On Analytical Achievable Rate for MIMO Linear Interference Alignment with Imperfect CSI

Abstract

This paper studies the impact of channel error on the achievable rate of symmetrical K-user multiple-input multiple-output linear interference alignment (IA) networks. The upper and lower bounds of the achievable sum rate are derived analytically with the assumption of orthonormal transmit precoders and receive filters designed from imperfect channel state information (CSI) over both the uncorrelated and correlated channels. For uncorrelated channels, quite tight lower and upper bounds are obtained. The impact of channel error on the degrees of freedom (DoF) and the DoF persistence conditions are also investigated. Results show that the DoF of IA networks persists only if the channel error decreases in an order higher than the signal-to-noise ratio. For correlated channel, the lower and upper bounds for one realization of IA are derived. The derived upper bound can be used to characterize the achievable rate approximately. Simulation results indicate that the achievable rate of IA network is influenced significantly by CSI uncertainty. The obtained analytical bounds provide an intuitive way to show the impact of channel error on the achievable rate and thus can help practical systems deign.

Appendices

Appendix 1: Properties of Complex Gaussian Matrix Variate

Lemma 1

If $\mathbf {A}$ is a Gaussian matrix variate with the elements of which are i.i.d ZMCSCG variables, i.e., $\mathbf {A} \sim {\mathcal {CN}}_{d,d}(\mathbf {0},{\mathbf {I}}_d \otimes {\mathbf {I}}_d)$ , then the linear transform $\mathbf {B}={\mathbf {U}}^{\text {H}} \mathbf {A} \mathbf {V}$ is still Gaussian matrix variate with ${{\mathbf {H}}} \sim {\mathcal {CN}}_{d,d}(\mathbf {0},\varvec{\Sigma } ^* \otimes \varvec{\Psi } )$ , where $\varvec{\Sigma }={\mathbf {U}}^{\text {H}}{\mathbf {U}}$ and $\varvec{\Psi }={\mathbf {V}}^{\text {H}}{\mathbf {V}}$.

Proof

Using relationships between matrix vectorization and Kronecker product [39], $\mathbf {B}$ can be vectorized as $\text {vec}( \mathbf {B} ^{\text {H}})=({\mathbf {U}}^\text {T}\otimes {\mathbf {V}}^{\text {H}}) \text {vec}( \mathbf {A} ^{\text {H}})$, which is a vector with zero mean and covariance matrix $({\mathbf {U}}^{\text {H}}{\mathbf {U}})^*\otimes ({\mathbf {V}}^{\text {H}}{\mathbf {V}})$. According to the definition of matrix variate Gaussian distribution [30], $\mathbf {B} \sim {\mathcal {CN}}_{d,d}(\mathbf {0},\varvec{\Sigma }^* \otimes \varvec{\Psi })$. $\square$

Appendix 2: Analytical Bound Gap for Correlated Channel

Using eigen-decomposion, $\mathbb{\mathbf {V}}_{j}^{\text {H}} \varvec{\Phi }_\text {t} {\mathbf {V}}_{j} = {{\mathbf {C}}}_{\text {t},j} \varvec{\Lambda }_{\text {t},j} {{\mathbf {C}}}_{\text {t},j} ^{\text {H}}$, ${\mathbf {U}}_{j}^{\text {H}} \varvec{\Phi }_{\text {r}} {\mathbf {U}}_{j} = {{\mathbf {C}}}_{{\text {r}},j} \varvec{\Lambda }_{{\text {r}},j} {{\mathbf {C}}}_{{\text {r}},j} ^{\text {H}}$, their trace can be derived as $\text {tr}[ {\mathbf {V}}_{j}^{\text {H}} \varvec{\Phi }_\text {t} {\mathbf {V}}_{j} ] = \text {tr}( \varvec{\Lambda }_{\text {t},j})$, $\text {tr}[{\mathbf {U}}_{j}^{\text {H}} \varvec{\Phi }_{\text {r}} {\mathbf {U}}_{j}] = \text {tr}( \varvec{\Lambda }_{{\text {r}},j})$. Given $\{{\mathbf {U}}_j,\forall j\}$, $\varvec{\Phi }_t$ and $\varvec{\Phi }_t$, the covariance matrix of the effective noise can be derived as

