Opportunistic cooperation in wireless ad hoc networks with interference correlation

Abstract

Compared with conventional direct transmissions, the cooperative transmissions are not always beneficial and redistribute the interference over the network coverage area due to relay transmissions. In this paper, we propose an opportunistic cooperation strategy for a wireless ad hoc network with randomly positioned single-hop source-destination pairs and relays, where each source-destination pair activates the cooperative transmission only when the number of potential relays is not smaller than a cooperation threshold. Such a threshold determines the proportion of concurrent cooperative transmissions and it can be adjusted to enhance the overall network performance. The correlation of nodes’ locations induces the correlation of interference power. Based on stochastic geometry, we derive the correlation coefficient of interference power at a destination during the transmission periods of the sources and relays. The outage probability of opportunistic cooperation is derived for selection combining, while taking into account the interference correlation, relay selection strategy, and spatial distributions of sources and relays. Extensive simulations are conducted to validate the performance analysis. The analytical results evaluate the effectiveness of opportunistic cooperation and provide useful insights on protocol design and parameter setting for large-scale networks.

Appendix Proof of Proposition 1

According to the definition of the correlation coefficient between two random variables, we have

$$ \rho\! =\! \frac{{\mathbb{E}\left[ {{I_{D_{0}:1}}\left( {\Phi}_{D},\! {\Phi}_{C} \right)\!{I_{D_{0}:2}}\left( {\Phi}_{D},\! {\Phi}_{F} \right)} \right]\! -\! \mathbb{E}\left[ {I_{D_{0}:1}}\left( {\Phi}_{D},\! {\Phi}_{C} \right) \right]\!\mathbb{E}\left[ {I_{D_{0}:2}}\left( {\Phi}_{D},\! {\Phi}_{F} \right) \right]}}{{\sqrt {\text{Var}\left( {I_{D_{0}:1}}\left( {\Phi}_{D},\! {\Phi}_{C} \right) \right)} \sqrt {\text{Var}\left( {I_{D_{0}:2}}\left( {\Phi}_{D},\! {\Phi}_{F} \right) \right)} }}, $$

(26)

where $\mathbb {E}\left [ X \right ]$ and Var(X) represent the mean and variance of random variable X, respectively.

Due to the unit mean of fading coefficients, the mean of interference power $I_{D_{0}:1}\left ({\Phi }_{D}, {\Phi }_{C} \right )$ is given by

$$\begin{array}{@{}rcl@{}} \mathbb{E}\left[ I_{D_{0}:1}\left( {\Phi}_{D}, {\Phi}_{C} \right) \right] &=& \mathbb{E}\left[ {\sum\limits_{{s_{i}} \in {{\Phi}_{D}}} {{H_{{S_{i}}{D_{0}}:1}}g\left( {{s_{i}}} \right)} + \sum\limits_{{s_{j}} \in {{\Phi}_{C}}} {{H_{{S_{j}}{D_{0}}:1}} g\left( {{s_{j}}} \right)} } \right] \\ &\overset{(a)}{=}& {\lambda_{S}} {\int}_{{\mathbb{R}^{2}}} {g\left( s \right)\mathrm{d}s},\\ \end{array} $$

(27)

where (a) follows from the Campbell’s Theorem [15].

Similarly, we have $\mathbb {E}\left [ {I_{D_{0}:2}}\left ({\Phi }_{D}, {\Phi }_{F} \right ) \right ] = \left ({\lambda _{D}} + {\lambda _{F}} \right ) {\int }_{{\mathbb {R}^{2}}} {g\left (s \right )\mathrm {d}s}$, where $\lambda _{F} = {\lambda _{C}} \cdot \mathbb {P}\left ({{{\Omega }_{0}} \ne \emptyset } \right )$ denotes the spatial density of PPP Φ _F . As the cooperative transmission is activated only when there exist at least θ _C potential relays, the probability of an empty relay set (i.e., no qualified relays), denoted as $q_{e} = \mathbb {P} \left ({\Omega }_{0} = \emptyset \right )$, is given by

$$ {q_{e}} = \sum\limits_{k = {\theta_{C}}}^{\infty} \mathbb{P}\left( {{K_{0}} = k} \right) \cdot \underbrace{\mathbb{P}\left( {{\gamma_{{S_{0}}{R_{1}}:1}} < {\beta_{2\nu}}, {\cdots} ,{\gamma_{{S_{0}}{R_{k}}:1}} < {\beta_{2\nu}}} \left| {{K_{0}} = k} \right. \right)}_{\mathcal{A}}, $$

