The demand of new wireless communication systems with much higher data
rates that allow, e.g., mobile wireless broadband Internet
connections inspires a quick advance in wireless transmission
technology. So far most systems rely on an approach where the channel
state is measured with the help of regularly transmitted training
sequences. The detection of the transmitted data is then done under
the assumption of perfect knowledge of the channel state. This
approach will not be sufficient anymore for very high data rate
systems since the loss of bandwidth due to the training sequences is
too large. Therefore, the research interest on joint estimation and
detection schemes has been increased considerably.

Apart from potentially higher data rates a further advantage of such a
system is that it allows for a fair analysis of the theoretical upper
limit, the so-called channel capacity. "Fair" is used here in
the sense that the capacity analysis does not ignore the estimation
part of the system, i.e., it takes into account the need of the
receiver to gain some knowledge about the channel state without
restricting it to assume some particular form (particularly, this
approach does also include the approach with training sequences!). The
capacity of such a joint estimation and detection scheme is often also
known as non-coherent capacity.

Recent studies investigating the non-coherent capacity of fading
channels have shown very unexpected results. In stark contrast to the
capacity with perfect channel knowledge at the receiver, it has been
shown that non-coherent fading channels become very power-inefficient
at high signal-to-noise ratios (SNR) in the sense that increasing the
transmission rate by an additional bit requires squaring the
necessary SNR (or doubling the SNR on a dB-scale)! Here,
depending on the channel in use, "high SNR" typically starts somewhere
between 30 to 80 dB. Since transmission in such a regime will be
highly inefficient, it is crucial to avoid a system operating at such
a rate. Hence we need to better understand this behavior and to be
able to give an estimation as to where the inefficient regime
starts. One parameter that provides a good approximation to such a
threshold between the power-efficient low-SNR and the
power-inefficient high-SNR regime is the so-called fading
which is defined as the second term in the high-SNR
asymptotic expansion of channel capacity.

The results of this report concern this fading number. We investigate
a channel model based on a flat fading assumption without inter-symbol
interference, which is a typical situation encountered when using a
system based on orthogonal frequency division multiplexing (OFDM) or
on orthogonal frequency division multiple-access (OFDMA). We assume
several users at the transmitter side, but only one receiver (typical
setup of a several mobile users communicating with one base-station),
where all users and the receiver might have multiple antennas
available. In the most general setup we do not restrict the fading to
have a particular distribution (i.e., it need not be Gaussian),
but we only ask it to be a stationary, ergodic, finite-energy, and
regular random process, possibly with memory both over time and
space. For the sake of simplicity, however, we will introduce further
restrictions or simplifying assumptions on the way.

The results can be grouped into two main chapters: firstly we
investigate a single-user setup where we allow multiple-antennas both
at transmitter and receiver. The fading is assumed to have no temporal
memory, but the different antennas are allowed to have arbitrary
correlation. The distribution of the fading is not restricted to be
Gaussian, and not specified apart from the stationarity, ergodicity
and regularity assumptions. In this setup we are able to derive the
fading number precisely, i.e., we are able to specify the exact
asymptotic channel capacity in the limit when the available power
tends to infinity, and we can give a good estimation of the threshold
between the efficient low- to medium-SNR regime and the highly
power-inefficient high-SNR regime.

This result is then specialized to the already known cases of
single-input multiple-output (SIMO), multiple-input single-output
(MISO), and single-input single-output (SISO) fading channels, as well
as to the situation of Gaussian fading. As a byproduct a new upper
bound is derived on g_m(.), the expected value of a logarithm of a
non-central chi-square random variable.

In a second chapter we investigate a simple two-user single-antenna
setup assuming no memory and Gaussian fading. We prove that under the
additional constraint that the users use circularly symmetric
signaling, the sum-rate capacity of this multiple-access channel
equals to the single-user capacity of the user with the better


Channel capacity, fading, fading number, flat fading channel, Gaussian
fading, general fading, high signal-to-noise ratio, high SNR, joint
estimation and detection, MAC, memory, MIMO, multiple-access channel,
multiple-antenna, multiple-input multiple-output, multiple-user,
non-central chi-square, non-coherent.

-||-   _|_ _|_     /    __|__   Stefan M. Moser
[-]     --__|__   /__\    /__   Senior Researcher & Lecturer, ETH Zurich, Switzerland
_|_     -- --|-    _     /  /   Adj. Professor, National Chiao Tung University (NCTU), Taiwan
/ \     []  \|    |_|   / \/    Web: http://moser-isi.ethz.ch/

Last modified: Fri Oct 27 08:47:07 2006