Abstract
In this paper, selection and incremental schemes are investigated for multi-input multi-output (MIMO) cooperative systems in which multiple amplify-and-forward relays and space–time block coding (STBC) are applied. In order to make an efficient use of the degrees of freedom of channel and provide full diversity order in the multi-relay STBC network, the selection and incremental protocols are employed by exploiting limited feedback from destination. Maximum ratio combination technique is applied at the destination. Outage probabilities of the both cooperative schemes are analytically derived at high signal to noise ratio regime. It is shown the incremental selection STBC scheme leads to a higher performance compared to the STBC selection relaying scheme and conventional STBC MIMO channels; so that the outage capacity provided by the incremental selection scheme is twice of the selection scheme at the low outage probabilities.
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Appendix
Appendix
Claim
Let Z = f(U, V) + f(U, W) where f(x, y) = xy/(x + y) and also U, V and W are independent random variables. In addition PDF of U is f U (u) = λ 2 ue −λu, and V and W have exponential distribution of μ parameter. Letting h > 0, the probability Pr [Z < h] satisfies
Proof of Claim
It is clear that the solving of the math problem \( \mathop{\lim }\limits_{z \to 0} \frac{1}{{z^{2} }}\Pr (Z < z) = \mathop{\lim }\limits_{z \to 0} \frac{1}{{z^{2} }}F_{Z} (z) \) is wanted where function F Z (z) indicates cumulative density function (CDF) of Z variable. So, \( \mathop{\lim }\limits_{z \to 0} \frac{1}{{z^{2} }}\Pr \left[ {Z \le z} \right] \) can be written as [24],
where f UVW (u, v, w) indicates PDF, and region D should satisfy
From (18) it is clear that for z → 0, v and w should be limited when u varies from 0 to infinite. And also, u should be limited when v and w vary from 0 to infinite. Hence, one can assume that D is made by two region as presented in Table 2. By this description, one can rewrite (17) as follows
For the first term in the right hand side of (19) one has
By taking \( X = \frac{uV}{u + V} \), \( Y = \frac{uW}{u + W} \) and using change of variable t = x/z, P[X + Y < z] satisfies
Since
so it can be derived that \( \frac{\Pr (Y < y)}{y} \to \mu \) as y → 0. Using this fact and after some manipulations, (21) can be simplified as follows
where in the last equality the fact that X variable has the same property as Y variable is applied. Finally one has
Second term in the right hand side of (19) can be calculated as
To derive exact integration bounds along u, one should solve (18) as
Clearly u should be in the region [u 0 , u 1 ] for z → 0, where
and Δ = 4v 2 w 2 + z 2(v − w)2. Since u 0 < 0 for z → 0, therefore one has u ∊ [0, u 1(z)] in (24). Now it can be derived that
By taking again differentiation from (26) one derive that
It can be derived that u 1(0) = 0, \( \frac{\partial }{\partial z}u_{1} (z)\left| {_{{_{z = 0} }} } \right. = 1/2 \), f U (0) = 0 and \( \frac{\partial }{\partial z}f_{U} (z)\left| {_{{_{z = 0} }} } \right. = \lambda^{2} \). Therefore, one has
To solve the third term of (19), it should be note that all of the limits of variables are upper bounded by z, so one has
where last inequality in above benefits from f UVW (u, v, w) ≤ 1. Therefore, clearly it can be derived that
Finally by applying (23), (28) and (30) in (19), claim is proved.
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Poursajadi, S., Madani, M.H. Outage Performance Analysis of Incremental Relay Selection for STBC AF Cooperative Networks. Wireless Pers Commun 83, 2317–2331 (2015). https://doi.org/10.1007/s11277-015-2523-y
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DOI: https://doi.org/10.1007/s11277-015-2523-y