Azimuth and elevation computation in high resolution DOA estimation

Abstract

In this paper, we discuss a number of high-resolution direction finding methods for determining the two-dimensional directions of arrival of a number of plane waves, impinging on a sensor array. The array consists of triplets of sensors that are identical, as an extension of the 1D ESPRIT scenario to two dimensions. New algorithms are devised that yield the correct parameter pairs while avoiding an extensive search over the two separate one-dimensional parameter sets. EDICS: 5.2.2., 5.2.4. PRELIMINARIES The data modelConsider m sensor triplets, each composed of three identical sensors with unknown gain and phase patterns, which may vary from triplet to triplet. For every triplet, the displacement vectors d xy and d xz between its components are required to be the same (and, for convenience, are assumed to be orthogonal). This way, assigning the three sensor of each triplet to each of the sensor-arrays X, Y, Z, respectively, three identical although displaced arrays are obtained. This is a direct extension of the 1D ESPRIT scenario to two dimensions (see also [9]). Impinging on every array are d narrowband non-coherent signals s k t, The subspace approachMatrix polynomials of the form E − αF;α ∈ | C ; are called matrix pencils. Forming the pencilsit is seen that, in the noise free case, numbers λ = λ i and µ = µ j , i; j = 1; 2; : : : ; d , that reduce the rank of the pencil by one are equal to φ −1 i and θ −1 j respectively. With square data matrices, these rank reducing numbers are the generalized eigenvalues of the matrix pairs X ; Y and X ; Z.With noise present, however, a large number of samples are taken to improve accuracy. As a result, X, Y and Z will not be square. Noise will also increase the rank of the pencils, and this will introduce new rank reducing numbers. By computing a Total Least Squares projection of the data matrices (see e.g., [10] on

Joint angle and delay estimation using shift-invariance techniques

“…Increasing usually results in better performance. This is because the angle-delay matrix [ ] in (11) or (13) gets taller so that its columns are more linearly independent from each other, resulting in increased accuracy of the angle and delay estimates. However, the delay matrix in (13) gets less wide, and as we increase past a certain point, the performance starts to degrade again because the rows are insufficiently independent.…”

“…However, this does not give a pairing of angles to corresponding delays and might result in poor resolution for closely spaced angles and delays. The algorithm derived in Section III provides a joint estimate, using ideas from 2-D DOA estimation (viz., [11], [12]). …”

“…These data matrices have the structure (10) where , . If dimensions are such that these are low-rank factorizations, then we can apply the 2-D ESPRIT algorithm [11], [12] to estimate and . In particular, since the are given by the rank reducing numbers of the pencil , whereas the are the rank reducing numbers of .…”

“…Finally, it is not easily modified to work with generalized eigenvalues; clearly, it is not correct to compute a generalized eigenvalue decomposition of . Method " ": A second method is described in [11] and was later called the algebraically coupled matrix pencil (ACMP). It works with the Schur decomposition (17) or generalized Schur decomposition (18).…”

“…This is, of course, a classical approach and has been considered, e.g., in [6]- [8] as well. New here is the observation that by stacking the result into a Hankel matrix, the problem is reduced to one that can be solved using 2-D ESPRIT techniques [11], [12], which were developed for joint azimuth-elevation estimation. Thus, the algorithm is closedform and computationally attractive, and angles and delays are jointly estimated and automatically paired.…”