In this section, we investigate the impact of imperfect relay error detection. Assume the source transmit messagesi, recall that the received signal can be written as
rd=√
We start the analysis with an observation about following equation
P(Ci → Cj|Ci) =
Sˆ1K
P( ˆS1K)P(Ci → Cj|Ci, ˆS1K), (3.30)
where we abbreviate ˆS1, ˆS2,· · · , ˆSKas ˆS1K, and the summation is taken over all possible Sˆ1K.
We can see that the diversity, the exponent of LHS, will be dominated by the one with worst exponent in the summand of RHS Thus, we should calculate the worst one of P( ˆS1K)P(Ci → Cj|Ci, ˆS1K) in exponent for different realization of ˆS1K. Now, we express
P( ˆS1K) as
r=1Sm,r is the number of active relay that transmit mth message and nS0 = R−K
m=1nSm is the number of silent relay nodes.
For obtaining equation (3.31), aside from the model of detection accuracy in section 2, we further need an assumption that if a relay decode incorrectly, then it decoded message would be uniformly distributed over {1, 2, · · · , K} \ {i}, i.e., K−11 for each. Such as-sumption rely on a symmetric design of the codebook used in phase I transmission which is consist of many SISO systems and is not of our focus. In more rigorous words, this assumption would not affect the exponent as long as there is no pair of codeword used in phase I has a different SNR-exponent of PEP than others.
To catch the diversity, we simplify (3.31) to
P( ˆS1K) .
= (p0|1ps)m=inSm(p1|0+ ps)nS0. (3.32)
As we can see, the parameters of detection accuracy p0|1and p1|0 will affect the exponent in different manners. Next, to have a thorough analysis, we need to evaluate P(Ci → Cj|Ci, ˆS1K) for all possible ˆS1K. Unfortunately, the math structure is too complicated and we failed to carry it out. Thus, we only investigate the impact of imperfect error detection on the system diversity by following two facts and one proposition.
Fact1 Even with ML decoding at receiver, fixed positive p1|0leads to zero diversity.
Fact2 Diversity order under fixed positive p0|1 do not exceed R2 L for any decoder.
Proposition3 Correlator-like decoder proposed in section 3.1.2 can only retrieve di-versity of order min{L + L01, RL10, RL} under detection accuracy p0|1 .
= P−L01 and p1|0 .
=P−L10.
Proposition3 indicates that correlator-like decoder might not attain diversity of order RL, especially when L01 is small. It would turn out that the argument of (3.21) won’t
work here. ML decoder would probably do better in diversity. Actually, the one that we are incapable to treat its involved math is the PEP using ML decoder under existence of harmful relays. Hence, unlike the case of correlator-like decoder, we are incapable to give a complete analysis for an ML decoder. For example, inFact2, we do not know whether or not using ML decoder can achieve the diversity R2 L under fixed positive p0|1. ML decoder takes the form
It is again a gaussian mixture and need to do the summation over all possible ˆS1K, which we are not able to tackle with both analytically and practically. For this part, we simulate it by computer and have some discussion in section 4.2. For the rest of this section, we give brief arguments about the two facts and one proposition.
Fixed positive p1|0leads to zero diversity We aims to prove
P(Ci → Cj|Ci) .
= ζ1 (3.34)
for some constant ζ1irrelevant to SNR, which implies the zero diversity. To see this, con-sider the particular summand in (3.30) that ˆS1K = (0RL, 0RL,· · · , 0RL)def= O in equation (3.30), which means that ˆS0 = IRL and nS0 = RL, i.e., no relay transmit to destination.
In this case, since the receiver receives only additive noise, we have P(Ci → Cj|Ci, O) .
= ζa
for some constant ζafor any type of decoder used in receiver end. Combing with equation (3.32), we reach
P( ˆS1K = O)P(Ci → Cj|Ci, O) .
= (p1|0+ ps)RLζa
= (p. 1|0)RLζa (3.35)
def= ζb,
where (3.35) follows from ps .
= P−L and p1|0 is a constant. And as mentioned, the diversity will be dominated by the worst exponent one summand of (3.30), hence equation (3.34) holds.
This fact demonstrate a result: we do a quite accurate error detection at each relay and have a very small probability of existence of useless relay. Even though, if we have fixed positive p1|0, it will be significant on the behavior of error probability at high SNR since the decoding error will be dominated by the event that all relays are useless relay in high SNR regime. And a simple conclusion can be made from equation (3.32) is that, to illuminate such effect, we need at least p1|0 .
