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# S

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.