Robust Interference Management via Linear Precoding and Linear/Non-Linear Equalization

Abstract

This work studies the robust design of linear precoding and linear/ non-linear equalization for multi-cell MIMO systems in the presence of imperfect channel state information (CSI). A worst-case design approach is adopted whereby the CSI error is assumed to lie within spherical sets of known radius. First, the optimal robust design of linear precoders is tackled for a MIMO interference broadcast channel (MIMO-IBC) with general unicast/multicast messages in each cell and operating over multiple time-frequency resources. This problem is formulated as the maximization of the worst-case sum-rate assuming optimal detection at the mobile stations (MSs). Then, symbol-by-symbol non-linear equalization at the MSs is considered. In this case, the problem of jointly optimizing linear precoding and decision-feedback (DF) equalization is investigated for a MIMO interference channel (MIMO-IC) with the goal of minimizing the worst-case sum-mean squared error (MSE). Both problems are addressed by proposing iterative algorithms with descent properties. The algorithms are also shown to be implementable in a distributed fashion on processors that have only local and partial CSI by means of the Alternating Direction Method of Multipliers (ADMM). From numerical results, it is shown that the proposed robust solutions significantly improve over conventional non-robust schemes in terms of sum-rate or symbol error rate. Moreover, it is seen that the proposed joint design of linear precoding and DF equalization outperforms existing separate solutions.

Appendices

Appendix A: Useful Lemmas

In this section, we review some useful lemmas that are used in the derivations presented in the text.

Lemma 3 ((Fenchel Conjugate Function [26]))

Consider a matrix $\mathbf {E}\in \mathbb {C}^{d\times d}$ with E≻0. Then, we have the equality

$$ \log\det\left( \mathbf{E}^{-1}\right)=\max\limits_{\mathbf{S}\succeq\mathbf{0}}\left\{ -\text{tr}\left( \mathbf{S}\mathbf{E}\right)+\log\det\left( \mathbf{S}\right)+d\right\} . $$

(35)

Lemma 4 (Sign Definiteness Lemma [9])

Let A, $\{\mathbf {P}_{i},\mathbf {Q}_{i}\}_{i=1}^{N}$ be given matrices with appropriate sizes and A=A ^‡ . Then, the condition

$$ \mathbf{A}\succeq\sum\limits_{i=1}^{N}\left( \mathbf{P}_{i}^{\dagger}\mathbf{X}_{i}\mathbf{Q}_{i}+\mathbf{Q}_{i}^{\dagger}\mathbf{X}_{i}^{\dagger}\mathbf{P}_{i}\right) $$

(36)

holds for all X _i satisfying ||X _i ||≤ε _i , i∈{1,…,N} if there exist real nonnegative numbers μ ₁ ,…,μ _N ≥0 that satisfy the condition

$$ \left[\begin{array}{cccc} \mathbf{A}-{\sum}_{i=1}^{N}\mu_{i}\mathbf{Q}_{i}^{\dagger}\mathbf{Q}_{i} & -\varepsilon_{1}\mathbf{P}_{1}^{\dagger} & {\cdots} & -\varepsilon_{N}\mathbf{P}_{N}^{\dagger}\\ -\varepsilon_{1}\mathbf{P}_{1} & \mu_{1}\mathbf{I} & {\cdots} & \mathbf{0}\\ {\vdots} & {\vdots} & {\ddots} & \vdots\\ -\varepsilon_{N}\mathbf{P}_{N} & \mathbf{0} & {\cdots} & \mu_{N}\mathbf{I} \end{array}\right]\succeq\mathbf{0}. $$

(37)

The converse is also true when N=1 [ 10, Sec. IV][31, Sec. 2.6.3].

