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) network into the system, we address the combined power constraints: 1) the peak interference power at the primary user and 2) the maximal transmit power at the secondary transmitters. 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. To evaluate the performance, we first derive the closed-form statistics of the end-to-end signal-to-noise (SNR) ratio, from which the exact outage probability, ergodic capacity and symbol error rate expressions are derived. To reveal further insights into the effective unreliable backhaul links and the diversity of fading parameters, the asymptotic expressions are also attained. The two interesting non-cooperative and Rayleigh fading scenarios are also investigated. Numerical results are conducted to verify the performance of the considered system via Monte-Carlo simulations.

Appendices

Appendix A: Proof of Lemma 1

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

$$\begin{array}{@{}rcl@{}} F_{{\gamma_{k}^s}}\left( {x}\right) &=& \Pr \left\lbrace \min \left( \bar{\gamma}_{\mathcal{P}}|{h_{k}^{s}}|^{2}, \frac{\bar{\gamma}_{\mathcal{I}}}{|{h_{k}^{p}}|^{2}}|{h_{k}^{s}}|^{2} \right) \leq x \right\rbrace \\ &=& \underbrace{\Pr \left\lbrace |{h_{k}^{s}}|^{2} \leq \frac{x}{\bar{\gamma}_{\mathcal{P}}};\frac{\bar{\gamma}_{\mathcal{I}}}{|{h_{k}^{p}}|^{2}} \geq \bar{\gamma}_{\mathcal{P}}\right\rbrace}_{\mathcal{J}_{1}}\\&& + \underbrace{\Pr \left\lbrace \frac{|{h_{k}^{s}}|^{2}}{|{h_{k}^{p}}|^{2}} \leq \frac{x}{\bar{\gamma}_{\mathcal{I}}};\frac{\bar{\gamma}_{\mathcal{I}}}{|{h_{k}^{p}}|^{2}} \leq \bar{\gamma}_{\mathcal{P}} \right\rbrace}_{\mathcal{J}_{2}}. \end{array} $$

(28)

Because the RV $|{h_{k}^{s}}|^{2}$ and $|{h_{k}^{p}}|^{2}$ are independent each other. We can derive the first term $\mathcal {J}_{1}$ as follows

$$\begin{array}{@{}rcl@{}} \mathcal{J}_{1} &=& \Pr \left\lbrace |{h_{k}^{s}}|^{2} \leq \frac{x}{\bar{\gamma}_{\mathcal{P}}}\right\rbrace \Pr\left\lbrace |{h_{k}^{p}}|^{2} \leq \frac{\bar{\gamma}_{\mathcal{I}}}{\bar{\gamma}_{\mathcal{P}}} \right\rbrace \\ &=& F_{|{h_{k}^{s}}|^{2}} \left( \frac{x}{\bar{\gamma}_{\mathcal{P}}} \right) F_{|{h_{k}^{p}}|^{2}} \left( \frac{\bar{\gamma}_{\mathcal{I}}}{\bar{\gamma}_{\mathcal{P}}} \right), \end{array} $$

(29)

where $F_{|{h_{k}^{s}}|^{2}}(.)$ and $F_{|{h_{k}^{p}}|^{2}}(.)$ are the CDF of Gamma RV $|{h_{k}^{s}}|$ and $|{h_{k}^{p}}|$, respectively. For the second term $\mathcal {J}_{2}$, we can derive by utilize the concept of probability theory, which can be expressed as

$$\begin{array}{@{}rcl@{}} \mathcal{J}_{2} &=&{\int}_{\frac{\bar{\gamma}_{\mathcal{I}}}{\bar{\gamma}_{\mathcal{P}}}}^{\infty} f_{|{h_{k}^{p}}|^{2}}(y) {\int}_{0}^{\frac{xy}{\bar{\gamma}_{\mathcal{I}}}} f_{|{h_{k}^{s}}|^{2}}(x) dxdy \\ &=&{\int}_{\frac{\bar{\gamma}_{\mathcal{I}}}{\bar{\gamma}_{\mathcal{P}}}}^{\infty} f_{|{h_{k}^{p}}|^{2}}(y) F_{|{h_{k}^{s}}|^{2}} \left( \frac{xy}{\bar{\gamma}_{\mathcal{I}}} \right)dy. \end{array} $$

