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Energy-efficient transceiver designs for multiuser MIMO cognitive radio networks via interference alignment

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Abstract

This paper studies the transceiver design for multiuser multiple-input multiple-output cognitive radio networks. Different from the conventional methods which aim at maximizing the spectral efficiency, this paper focuses on maximizing the energy efficiency (EE) of the network. First, we formulate the precoding and decoding matrix designs as optimization problems which maximize the EE of the network subject to per-user power and interference constraints. With a higher priority in accessing the spectrum, the primary users (PUs) can design their transmission strategies without awareness of the secondary user (SU) performance. Thus, we apply a full interference alignment technique to eliminate interference between the PUs. Then, the EE maximization problem for the primary network can be reformulated as a tractable concave-convex fractional program which can be solved by the Dinkelbach method. On the other hand, the uncoordinated interference from the PUs to the SUs cannot be completely eliminated due to a limited coordination between the PUs with the SUs. The secondary transceivers are designed to optimize the EE while enforcing zero-interference to the PUs. Since the EE maximization for the secondary network is an intractable fractional programming problem, we develop an iterative algorithm with provable convergence by invoking the difference of convex functions programming along with the Dinkelbach method. In addition, we also derive closed-form expressions for the solutions in each iteration to gain insights into the structures of the optimal transceivers. The simulation results demonstrate that our proposed method outperforms the conventional approaches in terms of the EE.

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Acknowledgements

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 102.04-2013.46.

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Correspondence to Ha Hoang Kha.

Appendices

Appendix 1: Proof of concavity and convexity of (34) and (35)

For a given \(\pmb {Q}_{-k}\), the proof the concavity of \(\mathcal {R}_k(\pmb {Q}_k,\pmb {Q}_{-k})\) in \(\pmb {Q}_k\) is easily derived by the property of restriction of convex function to a line as in [33]. Using this property, we next prove the convexity of \(f_k(\pmb {Q}_k,\pmb {Q}_{-k})\) in \(\pmb {Q}_{k}\) by showing that \(f_k(z)\triangleq (\pmb {A}_k+z\pmb {B}_k,\pmb {Q}_{-k})\) is convex in \(z\in [0,1]\) where \(\pmb {A}_k,\pmb {B}_k\in \{\pmb {Q}_k|(\hbox {29b})\}\). First, we recall some useful formulas for the matrix differential calculus of a matrix function \(\pmb {X}(z)\) as follows [34]

$$\begin{aligned}&\frac{\partial \langle \pmb {X}\rangle }{\partial z} =\Big \langle \frac{\partial \pmb {X}}{\partial z}\Big \rangle \end{aligned}$$
(51)
$$\begin{aligned} \frac{\partial \ln |\pmb {X}|}{\partial z}&\quad =\Big \langle \pmb {X}^{-1}\frac{\partial \pmb {X}}{\partial z}\Big \rangle \end{aligned}$$
(52)
$$\begin{aligned} \frac{\partial \pmb {X}^{-1}}{\partial z}&\quad =-\pmb {X}^{-1}\frac{\partial \pmb {X}}{\partial z}\pmb {X}^{-1}. \end{aligned}$$
(53)

Applying (52) and (53) to (35) yields

$$\begin{aligned}&\frac{\partial f_k(\pmb {A}_k+z\pmb {B}_k,\pmb {Q}_{-k})}{\partial z} =\frac{1}{\ln 2} \sum _{\ell \in \mathcal {K}_s\setminus k} \big \langle \nonumber \\&\qquad -(\pmb {I}+\pmb {X}_\ell \pmb {R}_\ell ^{-1})^{-1} \pmb {X}_\ell \pmb {R}_\ell ^{-1}\pmb {Y}_\ell \pmb {R}_\ell ^{-1} \big \rangle \nonumber \\&\quad =\frac{1}{\ln 2} \sum _{\ell \in \mathcal {K}_s\setminus k} \big \langle \nonumber \\&\qquad -(\pmb {R}_\ell +\pmb {X}_\ell )^{-1} \pmb {X}_\ell \pmb {R}_\ell ^{-1}\pmb {Y}_\ell \big \rangle \end{aligned}$$
(54)

where \(\pmb {X}_\ell ={\pmb {H}}_{\ell ,\ell }\pmb {Q}_\ell {\pmb {H}}_{\ell ,\ell }^H\) and \(\pmb {Y}_\ell ={\pmb {H}}_{\ell ,k}\pmb {B}_k{\pmb {H}}_{\ell ,k}^H, \forall \ell \in \mathcal {K}_s\setminus k\). Applying to (51), (52) and (53) to (54), we then obtain

