#
On the Capacity and the Optimal Sum-Rate of a Class of Dual-Band Interference Channels^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

- We show that the capacity region of the DCLIC can be decomposed into the capacity region of the underlying CLIC and two non-interfering direct links in the mm-wave band. This illustrates that the cross channels are actively involved in characterizing the capacity of the CLIC, whereas the direct channels improve the rates of individual users.
- We characterize the capacity of the strong CLIC, and observe that the strong interference condition in the microwave band is sufficient to characterize the capacity.
- For the weak CLIC, we characterize sufficient channel conditions under which its capacity is established. This shows that even if the GIC in the microwave band has weak interference, adequately strong cross channels in the mm-wave band are sufficient to characterize the capacity.
- We characterize the optimal power allocation in the direct and cross channels that maximizes the sum-rate of the DCLIC, and study channel conditions under which the optimal power allocation either assigns the entire power budget to a specific subset of channels, or shares the power budget among all channels. We establish a direct relation between the channel parameters and the optimal powers, from which we observe the following:
- -
- The optimal power allocation distributes the power budget among the direct and cross channels following two properties: a waterfilling-like property and a max-min property.
- -
- When the power budget is sufficiently small, the optimal allocation assigns power to either both direct channels, or both cross channels and at most one direct channel.
- -
- Due to the max-min property, the optimal allocation imposes a maximum limit on the cross-channel powers. When the power budget exceeds a certain threshold, the limit on the cross-channel powers are reached, and all additional increments to the power budget are then added only to the direct channels that do not have such limits.
- -
- If the underlying GIC in the microwave band has very strong interference, the optimal power allocation assigns the power budget entirely to the direct channels.
- -
- If the channel parameters satisfy one of the following criteria, then transmitting only in the direct channels is approximately optimal, in the sense that the difference between the sum-rates resulting from allocating to only direct channels and allocating optimally in all channels, is negligibly small: (a) the transmit powers in the underlying GIC in the microwave band is very small; or (b) the cross channel gains in the mm-wave band are very large.

## 2. System Model

## 3. Decomposition Result on the Capacity of the DCLIC

**Theorem**

**1.**

**Proof.**

**Lemma**

**1.**

**Corollary**

**1.**

## 4. Capacity of the Weak CLIC

**Lemma**

**2.**

**Proof.**

**Theorem**

**2.**

**Proof.**

## 5. The Optimal Sum-Rate Problem

- the underlying GIC of the DCLIC has strong interference, but not very strong interference, $1\le {a}_{12}^{2}<1+{Q}_{1}$ and $1\le {a}_{21}^{2}<1+{Q}_{2}$;
- the underlying GIC of the DCLIC satisfies: ${Q}_{2}+{a}_{12}^{2}{Q}_{1}={Q}_{1}+{a}_{21}^{2}{Q}_{2}$;
- $\beta $ and $\overline{\beta}$ are fixed a priori;
- the transmission power in the direct channel (${p}_{k}$) and cross channel (${q}_{k}$) from transmitter k ($T{x}_{k}$) satisfy the constraint, $\beta {p}_{k}+\overline{\beta}{q}_{k}=P,k=1,2$, where P is the power budget.

#### 5.1. Problem Formulation and Solution

#### 5.2. The Waterfilling-Like Nature of the Optimal Power Allocation

- If ${P}_{1}^{*}>0$: ${\mathbf{OA}}_{\mathbf{1}}$ follows the sequence ${\mathcal{S}}_{D}\to {\mathcal{S}}_{CD}\to {\mathcal{S}}_{\mathsf{sat}}$ (denoted by [S1]). Since the direct channels are stronger, ${\mathbf{OA}}_{\mathbf{1}}$ allocates all of P to them as in ${\mathcal{S}}_{D}$ when P is sufficiently small (i.e., $P<{P}_{1}^{*}$). However, as P increases, the additional benefit from transmitting only in the direct channels decreases, and thus, when $P\ge {P}_{1}^{*}$, ${\mathbf{OA}}_{\mathbf{1}}$ begins transmitting in both cross and direct channels as in ${\mathcal{S}}_{CD}$. This allocation follows from its WF-like property, and remains optimal for all ${P}_{1}^{*}\le P\le {P}_{3}^{*}$. Finally, when $P>{P}_{3}^{*}$, the max-min property of ${\mathbf{OA}}_{\mathbf{1}}$ comes into effect, and thus the cross channels become saturated and ${\mathbf{OA}}_{\mathbf{1}}$ starts following the allocation in ${\mathcal{S}}_{\mathsf{sat}}$. Note that, in this case, the saturation threshold for P is ${P}_{\mathsf{sat}}={P}_{3}^{*}$.
- If ${P}_{2}^{*}>0$ but $0<{P}_{3}^{*}<{P}_{4}^{*}<{P}_{2}^{*}$ is not satisfied: ${\mathbf{OA}}_{\mathbf{1}}$ follows the sequence ${\mathcal{S}}_{C}\to {\mathcal{S}}_{CD}\to {\mathcal{S}}_{\mathsf{sat}}$ ([S2]). This case is similar to [S1] above, except for the fact that now the cross channels are stronger. Hence, ${\mathbf{OA}}_{\mathbf{1}}$ transmits in the cross channels and the direct channel with gain ${d}_{2}^{2}$ as in ${\mathcal{S}}_{C}$ when P is sufficiently small (i.e., $P<min\{{P}_{2}^{*},{P}_{4}^{*}\}$). Next, following its WF-like property, ${\mathbf{OA}}_{\mathbf{1}}$ starts transmitting in all the direct and cross channels as in ${\mathcal{S}}_{CD}$ when $P\ge min\{{P}_{2}^{*},{P}_{4}^{*}\}$. Finally, when $P>max\{{P}_{3}^{*},{P}_{4}^{*}\},$ the cross channels become saturated, and ${\mathbf{OA}}_{\mathbf{1}}$ follows the allocation in ${\mathcal{S}}_{\mathsf{sat}}$ thereon. The saturation threshold for P in this case is ${P}_{\mathsf{sat}}=max\{{P}_{3}^{*},{P}_{4}^{*}\}$.Note that, whenever ${P}_{2}^{*}>0$, ${\mathbf{OA}}_{\mathbf{1}}$ follows [S2] irrespective of how ${P}_{2}^{*},{P}_{3}^{*}$, and ${P}_{4}^{*}$ compare, except in two cases: (a) $0<{P}_{3}^{*}<{P}_{4}^{*}<{P}_{2}^{*}$, where ${\mathbf{OA}}_{\mathbf{1}}$ follows [S3], described next, and (b) $0<{P}_{4}^{*}<{P}_{3}^{*}<{P}_{2}^{*}$, which is infeasible as they violate the mutual exclusiveness of ${\mathcal{S}}_{C}$ and ${\mathcal{S}}_{\mathsf{sat}}$.
- If $0<{P}_{3}^{*}<{P}_{4}^{*}<{P}_{2}^{*}$: ${\mathbf{OA}}_{\mathbf{1}}$ follows the sequence ${\mathcal{S}}_{C}\to {\mathcal{S}}_{\mathsf{sat}}$ ([S3]). In this case, the cross channels are much stronger than the direct channels. Hence, similar to [S2], ${\mathbf{OA}}_{\mathbf{1}}$ allocates power to both cross channels and a direct channel as in ${\mathcal{S}}_{C}$ when P is sufficiently small (i.e., $0\le P<{P}_{4}^{*}$). As P increases and $P\ge {P}_{4}^{*}$, the cross channels become saturated, and ${\mathbf{OA}}_{\mathbf{1}}$ begins assigning powers to all channels as in ${\mathcal{S}}_{\mathsf{sat}}$. Interestingly, in this case, ${\mathbf{OA}}_{\mathbf{1}}$ skips ${\mathcal{S}}_{CD}$. This shows that, since the cross channels are much stronger, it is optimal to allocate power as in ${\mathcal{S}}_{C}$ until they become saturated at ${P}_{\mathsf{sat}}={P}_{4}^{*}$, beyond which the allocation in ${\mathcal{S}}_{\mathsf{sat}}$ becomes optimal.

