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Energy efficiency of full-duplex cognitive radio in low-power regimes under imperfect spectrum sensing

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Abstract

This paper investigates the energy efficiency of a full duplex (FD) cognitive radio (CR) system in the low-power regime under a practical self-interference cancellation performance and imperfect spectrum sensing. We consider an opportunistic spectrum access network in which the secondary user (SU) is capable of self-interference suppression (SIS). The SIS ability enables the SU to work in simultaneous transmit and sense (TS) mode in order to increase the quality of channel sensing. Towards this goal, we first study the sensing performance, i.e., false-alarm and miss-detection probabilities, of the FD CR networks using TS, and compare the results with traditional half duplex (HD) CR systems using transmit only mode (TO). In the next step, we show that due to imperfect spectrum sensing, the secondary network channel is best characterized as a Gaussian-Mixture (GM) channel, which is widely used to capture the asynchronism in heterogeneous cellular networks. We then analytically characterize the low-signal-to-noise-ratio (low-SNR) metrics of the minimum energy per bit and wideband slope of the spectral-efficiency curve, and obtain these fundamental limits in closed-form. Furthermore, the characterization of these fundamental metrics allows us to identify practical signaling strategies that are optimally efficient in the low-SNR regime for the considered FD CR system. The benefits in terms of energy efficiency offered by FD CR over HD CR are also clearly demonstrated.

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Acknowledgements

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 102.04-2017.16.

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Appendix

Appendix

Proposition 2

Using the Central Limit Theorem, for a large number of samples (fsTs), the distribution of MFD given H0 can be approximated by Gaussian distribution with the following mean and variance

$$ E\left[M_{FD} \mid H_{0}\right]=\chi^{2}{\sigma_{s}^{2}}+N_{0}, $$
(82)
$$ Var\left[M_{FD} \mid H_{0}\right]=\frac{1}{f_{s}T_{s}}\left\{ \chi^{4} {\sigma_{s}^{4}}\left( 2\kappa_{s}-1\right) +{N_{0}^{2}}+2\chi^{2}{\sigma_{s}^{2}}N_{0}\right\}. $$
(83)

Proof

$$ \begin{array}{@{}rcl@{}} &&E\left[M_{FD} \mid H_{0}\right]=E\left[\left|\boldsymbol{y}_{_{FD}}\left( n\right)\right|^{2}\mid H_{0}\right]=E\left[\left|\boldsymbol{ h}_{si}\left( n\right) \boldsymbol{x}\left( n\right)+\boldsymbol{ n}\left( n\right)\right|^{2}\right]\\ &=&E\left[\left|\boldsymbol{h}_{si}\left( n\right)\right|^{2}\left|\boldsymbol{x}\left( n\right)\right|^{2}+\left|\boldsymbol{ n}\left( n\right)\right|^{2}+\boldsymbol{h}_{si}\left( n\right) \boldsymbol{x}\left( n\right)\boldsymbol{n}^{*}\left( n\right)+\boldsymbol{ h}_{si}^{*}\left( n\right) \boldsymbol{ x}^{*}_{s}\left( n\right)\boldsymbol{n}\left( n\right)\right]\\ &=&\chi^{2}{\sigma_{s}^{2}}+N_{0}. \end{array} $$
(84)

Moreover,

$$ \begin{array}{@{}rcl@{}} Var\left[M_{FD} \mid H_{0}\right]&=&\frac{1}{f_{s}T_{s}}Var\left[\left|\boldsymbol{y}_{_{FD}}\left( n\right)\right|^{2}\mid H_{0}\right]\\ &=&\frac{1}{f_{s}T_{s}}\left\{E\left[\left|\boldsymbol{y}_{_{FD}}\left( n\right)\right|^{4}\mid H_{0}\right]-\left( E\left[\left|\boldsymbol{y}_{_{FD}}\left( n\right)\right|^{2}\mid H_{0}\right]\right)^{2}\right\}. \end{array} $$
(85)

We already have \(E\left [\left |\boldsymbol {y}_{_{FD}}\left (n\right )\right |^{2}\mid H_{0}\right ]\) in Eq. 84. So we only need to calculate \(E\left [\left |\boldsymbol {y}_{_{FD}}\left (n\right )\right |^{4}\mid H_{0}\right ]\).

