Limited Feedback Strategy For MU-MIMO Systems Under Spatially Correlated Channels

Abstract

This paper studies limited feedback transmission strategy for downlink multiuser multiple-input multiple-output (MU-MIMO) systems under spatially correlated channels. To exploit the channel statistics for improving the performance of precoding at the base station (BS) with quantized channel direction information (CDI), we propose a two-level weighted DFT codebook (WDFT). It is composed of a long term DFT codebook representing the average CDIs and a short term codebook serving as an instantaneous weighting vector on them. The proposed codebook can adapt to various correlated channels without the need of channel correlation matrix at the BS. To mitigate the sum rate ceiling in high signal-to-noise ratio region caused by the quantized CDI with fixed size codebooks in less correlated channels, we propose a channel quality information (CQI) feedback strategy where each user helps the BS schedule users. By comparing its own CDI quantization error with a pre-determined threshold, each user feeds back one bit to indicate its preference of the transmission mode together with a corresponding CQI. With these information the BS first selects users separately from two user sets and finally selects users for downlink transmission. Simulation results show that the proposed limited feedback strategy including the WDFT codebook and the CQI feedback strategy outperforms existing schemes in various correlated channels.

Appendix

In the following, the average distortion led by codebook ${\mathcal {F}}$ is analyzed (the subscript k is omitted for simplicity). According to the definition of average distortion, we have

$$\begin{aligned} \mathbf {\mathcal {D}}({\mathbf {h}},{\mathcal {F}})= {\mathbb {E}}\left\{ \Vert {\mathbf {h}}\Vert ^2\right\} -{\mathbb {E}}\left\{|{{\mathbf {h}}}^H{\mathbf {v}}|^2\right\} \nonumber \\= M-{\mathbb {E}}\left\{ \max _{\phi _i}\left|\sum \nolimits _n{a^H_n{\mathbf {u}}^H(\beta _n)}\mathbf {f}({\phi _i})\right|^2\right\} , \end{aligned}$$

(28)

where channel energy ${\mathbb {E}}\{\Vert {\mathbf {h}}\Vert ^2\}=M$ is used.

Moreover, the second term

$$\begin{aligned}&{\mathbb {E}}\left\{ \max _{\phi _i}|\sum \limits _n{a^H_n{\mathbf {u}}^H(\beta _n)}\mathbf {f}({\phi _i})|^2\right\} \nonumber \\&\quad \mathop {\ge }^\text {a} {\mathbb {E}}_{\beta _n}\left\{ \max _{\phi _i} {\mathbb {E}}_{a_n|\beta _n}\left\{ |\sum \limits _n{a^H_n{\mathbf {u}}^H(\beta _n)}\mathbf {f}({\phi _i})|^2 \right\} \right\} \nonumber \\&\quad \mathop {=}^\text {b} {\mathbb {E}}_{\beta _n}\left\{ \max _{\phi _i} {\mathbb {E}}_{a_n|\beta _n}\left\{ \sum \limits _n|{a^H_n{\mathbf {u}}^H(\beta _n)}\mathbf {f}({\phi _i})|^2 \right\} \right\} \nonumber \\&\quad ={\mathbb {E}}_{\beta _n}\left\{ \max _{\phi _i}\sum \limits _n{p_n|{\mathbf {u}}^H(\beta _n)\mathbf {f}(\phi _i)|^2}\right\} , \end{aligned}$$

(29)

where (a) is due to the fact that the global optimization is reduced to a suboptimal optimization, (b) is based on the fact that the random variables $a_n$ and $a_m$ ($n\ne m$) are uncorrelated.

Fig. 1

Illustration of WDFT codebook design: using two DFT codewords to quantize a channel with two resolvable subpath clusters. $\phi _i$ are the samples of the average angle centers and $\Delta \phi _i$ are the derivations from $\phi _i$, $i=1,2$

Fig. 2

Sum rate of different codebooks versus $B_2$, $B_1=6$ bits, SNR = 25 dB and $\mathrm{AS}=15^{\circ }$

Fig. 3

Sum rate of different codebooks versus SNR, $B_1=6$, $B_2 = 6$ bits and $\mathrm{AS}=15^{\circ }$

By substituting (29) into (28), we get

$$\begin{aligned} \mathbf {\mathcal {D}}({\mathbf {h}},{\mathcal {F}})&\le M-{\mathbb {E}}\left\{ \max _{\phi _i}\sum \limits _n{p_n|{\mathbf {u}}^H(\beta _n)\mathbf {f}(\phi _i)|^2}\right\} \end{aligned}$$

(30)

$$\begin{aligned}&= M-\frac{1}{M}{\mathbb {E}}\left\{ \max _{\phi _i}\sum \limits _n{p_n \frac{\sin ^2(M\pi (\beta _n-\phi _i))}{\sin ^2(\pi (\beta _n-\phi _i))}}\right\} \end{aligned}$$

(31)

$$\begin{aligned}&= c_1 M {\mathbb {E}}\left\{ \max _{\phi _i}\sum \limits _n{p_n(\beta _n-\phi _i)^2}\right\} , \end{aligned}$$

(32)

in (30) the inequality $\frac{\sin ^2{(M\pi x)}}{\sin ^2{(\pi x)}}\ge M^2(1-c_1x^2)$ is applied ($c_1x^2 < 1$), where $c_1= -\frac{d^2}{dx^2}\frac{\sin ^2(M\pi x)}{ M^2\sin ^2(\pi x)} |_{x=0}=\frac{2(M^2-1)}{3}\pi ^2$, which is not hard to be derived (Table 1).

Fig. 4

Sum rate of different codebooks versus AS, SNR = 25 dB, $B_1 = 6$ and $B_2 = 6$ bits

Fig. 5

Sum rate of different codebooks with CDI indicator versus SNR, $B_1=6$, $B_2 = 6$ bits. The result of the proposed CQI with the threshold $\eta =1-2^{\frac{B_2}{M-1}}$ overlaps with the result of the purely MU-MIMO transmission strategy when $\mathrm{AS}=15^{\circ }$

Fig. 6

Average number of scheduled users versus SNR, $B_1=$ 6 , $B_2=6$ bits, $\mathrm{AS}=35^{\circ }$

Fig. 7

Sum rate of user number, $B_1=$ 6, $B_2=6$ bits, SNR = 25 dB

Table 1 The value of the average per-user rate loss upper bound with $\eta =1-2^{\frac{B}{M-1}}$ in different cases