3.3 Dual Diversity in Correlated Hoyt Fading channels
3.3.4 Selection Combining Receiver
Solving the integral using [4, (3.381.1)] (reproduced in Equation B.6), the above expression can be given as
Fα(α) = q2(1−ρ2)
∑
∞ k1=0∑
∞ k2=0∑
∞ k3=0∑
∞ k4=0(2k1−1)!!(2k2−1)!! k1+12
k3 k1+12
k4
4
i=1∏ki!
Γ2(k1+k2+1)(k1+k2+1)k3
× ρq2
2
2(k1+k2) 1−q2k3+k4
(k1+k2+1)k4g
k1+k2+k3+1, α2 2q2(1−ρ2)
×g
k1+k2+k4+1, α2 2q2(1−ρ2)
. (3.77)
An expression for the PDF ofαcan be obtained by the differentiation ofF(α)in Equation 3.77 w.r.t.
α. From the relationγsc= ENb0α2, the PDF of the combiner output SNR fγsc(γsc)can be obtained by the transformation of RV. Putting the values ofσx, σxand ENb0 as discussed in Section A.1 followed by necessary algebraic manipulations, an expression for the PDF of the correlated dual-SC output SNRγsc, can be given as
fγsc(γsc) = 1+q2 2πq2¯γ
∑
∞ k1=0∑
∞ k2=0∑
∞ k3=0∑
∞ k4=0ρ2(k1+k2)ζ2k1 1+2k2+k3+k4γ2ksc1+2k2+k3+k4+1e−
2ζ1 q2γsc
4
i=1∏ki!
k1+k3+12
k2+1
2 k1+k4+12
k2+1 2
×
1−q2 q2
k3+k4
1F1
1;k1+k2+k3+2;ζq12γsc
k1+k2+k3+1
+1 F1
1;k1+k2+k4+2;ζq12γsc
k1+k2+k4+1
, (3.78)
where ζ1=∆ 2¯γ1+q(1−ρ22). For ρ =0 and q =1, Equation 3.78 can be shown reduces to fγsc(γsc) =
2¯ γe−γscγ¯
1−e−γscγ¯
which is same as the result in [10, (7.60)]. Also, for independent fading channels i.e.,ρ=0, Equation 3.78 can be simplified to the obtained expression in Equation 3.6 forL=2.
Upper Bound on Truncation Error
It can be observed that Equation 3.78 consists of four infinite series. In the numerical evaluation, it is practicable to include a finite number of terms for each series in the sum. This results in a truncation error of the expression. Considering equal number of terms ‘K’ of each infinite series in the sum, it can be shown that the truncation error thus occurs has an upper bound. This upper bound is derived in Section A.10.4 in Appendix. This expression can be given as
EK ≤ 2(1−ρ2)[(1−q)ρ2]2KB2 2K+12,K+12
π(3K+1)q2K+2(K!)4Γ2(K+12) ζ61K+2γ6scK+1e−
2ζ1 q2γsc
×1F1
1 ζ1(1−q2q2)γsc K+1
2 1F1
1;3K+2;ζ1qγ2sc
1F1
1 (ρζ1γsc)2 K+1
×1F3
1 (ρζ1γsc)2
K+1 K+12 K+12
. (3.79)
Moments of Output Signal-to-Noise Ratio
The definition of moment is given in Equation 3.9. Putting Equation 3.78 into Equation 3.9, an expression for theNth moment ofγsc can be given as
E[γNsc] = ζ21(1−ρ2) πq2
∑
∞ k1=0∑
∞ k2=0∑
∞ k3=0∑
∞ k4=0ρ2(k1+k2)ζ2k1 1+2k2+k3+k4 4
i=1∏ki!
k1+k3+12
k2+1
2 k1+k4+12
k2+1 2
×
1−q2 q2
k3+k4
1
k1+k2+k4+1 Z∞
0
γ2ksc1+2k2+k3+k4+N+1e−
2ζ1 q2γsc
×1F1
1;k1+k2+k4+2;ζ1 q2γsc
dγsc+ 1
k1+k2+k3+1
× Z∞ 0
γ2ksc1+2k2+k3+k4+N+1e−
2ζ1 q2γsc
1F1
1;k1+k2+k3+2;ζ1 q2γsc
dγsc
. (3.80) The integration in Equation 3.80 can be solved using [4, 7.621.4] (reproduced in Equation B.10).
