IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 45, NO. 11, NOVEMBER 1997 1611 Performance of Radial-Basis Function Networks for Direction of Arrival Estimation with Antenna Arrays Ahmed H. El Zooghby, Student Member, IEEE, Christos G. Christodoulou, Senior Member, IEEE, and Michael Georgiopoulos, Member, IEEE Abstract—The problem of direction of arrival (DOA) estimation of mobile users using linear antenna arrays is addressed. To re- duce the computational complexity of superresolution algorithms, e.g. multiple signal classification (MUSIC), the DOA problem is approached as a mapping which can be modeled using a suitable artificial neural network trained with input output pairs. This paper discusses the application of a three-layer radial-basis function neural network (RBFNN), which can learn multiple source-direction findings of a six-element array. The network weights are modified using the normalized cumulative delta rule. The performance of this network is compared to that of the MUSIC algorithm for both uncorrelated and correlated signals. It is also shown that the RBFNN substantially reduced the CPU time for the DOA estimation computations. Index Terms—Antenna arrys, direction of arrival estimation. I. INTRODUCTION M OBILE satellite communication systems using fre- quency division multiple access (FDMA) are facing an increasing number of potential users to be served in the same allocated bandwidth. Multiple reuse of each channel, accomplished by the spatial separation of channels assigned the same narrow frequency band, is used to avoid co-channel interference. Cells with the same frequency are separated by the reuse distance which is directly related to the cluster size . Increasing allows more users to be served in the same geographic area, increases the carrier to interference ratio but also yields larger reuse distances. Closer proximity of cofrequency cells or beams allows additional frequency reuse [1]–[3] . This can be accomplished through two steps. First, a superresolution angle of arrival (DOA) algorithm, multiple signal classification (MUSIC) [4], is used to locate desired as well as cochannel mobile users. This algorithm has the advantage of high resolution for signals with small angular separation (few degrees to few tenths of a degree in many mobile satellite systems) and is known to perform well under low signal-to-noise ratios (SNR’s). Once the direction of the users are specified, this information can be used in conjunction with any adaptive array technique [5] so that the radiation pattern of the array is adapted to allocate the maximum toward the mobiles of interest while other sources of interference in the same frequency slot are nulled and the system is able to track these mobiles in real time. Manuscript received September 3, 1996; revised June 25, 1997. The authors are with the Electrical and Computer Engineering Department, University of Central Florida, Orlando, FL 32816 USA. Publisher Item Identifier S 0018-926X(97)08000-9. Superresolution algorithms have been successfully applied to the problem of DOA estimation to locate radiating sources with additive noise, uncorrelated, and correlated signals. One of the main disadvantages of the superresolution algorithms is that they require extensive computation and as a result they are difficult to implement in real-time. Recently, neural networks have been proposed as successful candidates to carry on the computational tasks required in several array processing applications [6], [7]. Also, in the DOA estimation problem [8], [9], neural network are used in the estimation of the noise subspace necessary for the computation of the MUSIC spectrum by mapping the problem to the quadratic energy function of the network. In this paper, the application of neural networks to handle the computational problem of the DOA estimation step is treated from a different point of view. The DOA problem is approached as a mapping which can be modeled using a suitable artificial neural network trained with input output pairs [10]. The network is then capable of estimating or predicting outputs not included in the learning phase through generalization. Moreover, one of the main advantages of neural networks is that they can be implemented in analog circuits with time constants in the order of nanoseconds [6], [14] and consequently they have fast convergence rates. In Section II, the architecture of a radial-basis function neural network (RBFNN) is presented as well as the input preprocessing and output post-processing. The MUSIC algorithm is briefly described in Section III. In Section IV the training algorithm used in this paper is discussed. Section V presents results obtained from the application of the RBFNN to the DOA estimation for multiple sources with comparisons to the performance of the MUSIC algorithm for uncorrelated and correlated signals. II. RADIAL-BASIS FUNCTION NEURAL NETWORK RBFNN’s [11], [12] are a member of a class of general- purpose method for approximating nonlinear mappings since the DOA problem is of nonlinear nature. Unlike the backprop- agation networks which can be viewed as an application of an optimization problem, RBFNN can be considered as designing neural networks as a curve fitting (or interpolation) problem in a high-dimensional space. The mapping from the input space to the output space may be thought of as a hypersurface representing a multidimensional function of the input. During the training phase, the input–output patterns presented to the network are used to perform a fitting for . The generalization phase represents an interpolation of the input data points along 0018–926X/97$10.00  1997 IEEE