Analysis and design of COFDM multicarrier modulation techniques in view of transmission stability in time variant channels
Final Report Abstract
In Multi-Carrier Modulation (MCM) communication schemes, the channel input signal is synthesized as a linear combination (superposition) of certain basis functions whose coefficients (weights) are bearing digital information. The performance of an MCM system depends largely on the choice of the basis functions which should allow the receiver to perform a fast and reliable recovery of the transmitted information from the distorted channel output. For years, Orthogonal Frequency Division Multiplexing (OFDM) has dominated MCM transmission systems, in particular in time invariant environments as present in DSL applications. In OFDM, the information carrying basis functions are of equal shape but shifted in the so-called time-frequency plane. It is well known that in stationary set-ups such as present in wired communications, these OFDM carrier signals are well suited for data transmission as they are approximate eigenfunctions of the channel operator, that is, the signals are attenuated by the channel but not reshaped. Within mathematics, or more precisely, within so-called time-frequency analysis, OFDM has been extensively studied under the disguise of Gabor analysis. In more theoretical terms, the stability of the synthesis map and corresponding algorithms have been discussed in great detail. Further, extensive effort has been invested into understanding the dependence of such algorithms on the shape of the basis functions in the time-frequency plane. In addition, time-frequency representations of operators developed into a powerful tool in the analysis of operators, in particular those operators representing communication channels. The central goal of this project was to establish sound footing for the novel results and techniques from the mathematical theory in the design of OFDM based communication systems for wireless and, in general, not stationary channels. In mobile communications environments for example, the transmitted signal reaches the receiver along a continuum of different signal paths, each featuring a path dependent time delay and a frequency shift caused by the doppler effect. Nevertheless, physical constraints imply that this time-frequency dispersion of the transmission signal is of limited extent. Hence, wireless channels can be modelled by operators, which are weighted superpositions of time and frequency shifts occupying a limited region (the so-called spreading support) in the time-frequency plane, a fact which is responsible for the good performance of OFDM in time-varying environments. The weight functions are the so-called spreading functions. One of the main achievements of this project is the development of numerical analysis tool which enables us to effectively compare the performance of different bases in a variety of communications channels. To this end, we showed that many narrowband finite lifelength systems such as wireless radio communications can be well modelled by smooth and compactly supported spreading functions. Further, we exploited this fact to derive a fast algorithm for computing the matrix representation of a channel operator with respect to pulseshaped OFDM bases. Hereby, we used a minimum of assumptions and simplifications on the channel.
Publications
- Time varying narrowband communications channels: analysis and implementation. Research report, 119 pages, 2006
Niklas Grip, Götz Pfander
- A discrete model for the efficient analysis of time-varying narrowband communications channels. Multidimensional Systems and Signal Processing, 38 pages, 2007
Niklas Grip, Götz Pfander
(See online at https://doi.org/10.1007/s11045-007-0032-1) - Uncertainty in time-frequency representations on finite abelian groups and applications. Applied and Computational Harmonic Analysis, 31 pages; 2007
Felix Krahmer, Götz Pfander, Peter Rashkov
(See online at https://doi.org/10.1016/j.acha.2007.09.008)
