Optimality Properties, Closed-Form Parameterizations and Distributed Strategy of the Two-User MISO Interference Channel

Abstract

In this paper, the closed-form parameterizations to the Pareto boundary for the two-user multiple-input single-output interference channel are studied. Firstly, for the equivalent channel model with each transmitter having only two antennas, the weighted sum-rate maximization (WSRMax) problem is reformulated with newly defined angle variables. Then, a centralized weighted leakage-plus-noise-to-signal ratio minimization (WLNSRMin) algorithm is proposed to find a locally optimal weighted sum-rate point. Each step of the algorithm is solved by evaluating closed-form expressions. A distributed algorithm is also given to avoid the exchange of the channel state information (CSI) between transmitters. Numerical results show that the centralized WLNSRMin algorithm converges to a local optimum of the WSRMax problem after a few iterations and the distributed algorithm achieves a performance very close to that of the centralized algorithm with only local CSI.

Appendices

Appendix 1: Proof of Proposition 2

Considering the second entry of $\mathbf{h }_{kk}$, one has

$$\begin{aligned}\frac{\tilde{\mathbf{h }}_{kk}^{H}\tilde{\mathbf{h }}_{kj}^{\perp }}{\Vert \tilde{\mathbf{h }}_{kj}^{\perp }\Vert \Vert \tilde{\mathbf{h }}_{kk}\Vert }& = \frac{\tilde{\mathbf{h }}_{kk}^{H}\left( \tilde{\mathbf{h }}_{kk}-\frac{\tilde{\mathbf{h }}_{kj}^H\tilde{\mathbf{h }}_{kk}}{\Vert \tilde{\mathbf{h }}_{kj}\Vert ^2}\tilde{\mathbf{h }}_{kj}\right) }{\Vert \tilde{\mathbf{h }}_{kj}^{\perp }\Vert \Vert \tilde{\mathbf{h }}_{kk}\Vert }\nonumber \\&\overset{\left( a\right) }{=}\frac{\Vert \tilde{\mathbf{h }}_{kk}\Vert ^2-\frac{\tilde{\mathbf{h }}_{kj}^H\tilde{\mathbf{h }}_{kk}\tilde{\mathbf{h }}_{kk}^H\tilde{\mathbf{h }}_{kj}}{\Vert \tilde{\mathbf{h }}_{kj}\Vert ^2}}{\Vert \tilde{\mathbf{h}}_{kj}^{\perp }\Vert \Vert \tilde{\mathbf{h }}_{kk}\Vert }\nonumber \\ & = \frac{\Vert \tilde{\mathbf{h }}_{kk}\Vert ^2-\frac{|\langle \tilde{\mathbf{h }}_{kk},\tilde{\mathbf{h }}_{kj}\rangle |^2}{\Vert \tilde{\mathbf{h }}_{kj}\Vert ^2}}{\Vert \tilde{\mathbf{h }}_{kj}^{\perp }\Vert \Vert \tilde{\mathbf{h }}_{kk}\Vert }\nonumber \\&\overset{\left( b\right) }{\ge }\frac{\Vert \tilde{\mathbf{h }}_{kk}\Vert ^2-\frac{\Vert \tilde{\mathbf{h }}_{kk}\Vert ^2\Vert \tilde{\mathbf{h }}_{kj}\Vert ^2}{\Vert \tilde{\mathbf{h }}_{kj}\Vert ^2}}{\Vert \tilde{\mathbf{h }}_{kj}^{\perp }\Vert \Vert \tilde{\mathbf{h }}_{kk}\Vert }\nonumber \\& = 0\nonumber \end{aligned}$$

(44)

where $\left( a\right)$ is because $\tilde{\mathbf{h }}_{kj}^H\tilde{\mathbf{h }}_{kk}$ is a scalar, $\left( b\right)$ is by the Cauchy–Schwartz inequality $|\langle \tilde{\mathbf{h }}_{kk},\tilde{\mathbf{h }}_{kj}\rangle |^2\le \Vert \tilde{\mathbf{h }}_{kk}\Vert ^2\Vert \tilde{\mathbf{h }}_{kj}\Vert ^2$, equality holds if and only if $\tilde{\mathbf{h }}_{kk}$ and $\tilde{\mathbf{h }}_{kj}$ are linearly dependent. Since the linearly independent channels are considered in this paper, the second entry of $\mathbf{h }_{kk}$ is a positive real number. Combining this property and Proposition 1, we can rewrite the equivalent channel vectors $\mathbf{h }_{kk}$ and $\mathbf{h }_{kj}$ as in (18) and (19) respectively. This completes the proof.

