Problems of Information Transmission, Vol. 39, No. 4, 2003, pp. 324–340. Translated from Problemy Peredachi Informatsii, No. 4, 2003, pp. 10–29.
Original Russian Text Copyright
2003 by Prelov, van der Meulen.
INFORMATION THEORY AND CODING THEORY
Higher-Order Asymptotics of Mutual Information
for Nonlinear Channels with Non-Gaussian Noise
V. V. Prelov
and E. C. van der Meulen
Institute for Information Transmission Problems, RAS, Moscow
Katholieke Universiteit Leuven, Belgium
Received October 24, 2002
Abstract—Nonlinear channels with non-Gaussian noise where the transmitted signal is a
random function of the input signal are considered. Under some assumptions on smoothness
and the behavior of tails of the noise density function, higher-order asymptotics of the mutual
information between the input and output signals in such channels is obtained, as the mean
power of the input signal (or, equivalently, the signal-to-noise ratio) goes to zero.
Consider the model of a nonlinear memoryless communication channel in which the output
signal Y equals the sum
Y = εf(X)+Z (1)
of the transmitted signal εf (X), which is a random function of the input signal X, and noise Z.
The parameter ε>0 characterizes the signal-to-noise ratio in channel (1). We assume that X,
f(X), and Z are real-valued random variables. Moreover, we always assume that the pair (X, f(X))
and Z are mutually independent.
In particular, if f(X)=ϕ(X, U), where ϕ(· , ·) is a nonrandom function and U is a real-valued
random variable independent of X and Z, then (1) reduces to the model
Y = εϕ(X, U)+Z (2)
of a channel with random parameter U. For the special cases where ϕ(X, U)=UX or ϕ(X,U)=
X + U, we obtain the models
Y = εU X + Z (3)
Y = εX + Z + εU, (4)
which can be considered as a one-dimensional real-case fading channel model and a channel model
with additional (contaminating) weak noise εU respectively. Note also that the simplest channel
model with additive noise and weak input signal
Y = εX + Z (5)
is a special case of channel model (1) with f(X)=X. It is important to note that, in all the
channel models above, we do not assume that the noise Z is Gaussian. We only assume that Z has
Supported in part by the Russian Foundation for Basic Research, project nos. 03–01–00592 and 03-01-
00098; INTAS, Grant no. 00-738; and Project GOA/98/06 of the Research Fund K. U. Leuven.
2003 MAIK “Nauka/Interperiodica”