Grid-free compressive beamforming

Abstract

The direction-of-arrival (DOA) estimation problem involves the localization of a few sources from a limited number of observations on an array of sensors, thus it can be formulated as a sparse signal reconstruction problem and solved efficiently with compressive sensing (CS) to achieve high-resolution imaging. On a discrete angular grid, the CS reconstruction degrades due to basis mismatch when the DOAs do not coincide with the angular directions on the grid. To overcome this limitation, a continuous formulation of the DOA problem is employed and an optimization procedure is introduced, which promotes sparsity on a continuous optimization variable. The DOA estimation problem with infinitely many unknowns, i.e., source locations and amplitudes, is solved over a few optimization variables with semidefinite programming. The grid-free CS reconstruction provides high-resolution imaging even with non-uniform arrays, single-snapshot data and under noisy conditions as demonstrated on experimental towed array data.

Sparse framework for hybrid TDoA/DoA multiple emitter localization

“…The matrices B i in (14) can be obtained from a T × T identity matrix I T by randomly selecting P rows from it, for instance. Note that we arrive at the Nyquist-rate relation (12) by simply choosing B i = I T .…”

“…In search for such sparse formulations, the majority of the work in this area considers one particular type of measure. Thus, the seminal work [4], representative of a number of related papers on super-resolution direction finding such as [13], [14] for instance, focuses on a sparse version of a DoA estimation problem. Sparsity-promoting formulations of the time difference of arrival (TDoA) based localization can be found in [7], [8], [11], while [5], [6] and, more recently, [9] provide an example of similar approaches based on received signal strength (RSS) measurements.…”

Sparse DOA estimation with polynomial rooting

“…The dual problem finds the best lower bound by maximizing the dual function over the dual variables; see Ref. [8] for a detailed derivation of the dual problem.…”

“…Opposed to cross-spectral methods, single snapshot processing allows accurate reconstruction of moving sources and coherent arrivals. Moreover, Root-CS can be used with non-uniform arrays [8], [9] which is not possible for the established root methods.…”

“…To achieve computational efficiency along with very fine resolution, we solve the CS optimization problem in the dual domain [6]- [8]. We use the optimal dual variables to achieve high-resolution DOA estimation with the method of polynomial rooting.…”