Kapazität von OFDM über Kanäle mit Spitzenleistungsbeschränkung und Signalformungsalgorithmen zum Erreichen der Kapazität
Final Report Abstract
In wireless communications systems, information is represented as a sequence of numbers. These numbers are assigned to a set of indivual transmission signals, and a sum of the signals is created, which is a more complicated signal. This signal is then transmitted, and the receiver decodes the original sequence from the composite signal. For this to work well, the signal must arrive at the receiver with the least possible amount of disruption. A very typical disruption is caused by amplifiers; in particular, amplifiers perform poorly at high amplitudes, and disrupt large peaks of the transmission signal. For a set of possible information numbers and a set of basic transmission signals, it is important to understand how the peak of the transmitted signal behaves. Additionally, current research is also concerned with reducing the peak values of such signals. This large area is called the Peak-to-Average Power Ratio problem. In this project we were able to address three aspects of this area. First, in the context of Orthogonal Frequency Domain Modulation (OFDM) we analyzed the scaling behavior of a fundamental approach to PAPR reduction problem. The fundamental approach, known as tone-reservation, uses one subset of the transmission signals to carry information and the remaining signals to reduce the peak value of the entire signal. Because this approach is so simple and mathematically fundamental, it has been quite important to understand the limits of its performance. We addressed the effectiveness of this approach as the size of the system grows and determined its scaling behavior. In particular, we showed that if a strict peak condition is imposed, that is all signals must have a peak smaller than a specific value, then as the size of the system grows, the percentage of signals allocated for information transmission must converge to zero. To do so, we relied on sophisticated combinatorics and, thus, provided another connection between this area of mathematics and information theory. In a second piece of work, we addressed the analogous questions for CDMA systems. Here it was widely believed that the peak behavior observed in the OFDM case does not occur, but we showed that it does and that the scaling law for tone reservation also holds in exactly the same way. Here we were able to prove this directly and obtain specific bounds. We accompanied this result with a handful of smaller results comparing the behavior of the OFDM and CDMA systems. Our third contribution was to determine the average peak behavior of a sum of sine functions. These functions are fundamental in signal processing, communications, sampling and information theory. If the coefficients in a linear combination of shifts of these functions have Gaussian distribution, then the peak is expected to grow like √log n, where n is the number of functions. We proved that this is in fact the case. However, it was also believed that if the coefficients are ± 1 , then the peak will behave in the same way. We showed, however, that in this case the expected value of the peak grows like log log n. This result should be quite valuable for the community. It is a rare example in this area of a property being dependent on the statistics of the underlying random variables. Additionally, it shows that signals on the unit interval, such as in the OFDM and CDMA cases, behave fundamentally differently from functions on the real line, i.e. the single carrier case, for certain coefficient statistics.
Publications
- Papr for OFDM and the proportion of information bearing signals for tone reservation. In Information Theory and its Applications (ISITA), 2010 International Symposium, p. 1058-1063, Oct. 2010
H. Boche and B. Farrell
- PAPR and the Density of Information Bearing Signals in OFDM. EURASIP Journal on Advances in Signal Processing, 2011
H. Boche and B. Farrell