$$ \begin{array}{@{}rcl@{}} &&E\left[\left|\boldsymbol{y}_{_{FD}}\left( n\right)\right|^{4}\mid H_{0}\right]=E\left[\left|\boldsymbol{h}_{si}\left( n\right) \boldsymbol{x}\left( n\right)+\boldsymbol{n}\left( n\right)\right|^{4}\right]\\ &=&E\left[\left|\left|\boldsymbol{h}_{si}\left( n\right)\right|^{2}\left|\boldsymbol{x}\left( n\right)\right|^{2}+\left|\boldsymbol{ n}\left( n\right)\right|^{2}+\boldsymbol{h}_{si}\left( n\right) \boldsymbol{x}\left( n\right)\boldsymbol{n}^{*}\left( n\right)+\boldsymbol{ h}_{si}^{*}\left( n\right) \boldsymbol{ x}^{*}_{s}\left( n\right)\boldsymbol{n}\left( n\right)\right|^{2}\right]\\ &=&E\left[\left|\boldsymbol{h}_{si}\left( n\right)\right|^{4}\left|\boldsymbol{x}\left( n\right)\right|^{4}+\left|\boldsymbol{ n}\left( n\right)\right|^{4}+\left|\boldsymbol{h}_{si}\left( n\right)\boldsymbol{x}\left( n\right)\boldsymbol{ n}^{*}\left( n\right)\right|^{2}\right.\\ &&\left.\mkern 80mu +\left|\boldsymbol{h}_{si}^{*}\left( n\right)\boldsymbol{x}^{*}_{s}\left( n\right)\boldsymbol{ n}\left( n\right)\right|^{2}+2\left|\boldsymbol{h}_{si}\left( n\right)\right|^{2}\left|\boldsymbol{x}\left( n\right)\right|^{2}\left|\boldsymbol{ n}\left( n\right)\right|^{2}\right]\\ &=&2 \chi^{4} \kappa_{s} {\sigma_{s}^{4}} +2{N_{0}^{2}}+4\chi^{2}{\sigma_{s}^{2}}N_{0}. \end{array} $$
(86)

In the above expression we have used the fact that for circularly symmetric complex random varibale X with mean zero we have

$$ \begin{array}{@{}rcl@{}} E\left[X^{2}\right]=E\left[\left( X_{r}+jX_{i}\right)^{2}\right]=E\left[\left|X_{r}\right|^{2}\right]-E\left[\left|X_{i}\right|^{2}\right]+2jE\left[X_{r}X_{i}\right]=0. \end{array} $$
(87)

By using Eqs. 8485, and 86, Eq. 83 is proved. □

Proposition 3

Using the Central Limit Theorem, for the large number of samples (fsTs), the distribution of MFD given H1 can be approximated by Gaussian distribution with the following mean and variance

$$ \begin{array}{@{}rcl@{}} E\left[M_{FD} \mid H_{1}\right]=2\sum\limits_{i=1}^{p}\gamma_{i}\sigma_{w,i}^{2}+\chi^{2}{\sigma_{s}^{2}}+N_{0}, \end{array} $$
(88)
$$ \begin{array}{@{}rcl@{}} &&Var\left[M_{FD} \mid H_{1}\right]=\frac{1}{f_{s}T_{s}}\left\{ 8\sum\limits_{i=1}^{p}\gamma_{i}\sigma_{w,i}^{4}+\chi^{4}{\sigma_{s}^{4}}\left( 2\kappa_{s}-1\right)\right.\\ &&\left.+4\sum\limits_{i=1}^{p}\gamma_{i}\sigma_{w,i}^{2}\left( -\sum\limits_{i=1}^{p}\gamma_{i}\sigma_{w,i}^{2}+N_{0}\right) +2\chi^{2}{\sigma_{s}^{2}}\left( 2\sum\limits_{i=1}^{p}\gamma_{i}\sigma_{w,i}^{2}+N_{0}\right)+{N_{0}^{2}}\right\}. \end{array} $$
(89)

Proof

$$ \begin{array}{@{}rcl@{}} &&E\left[M_{FD} \mid H_{1}\right]=E\left[\left|\boldsymbol{y}_{_{FD}}\left( n\right)\right|^{2}\mid H_{1}\right]=E\left[\left| \boldsymbol{ w}\left( n\right)+\boldsymbol{h}_{si}\left( n\right) \boldsymbol{x}\left( n\right)+\boldsymbol{ n}\left( n\right)\right|^{2}\right]\\ &&=E\left[\left|\boldsymbol{w}\left( n\right)\right|^{2}+\left|\boldsymbol{h}_{si}\left( n\right)\right|^{2}\left|\boldsymbol{ x}\left( n\right)\right|^{2}+\left|\boldsymbol{n}\left( n\right)\right|^{2}+\boldsymbol{w}\left( n\right)\boldsymbol{ h}_{si}^{*}\left( n\right) \boldsymbol{x}^{*}\left( n\right)\right.\\ &&\left. \mkern 80mu+\boldsymbol{w}\left( n\right)\boldsymbol{n}^{*}\left( n\right)+\boldsymbol{h}_{si}\left( n\right) \boldsymbol{ x}\left( n\right)\boldsymbol{w}^{*}\left( n\right)+\boldsymbol{h}_{si}\left( n\right) \boldsymbol{x}\left( n\right)\boldsymbol{ n}^{*}\left( n\right) \right.\\ &&\left. \mkern 80mu+\boldsymbol{w}^{*}\left( n\right)\boldsymbol{n}\left( n\right)+\boldsymbol{h}_{si}^{*}\left( n\right) \boldsymbol{ x}^{*}\left( n\right)\boldsymbol{n}\left( n\right)\right]\\ &&=2\sum\limits_{i=1}^{p}\gamma_{i}\sigma_{w,i}^{2}+\chi^{2}{\sigma_{s}^{2}}+N_{0}. \end{array} $$
(90)

