How Many Beamforming Vectors Generate the Same Beampattern?

TL;DR

This paper investigates the maximum number of distinct beamforming vectors that produce an identical beampattern, providing a method to generate all such vectors efficiently from a single 'mother' vector, with applications in MIMO radar.

Contribution

It establishes the maximum number of beamforming vectors sharing the same beampattern and proposes a method to generate all of them efficiently from a single vector.

Findings

Maximum of 2^{M-1}-1 vectors with same beampattern.

A procedure to generate all such vectors from a single mother vector.

Application demonstrated in MIMO radar transmit beamspace design.

Abstract

In this letter, we address the fundamental question of how many beamforming vectors exist which generate the same beampattern? The question is relevant to many fields such as, for example, array processing, radar, wireless communications, data compression, dimensionality reduction, and biomedical engineering. The desired property of having the same beampattern for different columns of a beamspace transformation matrix (beamforming vectors) often plays a key importance in practical applications. The result is that at most 2^{M-1}-1 beamforming vectors with the same beampattern can be generated from any given beamforming vector. Here M is the dimension of the beamforming vector. At the constructive side, the answer to this question allows for computationally efficient techniques for the beamspace transformation design. Indeed, one can start with a single beamforming vector, which gives a desired beampattern, and generate a number of other beamforming vectors, which give absolutely the same beampattern, in a computationally efficient way. We call the initial beamforming vector as the mother beamforming vector. One possible procedure for generating all possible new beamforming vectors with the same beampattern from the mother beamforming vector is proposed. The application of the proposed analysis to the transmit beamspace design in multiple-input multiple-output radar is also given.