Appendix B: Proof of Lemma 1

In this appendix, we show that the optimal solution of the problem (16a) is lower-bounded by that of the problem (17a) in Lemma 1. We first write an epigraph form of the problem (16a16) as

$$\begin{array}{@{}rcl@{}} \!\!&&\underset{\mathbf{V},R,\mathbf{U},\mathbf{S}\succeq\mathbf{0},\gamma}{\text{maximize}} \sum\limits_{i\in\mathcal{B}}\sum\limits_{m\in\mathcal{M}_{i}}R_{i,m} \end{array} $$

(38a)

$$\begin{array}{@{}rcl@{}} &&\quad\mathrm{s.t.}~ R_{i,m}\!\!\leq\!\!-\gamma_{i,m,k}\,+\,\log\det\left( \mathbf{S}_{i,m,k}\right)\,+\,d_{i,m},\text{ for all } i\!\in\!\mathcal{B},\, m\!\in\!\mathcal{M}_{i},\, k\!\in\!\mathcal{D}_{i,m},\qquad \end{array} $$

(38b)

$$\begin{array}{@{}rcl@{}} &&\qquad \gamma_{i,m,k}\!\geq\!\text{tr}\left( \!\mathbf{S}_{i,m,k}\mathbf{E}_{i,m,k}(\mathbf{V},\mathbf{U},\hat{\mathbf{H}}\,+\,\boldsymbol{\Delta})\!\right), \end{array} $$

(38c)

$$\begin{array}{@{}rcl@{}} &&\qquad\text{for all }\{\boldsymbol{\Delta}_{i,k,j}(l)\!\in\!\mathcal{U}_{i,k,j}(l)\}_{l\in\mathcal{L},j\in\mathcal{B}},\, i\!\in\!\mathcal{B},\, m\!\in\!\mathcal{M}_{i},\, k\!\in\!\mathcal{D}_{i,m}, \\ &&\qquad \sum\limits_{m\in\mathcal{M}_{i}}\text{tr}\left( \!\mathbf{V}_{i,m}\mathbf{V}_{i,m}^{\dagger}\!\right)\!\leq\! LP_{i},\text{ for all } i\in\mathcal{B}, \end{array} $$

(38d)

where we have defined the variables $\gamma \triangleq \{\gamma _{i,m,k}\}_{i\in \mathcal {B},m\in \mathcal {M}_{i},k\in \mathcal {D}_{i,m}}$ and $\tilde {\mathbf {S}}\triangleq \{\tilde {\mathbf {S}}_{i,m,k}\}_{i\in \mathcal {B},m\in \mathcal {M}_{i},k\in \mathcal {D}_{i,m}}$ with $\tilde {\mathbf {S}}_{i,m,k}\triangleq \mathbf {S}_{i,m,k}^{1/2}$.

From the MSE expression in Eq. 13, we can see that the constraints (38c) are equivalent to the conditions

$$\begin{array}{@{}rcl@{}} &&\gamma_{i,m,k}\!\!\geq\!\! \sum\limits_{j\in\mathcal{B}}\tau_{i,m,k,j}\,+\,\left\Vert \mathbf{\Sigma}_{i,k}^{1/2}\mathbf{U}_{i,m,k}\tilde{\mathbf{S}}_{i,m,k}\right\Vert_{F}^{2},\text{ for all } i\!\!\in\!\!\mathcal{B},~ m\!\in\!\mathcal{M}_{i},\, k\!\in\!\mathcal{D}_{i,m},\qquad \end{array} $$

(39)

$$\begin{array}{@{}rcl@{}} \!\!\!\!\!\!&&\text{and }\tau_{i,m,k,j}\!\geq\!\!\!\!\!\!\!\sum\limits_{q\in\mathcal{M}_{i,m,k,j}}\left\Vert \tilde{\mathbf{S}}_{i,m,k}^{\dagger}\left( \mathbf{U}_{i,m,k}^{\dagger}(\hat{\mathbf{H}}_{i,k,j}\,+\,\boldsymbol{\Delta}_{i,k,j})\mathbf{V}_{j,q}\,-\,\delta_{(i,m),(j,q)}\mathbf{I}\right)\right\Vert_{F}^{2},\, \end{array} $$