**Remark**

**1.**

#### 5.3. Optimum Power Allocation in the Symmetric DCLIC

- If ${c}^{2}\le 2{d}^{2}$ (direct channels are “stronger”): ${\mathbf{OA}}_{\mathbf{S}}$ follows the sequence ${\overline{\mathcal{S}}}_{D}\to {\overline{\mathcal{S}}}_{CD}\to {\overline{\mathcal{S}}}_{\mathsf{sat}}$. It transmits only in the direct channels as in ${\overline{\mathcal{S}}}_{D}$ when $P<{\overline{P}}_{1}^{*}$, then transmits in all channels as in ${\overline{\mathcal{S}}}_{CD}$ when ${\overline{P}}_{1}^{*}\le P\le {P}_{3}^{*}$, and finally starts following the allocation in ${\overline{\mathcal{S}}}_{\mathsf{sat}}$ when $P>{\overline{P}}_{3}^{*}$.
- If $2{d}^{2}<{c}^{2}<2{d}^{2}\overline{\gamma}$ (cross channels are “stronger”): ${\mathbf{OA}}_{\mathbf{S}}$ follows the sequence ${\overline{\mathcal{S}}}_{C}\to {\overline{\mathcal{S}}}_{CD}\to {\overline{\mathcal{S}}}_{\mathsf{sat}}$. It transmits only in the cross channels as in ${\overline{\mathcal{S}}}_{C}$ when $P<{\overline{P}}_{2}^{*}$, then transmits in all channels as in ${\overline{\mathcal{S}}}_{CD}$ when ${\overline{P}}_{2}^{*}\le P\le min\{{P}_{3}^{*},{P}_{4}^{*}\}$, and finally follows ${\overline{\mathcal{S}}}_{\mathsf{sat}}$ when $P>min\{{P}_{3}^{*},{P}_{4}^{*}\}$.
- If ${c}^{2}\ge 2{d}^{2}\overline{\gamma}$ (cross channels are much “stronger”): ${\mathbf{OA}}_{\mathbf{S}}$ follows the sequence ${\overline{\mathcal{S}}}_{C}\to {\overline{\mathcal{S}}}_{\mathsf{sat}}$. It follows ${\overline{\mathcal{S}}}_{C}$ when $P<{\overline{P}}_{4}^{*}$, and then follows ${\overline{\mathcal{S}}}_{\mathsf{sat}}$ when $P\ge {P}_{4}^{*}$, while skipping ${\overline{\mathcal{S}}}_{CD}$ altogether.

#### 5.4. Discussion and Insights

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

GIC | Gaussian interference channel |

CLIC | Cross-Link interference channel |

DLIC | Direct-Link interference channel |

DCLIC | Direct-and-Cross-Link interference channel |

PGIC | Parallel Gaussian interference channel |

BMF | Bandwidth mismatch factor |

DM | Discrete memoryless |

RV | Random variable |

WF | Waterfilling |

KKT | Karush–Kuhn–Tucker |

## Appendix A. Proof of Theorem 1

**Proof.**

**Outer bound:**We derive the bounds on ${R}_{1}$ and ${R}_{2}$ as follows. If transmitter $T{x}_{k}$ transmits the message ${M}_{k},k=1,2$, we have from Fano’s inequality

**Achievability:**We will code over t blocks of symbols together. We choose $(n,{n}_{1},{n}_{2})$, and define ${\mathbf{U}}_{k}:=({X}_{k}^{n},{\widehat{X}}_{k}^{{n}_{1}}),k=1,2,$ where $p({\mathbf{U}}_{1},{\mathbf{U}}_{2})=p({\mathbf{U}}_{1})p({\mathbf{U}}_{2})$, and define ${\overline{\mathbf{U}}}_{k}:={\overline{X}}_{k}^{{n}_{2}},k=1,2$ and choose ${\overline{X}}_{k\ell}\sim \mathcal{N}(0,{\overline{P}}_{k})$, i.i.d., for $\ell =1,\dots ,{n}_{2}$. Suppose $T{x}_{k}$ transmits ${M}_{k}\in {\mathcal{M}}_{k},k=1,2$. Thus, to encode ${M}_{k}$ over t blocks for the underlying CLIC, we generate ${2}^{tn{R}_{k}}$ i.i.d. sequences ${\mathbf{U}}_{k}^{t}({M}_{k})$, one for each ${M}_{k}\in {\mathcal{M}}_{k}$, where ${\mathbf{U}}_{k}^{t}({M}_{k})$ are distributed according to $p({\mathbf{u}}_{k}^{t})={\prod}_{\ell =1}^{t}p({\mathbf{u}}_{k,\ell})={\prod}_{\ell =1}^{t}p\left(\right)open="("\; close=")">{x}_{k,(\ell -1)n+1}^{n\ell}{\widehat{x}}_{k,(\ell -1){n}_{1}+1}^{{n}_{1}\ell}$. Similarly, for the two direct channels, we generate ${2}^{tn{R}_{k}}$ i.i.d. sequences ${\overline{\mathbf{U}}}_{k}^{t}({M}_{k})$, distributed as ${\prod}_{\ell =1}^{{n}_{2}t}p({\overline{x}}_{k\ell}),$ where ${\overline{x}}_{k\ell}\sim \mathcal{N}(0,{\overline{P}}_{k})$, for $k=1,2,$ and $\ell =1,\dots ,{n}_{2}$.

