Simultaneous Estimation of Azimuth and Elevation Angles and Frequency of Plane Wave Signals Using a Modified Matrix Pencil Method Muhammad Faisal Khan 1 and Muhammad Tufail 2 Department of Electrical Engineering, Pakistan Institute of Engineering And Applied Sciences PIEAS, Nilore, Islamabad, Pakistan E-mail : 1 faisalmk72@yahoo.com, 2 tufail pieas.yahoo.com Abstract: In this paper, the modified matrix pencil method [1] is extended from 2D to 3D for simultaneously estimat- ing azimuth and elevation angles and frequency of multiple signals impinging on a volumetric array of sensors. Conven- tional matrix pencil method, which is used to estimate un- known parameters, needs a separate algorithm to associate the estimated components with each other to get proper groups of azimuth and elevation angles and frequencies of incom- ing signals [2]. The method proposed here automatically esti- mates unknown parameters in a group form, thereby bypass- ing the computationally expensive pairing operation. More- over, simulation results show that grouping of unknown pa- rameters using the proposed method is always correct in con- trast to matrix pencil method whose results are sometimes er- roneous. 1. Introduction Many methods and techniques have been developed in the last few decades to estimate unknown parameters of plane wave signals using sensor arrays [3]. Supper resolution techniques like MUSIC [4] and ESPRIT [5] are widely used to estimate the unknown parameters by exploiting the eigen structure of the covariance matrix. These techniques are usually suitable for stationary environment and assume that the signals are not fully correlated or coherent. Another method for estimating the unknown parameters of sinusoids in noise is matrix pencil method [6]. The matrix pencil method is a direct data do- main method that analyzes the data on snapshot-by-snapshot basis; consequently, non-stationary environment can be han- dled easily. All the techniques mentioned above are basically one dimensional techniques and find only one direction cosine of incoming signals. To find azimuth and elevation angles along with frequency, all three direction cosines need to be estimated. For estimation of other direction cosines simulta- neously, these techniques are enhanced for multidimensional cases. In [7] matrix pencil method is enhanced for 2-D sig- nals and matrix enhancement matrix pencil method (MEMP) is developed. The MEMP method is further enhanced in [2] to simultaneously estimate azimuth and elevation angles and frequency of incoming plane wave signals. A major drawback in these enhanced techniques is that es- timated parameters of different signals are not grouped auto- matically. To properly group these parameters, an algorithm which exploits the orthogonal property between signal and noise subspaces is used. Although this algorithm separates es- timated parameters, it is computationally expensive and does not always render the correct grouping because it is a correla- tion maximization algorithm. Modified matrix enhancement matrix pencil (MMEMP) method [1] remedies this problem. This MMEMP method was originally developed to estimate the frequency components of a 2-dimensional signal. Here, we extend this method for 3-dimensional case and apply it on a uniform volumetric array to estimate azimuth and elevation angles and frequency of plane wave signals. 2. Problem Formulation Consider a 3-dimensional uniform array of sensors in space with axes oriented along the cartesian coordinates. The dis- tances between array elements are Δx, Δy and Δz along x, y and z axis, respectively, and the corresponding number of sensors are A, B and C. If I signals with azimuth and eleva- tion angles (φ i , θ i ) and wavelengths λ i (i =1...I) arrive at the input of this array, the voltage level at the output of the sensor located at cartesian coordinates (a, b, c) is given by v(a, b, c)= I  i=1 M i exp j (γ i + 2π λ i Δx cos φ i sin θ i a + 2π λ i Δy sin φ i sin θ i b + 2π λ i Δz cos θ i c) (1) where M i ’s are the amplitudes and γ i ’s are the phases of in- coming signals. If we define f 1i , f 2i and f 3i as f 1i = 1 λ i Δx cos φ i sin θ i , f 2i = 1 λ i Δy sin φ i sin θ i f 3i = 1 λi Δz cos θ i (2) then v(a, b, c) can be written as v(a, b, c)= I  i=1 M i exp j (γ i +2πf 1i a +2πf 2i b +2πf 3i c). (3) To find direction cosines of incoming signals, we have to es- timate the three frequencies of each 3-D sinusoid and group them appropriately. For further simplification we define α i = M i exp(jγ i ), x i = exp(j 2πf 1i ) y i = exp(j 2πf 2i ), z i = exp(j 2πf 3i ) (4) so that (3) becomes v(a, b, c)= I  i=1 α i x a i y b i z c i . (5) Hence, our target is to estimate x i , y i and z i , which can be used to find three direction cosines of the incoming signals. The 23rd International Technical Conference on Circuits/Systems, Computers and Communications (ITC-CSCC 2008) 873