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SCIENCE CHINA Information Sciences, Volume 62, Issue 2: 022303(2019) https://doi.org/10.1007/s11432-018-9413-6

Impacts of practical channel impairments on the downlink spectral efficiency of large-scale distributed antenna systems

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  • ReceivedJan 18, 2018
  • AcceptedMar 29, 2018
  • PublishedDec 18, 2018

Abstract

Channel impairments are major limiting factors in the performance of large-scale antenna systems.In this paper, we analyze the impacts of practical channel impairments caused by pilot contamination, Doppler shift, and phase noise on the downlink spectral efficiency of large-scale distributed antenna systems (L-DASs) with maximum ratio transmission (MRT) and zero-forcing (ZF) beamforming, in which per user power normalization is considered.Using a joint channel model that allows study of the simultaneous impacts of these channel impairments, we derive accurate and tractable closed-form approximations for the ergodic achievable downlink rate, thereby enabling spectral efficiency analysis of L-DASs and an efficient evaluation of the impacts of the channel impairments. It is shown that channel impairments reduce the downlink spectral efficiency and have a significant impact on ZF beamforming. The asymptotic user rate limit is also determined, from which we analyze the asymptotic performance of L-DASs with channel impairments. The analytical results show that MRT and ZF beamforming achieve the same asymptotic performance limit even with channel impairments. It is also found that the use of a large-scale antenna array at the base station sides can weaken the impacts of channel impairments.


Acknowledgment

This work was supported in part by National Natural Science Foundation of China (NSFC) (Grant Nos. 61501113, 61571120, 61271205, 61521061, 61372100,), and Jiangsu Provincial Natural Science Foundation (Grant Nos. BK20150630, BK20151415).


Supplement

Appendix

Proof of Theorem 1

For the signal power term $|{\rm~E}[{\boldsymbol~g}_{l,l,k}^{\text{H}}[t]~{\boldsymbol~w}_{l,k}^{\text{MRT}}[t]~]~|^2$, we have \begin{align} \big|{\rm E}\big[{\boldsymbol g}_{l,l,k}^{\text{H}}[t] {\boldsymbol w}_{l,k}^{\text{MRT}}[t] \big] \big|^2 \overset{({\rm a})}{=}&\, \big|{\rm E}\big[\|\hat{{\boldsymbol g}}_{l,l,k}[t] \|\big] \big|^2 \\ \overset{({\rm b})}=&\, \xi(\hat{k}_{l,l,k,\text{a}}[t])\hat{\theta}_{l,l,k,\text{a}}[t], \tag{39} \end{align} where $({\rm~a})$ is obtained because $\hat{{\boldsymbol~g}}_{l,l,k}[t]$ and $\tilde{{\boldsymbol~g}}_{l,l,k}[t]$ are independent, $({\rm~b})$ results from $\|\hat{{\boldsymbol~g}}_{l,l,k}[t]~\|~\sim~\text{Nakagami}(\hat{k}_{l,l,k,\text{a}}[t],\hat{k}_{l,l,k,\text{a}}[t]~\hat{\theta}_{l,l,k,\text{a}}[t])$ since $\|\hat{{\boldsymbol~g}}_{l,l,k}[t]~\|^2=|\hat{{\boldsymbol~g}}_{l,l,k}^{\text{H}}[t]{\boldsymbol~w}_{l,k}^{\text{MRT}}[t]|^2\sim~\Gamma(\hat{k}_{l,l,k,\text{a}}[t],~\hat{\theta}_{l,l,k,\text{a}}[t])$, which is obtained from Lemma 2 and (26).

