Outage performance of downlink communications in cognitive-based two-tier networks: cooperative and non-cooperative femtocells

Abstract

Femtocell deployment, which is a promising approach to the coverage and capacity improvement of indoor communications, suffers from cross-tier interference. Therefore to make the femtocell technology practical this issue needs to be addressed appropriately. One serious type of cross-tier interference occurs in downlink communication, in which a macrocell user is located far from its macro base station. In this setup, the communication of the adjacent femto access points with their users makes the macrocell user experience a low SINR. This paper considers this scenario and shows how cognitive-enabled femto access points can cope with cross-tier interference. More precisely, we compute the outage probability of macro users in a two-tier network when femto access points use the energy detection-based spectrum sensing technique to find the unoccupied frequency subband. To improve the outage probability of macro users, we also study the effectiveness of cooperation among neighbor femto access points. In all cases, the analytical expressions are validated by computer simulations which confirm the accuracy of the used approximations.

Appendices

Appendix 1: Derivation of missed detection probability at the FAP

A dual hypothesis testing that include $H_0$ and $H_1$ to represent absence and presence of the MU in a specific subband, respectively, is applied to our system model and defined as

$$\begin{aligned} \left\{ {\begin{array}{ll} {{H_0}:y_r^f = {n_f}}\\ {{H_1}:y_r^f = \frac{{\sqrt{{P_M}} {h_{Mf}}}}{{\sqrt{P{L_{Mf}}} }}{b_M} + {n_f}} \end{array}} \right. , \end{aligned}$$

(27)

where, $y_r^f$ is the received signal at the FAP during the spectrum sensing process and existence of the MU is detected based on this parameter. The parameter $b_M$ denotes the transmitted signal from MBS to the MU, $h_{M,f}$ is the channel fading of the link between MBS and FAP and finally $n_f$ refers to an AWGN with zero mean and variance $\sigma _n^2$.

The energy detector which has been used by each FAP, firstly creates an output statistic, $T\left[ {y_r^f} \right]$, using N samples of the received signal. Then the detector compares it with a certain threshold, $\lambda$. Hence, the result of spectrum sensing at the FAP can be expressed as

$$\begin{aligned} {{\hat{H}}_f} = \left\{ {\begin{array}{c} {{H_0},\quad T\left[ {y_r^f} \right] < \lambda }\\ {{H_1},\quad T\left[ {y_r^f} \right] > \lambda } \end{array}} \right. ,\quad T\left[ {y_r^f} \right] = \frac{1}{N}\sum \limits _{n = 1}^N {|y_r^f(n){|^2}} . \end{aligned}$$

(28)

For deterministic value of fading coefficients and path loss, according to the central limit theorem and also the results presented in [37, 38] $H_0$ and $H_1$ will be converged to the normal distribution for large N as

$$\begin{aligned} T\left[ {y_r^f\left| {{H_0}} \right. } \right] \sim {\mathcal {N}}\left( {\sigma _n^2,\frac{{{{\left( \sigma _n^2\right) }^2}}}{N}} \right) \end{aligned}$$

(29)

and

$$\begin{aligned} T\left[ {y_r^f\left| {{H_1}} \right. } \right] \sim {\mathcal {N}}\left[ {\left( {\frac{{|{h_{Mf}}{|^2}}}{{P{L_{Mf}}}}\cdot \frac{{{P_M}}}{{\sigma _n^2}} + 1} \right) \sigma _n^2,\left( {\frac{{|{h_{Mf}}{|^2}}}{{P{L_{Mf}}}}\cdot \frac{{{P_M}}}{{\sigma _n^2}} + 1} \right) \frac{{{{\left( \sigma _n^2\right) }^2}}}{N}} \right] . \end{aligned}$$

(30)

