Anti-interference distributed energy-efficient for multi-carrier millimeter-wave ultra-dense networks

Abstract

This paper investigates an anti-interference energy-efficient power allocation scheme in the multi-carrier millimeter-wave (mmWave) ultra-dense networks. To suppress the severe intercell-interference, this work proposes a novel interference minimization scheme based on the non-cooperative game theory for energy-efficiency maximization. In each best response, the non-convex problem is transformed into some convex subproblems, and each is solved by a low-complexity stair water-filling (SWF) algorithm over some subintervals. The interference minimization scheme, together with the SWF algorithm, has been proven to converge to a unique Nash equilibrium point. Simulation results and numerical analysis show that the scheme displays significant energy-efficiency performance advantages over other iterative water-filling methods.

Appendices

Appendix A

Let $g(\mu )=\frac{\log _{2} \mu -\frac{1}{L_{k, i}} \sum _{d_{k, t} \in \mathrm {Q_{k, i}}} \log _{2} d_{k, t}}{\frac{\mu }{B_{0}}-\left( \frac{\sum _{d_{k, t} \in \mathrm {Q}_{\mathrm{k}, \mathrm{i}}} d_{k, t}-p_{\mathrm {c}}}{B_{0} L_{k, i}}\right) }$, the first-order derivative of $g(\mu )$ is $g^{\prime }(\mu )=\frac{f(\mu )}{p(\mu )}$, $f(\mu )$ is shown as (14), $p(\mu )=\left( \frac{\mu }{B_{0}}-\left( \frac{\sum _{k_{k, t} \in \mathrm {Q_{k, i}}} d_{k, t}-p_{c}}{B_{0} L_{k, i}}\right) \right) ^{2}$.

Property3: $\mu ^{p}$ be the zero-point of $g^{\prime }(\mu )$ and $f(\mu )$. $\mu ^{q}$ is the zero-point of $f^{\prime }(\mu )$ and $p^{\prime }(\mu )$. When $\mu >\mu ^{q}, g^{\prime }(\mu )$ is monotonically decreasing. When $\mu <\mu ^{q}, g^{\prime }(\mu )$ is monotonically increasing. Therefore, $\mu ^{q}$ is the point of monotonicity change of $g^{\prime }(\mu )$.

The maximal value of the quasi-convex function $g(\mu )$ will appear at the bound point or the inner point $\mu ^{q}$. Based on $\mu ^{q}$, the feasible domain of $g(\mu )$ can be divided into the following cases:

• 1. When $\mu ^{q} \in \bar{C}_{k, i}$ and there is no zero-point of $g^{\prime }(\mu ),$ the maximal value of $g(\mu )$ is at the bound point, which can be obtained according to Property 3. When $\mu ^{q} \in \bar{C}_{k, i}$ and there is a zero-point of $g^{\prime }(\mu ),$ the maximal value of $g(\mu )$ only appears in the case of $\mu ^{q} \prec C_{k, i}$, i.e., $\mu _{k, i}^{\star } =\mu ^{p}$. This is because the maximal value of $g(\mu )$ appears only when $g^{\prime }(\mu )$ is greater than zero and then less than zero as $\mu$ increases, since $g^{\prime }(\mu )$ is monotonically decreasing. The value of $g^{\prime }(\mu )$ at the bound points $\left\{ d_{i}, d_{i+1}\right\}$ can be used to judge the existence of zero-point, i.e., when $\mathrm {g}^{\prime }\left( d_{i}\right) >0, g^{\prime }\left( d_{i+1}\right) <0,$ there is zero-point of $g^{\prime }(\mu )$, $\mu _{k, i}^{\star } =\mu ^{p}$.

• 2. When $\mu ^{q} \in C_{k, i}$, $g^{\prime }(\mu )$ is not the monotonic function on $C_{k, i}$. Therefore, $C_{k, i}$ is divided into the two intervals $D_{1}=\left[ d_{i}, \mu ^{q}\right]$ and $D_{2}=\left[ \mu ^{q}, d_{i+1}\right] .$ Similarly the analysis of case1, the maximal inner point of $g^{\prime }(\mu )$ appears at the monotonical decreasing domain of $g^{\prime }(\mu ),$ i.e., $D_{2}=\left[ \mu ^{q}, d_{i+1}\right] .$ When $g^{\prime }\left( \mu ^{q}\right) \ge 0, g^{\prime }\left( d_{i+1}\right) \le 0$, there is a unique zero-point $\mu ^{p}$. According to the residue property, there is ${\text {Res}}\left( g^{\prime }\left( \mu ^{q}\right) \right) =0$. Therefore, when $g^{\prime }\left( d_{i+1}\right) \le 0, g^{\prime }(\mu )$ is monotonical decreasing on $D_{2}$ and there is a unique zero-point $\mu _{k, i}^{\star } =\mu ^{p}=\mu ^{q}$.

