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This paper evaluates outage probability (OP) and intercept probability (IP) of physical-layer security based MIMO networks adopting cooperative jamming (Coop-Jam). In the considered scenario, a multi-antenna source communicates with a multi-antenna destination employing transmit antenna selection (TAS)/ selection combining (SC), in presence of a multi-antenna eavesdropper using SC. One of jammers appearing near the destination is selected for generating jamming noises on the eavesdropper. Moreover, the destination supports the wireless energy for the chosen jammer, and cooperates with it to remove the jamming noises. We consider two jammer selection approaches, named RAND and SHORT. In RAND, the destination randomly selects the jammer, and in SHORT, the jammer, which is nearest to the destination, is chosen. We derive exact and asymptotic expressions of OP and IP over Rayleigh fading, and perform Monte-Carlo simulations to verify the correction of our derivation. The results present advantages of the proposed RAND and SHORT methods, as compared with the corresponding one without using Coop-Jam.

In this paper, we propose a two-way relaying scheme using digital network coding in an underlay cognitive radio network. In the proposed scheme, the transmit antenna selection and selection techniques are combined using a primary transmitter and a primary receiver, respectively. In the secondary network, two source nodes that cannot directly communicate attempt to exchange their data with each other. As a result, the relaying technique using partial relay selection is applied to assist the data exchange. Particularly, at the first time slot, the selected secondary relay applies an interference cancellation technique to decode the data received from the secondary sources. Then, the selected relay uses digital network coding to send XOR-ed data to the sources at the second time slot. We first derive the outage probability of the primary network over block the Rayleigh fading channel. Then, the transmit power of the secondary transmitters including the source and relay nodes are calculated to guarantee the quality of service of the primary network. Finally, the exact closed-form formulas of the outage probability of the secondary sources over the block Rayleigh fading channel are derived, and then verified by computer simulations using the Monte Carlo method.

This paper measures end-to-end (e2e) throughput of power beacon-assisted multi-hop decode-and-forward (DF) relaying scheme adopting non-orthogonal multiple access (NOMA) and transmit antenna selection (TAS)/selection combining (SC). Particularly, TAS/SC and NOMA are adopted at each hop to relay different data of a source to multiple destinations. Moreover, the transmitters including source and relays have to harvest wireless energy from radio frequency (RF) signals from a power beacon. We also propose a simple and efficient power allocation method for the signals transmitted at each hop. For performance measurement and comparison, we provide closed-form formulas of the e2e throughput over Rayleigh fading channel. We then verify our derivations by computer simulations as well as compare the e2e throughput performance between our scheme and the corresponding one that does not use NOMA.

The aim of this paper is to characterize the nonnegative functions $\varphi$ defined on $(0,\infty)$ for which the Hausdorff operator $$\mathscr H_\varphi f(z)= \int_0^\infty f\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}dt$$ is bounded on the Hardy spaces of the upper half-plane $\mathcal H_a^p(\mathbb C_+)$, $p\in[1,\infty]$. The corresponding operator norms and their applications are also given., Comment: Proc. Roy. Soc. Edinburgh Sect. A (to appear)

We obtain necessary and sufficient conditions on functions s 1(t),..., s m (t) and ψ(t) such that the weighted multilinear Hardy–Cesaro operator $$\left( {f_1 , \ldots f_m } \right) \mapsto \int_{[0,1]^n } {\left( {\prod\limits_{k = 1}^m {f_k } \left( {s_k \left( t \right)x} \right)} \right)\psi (t)dt}$$ is bounded from $$\dot K_{q_1 }^{\alpha _1 ,p_1 } \left( {\omega _1 } \right) \times \cdots \times \dot K_{q_m }^{\alpha _m ,p_m } \left( {\omega _m } \right)$$ to $$\dot K_q^{\alpha ,p} \left( \omega \right)$$ and from $$M\dot K_{p_1 ,q_1 }^{\alpha _1 ,\lambda _1 } \left( {\omega _1 } \right) \times \cdots \times M\dot K_{p_m ,q_m }^{\alpha _m ,\lambda _m } \left( {\omega _m } \right)$$ to $$M\dot K_{p,q}^{\alpha ,\lambda } \left( \omega \right)$$ . Sharp bounds are also obtained for both cases 0 < p < 1 and 1 ≤ p < ∞. Provided b 1, ..., b m are Lipschitz functions we give a sufficient condition on functions s 1(t), ..., s m (t), ψ(t) such that the commutator of the weighted Hardy–Cesaro operator $$\left( {f_1 , \ldots f_m } \right) \mapsto \int_{[0,1]^n } {\left( {\prod\limits_{k = 1}^m {f_k \left( {s_k \left( t \right)x} \right)} } \right)} \left( {\prod\limits_{k = 1}^m {\left( {b_k \left( x \right) - b_k \left( {s_k \left( t \right)x} \right)} \right)} } \right)\gamma (t)dt$$ is bounded from $$M\dot K_{p_1 ,q_1 }^{\alpha _1 ,\lambda _1 } \left( {\omega _1 } \right) \times \cdots \times M\dot K_{p_m ,q_m }^{\alpha _m ,\lambda _m } \left( {\omega _m } \right)$$ to $$M\dot K_{p,q}^{\alpha ',\lambda } \left( \omega \right)$$ for both cases 0 < p < 1 and 1 ≤ p < ∞. As a consequence, when m = n = 1 and s 1(t) = t, we obtain an improvement of a recent result by Tang, Xue and Zhou.

