92 Introduction to Direction-of-Arrival Estimation

A brief summary of standard ESPRIT computation steps is shown in Table 5.1 [1].

As seen from Table 5.1, the standard ESPRIT computations involve complex number operations. To avoid this, an improved ESPRIT employing only real-valued computations has been developed. It is described in Section 5.4.

5.4 Real-Valued Transformation

The standard ESPRIT is comprised of complex-valued computations throughout the algorithm. This makes the algorithm computationally expensive. However, it is observed that if centro-symmetric arrays are used, the signal subspace can be estimated by real-valued computations. Exploiting this property for centro-symmetric arrays leads to the unitary ESPRIT with only real-valued computations. Centro-symmetric arrays introduced in Chapter 2 thus are essential for implementing unitary ESPRIT. Unitary ESPRIT is formulated and based on the following theorem [5].

Table 5.1

Summary of Standard ESPRIT Algorithm

1. Signal Subspace Estimation: Compute Us

as the d dominant left singular vectors of X (square-root approach) or the d dominant eigenvectors of XXH (covariance approach)

2.Solution of the Invariance Equation: Solve the following equation for

J1Us Ψ ≈ J2Us

by means of least-squares or total least-squares techniques

3.DOA Estimation: Calculate the eigenvalues of the resulting complex-valued solution

Ψ = TΦT−1

and then extract the angular information via

Theorem 5.1

Let Qp and Qq denote the left Π-real matrices of p × p and q × q, respectively, defined in (3.16). Then given any p × q centro-Hermitian matrix G, Q −p1GQ q is a real-valued p × q matrix.

Consider the extended data matrix defined in (4.8). It is written here again for ready reference.

It can be shown that Z is centro-Hermitian [6]. Also, similar to what was described in Section 4.2, the estimate R xxfb of the forward-back-

ward averaged covariance matrix is centro-Hermitian [6].

Now, the unitary transformation of the complex-valued data matrix to real-valued matrix can be obtained by applying Theorem 5.1 to the extended data matrix.

The transformation Γ(X) thus accomplishes forward-backward averaging by first extending the complex-valued data matrix X of size M × N to a complex-valued centro-Hermitian matrix Z of size M × 2N and then transforming Z into a real-valued matrix of the same size. As a result, we can obtain

As seen, the forward-backward averaging can be achieved automatically and implicitly by taking the real part of the transformed covariance