Hence, it is observed that beamspace array steering matrix B satisfies a shift invariance property similar to the element space array steering matrix given in (5.44), thus enabling the implementation of ESPRIT-based algorithms.

5.6.2DFT Beamspace in a Reduced Dimension

In the previous section, the beamspace transformation matrix was taken as Wb. Hence, each beamspace steering vector b(μi) has M components. If one has a priori knowledge of angle(s) of the impinging signal(s), then it is possible to choose only those components that span the desired DOA sector around the angle(s) [8], since each row of the matrix equation (5.80) relates two successive components of the DFT beamspace steering vectors b(μi) (i.e., they point to successive three angles).

This observation allows us to apply L ≤ M successive rows of Wb

beginning at row kmin, 1 ≤ kmin ≤ M, instead of all M rows to the data matrix X. In the reduced number of rows, denote the resulting

beamforming matrix that produces L consecutive beams as W Lb C B × M . The beamformer W Lb thus narrows the scope of the search for DOAs to the spatial sector specified by

where a steering angle γ > π is identical to the steering angle γ − 2π.

The number of rows B depends on the angle width of the DOA sector of interest and can be substantially less than the number of elements M. Consequently, the dimension of the SVD of (5.71), and therefore Es, along with that the real-valued invariance equation, will reduce to L × d. By selecting appropriate subblocks of Γ1 and Γ2, the beamspace processing performs the same as element space except for its reduced dimensionality. The resulting selection matrices of size (L − 1) × L shall be called Γ1( L ) and Γ2( L ) such that the B-dimensional DFT beamspace steering vectors bL (μi ) = W Lb ac (μi ) still satisfy the invariance relation.