading in theory is discussed in Section 4. In Section 5, the method for estimating the diagonal loading value is discussed and analyzed. The experimental results are shown in Section 6. Finally, we conclude this paper in Section 7.

#### 2. Beamforming Array Signal Model

Figure 1 presents uniform linear array signal model. There are narrow band multisignal sources (containing both desired signals and interference signals) radiating at the same time, whose wavelengths are all . The received antenna is an linear array with distance and . Taking the weather radar system as an example, the desired signals are usually the meteorological information with different desired directions and the interference signals are the various clutter information reflected by the ground. According to the Shannon communication system model, the noise is concentrated on the beamformer. In practice, the desired signals are much smaller than the interference signals and the noise, whose Signal-Noise Ratio and Signal/Interference-Noise Ratio are often up to −20 dB. Thus, the beamformer is used to receive narrow band multisignal source. It makes the desired signals without attenuation through adjusting weight factor . And at the same time, the interference signals and noise are greatly suppressed.