Outage Probability for Cognitive Heterogeneous Networks with Unreliable Backhaul Connections

Abstract

To enhance the spectrum scarcity of cooperative heterogeneous networks (HetNets) with unreliable backhaul connections, we examine the impact of cognitive spectrum sharing over multiple small-cell transmitters in Nakagami-m fading channels. In this system, the secondary transmitters are connected to macro-cell via wireless backhaul links and communicate with the secondary receiver by sharing the same spectrum with the primary user. Integrating cognitive radio (CR), we address the combined power constraints: (1) the peak interference power and (2) the maximal transmit power. In addition, to exclude the signaling overhead for exchanging channel-state-information (CSI) at the transmitters, the selection combining (SC) protocol is assumed to employ at the receivers. The closed-form statistics of the end-to-end signal-to-noise (SNR) ratio are derived to attain the exact formulas of outage probability and its asymptotic performance to reveal further insights into the effective unreliable backhaul links.

Appendices

Appendix A: Proof of Lemma 1

According to the definition of RV $\gamma _{k}^s$ at particular $\text {SU-T}_k$ , which was given as $\gamma _{k}^s= \min \left( \bar{\gamma }_\mathcal {P}|h_k^s|^2, \dfrac{\bar{\gamma }_\mathcal {I}}{|h_k^p|^2}|h_k^s|^2 \right)$, results the CDF as

$$\begin{aligned} F_{{\gamma _{k}^s}}\left( {x}\right)&= \Pr \left\{ \min \left( \bar{\gamma }_\mathcal {P}|h_k^s|^2, \dfrac{\bar{\gamma }_\mathcal {I}}{|h_k^p|^2}|h_k^s|^2 \right) \le x \right\} \nonumber \\&= \underbrace{\Pr \left\{ |h_k^s|^2 \le \dfrac{x}{\bar{\gamma }_\mathcal {P}};\dfrac{\bar{\gamma }_{\mathcal {I}}}{|h_k^p|^2} \ge \bar{\gamma }_\mathcal {P}\right\} }_{\mathcal {J}_1} \nonumber \\&\qquad \qquad \qquad + \underbrace{\Pr \left\{ \dfrac{|h_k^s|^2}{|h_k^p|^2} \le \dfrac{x}{\bar{\gamma }_{\mathcal {I}}};\dfrac{\bar{\gamma }_{\mathcal {I}}}{|h_k^p|^2} \le \bar{\gamma }_\mathcal {P}\right\} }_{\mathcal {J}_2}. \end{aligned}$$

(A.1)

where

$$\begin{aligned} \mathcal {J}_1&= F_{|h_k^s|^2} \left( \dfrac{x}{\bar{\gamma }_\mathcal {P}} \right) F_{|h_k^p|^2} \left( \dfrac{\bar{\gamma }_{\mathcal {I}}}{\bar{\gamma }_\mathcal {P}} \right) , \; \text {and} \end{aligned}$$

(A.2)

$$\begin{aligned} \mathcal {J}_2&=\int _{\dfrac{\bar{\gamma }_{\mathcal {I}}}{\bar{\gamma }_\mathcal {P}}}^{\infty } f_{|h_k^p|^2}(y) F_{|h_k^s|^2} \left( \dfrac{xy}{\bar{\gamma }_{\mathcal {I}}} \right) dy. \end{aligned}$$

(A.3)

After some manipulations, we obtain the CDF of $\gamma _{k}^s$ as follows.

