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. Since transmission in such a regime will be highly inefficient,
it is crucial 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 border between
the power-efficient low-SNR and the power-inefficient high-SNR regime
is the so-called fading number 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 restrict
ourselves to fading channels with multiple antennas at the
transmitter, but only one antenna at the receiver (a multiple-input
single-output (MISO) situation), however, we do allow memory.
Furthermore, the fading laws are not restricted to be Gaussian, but is
assumed to be a general regular law with spatial and temporal
memory. The main result of this report are a new upper bound and a new
lower bound on the fading number of this MISO fading channel with
memory. It can be seen as a further step towards the final goal of the
fading number of general multiple-inputs multiple-outputs (MIMO)
fading channels with memory.

In case of an isotropically distributed fading vector it is proven
that the upper and lower bound coincide, i.e., the general MISO
fading number with memory is known precisely.

The upper and lower bounds show that a type of beam-forming is
asymptotically optimal.


Beam-forming, channel capacity, fading, fading number, flat fading
channel, high SNR, joint estimation and detection, memory, MISO,
multiple-antenna, non-coherent detection.

-||-   _|_ _|_     /    __|__   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