SECCION VIII- ASPECTOS DIVERSOS Regla 51
CAPITULO 4 CONTENCION DE LA CARGA 4.1 Definiciones
The probability of bit error of the collaborative coded multiuser signals using BPSK mapping in flat slowly fading channels can be derived by doing some simple modifications to the tools developed for single user signals in [107, 108]. For this purpose, error metric associated with each codeword combination of the transmitted signals is derived. The absolute magnitude of distance between different composite codeword combinations are calculated and normalized with n to give an error metric zmas follows
zm = {C1x+ C2x} − {C1y+ C2y} 2 n 1 < m < M ; 1 < x 6= y < L, (4.22)
where {C1x, C2x} and {C1y, C2y} are any possible two composite codeword combinations of user
1 and 2, M =PL−1
Assuming that all the codewords are equally likely to be transmitted, it is appropriate to find the average error metric so that the tools developed for single user signals can be used. Averaging zm
over all M possible distances between the codeword combinations, the average error metric z , that is used for the bit error performance approximation of the CCMA scheme, is obtained as
z = PM
m=1zm
M . (4.23)
Using z, the probability of bit error of existing (non-cooperative) CCMA over fading channels is now calculated, both with and without diversity. For the ideal cooperation case, i.e. where each user perfectly decodes its partner’s codewords, the performance of the proposed scheme becomes identical to that of CCMA scheme with dual space diversity reception. The BER performance conditioned on the transmit channels of the cooperating users can also be written as
P (z|γ1d,γ2d) = Q r z γ1d+ γ2d ! , (4.24) where Q(x) = √1 2π R∞ x e −t2/2
dt is the well known Gaussian error function , γ1d and γ2dare the
instantaneous SNRs of the transmit channels of user 1 and 2, respectively. To obtain the average error probability P (z) in fading channels, calculation has to be done (4.24) over all the fading events of the users [108] and is given by
P (z) = Z ∞ 0 Z ∞ 0 P (z|γ1dγ2d)p(γ1d)p(γ2d)dγ1ddγ2d, (4.25)
where p(γ1d) and p(γ2d) are the PDF of fading distributions of user 1 and 2 respectively. In [108]
a unified approach to calculate the probability of error of linearly modulated single user digital signals in arbitrary fading channels is proposed. This approach, using moment generating func- tion (MGF) and alternate representation of Q function, is originally proposed by Craig in [107]. It allows the expressions within indefinite integrals of fading events in (4.25) to be accurately approximated using set of definite integrals. Using this approach P (z) can then be written as
P (z) = 1 π Z π/2 0 1 + zΓ1d sin2θ !−1 1 + zΓ2d sin2θ !−1 dθ, (4.26)
where Γ1d and Γ2d , are the ensemble average SNR of fading distributions of the user 1 and 2
with instantaneous SNR of γ1dand γ2d, respectively. The upper bound on the P (z) is obtained
by knowing the fact that the integrands in (4.26) are maximized when sin2θ = 1. Since average distance of codewords is used for our analysis, an approximation of probability of error of CCMA signals with dual receive diversity can be shown as
P (z) ≈ 1 2 1 1 + zΓ1d ! 1 1 + zΓ2d ! . (4.27)
For the case of the CCMA without diversity, using the error metric of (4.23), the the probability of error P (z) can be derived as follows:
P (z) ≈ 1 2 1 1 + zΓd ! , (4.28)
where Γd= 1/2(Γ1d+ Γ2d) is the averaged SNR of users’ channels to the base-station receiver.
Figure 4.7: Two user collaborative codeword combinations
4.3.4 Performance Results and Comparisons
Figure 4.8: BER performance in flat Rayleigh fading channels for cooperative and non-cooperative 2-user CCMA schemes
In this subsection, the performance bounds and simulation results of the cooperative and non- cooperative (with and without receive diversity) CCMA schemes are presented under different channel conditions. A simple 2-user CCMA system with BPSK mapping and two codewords per user each of length 3 is used. The modulated codewords of user 1 and user 2 are C1 =
{1, 1, 1}, {1, −1, 1} and C2 = {1, −1, 1}, {−1, 1, 1} , respectively and shown in Figure 4.7. It is
assumed that all the users’ and the base-station receivers have perfect knowledge of their received channels.
In Figure 4.8, the derived BER performance bounds of the non-cooperative CCMA using the codewords as given in equations (4.27) and (4.28)are shown. For the purpose of verification, the simulation results are also obtained and shown in Figure 4.8. It is noted that the derived BER bounds become tighter with the increase of diversity order. The performance of the cooperative CCMA with inter-user channel SNR gain of (β1= β2= 20dB) is shown as expected to be within
the range of the dual diversity and no diversity bounds. Also, the BER performance of a single user BPSK with dual diversity using maximum ratio combining (MRC) is shown for comparison. Figure 4.9 shows the BER simulation results of the proposed 2-user cooperative and the non-
Figure 4.9: BER performance in flat Rayleigh fading channels for the 2-user cooperative CCMA under different inter-user channel SNR gains
cooperative CCMA with same channel settings. The ratios β1and β2described in (4.16) are used
to quantify the degree of cooperation of the proposed scheme. The variances of all users transmit channels to the base-station are assumed to be equal to one (σ21d = σ2d2 = 1) and β1 = β2.
Then as expected, as β1 increases, the degree of cooperation increases and the proposed scheme
shows rapid improvement in BER performance. Also it is noted from the figure that even when all the channels have average equal variances i.e. β1 = 0dB, the scheme still offers several dB of
Eb/N0 gain compared to the non-cooperative scheme. When β1 = 20dB, at the BER of 10−3,
the performance of cooperative CCMA is only around 0.5dB worse to that of CCMA with dual receive diversity. Figure 4.10 shows the BER performance of CCMA schemes for the case of non-identical average variances of each cooperating users’ transmit channels to the base-station
Figure 4.10: BER performance in flat Rayleigh fading channels for the 2-user cooperative CCMA and non-cooperative CCMA with channel asymmetry condition of βd = 10dB and inter-user
SNR gains β1= 0dB, β2= 10dB
receiver. The condition, also termed as ‘channel asymmetry’, is defined by the ratio given below βd=
σ1d2
σ2d2 (4.29)
For the channel asymmetry ratio of βd= 10dB, when the inter-user channels have relative SNR
gain of β1 = 0dB, β2= 10dB, the cooperative CCMA scheme offers significant improvement in
error performance for both users compared to that with non-cooperative CCMA without diversity. This result is very beneficial for the mobile users, and particularly for the weaker users as their performances are now significantly less sensitive to the fading than in the non-cooperative case.
The transmit antenna diversity and user collaborative diversity techniques for CCMA have shown to be very effective for operation in fading environments. Unlike CCMA where all users are detected jointly, in CDMA each user is detected separately while other users contribute to MAI. Hence achieving full diversity in this case becomes more challenging, the work presented in next section addresses this problem.