The transmission in the second, and the third-time slots depend on the cooperation scheme, DF or AF. The three-time slots DF scenario is investigated in Section “DF orthogonal three-time slots scenario”, and the three-time slots AF scenario is investigated in Section “AF orthogonal three-time slots scenario”.
DF orthogonal three-time slots scenario
As illustrated in Table
1, the received signal
at the destination node in time slot T
kfor DF scheme is as follows:
(4)
where
is the received noise at the destination node in time slot T
k, and
is the jth user decoded symbol at the ith relay. The ith ordered relay transmits with power
equals half the source power (
) for each user. Defining γ
(0.5D)as the instantaneous SNR for the jth user at the destination node after using MRC, and assuming the relays R
b0 and R
b1 decoded the symbol x
jcorrectly (i.e.,
), then γ
(0.5) is obtained as:
(5)
where , is defined as , and , are the instantaneous end-to-end SNR of the best and the next-best relays, respectively. The factor in (5) is due to the fact that the best and next-best relays are shared between the two users with equal power, where for i∈{0,1}.
The best relay Rb0is the relay with the maximum instantaneous end-to-end SNR at the destination node, i.e., . The next-best relay Rb1is the relay with the next-maximum instantaneous end-to-end SNR at the destination node, i.e., , where l=1,…,N (l is used as an index for the relay without ordering). The selection of the best relay Rb0 and next-best relay Rb1from the N available relays is determined by ordering the instantaneous end-to-end SNRs from the N relays as follows .a In the following, the probability density function (PDF) and the MGF of the end-to-end SNR γ(0.5D) are derived in order to evaluate the BEP and outage probability performances of the proposed scenario.
In order to find the PDF
of the instantaneous end-to-end SNR
from the best and the next-best relays, we consider the following. Instead of dealing with the decoding set C as in [
5], we assume that the relay is selected from the N available relays. However, if a relay cannot decode the message correctly, it will not transmit and hence, the instantaneous end-to-end SNR will be set to zero [
25]. The lth relay can decode the message of the source S
j if
is greater than some threshold value Δ
TH, i.e., if
. Defining β as the probability of erroneously decoding the message, then β is computed as
, where
is an exponential random variable random variable with a parameter λ
SR. Assuming the identical random variables, then Δ
TH can be computed as the value of
that is sufficient to satisfy a given transmission rate R. In other words, Δ
TH is the threshold value satisfying the inequality
or equivalently
. The relay R
lthat satisfies this condition is considered in the decoding set C [
25,
26]. Defining the unordered instantaneous end-to-end SNR at the destination node from the
l th relay as
, ∀l={1,…,N},
is then obtained as:
The PDF
, and the cumulative density function (CDF)
of
are given as [
17]:
(6)
(7)
where μ(x) is the unit step function, δ(x) is Dirac’s delta function, and λRDis the parameter of the exponential random variable characterizing the received SNR at the destination node from the relay. All relay-destination links are assumed to be iid random variables with .
In order to find the PDF of the SNR of the best and next-best relays, we refer to the order statistics of random variables. Let X
1, X
2, …, X
N be defined as iid random variables with PDF f
X(x) and CDF F
X(x). In addition, define the ordered random variables Y
1<Y
2⋯<Y
N, where Y
1=min
lX
l, and Y
N=max
lX
l for l∈{1,…N} then the PDF of the kth ordered random variable Y
kis obtained as [
27]:
(8)
and the joint PDF of the two ordered random variables (Y
r and Y
s), where r<s is given by [
28]:
(9)
From (8) the PDFs of the received SNR
, and
from the best, and next-best relays are, respectively obtained as:
(10)
(11)
The joint PDF of the received SNR from the ordered relays (R
b0,R
b1) using (9) is obtained as:
(12)
Substituting (6) in (12), results in:
(13)
Note that,
, and
are dependent random variables due to the ordering of the instantaneous end-to-end SNR. The PDF of their sum
is computed as [
29]:
(14)
Defining c
k as:
(15)
then A
(DF), B
(DF), D
(DF), and E
(DF) are computed as given in column 2 of Table
2.
Table 2
Coefficients of the PDF, the MGF, the BEP, and the
P
out
for DF sharing scheme
A(DF) |
| a(DF) |
|
| + c2(1−β)βλRD | | |
B(DF) |
| b(DF) |
|
D(DF) |
| f(DF) |
|
| | |
|
|
|
|
|
The MGF
of the received SNR
is obtained using (14) as:
(16)
For the case of sharing the best and next-best relays, it is required to compute the MGF of
, which can be obtained simply from
as
. Since the received SNR from the source-destination link
is independent of the received SNR from the relay-destination link
, the MGF
of the received SNR at the destination node after using MRC is
. Applying the partial fraction expansion, we arrive at:
(17)
where a
(DF),
b(DF), f
(DF), and
(assuming that λ
SD ≠ λ
RD or multiple of it for simplicity, but the analysis can be easily extended), are given as in column 4 of Table
2.