## Appendix B. Proof of Theorem 2

**Proof.**

**Outer bound:**We derive the bound on ${R}_{1}$ first. Assuming that the transmitter $T{x}_{k}$ transmits the message ${M}_{k},k=1,2$, we have from Fano’s inequality

**Achievability**: We fix n and ${n}_{1}$, and choose independent input distributions for the codewords of two users in each channel, i.e., $p({x}_{1},{x}_{2})=p({x}_{1})p({x}_{2})$, and $\widehat{p}({\widehat{x}}_{1},{\widehat{x}}_{2})=\widehat{p}({\widehat{x}}_{1})\widehat{p}({\widehat{x}}_{2})$. The transmitter $T{x}_{k}$ encodes a message, ${M}_{k}\in {\mathcal{M}}_{k}$ into two codewords, ${x}_{k}^{n}({M}_{k})$ and ${\widehat{x}}_{k}^{{n}_{1}}({M}_{k})$, generated according to the i.i.d. distributions, ${\prod}_{\ell =1}^{n}{p}_{k}({x}_{k\ell})$ and ${\prod}_{\ell =1}^{{n}_{1}}{\widehat{p}}_{k}({\widehat{x}}_{k\ell}),k=1,2$. The codewords, ${x}_{k}^{n}({M}_{k})$ and ${\widehat{x}}_{k}^{{n}_{1}}({M}_{k})$, are then transmitted through the IC in the first band and the cross channels in the second band, respectively. Each receiver then estimates both messages, $({M}_{1},{M}_{2})$, from the signals observed over both bands as in a multiple access channel. We use standard error analysis technique [32] (Chapter 4.5) to find that the probability of decoding error becomes arbitrarily small as $n\to \infty $, if the rates achieved in the receiver $R{x}_{k}$ satisfies

## Appendix C. Derivation of the Optimal Power Allocation

**The KKT Conditions:**We show that $\left[\mathbf{P}\mathbf{1}\right]$ is convex, and formulate its KKT conditions. Denote by $\mathbf{x}:=({p}_{1},{q}_{1},{p}_{2},{q}_{2},R)\in {\mathbb{R}}_{+}^{5}$ a feasible point, satisfying the constraints in $\left[\mathbf{P}\mathbf{1}\right]$. Now, the objective of $\left[\mathbf{P}\mathbf{1}\right]$ is equivalent to minimizing$-R$, which is linear, and the equality constraints (30) and (31) are affine. Next, we note that the constraints (27)–(29) are convex. To illustrate, we define ${G}_{1}(\mathbf{x}):=R-{A}_{1}-\frac{\alpha \overline{\beta}\kappa}{2}ln(1+{c}_{21}^{2}{q}_{2})-\frac{\alpha \beta \kappa}{2}ln(1+{d}_{1}^{2}{p}_{1})-\frac{\alpha \beta \kappa}{2}ln(1+{d}_{2}^{2}{p}_{2})$ corresponding to (27) with $\kappa :=1/ln2$, and derive its Hessian, ${\nabla}^{2}{G}_{1}(\mathbf{x}):=\frac{\alpha \kappa}{2}\phantom{\rule{4.pt}{0ex}}\mathrm{diag}\left[\left(\right),\beta {d}_{1}^{4},/,{(1+{d}_{1}^{2}{p}_{1})}^{2},,,\phantom{\rule{0.277778em}{0ex}},\phantom{\rule{0.277778em}{0ex}},0,,,\phantom{\rule{0.277778em}{0ex}},\phantom{\rule{0.277778em}{0ex}},\beta {d}_{2}^{4},/,{(1+{d}_{2}^{2}{p}_{2})}^{2},,,\phantom{\rule{0.277778em}{0ex}},\phantom{\rule{0.277778em}{0ex}},\overline{\beta}{c}_{21}^{4},/,{(1+{c}_{21}^{2}{q}_{2})}^{2},,,\phantom{\rule{0.277778em}{0ex}},\phantom{\rule{0.277778em}{0ex}},0\right]$, where $D=\phantom{\rule{4.pt}{0ex}}\mathrm{diag}\phantom{\rule{4.pt}{0ex}}[{a}_{1},\dots ,{a}_{m}]$ is a diagonal matrix with elements ${a}_{1},\dots ,{a}_{m}$. Note that ${\nabla}^{2}{G}_{1}(\mathbf{x})$ is positive semidefinite, and thus (27) is convex. Likewise, (28) and (29) are found to be convex. In addition, (30) and (31) and $\mathbf{x}\u2ab0\mathbf{0}$ imply that the feasible set is compact for given $P>0$ and $\beta \in (0,1)$. Hence, $\left[\mathbf{P}\mathbf{1}\right]$ is a convex problem over a compact set. Furthermore, $\left[\mathbf{P}\mathbf{1}\right]$ satisfies Slater’s condition [35] (Chapter 5.2.3), since the point $\tilde{\mathbf{x}}:=\left(\right)open="("\; close=")">P-\overline{\beta}\u03f5/\beta ,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\u03f5,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}P-\overline{\beta}\u03f5/\beta ,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\u03f5,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{A}_{1}$ is strictly feasible for sufficiently small $\u03f5>0$. Therefore, $\left[\mathbf{P}\mathbf{1}\right]$ can be solved using the KKT conditions [35] (Chapter 5.5.3).