(30)

Expanding from Eq. 6 and the help of [36, Eq. (3.350.2)] , the expression in Eq. 30 can be written as

$$\begin{array}{@{}rcl@{}} \mathcal{J}_{2} &=& \frac{\Gamma\left( \mu_{p},\bar{\gamma}_{\mathcal{I}}/\bar{\gamma}_{\mathcal{P}} \eta_{p}\right)}{\Gamma(\mu_{p})}\\&& - \sum\limits_{l=0}^{\mu_{s}-1}\frac{x^{l}}{l!(\eta_{s}\bar{\gamma}_{\mathcal{I}})^{l} {\Gamma}(\mu_{p}) (\eta_{p})^{\mu_{p}}}\\&& {\int}_{\frac{\bar{\gamma}_{\mathcal{I}}}{\bar{\gamma}_{\mathcal{P}}}}^{\infty} y^{\mu_{p}+l-1} e^{-\left( \frac{1}{\eta_{p}}+\frac{x}{\eta_{s}\bar{\gamma}_{\mathcal{I}}} \right)y}dy \\ &=& \frac{\Gamma\left( \mu_{p},\bar{\gamma}_{\mathcal{I}}/\bar{\gamma}_{\mathcal{P}} \eta_{p}\right)}{\Gamma(\mu_{p})} \\&&- \sum\limits_{l=0}^{\mu_{s}-1}\frac{x^{l}{\Gamma}\left( \mu_{p}+l,\frac{\bar{\gamma}_{\mathcal{I}}}{\bar{\gamma}_{\mathcal{P}}}\left( \frac{1}{\eta_{p}}+\frac{x}{\eta_{s}\bar{\gamma}_{\mathcal{I}}} \right) \right)}{l!(\eta_{s}\bar{\gamma}_{\mathcal{I}})^{l} {\Gamma}(\mu_{p}) (\eta_{p})^{\mu_{p}} \left( \frac{1}{\eta_{p}}+\frac{x}{\eta_{s}\bar{\gamma}_{\mathcal{I}}} \right)^{\mu_{p}+l}}, \end{array} $$

(31)

where ${\Gamma }(\alpha ,x) \overset {\triangle }{=} {\int }_{x}^{\infty } e^{-t} t^{\alpha -1}dt$ denotes the upper incomplete Gamma function [36, Eq. (8.350.2)]. After some manipulations, we obtain the CDF of $\gamma _{k}^{s}$ as follows.

$$\begin{array}{@{}rcl@{}} F_{{\gamma_{k}^s}}\left( {x}\right) &=& 1- {\Phi} e^{-\left( \frac{x}{\bar{\gamma}_{\mathcal{P}} \eta_{s}}\right)} \sum\limits_{i=0}^{\mu_{s}-1} \frac{1}{i!}\left( \frac{x}{\bar{\gamma}_{\mathcal{P}} \eta_{s}}\right)^{i} \\ & -& \sum\limits_{j=0}^{\mu_{s}-1} \binom{\mu_{p}+j-1}{\mu_{p}-1} \epsilon^{\mu_{p}} e^{-\left( \frac{\bar{\gamma}_{\mathcal{I}}}{\bar{\gamma}_{\mathcal{P}} \eta_{p}} \right)}\\&& \frac{x^{j}e^{- \left( \frac{x}{\bar{\gamma}_{\mathcal{P}} \eta_{s}}\right)} {\sum}_{g=0}^{\mu_{p}+j-1}\frac{1}{g!(\bar{\gamma}_{\mathcal{P}} \eta_{s})^{g}}\left( x + \epsilon \right)^{g}}{\left( x + {\epsilon} \right)^{\mu_{p} +j}},\end{array} $$

(32)

with the help of [36, Eq. (8.352.4)]. Hence, the PDF of a particular RV $\gamma _{k}^s \mathbb {I}_{k}$ is modeled by the mixed distribution

$$ f_{\gamma_{k}^s \mathbb{I}_{k}}(x)= (1-{\Lambda})\delta(x)+{\Lambda} \frac{\partial F_{{\gamma_{k}^s}}\left( {x}\right)}{\partial x} , $$