Appendix 2: Proof of Theorem 1

Firstly, we reformulate the optimization problem (17) with the newly defined angle variable $\varphi _k$. For linearly independent channels, the original non-convex optimization problem (17) is equivalent to the following optimization problem for the same rate weight $\mu _k$ [5]:

$$\begin{array}{ll} {\mathop {{\mathrm {max}}}\limits _{{\mathbf{S }}_k,\;{\mathbf{S }}_j}}&{}\quad \sum\limits_{k=1}^{2}\mu _k \log _2\left( 1+\frac{u_k^2}{\mathbf{h }_{jk}^{H}{{\mathbf{S }}}_{j}\mathbf{h }_{jk}+\sigma ^2}\right) \\ \text {s.t.}&{}\quad \mathbf{h }_{kk}^{H}{{\mathbf{S }}}_k\mathbf{h }_{kk}=u_k^2, \\ &{}\quad {{\text {tr}}}({{\mathbf{S }}}_k)= P_k,\quad j\ne k,\;\; k, j\in \{1,2\}, \end{array}$$

(45)

where ${{\mathbf{S }}}_k=\mathbf{w }_k\mathbf{w }_k^H$, and the power constraint at the ${\text {BS}}_k$ is ${\text {tr}}({{\mathbf{S }}}_k)= P_k$ which results from that full transmit power should be used to obtain the optimal solution to the WSRMax problem (17) when the channels are linearly independent [3, 11].

• (1) Useful signal power From the definition (22), the useful signal power term of (45) can be expressed with $\varphi _k$ as

  $$\begin{aligned} u_k^2=\mathbf{h }_{kk}^H {{\mathbf{S }}}_k \mathbf{h }_{kk}=P_k\sin ^2{\varphi _k}. \end{aligned}$$

  (46)

• (2) Interference power Assume that the singular-value decompositions (SVDs) of $\mathbf{h }_{kk}$

and $\mathbf{h }_{kj}$ are

$$\begin{aligned} \mathbf{h }_{kk}=\mathbf{U }_{kk}\left[ {\begin{array}{l}{\Vert \mathbf{h }_{kk} \Vert }\\ {0}\end{array}}\right] =\mathbf{U }_{kk}\left[ {\begin{array}{l}{1}\\ {0}\end{array}}\right] , \end{aligned}$$

(47)

and

$$\begin{aligned} \mathbf{h }_{kj}=\mathbf{U }_{kj}\left[ {\begin{array}{l}{\Vert \mathbf{h }_{kj} \Vert }\\ {0}\end{array}}\right] =\mathbf{U }_{kj}\left[ {\begin{array}{l}{d}\\ {0}\end{array}}\right] , \end{aligned}$$

(48)

where $\mathbf{U }_{kk}$ and $\mathbf{U }_{kj}$ are $2\times 2$ unitary matrices. Define

$$\begin{aligned}\hat{\mathbf{h }}_{kj}\,\triangleq\,\mathbf{U }_{kk}^H\mathbf{h }_{kj}=\left[ {\begin{array}{l}{\alpha }\\ {\beta }\end{array}}\right] ,\end{aligned}$$

(49)

$$\begin{aligned}&\hat{\mathbf{S }}_k\,\triangleq\,{\mathbf{U}} _{kk}^H{\mathbf{S}} _k{\mathbf{U}} _{kk}=\left[ {\begin{array}{*{20}c}{\hat{S}_{11}}&{}{\hat{S}_{21}^H}\\ {\hat{S}_{21}}&{}{\hat{S}_{22}}\end{array}}\right], \end{aligned}$$

(50)

where all $\alpha ,\, \beta ,\, \hat{S}_{11},\, \hat{S}_{21}$ and $\hat{S}_{22}$ are scalars. Considering the useful signal power term in (46), we have

$$\begin{aligned} {\mathbf{h}} _{kk}^H{\mathbf{S}}_k {\mathbf{h}} _{kk}=\left[ {\begin{array}{*{20}l}{1}&{0}\end{array}}\right] {\mathbf{U}}_{kk}^H{\mathbf{S}} _k{\mathbf{U}} _{kk}\left[ {\begin{array}{*{20}c}{1}\\ {0}\end{array}}\right] =\hat{S}_{11}= u_k^2. \end{aligned}$$

(51)

