In the future, we can potentially extend our research to the following directions:
Partially correlated sources: Throughout this dissertation, we assumed that the sources are uncorrelated. In practice, this assumption may not hold in every environment, due to multi-path effects [97, 98]. Consequently, the sources will be partially correlated. Recently, using sparse linear arrays, new algorithms have been proposed to resolve more correlated sources than the number of sensors [99, 100]. It would be interesting to extend our analysis in Chapter 3 to cases of partially correlated sources and to analyze the maximum number of identifiable partially correlated sources.
Optimal sparse linear array design: In this dissertation, we derived the closed-form asymptotic MSE expression of DA-MUSIC and SS-MUSIC, as well as the CRB. We also analyzed the performance of sparse linear arrays in the presence of sensor location errors. These results enabled us to formulate optimal array design problems. Instead of using pre- configured array geometries, it would be interesting to be able to set constraints on metrics such as the MSE and sensitivity to model errors, and to obtain the optimal array geometries for specific application scenarios by solving the resulting optimization problems.
Extension of the stochastic error model: In our analysis of the stochastic error model, we simply assumed that the time-variant sensor location errors are i.i.d. This assumption, while convenient in statistical analysis, may not hold in practice. In the future, our analysis of the stochastic error model could be extended by introducing motion models for the sensor location errors. Then robust DOA estimation algorithms could be developed based on such models.
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Appendix A
Proof of Theorem 3.1
We first derive the first-order expression of DA-MUSIC. Denote the eigendecomposition of
Rv1 by
Rv1= EsΛs1EsH + EnΛn1EnH,
where Es and En are eigenvectors of the signal subspace and noise subspace, respectively,
and Λs1, Λn1 are the corresponding eigenvalues. Specifically, we have Λn1 = σ2I.
Let ˜Rv1 = Rv1+ ∆Rv1, ˜En1 = En+ ∆En1, and ˜Λn1 = Λn1+ ∆Λn1 be the perturbed versions
of Rv1, En, and Λn1. The following equality holds:
(Rv1+ ∆Rv1)(En+ ∆En1) = (En+ ∆En1)(Λn1 + ∆Λn1).
If the perturbation is small and the SNR is high, we can omit high-order terms and obtain [53, 64, 67]
AHco∆En1
.
Because P is diagonal, for a specific θk, we have
aH(θk)∆En1
.
= −p−1k eTkA†co∆Rv1En, (A.2)
where ek is the k-th column of the identity matrix IK×K. Based on the conclusion in
Appendix B of [61], under sufficiently small perturbations, the error expression of DA-MUSIC for the k-th DOA is given by
ˆ θ(1)k − θk . = −<[a H co(θk)∆En1EHn a˙co(θk))] ˙ aH co(θk)EnEnHa˙co(θk) , (A.3) where ˙aco(θk) = ∂aco(θk)/∂θk.
Substituting (A.2) into (A.3) gives
ˆ θk(1)− θk=. <[eT kA † co∆Rv1EnEnHa˙co(θk)] pka˙Hco(θk)EnEnHa˙co(θk) . (A.4)
Because vec(AXB) = (BT ⊗ A) vec(X) and E
nEnH = Π ⊥
Aco, we can use the notations
introduced in (3.2b)–(3.2d) to express (A.4) as
ˆ
θ(1)k − θk
.
= −(γkpk)−1<[(βk⊗ αk)T∆rv1], (A.5)
where ∆rv1 = vec(∆Rv1).
Note that ˜Rv1 is constructed from ˜R. It follows that ∆Rv1 actually depends on ∆R, which