Hilbert space frames and bases a comparison of Gabor and wavelet frames and applications to multicarrier digital communications

Abstract: Several signal processing applications today are based on the use of different transforms. The signals under consideration are written as a linear combination (or series) of some predefined set of functions. Traditionally, orthogonal bases have been used for this purpose, for example, in the discrete Fourier transform. The theory for orthogonal bases for Hilbert spaces can, however, be generalized to other sequences of functions, called frames. The first part of this thesis begins with an application-oriented introduction to the theory of frames and bases for separable Hilbert spaces. We explain similarities with and differences from the theory of orthogonal bases. Special attention is given to the relatively new theory of Gabor and Wavelet frames. We explain how they can be used for so-called time-frequency analysis. The main emphasis is on explaining fundamental similarities and differences between Gabor and wavelet frames. We also give an example of an application (OFDM) related to the second part of the thesis, for which nonorthogonal Gabor frames are superior to any orthogonal basis. The second part of this thesis concerns the current development of a standard for very high speed digital communication in ordinary telephone copper wires. It is the result of a cooperation with the Division of Signal Processing and Telia Research. We present a novel duplex method for Very high bitrate Digital Subscriber Lines (VDSL), called Zipper. It is intended to provide bit rates up to 52 Mbit per second, about 1000 times faster than the most common modems today. Zipper is based on Discrete Multi Tone (DMT) modulation. It uses an orthogonal basis of Gabor type for the signal transmission. Certain cyclical extensions are used to ensure the orthogonality between the basis functions. Zipper is proposed as a standard for VDSL to both the American National Standards Institute (ANSI) T1E1.4 group and in the European Telecommunication Standards Institute (ETSI) TM6 group. It will also be presented for the International Telecommunication Union (ITU). Telia Research is currently building a prototype together with ST Microelectronics (former SGS-Thomson), France. The first Zipper-VDSL modems are expected to be available on the mass market in the year 2001. The second part of this thext consists of a brief introduction to Zipper, an ANSI standard contribution and three conference papers. The standard contribution compares Zipper performance with competing standard proposals at that time: TDD and FDD. In the first conference paper we present a new and patented method for reducing the interference that the unshielded copper wires experience from radio transmissions. The two last conference papers present a low complexity method for reducing the so-called Peak to Average power Ratio (PAR) of the transmitted signal. PAR is a measure for the amount of rare but very high peaks in the signal. A reduced PAR allows for using a cheaper digital-to-analog converter and amplifier in the transmitter.