$$\begin{aligned} \mathbf {Q}_{\bar{{\mathbf {n}}}_{k}}&= \sum _{j=1}^K {{\mathbf {U}}}_k^{\text {H}} \varvec{\Phi }_{\text {r}}^{\frac{1}{2}} {\mathbb{E}}\big [{\mathbf {E}}_{kj}^{\text {w}} \varvec{\Phi }_\text {t}^{\frac{1}{2}} {\mathbf {V}}_{j} \mathbf {Q}_{{\mathbf {x}}} {\mathbf {V}}_{j}^{\text {H}} \varvec{\Phi }_\text {t}^{\frac{1}{2}} \cdot ({\mathbf {E}}_{kj}^{\text {w}})^{\text {H}} \big ] \varvec{\Phi }_{\text {r}}^{\frac{1}{2}} {\mathbf {U}}_k + N_0 {\mathbf {I}}_d \nonumber \\&= \sum _{j=1}^K {\mathbf {U}}_k^{\text {H}} \varvec{\Phi }_{\text {r}}^{\frac{1}{2}} \text {tr}[\varvec{\Phi }_\text {t}^{\frac{1}{2}} {\mathbf {V}}_{j} \mathbf {Q}_{{\mathbf {x}}} {\mathbf {V}}_{j}^{\text {H}} \varvec{\Phi }_\text {t}^{\frac{1}{2}}] \sigma _{{\mathbf {E}}^{\text {w}}}^2 \varvec{\Phi }_{\text {r}}^{\frac{1}{2}} {\mathbf {U}}_k + N_0 {\mathbf {I}}_d \nonumber \\&= {\mathbf {C}}_{{\text {r}},k}\Big (\sum _{j=1}^K \sum _{i=1}^d \frac{P}{d} \lambda _{j,i}^\text {t} \sigma _{{\mathbf {E}}^{\text {w}}}^2 \varvec{\Lambda }_{{\text {r}},k} + N_0 {\mathbf {I}}_d \Big ){\mathbf {C}}_{{\text {r}},k}^{\text {H}} \nonumber \\&= N_0{\mathbf {C}}_{{\text {r}},k}\varvec{\Lambda }_{k}{\mathbf {C}}_{{\text {r}},k}^{\text {H}}, \end{aligned}$$

(49)

where the lth diagonal element of $\varvec{\Lambda }_k$ denotes as

$$\begin{aligned} \lambda _{k,l}={\rho \sigma _{{\mathbf {E}}^{\text {w}}}^2} \sum _{j=1}^K\sum _{i=1}^d \lambda _{j,i}^{\text {t}} \lambda _{k,l}^{{\text {r}}}+1, \end{aligned}$$

(50)

with $\lambda _{j,i}^{\text {t}}$ denotes the ith diagonal element of $\varvec{\Lambda }_{\text {t},j}$ and $\lambda _{j,i}^{{\text {r}}}$ denotes the ith diagonal element of $\varvec{\Lambda }_{{\text {r}},j}$.

Given ${\mathbf {x}}_1,{\mathbf {x}}_2,\ldots ,{\mathbf {x}}_K$, the condition correlation matrix for the effective noise is

$$\begin{aligned} \mathbf {Q}_{\bar{\mathbf {n}}_{k}|{\mathbf {x}}_1,\ldots ,{\mathbf {x}}_K}&= \sum _{j=1}^K {\mathbf {U}}_k^{\text {H}} \varvec{\Phi }_{\text {r}}^{\frac{1}{2}} {\mathbb{E}}\big [{\mathbf {E}}_{kj}^{\text {w}} \varvec{\Phi }_\text {t}^{\frac{1}{2}} \mathbf {V}_{j} {\mathbf {x}}_j{{\mathbf {x}}_j^{\text {H}}} \mathbf {V}_{j}^{\text {H}} \varvec{\Phi }_\text {t}^{\frac{1}{2}} \cdot ({\mathbf {E}}_{kj}^{\text {w}})^{\text {H}} \big ] \varvec{\Phi }_{\text {r}}^{\frac{1}{2}} {\mathbf {U}}_k + N_0 {\mathbf {I}}_d \nonumber \\&= {\mathbf {C}}_{{\text {r}},k}\Big [\sum _{j=1}^K \text {tr}( {\mathbf {x}}_j{\mathbf {x}}_j^{\text {H}} \hat{\mathbf {V}}_j^{\text {H}} \varvec{\Phi }_\text {t}\hat{\mathbf {V}}_j) \sigma _{{\mathbf {E}}^{\text {w}}}^2 \varvec{\Lambda }_{{\text {r}},k} + N_0 {\mathbf {I}}_d \Big ]{\mathbf {C}}_{{\text {r}},k}^{\text {H}} \nonumber \\&= {\mathbf {C}}_{{\text {r}},k}\Big [\sum _{j=1}^K \text {tr}({{\mathbf {C}}}_{\text {t},j}^{\text {H}} {\mathbf {x}}_j{\mathbf {x}}_j^{\text {H}} {{\mathbf {C}}}_{\text {t},j} \varvec{\Lambda }_{\text {t},j} ) \sigma _{{\mathbf {E}}^{\text {w}}}^2 \varvec{\Lambda }_{{\text {r}},k} + N_0 {\mathbf {I}}_d \Big ]{\mathbf {C}}_{{\text {r}},k}^{\text {H}} \nonumber \\&= {\mathbf {C}}_{{\text {r}},k}\Big [\sum _{j=1}^K \sum _{i=1}^d \sigma _{{\mathbf {E}}^{\text {w}}}^2 \lambda _{j,i}^\text {t} |x_{j,i}|^2 \varvec{\Lambda }_{{\text {r}},k} + N_0 {\mathbf {I}}_d \Big ]{\mathbf {C}}_{{\text {r}},k}^{\text {H}} \nonumber \\&= N_0{\mathbf {C}}_{{\text {r}},k}\varvec{\Lambda }_k^\prime {\mathbf {C}}_{{\text {r}},k}^{\text {H}}. \end{aligned}$$