Let \(\varphi \) be a nonnegative integrable function on \((0,\infty )\). It is well-known that the Hausdorff operator \({{\mathcal {H}}}_\varphi \) generated by \(\varphi \) is bounded on the real Hardy space \(H^1({{\mathbb {R}}})\). The aim of this paper is to give the exact norm of \({{\mathcal {H}}}_\varphi \). More precisely, we prove that $$\begin{aligned} \Vert {{\mathcal {H}}}_\varphi \Vert _{H^1({{\mathbb {R}}})\rightarrow H^1({{\mathbb {R}}})}= {\mathop {\int }\limits _{0}^{\infty }} \varphi (t)dt. \end{aligned}$$

This paper evaluates trade-off between reliable and secure communication for multi-hop low-energy adaptive clustering hierarchy (LEACH) networks under impact of hardware imperfection. In the proposed scheme, a source communicates with a destination via a multi-hop transmission, where the source, destination and intermediate relay nodes are cluster heads (CHs). To protect the data at each hop, harvest-to-jam and randomize-and-forward methods are employed by CHs and cooperative jammers, respectively. Each jammer which is randomly selected at each cluster harvests energy from the previous CHs to transmit noises to an eavesdropper. We derive closed-form expressions of the end-to-end outage probability (OP) and intercept probability (IP) for the proposed scheme over Rayleigh fading channels. Then, the theoretical derivations are verified by Monte-Carlo simulations.

In this paper we aim to investigate the boundedness of the p-adic weighted Hardy-Cesaro operators and their commutators on weighted functional spaces of Morrey type. In each case, we obtain the corresponding operator norms.

This paper deals with a general class of weighted multilinear Hardy-Cesaro operators that acts on the product of Lebesgue spaces and central Morrey spaces. Their sharp bounds are also obtained. In addition, we obtain sufficient and necessary conditions on weight functions so that the commutators of these weighted multilinear Hardy-Cesaro operators (with symbols in central BMO spaces) are bounded on the product of central Morrey spaces. These results extends known results on multilinear Hardy operators.

In this paper, we evaluate secrecy performance of cooperative jamming (CJ) aided secure transmission for Multi-hop Low-Energy Adaptive Clustering Hierarchy (M-LEACH) networks, where data transmission between a source and a destination is performed via cluster heads (CHs) of intermediate clusters, and is overheard by an eavesdropper. To protect the source information at each hop, each CH selects one or many cluster nodes to realize the CJ technique. We propose energy harvesting (EH) based jammer selection methods named MAX-EH, MAX-IP and ALL-CJ, where the selected jammer(s) has to harvest wireless energy from a dedicated power beacon to generate jamming noises. In MAX-EH, the cluster node obtained the MAXimum Energy Harvested from the power beacon is chosen. In MAX-IP, the selected jammer is the cluster node providing the MAXimum Interference Power to the eavesdropper. In ALL-CJ, ALL cluster nodes in the clusters take part in the CJ operation. For performance evaluation, we derive expressions of the end-to-end (e2e) secrecy outage probability (SOP) for the proposed methods over Rayleigh fading channels. We then perform Monte-Carlo based computer simulations to verify the theoretical results. The results presented that the proposed methods outperform the RANDom jammer selection (RAND) and the corresponding method WithOut using the CJ technique (WO-CJ).