  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.