(40)

$$\begin{array}{@{}rcl@{}} &&\qquad \text{ for all }\{\boldsymbol{\Delta}_{i,k,j}(l)\!\in\!\mathcal{U}_{i,k,j}(l)\}_{l\in\mathcal{L}},~i,j\!\in\!\mathcal{B},\, m\!\in\!\mathcal{M}_{i},\, k\!\in\!\mathcal{D}_{i,m}, \end{array} $$

where we have introduced auxiliary variables τ _{i m,,k,j} to simplify the expression and the sets $\mathcal {M}_{i,m,k,j}$ are defined in Eq. 13. After some manipulations, the constraint (40) can be rewritten as

$$\begin{array}{@{}rcl@{}} \tau_{i,m,k,j}&\!\!\geq\!\! & \left\Vert \mathbf{c}_{i,m,k,j}\,+\,\sum\limits_{l\in\mathcal{L}}\mathbf{C}_{i,m,k,j}(l)\mathbf{d}_{i,k,j}(l)\right\Vert^{2},\\ && \text{for all }\left\{\left\Vert \mathbf{d}_{i,,k,j}(l)\right\Vert \!\!\leq\!\!\varepsilon_{i,k,j}(l)\right\}_{l\in\mathcal{L}},\, i,j\!\in\!\mathcal{B},\, m\!\in\!\mathcal{M}_{i},~ k\!\in\!\mathcal{D}_{i,m},\\ \end{array} $$

(41)

where we have defined the vector $\mathbf {d}_{i,k,j}(l)\triangleq \text {vec}(\boldsymbol {\Delta }_{i,k,j}(l))$ and the notations c _i,m,k,j and C _i,m,k,j (l) are defined in Eqs. 19–22.

Applying the Schur complement Lemma [29, Appendix C] to the constraint (41), we obtain the following equivalent linear matrix inequality.

$$\begin{array}{@{}rcl@{}} && \left[\begin{array}{cc} \tau_{i,m,k,j} & \mathbf{c}_{i,m,k,j}^{\dagger}\\ \mathbf{c}_{i,m,k,j} & \mathbf{I} \end{array}\right]+\sum\limits_{l\in\mathcal{L}}\left[\begin{array}{cc} 0 & \mathbf{d}_{i,k,j}^{\dagger}(l)\mathbf{C}_{i,m,k,j}^{\dagger}(l)\\ \mathbf{C}_{i,m,k,j}(l)\mathbf{d}_{i,k,j}(l) & \mathbf{0} \end{array}\right]\succeq\mathbf{0},\\ &&\quad \text{ for all }\left\{\left\Vert \mathbf{d}_{i,k,j}(l)\right\Vert \leq\varepsilon_{i,k,j}(l)\right\}_{l\in\mathcal{L}},\, i,j\in\mathcal{B},\, m\in\mathcal{M}_{i},\, k\in\mathcal{D}_{i,m}. \end{array} $$

(42)

From Lemma 4, we can see that the constraint (42) holds if the condition

$$\begin{array}{@{}rcl@{}} \left[\begin{array}{ccc} \tau_{i,m,k,j}\,-\,{\sum}_{l\in\mathcal{L}}\mu_{i,m,k,j}(l) & \mathbf{c}_{i,m,k,j}^{\dagger} & \mathbf{0}\\ \mathbf{c}_{i,m,k,j} & \mathbf{I} & \!-\mathbf{C}_{i,m,k,j}\\ \mathbf{0} & -\mathbf{C}_{i,m,k,j}^{\dagger} & \text{diag}\left( \{\mu_{i,m,k,j}(l)\}_{l\in\mathcal{L}}\right)\otimes\mathbf{I} \end{array}\right]\!\succeq\!\mathbf{0},\\ \end{array} $$

(43)

is satisfied for some $\{\mu _{i,m,k,j}(l)\geq 0\}_{l\in \mathcal {L}}$ and for all $i,j\in \mathcal {B}$, $m\in \mathcal {M}_{i}$ and $k\in \mathcal {D}_{i,m}$ with the notation $\mathbf {C}_{i,m,k,j}\triangleq -[\varepsilon _{i,k,j}(1)\mathbf {C}_{i,m,k,j}(1),\,\ldots ,\,\varepsilon _{i,k,j}(L)\mathbf {C}_{i,m,k,j}(L)]$. Also, the converse is true when L=1 (see, e.g., [10, Sec. IV][31, Sec. 2.6.3]). Note that the condition (43) implies (42) but is not equivalent unless L=1. Therefore, replacing the condition (40) with Eq. 43 leads to a problem whose solution lower-bounds that of the problem (38a). Substituting the conditions (39), (40) and (43) into the problem (38a) results in the problem (17a) in Lemma 1, which concludes the proof.