Information Theoretic Reliability Function for Flat Fading Channels.

Walid K. M. Ahmed.

Degree: Ph.D.
Supervisor: Prof. Peter J. McLane.
Department: Electrical and Computer Engineering. Queen's University at Kingston, Ontario, Canada.
Date of Graduation: September 1997.


With the emergence of wireless and mobile communication systems and technology, new channel models and impairment phenomena have appeared. Fading and multipath effects are the severest among these phenomena. The techniques developed for communications over bandlimited AWGN channels have been applied to fading channels. However, performance of these techniques in fading is significantly inferior to additive noise channel results. Much of the research currently being performed deals with the modification of known channel coding techniques for use over fading channels. So far, some great improvements have been made, but data rates achievable in fading are still quite modest in comparison to those for wireline channels.

Due to the extreme difference in reliability between fading and noise channels, some fundamental questions arise. Must performance in fading be so inferior to the results obtained for the additive noise channel? Although a new code may result in an improvement over all other known codes, does this mean that the new one is actually a good one? To answer these questions, the limits to communications over multipath fading channels should be determined.

The channel capacity is a crucial asset that can be found by the use of information theory to determine the achievable transmission rates for reliable communication. That is to say, if the transmission rate is higher than the channel capacity, reliable communication and arbitrarily small decoding error probability can not be achieved. However, if the transmission rate is to be kept below the capacity, we know that reliable communication is possible. The question here becomes, at a rate below channel capacity, how reliable is the communication process? Another question is how complex are the codes required to achieve a certain level of reliability? And finally, how rapid is the decay of decoding error probability as a function of the code length? To answer these questions, a measure of reliability has to be defined. This measure, in fact, exists and is known to information theorists as the ''Random Coding Reliability Function''. Also, known as the ''Random Coding Error Exponent'', due to Gallager [1965]. Gallager has shown that the probability of error of the best block codes of length N and rate R decreases exponentially with block length. Gallager's bound determines the behavior of the probability of error in terms of the transmission rate as well as the code length N, which reflects coding complexity, thus, making the bound an attractive measure of reliability.

In this thesis, the random coding reliability function has been derived for various fading channels that model current wireless and mobile communication channels. The effects of amplitude fading, non-ideal channel state information, space diversity and fading time correlation on the reliability function have been considered. Channels with average and peak-power-constrained inputs as well as channels with discrete and continuous inputs have also been studied. Comparison with the additive white Gaussian noise (AWGN) channel has been made in order to determine the amount of degradation due to fading. In addition, estimates of the required code lengths for a certain probability of error have often been calculated in order to aid in the assessment of the required coding complexity over fadingchannels.