a suitable estimate for . Two approaches generally employed for problems of this nature are least squares (LS) [4] and total least squares (TLS) [5], giving rise to two versions of ESPRIT.

The two approaches can be explained by considering (5.6) as a simple matrix model AX = B. To estimate X, the least squares technique assumes that the matrix A is known and any errors in the problem are attributed to noise in B. The least squares solution is then given as

X = [AA H ]−1 A H B

where X is the estimate of X.

However, A is also determined from the received data containing noises or disturbances, and hence it may have errors too. Consequently, the total least squares (TLS) technique is preferred and used to solve the invariance equation that takes into account the noises in both A and B. The TLS criterion can be stated as finding residual matrices RA and RB of minimum Frobenius norm such that

[A + R A ]X = B + R B

The actual computation procedure for TLS can be found in many open literatures including [5].

For comparison purposes, both ESPRIT approaches are considered and elaborated upon in the following sections.

5.3 Standard ESPRIT

With the basic concepts and properties of ESPRIT presented earlier, we now describe the standard ESPRIT algorithm. For clarity, we take the uniform linear array (ULAs) as an example to illustrate the standard ESPRIT procedure. Consider a ULA consisting of M elements. A few different configurations of the array suitable for ESPRIT are shown in Figure 5.2.

Let the two subarrays be chosen with maximum overlap, that is, each subarray contains m = M − 1 elements [see Figure 5.2(a)]. The overall array steering matrix A of a ULA has the following Vandermonde structure:

Figure 5.2 Centro-symmetric linear arrays and the corresponding subarrays for ESPRIT.

and x(t) = As(t) + n(t). Each row in the steering matrix A corresponds to each element of the linear array. Choosing a particular subarray configuration can be made mathematically by applying a selection matrix to the overall steering matrix (5.7); it picks m rows of A.

For example, consider a six-element ULA as shown in Figure 5.2. In the case of the maximum overlap, the first five elements form the first subarray and the last five elements form the second subarray. The selection matrices can then be formed by J1, which picks up the first m = M − 1 = 5 rows of A and J2, which selects the last m = M − 1 = 5 rows of the array steering matrix A. The selection matrices are given by

where Im is an m × m identity matrix. In general, the subarrays or equivalently the two selection matrices are chosen to be centro-symmetric with respect to one another. That is,

Here Πp is a p × p exchange matrix (p = m or M ). Because of the fixed displacement between the first and second subarrays, the array steering vector of the second subarray J2a(μi) is just a scaled version of the array steering vector of the first subarray J1a(μi):

This shift invariance property of all the d steering vectors a(μi) can be expressed in a compact matrix form as:

where Φ = diag[e j μ1 , ,e j μd ] is a unitary diagonal d × d matrix whose

diagonal elements contain the desired DOA information in their phases, μi. ESPRIT type algorithms are based on this shift invariance property of the array steering matrix A. However, as discussed in previous sections, instead of finding directly, ESPRIT algorithms estimate the subspace rotating operator , which is directly related to . In doing so, an estimation of the signal subspace is needed as described next.

5.3.1Signal Subspace Estimation

5.3.1.1 True Signal Subspace

Due to the assumption that the d impinging signal wavefronts are not correlated (or have been “decorrelated” as described in Section 4.2), the signal covariance matrix Rss = E{s(t)sH(t)} has full rank, namely rank {Rss} = d. Thus the covariance matrix of the noise-corrupted signals or data [4] is

In this case, its M − d smallest eigenvalues are equal to zero, and an eigenvalue decomposition of Rxx can be expressed as