Based on the independence of $\hat{{\boldsymbol~g}}_{l,l,k}[t]$ and considering the effect of pilot contamination, we decompose the interference power terms $\text{var}[{\boldsymbol~g}_{l,l,k}^{\text{H}}[t]{\boldsymbol~w}_{l,k}^{\text{MRT}}[t]]$, and $\sum_{(i,j)\neq~{(l,k)}}~{\rm~E}[~|{\boldsymbol~g}_{i,l,k}^{\text{H}}[t]~{\boldsymbol~w}_{i,j}^{\text{MRT}}[t]~|^2~]$ as follows: \begin{align} \text{var}\big[{\boldsymbol g}_{l,l,k}^{\text{H}}[t]{\boldsymbol w}_{l,k}^{\text{MRT}}[t]\big] = {\rm E}\big[ \big|\hat{{\boldsymbol g}}_{l,l,k}^{\text{H}}[t] {\boldsymbol w}_{l,k}^{\text{MRT}}[t]\big|^2 \big] + {\rm E}\big[ \big|\tilde{{\boldsymbol g}}_{l,l,k}^{\text{H}}[t] {\boldsymbol w}_{l,k}^{\text{MRT}}[t]\big|^2 \big] - \big|{\rm E}\big[\hat{{\boldsymbol g}}_{l,l,k}^{\text{H}}[t]{\boldsymbol w}_{l,k}^{\text{MRT}}[t] \big] \big|^2, \tag{40} \end{align} and \begin{align} \sum\limits_{(i,j)\neq {(l,k)}} {\rm E}\big[ \big|{\boldsymbol g}_{i,l,k}^{\text{H}}[t] {\boldsymbol w}_{i,j}^{\text{MRT}}[t] \big|^2 \big] =& \sum\limits_{j\neq k} {\rm E} \big[\big|{\boldsymbol g}_{l,l,k}^{\text{H}}[t] {\boldsymbol w}_{l,j}^{\text{MRT}}[t] \big|^2 \big] + \sum\limits_{i\neq l}\sum\limits_{j\neq k} {\rm E} \big[\big|\hat{{\boldsymbol g}}_{i,l,k}^{\text{H}}[t] {\boldsymbol w}_{i,j}^{\text{MRT}}[t] \big|^2 \big] \\ &+\sum\limits_{i\neq l}{\rm E} \big[\big|\hat{{\boldsymbol g}}_{i,l,k}^{\text{H}}[t] {\boldsymbol w}_{i,k}^{\text{MRT}}[t] \big|^2\big] + \sum\limits_{i\neq l}\sum\limits_{j=1}^{K}{\rm E}\big[\big|\tilde{{\boldsymbol g}}_{i,l,k,}^{\text{H}}[t]{\boldsymbol w}_{i,j}^{\text{MRT}}[t] \big|^2\big]. \tag{41} \end{align} By applying Lemma 2 to approximate the distributions of the terms in $\left(\text{\ref{var}}~\right)$ and $\left(\text{\ref{com}}~\right)$, we have \begin{align} &\big|\hat{{\boldsymbol g}}_{l,l,k}^{\text{H}}[t] {\boldsymbol w}_{l,k}^{\text{MRT}}[t]\big|^2 \sim \Gamma(\hat{k}_{l,l,k,\text{a}}[t],\hat{\theta}_{l,l,k,\text{a}}[t]), \tag{42} \\ &\big|\tilde{{\boldsymbol g}}_{l,l,k}^{\text{H}}[t] {\boldsymbol w}_{l,k}^{\text{MRT}}[t]\big|^2 \sim \Gamma\left(\frac{1}{MN}\tilde{k}_{l,l,k,\text{a}}[t],\tilde{\theta}_{l,l,k,\text{a}}[t]\right), \tag{43} \\ &\big|{\boldsymbol g}_{l,l,k}^{\text{H}}[t] {\boldsymbol w}_{l,j}^{\text{MRT}}[t] \big|^2 \sim \Gamma\left(\frac{1}{MN}k_{l,l,k,\text{a}}[t], \theta_{l,l,k,\text{a}}[t]\right), \tag{44} \\ &\big|\hat{{\boldsymbol g}}_{i,l,k}^{\text{H}}[t] {\boldsymbol w}_{i,k}^{\text{MRT}}[t] \big|^2 \sim \Gamma(\hat{k}_{i,l,k,\text{a}}[t], \hat{\theta}_{i,l,k,\text{a}}[t]), \tag{45} \\ &\big|\hat{{\boldsymbol g}}_{i,l,k}^{\text{H}}[t] {\boldsymbol w}_{i,j}^{\text{MRT}} [t]\big|^2 \sim \Gamma\left(\frac{1}{MN}\hat{k}_{i,l,k,\text{a}}[t], \hat{\theta}_{i,l,k,\text{a}}[t]\right), \tag{46} \\ &\big|\tilde{{\boldsymbol g}}_{i,l,k,}^{\text{H}}[t]{\boldsymbol w}_{i,j}^{\text{MRT}}[t] \big|^2 \sim \Gamma\left(\frac{1}{MN}\tilde{k}_{i,l,k,\text{a}}[t], \tilde{\theta}_{i,l,k,\text{a}}[t]\right). \tag{47} \end{align} Substituting (39) and (42)–(47) into (19) concludes the proof.

