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Fading Margin Analysis for Asynchronous CDMA Systems

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In this paper, a detailed theoretical analysis of fading margin in an asynchronous code division multiple access (A-CDMA) system is discussed. Rayleigh and Rician frequency-selective slowly fading channels are considered. Probability distribution and density functions of the probability of error are derived for Rayleigh and Rician fading channels. The fluctuations in the channel capacity are proved to be directly proportional to the signal-to-noise ratio (SNR) variations. Fading margin is calculated for both Rayleigh and Rician fading channels as a function of the probability of error specification and the probability of unsatisfactory operation.

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Author information

Authors and Affiliations

  1. Departement Genie des Systemes d’Information et de Telecommunication, Universite de Technologie de Troyes, 12 Rue Marie Curie, 10010, Troyes Cedex, France

    Vidhyacharan Bhaskar

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  1. Vidhyacharan Bhaskar
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Correspondence to Vidhyacharan Bhaskar.

Appendix

Appendix

For a coherent BPSK system, the probability of bit error is given by [17]

$$y=g(\gamma_{c})={1\over 2}\left[1-\sqrt{{\gamma_{c}\over 1+\gamma_{c}}}\right], $$
(A.1)

where γ c is the signal-to-noise ratio. In our case, we can consider γ c ≡ SNR. So, g −1(y)≡ SNR. Let x≡γ c . Then \({\partial g^{-1}(y)\over \partial y}={\partial x\over \partial y}\). From (A.1), we have

$$\gamma_{c}={(1-2y)^{2}\over 4(y-y^{2})}. $$
(A.2)

From (A.2), we have

$$\eqalign{{\partial \hbox{SNR}\over \partial y} &= {\partial\over \partial y}\left\{{(1-2y)^{2}\over 4(y-y^{2})}\right\}\cr &={1\over 4}\left\{{-4(y-3y^{2}+2y^{3})- (1-8y^{3}-6y+12y^{2})\over (y-y^{2})^{2}}\right\}\cr &={1\over 4}\left\{{2y-1\over (y-y^{2})^{2}}\right\}=-c(y)}$$
(A.3)

since SNR is a decreasing function of y. Substituting the result of (A.3) for c(y) in (20), we have

$$ f_{P_{e}(t),\rm{RaF}}(y)={1\over 8\sigma^{2}}\exp\left[{-g^{-1}(y)\over 2\sigma^{2}}\right] \left[{1-2y\over (y-y^{2})^{2}}\right].$$
(A.4)

Equation A.4 gives the density function of the probability of error for a Rayleigh fading channel.

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Bhaskar, V. Fading Margin Analysis for Asynchronous CDMA Systems. Int J Wireless Inf Networks 12, 159–168 (2005). https://doi.org/10.1007/s10776-005-0021-y

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