Optimal Training for Time-Selective Wireless Fading Channels Using Cutoff Rate

  • Saswat MisraEmail author
  • Ananthram Swami
  • Lang Tong
Open Access
Research Article
Part of the following topical collections:
  1. Reliable Communications over Rapidly Time-Varying Channels


We consider the optimal allocation of resources—power and bandwidth—between training and data transmissions for single-user time-selective Rayleigh flat-fading channels under the cutoff rate criterion. The transmitter exploits statistical channel state information (CSI) in the form of the channel Doppler spectrum to embed pilot symbols into the transmission stream. At the receiver, instantaneous, though imperfect, CSI is acquired through minimum mean-square estimation of the channel based on some set of pilot observations. We compute the ergodic cutoff rate for this scenario. Assuming estimator-based interleaving and Open image in new window -PSK inputs, we study two special cases in-depth. First, we derive the optimal resource allocation for the Gauss-Markov correlation model. Next, we validate and refine these insights by studying resource allocation for the Jakes model.


Resource Allocation Fading Channel Channel State Information Pilot Symbol Doppler Spectrum 


  1. 1.
    Cavers JK: An analysis of pilot symbol assisted modulation for Rayleigh fading channels [Mobile Radio]. IEEE Transactions on Vehicular Technology 1991, 40(4):686–693. 10.1109/25.108378CrossRefGoogle Scholar
  2. 2.
    Tong L, Sadler BM, Dong M: Pilot-assisted wireless transmissions. IEEE Signal Processing Magazine 2004, 21(6):12–25. 10.1109/MSP.2004.1359139CrossRefGoogle Scholar
  3. 3.
    Garcia MJ, Páez-Borrallo JM: Tracking of time misalignments for OFDM systems in multipath fading channels. IEEE Transactions on Consumer Electronics 2002, 48(4):982–989.CrossRefGoogle Scholar
  4. 4.
    Kuo W, Fitz MP: Frequency offset compensation of pilot symbol assisted modulation in frequency flat fading. IEEE Transactions on Communications 1997, 45(11):1412–1416. 10.1109/26.649760CrossRefGoogle Scholar
  5. 5.
    Cai X, Giannakis G: Adaptive PSAM accounting for channel estimation and prediction errors. IEEE Transactions on Wireless Communications 2005, 4(1):246–256.CrossRefGoogle Scholar
  6. 6.
    Dong M, Tong L, Sadler BM: Optimal insertion of pilot symbols for transmissions over time-varying flat fading channels. IEEE Transactions on Signal Processing 2004, 52(5):1403–1418. 10.1109/TSP.2004.826182MathSciNetCrossRefGoogle Scholar
  7. 7.
    Dong X, Xiao L: Symbol error probability of two-dimensional signaling in Ricean fading with imperfect channel estimation. IEEE Transactions on Vehicular Technology 2005, 54(2):538–549. 10.1109/TVT.2004.841537MathSciNetCrossRefGoogle Scholar
  8. 8.
    Hassibi B, Hochwald BM: How much training is needed in multiple-antenna wireless links? IEEE Transactions on Information Theory 2003, 49(4):951–963. 10.1109/TIT.2003.809594CrossRefGoogle Scholar
  9. 9.
    Adireddy S, Tong L, Viswanathan H: Optimal placement of training for frequency-selective block-fading channels. IEEE Transactions on Information Theory 2002, 48(8):2338–2353. 10.1109/TIT.2002.800466MathSciNetCrossRefGoogle Scholar
  10. 10.
    Ma X, Giannakis GB, Ohno S: Optimal training for block transmissions over doubly selective wireless fading channels. IEEE Transactions on Signal Processing 2003, 51(5):1351–1366. 10.1109/TSP.2003.810304MathSciNetCrossRefGoogle Scholar
  11. 11.
    Ohno S, Giannakis GB: Capacity maximizing MMSE-optimal pilots for wireless OFDM over frequency-selective block Rayleigh-fading channels. IEEE Transactions on Information Theory 2004, 50(9):2138–2145. 10.1109/TIT.2004.833365MathSciNetCrossRefGoogle Scholar
  12. 12.
    Baltersee J, Fock G, Meyr H: An information theoretic foundation of synchronized detection. IEEE Transactions on Communications 2001, 49(12):2115–2123. 10.1109/26.974258CrossRefGoogle Scholar
  13. 13.
