Signal Processing - November 2017 - 143

©ISTOCKPHOTO.COM/MIAKIEVY

bound is often a prerequisite for any feasibility study. The
availability of a lower bound for the estimation problem at
hand makes the SP practitioner aware of the practical impossibility to achieve better estimation accuracy than the one indicated by the bound itself. Another fundamental feature of a
performance bound is its ability to capture and reveal the complex dependences among the various parameters of interest,
thus offering the opportunity to more deeply understand the
estimation problem at hand and, ultimately, to identify an
appropriate design choice of parameters and criteria for an
estimator [23].
Before describing specific performance bounds, it is worth
mentioning that estimation theory explores two different
frameworks: one is deterministic and one is Bayesian. In the
classical deterministic approach, the parameters to be estimated are modeled as deterministic but unknown variables. This
implies that no a priori information is available that would
suggest that one outcome is more or less likely than another.
In the Bayesian framework, the parameters of interest are
assumed to be random variables, and the goal is to estimate
their particular realizations. Unlike the classical deterministic approach, the Bayesian approach exploits this random
characterization of the unknown parameters by incorporating a priori information about the unknown parameters in the
derivation of an estimation algorithm. The joint pdf of the
unknown parameters is assumed to be known and, therefore,
can be taken into account in the estimation process through
Bayes' theorem [23].

Basics about performance bounds
When discussing lower bounds, the first distinction that needs
to be made is between local (small-error) bounds and global
(large-error) bounds. A bound can be considered a local-error
bound if its calculation relies exclusively on the behavior of
the pdf of the data at a single point value of the parameter (or
perhaps a very small local neighborhood around this point). If
the calculation of a bound requires knowledge of the pdf
behavior at multiple (more than one) distinct and well-separated (nonlocal) points, then the bound can be characterized as a
global-error bound. Local-error bounds at best determine the
limits of the asymptotics of optimal algorithms like ML,
whereas the characterization of nonasymptotic performance
must somehow take into account the possible influence of
parameter values other than the true value.
A bound is said to be tight if it reasonably predicts the
performance of the ML estimator. If a bound is only asymptotically tight, then it is reliable only in the presence of a high
signal-to-noise ratio (SNR) or a sufficiently large number of
measurements. However, if a bound is globally tight, then
it is a reliable bound for the error covariance of an ML estimator, irrespective of the SNR level or of the amount of available data. The deterministic bound that can be regarded as the
most g- eneral representative of the class of global bounds is the
Barankin bound (BB) [3]. However, due to its generality, the
calculation of the BB is not straightforward, and it usually does
not admit a closed-form representation. The most popular local
bound is the aforementioned CRB. Unlike the BB, the CRB

IEEE SIGNAL PROCESSING MAGAZINE

November 2017

143

http://www.ISTOCKPHOTO.COM/MIAKIEVY

Table of Contents for the Digital Edition of Signal Processing - November 2017