(33)

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

$$ F_{\gamma_{k}^s \mathbb{I}_{k}}(x) = {\int}_{0}^{\infty}f_{\gamma_{k}^s \mathbb{I}_{k}}(x)dx = 1-{\Lambda}({\Theta}_{1}(x) + {\Theta}_{2}(x)). $$

(34)

Appendix B: Proof of Theorem 1

From the definition of S-SNR γ _S in Eq. 5, which is given by

$$ \gamma_{S} = \max_{k \in K} \left( {\gamma_{1}^{s}} \mathbb{I}_{1}, {\gamma_{2}^{s}} \mathbb{I}_{2},...,{\gamma_{k}^{s}} \mathbb{I}_{k},...,{\gamma_{K}^{s}} \mathbb{I}_{K} \right). $$

(35)

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

$$\begin{array}{@{}rcl@{}} F_{\gamma_{S}}(x) &=&F_{{\gamma_{k}^s \mathbb{I}_{k}}}^{K} (x) \\ &=&1+\sum\limits_{k=1}^{K}\binom{K}{k}(-1)^{k} {\Lambda}^{k} ({\Theta}_{1}(x) +{\Theta}_{2}(x))^{k} \\ &=&1+\sum\limits_{k=1}^{K}\binom{K}{k}(-1)^{k} {\Lambda}^{k} \sum\limits_{l=0}^{k}\binom{k}{l}{\Theta}_{1}(x)^{k-l}{\Theta}_{2}(x)^{l}. \end{array} $$

(36)

Applying multinomial theorem provides the following expression

$$\begin{array}{@{}rcl@{}} {\Theta}_{1}(x)^{k-l} &=&\left( {\Phi} e^{-\left( \frac{x}{\bar{\gamma}_{\mathcal{P}} \eta_{s}}\right)} \sum\limits_{i=0}^{\mu_{s}-1} \frac{1}{i!}\left( \frac{x}{\bar{\gamma}_{\mathcal{P}} \eta_{s}}\right)^{i} \right)^{k-l} \\ &=&\sum\limits_{u_{1}...u_{\mu_{s}}}^{k-l} \frac{(k-l)!}{u_{1}!...u_{\mu_{s}}!} \frac{{\Phi}^{k-l} e^{-\left( (k-l)/{\bar{\gamma}_{\mathcal{P}} \eta_{s}}\right)x} x^{{\sum}_{\vartheta=0}^{\mu_{s}-1}\vartheta u_{\vartheta+1}}}{{\prod}_{\vartheta=0}^{\mu_{s}-1}\left( \vartheta!(\bar{\gamma}_{\mathcal{P}}\eta_{s})^{\vartheta} \right)^{u_{\vartheta+1}}}.\\ \end{array} $$

(37)

Again multinomial and binomial theorem give the following expression for Θ₂(x)^l as

$$\begin{array}{@{}rcl@{}} {\Theta}_{2}(x)^{l} &=&\sum\limits_{w_{1}...w_{\mu_{s}}}^{l} \frac{l!}{w_{1}!...w_{\mu_{s}}!} \prod\limits_{t=0}^{\mu_{s}-1} {\binom{\mu_{p}+t-1}{\mu_{p}-1}}^{w_{t+1}} e^{-\left( \bar{\gamma}_{\mathcal{I}} l/{\bar{\gamma}_{\mathcal{P}} \eta_{p}} \right)} \epsilon^{\mu_{p} l} e^{-\left( l/{\bar{\gamma}_{\mathcal{P}} \eta_{s}}\right)x} x^{{\sum}_{t=0}^{\mu_{s}-1} tw_{t+1}} \\ &&\underbrace{ \prod\limits_{t=0}^{\mu_{s}-1} \left( \sum\limits_{g=0}^{\mu_{p}+t-1} \frac{1}{g!(\bar{\gamma}_{\mathcal{P}} \eta_{s})^{g}}(x+\epsilon)^{g} \right)^{w_{t+1}}}_{\mathcal{J}_{3}} \left( \underbrace{ \prod\limits_{t=0}^{\mu_{s}-1}\left( (x+\epsilon)^{\mu_{p} +t}\right)^{w_{t+1}}}_{\mathcal{J}_{4}} \right)^{-1} . \end{array} $$