Since MRC is used at the destination node, the SEP can be calculated by averaging the multichannel conditional SEP over the PDF of the random variable representing the received end-to-end SNR at the destination node [
14]. The SEP for MPSK and MQAM are respectively obtained by using the MGF Ψ(s) of the received end-to-end SNR as follows [
30]:
(18)
(19)
where
. In this article, the SEP is calculated only for BPSK modulation by substituting M=2 in (18), but it can be easily extended to MPSK, and MQAM using (18) and (19), respectively, and can be expressed in a closed-form using the hypergeometric function [
30]. In the following we obtain the BEP for different sharing scenarios. The BEP is computed using (18) as:
(20)
Defining the function F
n(u) as:
(21)
where
and n is an integer. The integral for n=1 simplifies to
. The
is obtained by substituting (17) into (20), which can be computed as follows:
(22)
The outage probability P
out is defined as the probability of the end-to-end SNR Γ when it falls below a threshold value γ
TH, and computed as:
(23)
For the two best ordered relays DF sharing scenario, the CDF
of the end-to-end SNR γ
(0.5DF) can be easily obtained from the MGF
given in (17). The outage probability
can then be formulated as:
(24)
The diversity order of sharing the two ordered best relays can be investigated using asymptotic analysis of the BEP or the outage probability P
outat high SNR values [
3,
13]. Another approach, is to use the asymptotic analysis of the PDF or the MGF of the end-to-end SNR [
31‐
33]. We follow the latter approach using the MGF of the end-to-end SNR at the output of the MRC. Using the results of [
31], the MGF can be approximated as s→∞ by b|s|
−d + O(|s|
−d),
b where d is the diversity order, and b is related to the coding gain. Writing
as a division of two polynomials
, where B(s) and A(s) are the numerator and denominator polynomials, respectively. A(s) can be written as,
, which can be approximated for s→∞ as
. The numerator polynomial can be found by collecting and combining the corresponding terms, which is clearly of degree less than the denominator polynomial. Taking only the constant term of the numerator polynomial, and divide this term by the approximation of the denominator polynomial results in the term b|s|
−(N + 1). This means that the diversity order is N + 1. Other terms which result from the division of the numerator polynomial with the approximation of the denominator polynomial contribute to O(|s|
−(N + 1)).
AF orthogonal three-time slots scenario
As illustrated in Table
1, the received signal
at the destination node in time slot T
kfor AF scheme is as follows:
(25)
where is the additive noise at the destination, and is the normalizing factor at the relay, which depends on the instantaneous CSI between the jth source and the ith ordered best relay. Assuming that each relay knows its instantaneous channel information , the normalizing factor using (3) is .
The end-to-end received SNR at the destination node for AF scheme (with instantaneous CSI at the lth relay) with
, and using the normalizing factor
is obtained as:
(26)
which can be upper and lower bounded as [
7,
34]:
(27)
The upper bound in (27) is shown to be tight, and can be used to simplify the analysis [
22]. It is easy to show that the PDF of the upper bound in (27) for Raleigh flat fading channels is an exponential random variable with parameter
. The upper bound of the received end-to-end SNR γ
(0.5AF)at the destination node after using MRC, and using (25) and (3) is given as:
(28)
where with and are the upper bound of the end-to-end SNR from the best, and the next-best relays respectively. The best relay Rb0is selected as the relay with the maximum upper bound of the end-to-end SNR at the the destination node, i.e., , where l=1,…,N. Similarly, the next-best relay Rb1is selected as the relay with the next-maximum upper bound of the end-to-end SNR at the the destination node, i.e., . The selection of the best and next-best relays Rb0and Rb1, respectively from the N available relays is done using the ordering of the upper bound of end-to-end SNRs from the N available relays as: .
The joint PDF of the upper bound received SNRs from the ordered relays R
b0 and R
b1 using (9) can then be obtained as:
(29)
Using (29), and following a similar procedure to that followed in Section “DF orthogonal three-time slots scenario”, the BEP
γ(0.5AF) can be derived as:
(30)
where A
(AF), B
(AF), and
are defined in column 2 of Table
3, and a
(AF), b
(AF),
and f
(AF) are defined in column 4 of Table
3. The outage probability
of sharing the two ordered best relays for AF scheme is then found using (23) as:
(31)
Table 3
Coefficients of the PDF, the MGF, the BEP, and the
P
out
for AF sharing scheme
A(AF) |
| a(AF) |
|
B(AF) |
| b(AF) |
|
|
|
|
|
| | f(AF) |
|
The diversity order of sharing the two ordered best relays for AF scheme is also N + 1, which can be found from the similarity between BEPγ(0.5AF) for AF sharing (30) and the BEPγ(0.5DF)for DF sharing (22).
It is worth noting that, for AF sharing scenario, the ordering of the best, and next-best relays depends on the relay’s transmitted power with the assumption that the relays transmit with the same power level. Therefore, the best and next-best relays in this scenario are different from the best and next-best relays without sharing. In the sharing scenario, the best relay is selected as
but the best relay without sharing is selected as
. It is clear that the factor
affects the ordering of the relays. The same result holds for the next-best relay. In general the ordered best relays for sharing and without sharing AF are different (even the relays transmit with equal power). Whereas for DF scenario the relays are ordered depending on the received SNR at the destination as given by
if the relays transmit with full power, and as
if the relays transmit with half power. The factor
in the last expression does not affect the ordering of the relays, which can be removed from the expression without affecting the ordering. The different cases for AF ordering are illustrated in Table
4, the rows 3 and 4 illustrate that the sharing scenario may use different relays from the best and next-best relays which were ordered based on full power transmission. Simulation results show that AF sharing scenario based on half power allocation achieves full diversity order and outperforms the BEP performance of the best relay (alone) scenario. The ordering based on relay half power allocation is used for two reasons: First, the BEP performance of the sharing based on half power ordering outperforms the sharing based on full power ordering. Second, using half power ordering simplifies deriving the PDF expression (29).
Table 4
Best relay selection criterion for AF scheme
|
|
|
Same
|
|
|
| Different |
|
|
| Different |