5.7 Unitary ESPRIT in DFT Beamspace

rows of the FFT matrix, followed by an appropriate scaling of the rows of resulting matrix.

5.7.1.2 Real-Valued Signal Subspace Estimation

The unitary transformation is now applied as discussed in Section 5.4. An SVD of the real-valued matrix

is computed to estimate the d left singular vectors that correspond to the d largest singular values of (5.84); otherwise, an EVD of

can also be computed (i.e., covariance approach). This transformation, as described in Section 5.4, automatically achieves forward-backward averaging.

5.7.1.3 Real-Valued Invariance Equation

Let the d singular vectors corresponding to the d largest singular values of (5.84) be denoted by Es R M × d . Asymptotically, the real-valued matri-

ces Es and B span the same d-dimensional signal subspace, so there is a nonsingular matrix TA such that

which is solved by LS or TLS as explained in previous sections.

5.7.1.3 DOA Estimation

The spatial frequencies or DOAs are then estimated by taking eigenvalues of the solution of the invariance equation given by

Γ = T A ΩT A−1

where

Ω = diag{ω1 , ω 2 , , ωi , , ω d }

contains the desired DOA information. The whole algorithm that operates in an L-dimensional DFT beamspace is summarized in Table 5.4 and its performance is analyzed in Chapter 6 [1].

In the next section, the concept of DFT beamspace is extended to two-dimensional arrays.

5.7.2Two-Dimensional Unitary ESPRIT in DFT Beamspace

In this section, DFT beamspace technique is applied to two-dimensional arrays, more specifically to uniform rectangular arrays. Consider a uniform rectangular array of size M = Mx × My. Assume the signal and data

Table 5.4

Summary of One-Dimensional Unitary ESPRIT in DFT Beamspace

1.Transformation to Beamspace: Y = WbX

2.Signal Subspace Estimation: Compute Es

as the d dominant left singular vectors of [Re{Y} Im{Y}] (square-root approach)

or as the d dominant eigenvectors of Re{X}Re{Y}H + Im{X}Im{Y}H (covariance approach)

3. Solution of the Invariance Equation: Solve the following equation for ϒ

γ1(B )Es ϒ ≈ Γ2(B )Es

by means of LS or TLS

4. DOA Estimation: Calculate the eigenvalues of the real-valued solution

model discussed in Chapter 3. By following the similar line for beamspace processing of one-dimensional ULA, we can compute the two-dimen- sional DFT beamspace ESPRIT in a straightforward manner.

5.7.2.1 Transformation to Beamspace

Consider the case of reduced dimensionality directly and assume that Lx out of Mx beams in the x direction and Ly out of My beams in the y direc-

Let the noise corrupted signals or data received by the array at any given time tn be given by the matrix χ(t n ) C M x × M y . Premultiplying this

We place this column vector as a column of the matrix Y C L×N. The vec{•} operator maps a Bx × By matrix to a B × 1 vector by stacking the columns of the matrix. Again by applying the property of Kronecker products (Appendix, Section A.1) to (5.88), we get

Signal subspace now can be estimated from this data.

5.7.2.2 Real-Valued Subspace Estimation

Let Es denote the real-valued signal subspace. The columns of Es RL×d contain the d left singular vectors of

corresponding to its d largest singular values. Otherwise, using EVD, the d dominant eigenvectors of

spans the real-valued signal space.

5.7.2.3 Real-Valued Invariance Equation

The transformed real-valued array manifold matrix is now given by

are the one-dimensional DFT beamspace manifold vectors in a reduced

(2.24).

Consider the one-dimensional DFT beamspace vectors in the x direction. As bL x (μi ) satisfies the invariance relationship in (5.82), it follows that B(μi, νi) satisfies

where the one-dimensional selection matrices in the x direction, Γ1( L x ) and Γ2( L x ) , were defined as before for the one-dimensional DFT beamspace. By using the property of the vec{•} operator in the Appendix, Section A.1, L = Lx × Ly dimensional DFT beamspace steering vectors b(μi, νi) = vec{B(μi, νi)}satisfy

are of size bx × L with bx = (Lx − 1) Ly. Similarly, the one-dimensional DFT beamspace manifold vectors in the y direction, bL y (ν i ), satisfy

are of size by × L with by = Lx (Ly −1).

The real-valued two-dimensional DFT beamspace steering matrix is then given as

= [b ( μ1 ,v1 ) b ( μ 2 ,v 2 ) b ( μ d ,v d )] R L × d

Hence, B satisfies the following two invariance properties,