Now, $P_{md}$ is easily obtained as

$$\begin{aligned} {P_{md}} = 1-\Pr \left\{ {{{{\hat{H}}}_f} = {H_1}\left| {{H_M} = {H_1}} \right. } \right\} = 1 - Q\left( {\frac{{\lambda - \left( {\frac{{|{h_{Mf}}{|^2}}}{{P{L_{Mf}}}}\cdot \frac{{{P_M}}}{{\sigma _n^2}} + 1} \right) \sigma _n^2}}{{\frac{{\sigma _n^2}}{{\sqrt{N} }}\sqrt{\left( {\frac{{|{h_{Mf}}{|^2}}}{{P{L_{Mf}}}}\cdot \frac{{{P_M}}}{{\sigma _n^2}} + 1} \right) } }}} \right) . \end{aligned}$$

(31)

It is worthwhile to mention that the probability of false alarm (declaring that a subband is occupied while it is free) is equal to

$$\begin{aligned} {P_{fa}} = \Pr \left\{ {{{{\hat{H}}}_f} = {H_1}\left| {{H_M} = {H_0}} \right. } \right\} = 1 - Q\left( \frac{\lambda -\sigma _n^2}{\frac{\sigma _n^2}{{\sqrt{N}}}} \right) . \end{aligned}$$

(32)

Therefore in a desired specific value for the false alarm probability, the threshold level $(\lambda )$ in the energy detector can be computed as

$$\begin{aligned} \lambda = \frac{\sigma _n^2}{{\sqrt{N}}}Q^{-1}\left( P_{fa}+\sigma _n^2\right) . \end{aligned}$$

(33)

Appendix 2: Average of missed detection probability at FC in the cooperative scheme

As mentioned previously, in the case of cooperation among neighboring FAPs the missed detection probability is equal to

$$\begin{aligned} P_{md {\big | {h_{M,f}^{(i)}}, N_{FAP}}}^c = \prod _{i=1}^{N_{FAP}} P_{md {\big | {h_{M,f}^{(i)}}}}^{(i)}, \end{aligned}$$

(34)

where

$$\begin{aligned} P^{(i)}_{md {\big | {h_{M,f}^{(i)}}}} = 1- Q \left( \sqrt{N} \frac{\left( {\lambda \over \sigma _n^2}-1-\eta \left| h_{M,f}^{(i)}\right| ^2\right) }{\sqrt{\eta |h_{M,f}^{(i)}|^2+1}} \right) . \end{aligned}$$

(35)

Considering that channel coefficients ${h_{M,f}^{(i)}}$s for $i=1,\ldots N_{FAP}$ are independent from each other and follow Rayleigh distribution with parameter $\sigma _{M,f}$ we can remove the effect of channel coefficient as

$$\begin{aligned} P_{md}^{(i)} = \int _0^\infty \left( 1- Q \left( \sqrt{N} \frac{\left( {\lambda \over \sigma _n^2}-1-\eta x\right) }{\sqrt{\eta x+1}} \right) \frac{1}{{\sigma _{M,f}^2}}{e^{ - {{{{x}} \over {\sigma _{M,f}^2}}}}}\right) d{x}. \end{aligned}$$

(36)

Applying GQR technique to the above integral resulting in

$$\begin{aligned} P_{m{d}}^{(i)} = \sum \limits _{j = 1}^{n} {w'_j} \left( 1- Q \left( \sqrt{N} \frac{\left( {\lambda \over \sigma _n^2}-1-\eta \sigma _{M,f}^2 x'_j\right) }{\sqrt{\eta \sigma _{M,f}^2 x'_j+1}} \right) \right) \triangleq \kappa , \end{aligned}$$

(37)

where $\lambda$ is the threshold level of energy detector. Now from (34),

$$\begin{aligned} P_{md\big | N_{FAP}}^c = \left( P_{md}^{(i)} \right) ^{N_{FAP}}. \end{aligned}$$

(38)

Finally as FAPs are distributed according to a PPP with density $\lambda _f$ in a cooperation circle with radius $R_c$, the average missed detection probability at FC is obtained as

$$\begin{aligned} {\overline{P_{md}^c}} = e^{\lambda _f\pi R_c^2\left( \kappa -1\right) }, \end{aligned}$$

(39)

where $\kappa$ was defined in (37).