(38)

Let denotes $\widetilde {L_{a_{n}}} = {\sum }_{b_{n}=0}^{\mu _{p}+n-2} b_{n}a_{b_{n} +1}$, we obtain $\mathcal {J}_{3}$ as in Eq. 39 in the top of next page and

$$\begin{array}{@{}rcl@{}} \mathcal{J}_{3} &=& \left( \sum\limits_{g_{1}=0}^{\mu_{p}-1} \frac{1}{g_{1}!(\bar{\gamma}_{\mathcal{P}} \eta_{s})^{g_{1}}}(x+\epsilon)^{g_{1}} \right)^{w_{1}} \left( \sum\limits_{g_{2}=0}^{\mu_{p}} \frac{1}{g_{2}!(\bar{\gamma}_{\mathcal{P}} \eta_{s})^{g_{2}}}(x+\epsilon)^{g_{2}} \right)^{w_{2}} ... \left( \sum\limits_{g_{\mu_{s}}=0}^{\mu_{p}+\mu_{s}-2} \frac{1}{g_{\mu_{s}}!(\bar{\gamma}_{\mathcal{P}} \eta_{s})^{g_{\mu_{s}}}}(x+\epsilon)^{g_{\mu_{s}}} \right)^{w_{\mu_{s}}} \\&=& \sum\limits_{a_{1,1}...a_{1,\mu_{p}}}^{w_{1}} \sum\limits_{a_{2,1}...a_{2,\mu_{p}+1}}^{w_{2}} ... \sum\limits_{a_{\mu_{s},1}...a_{\mu_{s},\mu_{p} + \mu_{s}-1}}^{w_{\mu_{s}}} \frac{w_{1}!}{a_{1,1}!...a_{1,\mu_{p}}!} \frac{w_{2}!}{a_{2,1}!...a_{2,\mu_{p}+1}!} ... \frac{w_{\mu_{s}}!}{a_{\mu_{s},1}!...a_{\mu_{s},\mu_{p} + \mu_{s}-1}!} \\&& \frac{1}{{\prod}_{b_{1}=0}^{\mu_{p}-1}\left( b_{1}!(\bar{\gamma}_{\mathcal{P}}\eta_{s})^{b_{1}} \right)^{a_{1,b_{1}+1}}} \frac{1}{{\prod}_{b_{2}=0}^{\mu_{p}}\left( b_{2}!(\bar{\gamma}_{\mathcal{P}}\eta_{s})^{b_{2}} \right)^{a_{2,b_{2}+1}}}...\frac{1}{{\prod}_{b_{\mu_{s}}=0}^{\mu_{p}+\mu_{s}-2}\left( b_{\mu_{s}}!(\bar{\gamma}_{\mathcal{P}}\eta_{s})^{b_{\mu_{s}}} \right)^{a_{\mu_{s},b_{\mu_{s}}+1}}} \\ && \sum\limits_{c_{1}=0}^{\widetilde{L_{a_{1}}}} \sum\limits_{c_{2}=0}^{\widetilde{L_{a_{2}}}} ...\sum\limits_{c_{\mu_{s}}=0}^{\widetilde{L_{a_{\mu_{s}}}}} \binom{\widetilde{L_{a_{1}}}}{c_{1}} \binom{\widetilde{L_{a_{2}}}}{c_{2}}... \binom{\widetilde{L_{a_{\mu_{s}}}}}{c_{\mu_{s}}} \epsilon^{\left( \widetilde{L_{a_{1}}}+\widetilde{L_{a_{2}}}+...+ \widetilde{L_{a_{\mu_{s}}}} -(c_{1}+c_{2}+...+c_{\mu_{s}})\right)} x^{\left( c_{1}+c_{2}+...+c_{\mu_{s}} \right)}. \end{array} $$

(39)

$$ \mathcal{J}_{4} = (x+\epsilon)^{{\sum}_{t=0}^{\mu_{s}-1}(\mu_{p}+t)w_{t+1}}. $$

(40)

By pulling (36), (37), (38) together, yields (9).