$$\begin{aligned} F_{{\gamma _{k}^s}}\left( {x}\right)&= 1- \varPhi e^{-\left( \dfrac{x}{\bar{\gamma }_\mathcal {P}\eta _{s}}\right) } \sum _{i=0}^{\mu _{s}-1} \dfrac{1}{i!}\left( \dfrac{x}{\bar{\gamma }_\mathcal {P}\eta _{s}}\right) ^i \nonumber \\&- \sum _{j=0}^{\mu _{s}-1} \left( {\begin{array}{c}\mu _{p}+j-1\\ \mu _{p}-1\end{array}}\right) \epsilon ^{\mu _{p}} e^{-\left( \dfrac{\bar{\gamma }_{\mathcal {I}}}{\bar{\gamma }_\mathcal {P}\eta _{p}} \right) }\nonumber \\&\dfrac{x^{j}e^{- \left( \dfrac{x}{\bar{\gamma }_\mathcal {P}\eta _{s}}\right) } \sum _{g=0}^{\mu _{p}+j-1}\dfrac{1}{g!(\bar{\gamma }_\mathcal {P}\eta _{s})^g}\left( x + \epsilon \right) ^g}{\left( x + {\epsilon } \right) ^{\mu _{p}+j}}, \end{aligned}$$

(A.4)

with the help of [24, Eq. (8.352.4)]. The PDF of a particular RV $\gamma _{k}^s\mathbb {I}_k$ is modeled by the mixed distribution as

$$\begin{aligned} f_{\gamma _{k}^s\mathbb {I}_k}(x)= (1-\varLambda )\delta (x)+\varLambda \dfrac{\partial F_{{\gamma _{k}^s}}\left( {x}\right) }{\partial x} , \end{aligned}$$

(A.5)

where $\delta (x)$ indicates the Dirac delta function. Hence, the CDF of the RV $\gamma _{k}^s\mathbb {I}_k$ can be written as follows

$$\begin{aligned} F_{\gamma _{k}^s\mathbb {I}_k}(x) = \int _{0}^{\infty }f_{\gamma _{k}^s\mathbb {I}_k}(x)dx = 1-\varLambda (\varTheta _1(x) + \varTheta _2(x)). \end{aligned}$$

(A.6)

Appendix B: Proof of Theorem 1

From the definition of S-SNR $\gamma _S$ in (5), which is given by

$$\begin{aligned} \gamma _S = \max _{k \in K} \left( \gamma _1^s \mathbb {I}_1, \gamma _2^s \mathbb {I}_2,...,\gamma _k^s \mathbb {I}_k,...,\gamma _K^s \mathbb {I}_K \right) . \end{aligned}$$

(B.1)

Since all RVs $\gamma _k^s \mathbb {I}_k$ are independent and identically distributed with each other, the CDF of SNR $\gamma _S$ can be written as

$$\begin{aligned} F_{\gamma _S}(x)&=F_{{\gamma _{k}^s\mathbb {I}_k}}^{K} (x)\nonumber \\&=1+\sum _{k=1}^{K}\left( {\begin{array}{c}K\\ k\end{array}}\right) (-1)^k \varLambda ^k (\varTheta _1(x) +\varTheta _2(x))^k \nonumber \\&=1+\sum _{k=1}^{K}\left( {\begin{array}{c}K\\ k\end{array}}\right) (-1)^k \varLambda ^k \sum _{l=0}^{k}\left( {\begin{array}{c}k\\ l\end{array}}\right) \varTheta _1(x)^{k-l}\varTheta _2(x)^l. \end{aligned}$$

(B.2)

Applying multinomial theorem provides the following expression

$$\begin{aligned} \varTheta _1(x)^{k-l}&=\left( \varPhi e^{-\left( \dfrac{x}{\bar{\gamma }_\mathcal {P}\eta _{s}}\right) } \sum _{i=0}^{\mu _{s}-1} \dfrac{1}{i!}\left( \dfrac{x}{\bar{\gamma }_\mathcal {P}\eta _{s}}\right) ^i \right) ^{k-l} \nonumber \\&=\sum _{u_1...u_{\mu _{s}}}^{k-l} \dfrac{(k-l)!}{u_1!...u_{\mu _{s}}!} \dfrac{\varPhi ^{k-l} e^{-\left( (k-l)/{\bar{\gamma }_\mathcal {P}\eta _{s}}\right) x} x^{\sum _{\vartheta =0}^{\mu _{s}-1}\vartheta u_{\vartheta +1}}}{\prod _{\vartheta =0}^{\mu _{s}-1}\left( \vartheta !(\bar{\gamma }_\mathcal {P}\eta _{s})^\vartheta \right) ^{u_{\vartheta +1}}}. \end{aligned}$$