    Medard M, Abou-Faycal I, Madhow U: Adaptive coded modulation without channel feedback for pilot symbol assisted modulation. Proceedings of the 38th Annual Allerton Conference on Communication, Control and Computing, October 2002, Monticello, Ill, USAGoogle Scholar
  14. 14.
    Ohno S, Giannakis G: Average-rate optimal PSAM transmissions over time-selective fading channels. IEEE Transactions on Wireless Communications 2002, 1(4):712–720. 10.1109/TWC.2002.804183CrossRefGoogle Scholar
  15. 15.
    Arikan E: Upper bound on the cutoff rate of sequential decoding. IEEE Transactions on Information Theory 1988, 34(1):55–63. 10.1109/18.2601MathSciNetCrossRefGoogle Scholar
  16. 16.
    Biglieri E, Proakis J, Shamai S: Fading channels: information-theoretic and communications aspects. IEEE Transactions on Information Theory 1998, 44(6):2619–2692. 10.1109/18.720551MathSciNetCrossRefGoogle Scholar
  17. 17.
    Jamali S, Le-Ngoc T: Coded-Modulation Techniques for Fading Channels. Kluwer Academic, Dordrecht, The Netherlands; 1994.CrossRefGoogle Scholar
  18. 18.
    Leeuwin-Boulle K, Belfiore JC: Cutoff rate of time correlated fading channels. IEEE Transactions on Information Theory 1993, 39(2):612–617. 10.1109/18.212291CrossRefGoogle Scholar
  19. 19.
    McEliece RJ, Stark WE: Channels with block interference. IEEE Transactions on Information Theory 1984, 30(1):44–53. 10.1109/TIT.1984.1056848CrossRefGoogle Scholar
  20. 20.
    Hero AO, Marzetta TL: Cutoff rate and signal design for the quasi-static Rayleigh-fading space-time channel. IEEE Transactions on Information Theory 2001, 47(6):2400–2416. 10.1109/18.945254MathSciNetCrossRefGoogle Scholar
  21. 21.
    Misra S, Swami A, Tong L: Cutoff rate of the Gauss-Markov channel with adaptive energy allocation. Processing of IEEE Workshop on Signal Processing Advances in Wireless Communications, June 2003, Rome, ItalyGoogle Scholar
  22. 22.
    Kay S: Fundamentals of Statistical Signal Processing. Prentice-Hall, Englewood Cliffs, NJ, USA; 1993.zbMATHGoogle Scholar
  23. 23.
    Massey J: Coding and moulation in digital communications. Proceedings of International Zurich Seminar on Digital Communications, March 1974Google Scholar
  24. 24.
    Wozencraft JM, Jacobs IM: Principles of Communication Engineering. John Wiley & Sons, New York, NY, USA; 1965.Google Scholar
  25. 25.
    Proakis J: Digital Communications. McGraw-Hill, New York, NY, USA; 2001.zbMATHGoogle Scholar
  26. 26.
    Wu PH-Y, Duel-Hallen A: Multiuser detectors with disjoint Kalman channel estimators for synchronous CDMA mobile radio channels. IEEE Transactions on Communications 2000, 48(5):752–756. 10.1109/26.843185CrossRefGoogle Scholar
  27. 27.
    Stojanovic M, Proakis JG, Catipovic JA: Analysis of the impact of channel estimation errors on the performance of a decision-feedback equalizer in fading multipath channels. IEEE Transactions on Communications 1995, 43(2):877–886. 10.1109/26.380120CrossRefGoogle Scholar
  28. 28.
    Medard M: The effect upon channel capacity in wireless communications of perfect and imperfect knowledge of the channel. IEEE Transactions on Information Theory 2000, 46(3):933–946. 10.1109/18.841172MathSciNetCrossRefGoogle Scholar
  29. 29.
    Kailath T, Sayed A, Hassibi B: Linear Estimation. Prentice-Hall, Englewood Cliffs, NJ, USA; 2000.zbMATHGoogle Scholar
  30. 30.
    Jakes WC Jr.: Microwave Mobile Communication. John Wiley & Sons, New York, NY, USA; 1974.Google Scholar
  31. 31.
    Rappaport T: Wireless Communications: Principles & Practice. Prentice-Hall, Upper Saddle River, NJ, USA; 1996.zbMATHGoogle Scholar
  32. 32.
    Baccarelli E: Bounds on the symmetric cutoff rate for QAM transmissions over time-correlated flat-faded channels. IEEE Communications Letters 1998, 2(10):279–281. 10.1109/4234.725223CrossRefGoogle Scholar

Copyright information

© Saswat Misra et al. 2006

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.The Army Research LaboratoryAdelphiUSA
  2. 2.Department of Electrical and Computer EngineeringCornell UniversityIthacaUSA

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