(28)

where ${\gamma _{{S_{0}}{R_{k}}:1}} = {{{H_{{S_{0}}{R_{k}}:1}}g\left ({{s_{0}} - {r_{k}}} \right )}{\left / {\vphantom {{{H_{{S_{0}}{R_{k}}:1}}g\left ({{s_{i}} - {r_{k}}} \right )} {{I_{{R_{k}}:1}}\left ({{{\Phi }_{D}},{{\Phi }_{C}}} \right )}}} \right . \kern -\nulldelimiterspace } {{I_{{R_{k}}:1}}\left ({{{\Phi }_{D}},{{\Phi }_{C}}} \right )}}$ denotes the received SIR at relay R _k when receiving a packet from source S ₀.

The probability that k potential relays fail to decode the packet from source S ₀ is given by

$$\begin{array}{@{}rcl@{}} \mathcal{A} &\overset{(a)}{=}& \mathbb{E}\left[ {\prod\limits_{i = 1}^{k} {\left( {1 - \exp \left[ { - {\beta_{2\nu}}{g^{-1}(s_{0}-r_{i}) }\left( {{I_{DR_{i}:1}}\left( {\Phi}_{D} \right) + {I_{CR_{i}:1}}\left( {\Phi}_{C} \right)} \right)} \right]} \right)} } \right]\\ &\overset{(b)}{=}& \mathbb{E}\left[ {{{\left( {1\! - \prod\limits_{{s_{j}} \in {{\Phi}_{S}}} {\frac{1}{{1\! +\! {\beta_{2\nu}}{g^{-1}(s_{0}-r_{i}) }g\left( {{s_{j}}\! -\! {r_{i}}} \right)}}} } \right)}^{k}}} \right]\\ &\overset{(c)}{=}& \sum\limits_{m = 0}^{k} { \binom {k} {m} {{\left( { - 1} \right)}^{m}}} \underbrace{ {\mathbb{E}}\left[ {\prod\limits_{{s_{j}} \in {{\Phi}_{S}}} {\frac{1}{{{{\left( {1 + {\beta_{2\nu}}{g^{-1}(s_{0}-r_{i}) }g\left( {{s_{j}} - {r_{i}}} \right)} \right)}^{m}}}}} } \right]}_{{\mathcal{A}_{1}}}, \end{array} $$

(29)

where (a) follows by taking expectations over independent exponential channel fading between source S ₀ and potential relays, (b) follows by taking Laplace transforms of independent channel fading between the interferers and potential relays, and (c) follows from the binomial expansion. Note that the spatial correlation of interference power at potential relays is considered by taking a joint expectation over the spatial locations of the same set of interferers.

Via applying the PGFL of the PPP [16] and performing a coordinate transformation, we have

$$\begin{array}{@{}rcl@{}} \mathcal{A}_{1} &=& \frac{1}{{{A_{R}}}}{\int}_{C{R_{0}}} \exp \Big({ - 2\pi {\lambda_{S}}}\\ &&\quad\quad\quad\quad\quad\quad \left.\times {\int}_{0}^{\infty}{\left[ {1 - {{{{\left( {1 + {\beta_{2\nu}}{g^{-1}(s_{0}-r) }{l^{- \alpha }}} \right)}^{-m}}}}} \right]l\mathrm{d}l}\right) \mathrm{d}r\\ &=& \frac{1}{{{A_{R}}}}{\int}_{C{R_{0}}} \exp \left( { - {\lambda_{S}} Q g^{-\delta}(s_{0}-r) } \right) \mathrm{d}r \end{array} $$

(30)

where Q is defined in Eq. 9.

By substituting Eqs. 29 and 30 into Eq. 28, λ _F and $\mathbb {E}\left [ {I_{{D_{0}:2}}}\left ({\Phi }_{D}, {\Phi }_{F} \right ) \right ]$ can be derived.