Substituting (49), (50) and (51) into the interference power term of (45), we have

$$\begin{aligned} {\mathbf{h}} _{kj}^H{\mathbf{S}} _k {\mathbf{h}} _{kj}& = \left[ {\begin{array}{*{20}l}{\alpha ^H}&{\beta ^H}\end{array}}\right] {\mathbf{U}} _{kk}^H{\mathbf{S}} _k{\mathbf{U}} _{kk}\left[ {\begin{array}{*{20}l}{\alpha }\\ {\beta }\end{array}}\right] \nonumber \\& = |\alpha |^2 u_k^2 + \hat{S}_{21}\alpha \beta ^H+\hat{S}_{21}^H\alpha ^H\beta + \hat{S}_{22}|\beta |^2 \nonumber \\&\overset{(a)}{=} |\alpha |^2 u_k^2 + \hat{S}_{21}\alpha \beta ^H+\hat{S}_{21}^H\alpha ^H\beta +|\beta |^2(P_k - u_k^2), \end{aligned}$$

(52)

where (a) results from ${\text {tr}}({{\mathbf{S }}}_k)={\text {tr}}(\hat{{{\mathbf{S }}}}_{k})=\hat{S}_{11}+\hat{S}_{22}=P_k$. With the definition of

$$\begin{aligned} \tilde{S}_{21}\,\triangleq\,\frac{\beta ^H}{|\beta |}\hat{S}_{21}, \end{aligned}$$

(53)

we have

$$\begin{aligned} \mathbf{h }_{kj}^H{{\mathbf{S }}}_k \mathbf{h }_{kj}& = |\alpha |^2 u_k^2 + \hat{S}_{21}\alpha \beta ^H+\hat{S}_{21}^H\alpha ^H\beta +|\beta |^2(P_k - u_k^2) \nonumber \\& = |\alpha |^2u_k^2 + |\beta |\left( \tilde{S}_{21} \alpha + \alpha ^H\tilde{S}_{21}^H\right) + |\beta |^2(P_k - u_k^2) \nonumber \\&\overset{(a)}{\ge }|\alpha |^2u_k^2 -2|\beta ||\tilde{S}_{21}||\alpha | + |\beta |^2(P_k - u_k^2) \nonumber \\&\overset{(b)}{\ge }|\alpha |^2u_k^2 -2|\beta ||\alpha | u_k\sqrt{P_k-u_k^2} + |\beta |^2(P_k - u_k^2)2 \nonumber \\& = \left( |\alpha |u_k - |\beta |\sqrt{P_k- u_k^2} \right) ^2, \end{aligned}$$

(54)

where (a) results from $\tilde{S}_{21}\alpha + \alpha ^H\tilde{S}_{21}^H \ge -2|\tilde{S}_{21}||\alpha |$ since $\left( \tilde{S}_{21}\alpha + \alpha ^H\tilde{S}_{21}^H \right) ^2=\left( 2\text {Re}(\tilde{S}_{21}\alpha ) \right) ^2 \le 4|\tilde{S}_{21}|^2|\alpha |^2$, and (b) is due to $\hat{S}_{11}\hat{S}_{22}= u_k^2(P_k-u_k^2) \ge |\hat{S}_{21}|^2=|\tilde{S}_{21}|^2$ since $\hat{{{\mathbf{S }}}}_k\succeq 0$. The equality of (a) holds if and only if $\tilde{S}_{21}\alpha$ is non-positive real number, i.e.,

$$\begin{aligned} \tilde{S}_{21}\alpha = (\tilde{S}_{21}\alpha )^H = -|\tilde{S}_{21}||\alpha |. \end{aligned}$$

(55)

The equality of (b) holds under the condition

$$\begin{aligned} \tilde{S}_{21} = \frac{\beta ^H \alpha ^H}{|\beta ||\alpha |}u_k\sqrt{P_k - u_k^2}, \end{aligned}$$

(56)

where the choice of $\tilde{S}_{21}$ ensures that (55) is satisfied. Under the conditions of (55) and (56), the minimum of the interference power term in (54) can be given by

$$\begin{aligned} \left( \mathbf{h }_{kj}^H{{\mathbf{S }}}_k \mathbf{h }_{kj} \right) _{\text {min}}&\overset{(a)}{=}\,d_k^2\left( \sqrt{a_k^2+b_k^2} u_k - \sqrt{1-(a_k^2+b_k^2)}\sqrt{P_k - u_k^2} \right) ^2 \nonumber \\&\overset{(b)}{=}\,P_k d_k^2 \left( \cos {\theta _k} \frac{u_k}{\sqrt{P_k}} - \sin {\theta _k} \sqrt{\frac{P_k - u_k^2}{P_k}} \right) ^2 \nonumber \\&\overset{(c)}{=}\,P_k d_k^2 \left( \cos {\theta _k}\sin {\varphi _k} - \sin {\theta _k}\cos {\varphi _k} \right) ^2 \nonumber \\ &= P_k \Vert \mathbf{h }_{kj} \Vert ^2 \sin ^2{\left( \varphi _k - \theta _k\right) } \end{aligned}$$

(57)

where (a) results from $\alpha = d_k(a_k-i b_k)$ and $|\beta | = \sqrt{\Vert \mathbf{h }_{kj} \Vert ^2 - |\alpha |^2}$ which are obtained from (49), (b) and (c) result from (20) and (22), respectively.

