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Theoretical Performance of MB-OFDM in Fading Channels

Examining theoretical upper bounds on convolutional codes in MB-OFDM for fading channels, providing insight into real bit error probabilities.

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Theoretical Performance of MB-OFDM in Fading Channels

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  1. Project: IEEE P802.15 Working Group for Wireless Personal Area Networks (WPAN’s)Submission Title: [Theoretical performance of MB-OFDM in the fading channel] Date Submitted: [September, 2004] Source: [Brian Gaffney] Company [decaWave Ltd]E-Mail:[brian.gaffney@ee.ucd.ie; brian.gaffney@decaWave.com] Abstract: [Examines the theoretical upper bounds on the performance of the convolutional codes used in the MB-OFDM proposal in the fading channel]Purpose: [Provide technical information to the TG3a voters regarding PHY proposals.]Notice: This document has been prepared to assist the IEEE P802.15. It is offered as a basis for discussion and is not binding on the contributing individual(s) or organization(s). The material in this document is subject to change in form and content after further study. The contributor(s) reserve(s) the right to add, amend or withdraw material contained herein.Release: The contributor acknowledges and accepts that this contribution becomes the property of IEEE and may be made publicly available by P802.15. Brian Gaffney, decaWave

  2. Overview of presentation • Examine the theoretical upper bounds on the bit error probability in the fading channel. • Demonstrate how these upper bounds give a good example of the real bit error probabilities at the BER’s of interest. Brian Gaffney, decaWave

  3. Channel Model • Channel model 1 was used in both theoretical analysis and simulation. • The shadowing term was neglected, such that all the energy captured was assumed to be unity. Brian Gaffney, decaWave

  4. Upper bounds on the probability of error • With repetition • With no repetition Brian Gaffney, decaWave

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  13. Summary • Theoretical upper bounds on the probability of error of some of the codes used in the MB-OFDM proposal were calculated. • The performance was shown to suffer at the higher coding rates and without any repetition. • It was then demonstrated that these upper bounds give a good idea on the real probability of error for the lower BER’s. Brian Gaffney, decaWave

  14. Additional slides Brian Gaffney, decaWave

  15. Glossary d are the weight spectra of the code.  is the Eb/No ratio. R = k/n is the rate of the convolutional code. dfreeis the minimum free distance of the code. P2(d) is the probability of decoding a path with a hamming distance of d from the code. Brian Gaffney, decaWave

  16. Upper bound on probability of error For the case with no repetition, from [1], the probability of error can be upper bounded The pairwise probability of error, conditioned on the sum of d fading amplitudes all squared, xd, is The expected P2(d) is given by Where p(xd) is the probability distribution function of the sum of the fading amplitudes all squared. Brian Gaffney, decaWave

  17. From [2], the normalised distribution function is where where 2 is the variance of , (2d-1)!! = (2d-1) (2d-3). . . . . . (3)(1), and t2=xd2/d. By substituting this distribution into the above integral, and upper bounding the Q function with Brian Gaffney, decaWave

  18. The expected value of the pairwise probability of error can be upper bounded as Substituting into the expression for the upper bound on the probability of error For the case of repetition with Maximal ratio combining, the pairwise probability of error conditioned on the sum of d combined variables, where each combined variable is the sum of L squared fading amplitudes, sd, is The probability distribution function of sd is Brian Gaffney, decaWave

  19. As before, the expected pairwise probability of error can be upper bounded by So, the bit error probability is upper bounded by Brian Gaffney, decaWave

  20. References [1] Proakis, J.G., Digital Communications. McGraw-Hill International edition, 2001. [2] Beaulieu, N.C., “An infinite series for the computation of the complementary probability distribution function of a sum of independent random variables and its application to the sum of Rayleigh random variables,” IEEE Transactions on Communications, Vol. 38, No. 9, Sept. 1990, pp.1463-1474. Brian Gaffney, decaWave

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