**Partitioning the Set of the Optimal Lagrange Multipliers:**First, note that ${\rho}_{1}$ and ${\rho}_{2}$ do not satisfy ${\rho}_{1}>0$ and ${\rho}_{2}>0$, as this implies ${p}_{1}={q}_{1}=0$, which violates (A12). Similarly, ${\rho}_{3}$ and ${\rho}_{4}$ do not satisfy ${\rho}_{3}>0$ and ${\rho}_{4}>0$, as this violates (A13). Therefore, we partition the set of $({\rho}_{1},{\rho}_{2})\u2ab0\mathbf{0}$ into the disjoint subsets, ${\mathcal{B}}_{1}:=\{({\rho}_{1},{\rho}_{2}):{\rho}_{1}>0,{\rho}_{2}=0\}$, ${\mathcal{B}}_{2}:=\{({\rho}_{1},{\rho}_{2}):{\rho}_{1}=0,{\rho}_{2}>0\}$, and ${\mathcal{B}}_{3}:=\{({\rho}_{1},{\rho}_{2}):{\rho}_{1}=0,{\rho}_{2}=0\}$. Similarly, we partition the set $({\rho}_{3},{\rho}_{4})\u2ab0\mathbf{0}$ into three disjoint subsets, ${\mathcal{C}}_{k},k=1,2,3$. Therefore, any tuple $({\rho}_{1},{\rho}_{2},{\rho}_{3},{\rho}_{4})$ must be in one of the nine resulting cases, ${\mathcal{B}}_{k}\cap {\mathcal{C}}_{l},k,l=1,2,3$. For conciseness, we denote the tuples $({\rho}_{1},{\rho}_{2},{\rho}_{3},{\rho}_{4})$ and $({\lambda}_{1},{\lambda}_{2},{\lambda}_{3})$ by the vectors $\mathbf{\rho}$ and $\mathbf{\lambda}$, respectively.

- ${\mathsf{R}}_{1}$: For any $k\in \{1,3\}$, $l\in \{1,2,3\}$, if $\mathbf{\rho}\in {\mathcal{B}}_{k}\cap {\mathcal{C}}_{l}$ then $\mathbf{\lambda}\notin {\mathcal{D}}_{j},j=1,3.$
- ${\mathsf{R}}_{2}$: For any $k\in \{1,2,3\}$, $l\in \{1,3\}$, if $\mathbf{\rho}\in {\mathcal{B}}_{k}\cap {\mathcal{C}}_{l}$ then $\mathbf{\lambda}\notin {\mathcal{D}}_{j},j=2,3.$
- ${\mathsf{R}}_{3}$: If $\mathbf{\rho}\in {\mathcal{B}}_{1}\cap {\mathcal{C}}_{2}$ then $\mathbf{\lambda}\notin {\mathcal{D}}_{j}$, $j=2,4,5$.
- ${\mathsf{R}}_{4}$: If $\mathbf{\rho}\in {\mathcal{B}}_{2}\cap {\mathcal{C}}_{1}$ then $\mathbf{\lambda}\notin {\mathcal{D}}_{j}$, $j=1,4,5$.
- ${\mathsf{R}}_{5}$: If $\mathbf{\rho}\in {\mathcal{B}}_{2}\cap {\mathcal{C}}_{3}$ then $\mathbf{\lambda}\notin {\mathcal{D}}_{j}$, $j=1,4,5$.
- ${\mathsf{R}}_{6}$: If $\mathbf{\rho}\in {\mathcal{B}}_{3}\cap {\mathcal{C}}_{2}$ then $\mathbf{\lambda}\notin {\mathcal{D}}_{j}$, $j=2,4,5$.
- ${\mathsf{R}}_{7}$: If $\mathbf{\rho}\in {\mathcal{B}}_{2}\cap {\mathcal{C}}_{2}$ then $\mathbf{\lambda}\notin {\mathcal{D}}_{j}$, $j=1,2,3,5$.

${\mathcal{B}}_{\mathit{k}}\cap {\mathcal{C}}_{\mathit{l}}$ | ${\mathcal{D}}_{1}$ | Cond. | ${\mathcal{D}}_{2}$ | Cond. | ${\mathcal{D}}_{3}$ | Cond. | ${\mathcal{D}}_{4}$ | Cond. | ${\mathcal{D}}_{5}$ | Cond. |
---|---|---|---|---|---|---|---|---|---|---|

${\mathcal{B}}_{1}\cap {\mathcal{C}}_{1}$ | × | ${\mathsf{R}}_{1}$ | × | ${\mathsf{R}}_{2}$ | × | ${\mathsf{R}}_{1}$ | × | ${c}_{12}^{2}<{c}_{21}^{2}$ | × | ${c}_{12}^{2}<{c}_{21}^{2}$ |

${\mathcal{B}}_{1}\cap {\mathcal{C}}_{2}$ | × | ${\mathsf{R}}_{1}$ | × | ${\mathsf{R}}_{3}$ | × | ${\mathsf{R}}_{1}$ | × | ${\mathsf{R}}_{3}$ | × | ${\mathsf{R}}_{3}$ |

${\mathcal{B}}_{1}\cap {\mathcal{C}}_{3}$ | × | ${\mathsf{R}}_{1}$ | × | ${\mathsf{R}}_{2}$ | × | ${\mathsf{R}}_{1}$ | √ | [C1] | √ | [C3] |

${\mathcal{B}}_{2}\cap {\mathcal{C}}_{1}$ | × | ${\mathsf{R}}_{4}$ | × | ${\mathsf{R}}_{2}$ | × | ${\mathsf{R}}_{2}$ | × | ${\mathsf{R}}_{4}$ | × | ${\mathsf{R}}_{4}$ |

${\mathcal{B}}_{2}\cap {\mathcal{C}}_{2}$ | × | ${\mathsf{R}}_{7}$ | × | ${\mathsf{R}}_{7}$ | × | ${\mathsf{R}}_{7}$ | √ | [C2] | × | ${\mathsf{R}}_{7}$ |

${\mathcal{B}}_{2}\cap {\mathcal{C}}_{3}$ | × | ${\mathsf{R}}_{5}$ | × | ${\mathsf{R}}_{2}$ | × | ${\mathsf{R}}_{2}$ | × | ${\mathsf{R}}_{5}$ | × | ${\mathsf{R}}_{5}$ |

${\mathcal{B}}_{3}\cap {\mathcal{C}}_{1}$ | × | ${\mathsf{R}}_{1}$ | × | ${\mathsf{R}}_{2}$ | × | ${\mathsf{R}}_{2}$ | × | ${c}_{12}^{2}<{c}_{21}^{2}$ | × | ${c}_{12}^{2}<{c}_{21}^{2}$ |

${\mathcal{B}}_{3}\cap {\mathcal{C}}_{2}$ | × | ${\mathsf{R}}_{1}$ | × | ${\mathsf{R}}_{6}$ | × | ${\mathsf{R}}_{1}$ | × | ${\mathsf{R}}_{6}$ | × | ${\mathsf{R}}_{6}$ |

