Signal Processing - November 2017 - 154
expressions can be found in [37] for the case where both the
true and the assumed conditional distributions are complex
Gaussian, for example. The resulting lower bound on the MSE
follows from (24) and is given by
MSE p X, ^it (x)h $ 1 E p " N T (i) , E -p 1 " J (i) , E p " N (i) ,
M
(26)
+ E p " rr T , .
i
i
i
i
i
The class of estimators to which the above MBCRB applies
is that which has a mean and an estimator-score function correlation that, respectively, satisfy the following constraints:
E p X, " it (x) , = E p " n (i) ,,
E p X, " h (x, i) [it (x) - n (i)] T , = E p " N (i) , .
i
i
i
i
(27)
These constraints follow from the covariance inequality
[37, Sect. III-C] and the choice of score function. This limits
the applicability of the bound in contrast to bounds obtained
when the model is perfectly specified. Thus, an obvious area
of future effort is the development of Bayesian bounds under
misspecified models with fewer constraints and broader applicability. To conclude, we note that an example demonstrating
the applicability of this Bayesian bound to a DOA estimation
for sparse arrays is given in [22].
F = " fX fX (x s, i)
= N ^sv (i), I N + v 2j v (i j) v (i j) H h, 6s ! C, i ! [0, 2r) , .
(28)
Examples of applications
In this section, we describe some examples related to the
problems of DOA estimation and data covariance/scatter
matrix estimation. These problems are relevant in many array
processing and adaptive radar applications.
DOA estimation under model misspecification
The estimation of the DOAs of plane-wave signals by means of
an array of sensors has been the core research area within the SP
array community for years [42]. The fundamental prerequisite for
any DOA estimation algorithm is that the positions of the sensors
in the array are known exactly, i.e., known geometry. Many
authors have investigated the impact of imperfect knowledge of
the sensor positions on the DOA estimation performance or of
the miscalibration of the array itself (see, e.g., [15] and [42], just
to name two). Other authors have proposed hybrid or modified
CRBs with the aim to predict the MSE of the DOA estimators in
the presence of the position uncertainties [31], [39]. The goal of
this section is to show that the misspecified estimation framework presented in this article is a valuable and general tool to
deal with modeling errors in the array manifold. The application
of the MCRB and the MML estimator to the DOA estimation
problem has recently been investigated in [35] for MIMO radar
systems and in [37] for uniform linear arrays (ULAs).
Following [37], consider a ULA of N sensors and a single
plane-wave signal impinging on the array from a conic angle
ir. Moreover, suppose that, due to an array misscalibration, the
true position vector pn of the nth sensor, defined in a threedimensional Cartesian coordinate frame, is known up to an
error term modeled as a zero-mean, Gaussian random vector,
154
i.e., e n + N (0, v 2e I 3). Then, the received data can be expressed
as x n = s [d (ir)] n + [c] n, where [d (ir)] n = exp ( jk Tir (p n + e n))
is the nth element of the true (perturbed) steering vector
and k ir = (2r m) u (ir), where u (ir) is a unit vector pointing
at the direction defined by ir and m is the wavelength of the
-transmitted signal. Moreover, s is an unknown deterministic
complex scalar that accounts for the transmitted power, the
source scattering characteristics, and the two-way path loss,
while c = n + j is the disturbance noise term composed of
white Gaussian noise n and of interference signal (or jammer) j.
Given particular realizations of the position errors en, the clutter vector is usually modeled as a zero-mean complex Gaussian
random vector c + N ^0, I N + v 2j d (i j) d (i j) H h, where v 2j and
i j represent the power and the DOA of the jamming signal.
The DOA estimation problem is clearly the estimation of ir,
given the complex data vector x. Since, in practice, it is impossible to be aware of the particular realizations of the position
error vectors en, the user may decide to derive a DOA estimator
starting from the nominal steering vector v (ir), whose components are [v (ir)] n = exp (jk Tir p n), i.e., the user neglects the sensor position errors. The true (unknown) data model is given by
the pdf p X (x) = N ^sr d (ir), I N + v 2j d (i j) d (i j) H h ! P, while
the assumed parametric model is
The true pdf p X (x) does not belong to F ; in other words, the
assumed parametric pdf fX (x | s, i) differs from p X (x) for every
value of i ! [0, 2r) . This is because, even if both the true and
the assumed pdfs are complex Gaussian, by neglecting the position errors in the assumed steering vector, we are choosing the
wrong parameterization for the mean value and the covariance
matrix of the assumed Gaussian model. Therefore, how large is
the performance loss due to this model mismatch? The MCRB
presented in the -section "The Misspecified CRB" answers this
question. We omit the details of the calculation of the MCRB
and the derivation of the joint MML estimator of the DOA and
of the scalar s, and we refer readers to [37]. However, to provide
some insights about this mismatched estimation problem, Figure 1
illustrates the matched CRB in the estimation of ir, i.e., the CRB
on the DOA estimation evaluated by considering the true data
pdf p X (x), the MCRB, and the MSE of the MML estimator
obtained from the assumed and misspecified pdf fX (x | s, i) .
Figure 1 plots the square roots of the bounds and of the MSE
(RMSE) in units of beamwidths as a function of element-level
SNR. The MCRB accurately predicts the performance of the
MML estimator. If the system goal is a ten-to-one beamsplit
ratio, i.e., -10 dB RMSE in beamwidths, then this could be
accomplished with an SNR of 9.28 dB when the model is perfectly known, but not precisely knowing the true sensor positions requires an additional ~10 dB of SNR to achieve the same
goal ( MCRB , -10 dB for SNR , 19.4 dB). However, if the
system receives an SNR , 9.3 dB, then the minimum achievable beamsplit ratio in the presence of array errors is three to
one, i.e., the MCRB , -5 dB RMSE in beamwidths. This
IEEE SIGNAL PROCESSING MAGAZINE
|
November 2017
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Table of Contents for the Digital Edition of Signal Processing - November 2017
Signal Processing - November 2017 - Cover1
Signal Processing - November 2017 - Cover2
Signal Processing - November 2017 - 1
Signal Processing - November 2017 - 2
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Signal Processing - November 2017 - Cover3
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