=P−L of error detection accuracy.
Diversity under fixed positive p0|1do not exceed R2 L In this part, we consider the summand in (3.30) that
Sˆi = IL⊗ diag{1, · · · , 1
and for convenience, we abbreviate such case as ˆS1K = Vij. By symmetry, the receiver will have same favor ofCiandCj. Hence we have
P(Ci → Cj|Ci, Vij) = 1 2.
And again together with the probability of such relay status from equation (3.32), we can obtain
where the dot equal in (3.36) follows from ps .
=P−Land p0|1 > 0 is a constant. Similarly, since diversity is dominated by worst one summand, we conclude that diversity do not exceed R2L under fixed positive p0|1, the possibility for existence of harmful relay.
Correlator-like decoder proposed in section 3.1.2 can only retrieve diversity of order min{L + L01, RL10, RL} under detection accuracy p0|1 .
=P−L01 and p1|0 .
=P−L10 Recall the correlator-like decoder proposed in Sec. 3.1.2,
Cˆsubopt= arg max
And the corresponding pairwise error event,
{rHdEjEHj rd− rHdEiEHi rd > 0}. Proof: We write the PEP as
P(Ci →Cj|Ci, Uj) Then, the claim is equivalent to
P→∞lim P(wP > 0|Ci, Uj) > 0
Define w∞= ˜rHdEjEHj ˜rd− ˜rHdEiEHi ˜rd, where ˜rd= EiSˆih + EjSˆjh. Then obviously, wP −→ w∞ almost surely asP −→ ∞.
Thus, we can prove the claim by showing
P(w∞> 0|Ci, Uj) > 0.
Expand w∞by using ˜rd= EiSˆih + EjSˆjh, through some manipulation, w∞= ˜rHdEjEHj ˜rd− ˜rHdEiEHi ˜rd
= hHSˆj(I − EHj EiEHi Ej) ˆSjh
− hHSˆi(I − EHi EjEHj Ei) ˆSih
X − Y,
where X hHSˆj(IRL− EHj EiEHi Ej) ˆSjh and Y hHSˆi(IRL− EHi EjEHj Ei) ˆSih.
Without loss of generality, we assume Ei = Ej. (Otherwise, codewords i and j cannot be distinguished by receiver.) Then we have IRL − EHj EiEHi Ej being positive definite.
As we can see, X and Y both follow a generalized chi-square distribution with support [0,∞). Further note that X and Y are independent, we obtain
P(w∞> 0|Ci, Uj) = P(X− Y > 0|Ci, Uj)
≥ P({X > 1} ∩ {Y < 1} |Ci, Uj)
= P(X > 1|Ci, Uj) P(Y < 1|Ci, Uj)
> 0 Hence the claim is proved.
This claim indicate that under any existence of harmful relay, an error floor occur when employing correlator-like decoder. When there does not exist harmful relay, on the other hand, by the result established in Sec. 3.1.2, the diversity order will equal to the number of useful relays in the system. Again, reexamining (3.30) and using (3.32), the worst summand of (3.30) could be found as
min{L + L + 01, min
1≤k≤R(R− k)L + kL10}.
Solve the minimization, we can conclude that correlator-like decoder can only retrieve diversity of order min{L + L01, RL10, RL} under detection accuracy p0|1 .
= P−L01 and p1|0 .
= P−L10. Note that when L + L01 dominate, the diversity do not scale with R, the number of relays. This gives us some clue how p0|1 and p1|0 affect the diversity order differently.
Chapter 4 Simulations
4.1 Simulation Setup
We present simulation results in this section. We simulated the system with channel taps L = 2 and number of subcarriers N = 16. And we assume a symmetry power delay profile between relays, i.e., σr,l2 = σr2,l for l = 0, 1 and for all r = r . Setting the noise power to one, we define the SNR as the total transmit power from R relays per unit frequency, i.e. SNR = PR/N. Since our analysis excludes the source node coding, we give the source node an extra power equal to that of each relay in the simulation in order to determine the probability psat relay nodes. Accordingly, we assume ps = β×P−Land the constant β is irrelevant to the diversity order. Hence we just set it a particular number that ensures 0 < ps< 1.