Communication Rates for Fading Channels with Imperfect Channel-State Information

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 corre- lation, 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.