${\mathcal{B}}_{3}\cap {\mathcal{C}}_{3}$ | × | ${\mathsf{R}}_{1}$ | × | ${\mathsf{R}}_{2}$ | × | ${\mathsf{R}}_{1}$ | √ | [C4] | √ | [C3] |

**Power Allocation in**${\mathbf{OA}}_{\mathbf{1}}$

**:**We associate the five compatible subsets in Table A1 (denoted by √) to the four sets ${\mathcal{S}}_{(.)}$ in Table A1 according to the resulting power allocations: channel parameters and P are in (a) ${\mathcal{S}}_{C}$ if $\mathbf{\rho}\in {\mathcal{B}}_{1}\cap {\mathcal{C}}_{3}$, $\mathbf{\lambda}\in {\mathcal{D}}_{4}$; (b) ${\mathcal{S}}_{D}$ if $\mathbf{\rho}\in {\mathcal{B}}_{2}\cap {\mathcal{C}}_{2}$, $\mathbf{\lambda}\in {\mathcal{D}}_{4}$; (c) ${\mathcal{S}}_{\mathsf{sat}}$ if $\mathbf{\rho}\in {\mathcal{B}}_{3}\cap {\mathcal{C}}_{3}$ or ${\mathcal{B}}_{1}\cap {\mathcal{C}}_{3}$, $\mathbf{\lambda}\in {\mathcal{D}}_{5}$; and (d) ${\mathcal{S}}_{CD}$ if $\mathbf{\rho}\in {\mathcal{B}}_{3}\cap {\mathcal{C}}_{3}$, $\mathbf{\lambda}\in {\mathcal{D}}_{4}$. Next, we characterize the conditions, [Cm]$,m=1,\cdots ,4$, in terms of the channel parameters, which give the conditions of sets ${\mathcal{S}}_{(.)}$.