$$\begin{aligned}&\frac{\partial ^2 f_k(\pmb {A}_k+z\pmb {B}_k,\pmb {Q}_{-k})}{\partial z^2} \nonumber \\&\quad =\frac{1}{\ln 2} \sum _{\ell \in \mathcal {K}_s\setminus k} \Big \langle (\pmb {R}_\ell +\pmb {X}_\ell )^{-1} \pmb {Y}_\ell (\pmb {R}_\ell +\pmb {X}_\ell )^{-1} \pmb {X}_\ell \pmb {R}_\ell ^{-1}\pmb {Y}_\ell \nonumber \\&\qquad \qquad +(\pmb {R}_\ell +\pmb {X}_\ell )^{-1} \pmb {X}_\ell \pmb {R}_\ell ^{-1}\pmb {Y}_\ell \pmb {R}_\ell ^{-1}\pmb {Y}_\ell \Big \rangle \nonumber \\&\quad =\frac{1}{\ln 2} \sum _{\ell \in \mathcal {K}_s\setminus k} \big \langle (\pmb {R}_\ell +\pmb {X}_\ell )^{-1}\pmb {Y}_\ell \pmb {C}_\ell \pmb {Y}_\ell \big \rangle \end{aligned}$$
(55)

where \(\pmb {C}_\ell \triangleq (\pmb {R}_\ell +\pmb {X}_\ell )^{-1}\pmb {X}_\ell \pmb {R}_\ell ^{-1} =\pmb {R}_\ell ^{-1}-(\pmb {R}_\ell +\pmb {X}_\ell )^{-1}\). Since \(\pmb {R}_\ell \succeq 0\) and \(\pmb {X}_\ell \succeq 0\), \(\pmb {R}_\ell +\pmb {X}_\ell \succeq 0\), \(\pmb {R}_\ell +\pmb {X}_\ell \succeq \pmb {R}_\ell \), we have \((\pmb {R}_\ell +\pmb {X}_\ell )^{-1}\succeq 0\) and \(\pmb {C}_\ell =\pmb {R}_\ell ^{-1}-(\pmb {R}_\ell +\pmb {X}_\ell )^{-1}\succeq 0\). Therefore, there always exist matrices \(\pmb {M}_\ell \), \(\pmb {N}_\ell \) and \(\pmb {K}_\ell \) such that \(\pmb {C}_\ell =\pmb {M}_\ell \pmb {M}_\ell ^H\), \((\pmb {R}_\ell +\pmb {X}_\ell )^{-1}=\pmb {N}_\ell \pmb {N}_\ell ^H\) and \(\pmb {R}_\ell ^{-1}=\pmb {K}_\ell \pmb {K}_\ell ^H\). Thus, we have

$$\begin{aligned}&\big \langle (\pmb {R}_\ell +\pmb {X}_\ell )^{-1}\pmb {Y}_\ell \pmb {C}_\ell \pmb {Y}_\ell \big \rangle = \big \langle (\pmb {N}_\ell ^H\pmb {Y}_\ell \pmb {M}_\ell )(\pmb {N}_\ell ^H\pmb {Y}_\ell \pmb {M}_\ell )^H \big \rangle \ge 0 \\&\big \langle \pmb {C}_\ell \pmb {Y}_\ell \pmb {R}_\ell ^{-1}\pmb {Y}_\ell \big \rangle = \big \langle (\pmb {M}_\ell ^H\pmb {Y}_\ell \pmb {K}_\ell )(\pmb {M}_\ell ^H\pmb {Y}_\ell \pmb {K}_\ell )^H \big \rangle \ge 0 \end{aligned}$$

because \((\pmb {N}_\ell ^H\pmb {Y}_\ell \pmb {M}_\ell )(\pmb {N}_\ell ^H\pmb {Y}_\ell \pmb {M}_\ell )^H \succeq 0\) and \((\pmb {M}_\ell ^H\pmb {Y}_\ell \pmb {K}_\ell )(\pmb {M}_\ell ^H\pmb {Y}_\ell \pmb {K}_\ell )^H \succeq 0\). Therefore, \(\frac{\partial ^2 f_k(\pmb {A}_k+z\pmb {B}_k,\pmb {Q}_{-k})}{\partial z^2}\ge 0\), which means that \(f_k(\pmb {A}_k+z\pmb {B}_k,\pmb {Q}_{-k})\) is convex in z and hence, \(f_k(\pmb {Q}_{k},\pmb {Q}_{-k})\) is a convex function in \(\pmb {Q}_k\). This completes the proof.