(51)

In (51), we have used the property $\text {tr}({{\mathbf {C}}}_{\text {t},j}^{\text {H}} {\mathbf {x}}_j{\mathbf {x}}_j^{\text {H}} {{\mathbf {C}}}_{\text {t},j} \varvec{\Lambda }_{\text {t},j} )=\sum _{i=1}^d |x_{j,i}|^2 \lambda _{k,i}^\text {t}$.

From (49) and (51), the bound gap for the correlated channels for one channels realization is represented as

$$\begin{aligned} \varDelta _k^{\text {cond}} ={\mathbb{E}}\Big [ \log _2\frac{ | \varvec{\Lambda }_k |}{| \varvec{\Lambda }_k^\prime |} \Big ] = \sum _{l=1}^d {\log _2\lambda }_{k,l} - \sum _{l=1}^d {\mathbb{E}} [ {\log _2\lambda }_{k,l}^\prime ], \end{aligned}$$

(52)

where

$$\begin{aligned} \lambda _{k,l}^\prime& = {} \frac{\sigma _{{\mathbf {E}}^{\text {w}}}^2 }{N_0} \sum _{j=1}^K \sum _{i=1}^d \lambda _{j,i}^\text {t} |x_{j,i}|^2 {\lambda }_{k,l} ^{\text {r}} + 1 \nonumber \\& = {} \sum _{j=1}^K \sum _{i=1}^d \rho \sigma _{{\mathbf {E}}^{\text {w}}}^2 \lambda _{j,i}^\text {t} {\lambda }_{k,l} ^{\text {r}} |z_{j,i}|^2 + 1 \nonumber \\& = {} \sum _{n=1}^{Kd} z_{n,l} + 1 \end{aligned}$$

(53)

where $z_{j,i}=\frac{x_{j,i}}{\sqrt{P/d}} \sim {\mathcal {CN}}(0,1)$, $z_{n,l}= \rho \sigma _{{\mathbf {E}}^{\text {w}}}^2 \alpha _{n,l}|z_{j,i}|^2$ with $\alpha _{n,l}= \lambda _{j,i}^\text {t} {\lambda }_{k,l} ^{\text {r}},n=(j-1)d+i, \forall j\in \{1,\ldots ,K\}, i\in \{1,\ldots ,d\}$. Then, $z_{n,l}$ is an exponential distributed variable with parameter $\rho \sigma _{{\mathbf {E}}^{\text {w}}}^2\alpha _{n,l}$. We assume the channels are generic, which means the probability of $\alpha _{n,l}=\alpha _{m,l}, \forall n \ne m$ is 0, then (53) is a sum of Kd independent non-identically distributed (i.n.i.d.) exponential random variables. With the probability density function proposed in [40] for i.n.i.d. exponential random variables, the logarithmic expectation $\sum _{l=1}^d {\mathbb{E}} [ {\log _2\lambda }_{k,l}^\prime ]$ can be derived as

$$\begin{aligned} {\mathbb{E}} [\log _2 \lambda _{k,l}^\prime ]& = {} \frac{1}{\ln 2} \sum _{n=1}^{Kd} \left( \prod _{m\ne n} \frac{\alpha _{n,l}}{\alpha _{n,l}-\alpha _{m,l}} \right) e^{\frac{1}{\rho \sigma _{{\mathbf {E}}^{\text {w}}}^2 \alpha _{n,l}}} \text {Ei}\left( \frac{1}{\rho \sigma _{{\mathbf {E}}^{\text {w}}}^2 \alpha _{n,l}}\right). \end{aligned}$$

(54)

Substitute (54) into (52), we obtain a closed-form bound gap.