On the other hand,

$$ \begin{array}{@{}rcl@{}} Var\left[M_{FD} \mid H_{1}\right]&=&\frac{1}{f_{s}T_{s}}Var\left[\left|\boldsymbol{y}_{_{FD}}\left( n\right)\right|^{2}\mid H_{1}\right]\\ &=&\frac{1}{f_{s}T_{s}}\left\{E\left[\left|\boldsymbol{y}_{_{FD}}\left( n\right)\right|^{4}\mid H_{1}\right]-\left( E\left[\left|\boldsymbol{y}_{_{FD}}\left( n\right)\right|^{2}\mid H_{1}\right]\right)^{2}\right\}. \end{array} $$
(91)

We already have \(E\left [\left |\boldsymbol {y}_{_{FD}}\left (n\right )\right |^{2}\mid H_{1}\right ]\) in Eq. 90. So we only need to calculate \(E\left [\left |\boldsymbol {y}_{_{FD}}\left (n\right )\right |^{4}\mid H_{1}\right ]\).

$$ \begin{array}{@{}rcl@{}} &&E\left[\left|\boldsymbol{y}_{_{FD}}\left( n\right)\right|^{4}\mid H_{1}\right]=E\left[\left| \boldsymbol{w}\left( n\right)+\boldsymbol{ h}_{si}\left( n\right) \boldsymbol{x}\left( n\right)+\boldsymbol{ n}\left( n\right)\right|^{4}\right]\\ &=&E\left[\left|\left|\boldsymbol{w}\left( n\right)\right|^{2}+\left|\boldsymbol{h}_{si}\left( n\right)\right|^{2}\left|\boldsymbol{ x}\left( n\right)\right|^{2}+\left|\boldsymbol{n}\left( n\right)\right|^{2}+\boldsymbol{w}\left( n\right)\boldsymbol{ h}_{si}^{*}\left( n\right)\boldsymbol{x}^{*}\left( n\right)\right.\right.\\ &&\left. \left.\mkern 80 mu +\boldsymbol{w}\left( n\right)\boldsymbol{n}^{*}\left( n\right)+\boldsymbol{h}_{si}\left( n\right) \boldsymbol{ x}\left( n\right)\boldsymbol{w}^{*}\left( n\right)+\boldsymbol{h}_{si}\left( n\right) \boldsymbol{x}\left( n\right)\boldsymbol{ n}^{*}\left( n\right)\right.\right.\\ &&\left.\left. \mkern 80 mu +\boldsymbol{w}^{*}\left( n\right)\boldsymbol{n}\left( n\right)+\boldsymbol{h}_{si}^{*}\left( n\right) \boldsymbol{x}^{*}\left( n\right)\boldsymbol{n}\left( n\right)\right|^{2}\right]\\ &=&E\left[\left|\boldsymbol{w}\left( n\right)\right|^{4}+\left|\boldsymbol{h}_{si}\left( n\right) \right|^{4}\left|\boldsymbol{ x}\left( n\right)\right|^{4}+\left|\boldsymbol{n}\left( n\right)\right|^{4}+4 \left|\boldsymbol{w}\left( n\right)\right|^{2}\left|\boldsymbol{h}_{si}\left( n\right)\boldsymbol{ x}\left( n\right)\right|^{2}\right.\\ &&\left.\mkern 80 mu+4\left|\boldsymbol{w}\left( n\right)\right|^{2}\left|\boldsymbol{n}\left( n\right)\right|^{2} +4\left|\boldsymbol{ h}_{si}\left( n\right) \boldsymbol{x}\left( n\right)\right|^{2}\left|\boldsymbol{ n}\left( n\right)\right|^{2}\right]\\ &=&8\sum\limits_{i=1}^{p}\gamma_{i}\sigma_{w,i}^{4}+2\chi^{4}\kappa_{s}{\sigma_{s}^{4}}+2{N_{0}^{2}}+8\left( \chi^{2}{\sigma_{s}^{2}}+N_{0}\right)\sum\limits_{i=1}^{p}\gamma_{i}\sigma_{w,i}^{2}+4\chi^{2}{\sigma_{s}^{2}}N_{0}. \end{array} $$
(92)

In the above expression we have used Eq. 87. By combining Eqs. 9091 and 92 proves Eq. 89. □

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Ranjbar, M., Nguyen, H.L., Tran, N.H. et al. Energy efficiency of full-duplex cognitive radio in low-power regimes under imperfect spectrum sensing. Mobile Netw Appl 26, 1750–1764 (2021). https://doi.org/10.1007/s11036-021-01755-z

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