- ${\mathcal{S}}_{C}$: Since $\mathbf{\rho}\in {\mathcal{B}}_{1}\cap {\mathcal{C}}_{3}$, they satisfy ${\rho}_{1}>0,{\rho}_{2}=0,{\rho}_{3}=0,{\rho}_{4}=0$, which implies ${p}_{1}=0,$${q}_{1}=P/\overline{\beta}$ from (A12), (A20) and (A21), and ${p}_{2}>0,{q}_{2}>0$ from (A22) and (A23). In addition, $\mathbf{\lambda}\in {\mathcal{D}}_{4}$ implies ${\Sigma}_{1}={\Sigma}_{2}$ from (A17) and (A18), which, from the expressions of ${\Sigma}_{1}$ and ${\Sigma}_{2}$ in (24) and (25) gives ${c}_{21}^{2}{q}_{2}={c}_{12}^{2}P/\overline{\beta}$. Thus, we have ${q}_{2}=P{c}_{12}^{2}/({c}_{21}^{2}\overline{\beta})$, and therefore, ${p}_{2}=P(1-{c}_{12}^{2}/{c}_{21}^{2})/\beta $ from (A13). Note that $P>0$ and ${c}_{12}^{2}<{c}_{21}^{2}$ are sufficient for $({q}_{1},{p}_{2},{q}_{2})\succ \mathbf{0}$. Moreover, ${\lambda}_{3}=0$ implies ${\Sigma}_{2}<\Sigma $, resulting in $P<\overline{\beta}(\gamma -1)/{c}_{12}^{2}$, i.e., $P<{P}_{4}^{*}$ where $\gamma $, defined in (38), is $\gamma >1$ due to assumption [A1]. Next, from ${\rho}_{3}=0,{\rho}_{4}=0$ and (A10) and (A11), we have ${\lambda}_{1}=(1+P{c}_{12}^{2}/\overline{\beta})/(P({c}_{21}^{2}-{c}_{12}^{2})/\beta +{c}_{12}^{2}/{d}_{2}^{2})$. In addition, from ${\rho}_{2}=0$ and (A8), the condition for ${\rho}_{1}>0$ is ${\lambda}_{2}>{d}_{1}^{2}(1+P{c}_{12}^{2}/\overline{\beta})/{c}_{12}^{2}$. Since ${\lambda}_{3}=0,$ from (A7) we have ${\lambda}_{2}+{\lambda}_{1}=1$, which subsequently gives ${d}_{1}^{2}(1+P{c}_{12}^{2}/\overline{\beta})/{c}_{12}^{2}+(1+P{c}_{12}^{2}/\overline{\beta})/(P({c}_{21}^{2}-{c}_{12}^{2})/\beta +{c}_{12}^{2}/{d}_{2}^{2})<1$, i.e., ${g}_{2}(P)<1$ as in Table 1. Thus, the condition of ${\mathcal{S}}_{C}$ is$$\begin{array}{c}\hfill \left[\mathrm{C}1\right]\text{:}\phantom{\rule{1.em}{0ex}}P<{P}_{4}^{*},\phantom{\rule{1.em}{0ex}}{g}_{2}(P)<1.\end{array}$$
- ${\mathcal{S}}_{D}$: Since $\mathbf{\rho}\in {\mathcal{B}}_{2}\cap {\mathcal{C}}_{2}$, they satisfy ${\rho}_{1}=0,{\rho}_{2}>0,{\rho}_{3}=0,{\rho}_{4}>0$, and imply ${p}_{1}=P/\beta ,$${q}_{1}=0,{p}_{2}=P/\beta ,{q}_{2}=0$, following (A20)–(A23) and (A12) and (A13). In addition, $\mathbf{\lambda}\in {\mathcal{D}}_{4}$ implies ${\Sigma}_{1}={\Sigma}_{2}<\Sigma $ from (A17)–(A19), for which assumption [A1] is sufficient. Next, using ${\rho}_{1}={\rho}_{3}=0$ and (A8)–(A11) the sufficient conditions for ${\rho}_{2}>0$ and ${\rho}_{4}>0$ are found to be ${\lambda}_{2}<{d}_{1}^{2}/({c}_{12}^{2}(1+{d}_{1}^{2}P/\beta ))$ and ${\lambda}_{1}<{d}_{2}^{2}/({c}_{21}^{2}(1+{d}_{2}^{2}P/\beta ))$ respectively. Since ${\lambda}_{3}=0$, and thus ${\lambda}_{1}+{\lambda}_{2}=1$ from (A7), the bounds on ${\lambda}_{1}$ and ${\lambda}_{2}$ are combined, which gives ${d}_{1}^{2}/({c}_{12}^{2}(1+{d}_{1}^{2}P/\beta ))+{d}_{2}^{2}/({c}_{21}^{2}(1+{d}_{2}^{2}P/\beta ))>1$, i.e., ${g}_{1}(P)>1$ as in Table 1. Finally, $P>0$ is sufficient for $({p}_{1},{p}_{2})\succ \mathbf{0}$. Thus, the condition of ${\mathcal{S}}_{D}$ is$$\begin{array}{c}\hfill \left[\mathrm{C}2\right]\text{:}\phantom{\rule{1.em}{0ex}}{g}_{1}(P)>1.\end{array}$$
- ${\mathcal{S}}_{\mathsf{sat}}$: Since $\mathbf{\rho}\in {\mathcal{B}}_{3}\cap {\mathcal{C}}_{3}$, they satisfy ${\rho}_{1}={\rho}_{2}={\rho}_{3}={\rho}_{4}=0$, and imply $({p}_{1},{q}_{1},{p}_{2},{q}_{2})\succ \mathbf{0}$, following (A20)–(A23). In addition, $\mathbf{\lambda}\in {\mathcal{D}}_{5}$ implies ${\Sigma}_{1}=\Sigma $ and ${\Sigma}_{2}=\Sigma $ from (A17)–(A19), which gives ${q}_{1}=\frac{\gamma -1}{{c}_{12}^{2}}$ and ${q}_{2}=\frac{\gamma -1}{{c}_{21}^{2}},$ respectively, where $\gamma >1$. Thus, we have ${p}_{1}=\frac{P}{\beta}-\frac{\overline{\beta}(\gamma -1)}{\beta {c}_{12}^{2}}$ and ${p}_{2}=\frac{P}{\beta}-\frac{\overline{\beta}(\gamma -1)}{\beta {c}_{21}^{2}}$ from (A12)–(A13). Note that the condition for ${p}_{1}>0$ is $P>\overline{\beta}(\gamma -1)/{c}_{12}^{2}$, i.e., $P>{P}_{4}^{*}$, and it is also sufficient for ${p}_{2}>0$ due to ${c}_{21}^{2}>{c}_{12}^{2}$.Now, from $\mathbf{\rho}=\mathbf{0}$ and (A8)–(A11), and using the expressions of ${p}_{l},{q}_{l},l=1,2,$ above we find ${\lambda}_{1}=\gamma \beta /(P{c}_{21}^{2}+\beta {c}_{21}^{2}/{d}_{2}^{2}-\overline{\beta}(\gamma -1))$, and ${\lambda}_{2}=\gamma \beta /(P{c}_{12}^{2}+\beta {c}_{12}^{2}/{d}_{1}^{2}-\overline{\beta}(\gamma -1))$. We also note that $P>{P}_{4}^{*}$ is sufficient for ${\lambda}_{l}>0,l=1,2$. Next, to ensure ${\lambda}_{3}>0$, ${\lambda}_{1}$ and ${\lambda}_{2}$ must satisfy ${\lambda}_{1}+{\lambda}_{2}<1$, which gives the condition ${g}_{3}(P)<1$ as in ${\mathcal{S}}_{\mathsf{sat}}$ in Table A1. In addition, note that $\gamma >1$ is sufficient for $({q}_{1},{q}_{2})\succ \mathbf{0}$.Finally, the case with $\mathbf{\rho}\in {\mathcal{B}}_{1}\cap {\mathcal{C}}_{3}$ only differ from that with ${\mathcal{B}}_{3}\cap {\mathcal{C}}_{3}$ in that now ${\rho}_{1}>0$, and thus ${p}_{1}=0$. We note that $P={P}_{4}^{*}$ is sufficient for ${p}_{1}=0$. We also note that the other condition, which follows from the conditions on ${\lambda}_{1}$ and ${\lambda}_{2}$ as in ${\mathcal{B}}_{3}\cap {\mathcal{C}}_{3}$, is expressed by evaluating ${g}_{3}(P)<1$ above at $P={P}_{4}^{*}$. Therefore, the conditions of the two cases are combined as$$\begin{array}{c}\hfill \left[\mathrm{C}3\right]\text{:}\phantom{\rule{1.em}{0ex}}P\ge {P}_{4}^{*},\phantom{\rule{1.em}{0ex}}{g}_{3}(P)<1.\end{array}$$
- ${\mathcal{S}}_{CD}$: Since $\mathbf{\rho}\in {\mathcal{B}}_{3}\cap {\mathcal{C}}_{3}$, they satisfy $\mathbf{\rho}=\mathbf{0}$, and imply $({p}_{1},{q}_{1},{p}_{2},{q}_{2})\succ \mathbf{0}$, following (A20)–(A23. In addition, $\mathbf{\lambda}\in {\mathcal{D}}_{4}$ imply ${\Sigma}_{1}={\Sigma}_{2}$. Next, from $\mathbf{\rho}=\mathbf{0}$, (A8)–(A11), (A12) and (A13), we find $\frac{\alpha \kappa}{2{\mu}_{1}}(\beta +\overline{\beta}{\lambda}_{2})=P+\frac{\overline{\beta}}{{c}_{12}^{2}}+\frac{\beta}{{d}_{1}^{2}}$, and $\frac{\alpha \kappa}{2{\mu}_{2}}(\beta +\overline{\beta}{\lambda}_{1})=P+\frac{\overline{\beta}}{{c}_{21}^{2}}+\frac{\beta}{{d}_{2}^{2}}$. Also, since ${\Sigma}_{1}={\Sigma}_{2}$, we have from (24) and (25), ${\lambda}_{2}{\mu}_{2}{c}_{12}^{2}={\lambda}_{1}{\mu}_{1}{c}_{21}^{2}$. In addition, since ${\lambda}_{3}=0$ we have ${\lambda}_{1}=1-{\lambda}_{2}$. Combining these conditions, we get a quadratic equation of ${\lambda}_{2}$, $\widehat{A}{\lambda}_{2}^{2}-\widehat{B}{\lambda}_{2}+\widehat{C}=0$, where $\widehat{A}:=\overline{\beta}({E}_{1}-{E}_{2}),$$\widehat{B}:={E}_{1}-{E}_{2}+2\beta {E}_{2}$, and $\widehat{C}:=\beta {E}_{2},$ with ${E}_{1}:=P{c}_{12}^{2}+\overline{\beta}+\beta \frac{{c}_{12}^{2}}{{d}_{1}^{2}}$ and ${E}_{2}:=P{c}_{21}^{2}+\overline{\beta}+\beta \frac{{c}_{21}^{2}}{{d}_{2}^{2}}$ as defined in (37). One of its roots, ${\lambda}_{2}^{(1)}:=(\widehat{B}+\sqrt{{\widehat{B}}^{2}-4\widehat{A}\widehat{C}})/2\widehat{A}$, is infeasible as it violates (A7) and the nonnegativity of ${\lambda}_{2}$, respectively, when $\widehat{A}>0$ and $\widehat{A}<0$. Therefore, the valid solution is ${\lambda}_{2}={\lambda}_{2}^{(2)}:=(\widehat{B}-\sqrt{{\widehat{B}}^{2}-4\widehat{A}\widehat{C}})/2\widehat{A}$. Next, from (A9) and substituting ${\lambda}_{2}^{(2)}$ in $\frac{\alpha \kappa}{2{\mu}_{1}}(\beta +\overline{\beta}{\lambda}_{2})=P+\frac{\overline{\beta}}{{c}_{12}^{2}}+\frac{\beta}{{d}_{1}^{2}}$, and with some algebraic simplification, we have ${q}_{1}=F(P)/{c}_{12}^{2}$ where $F(P):=\left(\right)open="("\; close=")">\frac{{E}_{1}+{E}_{2}-\sqrt{{({E}_{1}-{E}_{2})}^{2}+4{\beta}^{2}{E}_{1}{E}_{2}}}{2\overline{\beta}(1+\beta )}-1$, and ${E}_{1}$ and ${E}_{2}$ defined in (37). Finally, from the mutual exclusiveness of the sets ${\mathcal{S}}_{(.)}$, the condition of ${\mathcal{S}}_{CD}$ is given by$$\begin{array}{c}\hfill \left[\mathrm{C}4\right]\text{:}\phantom{\rule{4.pt}{0ex}}\mathrm{complement}\phantom{\rule{4.pt}{0ex}}\mathrm{of}\phantom{\rule{4.pt}{0ex}}([\mathrm{C}1]\cup [\mathrm{C}2]\cup [\mathrm{C}3]).\end{array}$$

