a la physique de l'information - Lisa - Université d'Angers

Raising the noise to improve performance in optimal processing
observable signal x(t) =w[νt + ξ(t)+η(t)], with the augmented noise ξ(t)+η(t) which
remains Gaussian, as in figure 4, while its rms amplitude increases. In practice, for the
periodic wave traveling through a fluctuating medium, as evoked just after equation (18),
the initial phase noise ξ(t) in equation (18) is due to random fluctuations in the
propagating medium. Then, a receiving sensor subjected to random vibrations according
to the additional noise η(t) will produce the observable signal x(t) =w[νt + ξ(t)+η(t)].
In outline, optimal estimation on a periodic wave traveling through a fluctuating medium
could be improved by randomly shaking the receiver in an appropriate way.
The results of this section 3.2 demonstrate, with an example, the possibility of
improving the performance of an optimal estimator when the noise level increases.
Section 3.2 addresses the estimation of a deterministic unknown parameter ν and it gives
new results on noise-aided optimal estimation. A comparable property of improvement by
noise in optimal estimation was demonstrated with another example in [31]. Reference [31]
addresses estimation of a stochastic unknown parameter ν, with a classic Bayesian
estimator minimizing the rms error of equation (13), when the expectation E(·) in
equation (13) is according to the probabilization established in conjunction by the noise
ξ(t) and the prior probability on ν.
4. The basic mechanism
Sections 2.3 and 3.2, aswellas[27]–[29], [31], report examples of improvement by noise
in optimal processing. We are dealing here with optimal processing (optimal detection
and optimal estimation) in a classical sense, as defined by classical optimal detection
and estimation theories [23, 24, 30]. The measures of performance which are analyzed are
standard measures for detection and estimation, i.e. the probability of detection error
Per defined in equation (1) and the rms estimation error E of equation (13). These
quantities are measures commonly used for performance evaluation in the classical theories
of statistical detection and estimation, and they represent the performance that is reached
in practice through ensemble average over a large number of realizations of the random
signal x(t) which is optimally processed. The results of figures 2 and 4 are ensemble
averages in this sense, and they demonstrate in concrete examples the possibility of
improvement by noise of the performance of optimal processors. In practice, as we briefly
indicated, the example of figure 2 could be relevant, for instance, to the detection of signals
in bimodal ‘logical’ noise, while the example of figure 4 could be relevant, for instance,
to the estimation on periodic waves traveling through fluctuating media. However, our
main purpose here is not to argue about the practical usefulness of these examples, but
rather we want to focus on their meaning in principle. These examples stand as proofs of
feasibility in principle that it is possible to improve the performance of optimal processors
by increasing the noise. It may then seem paradoxical that optimal processors, in the
classical sense, can be improved by raising the noise. If they are truly optimal, how can
they be improved?
The point is that these processors are optimal in the sense that they represent the
best possible deterministic processing that can be done on the data x to optimize a fixed
given measure of performance Q (for instance, Per of equation (1) orE of equation (13)).
In the classical theory of these optimal processors, the performance Q is a functional of the
probabilization established by the initial noise ξ(t). This probabilization is expressed by
doi:10.1088/1742-5468/2009/01/P01003 11
125/197
J. Stat. Mech. (2009) P01003