The mean product of $I_{D_{0}:1}\left ({\Phi }_{D}, {\Phi }_{C} \right )$ and $I_{D_{0}:2}\left ({\Phi }_{D}, {\Phi }_{F} \right )$ is given by

$$\begin{array}{@{}rcl@{}} &&\mathbb{E} \left[ {I_{D_{0}:1}\left( {\Phi}_{D}, {\Phi}_{C} \right) I_{D_{0}:2}\left( {\Phi}_{D}, {\Phi}_{F} \right) } \right]\\ &&\quad= \;\; \mathbb{E}\left[ {{I_{DD_{0}:1}}\left( {\Phi}_{D} \right){I_{DD_{0}:2}}\left( {\Phi}_{D} \right)} \right]\\ &&\quad\quad+ \mathbb{E}\left[ {{I_{DD_{0}:1}}\left( {\Phi}_{D} \right){I_{FD_{0}:2}}\left( {\Phi}_{F} \right)} \right]\\ &&\quad\quad+ \;\; \mathbb{E}\left[ {{I_{CD_{0}:1}}\left( {\Phi}_{C} \right){I_{DD_{0}:2}}\left( {\Phi}_{D} \right)} \right]\\ &&\quad\quad+\mathbb{E} \left[ {{I_{CD_{0}:1}}\left( {\Phi}_{C} \right){I_{FD_{0}:2}}\left( {\Phi}_{F} \right)} \right].\\ \end{array} $$

(31)

As PPP Φ _D is independent of PPP Φ _C and PPP Φ _F , we have

$$\begin{array}{@{}rcl@{}} \mathbb{E} \left[ {{I_{DD_{0}:1}}\left( {\Phi}_{D} \right){I_{FD_{0}:2}}\left( {\Phi}_{F} \right)} \right] &=& {\lambda_{D}}{\lambda_{F}} {\left( {{\int}_{{\mathbb{R}^{2}}} {{g }\left( s \right)\mathrm{d}s} } \right)^{2}}\\ \mathbb{E} \left[ {{I_{CD_{0}:1}}\left( {\Phi}_{C} \right){I_{DD_{0}:2}}\left( {\Phi}_{D} \right)} \right] &=& {\lambda_{C}}{\lambda_{D}} {\left( {{\int}_{{\mathbb{R}^{2}}} {{g }\left( s \right)\mathrm{d}s} } \right)^{2}}. \end{array} $$

(32)

As PPP Φ _C and PPP Φ _F are not independent of each other, the mean product of $I_{CD_{0}:1}\left ({\Phi }_{C} \right )$ and $I_{FD_{0}:2}\left ({\Phi }_{F} \right )$ is given by

$$\begin{array}{@{}rcl@{}} &&\mathbb{E} \left[ {{I_{{CD_{0}:1}}}\left( {\Phi}_{C} \right){I_{FD_{0}:2}}\left( {\Phi}_{F} \right)} \right]\\ &&\overset{(a)}{=} \mathbb{E}\left[ {\left( {\sum\limits_{{s_{j}} \in {{\Phi}_{C}}} {{g }\left( {{s_{j}}} \right)} } \right)\left( {\sum\limits_{{r_{m}} \in {{\Phi}_{F}}} {{g }\left( {{r_{m}}} \right)} } \right)} \right] \\ &&\overset{(b)}{=} \mathbb{E} \left[ {\left( {\sum\limits_{{s_{j}} \in {{\Phi}_{C}}} {{g }\left( {{s_{j}}} \right)} } \right)\left( {\sum\limits_{{s_{j}} \in {{\Phi}_{C}}} {{\left( 1 - q_{e}\right)} \cdot {g}\left( {{s_{j}} + \tau} \right)} } \right)} \right] \\ &&= \mathbb{E} \left[ {\sum\limits_{{s_{j}} \in {{\Phi}_{C}}} {{\left( 1\! -\! q_{e}\right)} \cdot {g }\left( {{s_{j}}} \right){g }\left( {{s_{j}}\! +\! \tau } \right)} } \right]\\ &&\quad\! +\! \mathbb{E} \left[ {\sum\limits_{{s_{i}},{s_{j}} \in {{\Phi}_{C}}}^{{s_{i}} \ne {s_{j}}} {{\left( 1\! -\! q_{e}\right)} \cdot {g }\left( {{s_{j}}} \right){g }\left( {{s_{j}}\! +\! \tau } \right)} } \right] \\ &&\overset{(c)}{=} {\lambda_{F}}{\int}_{{\mathbb{R}^{2}}} {{g }\left( s \right) \mathbb{E}_{\tau}\left[ {g }\left( {s + \tau } \right) \right] \mathrm{d}s} + {\lambda_{C}}{\lambda_{F}}{\left( {{\int}_{{\mathbb{R}^{2}}} {{g }\left( s \right)\mathrm{d}s} } \right)^{2}}, \\ \end{array} $$

(33)

where (a) holds as the fading coefficients of different links are independent random variables with unit mean, (b) follows from the transformation between PPP Φ _C and PPP Φ _F and τ is the coordinate difference between a source and its selected relay (i.e., τ = r _m − s _j ), and (c) follows from the Campbell’s Theorem and second-order product density formula of the PPP [15].