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**Figure 1.**System model of the Gaussian DCLIC, which consists of an underlying GIC in the microwave band and the set of direct channels and cross channels in the mm-wave band.

**Figure 2.**In (

**a**,

**b**), we plot ${c}_{\mathsf{min}}^{2}$ and ${\alpha}_{1,\mathsf{min}}$, respectively. In (

**c**), the channel gains of a symmetric CLIC is partitioned based on whether its capacity has been characterized in each set.

**Figure 3.**Due to its WF-like property, ${\mathbf{OA}}_{\mathbf{1}}$ follows one of the three sequences depending on ${P}_{k}^{*}$. The saturation levels in the cross channels are due to its max-min property.

**Figure 4.**Optimal power allocation that follows the sequence [S1] ${\mathcal{S}}_{D}\to {\mathcal{S}}_{CD}\to {\mathcal{S}}_{\mathsf{sat}}.$ (

**a**) optimal power allocation $({p}_{1},{q}_{1},{p}_{2},{q}_{2})$; (

**b**) the resulting sum-rate constraints.

**Figure 5.**Optimal power allocation that follows the sequence [S2] ${\mathcal{S}}_{C}\to {\mathcal{S}}_{CD}\to {\mathcal{S}}_{\mathsf{sat}}.$ (

**a**) optimal power allocation $({p}_{1},{q}_{1},{p}_{2},{q}_{2})$; (

**b**) the resulting sum-rate constraints.

**Figure 6.**The set of all ${c}^{2}$ and ${d}^{2}$ is partitioned depending on whether the cross or the direct channels are “stronger”.

Set | Optimal Powers | Condition | |||
---|---|---|---|---|---|

${\mathcal{S}}_{D}$ | ${p}_{1}={\displaystyle \frac{P}{\beta}},$ | ${q}_{1}=0,$ | ${p}_{2}={\displaystyle \frac{P}{\beta}},$ | ${q}_{2}=0,$ | ${g}_{1}(P)>1$ |

${\mathcal{S}}_{C}$ | ${p}_{1}=0,$ | ${q}_{1}={\displaystyle \frac{P}{\overline{\beta}}},$ | ${p}_{2}={\displaystyle \frac{P({c}_{21}^{2}-{c}_{12}^{2})}{\beta {c}_{21}^{2}}},$ | ${q}_{2}={\displaystyle \frac{{c}_{12}^{2}}{{c}_{21}^{2}}}{\displaystyle \frac{P}{\overline{\beta}}},$ | $P<{P}_{4}^{*},\phantom{\rule{1.em}{0ex}}{g}_{2}(P)<1$ |

${\mathcal{S}}_{\mathsf{sat}}$ | ${p}_{1}={\displaystyle \frac{P}{\beta}}-{\displaystyle \frac{\overline{\beta}(\gamma -1)}{\beta {c}_{12}^{2}}},$ | ${q}_{1}={\displaystyle \frac{\gamma -1}{{c}_{12}^{2}}},$ | ${p}_{2}={\displaystyle \frac{P}{\beta}}-{\displaystyle \frac{\overline{\beta}(\gamma -1)}{\beta {c}_{21}^{2}}},$ | ${q}_{2}={\displaystyle \frac{\gamma -1}{{c}_{21}^{2}}},$ | $P\ge {P}_{4}^{*},\phantom{\rule{1.em}{0ex}}{g}_{3}(P)<1$ |

${\mathcal{S}}_{CD}$ | ${p}_{1}={\displaystyle \frac{P-\overline{\beta}{q}_{1}}{\beta}},$ | ${q}_{1}={\displaystyle \frac{F(P)}{{c}_{12}^{2}}},$ | ${p}_{2}={\displaystyle \frac{P-\overline{\beta}{q}_{2}}{\beta}},$ | ${q}_{2}={\displaystyle \frac{{c}_{12}^{2}}{{c}_{21}^{2}}}{q}_{1},$ | complement of other conditions |

Set | Optimal Powers | |||
---|---|---|---|---|

${\mathcal{S}}_{D}$ | ${p}_{1}={\displaystyle \frac{P}{\beta}},$ | ${q}_{1}=0,$ | ${p}_{2}={\displaystyle \frac{P}{\beta}},$ | ${q}_{2}=0,$ |

