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40 ADVANCES IN ELECTRONICS AND TELECOMMUNICATIONS, VOL. 1, NO. 1, APRIL 2010 [2] H. Sari, G. Karam, and I. Jeanclaude, “Channel Equalization and Carrier Synchronization in OFDM systems,” in Int. Tirrenia Workshop on Digital Communications, Tirrenia, Italy, Sep. 1993, in Audio and Video Digital Radio Broadcasting Systems and Techniques, R. de Gaudenzi and M. Luise (Editors), pp. 191-202, Elsevier Science Publishers, Amsterdam, 1994. [3] ——, “Transmission Techniques for Digital Terrestrial TV Broadcasting,” IEEE Commun. Mag., vol. 33, pp. 100–109, Feb. 1995. [4] H. Sari, Y. Levy, and G. Karam, “Orthogonal Frequency-Division Multiple Access for the Return Channel on CATV Networks,” in ICT’96 Conf. Rec, vol. 1, Istanbul, Turkey, Apr. 1996, pp. 52–57. [5] H. Sari, “Orthogonal Frequency-Division Multiple Access with Frequency Hopping and Diversity,” in Multi-Carrier Spread Spectrum, K. Fazel and G. P. Fettweis, Eds. Kluwer Academic Publishers, The Netherlands, 1997, pp. 57–68. [6] H. Sari and G. Karam, “Orthogonal Frequency-Division Multiple Access and its Application to CATV Networks,” European Transactions on Telecommunications (ETT), vol. 9, no. 6, pp. 507–516, Nov.-Dec. 1998. [7] M. Moeneclaey, M. V. Bladel, and H. Sari, “Sensitivity of Multiple Access Techniques to Narrowband Interference,” IEEE Trans. Commun., vol. 49, no. 3, pp. 497–505, Mar. 2001. [8] U. Sorger, I. D. Broeck, and M. Schnell, “Interleaved FDMA A new spread-spectrum multiple-access scheme,” in IEEE International Conference on Communications, ICC98, Atlanta, Georgia, USA, Jun. 1998. [9] T. Frank, A. Klein, E. Costa, and E. Schultz, “Robustness of IDFMA as Air Interface Candidate for Future High Rate Mobile Radio Systems,” Advances in Radio Science, vol. 3, pp. 265–270, 2005. [10] 3rd Generation Partnership Project; Technical Specification Group Radio Access Network, Physical Layer Aspects for Evolved Universal Terrestrial Radio Access (UTRA), 3GPP TR 25.814 V7.1.0, Sep. 2009. [11] Evolved Universal Terrestrial Radio Access (E-UTRA); User Equipment (UE) radio transmission and reception (Release 8), 3GPP TS 36.101 v8.7.0, Sep. 2009. [12] C. Rapp, “Effects of the HPA-nonlinearity on a 4-DPSK/OFDM signal for a digital sound broadcasting system,” in European Conference on Sattelite Communications ECSC91, Liège, Belgium, Oct. 1991. [13] A. Saleh, “Frequency independent and frequency dependent nonlinear models of TWT amplifiers,” IEEE Trans. Commun., vol. 29, pp. 1715– 1720, 1981. [14] C. Ciochina, “Physical Layer Design for the Uplink of Future Mobile Communications Systems,” Ph.D. Thesis, Ecole Supérieure d´Electricité, France, Jul. 2009. [15] C. Ciochina, D. Mottier, and H. Sari, “An Analysis of Three Multiple Access Techniques for the Uplink of Future Cellular Mobile Systems,” European Transactions on Telecommunications (ETT), vol. 19, no. 5, p. 581588, Aug. 2008. Cristina Ciochina was born in Bucarest, Romania, in 1980. She received her Engineering degree in Electronics and Telecommunications from the Polytechnic University of Bucarest, Romania, in 2004. She received her M.Sc. degree from the Ecole Supérieure d´Electricité (SUPELEC), France, in 2005 and her Ph.D. degree from the Université de Paris-Sud 11, France in 2009, both in Telecommunications. Since 2005, she has been with Mitsubishi Electric R&D Centre Europe (MERCE), France. Her technical interests include physical layer design, multicarrier systems, signal processing for future wireless communications systems, and 3GPP/LTE. Dr. Ciochina has published 18 technical papers and is the author or co-author of 8 patent applications. Hikmet Sari (S´78 - M´81 - SM´88 - F´95) received his Diploma (M.S.) and Ph.D. in Telecommunications Engineering from the ENST, Paris, France, in 1978 and 1980, respectively, and the Habilitation degree from the University of Paris-Sud, Orsay in 1992. He was with Philips Research Laboratories from 1978 to 1989, first as Researcher and then as Group Supervisor. From 1989 to 1996, he was R&D Department Manager at SAT (SAGEM Group), and from 1996 to 2000, he was Technical Director at Alcatel. In May 2000, he became Chief Scientist of the newly-founded Pacific Broadband Communications, which was acquired by Juniper Networks in December 2001. Since April 2003, he has been a Professor and Head of the Telecommunications Department at SUPELEC, and since December 2004 he is also Chief Scientist of Sequans Communications. Dr. Sari was an Editor of the IEEE Transactions on Communications from 1987 to 1991, a Guest Editor of the European Transactions on Telecommunications (ETT) in 1993, a Guest Editor of the IEEE JSAC in 1999, an Associate Editor of the IEEE Communications Letters from 1999 to 2002, and a Guest Editor of the EURASIP Journal on Wireless Communications and Networking in 2007. He was also Chair of the Communication Theory Symposium of the 2002 IEEE International Conference on Communications (ICC 2002), April 2002, New York, Technical Program Chair of ICC 2004, June 2004, Paris, and Vice General Chair of ICC 2006, June 2006, Istanbul. He is currently serving as General Chair for the forthcoming 2010 Annual IEEE Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC 2010) and the 2012 IEEE Wireless Communications and Networking Conference (WCNC 2012). He was elevated to the IEEE Fellow Grade and received the Andre Blondel Medal from the SEE (France) in 1995 and he received the Edwin H. Armstrong Achievement Award from the IEEE Communications Society in 2003.
