Theses: Communication Rates for Fading Channels with Imperfect Channel-State InformationPastore
An important specificity of wireless communication channels are the rapid fluctuations of propagation
coefficients. This effect is called fading and is caused by the motion of obstacles, scatterers
and reflectors standing along the different paths of electromagnetic wave propagation between
the transmitting and the receiving terminal. These changes in the geometry of the wireless
channel prompt the attenuation coefficients and the relative phase shifts between the multiple
propagation paths to vary. This suggests to model the channel coefficients (the transfer matrix)
as random variables.
The present thesis studies information rates for reliable transmission of information over fading
channels under the realistic assumption that the receiver has only imperfect knowledge of the
random fading state. While the over-idealized assumption of perfect channel-state information
at the receiver (CSIR) gives rise to many simple expressions and is fairly well understood, the
settings with imperfect CSIR or downright absence of CSIR are significantly more complex to
treat, and less is known about theoretical limits of communication in these circumstances.
Of particular interest are analytical expressions of achievable transmission rates under imperfect
and no CSI, that is, lower bounds on the mutual information and on the Shannon capacity.
A well-known mutual information lower bound for Gaussian codebooks is based on the notion
that the Gaussian distribution is the “worst-case” additive noise distribution in that it minimizes
the input-output mutual information. By conflating the additive noise (induced by thermal
noise in amplifiers at the receiver) with the multiplicative noise term due to the imperfections
of the CSIR into a single effective noise term, one can exploit the extremal property of the
Gaussian noise distribution to construct a “worse” channel by assuming that the effective noise
is Gaussian. This worst-case-noise approach allows to derive a strikingly simple lower bound
on the mutual information of the channel. This lower bound is well-known in literature and is
frequently used to provide simple expressions of achievable rates.
A first part of this thesis proposes a simple way to improve this worst-case-noise bound by
means of a rate-splitting approach: by expressing the Gaussian channel input as a sum of several
independent Gaussian inputs, and by assuming that the receiver performs successive decoding
of the corresponding information streams (as if multiple virtual users were transmitting over the
same physical link in a multiple-access fashion), we show how to derive a larger lower bound
on the channel’s mutual information. On channels with a single transmit antenna, the optimal
allocation of transmit power across the different inputs is found to be approached as the number
of inputs (so-called layers) tends to infinity, and the power assigned to each layer tends to zero
(i.e., becomes infinitesimally small). This infinite-layering limit gives rise to a mutual information
bound expressible as an integral. On channels with multiple transmit antennas, an analogous
result is derived. However, since multiple transmit antennas open up more possibilities for spatial
multiplexing, this leads to a higher-dimensional allocation problem, thus giving rise to a whole
family of infinite-layering mutual information bounds.
This family of bounds is closely studied for independent and identically zero-mean Gaussian
distributed fading coefficients (so-called i.i.d. Rayleigh fading). Several properties of the family
of bounds are derived. Most notably, it is shown that for asymptotically perfect CSIR, any
member of the family of bounds is asymptotically tight at high signal-to-noise ratios (SNR).
Specifically, this means that the difference between the mutual information and its lower bound
tends to zero as the SNR tends to infinity, provided that the CSIR tends to be exact as the SNR
tends to infinity.
A second part of this thesis proposes a framework for the optimization of a class of utility
functions in block-Rayleigh fading multiple-antenna channels with transmit-side antenna correlation,
and no CSI at the receiver. A fraction of each fading block is reserved for transmitting
a sequence of training symbols, while the remaining time instants are used for transmission of
data. The receiver estimates the channel matrix based on the noisy training observation and
then decodes the data signal using this channel estimate. The class of utility functions under
study consists of symmetric functions of the eigenvalues of the matrix-valued effective SNR.
Most notably, a simple achievable rate expression based on the worst-case-noise bound belongs
to this class.
The problems consisting in optimizing the pilot sequence and the linear precoder are cast
into convex (or quasi-convex) problems for concave (or quasi-concave) utility functions. We also
study an important subproblem of the joint optimization, which consists in computing jointly
Pareto-optimal pilot sequences and precoders. By wrapping these optimization procedures into a
cyclic iteration, we obtain an algorithm which converges to a local joint optimum for any utility.
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