${\mathcal{S}}_{C}$ | ${p}_{1}=0,$ | ${q}_{1}={\displaystyle \frac{P}{\overline{\beta}}},$ | ${p}_{2}={\displaystyle \frac{P{c}_{21}^{2}+\overline{\beta}-(P{c}_{12}^{2}+\overline{\beta})/\eta}{\beta {c}_{21}^{2}}},$ | ${q}_{2}={\displaystyle \frac{P{c}_{12}^{2}/\overline{\beta}+1-\eta}{\eta {c}_{21}^{2}}},$ |

${\mathcal{S}}_{\mathsf{sat}}$ | ${p}_{1}={\displaystyle \frac{P}{\beta}}-{\displaystyle \frac{\overline{\beta}(\gamma -1)}{\beta {c}_{12}^{2}}},$ | ${q}_{1}={\displaystyle \frac{{\gamma}_{1}-1}{{c}_{12}^{2}}},$ | ${p}_{2}={\displaystyle \frac{P}{\beta}}-{\displaystyle \frac{\overline{\beta}({\gamma}_{2}-1)}{\beta {c}_{21}^{2}}},$ | ${q}_{2}={\displaystyle \frac{{\gamma}_{2}-1}{{c}_{21}^{2}}},$ |

${\mathcal{S}}_{CD}$ | ${p}_{1}={\displaystyle \frac{P-\overline{\beta}{q}_{1}}{\beta}},$ | ${q}_{1}={\displaystyle \frac{\widehat{F}(P)}{{c}_{12}^{2}}},$ | ${p}_{2}={\displaystyle \frac{P-\overline{\beta}{q}_{2}}{\beta}},$ | ${q}_{2}={\displaystyle \frac{1+{c}_{12}^{2}{q}_{1}}{\eta {c}_{21}^{2}}}-{\displaystyle \frac{1}{{c}_{21}^{2}}},$ |

Set | Optimal Powers | Condition | |||
---|---|---|---|---|---|

${\widehat{\mathcal{S}}}_{1}$ | ${p}_{1}=0,$ | ${q}_{1}={\displaystyle \frac{P}{\overline{\beta}}},$ | ${p}_{2}={\displaystyle \frac{P}{\beta}},$ | ${q}_{2}=0,$ | $P\le {P}_{5}^{*},\phantom{\rule{1.em}{0ex}}P<{P}_{6}^{*}$ |

${\widehat{\mathcal{S}}}_{2}$ | ${p}_{1}=P+{\displaystyle \frac{\overline{\beta}}{{c}_{12}^{2}}}-{\displaystyle \frac{\overline{\beta}}{{d}_{1}^{2}}},$ | ${q}_{1}=P+{\displaystyle \frac{\beta}{{d}_{1}^{2}}}-{\displaystyle \frac{\beta}{{c}_{12}^{2}}},$ | ${p}_{2}={\displaystyle \frac{P}{\beta}},$ | ${q}_{2}=0,$ | $P>{P}_{6}^{*},\phantom{\rule{1.em}{0ex}}P>{P}_{7}^{*},\phantom{\rule{1.em}{0ex}}P<{P}_{8}^{*}$ |

${\widehat{\mathcal{S}}}_{3}$ | ${p}_{1}={\displaystyle \frac{P}{\beta}}-{\displaystyle \frac{\overline{\beta}(\eta -1)}{\beta {c}_{12}^{2}}},$ | ${q}_{1}={\displaystyle \frac{\eta -1}{{c}_{12}^{2}}},$ | ${p}_{2}={\displaystyle \frac{P}{\beta}},$ | ${q}_{2}=0,$ | $P>{P}_{5}^{*},\phantom{\rule{1.em}{0ex}}P\ge {P}_{8}^{*},\phantom{\rule{1.em}{0ex}}{f}_{3}(P)>1$ |

Set | Optimal Powers | Condition | |
---|---|---|---|

${\overline{\mathcal{S}}}_{D}$ | $p={\displaystyle \frac{P}{\beta}},$ | $q=0,$ | $P<{\overline{P}}_{1}^{*}$ |

${\overline{\mathcal{S}}}_{C}$ | $p=0,$ | $q={\displaystyle \frac{P}{\overline{\beta}}},$ | $P<{\overline{P}}_{2}^{*},\phantom{\rule{1.em}{0ex}}P<{\overline{P}}_{4}^{*}$ |

${\overline{\mathcal{S}}}_{CD}$ | $p={\displaystyle \frac{2}{1+\beta}}\left(\right)open="("\; close=")">P+{\displaystyle \frac{\overline{\beta}}{{c}^{2}}}-{\displaystyle \frac{\overline{\beta}}{2{d}^{2}}}$ | $q={\displaystyle \frac{1}{1+\beta}}\left(\right)open="("\; close=")">P+{\displaystyle \frac{\beta}{{d}^{2}}}-{\displaystyle \frac{2\beta}{{c}^{2}}}$ | $P\ge {\overline{P}}_{1}^{*},\phantom{\rule{1.em}{0ex}}P\ge {\overline{P}}_{2}^{*},\phantom{\rule{1.em}{0ex}}P\le {\overline{P}}_{3}^{*}$ |

${\overline{\mathcal{S}}}_{\mathsf{sat}}$ | $p={\displaystyle \frac{P}{\beta}}-{\displaystyle \frac{\overline{\beta}(\overline{\gamma}-1)}{\beta {c}^{2}}},$ | $q={\displaystyle \frac{\overline{\gamma}-1}{{c}^{2}}}$, | $P\ge {\overline{P}}_{4}^{*},\phantom{\rule{1.em}{0ex}}P>{\overline{P}}_{3}^{*}$ |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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Majhi, S.; Mitran, P.
On the Capacity and the Optimal Sum-Rate of a Class of Dual-Band Interference Channels. *Entropy* **2017**, *19*, 495.
https://doi.org/10.3390/e19090495

**AMA Style**

Majhi S, Mitran P.
On the Capacity and the Optimal Sum-Rate of a Class of Dual-Band Interference Channels. *Entropy*. 2017; 19(9):495.
https://doi.org/10.3390/e19090495

**Chicago/Turabian Style**

Majhi, Subhajit, and Patrick Mitran.
2017. "On the Capacity and the Optimal Sum-Rate of a Class of Dual-Band Interference Channels" *Entropy* 19, no. 9: 495.
https://doi.org/10.3390/e19090495