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  • Research Article
  • Open Access

Perfect Reconstruction Conditions and Design of Oversampled DFT-Modulated Transmultiplexers

  • 1,
  • 2 and
  • 3
EURASIP Journal on Advances in Signal Processing20062006:015756

  • Received: 1 September 2004
  • Accepted: 19 July 2005
  • Published:


This paper presents a theoretical analysis of oversampled complex modulated transmultiplexers. The perfect reconstruction (PR) conditions are established in the polyphase domain for a pair of biorthogonal prototype filters. A decomposition theorem is proposed that allows it to split the initial system of PR equations, that can be huge, into small independent subsystems of equations. In the orthogonal case, it is shown that these subsystems can be solved thanks to an appropriate angular parametrization. This parametrization is efficiently exploited afterwards, using the compact representation we recently introduced for critically decimated modulated filter banks. Two design criteria, the out-of-band energy minimization and the time-frequency localization maximization, are examined. It is shown, with various design examples, that this approach allows the design of oversampled modulated transmultiplexers, or filter banks with a thousand carriers, or subbands, for rational oversampling ratios corresponding to low redundancies. Some simulation results, obtained for a transmission over a flat fading channel, also show that, compared to the conventional OFDM, these designs may reduce the mean square error.


  • Quantum Information
  • Fading Channel
  • Design Criterion
  • Filter Bank
  • Initial System

Authors’ Affiliations

Laboratoire des Images et des Signaux (LIS), Université Joseph Fourier, Saint Martin d'Hères Cedex, 38402, France
Laboratoire RESA/BWA, Division Recherche et Développement, France Télécom, 4 rue du Clos Courtel, Cesson Sévigné Cedex, 35512, France
Laboratoire Mathématiques pour l'Industrie et la Physique (MIP), Université Paul Sabatier, Toulouse 3, Toulouse Cedex 9, 31062, France


