Estudio de la Subrasante

This part is a review of the basic ideas and concepts used for the derivation in [139], which also includes some necessary prerequisites for subsequent derivations.

5.4 Theoretical Performance Limits: The Cram´er-Rao Bound 65

in (5.2), constrained to a certain class of valid estimators. It is assumed that the dictionary parameter, θ, is known and only x has to be estimated. The constrained CRB is a local bound that is only valid for estimating vectors in a certain neighborhood of a target point, x0 ∈ RN. This neighborhood is specified by an -environment around

x0, i.e. B(x0) = {x ∈ RN| kx − x0k2 < }, and by a ‘locally balanced’ constraint

set, Xc. According to [139], the constrained set in the sparse setting cannot be written

in the form used for the classical constrained CRB. Therefore, the locally balanced constrained set is introduced. The term ‘locally balanced’ means that, for a certain point x0 ∈ Xc, and another point x ∈ eXc ⊂ Xc, this set is locally defined at x0, such

that

x0 = x0+ λ(x − x0) ∈ Xe_c ∀ |λ| ≤ 1 . (5.4) Herein, the vectors (x−x0)/k(x−x0)k2, x ∈ eXc, belong to the set of ‘feasible directions’

at the point x0. These are the directions in which one can move without violating

the constraints. An estimator for which the constrained CRB is a valid bound must satisfy unbiasedness within the neighborhood {x ∈ Xc| kx − x0k2 < }, i.e. in the

-environment around x0 where the constraints Xc are fulfilled. Therefore, the bias

gradient has to vanish with respect to the feasible directions, where the constraints are not violated.

The feasible directions are important for the concept of Xc-unbiasedness. It is shown in

[139] that Xc-unbiasedness is in fact a property of the subspace spanned by the feasible

directions. Any orthonormal basis can be chosen to describe this subspace but its dimensionality may change for different points x0. It is convenient to use the canonical

basis in a finite-dimensional Euclidean space, defined by vectors {ed}Dd=1, which can be

collected in a matrix Uf = [e1, . . . , eD]. For notational convenience, the dependence of

Uf on x0 is not explicitly mentioned below. Then, for an estimator, ˆx, with associated

bias function, b(x) = Exˆx − x, x ∈ Rˆ N, the term ‘Xc-unbiasedness’ means that the

bias gradient vanishes with respect to the subspace spanned by the feasible directions, i.e. Uf(∂b(x)_∂x ) = 0 [139]. This is a requirement for all points x ∈ { Xc ∩ B(x0) }. In

other words, the class of valid estimators for which the constrained CRB applies is required to have a vanishing bias gradient for all x ∈ { Xc ∩ B(x0) }. Such estimators

are called Xc-unbiased [139]. For a vanishing bias gradient, and for a K-sparse target

vector, x, the constraint CRB takes the form [139] Cov(ˆx)  Uf U>fI(x)Uf

−1

U>_f = UfI−1K (x)U >

f , kxk0 = K, (5.5)

where I(x) is the Fisher Information matrix (FIM), IK(x) = U>fI(x)Uf is the K-

reduced FIM, and Uf contains the directions corresponding to the non-zero entries in

x0 at the indices in S. A detailed derivation of I(x) can be found in Appendix A.2.

According to [139, 140], the FIM and the K-reduced FIM are given by I(x) = 1 σ2 n B>B, and IK(x) = 1 σ2 n B>_SBS, (5.6)

where σ_n2 is the noise variance, B is the combined sensing matrix in (5.2), and BS is

a sub-matrix of B that is composed of the columns with indices in S. The dictionary parameters, θ, are omitted, since they are assumed to be fixed in this context.

In the considered application of FBG-based fiber-optic sensing, it is assumed that the sparsity level, K, is known. This is a valid assumption in the absence of noise, since K is the number of reflections, i.e. the number of FBGs. When this assumption is relaxed and the signal is allowed to be s-sparse with 0 ≤ s ≤ K, then the constrained CRB is the same as in the unconstrained case [139]. This result is related to the fact that inequality constraints do not alter the value of the CRB, as stated in [141].

Some properties of the constrained CRB are stated in [139] and summarized below: (i) The constrained CRB can be lower than the unconstrained version, where the

bias gradient has to vanish for all possible directions. The latter is a stronger requirement, imposing additional restrictions on the class of estimators.

(ii) A requirement for the existence of (5.5) is that the range space R(UfU>f)

has to be a subset of R(UfU>fI(x)UfU>f). When spark(B) > 2K, then B

has unique reconstruction properties for any K-sparse vector and U>_fI(x)Uf

is invertible [9, 139]. Hence, the range spaces are equal and the requirement is fulfilled.

As stated in [139], the best achievable performance of Xc-unbiased estimators is that

of the oracle estimator, which has perfect knowledge of the support. Therefore, this estimator would asymptotically achieve the variance proposed by the constrained CRB.

5.4.1.1 The constrained CRB for orthonormal sub-matrices

According to (5.5)-(5.6), the constrained CRB yields a lower bound for the MSE, given by [139]

MSE(ˆ_{x, x) = E kx − ˆ}xk2₂ ≥ σ2

n Tr ((B >

SBS)−1), kxk0 = K, (5.7)

where ‘Tr’ is the trace-operator. When the columns of BS are orthogonal, the

constrained CRB takes the smallest value.

The proof of this statement is taken from [30]5 _{and given below.}

5_{C. Weiss and A. M. Zoubir, “A Compressed Sampling and Dictionary Learning Framework for}

Wavelength-Division-Multiplexing-Based Distributed Fiber Sensing,” accepted for publication in Jour- nal of the Optical Society of America A, 2017 (assigned issue: vol. 34, no. 5).

5.4 Theoretical Performance Limits: The Cram´er-Rao Bound 67

Proof:

The eigenvalues of the positive semi-definite matrix H = B>_SBS are denoted by

λk ≥ 0, k = 1, . . . , K. Then, using ‘Hadamard’s inequality’ (c.f. [142]),

det(H) = K Y k=1 λk ≤ K Y k=1 hkk = K Y k=1 e λk= det( diag(eλ1, . . . , eλK) ), (5.8)

where hkk are the diagonal entries of H and eλk are the eigenvalues of some diagonal

matrix, diag(eλ1, . . . , eλK). The inverse of the matrix H fulfills the equation

det(H−1) = K Y k=1 1 λk ≥ K Y k=1 1 e λk = det( [diag(eλ1, . . . , eλK)]−1) . (5.9)

Therefore, a lower bound is obtained when H is a diagonal, such that equality holds in (5.8) and (5.9). This happens when the columns of B are orthogonal. _ Usually, the columns of B are normalized, i.e. kbkk22 = eλk = 1. Then, a lower bound

for the MSE is found by

MSE(ˆx, x) ≥ σ_n2 K X k=1 1 e λk = K σ2_n, kxk0 = K. (5.10)