## Abstract

This paper provides a general outage analysis framework for cooperative cognitive networks with proactive relay selection over non-identical Rayleigh fading channels and under both maximum transmit power and interference power constraints. We firstly propose an exact closed-form outage probability expression, which is then exploited for determining the diversity order and coding gain for proactive relay selection scenarios as well as deriving system performance limits at either large maximum transmit power or large maximum interference power. The derived performance metrics bring several insights into system performance behavior without the need of time-consuming Monte-Carlo simulations. Various results confirm the validity of the proposed derivations and show that cooperative cognitive networks with proactive relay selection incur performance saturation and their performance depends considerably on the number of involved relays. In addition, cooperative cognitive networks are significantly better than dual-hop counterparts without any cost of system resources.

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## Notes

Due to the nature of regenerating the received information without noise enhancement, the DF relays are investigated in this paper and hence, literature survey relevant to the AF relays are not necessarily presented (e.g., [2]).

This means that \({\uplambda }_{tr}\) is the inverse of the fading power of the channel between the transmitter

*t*and the receiver*r*, i.e., \({\uplambda }_{tr}=1/\mathcal {E}_{\left| h_{tr}\right| ^2}\{{\left| h_{tr} \right| ^2}\}\) where \(\mathcal {E}_X\{\cdot \}\) denotes the statistical expectation over the random variable*X*.Some authors (e.g., [10]) also refer this scheme as the max-min relay selection.

Due to the nature of two-stage cooperative communications, the system capacity is given as \(C=\frac{1}{2}\log _2(1+\gamma _{SC})\). The capacity outage happens if \(C<U\) where

*U*is the required transmission rate. Equivalently, the outage event happens if \(\gamma _{SC}<z\) where \(z=2^{2U}-1\), which shows the relation between the required transmission rate*U*and the SNR threshold*z*. For example, \(z=3,15\) corresponds \(U=1,2\) bps/Hz, respectively.The Monte-Carlo simulation is well-known, e.g. [31], and hence, a description of how it is conducted is omitted.

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## Acknowledgments

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 102.04-2014.42

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## Appendices

### Appendix 1: Proof of Lemma 1

We rewrite \(\mathcal {G}_1\) in (21) as

Using the equality \(\prod \limits _{k = 1}^K {\left( {1 + {a_k}} \right) } = 1 + \sum \limits _{u = 1}^{K - 1} {\sum \limits _{{s_1} = 1}^{K - u + 1} {\sum \limits _{{s_2} = {s_1} + 1}^{K - u + 2} { \cdots \sum \limits _{{s_u} = {s_{u - 1}} + 1}^K {\prod \limits _{l = 1}^u {{a_{{s_l}}}} } } } } + \prod \limits _{k = 1}^K {{a_k}}\), one obtains

where

It is noted that the integral in (28) is completely solved in closed-form by integrating by parts. Plugging the above into (27), one obtains (22) where

Substituting (28) in the above, we reduce (29) to

Using the fact that \({f_{{Y_q}}}\left( {{y_q}} \right) = {{\uplambda }_{{R_q}L}}{e^{ - {{\uplambda }_{{R_q}L}}{y_q}}}\), one can solve the integral in (30) in closed-form as follows

where the second integral is solved by integrating by parts.

Inserting (31) into (30), the closed form of \(\tilde{\mathcal {G}}_{1i}\) can be expressed as (23), which completes the proof.

### Appendix 2: Proof of Lemma 2

We rewrite \(\mathcal {G}_2\) in (21) as

It is straightforward to infer that \(\int \limits _0^\tau {{f_X}\left( x \right) dx} = 1 - {e^{ - {{\uplambda }_{SL}}\tau }}\) and \({\int \limits _0^\infty {{f_{{Y_k}}}\left( {{y_k}} \right) d{y_k}} }=1\) while \(\int \limits _0^\infty {\frac{{{f_{{Y_k}}}\left( {{y_k}} \right) }}{{\min \left( {y_k^{ - 1}\tau ,1} \right) }}d{y_k}} = 1 + \frac{{{e^{ - {{\uplambda }_{{R_k}L}}\tau }}}}{{{{\uplambda }_{{R_k}L}}\tau }}\) with the aid of (31). Plugging all these results in (32), one obtains (24) which completes the proof.

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Ho-Van, K. On the performance of cooperative cognitive networks with proactive relay selection.
*Wireless Netw* **22**, 2131–2141 (2016). https://doi.org/10.1007/s11276-015-1090-1

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DOI: https://doi.org/10.1007/s11276-015-1090-1