# Linear Precoders for OFDM Wireless Communication

## Merouane Debbah

Degree: Ph.D.
Supervisor: Prof. Philippe Loubaton
Department: Laboratoire Systemes de CommunicationsoInstitution:

Institution: Ecole Normale Superieure de Cachan/Universite de Marne La Valle

Contact Information:
Prof. Merouane Debbah
Mobile Communication Group
Institut Eurecom
2229 Route des Cretes
B.P. 193
06904 SOPHIA ANTIPOLIS CEDEX
France
Email: debbah@eurecom.fr
Website: http://www.eurecom.fr/~debbah

Abstract

A multi-carrier {\bf OFDM} (Orthogonal Frequency Division Multiplexing) system using a Cyclic Prefix for preventing inter-block interference is known to be equivalent to a system based on multiple flat fading parallel transmission channels in the frequency domain. In such a system, the information sent on some carriers might be subject to strong attenuations and could be unrecoverable at the receiver. The aim of this thesis is to investigate a transmission scheme known as {\bf Linear Precoded OFDM} to overcome these drawbacks.

Linear Precoding consists in spreading the information contained in a $K$-dimensional vector onto $N$ carriers. Therefore , the $K \times 1$ infomation vector to be transmitted is multiplied by an $N \times K$ matrix. The precoder is intended to increase the overall frequency diversity of the modulator so that unreliable carriers can still be recovered by taking advantage of the subbands enjoying a high signal to noise ratio. Using new tools, borrowed from the so-called {\bf free probability theory}, the Signal to Interference plus Noise Ratio (SINR) at the output of linear and non-linear equalizers are derived when $N \to \infty$ and $K/N \to \alpha$ assuming both ergodic and non ergodic channels. Based on the assumption of asymptotic Gaussian interference, the SINR is then used as a benchmark measure of performance for different criterias.

These asymptotic results aim at adjusting the different parameters involved in the detection process : nature and impact of the orthogonality of the precoding matrix columns, optimal amount of redundancy of the precoding matrix when coding is applied (ratio $\frac{K}{N}$, values of the coefficients of the non-linear equalizers). Interestingly, the results have very simple interpretations and determine the useful parameters to be considered in the design of these transmissions schemes. Finally, conclusions are drawn on the future research topics in connection with our work.