Thus, an expression for theNth moment of the output SNR of the dual correlated SC combiner can
be obtained as E
γNsc
= q2N+2(1−ρ2) 2N+2πζ1N
∑
∞ k1=0∑
∞ k2=0∑
∞ k3=0∑
∞ k4=0ρq22(k1+k2)Γ(2k1+2k2+k3+k4+N+2) 4
i=1∏ki!
k1+k3+12
k2+1
2 k1+k4+12
k2+1 2
× 1−q2k3+k4
(2F1 1,2k1+2k2+k3+k4+N+2;k1+k2+k4+2;12 k1+k2+k4+1
+2F1 1,2k1+2k2+k3+k4+N+2;k1+k2+k3+2;12 k1+k2+k3+1
)
. (3.81)
It can be verified that for ρ=0, Equation 3.81 simplifies to Equation 3.11 withL=2. For N=1, Equation 3.81 gives ¯γsc, the average SNR for dual correlated SC receivers. It can be shown that for q=1 andρ=0 i.e for Rayleigh fading case, ¯γsc simplifies to [31, (12)] (¯γsc= 32γ).¯
Outage Probability
The outage probability is defined in Section 2.3. Thus, substituting Equation 3.78 in Equation 2.16, an expression for the outage probability for correlated dual-SC combiner can be given as
Pout(γth) = ζ21(1−ρ2) πq2
∑
∞ k1=0∑
∞ k2=0∑
∞ k3=0∑
∞ k4=0ρ2(k1+k2)ζ2k11+2k2+k3+k4 4
i=1∏ki!
k1+k3+12
k2+1
2 k1+k4+12
k2+1 2
×
1−q2 q2
k3+k4
γth
R
0 γ2ksc1+2k2+k3+k4+1e−
2ζ1 q2γsc
1F1
1;k1+k2+k4+2;ζq12γsc
dγsc
k1+k2+k4+1
+
γth
R
0 γ2ksc1+2k2+k3+k4+1e−
2ζ1 q2 γsc
1F1
1;k1+k2+k3+2;ζq12γsc
dγsc
k1+k2+k3+1
. (3.82)
Expressing the hypergeometric function in infinite series, the integrations in Equation 3.82 can be solved applying [4, (3.381.1)] (reproduced in Equation B.6). Thus, an expression for the outage
probability can be given as Pout(γth) = q2(1−ρ2)
4π
∑
∞ k1=0∑
∞ k2=0∑
∞ k3=0∑
∞ k4=0ρq22(k1+k2)
1−q2k3+k4
4
i=1∏ki!
22k1+2k2+k3+k4 k1+k3+12
k2+1 2
× 1
k1+k4+12
k2+1 2
∑
∞ k5=0g
2k1+2k2+k3+k4+k5+2,2ζq21γth
2k5(k1+k2+k4+1)(k1+k2+k4+2)k5
+
∑
∞ k6=0g
2k1+2k2+k3+k4+k6+2,2ζq21γth
2k6(k1+k2+k3+1)(k1+k2+k3+2)k6
. (3.83)
Average Bit Error Rate
ABER is given in Equation 2.17, which need the of PDF of γsc and the conditional bit error rate pe,coh(ε|γ) for evaluation. The conditional bit error rate (conditioned on the received SNR) for different digital modulation schemes are listed in Table 2.1. In this work, for binary coherent and noncoherent modulations, we obtain expressions for ABER.
1. Binary Coherent Modulations
For binary coherent modulation i. e., BPSK and BFSK, an expression for pe,coh(ε|γ)can be obtained by evaluating the entries for MPSK and MFSK modulations in Table 2.1, forM=2.
A simplified combined expression for this conditional BER is also given in Equation 2.24.