• (3) Value range of $\varphi _k$ The value range of $u_k$ is

  $$\begin{aligned} u_k\in \left[ 0,\sqrt{P_k}\Vert \mathbf{h }_{kk}\Vert \right] =\left[ 0,\sqrt{P_k}\right] , \end{aligned}$$

  (58)

which results from $\Vert \mathbf{h }_{kk}\Vert =1$. Combined with the definition (22), it can be seen that the value range of $\varphi _k$ is

$$\begin{aligned} \varphi _k\in \left[ 0,\pi /2\right] . \end{aligned}$$

(59)

Combining (45), (46), (57) and (59), the WSRMax problem (17) can be reformulated with the angle variables $\varphi _k$ as in (23). Next, we prove that the optimal beamforming vector $\mathbf{w }_k^{*}$ has the form as in (24). From (47), we have

$$\begin{aligned} u_{11}=a_k+ib_k,\quad {\text {and}}\quad u_{21}= c_k, \end{aligned}$$

(60)

where $u_{ij}=\left( \mathbf{U }_{kk}\right) _{ij}$. Due to the properties of unitary matrix, the unitary matrix $\mathbf{U }_{kk}$ can be given by

$$\begin{aligned} \mathbf{U }_{kk}=\left[ {\begin{array}{ll}{a_k+ib_k}&{}{-c_k}\\ {c_k}&{}{a_k-ib_k}\end{array}}\right] . \end{aligned}$$

(61)

From (53) and (56), we obtain

$$\begin{aligned} \hat{S}_{21}=\frac{\alpha ^H }{|\alpha |}u_k\sqrt{P_k - u_k^2}. \end{aligned}$$

(62)

Combining (50), (51), (52) and (62), we have

$$\begin{aligned} \hat{{{\mathbf{S }}}}_{k}=\left[ {\begin{array}{ll}{u_k^2}&{}{\frac{\alpha }{|\alpha |}u_k\sqrt{P_k - u_k^2}}\\ {\frac{\alpha ^H }{|\alpha |}u_k\sqrt{P_k - u_k^2}}&{}{P_k-u_k^2}\end{array}}\right] . \end{aligned}$$

(63)

Then, ${{\mathbf{S }}}_k$ can be obtained from (50), (61) and (63) as

$$\begin{aligned} {{\mathbf{S }}}_{k}=\mathbf{U }_{kk}\hat{{{\mathbf{S }}}}_{k}\mathbf{U }_{kk}^H \overset{(a)}{=} P_k \left[ {\begin{array}{ll}{\sin ^2{(\varphi _k-\theta _k)} }&{}{B\left( \cos {\eta _k}+i\sin {\eta _k} \right) }\\ {B\left( \cos {\eta _k}-i\sin {\eta _k} \right) }&{}{\cos ^2{(\varphi _k-\theta _k )}} \end{array}}\right] , \end{aligned}$$

(64)

where $B=\sin {(\varphi _k-\theta _k)}\cos {(\varphi _k-\theta _k)},\, (a)$ results from $\cos {\theta _k}=\sqrt{a_k^2+b_k^2},\, \sin {\theta _k}=c_k,\, \sin {\eta _k}=b_k/\sqrt{a_k^2+b_k^2},\, \cos {\eta _k}=a_k/\sqrt{a_k^2+b_k^2},\, \sin {\varphi _k}=u_k/\sqrt{P_k}$ and $\cos {\varphi _k}=\sqrt{(P_k-u_k^2)/P_k}$, which can be obtained from (20), (21) and (22), respectively. Moreover, since it has been proved in [1] that for the two-user MISO IC any point on the Pareto boundary is achievable with the weighted sum of the MRT and ZF beamforming vectors with some set of real-valued parameters which are in the range of [0, 1], and the MRT and ZF beamforming vectors at ${\text {BS}}_k$ can be given by

$$\begin{aligned} {\mathbf{w }}_k^{\text {MRT}} = \frac{{\mathbf{h} }_{kk}}{\Vert {\mathbf{h }}_{kk}\Vert }\;=\; \left[ a_k+ib_k,\; c_k\right] ^T,\end{aligned}$$

