The expression presented in (11) is only valid for a fixed (time-invariant) channel, that is, the fading coefficients K
ij
are fixed. The aim of this paper is to obtain an expression for the bit error probability when the fading coefficients vary according to a Rayleigh distribution.
A. Bit Error Probability
The bit error probability associated with the signal from user 1, at user 2, for a fixed gain is described in (6). Now assuming a nonstatic situation, the average bit error probability can be computed averaging (6) with respect to a Rayleigh distribution
(12)
where γ12 is the average signal-to-noise ratio, defined as
(13)
From (11), we can define two random variables U1 and U2, respectively, as
(14)
(15)
since K10 and K20 are Rayleigh distributed, the support of (14) will be always greater than zero. On the other hand, since we have negative values in the numerator of (15), its support will be all the real line. Taking this into account, we can rewrite (11) as
(16)
To obtain the error probability, we must average , over the probability density function (PDF) of U1 and U2[7]. Thus, we have to evaluate the integral
(17)
In order to calculate P
ef
, we have to know the distribution of U1 and U2, thus to facilitate the calculations, we assume an equal power allocation situation, where a12 = a13 = a23 = a. With this assumption the random variables U1 and U2 will be simplified to
(18)
(19)
Since U1 depends on K10 and K20, it is possible to write the cumulative distribution function (CDF) and the PDF of U1, respectively, as
(20)
(21)
In this case, is the region of the K10 × K20 plane where
(22)
Note that this region is very similar to a rotated ellipse but not exactly an ellipse.
Since K10 and K20 are independent Rayleigh distribution with parameters α10 and α20, respectively, we have
(23)
where
(24)
(25)
(27)
now it is possible to derive the PDF of U1 easily as
(28)
and unfortunately, it is not possible to evaluate (28) in a closed-form solution.
In order to validate the above formulation, Figure 1 shows the analytical and simulated PDF of U1. Note the excellent agreement between them showing the correctness of our formulation.
Following similar rationale, we now find the CDF and PDF of U2. Note that in this case, the region of integration, , will be given by
(29)
leading to the following CDF and PDF, respectively, as
(30)
and
(31)
where
(32)
(33)
where denotes the real part of a number, and R1 and R2 are described in the Appendix.
In the same way as in the first case, (31) cannot be obtained in a closed-form solution. Figure 2 compares the analytical and simulated PDF of U2 in order to validate our formulation.
Once that the PDFs of U1 and U2 were exactly computed, it is possible to obtain the average bit error probability by simply substituting (28) and (31) into (17). Figure 3 shows the simulation result of the bit error probability and the result of our theoretical expression given in (17), where we can observe that both curves are almost coincident. In this figure, . According to Section II-B, we consider three symbols periods, each period with an average power of P. Also, for simplicity, we consider that and are identical.
Although (17) presents the exact solution to the average bit error probability, in some cases, the complexity to compute this expression can be prohibitive. For this reason, we found a very accurate approximation for the bit error probability presented in the sequel.
B. Approximate Bit error Probability
The main problem in order to obtain a simpler expression for the bit error probability is to simplify the PDFs of U1 and U2 given, respectively, in (14) and (15). In order to obtain an approximation, the expressions (14) and (15) can be reduced when λ = 1, σ0 = 1 and a12 = a13 = a23 = 1. Therefore, the new random variables are given by
(35)
(36)
Considering as the region of the plane K10 × K20 where , it can be seen that corresponds to the area of an ellipse whose center is in the origin (0, 0). Unfortunately, the evaluation of the integral (20) is rather complex for the domain . For this reason, we consider a simplified version of , as being the area of a circle expressed as . This simplification can be applied since a circle corresponds to a particular case of the general ellipse. Hence
(37)
This gives
(38)
Since K10 and K20 are independent Rayleigh distributed with parameters α1 and α2, respectively, the PDF of U'
1
given in (38) will result in a chi-square probability distribution with four degrees of freedom [8]. Therefore, our approximation of U1 will be given by
(39)
where γ1 is the mean of U1 given in (14)
Figure 4 shows the comparison between our approximate PDF given in (39) and the computer simulation for the PDF of U1 given in (14) for two different values of λ keeping the same values for a12 = 1, a13 = 2, and a23 = 3. We observe that the curves are very close for both values of λ. Although only these two cases are presented here, many other cases were compared and the approximation still remains very good.
A similar rationale can be applied in order to find a good approximation for U2. The region of the K10 × K20 plane where is similar to (29). Note that the range of varies from , discarding all the distributions with positive support. In order to observe the behavior of the PDF of , a large number of simulations were performed, and the Gaussian distribution proves to fit extremely well in all the cases. Therefore, assuming a Gaussian distribution, the following can be written
(41)
(42)
where
Figure 5 shows the comparison between the approximate PDF given in (42) and the computer simulation for the PDF of U2 given in (15), for two different values of λ. Note that the approximation is less accurate for small values of λ, but this inaccuracy does not have a significant influence in the bit error probability. In all the cases, the approximation fits very well the exact PDF of U2.
Using (39) and (42) into (17), it is possible to obtain a very accurate approximate bit error probability. Fortunately, both integrals can be found in a closed-form solution as
(45)
and
(46)
All these calculations lead to the approximate bit error probability for the λ-MRC detector as shown in (41), where γ12 is given in (13), γ1 is given in (40), γ2 is given in (43), and ν
2
is given in (44).
Assuming an equal power allocation scheme (a12 = a13 = a23 = a), Figure 6 shows the comparison between the theoretical bit error probability presented in (17) using the exact PDFs (28) and (31) and our approximation given in (41). We can observe that both curves are almost the same, validating our approximation.
Our results are quite exact for a different power allocation scheme as well. This can be seen in Figure 7, where a comparison between the exact simulated bit error probability and our approximation given in (41) was performed. In this figure, the following parameters were used α10 = α20 = 1 and α12 = 0.8.
The final approximate expression allows us to determine the optimal value for λ in each case. As stated in [1], when the BS believes that the inter-user channel is "perfect", then λ = 1 and the optimal detector turns out to be the maximal ratio combining [7]. As the inter-user channel becomes more unreliable, i.e., as increases, the value of the best λ decreases toward to zero. In order to demonstrate this behavior, Figure 8 shows the optimized λ* versus the inter-user channel parameter α12. This curve was obtained using computational optimization techniques that minimizes our approximate bit error probability (41) with respect to λ for each value of the inter-user channel parameter, α12. The direct channel parameters α10 = α20 = 1 were kept constant, and the equal power allocation scheme (a12 = a13 = a23 = 1) was adopted.