**Author:** Yi Sun, University of Minnesota

**Advisor:** Prof. John Kieffer

**Co-advisor:** Prof. Laurie Nelson

**Date:** Jan. 1997

**Abstract**

This thesis consists of two parts. In the first part, the stochastic iteration
approach is proposed for signal set design. A fundamental stochastic iterative
algorithm without energy constraint is proposed, which models a stochastic
dynamic system using the detection probability of a signal set as energy
function. Based on this fundamental algorithm, four practical stochastic
iterative algorithms are proposed with respect to two energy constraints
(the average energy constraint and the equal energy constraint) and two
operation modes (sequential mode and batch mode). To study the performance
of these proposed algorithms, many simulation experiments are carried out.
Among these simulation experiments are the studies of almost all existing
theoretical results, which include the optimality of the L1 signal set (consisting
of a pair of antipodal signals and some zero signals), the optimality of
the L2 signal set (consisting of three signals located at the vertices of
an equilateral triangle and some zero signals), the truth of the weak simplex
conjecture and two of Dunbridge’s theorems. The L2 signal set is newly discovered
by the simulation and its optimality is shown in this thesis. Given the
same conditions as those in the above theoretical results, the proposed
algorithms always converge to the signal set proved optimal in theory. The
influence of SNR and a priori probabilities on signal set is investigated
via simulation. As an example of application of the proposed algorithms
to practical communication system design, in the scenario of satellite communications
in which SNR is very low, the signal sets of eight 2-D signals are studied
by simulation. Two signal sets better than the practically used 8-PSK set
are found in the SNR range of practical satellite communications. All simulation
results show promise of the proposed algorithms.

In the second part of this thesis, optimal properties of the L2 signal set
are analyzed in the SSC condition (equal a priori probability and average
energy constraint) at low signal-to-noise ratios. We first discuss the properties
of the mean width of the polytope generated by a signal set. Then two classes
of signal sets are analyzed. The first to be analyzed is the class of 2-D
signal sets E(M,K) (consisting of K signals equally spaced on a circle and
M-K zero signals). The L2 signal set is proved to be unique optimal in the
class of signal sets E(M,K) and further proved to be unique optimal in 2-D
space. The class of signal sets S(M,K) (consisting of a regular simplex
set of K signals and M-K zero signals) then is analyzed. It is shown that
the strong simplex conjecture for M>=4 is disproved by any of the signal
sets S(M,K) for 3 <= K <= M-1 <=K<=M-1 and is also disproved by S(M,2) (i.e., the L1 signal set) if M>=7. It is proved that the L2 signal set is the unique optimal signal set
in this class of signal sets S(M,K) for all M>=4. This disproves the strong
simplex conjecture for all M>=4 and also leads to the extension of the following
results obtained by Steiner for all M>=7 to all M>=4: (1) there is no signal
set which is optimal at all signal-to-noise ratios; (2) with average energy
constraint, the optimal solution as SNR approaches zero is not an equal
energy solution. Several other results are also obtained. Finally, we show
that for M>=7, there exists an integer K'<=M-1 such that any of the signal sets E(M,K) for
4<=K<=K' disproves the strong simplex conjecture.

In this thesis, we found that many signal sets can disprove the strong
simplex conjecture for M>=4, although the strong simplex conjecture is long-standing
and was not disproved for many years.

Yi Sun

Department of Electrical Engineering

University of Minnesota

200 Union Street, SE

Minneapolis, MN 55455

Phone: (612)626-1769 (W) (612)645-3743 (H)

Fax: (612)625-4583

E-mail: yisun@ee.umn.edu