(65)

$${\mathbf{w }}_k^{\text {ZF}} = \frac{\varPi _{\mathbf{h}_{kj}}^{\perp } {\mathbf{h }_{kk}}}{\Vert \varPi _{\mathbf{h }_{kj}}^{\perp } {\mathbf{h }}_{kk}\Vert }\;= \; \left[ 0, \; 1\right]^T,$$

(66)

where $\varPi _{{\mathbf{h }}_{kj}}^{\perp }\,\triangleq\,{\mathbf{I}} -{\mathbf{h}}_{kj}\left( {\mathbf{h }}_{kj}^H {\mathbf{h }}_{kj} \right) ^{-1} {\mathbf{h }}_{kj}^H$ is the orthogonal projection onto the orthogonal complement of $\mathbf{h }_{kj}$, the optimal beamforming vectors can be expressed as the following form without loss of optimality

$$\begin{aligned} \mathbf{w }_k^{*} = \sqrt{P_k}\left[ x_k+iy_k,\; z_k \right] ^T, \end{aligned}$$

(67)

where $x_k,\, y_k$ and $z_k$ are real numbers and $z_k > 0$ which results from $c_k> 0$. Therefore, the signal correlation matrix ${{\mathbf{S }}}_k^{*}$ can be given with $\mathbf{w }_k^{*}$ by

$$\begin{aligned} {{\mathbf{S }}}_k^{*} = \mathbf{w }_k^{*} \left( \mathbf{w }_k^{*}\right) ^H = P_k \left[ {\begin{array}{ll}{x_k^2+y_k^2}&{}{x_k z_k+i y_k z_k}\\ {x_k z_k-i y_k z_k}&{}{z_k^2}\end{array}}\right] . \end{aligned}$$

(68)

Combining (64) and (68), we have

$$\begin{aligned} \left\{ \begin{array}{llll} &{}x_k^2+y_k^2 = \sin ^2{(\varphi _k-\theta _k)}\\ &{}x_k z_k = \sin {(\varphi _k-\theta _k)}\cos {(\varphi _k-\theta _k)}\cos {\eta _k} \\ &{}y_k z_k = \sin {(\varphi _k-\theta _k)}\cos {(\varphi _k-\theta _k)}\sin {\eta _k} \\ &{}z_k^2 = \cos ^2{(\varphi _k - \theta _k)}\\ \end{array} \right. \end{aligned}$$

(69)

The remaining challenge is to solve the Eq. (69). From [14, 19], the range of useful signal power $u_k^2$ obtained by the optimal beamforming vector is given by

$$\begin{aligned} u_k^2 \in \left[ P_k \left| \mathbf{h }_{kk}^H \mathbf{w }_k^{\text {ZF}}\right| ^2,\; P_k \left| \mathbf{h }_{kk}^H \mathbf{w }_k^{\text {MRT}}\right| ^2 \right] =\left[ P_k c_k^2,\; P_k \right] . \end{aligned}$$

(70)

Substituting (70) into (22) yields the range of the optimal angle variable $\varphi _k^{*}$,

$$\begin{aligned} \varphi _k^{*} \in \left[ {\text {atan}}\left( \sqrt{\frac{c_k^2}{1-c_k^2} }\right) , \; \frac{\pi }{2} \right]\,\overset{(a)}{=}\,\left[ {\text {asin}}(c_k), \; \frac{\pi }{2} \right] , \end{aligned}$$

(71)

where (a) results from $\Vert \mathbf{h }_{kk}\Vert ^2 = a_k^2 + b_k^2 + c_k^2 =1$. Since $\theta _k = {\text {asin}}{(c_k)}$ which can be obtained from (20), we have $0\le \varphi _k^{*} - \theta _k \le \frac{\pi }{2} - {\text {asin}}(c_k)$. Hence, $\cos (\varphi _k^{*}-\theta _k)\ge 0$. Recalling that $z_k> 0$, by solving the Eq. (70), we obtain

$$\begin{aligned} \left\{ \begin{array}{lll} &{x_{k}} = \sin {(\varphi _k^{*}-\theta _k)}\cos {\eta _k} \\ &{y_{k}} = \sin {(\varphi _k^{*}-\theta _k)}\sin {\eta _k} \\ &{z_{k}} = \cos {(\varphi _k^{*}-\theta _k)} \end{array} \right. \end{aligned}$$

(72)

Consequently, the beamforming vector $\mathbf{w }_k^{*}$ can be expressed with $\varphi _k^{*}$ as in (24).