Proof of Theorem 2

First, given the distributions of $\hat{{\boldsymbol~g}}_{i,l,k}^{\text{H}}[t]~\hat{{\boldsymbol~g}}_{i,l,k}[t]$ in (26), the non-isotropic achievable CSI $\hat{{\boldsymbol~g}}_{i,l,k}[t]$ can be approximated as an isotropic vector $\hat{{\boldsymbol~g}}_{i,l,k,\text{a}}[t]$ with i.i.d. $\mathcal{CN}(0,\hat{\theta}_{i,l,k,\text{a}})$ elements [29]. Then, with the definition of ${\boldsymbol~F}_{l,\text{a}}[t]\triangleq[\hat{{\boldsymbol~g}}_{l,l,1,\text{a}}[t],~\ldots,$$\hat{{\boldsymbol~g}}_{l,l,K,\text{a}}[t]~]$, the useful signal power term $|{\rm~E}[{\boldsymbol~g}_{l,l,k}^{\text{H}}[t]~{\boldsymbol~w}_{l,k}^{\text{ZF}}[t]~]~|^2$ can be calculated by \begin{align} \big|{\rm E}\big[{\boldsymbol g}_{l,l,k}^{\text{H}}[t] {\boldsymbol w}_{l,k}^{\text{ZF}}[t] \big] \big|^2 &\overset{({\rm a})}{=} \big|{\rm E}[ 1 / \|{\boldsymbol f}_{l,l,k}[t] \|] \big|^2 \\ & \overset{({\rm b})}= \Big|{\rm E}\Big[\big( \big[\big({\boldsymbol F}_{l}^{\text{H}}[t] {\boldsymbol F}_l[t] \big)^{-1} \big]_{k,k} \big)^{-1/2} \Big] \Big|^2 \\ & \overset{({\rm c})}= \xi(\rho\hat{k}_{l,l,k,\text{a}}[t])\hat{\theta}_{l,l,k,\text{a}}[t], \tag{48} \end{align} where $({\rm~a})$ is obtained because of the independence of ${\boldsymbol~w}_{l,k}^{\text{ZF}}[t]$ and $\tilde{{\boldsymbol~g}}_{l,l,k}[t]$ and $\hat{{\boldsymbol~g}}_{l,l,k}^{\text{H}}[t]{\boldsymbol~w}_{l,k}^{\text{ZF}}[t]=1/\|{\boldsymbol~f}_{l,l,k}[t]~\|$, $({\rm~b})$ results from $\|{\boldsymbol~f}_{l,l,k}[t]\|^2~=~[({\boldsymbol~F}_{l}^{\text{H}}[t]~{\boldsymbol~F}_l[t]~)^{-1}~]_{k,k}~$, $({\rm~c})$ results from $1/\|{\boldsymbol~f}_{l,l,k}[t]\|\sim~\text{Nakagami}(\rho\hat{k}_{l,l,k,\text{a}}[t],~\rho\hat{k}_{l,l,k,\text{a}}[t]~\hat{\theta}_{l,l,k,\text{a}}[t])$ where we have applied Lemma 2 to approximate the distribution of $[({\boldsymbol~F}_{l}^{\text{H}}[t]~{\boldsymbol~F}_l[t]~)^{-1}~]_{k,k}$ as $\Gamma(\rho\hat{k}_{l,l,k,\text{a}}[t],~\hat{\theta}_{l,l,k,\text{a}}[t]~)$ since $[({\boldsymbol~F}_{l,\text{a}}^{\text{H}}[t]~{\boldsymbol~F}_{l,\text{a}}[t]~)^{-1}~]_{k,k}$ $\sim~\Gamma(MN-K+1,~\hat{\theta}_{l,l,k,\text{a}}[t]~)$ 