Similarly, by replacing τ with (0, 0), we have

$$ \mathbb{E} \left[ {{I_{DD_{0}:1}}\left( {\Phi}_{D} \right){I_{DD_{0}:2}}\left( {\Phi}_{D} \right)} \right] = {\lambda_{D}}{{\int}_{{\mathbb{R}^{2}}} { {g^{2} \left( s \right)} }}\mathrm{d}s + {\lambda_{D}^{2}}{\left( {{\int}_{{\mathbb{R}^{2}}} {g\left( s \right)\mathrm{d}s} } \right)^{2}}. $$

(34)

By substituting Eqs. (32-34) into Eq. 31, the numerator of (26), denoted as N, is given by

$$ N = {\lambda_{D}} {\int}_{{\mathbb{R}^{2}}} {g^{2}\left( s \right)\mathrm{d}s} + {\lambda_{F}} {\int}_{{\mathbb{R}^{2}}} {{g }\left( s \right) \mathbb{E}_{\tau}\left[ {g }\left( {s + \tau } \right) \right] \mathrm{d}s}. $$

(35)

The second moment of $I_{DD_{0}:1}\left ({\Phi }_{D} \right )$ is given by

$$ {\mathbb{E}} \left[ {I_{DD_{0}:1}^{2}\left( {\Phi}_{D} \right)} \right] \!\overset{(a)}{=}\! 2 {{\lambda_{D}}{\int}_{{\mathbb{R}^{2}}} {g^{2}\left( s \right)\mathrm{d}s} \,+\, {\lambda_{D}^{2}}{{\left( {{\int}_{{\mathbb{R}^{2}}} {{g }\left( s \right)\mathrm{d}s} } \right)}^{2}}}, $$

(36)

where (a) follows from the similar arguments in Eq. 33 and $\mathbb {E}\left [ {{H^{2}}} \right ] = 2$ for Rayleigh fading channels.

Using Eq. 36, the variance of $I_{D_{0}:1}\left ({\Phi }_{D}, {\Phi }_{C} \right )$ and $I_{D_{0}:2}\left ({\Phi }_{D}, {\Phi }_{F} \right )$ can be expressed respectively as

$$\begin{array}{@{}rcl@{}} \text{Var} \left( {{I_{{D_{0}:1}}}\left( {\Phi}_{D}, {\Phi}_{C} \right)} \right) &=& \mathbb{E}\left[ {I_{{D_{0}:1}}^{2}\left( {\Phi}_{D}, {\Phi}_{C} \right)} \right]\\ &&- \mathbb{E}{\left[ {{I_{{D_{0}:1}}}\left( {\Phi}_{D}, {\Phi}_{C} \right)} \right]^{2}}\\ &=& 2 {{\lambda_{S}}} {\int}_{{\mathbb{R}^{2}}} {g^{2}\left( s \right)\mathrm{d}s}\\ \text{Var} \left( {{I_{{D_{0}:2}}}\left( {\Phi}_{D}, {\Phi}_{F} \right)} \right) &=& 2 \left( {{\lambda_{D}} + {\lambda_{F}}} \right) {\int}_{{\mathbb{R}^{2}}} {g^{2}\left( s \right)\mathrm{d}s}. \end{array} $$

(37)

For the non-singular path loss model defined in Eq. 1, we have

$$ {\int}_{{\mathbb{R}^{2}}} {g^{2}\left( s \right)\mathrm{d}s} = \frac{{ \delta \left( {1 - \delta } \right){\pi^{2}}}}{{{\epsilon^{2 - \delta }}\sin \left( \pi \delta \right)}}. $$

(38)

By substituting Eqs. 35, 37, and 38 into Eq. 26, we obtain Eq. 7.