(B.3)

Again multinomial and binomial theorem give the following expression for $\varTheta _2(x)^l$ as

$$\begin{aligned} \varTheta _2(x)^l&=\sum _{w_1...w_{\mu _{s}}}^{l} \dfrac{l!}{w_1!...w_{\mu _{s}}!} \prod _{t=0}^{\mu _{s}-1} {\left( {\begin{array}{c}\mu _{p}+t-1\\ \mu _{p}-1\end{array}}\right) }^{w_{t+1}} \nonumber \\&e^{-\left( \bar{\gamma }_{\mathcal {I}}l/{\bar{\gamma }_\mathcal {P}\eta _{p}} \right) } \epsilon ^{\mu _{p}l} e^{-\left( l/{\bar{\gamma }_\mathcal {P}\eta _{s}}\right) x} x^{\sum _{t=0}^{\mu _{s}-1} tw_{t+1} } \nonumber \\&\underbrace{ \prod _{t=0}^{\mu _{s}-1} \left( \sum _{g=0}^{\mu _{p}+t-1} \dfrac{1}{g!(\bar{\gamma }_\mathcal {P}\eta _{s})^g}(x+\epsilon )^g \right) ^{w_{t+1}}}_{\mathcal {J}_3} \left( \underbrace{ \prod _{t=0}^{\mu _{s}-1}\left( (x+\epsilon )^{\mu _{p}+t}\right) ^{w_{t+1}}}_{\mathcal {J}_4} \right) ^{-1} . \end{aligned}$$

(B.4)

Let denotes $\widetilde{L_{a_n}} = \sum _{b_n=0}^{\mu _{p}+n-2} b_na_{b_n +1}$. By expanding $\mathcal {J}_3$ and $\mathcal {J}_4$, together with (B.2), (B.3), (B.4), yields (9).

Appendix C: Proof of Theorem 3

From (7), we can rewrite it as the Gamma form as

$$\begin{aligned} F_{\gamma _{k}^s\mathbb {I}_k}(x) = 1-&\varLambda \varPhi \dfrac{\varUpsilon \left( \mu _{s},\dfrac{x}{\bar{\gamma }_\mathcal {P}\eta _{s}}\right) }{\varGamma (\mu _{s})} \nonumber \\&- \varLambda \sum _{j=0}^{\mu _{s}-1}\dfrac{\epsilon ^{\mu _{p}} x^j \varGamma \left( \mu _{p}+j,\dfrac{x+\epsilon }{\bar{\gamma }_\mathcal {P}\eta _{s}} \right) }{j!\varGamma (\mu _{p}) \left( x+\epsilon \right) ^{\mu _{p}+j}}. \end{aligned}$$

(C.1)

It can be easily seen that as y goes to infinity,

$$\begin{aligned}&\lim _{y\rightarrow \infty } \dfrac{\varUpsilon (\mu _\chi ,x/y)}{\varGamma (\mu _\chi )} \approx 0 \; \text {and} \nonumber \\&\lim _{y\rightarrow \infty } \dfrac{\varGamma (\mu _\chi ,x/y)}{\varGamma (\mu _\chi )} \approx 1. \end{aligned}$$

(C.2)

Substituting (C.2) into (C.1) with the given outage threshold $\gamma _\text {th}$, we can obtain

$$\begin{aligned} \mathcal {P}_{out}^{Asy}(\gamma _\text {th})&\overset{\bar{\gamma }_\mathcal {P}\rightarrow \infty }{=} \prod _{k=1}^{K} \left( 1-\varLambda \dfrac{1}{ \left( 1+\dfrac{x}{\epsilon }\right) ^{\mu _{p}}} \right) \nonumber \\&\overset{\bar{\gamma }_\mathcal {P}\rightarrow \infty }{=} \prod _{k=1}^{K} \left( 1-\varLambda \right) , \end{aligned}$$

(C.3)

where $\sum _{j=0}^{\mu _{s}-1}(.)$ is dominated by $j=0$ as $\bar{\gamma }_\mathcal {P}\rightarrow \infty$.