1).Similar to the analysis in the proof given for Theorem 1 and considering that $\hat{{\boldsymbol~g}}_{l,l,k}[t]~{\boldsymbol~w}_{l,j}^{\text{ZF}}[t]~=~0$ for $j\neq~k$, the interference power terms $\text{var}[{\boldsymbol~g}_{l,l,k}^{\text{H}}[t]{\boldsymbol~w}_{l,k}^{\text{ZF}}[t]]$ and $\sum_{(i,j)\neq~{(l,k)}}~{\rm~E}[~|{\boldsymbol~g}_{i,l,k}^{\text{H}}[t]~{\boldsymbol~w}_{i,j}^{\text{ZF}}[t]~|^2~]$ can be decomposed as \begin{align} \text{var}\big[{\boldsymbol g}_{l,l,k}^{\text{H}}[t]{\boldsymbol w}_{l,k}^{\text{ZF}}[t]\big] ={\rm E}\big[ \big|\hat{{\boldsymbol g}}_{l,l,k}^{\text{H}}[t] {\boldsymbol w}_{l,k}^{\text{ZF}}[t]\big|^2 \big] + {\rm E}\big[ \big|\tilde{{\boldsymbol g}}_{l,l,k}^{\text{H}}[t] {\boldsymbol w}_{l,k}^{\text{ZF}}[t]\big|^2 \big] - \big|{\rm E}\big[\hat{{\boldsymbol g}}_{l,l,k}^{\text{H}}[t]{\boldsymbol w}_{l,k}^{\text{ZF}}[t] \big] \big|^2, \tag{49} \end{align} and \begin{align} \sum_{(i,j)\neq {(l,k)}} {\rm E}\big[ \big|{\boldsymbol g}_{i,l,k}^{\text{H}}[t] {\boldsymbol w}_{i,j}^{\text{ZF}}[t] \big|^2 \big] =& \sum_{j\neq k} {\rm E} \big[\big|\tilde{{\boldsymbol g}}_{l,l,k}^{\text{H}}[t] {\boldsymbol w}_{l,j}^{\text{ZF}}[t] \big|^2 \big]+ \sum_{i\neq l}\sum_{j\neq k} {\rm E} \big[\big|\hat{{\boldsymbol g}}_{i,l,k}^{\text{H}}[t] {\boldsymbol w}_{i,j}^{\text{ZF}}[t] \big|^2 \big] \\ &+ \sum_{i\neq l}{\rm E} \big[\big|\hat{{\boldsymbol g}}_{i,l,k}^{\text{H}}[t] {\boldsymbol w}_{i,k}^{\text{ZF}}[t] \big|^2 \big]+ \sum_{i\neq l}\sum_{j=1}^{K}{\rm E}\big[\big|\tilde{{\boldsymbol g}}_{i,l,k,}[t]{\boldsymbol w}_{i,j}^{\text{ZF}} \big|^2\big]. \tag{50} \end{align} The distributions of the terms in $\left(\text{\ref{var_zf}}~\right)$ and $\left(\text{\ref{inf2c_zf}}~\right)$ can be obtained by applying Lemma 2, \begin{align} \big|\hat{{\boldsymbol g}}_{l,l,k}^{\text{H}}[t] {\boldsymbol w}_{l,k}^{\text{ZF}}[t]\big|^2& \sim\Gamma(\rho\hat{k}_{l,l,k,\text{a}}[t],\hat{\theta}_{l,l,k,\text{a}}[t]), \tag{51} \\ \big|\tilde{{\boldsymbol g}}_{l,l,k}^{\text{H}}[t] {\boldsymbol w}_{l,k}^{\text{ZF}}[t]\big|^2 &\sim \Gamma\left(\frac{1}{MN}\tilde{k}_{l,l,k,\text{a}}[t],\tilde{\theta}_{l,l,k,\text{a}}[t]\right), \tag{52} \\ \big|\tilde{{\boldsymbol g}}_{l,l,k}^{\text{H}}[t] {\boldsymbol w}_{l,j}^{\text{ZF}}[t] \big|^2& \sim\Gamma\left(\frac{1}{MN}\tilde{k}_{l,l,k,\text{a}}[t],\tilde{\theta}_{l,l,k,\text{a}}[t]\right), \tag{53} \\ \big|\hat{{\boldsymbol g}}_{i,l,k}^{\text{H}}[t] {\boldsymbol w}_{i,k}^{\text{ZF}}[t] \big|^2&\sim \Gamma \big(\rho\hat{k}_{i,l,k,\text{a}}[t], \hat{\theta}_{i,l,k,\text{a}}[t]\big), \tag{54} \\ \big|\hat{{\boldsymbol g}}_{i,l,k}^{\text{H}}[t] {\boldsymbol w}_{i,j}^{\text{ZF}}[t] \big|^2&\sim \Gamma\left(\frac{1}{MN}\hat{k}_{i,l,k,\text{a}}[t], \hat{\theta}_{i,l,k,\text{a}}[t]\right), \tag{55} \\ \big|\tilde{{\boldsymbol g}}_{i,l,k}[t]{\boldsymbol w}_{i,j}^{\text{ZF}}[t] \big|^2&\sim \Gamma\left(\frac{1}{MN}\tilde{k}_{i,l,k,\text{a}}[t],\tilde{\theta}_{i,l,k,\text{a}}[t]\right). \tag{56} \end{align} Substituting (48) and (51)–(56) into (19) yields the closed-form approximation (34). This completes the proof.

Khansefid A, Minn H. Achievable downlink rates of MRC and ZF precoders in massive MIMO with uplink and downlink pilot contamination. IEEE Trans Commun, 2015, 63: 4849–4864.


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  • Figure 1

    (Color online) Fluctuation of the norms of MRT beamforming vectors around their means versus the total number of transmit antennas.

  • Figure 2

    (Color online) Cumulative distribution function of the average achievable rate per user at time $t=1$ for $L=7$, $T_{\text{s}}=10^{-5}$ s, $T_{\text{c}}=100$, $\sigma_{\varphi_{l,k}}=\sigma_{\phi_{i,m}}=0.72^{\circ}$ with different $K$, $M$ and $N$.

  • Figure 3

    (Color online) Average achievable rate per user versus phase noise increment standard deviations for different values of $f_{\text{D}}T_{\text{s}}$, $L=7$, $M=10$, $N=20$, $K=8$, $T_{\text{s}}=10^{-5}$ s, $T_{\text{c}}=100$.

  • Figure 4

    (Color online) Average achievable rate per user with channel impairments versus the total number of transmit antennas, $T_{\text{c}}=100$, $L=7$, $M=5$, $K=8$.

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