Abstract
This dissertation studies a new family of channel models for noncoherent com
munications, the additive energy channels. By construction, the additive en
ergy channels occupy an intermediate region between two widely used channel
models: the discretetime Gaussian channel, used to represent coherent com
munication systems operating at radio and microwave frequencies, and the
discretetime Poisson channel, which often appears in the analysis of intensity
modulated systems working at optical frequencies. The additive energy chan
nels share with the Gaussian channel the additivity between a useful signal and
a noise component. However, the signal and noise components are not complex
valued quadrature amplitudes but, as in the Poisson channel, nonnegative real
numbers, the energy or squared modulus of the complex amplitude.
The additive energy channels come in two variants, depending on whether
the channel output is discrete or continuous. In the former case, the energy is a
multiple of a fundamental unit, the quantum of energy, whereas in the second
the value of the energy can take on any nonnegative real number. For con
tinuous output the additive noise has an exponential density, as for the energy
of a sample of complex Gaussian noise. For discrete, or quantized, energy the
signal component is randomly distributed according to a Poisson distribution
whose mean is the signal energy of the corresponding Gaussian channel; part
of the total noise at the channel output is thus a signaldependent, Poisson
noise component. Moreover, the additive noise has a geometric distribution,
the discrete counterpart of the exponential density.
Contrary to the common engineering wisdom that not using the quadrature
amplitude incurs in a signi¯cant performance penalty, it is shown in this dis
sertation that the capacity of the additive energy channels essentially coincides
with that of a coherent Gaussian model under a broad set of circumstances.
Moreover, common modulation and coding techniques for the Gaussian chan
nel often admit a natural extension to the additive energy channels, and their
performance frequently parallels those of the Gaussian channel methods.
Four informationtheoretic quantities, covering both theoretical and practi
cal aspects of the reliable transmission of information, are studied: the channel
capacity, the minimum energy per bit, the constrained capacity when a given
digital modulation format is used, and the pairwise error probability. Of these
quantities, the channel capacity sets a fundamental limit on the transmission
capabilities of the channel but is sometimes di±cult to determine. The min
imum energy per bit (or its inverse, the capacity per unit cost), on the other
hand, turns out to be easier to determine, and may be used to analyze the
performance of systems operating at low levels of signal energy. Closer to
a practical ¯gure of merit is the constrained capacity, which estimates the
largest amount of information which can be transmitted by using a speci¯c
digital modulation format. Its study is complemented by the computation of
the pairwise error probability, an e®ective tool to estimate the performance of
practical coded communication systems.
Regarding the channel capacity, the capacity of the continuous additive
energy channel is found to coincide with that of a Gaussian channel with iden
tical signaltonoise ratio. Also, an upper bound the tightest known to
the capacity of the discretetime Poisson channel is derived. The capacity of
the quantized additive energy channel is shown to have two distinct functional
forms: if additive noise is dominant, the capacity is close to that of the continu
ous channel with the same energy and noise levels; when Poisson noise prevails,
the capacity is similar to that of a discretetime Poisson channel, with no ad
ditive noise. An analogy with radiation channels of an arbitrary frequency, for
which the quanta of energy are photons, is presented. Additive noise is found
to be dominant when frequency is low and, simultaneously, the signaltonoise
ratio lies below a threshold; the value of this threshold is well approximated
by the expected number of quanta of additive noise.
As for the minimum energy per nat (1 nat is log2 e bits, or about 1.4427 bits),
it equals the average energy of the additive noise component for all the stud
ied channel models. A similar result was previously known to hold for two
particular cases, namely the discretetime Gaussian and Poisson channels.
An extension of digital modulation methods from the Gaussian channels
to the additive energy channel is presented, and their constrained capacity
determined. Special attention is paid to their asymptotic form of the capacity
at low and high levels of signal energy. In contrast to the behaviour in the
vi
Gaussian channel, arbitrary modulation formats do not achieve the minimum
energy per bit at low signal energy. Analytic expressions for the constrained
capacity at low signal energy levels are provided. In the highenergy limit
simple pulseenergy modulations, which achieve a larger constrained capacity
than their counterparts for the Gaussian channel, are presented.
As a ¯nal element, the error probability of binary channel codes in the ad
ditive energy channels is studied by analyzing the pairwise error probability,
the probability of wrong decision between two alternative binary codewords.
Saddlepoint approximations to the pairwise error probability are given, both
for binary modulation and for bitinterleaved coded modulation, a simple and
e±cient method to use binary codes with nonbinary modulations. The meth
ods yield new simple approximations to the error probability in the fading
Gaussian channel. The error rates in the continuous additive energy channel
are close to those of coherent transmission at identical signaltonoise ratio.
Constellations minimizing the pairwise error probability in the additive energy
channels are presented, and their form compared to that of the constellations
which maximize the constrained capacity at high signal energy levels.
Original language  English 

Qualification  Doctor of Philosophy 
Awarding Institution 

Supervisors/Advisors 

Award date  28 Jan 2008 
Place of Publication  Eindhoven 
Publisher  
Print ISBNs  9789038617541 
DOIs  
Publication status  Published  2008 