ADVANCES IN ELECTRONICS AND TELECOMMUNICATIONS, VOL. 1, NO. 1, APRIL 2010 41 Bayesian Foundations of Channel Estimation for Smart Radios Romain Couillet, Andrea Ancora, and Mérouane Debbah Abstract—In this paper, we revisit the philosophical foundations of the field of channel estimation. Our main intention is to come up with a partial answer to the question: “given some available sensed signals, how should cognitive radios ideally perform channel estimation?”. We specifically introduce a general framework to provide optimal channel estimates under any prior knowledge at the sensing device. Our discussion is articulated as a top-down approach, introducing successively (i) a discussion on the philosophical foundations of channel estimation as a simplification means for the general problem of wireless detection, (ii) an information theoretically optimal approach to channel detection assuming the sensing device has infinite memory, and (iii) a derived optimal approach when limited memory size is accounted for. The key mathematical tools used in this discussion emerge from Bayesian probability theory and are known as the maximum entropy principle and the minimum update principle. Derivations are carried out for the particular case of channel estimation in orthogonal frequency division multiplexing (OFDM) systems. While some theoretical results will be proven to match already known techniques, such as Kalman filters, another set of novel results will be shown by simulations to perform better than known channel estimation schemes. I. INTRODUCTION CHANNEL estimation, along with most synchronization procedures, is itself a major and historical field of research in the realm of wireless communications . As such, thousands of novel channel estimators are proposed and compared to previous estimators every year. One of the obvious reasons for such an activity around channel estimation is that there does not exist a universal measure of performance to rate any given scheme with respect to any other; instead, several different selection criteria are considered, such as computational complexity, mean square error of the estimate, robustness against outage channel conditions, required processing memory etc. In this work, we wish to propose a framework to encompass the aforementioned selection conditions into a unique channel estimation framework. In some cases, we shall generate the optimal channel estimators for this framework. The sought for a general framework for channel estimation is motivated by the recent trend towards cognitive radios, and in particular by the trend towards developing terminalcentered intelligence in future wireless flexible networks. In the framework of cognitive radios, terminals are required to learn from their environment and take optimal decisions Romain Couillet is with ST-Ericsson, Supélec, 635 Route des Lucioles, 06560 Sophia Antipolis, France (email: romain.couillet@supelec.fr) Andrea Ancora is with ST-Ericsson, Eurécom, 635 Route des Lucioles, 06560 Sophia Antipolis, France (email: andrea.ancora@eurecom.fr) Mérouane Debbah is with Alcatel-Lucent Chair, Supélec, Plateau de Moulon, 3 rue Joliot-Curie, 91192 Gif sur Yvette, France (email: merouane.debbah@supelec.fr) based on a constantly updated knowledge: performing optimal channel estimation from limited side information is part of the requirements demanded of cognitive radios. The prior requirement for channel estimators is to help signal decoders as best as possible. The decision to consider a particular measure of performance for a given channel estimators should only reflect the subsequent effects on the eventual signal decoding process. For instance, minimizing the mean square error of channel estimators has no theoretical ground; it is merely a convenient mathematical way to compare estimators. In Section II, we shall briefly remind the foundations of signal decoding and channel estimation, which we shall review on a cognitive radio viewpoint. This study naturally follows the theoretical ideas introduced in [5], in which the authors introduce a new look on cognitive radios, and in [6], where the specific problem of blind source detection is addressed. Section II will conclude that today’s theoretical and technological advances does not yet allow smart devices to perform optimal signal decoding without considering channel estimation as an independent entity. In Section IV, we shall therefore treat channel estimation as a self-contained process, independent of the problem of signal detection, as is conventionally the case. We shall introduce a complete framework to derive optimal channel estimators under any prior state of knowledge at the signal receiver. While the conventional approach is to treat specific channel models and develop estimators for those models, we shall here instead consider prior knowledge about the environment at the receiver, and develop consistent estimators for this knowledge. We wish indeed to insist on the fact that smart devices should be able to come up with an ideal channel estimator for any given prior information on the channel. The approach addressed here is based on conventional Bayesian probability theory and on the maximum entropy principle [8]. In this section, we shall essentially remind the results originally derived in [7]. However, the practical finite memory size of the processing devices will not be taken into account in this section. To answer the problem of optimal channel estimation under finite memory-size constraint, we shall subsequently introduce a novel aspect of the maximum entropy principle, known as the minimal update principle [9]. The major difference between both is the fact that the maximum entropy principle assumes an initial starting point and performs optimal decisions for data collected from this starting point on, while the minimal update principle assumes that one might be oblivious of old data and performs optimal decisions for a finite window of “remembrance”. In our specific channel estimation considerations, this means that estimation based on the maximum
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