International Journal of Antennas and Propagation

Volume 2018, Article ID 7175653, 6 pages

https://doi.org/10.1155/2018/7175653

## A Low-Complexity GA-WSF Algorithm for Narrow-Band DOA Estimation

^{1}College of Computer and Communication Engineering, China University of Petroleum, Qingdao, Shandong 266580, China^{2}Graduate School of Engineering, Kitami Institute of Technology, Kitami, Hokkaido 090-8507, Japan

Correspondence should be addressed to Haihua Chen; nc.ude.cpu@auhiahnehc

Received 6 July 2018; Accepted 13 September 2018; Published 4 November 2018

Academic Editor: Stefania Bonafoni

Copyright © 2018 Haihua Chen 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

This paper proposes a low-complexity estimation algorithm for weighted subspace fitting (WSF) based on the Genetic Algorithm (GA) in the problem of narrow-band direction-of-arrival (DOA) finding. Among various solving techniques for DOA, WSF is one of the highest estimation accuracy algorithms. However, its criteria is a multimodal nonlinear multivariate optimization problem. As a result, the computational complexity of WSF is very high, which prevents its application to real systems. The Genetic Algorithm (GA) is considered as an effective algorithm for finding the global solution of WSF. However, conventional GA usually needs a big population size to cover the whole searching space and a large number of generations for convergence, which means that the computational complexity is still high. To reduce the computational complexity of WSF, this paper proposes an improved Genetic algorithm. Firstly a hypothesis technique is used for a rough DOA estimation for WSF. Then, a dynamic initialization space is formed around this value with an empirical function. Within this space, a smaller population size and smaller amount of generations are required. Consequently, the computational complexity is reduced. Simulation results show the efficiency of the proposed algorithm in comparison to many existing algorithms.

#### 1. Introduction

The narrow-band direction-of-arrival (DOA) estimation is a basic and important problem in sensor array signal processing. So far, a set of classical algorithms have been proposed, such as those in [1–7]. Based on this basic narrow-band model and the classical algorithms above, many innovative algorithms have been developed according to different models of signals, noise, and array manifolds, such as those in [8–12].

Among these techniques, the weighted subspace fitting (WSF) is one of the highest estimation accuracy algorithms and it can also deal with coherent signals directly. However, its criteria is a multimodal nonlinear multivariate optimization problem. As a result, the computational complexity of WSF is very high, which prevents its application to real systems.

Artificial intelligence algorithms such as Genetic Algorithm (GA) [13], Particle Swarm Optimization (PSO) [14], Joint-PSO [15], and Bee Colony [16] algorithms are considered to be general and efficient ways for such a problem. However, conventional artificial intelligence algorithms usually need a big population size to cover the whole searching space and a large number of iteration times for convergence. Although Joint-PSO is a rather efficient algorithm for SML estimation, it requires some preprocessing techniques which may make the system more complex.

Based on the Genetic Algorithm, this paper proposes an improved low-complexity Genetic algorithm for WSF estimation. Firstly, it uses a hypothesis technique for a rough DOA estimation for WSF. Then, a dynamic initialization space is formed around this value with an empirical function with respect to signal-to-noise ratio (SNR). Compared to the original whole searching space, this initialization space is much smaller and can be considered to be close to the solution of WSF. Then, a smaller population size and smaller amount of generations are required. Consequently, the computational complexity is reduced. At last, simulation results show the efficiency of the proposed algorithm in comparison to many other algorithms.

The rest of this paper is organized as follows. In Section 2, we introduce the problem of DOA and the formulation of WSF. In Section 3, we introduce the proposed algorithm. Simulation results are shown in Section 4, and the conclusion is drawn in Section 5.

#### 2. System Model and Problem Formulations

To make the article more compact, this section just shows a brief system model and problem formulation. For detailed information, the readers should refer to [5, 6].

Consider that there are sensors (the array configuration can be arbitrary) receiving narrow-band signal waves. and are known. Sensors and signals are in the same far-field plane. All the signals have distinct directions . Note that the number of sensors should be greater than the number of signals, that is, . Furthermore, we have assumed that the sensors are omnidirectional and the array response is ideal, otherwise some calibration techniques should be used, such as those in [17, 18]. Then, the output of the array is as follows: where is the -dimension output vector, is the -dimension signal vector, is the noise vector, and is the steering vector parameterized by . Taking snapshots of the array, the observed data is . Then, we calculate the covariance matrix of the observed data and make an eigen decomposition of it as follows: where and are eigenvalues and eigenvectors, is a diagonal matrix and constructed by the largest eigenvalues, is constructed by the corresponding eigenvectors of , is the orthogonal complement of , .

Then, the WSF criterion is shown as follows: where

From (3) to (4), it is clear that the estimation of WSF is to find a set of to minimize which is a multimodal nonlinear multivariate optimization problem.

#### 3. Improved Genetic Algorithm for WSF

The Genetic Algorithm (Algorithm 1) is considered to be a general and effective way for such a multimodal nonlinear multivariate optimization problem.

However, conventional GA usually needs a big population size to cover the whole searching space (every direction varies from −90 to 90 degrees) and a large number of generations for convergence, which means that the computational complexity is still high. To reduce computational complexity of WSF, this paper proposes an improved Genetic algorithm. The improved GA applied for WSF is shown as follows.

*Algorithm 1. *Genetic Algorithm
(1)Rough DOA searchA hypothesis technique is used in this step to find a rough DOA for WSF. Let be a cost function of in (4) where is the assuming signal number and .
(i)Assuming calculate the corresponding steering vector , the covariance matrix of observed data , and in turn. Obviously, the cost function is a one-dimensional optimization problem with respect to . Find which minimizes .(ii)Assuming and fixing obtained above, calculate . Now is also a one-dimensional optimization problem with respect to . Find which minimizes …. Assuming , find in the same manner.

Define obtained above. This is a rough search DOA for WSF. Although it is hard to prove that is close to the solution of WSF in theory, simulation results show that this hypothesis technique provides a rather good rough search value. Figure 1 shows the positions of a rough DOA search value, the true DOA, and the solution of WSF for both coherent and noncoherent cases. In the noncoherent case, the rough search DOA is rather close to the solution of WSF as Figure 1(a) shows, while in the coherent case, they are a little far apart (when , less than 5 degrees apart) as Figure 1(b) shows. It is clear that the rough search DOA can be considered to be rather close to the solution of WSF for both cases.
(2)Initialization spaceThen around this value, we should use a “scale” to span the initialization space. This initialization space should be close or even contain the solution of WSF. Then, a smaller population size is needed and all the individuals which are randomly initiated in this space could converge quickly to the solution of WSF.

We define the “scale” empirically as follows, which is a function with respect to SNR:
Note that the “scale” is different for coherent and noncoherent cases. This is because in the noncoherent case, the rough search DOA is more closer to the solution of WSF (as shown in Figure 1). As a result, the initialization space could be smaller. The initialization space is defined as a set of in which
where is the signal-to-noise ratio of the signal. Obviously, and . Note that this initialization space is dynamic. When SNR gets higher, the initialization space gets smaller because the rough search DOA is closer to the solution of WSF. Then, only a small population size is needed and all the individuals will converge quickly to the solution of WSF.