# Forming the estimation manifold for a 2-D beamformer

Given an array of L sensors in a uniform planar array, with known positions in space, I would like to estimate both the angles of arrival and delays from a point source under multipath conditions simultaneously, initially using a Bartlett beamformer. The data from the array is in the frequency domain, taking the form of an L x Nf matrix, with Nf bins around a carrier frequency f.

Based on this paper, I have an L x M angular response vector a, and an Nf x K delay response vector d, where M and K are the resolutions in angle and delay respectively. From these, I wish to create an estimation manifold (defined as U in the paper), which will become the 2-D analogue of the array response vector in the case of 1-D beamforming.

From equation 1 in the paper, it appears as if I need to set M = K, with their product becoming the estimation manifold U, however this produces a level surface when plotted. Alternatively, letting M != K and taking the Kronecker product means this manifold will quickly grow beyong a reasonably storable size, especially when M, K or L are large.

I have also tried slicing the input data into L x 1 slices and treating this as a 1-D estimation problem, which was then projected into 2-D using the delay response vector d, however as this removes the temporal relationships, this did not yield relevant results either.

Is there a better way of forming this estimation manifold, and eventually applying it to a Bartlett beamformer?

Also, for full disclosure this is part of university coursework.

• You might want to duplicate the equation 1 here as it is behind a paywall. That way more people could potentially answer. – IanJ Feb 15 at 16:02
• Could you mathematically write down what you like to solve for? There could be interested audience who are not really familiar with beamforming. – Tolga Birdal Feb 17 at 5:47