Thus, puttingpe,coh(ε|γ)from Equation 2.24 and fγsc(γsc)from Equation 3.78 into Equation 2.17, binary coherent ABER can be given as
Pe,ch(¯γ) = ζ21(1−ρ2) πq2
∑
∞ k1=0∑
∞ k2=0∑
∞ k3=0∑
∞ k4=0ρ2(k1+k2)ζ2k1 1+2k2+k3+k41−
q2 q2
k3+k4
4
i=1∏ki!
k1+k3+12
k2+12 k1+k4+12
k2+12
×
R∞
0 γ2ksc1+2k2+k3+k4+1Q(√2aγsc)e−
2ζ1 q2γsc
1 F1
1;k1+k2+k4+2;ζq12γsc
dγsc
k1+k2+k4+1
+ R∞
0γ2ksc1+2k2+k3+k4+1Q(√2aγsc)e−
2ζ1 q2 γsc
1 F1
1;k1+k2+k3+2;ζq12γsc
dγsc
k1+k2+k3+1
.
(3.84) Expressing the hypergeometric function in infinite series [4, 9.14.1] (reproduced in Equation B.11) andQ(·)function in terms of incomplete gamma function (using [1, A-(8a)], reproduced in Equation B.14 ) the above equation can be written as
Pe,ch(¯γ) = ζ21(1−ρ2) 2q2π√π
∑
∞ k1=0∑
∞ k2=0∑
∞ k3=0∑
∞ k4=0ρ2(k1+k2)ζ2k11+2k2+k3+k4 4
i=1∏ki!
k1+k3+12
k2+12 k1+k4+12
k2+12
×
1−q2 q2
k3+k4(
1
k1+k2+k4+1
∑
∞ k5=0ζ1 q2
k5 1
(k1+k2+k4+2)k5
× Z∞ 0
γ2ksc1+2k2+k3+k4+k5+1e−
2ζ1 q2γsc
Γ1 2,aγsc
dγsc+ 1
k1+k2+k3+1
∑
∞ k6=0ζ1
q2 k6
× 1
(k1+k2+k3+2)k6
∞
Z
0
γ2ksc1+2k2+k3+k4+k6+1e−
2ζ1 q2γsc
Γ1 2,aγsc
dγsc
. (3.85) Solving the integral (using [1, A-6]) (reproduced in Equation B.13), an expression for ABER can be given as
Pe,ch(¯γ) =
=
√aζ12q3(1−ρ2) 2π√π(aq2+2ζ1)52
∑
∞ k1=0∑
∞ k2=0∑
∞ k3=0∑
∞ k4=0ρq22(k1+k2)(1−q2)k3+k4 4
i=1∏ki!
k1+k3+12
k2+1
2 k1+k4+12
k2+1 2
×
ζ1 aq2+2ζ1
2k1+2k2+k3+k4( ∞
k
∑
5=0Γ 2k1+2k2+k3+k4+k5+52ζk15 (k1+k2+k4+1)(k1+k2+k4+2)k5
×2 F1
1,2k1+2k2+k3+k4+k5+52;2k1+2k2+k3+k4+k5+3;aq22ζ+2ζ1 1
(2k1+2k2+k3+k4+k5+1)(aq2+2ζ1)k5
+
∑
∞ k6=0Γ 2k1+2k2+k3+k4+k6+52 ζk16
(k1+k2+k3+2)k6(k1+k2+k3+1)(2k1+2k2+k3+k4+k6+2)(aq2+2ζ1)k6
×2F1
1,2k1+2k2+k3+k4+k6+5
2; 2k1+2k2+k3+k4+k6+3; 2ζ1
aq2+2ζ1
. (3.86) It can be shown that for independent channels i.e.,ρ=0, Equation 3.86 simplifies to Equation 3.16 withL=2.
2. Binary Non-coherent Modulations
As discussed in previous sections, substituting pe,nch(ε|γ) from Equation 2.25 and fγsc(γsc) from Equation 3.78 into Equation 2.17, the ABER expression can be given as
Pe,nch(¯γ) = ζ21(1−ρ2) 2πq2
∑
∞ k1=0∑
∞ k2=0∑
∞ k3=0∑
∞ k4=0ρ2(k1+k2)ζ2k1 1+2k2+k3+k4 4
i=1∏ki!
