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ETH Series in Information Theory and its Applications,
Vol. 5
edited by Amos Lapidoth
Tobias Koch,
On Heating Up and Fading in
Communication Channels.
1st edition 2009. XVIII, 206 pages, € 64,00.
ISBN 3-86628-261-3, 978-3-86628-261-2
Abstract
This dissertation studies two
phenomena that affect the transmission of data: heating up and fading.
In particular, the effect of these phenomena on channel capacity, which is the
largest rate at which data transmission with arbitrarily lower error probability
is possible, is investigated.
Heating
up is relevant in on-chip communication, where multiple terminals that are
located on the same microchip wish to communicate with each other. It accounts
for thermal coupling of data and noise. Indeed, the data to be transmitted are
corrupted by thermal noise, whose variance depends on the local temperature of
the chip. Furthermore, the transmission of data is associated with dissipation
of energy into heat and raises therefore the local temperature of the chip. This
gives rise to a channel model where the variance of the additive noise is datadependent. The capacity of this
channel is studied at low and at high transmit powers. At low transmit
powers, the slope of the capacity-vspower curve at
zero is computed, and it is shown that the heating-up effect is beneficial. At
high transmit powers, it is demonstrated that the heating-up effect is
detrimental. In fact, if the heat dissipates slowly then the capacity is
bounded in the transmit power, i.e., the capacity does not tend to infinity as
the allowed average power tends to infinity. A sufficient condition and a
necessary condition for the capacity to be bounded is
derived.
The
results of the above analyses suggest that at low transmit powers heat sinks
are not only unnecessary, but they even reduce the capacity by dissipating
heat, which contains information about the transmitted signal. The results
further accentuate the importance of an efficient heat sink at large transmit
powers.
Fading
occurs in wireless communication channels. In such channels the transmitted
signal is not only corrupted by additive noise, but also by multiplicative
noise, which accounts for the variation of the signal’s attenuation. This
multiplicative noise is referred to as fading. In contrast to many other
information-theoretic studies, where it is assumed that the receiver has
perfect knowledge of the fading, in this dissertation it is assumed that the
transmitter and the receiver only know the statistics of the fading but not its
realization.
First,
the capacity of multiple-input multiple-output (MIMO) Gaussian flat-fading
channels with memory is considered. Nonasymptotic
upper and lower bounds on the capacity are derived, and their asymptotic behavior is analyzed in the limit
as the signal-to-noise ratio (SNR) tends to infinity. In particular, upper
bounds on the fading number (which is defined as the second-order term in the
high-SNR expansion of capacity) and on the capacity pre-log (which is defined
as the limiting ratio of capacity to log SNR as SNR tends to infinity) are
computed. Furthermore, an approach to derive lower bounds on the fading number
is proposed. This lower bound is applied to derive a lower bound on the fading
number of spatially IID, zero-mean, MIMO Gaussian fading channels with memory.
The derived upper and lower bounds on the fading number demonstrate that when
the number of receive antennas does not exceed the number of transmit antennas,
the fading number of spatially IID, zero-mean, slowly-varying, Gaussian fading
channels is proportional to the number of degrees of freedom, i.e., to the
minimum of the number of transmit and receive antennas.
Second,
the capacity pre-log of single-input single-output (SISO) flatfading
channels with memory is studied. It is shown that, among all stationary and
ergodic fading processes of a given spectral distribution function and whose
law has no mass point at zero, the Gaussian process gives rise to the smallest
pre-log. It is further demonstrated that the assumption that the fading law has
no mass point at zero is essential in the sense that there exist stationary and
ergodic fading processes of some spectral distribution function (and whose law
has a mass point at zero) that give rise to a smaller pre-log than the Gaussian
process of equal spectral distribution function. These results are then
extended to multiple-input single-output (MISO) fading channels with memory.
Finally,
the capacity of multipath (frequency-selective) fading channels is studied. It
is shown that if the delay spread is large in the sense that the variances of
the path gains decay exponentially or slower, then the capacity is bounded in
the SNR. Thus, in this case the capacity does not grow to infinity as the SNR
tends to infinity. In contrast, if the variances of the path gains decay faster
than exponentially, then the capacity is unbounded in the SNR. It is further
demonstrated that if the number of paths is finite, then the capacity pre-loglog, which is defined as the limiting ratio of capacity
to log log SNR as SNR tends to infinity, is 1,
irrespective of the number of paths.
The
conclusions that can be drawn from the above described analyses of fading
channels are manifold. First, the presence of multiple antennas at the
transmitter and receiver is very beneficial, even if the receiver does not know
the realization of the fading. Second, the Gaussian fading assumption in the
analysis of fading channels at high SNR is conservative in the sense that for a
large class of fading
processes the Gaussian process gives rise to the smallest
capacity pre-log. Third, at high SNR multipath fading channels with an infinite
number of paths should
not be approximated by multipath fading channels with a finite number of paths,
since these channels possess completely different high-SNR capacity behaviors. And last but not least, the high-SNR asymptotic behavior of the capacity of fading channels is very
sensitive to the employed channel model. Thus, in the information-theoretic
analysis of fading channels at high SNR and in the evaluation of the results
thereof, one should attach great importance to the channel model. .
Keywords: Information theory, channel
capacity, capacity per unit, cost, channels with memory, high signal-to-noise
ratio, on-chip communication, wireless communication, .at-fading channels,
multipath fading channels.
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