  1. Bölcskei H, Hlawatsch F, Feichtinger HG: Frame-theoretic analysis of oversampled filter banks. IEEE Transactions on Signal Processing 1998, 46(12):3256–3268. 10.1109/78.735301View ArticleGoogle Scholar
  2. Hleiss R, Duhamel P, Charbit M: Oversampled OFDM systems. Proceedings of 13th IEEE International Conference on Digital Signal Processing (DSP '97), July 1997, Santorini, Greece 1: 329–332.View ArticleGoogle Scholar
  3. Hleiss R: Conception et égalisation de nouvelles structures de modulations multiporteuses, M.S. thesis. École Nationale Supérieure des Télécommunications de Paris (ENSTP), Paris, France; 2000.Google Scholar
  4. Lin Y-P, Phoong S-M: ISI-free FIR filterbank transceivers for frequency-selective channels. IEEE Transactions on Signal Processing 2001, 49(11):2648–2658. 10.1109/78.960412View ArticleGoogle Scholar
  5. Siclet C, Siohan P, Pinchon D: Analysis and design of OFDM/QAM and BFDM/QAM oversampled orthogonal and biorthogonal multicarrier modulations. Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP '02), May 2002, pp. IV–4181, Orlando, Fla, USA 4:Google Scholar
  6. Phoong S-M, Chang Y, Chen C-Y: DFT-modulated filterbank transceivers for multipath fading channels. IEEE Transactions on Signal Processing 2005, 53(1):182–192.MathSciNetView ArticleGoogle Scholar
  7. Vetterli M: Perfect transmultiplexers. Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP '86), April 1986, Tokyo, Japan 11: 2567–2570.View ArticleGoogle Scholar
  8. Bölcskei H: Efficient design of pulse-shaping filters for OFDM systems. Wavelet Applications in Signal and Image Processing VII, July 1999, Denver, Colo, USA, Proceedings of SPIE 3813: 625–636.View ArticleGoogle Scholar
  9. Vetterli M: Filter banks allowing perfect reconstruction. Signal Processing 1986, 10(3):219–244. 10.1016/0165-1684(86)90101-5MathSciNetView ArticleGoogle Scholar
  10. Louveaux J: Filter bank based multicarrier modulation for xDSL transmission, M.S. thesis. Laboratoire de Télécommunications et Télédétection, Université Catholique de Louvain (UCL), Louvain-la-Neuve, Belgium; May 2000.Google Scholar
  11. Le Floch B, Alard M, Berrou C: Coded orthogonal frequency division multiplex. Proceedings of IEEE 1995, 83(6):982–996. 10.1109/5.387096View ArticleGoogle Scholar
  12. Vahlin A, Holte N: Optimal finite duration pulses for OFDM. IEEE Transactions on Communications 1996, 44(1):10–14. 10.1109/26.476088View ArticleGoogle Scholar
  13. Haas R, Belfiore J-C: A time-frequency well-localized pulse for multiple carrier transmission. Wireless Personal Communications 1997, 5(1):1–18. 10.1023/A:1008859809455View ArticleGoogle Scholar
  14. Bölcskei H: Orthogonal frequency division multiplexing based on offset-QAM. In Advances in Gabor Analysis. Birkhäuser, Boston, Mass, USA; 2002:321–352.Google Scholar
  15. Siohan P, Siclet C, Lacaille N: Analysis and design of OFDM/OQAM systems based on filterbank theory. IEEE Transactions on Signal Processing 2002, 50(5):1170–1183. 10.1109/78.995073View ArticleGoogle Scholar
  16. Siclet C: Application de la théorie des bancs de filtres à l'analyse et à la conception de modulations multiporteuses orthogonales et biorthogonales, M.S. thesis. Université de Rennes I (URI), Rennes, France; 2002.Google Scholar
  17. Kozek W, Molisch AF: Nonorthogonal pulseshapes for multicarrier communications in doubly dispersive channels. IEEE Journal on Selected Areas in Communications 1998, 16(8):1579–1589. 10.1109/49.730463View ArticleGoogle Scholar
  18. Schafhuber D, Matz G, Hlawatsch F: Pulse-shaping OFDM/BFDM systems for time-varying channels: ISI/ICI analysis, optimal pulse design, and efficient implementation. Proceedings of 13th IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC '02), September 2002, Lisbon, Portugal 3: 1012–1016.View ArticleGoogle Scholar
  19. Strohmer T, Beaver S: Optimal OFDM design for time-frequency dispersive channels. IEEE Transactions on Communications 2003, 51(7):1111–1122. 10.1109/TCOMM.2003.814200View ArticleGoogle Scholar
  20. Cvetković Z, Vetterli M:Tight Weyl-Heisenberg frames in . IEEE Transactions on Signal Processing 1998, 46(5):1256–1259. 10.1109/78.668789MathSciNetView ArticleGoogle Scholar
  21. Pinchon D, Siclet C, Siohan P: A design technique for oversampled modulated filter banks and OFDM/QAM modulations. Proceedings of 11th International Conference on Telecommunications (ICT '04), August 2004, Fortaleza, Brazil 578–588.MATHGoogle Scholar
  22. Pinchon D, Siohan P, Siclet C: Design techniques for orthogonal modulated filter banks based on a compact representation. IEEE Transactions on Signal Processing 2004, 52(6):1682–1692. 10.1109/TSP.2004.827193MathSciNetView ArticleGoogle Scholar
  23. Bölcskei H, Hlawatsch F: Oversampled modulated filter banks. In Gabor Analysis: Theory, Algorithms, and Applications. Birkhäuser, Boston, Mass, USA; 1998:295–322. chapter 9View ArticleGoogle Scholar
  24. Vaidyanathan PP: Multirate Systems and Filter Banks. Prentice-Hall, Englewoods Cliffs, NJ, USA; 1993.MATHGoogle Scholar
  25. Strohmer T: Finite and infinite-dimensional models for oversampled filter banks. In Modern Sampling Theory: Mathematics and Applications. Birkhäuser, Boston, Mass, USA; 2000:297–320.MATHGoogle Scholar
  26. Doroslovački MI: Product of second moments in time and frequency for discrete-time signals and the uncertainty limit. Signal Processing 1998, 67(1):59–76. 10.1016/S0165-1684(98)00022-XView ArticleGoogle Scholar
  27. Jakes WC (Ed): Microwave Mobile Communications. 2nd edition. Wiley-IEEE Press, Piscataway, NJ, USA; 1994.Google Scholar


© Cyrille Siclet et al. 2006

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