Appendix 2: Derivation of (37)

Let us recall some useful formulas in the matrix differential calculus for given matrix function \(\pmb {X}\) as [34]

$$\begin{aligned}&\partial (\pmb {X}^{-1})=-\pmb {X}^{-1}(\partial \pmb {X})\pmb {X}^{-1} \end{aligned}$$
(56)
$$\begin{aligned}&\partial (\ln |\pmb {X}|)= \big \langle \pmb {X}^{-1}\partial \pmb {X}\big \rangle \end{aligned}$$
(57)

Applying (56) and (57) to (37), we have

$$\begin{aligned}&\partial (f_k(\pmb {Q}_k))=\partial \Big ( \sum _{\ell \in \mathcal {K}_s\setminus k}\log _2|\pmb {I}+\pmb {X}_\ell \pmb {R}_\ell ^{-1}| \Big ) \nonumber \\&\quad = \frac{1}{\ln 2} \Big \langle \sum _{\ell \in \mathcal {K}_s\setminus k} (\pmb {I}+\pmb {X}_\ell \pmb {R}_\ell ^{-1})^{-1} \partial (\pmb {I}+\pmb {X}_\ell \pmb {R}_\ell ^{-1}) \Big \rangle \nonumber \\&\quad = \frac{1}{\ln 2} \Big \langle \sum _{\ell \in \mathcal {K}_s\setminus k} (\pmb {I}+\pmb {X}_\ell \pmb {R}_\ell ^{-1})^{-1} \pmb {X}_\ell \partial (\pmb {R}_\ell ^{-1}) \Big \rangle \nonumber \\&\quad = \frac{1}{\ln 2} \Big \langle -\sum _{\ell \in \mathcal {K}_s\setminus k} (\pmb {I}+\pmb {X}_\ell \pmb {R}_\ell ^{-1})^{-1} \pmb {X}_\ell \pmb {R}_\ell ^{-1} {\pmb {H}}_{\ell ,k}\partial (\pmb {Q}_k){\pmb {H}}_{\ell ,k}^H \pmb {R}_\ell ^{-1} \Big \rangle \nonumber \\&\quad = \frac{1}{\ln 2} \Big \langle -\sum _{\ell \in \mathcal {K}_s\setminus k} {\pmb {H}}_{\ell ,k}^H (\pmb {R}_\ell +\pmb {X}_\ell )^{-1} \pmb {X}_\ell \pmb {R}_\ell ^{-1} {\pmb {H}}_{\ell ,k} \partial (\pmb {Q}_k) \Big \rangle \nonumber \\&\quad = \Big \langle -\frac{1}{\ln 2}\sum _{\ell \in \mathcal {K}\setminus k} {\pmb {H}}_{\ell ,k}^H \big [ \pmb {R}_\ell ^{-1}-(\pmb {R}_\ell +\pmb {X}_\ell )^{-1} \big ] {\pmb {H}}_{\ell ,k} \partial (\pmb {Q}_k) \Big \rangle \end{aligned}$$
(58)
$$\begin{aligned}&= \Big \langle \nabla _{Q_k}f_k(\pmb {Q}_k) \partial (\pmb {Q}_k) \Big \rangle . \end{aligned}$$
(59)

By comparing (58) and (59), the proof is completed.

Appendix 3: Proof of Theorem 1

We note that since \(\pmb {R}_\ell ^{-1}-(\pmb {R}_\ell +\pmb {X}_\ell )^{-1}\succeq 0\) (see “Appendix 1”), it is readily to prove that \(-\pmb {D}_k({\tilde{\pmb {Q}}}_k)\succeq 0\) and then \(\pmb {\varUpsilon }_k\succ 0\). Therefore, \(\pmb {\varUpsilon }_k\) is invertible. Let us define \({\bar{\pmb {Q}}}_k=\pmb {\varUpsilon }_k^{1/2}\pmb {Q}_k\pmb {\varUpsilon }_k^{1/2}\in \mathbb {C}^{d_k\times d_k}\). Problem (47) can be rewritten as

$$\begin{aligned}&\underset{\pmb {Q}_k\succeq 0}{\max }\,\,\, \log _2 \Big | \pmb {I}+ \left( \pmb {R}_{k}^{-1/2}{\tilde{\pmb {H}}}_{k,k}\pmb {\varUpsilon }_k^{-1/2} \right) \nonumber \\&\quad {\bar{\pmb {Q}}}_k \left( \pmb {R}_{k}^{-1/2}{\tilde{\pmb {H}}}_{k,k}\pmb {\varUpsilon }_k^{-1/2} \right) ^H \Big | - \langle {\bar{\pmb {Q}}}_k\rangle . \end{aligned}$$
(60)