Appendix C: Proof of Theorem 3

From Eq. 7, we can rewrite it as the Gamma form as

$$\begin{array}{@{}rcl@{}} F_{\gamma_{k}^s \mathbb{I}_{k}}(x) &=& 1- {\Lambda} {\Phi} \frac{\Gamma\left( \mu_{s},\frac{x}{\bar{\gamma}_{\mathcal{P}} \eta_{s}}\right)}{\Gamma(\mu_{s})} \\ &&- {\Lambda} \sum\limits_{j=0}^{\mu_{s}-1}\frac{\epsilon^{\mu_{p}} x^{j} {\Gamma}\left( \mu_{p}+j,\frac{x+\epsilon} {\bar{\gamma}_{\mathcal{P}} \eta_{s}} \right)}{j!{\Gamma}(\mu_{p}) \left( x+\epsilon\right)^{\mu_{p}+j}}. \end{array} $$

(41)

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

$$\begin{array}{@{}rcl@{}} && \lim_{y\rightarrow \infty} \frac{\Upsilon(\mu_{\chi},x/y)}{\Gamma(\mu_{\chi})} \approx 0 \; \text{and} \\ && \lim_{y\rightarrow \infty} \frac{\Gamma(\mu_{\chi},x/y)}{\Gamma(\mu_{\chi})} \approx 1. \end{array} $$

(42)

Substituting (42) into (41) with the given outage threshold γ _th, we can obtain

$$\begin{array}{@{}rcl@{}} \mathcal{P}_{out}^{Asy}(\gamma_{\text{th}}) &&\overset{\bar{\gamma}_{\mathcal{P}} \rightarrow \infty}{=} \prod\limits_{k=1}^{K} \left( 1-{\Lambda} \frac{1}{ \left( 1+\frac{x}{\epsilon}\right)^{\mu_{p}}} \right) \\ &&\overset{\bar{\gamma}_{\mathcal{P}} \rightarrow \infty}{=} \prod\limits_{k=1}^{K} \left( 1-{\Lambda}\right), \end{array} $$

(43)

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

Appendix D: Proof of Corollary 3

The symbol error rate is given by

$$\begin{array}{@{}rcl@{}} P_{e} &=& \frac{A \sqrt{B}}{2\sqrt{\pi}} {\int}_{0}^{\infty} x^{-1/2}e^{-B x} \left( 1+ \sum\limits_{k=1}^{K} \binom{K}{k} (-1)^{k} \widehat{\sum\limits_{k,\mu_{s},\mu_{p},{\Lambda},{\Phi}}} \frac{x^{\widetilde{\varphi_{1}}} e^{-\beta x}} {(x+\epsilon)^{\widetilde{\varphi_{2}}}} \right)dx \end{array} $$

(44)

$$\begin{array}{@{}rcl@{}} &=& \frac{A}{2} + \frac{A \sqrt{B}}{2\sqrt{\pi}} \sum\limits_{k=1}^{K} \binom{K}{k} (-1)^{k} \widehat{\sum\limits_{k,\mu_{s},\mu_{p},{\Lambda},{\Phi}}} \underbrace{{\int}_{0}^{\infty} \frac{x^{\widetilde{\varphi_{1}}-1/2} e^{-(\beta+B) x}} {(x+\epsilon)^{\widetilde{\varphi_{2}}}} dx}_{\mathcal{J}_{5}}, \end{array} $$

(45)

where the integral $\mathcal {J}_{5}$ can be evaluated with the help of [38, Eq. (2.3.6.9)] as

$$ \mathcal{J}_{5} = \epsilon^{\widetilde{\varphi_{1}}+\frac{1}{2}-\widetilde{\varphi_{2}}} {\Gamma} \left( \widetilde{\varphi_{1}}+\frac{1}{2}\right) {\Psi} \left( \widetilde{\varphi_{1}}+\frac{1}{2}; \widetilde{\varphi_{1}}+\frac{3}{2}-\widetilde{\varphi_{2}}; \epsilon (\beta +B)\right)\!, $$

(46)

so that the expression (44) can be written as in Eq. 22.