DOA Estimations with ESPRIT Algorithms |
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where the diagonal matrix 
R d×d contains the d largest singular values and Us spans the signal subspace.
5.3.2Solution of Invariance Equation
After the estimation of the matrix Us that spans the estimated signal subspace, based on (5.19), the two known selection matrices J1 and J2 are applied to form the following invariance equation.
J1U s Ψ ≈ J 2U s C m × d |
(5.22) |
where 
is the signal subspace rotating operator, defined by (5.17) or (5.6). In contrast to (5.19), this invariance equation might not have an exact solution; this is because the signal subspace is estimated from an estimated data covariance matrix (5.21) that is not the true or exact signal covariance matrix. Also, the size of the subarrays M − 1 should be at least equal to d in order to compute all the DOAs; otherwise, the invariance system (5.22) would be underdetermined.
Equation (5.22) is solved by using the least squares (LS) or total least squares (TLS) solutions to get an estimate of the subspace rotating operator 
as explained in the previous section. In real time, while implementing these solutions, the more efficient algorithm called the QR decomposition may also be used to solve the least squares. This is because the direct matrix inverse calculations are prone to error and consume lot of time in computations [5].
5.3.3Spatial Frequency and DOA Estimation
Once 
C d×d is found, the desired DOA information can be estimated from it. The eigenvalues of the estimated 
C d×d can be calculated by its eigendecomposition; this is because
Ψ = TΦT −1 where Φ = diag [e j μ1 , , e j μd ] |
(5.23) |
Therefore, the eigenvalue of 
, φi, represents estimates of the phase factors e j μi . Once the estimates of the spatial frequencies μi are found, the corresponding DOAs θi are obtained via the relationships