## Optimal stochastic approaches for signal detection and estimation under inequality constraints

##### Author(s)

##### Advisor

##### Date

2012##### Publisher

Bilkent University

##### Language

English

##### Type

Thesis##### Item Usage Stats

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##### Abstract

Fundamental to the study of signal detection and estimation is the design of
optimal procedures that operate on the noisy observations of some random phenomenon.
For detection problems, the aim is to decide among a number of
statistical hypotheses, whereas estimating certain parameters of the statistical
model is required in estimation problems. In both cases, the solution depends
on some goodness criterion by which detection (or estimation) performance is
measured. Despite being a well-established field, the advances over the last several
decades in hardware and digital signal processing have fostered a renewed
interest in designing optimal procedures that take more into account the practical
considerations. For example, in the detection of binary-valued scalar signals
corrupted with additive noise, an analysis on the convexity properties of the error
probability with respect to the transmit signal power has suggested that the error
performance cannot be improved via signal power randomization/sharing under
an average transmit power constraint when the noise has a unimodal distribution
(such as the Gaussian distribution). On the contrary, it is demonstrated that performance
enhancement is possible in the case of multimodal noise distributions
and even under Gaussian noise for three or higher dimensional signal constellations.
Motivated by these results, in this dissertation we adopt a structured approach
built on concepts called stochastic signaling and detector randomization,
and devise optimal detection procedures for power constrained communications
systems operating over channels with arbitrary noise distributions.
First, we study the problem of jointly designing the transmitted signals, decision
rules, and detector randomization factors for an M-ary communications
system with multiple detectors at the receiver. For each detector employed at the
receiver, it is assumed that the transmitter can randomize its signal constellation
(i.e., transmitter can employ stochastic signaling) according to some probability
density function (PDF) under an average transmit power constraint. We show
that stochastic signaling without detector randomization cannot achieve a smaller
average probability of error than detector randomization with deterministic signaling
for the same average power constraint and noise statistics when optimal
maximum a-posteriori probability (MAP) detectors are employed in both cases.
Next, we prove that a randomization between at most two MAP detectors corresponding
to two deterministic signal vectors results in the optimal performance.
Sufficient conditions are also provided to conclude ahead of time whether the
correct decision performance can or cannot be improved by detector randomization.
In the literature, the discussions on the benefits of stochastic signaling and
detector randomization are severely limited to the Bayesian criterion. Therefore,
we study the convexity/concavity properties for the problem of detecting the
presence of a signal emitted from a power constrained transmitter in the presence
of additive Gaussian noise under the Neyman-Pearson (NP) framework.
First, it is proved that the detection probability corresponding to the α−level
likelihood ratio test (LRT) is either concave or has two inflection points such that
the function is concave, convex and finally concave with respect to increasing values
of the signal power. Based on this result, optimal and near-optimal power
sharing/randomization strategies are proposed for average and/or peak power
constrained transmitters. Using a similar approach, the convexity/concavity
properties of the detection probability are also investigated with respect to the
jammer power. The results indicate that a weak Gaussian jammer should employ
on-off time sharing to degrade the detection performance.
Next, the previous analysis for the NP criterion is generalized to channels with
arbitrary noise PDFs. Specifically, we address the problem of jointly designing
the signaling scheme and the decision rule so that the detection probability is
maximized under constraints on the average false alarm probability and average
transmit power. In the case of a single detector at the receiver, it is shown that
the optimal solution can be obtained by employing randomization between at
most two signal values for the on-signal and using the corresponding NP-type
LRT at the receiver. When multiple detectors are available at the receiver, the
optimal solution involves a randomization among no more than three NP decision
rules corresponding to three deterministic signal vectors.
Up to this point, we have focused on signal detection problems. In the following,
the trade-offs between parameter estimation accuracy and measurement
device cost are investigateed under the influence of noise. First, we seek to determine
the most favorable allocation of the total cost to measurement devices so
that the average Fisher information of the resulting measurements is maximized
for arbitrary observation and measurement statistics. Based on a recently proposed
measurement device cost model, we present a generic optimization problem
without assuming any specific estimator structure. Closed form expressions are
obtained in the case of Gaussian observations and measurement noise.
Finally, a more elaborate analysis of the relationship between parameter estimation
accuracy and measurement device cost is presented. More specifically,
novel convex measurement cost minimization problems are proposed based on
various estimation accuracy constraints assuming a linear system subject to additive
Gaussian noise for the deterministic parameter estimation problem. Robust
allocation of the total cost to measurement devices is also considered by assuming
a specific uncertainty model on the system matrix. Closed form solutions are
obtained in the case of an invertible system matrix for two estimation accuracy
criteria. Through numerical examples, various aspects of the proposed optimization
problems are compared. Lastly, the discussion is extended to the Bayesian
framework assuming that the estimated parameter is Gaussian distributed.

##### Keywords

DetectionStochastic Signaling

Detector Randomization

Probability of Error

Neyman-Pearson (NP)

Convexity

Gaussian Noise

Multimodal Noise

Power Constraint

Jamming

Parameter Estimation

Measurement Cost

CramerRao Bound (CRB)

Wireless Sensor Networks (WSN)