Applying SVD \(\pmb {R}_{k}^{-1/2}{\tilde{\pmb {H}}}_{k,k}\pmb {\varUpsilon }_k^{-1/2} =\pmb {\varXi }_k\pmb {\varPhi }_k\pmb {\varDelta }_k^H\), where \(\pmb {\varXi }_k\in \mathbb {C}^{d_k\times d_k}\), \(\pmb {\varDelta }_k\in \mathbb {C}^{d_k\times d_k}\) and \(\pmb {\varPhi }_k=[\mathrm {diag}(\phi _{k,1},\ldots ,\phi _{k,d_k})]\), \(\phi _{k,1}\ge \ldots \ge \phi _{k,d_k}\). Substituting these results into (47) and exploiting the Hadamard inquality, the optimal solution to problem (47) has the form \({\bar{\pmb {Q}}}_k^{*}=\pmb {\varDelta }_k\pmb {\varGamma }_k\pmb {\varDelta }_k^H\) [15], where \(\pmb {\varGamma }_k=\mathrm {diag}(\gamma _{k,1},\ldots ,\gamma _{k,d_k})\) where \(\gamma _{k,t}=\left[ \frac{1}{\ln 2}-\frac{1}{\phi _{k,t}^2}\right] ^{+}, \forall t\in \mathcal {S}_k=\{1,\ldots ,d_k\}\). Therefore, the optimal solution to problem (47) has the form

$$\begin{aligned} \pmb {Q}_k^{*}=\pmb {\varUpsilon }_k^{-1/2}\pmb {\varDelta }_k\pmb {\varGamma }_k\pmb {\varDelta }_k^H\pmb {\varUpsilon }_k^{-1/2}. \end{aligned}$$
(61)

This finishes the proof.

Appendix 4: Proof of the convergence of Algorithm 2

Since the Dinkelbach method was proved to be converged [21], the convergence of Algorithm 2 relies on the convergence of (42). Suppose that \({\tilde{\pmb {Q}}}_k\) is an optimal solution from the previous iteration and \(\pmb {Q}_k^\star \) is the optimal solution at the current solution to (42). Then, at the current iteration, one has

$$\begin{aligned} \eta _{\text {SEE}}(\pmb {Q}_k^{\star }, {\tilde{\pmb {Q}}}_{-k})&\ge {\tilde{\eta }}_{\text {SEE}}(\pmb {Q}_k^{\star }, {\tilde{\pmb {Q}}}_{-k}) \end{aligned}$$
(62a)
$$\begin{aligned}&\ge {\tilde{\eta }}_{\text {SEE}}({\tilde{\pmb {Q}}}_k, {\tilde{\pmb {Q}}}_{-k}) \end{aligned}$$
(62b)
$$\begin{aligned}&= \eta _{\text {SEE}}({\tilde{\pmb {Q}}}_k, {\tilde{\pmb {Q}}}_{-k}) \end{aligned}$$
(62c)

where inequality (62a) is the result of (41); (62b) holds because \({\tilde{\pmb {Q}}}_k^{\star }\) is the optimal solution to problem (42) at the current iteration; and (62c) is due to the fact that (41) is meet with equality at . This means that function \(\eta _{\text {SEE}}(\pmb {Q}_k,\pmb {Q}_{-k})\) is nondecreasing after updating \(\pmb {Q}_k\) for given \(\pmb {Q}_{-k}\) at each link k iteratively. In addition, the objective \(\eta _{\text {SEE}}\) is upper-bounded and, hence, Algorithm 2 must converge. This completes the proof.

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Kha, H.H., Vu, T.T. & Do-Hong, T. Energy-efficient transceiver designs for multiuser MIMO cognitive radio networks via interference alignment. Telecommun Syst 66, 469–480 (2017). https://doi.org/10.1007/s11235-017-0300-9

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