• 3. Otherwise, the maximal point of $g(\mu )$ appears on the bound points $\left\{ d_{i}, d_{i+1}\right\}$.

Appendix B

Defining $\mathbf {Z}_{k}\left( \mathbf {p}_{k}^{\star }, \mathbf {p}_{{-k}}\right) =a \mathbf {p}_{k}^{\star }+(1-a)\left[ \mu _{k}-\tilde{\mathbf {I}}_{k}-\tilde{\varvec{\sigma }}_{k}^{2}\right] ^{\dagger }$ with $\tilde{\mathbf {I}}_{k}=\left[ \tilde{I}_{k, 1}, \ldots , \tilde{I}_{k, N}\right] ,$ where $\tilde{I}_{k, n}=\sum _{m \ne k} g_{k m, n} / g_{k k, n} \times p_{m, n}$, $\tilde{\varvec{\sigma }}_{k}^{2}=\left[ \sigma ^{2} / g_{k k, 1}, \ldots , \sigma ^{2} / g_{k k, N}\right]$. The interference channel gain at the Nash equalization point is divided into two cases:

• 1. Assuming the interference channel gain $\tilde{\mathbf {I}}_{k}^{(1)}$ can make the transmitted power greater than zero, i.e., $\left[ \mu _{k}^{(1)}-\tilde{\mathbf {I}}_{k}^{(1)}-\tilde{\varvec{\sigma }}_{k}^{2}\right] ^{\dagger }>0$. If there is a larger interference channel gain $\tilde{\mathbf {I}}_{k}^{(2)}$, which makes the transmitted power greater than zero, i.e., $\left[ \mu _{k}^{(2)}-\tilde{\mathbf {I}}_{k}^{(2)}-\tilde{\varvec{\sigma }}_{k}^{2}\right] >0$, then

  $$\begin{aligned} \begin{array}{l} e_{T_{k}} \triangleq \left\| \mathbf {Z}_{k}\left( \mathbf {p}^{(1)}\right) -\mathbf {Z}_{k}\left( \mathbf {p}^{(2)}\right) \right\| _{2} \\ \le (1{-}\alpha )\left\| \left[ \mu _{k}^{(1)}{-}\tilde{\mathbf {I}}_{k}^{(1)}{-}\tilde{\varvec{\sigma }}_{k}^{2}\right] ^{\dagger }{-}\left[ \mu _{k}^{(2)}{-}\tilde{\mathbf {I}}_{k}^{(2)}{-}\tilde{\varvec{\sigma }}_{k}^{2}\right] ^{\dagger }\right\| _{2} \\ \le (1-\alpha )\left\| \left[ \Delta \mu _{k}-\Delta \tilde{\mathbf {I}}_{k}\right] \right\| _{2}, \end{array} \end{aligned}$$

  where the first inequality is due to the triangle inequality and the second inequality is due to $\left[ \mu _{k}^{(i)}-\tilde{\mathbf {I}}_{k}^{(i)}-\tilde{\varvec{\sigma }}_{k}^{2}\right] ^{\dagger }=\left[ \mu _{k}^{(i)}-\tilde{\mathbf {I}}_{k}^{(i)}-\tilde{\varvec{\sigma }}_{k}^{2}\right] , i=1,2$. Since the water-level is only confirmed by the interference channel gain, then only when $\Delta \tilde{\mathbf {I}}_{k}=0$, there is a unique interference channel gain so that $e_{T_{k}} \le 0$, which indicates $\mathbf {Z}_{k}\left( \mathbf {p}_{k}^{\star }, \mathbf {p}_{-k}\right)$ converge.

• 2. When $\left[ \mu _{k}^{(1)}-\tilde{\mathbf {I}}_{k}^{(1)}-\tilde{\varvec{\sigma }}_{k}^{2}\right] ^{\dagger }=0$, the interference channel gain $\tilde{\mathbf {I}}_{k}^{(1)}$ is too much large to make $\mathrm {WF}\left( \mathbf {p}_{\mathrm {k}}\right) =0$. The larger interference channel gain will allocate less power to the SUE, i.e., $\left[ \mu _{k}^{(2)}-\tilde{\mathbf {I}}_{k}^{(2)}-\tilde{\varvec{\sigma }}_{k}^{2}\right] ^{\dagger }=0$, which indicates $\mathbf {Z}_{k}\left( \mathbf {p}_{k}^{\star }, \mathbf {p}_{-k}\right)$ converge.

The interference of case 1 is smaller than that of case 2. In order to guarantee the system converge to the minimal interference channel gain, the interference channel gain should gradually increase until the system achieves the Nash equalization of case 1.