k1+k3+12
k2+12
× 1
k1+k4+12
k2+12
1−q2 q2
k3+k4
R∞
0 γ2sck1+2k2+k3+k4+2e−
2ζ1+aq2 q2 γsc
k1+k2+k4+1
×1F1
1;k1+k2+k4+2;ζ1
q2γsc
dγsc+
R∞
0 γ2ksc1+2k2+k3+k4+2e−
2ζ1+aq2 q2 γsc
k1+k2+k3+1
×1F1
1;k1+k2+k3+2;ζ1 q2γsc
dγsc
. (3.87)
The integral can be solved using [4, (7.621.4)] (reproduced in Equation B.10), and the ABER expression can be obtained as
Pe,nch(¯γ) =
=(1−ρ2) 2π
qζ1
aq2+2ζ1 2 ∞
k
∑
1=0∑
∞ k2=0∑
∞ k3=0∑
∞ k4=0ρ2(k1+k2)Γ(2k1+2k2+k3+k4+2) 4
i=1∏ki!
k1+k4+12
k2+1
2 k1+k3+12
k2+1 2
×
q2ζ1
aq2+2ζ1
2k1+2k2+k3+k4
2F1
1,2k1+2k2+k3+k4+2;k1+k2+k4+2;aq2ζ+2ζ1
1
k1+k2+k4+1
+2 F1
1,2k1+2k2+k3+k4+2;k1+k2+k3+2;aq2ζ+2ζ1 1
k1+k2+k3+1
1−q2 q2
k3+k4
. (3.88) For independent Rayleigh fading channels i.e.,ρ=0 andq=1, Equation 3.88 can be simpli- fied toPe,nch(¯γ) =(2+a¯γ)(1+a¯1 γ) as in [52, (13)].
Results and Discussion
The outage probability expression Equation 3.83, is numerically evaluated and the curves forPout(¯γN) vs. ¯γN are shown in Figure 3.22. In the figure, the variation in the outage probability for different values ofρcan be observed. The variation is more forρ→1 relative to ρ=0. In Figures 3.23 and 3.24, ABER vs. ¯γfor coherent and noncoherent modulations are plotted, as a function ofρ andq. It can be observed that ABER increases with increase inρ, for any value ofq. The required excess SNR (The difference in SNR between the casesρ6=0 andρ=0) vs.ρfor an ABER of 10−3is also plotted in Figure 3.25, for CPSK and DPSK modulations. From this figure it can be observed that the excess SNR increases slowly forρ<0.4 but the rate of increased becomes high forρ>0.4. For example, for CPSK modulation, the excess SNR is close to 0.3 dB forρ=0.4 whereas it is nearly 2 dB forρ=0.8. It can also be observed from Figure 3.23 forρ=0.8, the ABER curve of the PSK modulation coincides withρ=0 ABER curve of FSK modulation, above the SNR of 15dB. It may be due to the high correlation which nullifies the performance advantage of PSK.
q =
{
0.51.0γ− Normalized SNR per branch, N dB
ρ =0.5 10
10 10−2
−1
10−3
10−4 10
0
−5
PoutOutage Probability, ρ =0
ρ =0.8 Outage probability
Simulation
0 5 10 15 20 25
Figure 3.22: Outage probability of correlated dual-SC receiver.
q =
{
0.51.010 10 10−2
−1
10−3
10−4 10
0
−5
γ dB Average SNR per branch, −
ρ =0.5 ρ =0.8
ρ =0 a =0.5
a = 1
Coherent modulation (CFSK)
ABER
(CPSK)
Simulation
0 5 10 15 20 25
Figure 3.23: ABER of correlated dual-SC receiver for CPSK and CFSK modulations.
γ dB Average SNR per branch, − a = 1
ρ =0
ρ =0.8 a =0.5
10 10 10−2
−1
10−3
10−4 10
0
−5
ABER
Simulation (NCFSK) q =
(DPSK)
Non−coherent modulation
{
0.51.00 5 10 15 20 25
Figure 3.24: ABER of correlated dual-SC receiver for NCFSK and DPSK modulations.
Excess SNR, dB
Correlation coefficient, ρ DPSK
CPSK q =
{
0.50.2 Excess SNR for an ABER of 10−30
1 2 3
0 0.2 0.4 0.6 0.8 1
Figure 3.25: Excess SNR required for an ABER of 10−3for dual-SC receiver.