Iterative Channel Estimation and Detection for High-Mobility MIMO-OFDM Systems: Mitigating Error Propagation by Exploiting Error Information

Abstract

Joint channel estimation and data detection are always challenging problems for high mobility multiple-input multiple-output orthogonal frequency division multiplexing systems. Existing iterative schemes focus on the performance improvement from the correct ones, ignoring error propagation from the incorrect ones. To deal with this problem, we propose a new iterative channel estimation and detection scheme that reduces the unexpected effects of both detection errors and channel estimation errors. The interference in channel estimation induced by detection errors is analyzed and transformed as part of the noise, which is filtered out by Kalman estimator using the derived covariance of both the channel and data errors in detection. Besides, to minimize the detection error caused by channel estimation errors, we propose a new detection algorithm with an optimized weight, which is obtained by the error covariance of the estimated channels and the detected data. Furthermore, matrix calculations and parameter approximations are performed to reduce computational complexity. Extensive simulation results are also presented to demonstrate the significant performance improvement in joint channel estimation and data detection with the proposed iterative schemes.

Appendix

Based on the independence between antennas, we have the covariance of channel matrix $\mathbf{{H}}_k^{(i)}$ in the $i{\text {th}}$ step of the SIC detection written as

$$\begin{aligned} {{\varvec{\sigma }}}_{\mathbf{{H}}_k^{(i)}}^2 = {\text{E}}\left\{ {\mathbf{{H}}_k^{(i)}{{\left( \mathbf{{H}}_k^{(i)}\right) }^H}} \right\} = {{\varvec{\sigma }}}_{\mathbf{{H}}_k^{(i - 1)}}^2 - {{\varvec{\sigma }}}_{{\mathbf {h}}_{k,{\beta _i}}^{}}^2 \end{aligned}$$

(44)

where the covariance is initiated as ${{\varvec{\sigma }}}_{\mathbf{{H}}_k^{(1)}}^2 = {{\varvec{\sigma }}}_{{\mathbf{{H}}_k}}^2$ and taken off ${{\varvec{\sigma }}}_{{\mathbf {h}}_{k,{\beta _i}}^{}}^2={\text{E}}\left\{ {{\mathbf {h}}_{k,{\beta _i}}^{}{\mathbf {h}}_{k,{\beta _i}}^H} \right\}$ in each detection order step along with the order $i$ increasing

$$\begin{aligned} {{\varvec{\sigma }}}_{{\mathbf{{H}}_k}}^2&= {} {\text{E}}\left\{ {{\mathbf{{H}}_k}{{\left( {{\mathbf{{H}}_k}} \right) }^H}} \right\} = \sum \limits _{{n_t} = 1}^{{N_T}} {\text{{diag}}\left\{ {{{\varvec{\sigma }}}_{\mathbf{{H}}_k^{(1,{n_t})}}^2, \ldots ,{{\varvec{\sigma }}}_{\mathbf{{H}}_k^{({N_R},{n_t})}}^2} \right\} } \nonumber \\ {{\varvec{\sigma }}}_{{\mathbf {h}}_{k,{\beta _i}}^{}}^2&= {} {\text{ E}}\left\{ {{\mathbf {h}}_{k,{\beta _i}}^{}{\mathbf {h}}_{k,{\beta _i}}^H} \right\} = \sum \limits _{{n_t} = 1}^{{N_T}} {\text{{diag}}\left\{ {{{\varvec{\sigma }}}_{{\mathbf {h}}_{k,{\beta _i}}^{(1,{n_t})}}^2, \ldots ,{{\varvec{\sigma }}}_{{\mathbf {h}}_{k,{\beta _i}}^{({N_R},{n_t})}}^2} \right\} } \end{aligned}$$

(45)

The $(n,n'){\text {th}}$ entry of ${{\varvec{\sigma }}}_{\mathbf{{H}}_k^{({n_r},{n_t})}}^2$ and ${{\varvec{\sigma }}}_{{\mathbf {h}}_{k,{\beta _i}}^{({n_r},{n_t})}}^2$ are given by $\sum \limits _{m = 1}^{{N_C}} {{\text{E}}\left\{ {h_{k,n,m}^{({n_r},{n_t})}{{(h_{k,n',m}^{({n_r},{n_t})})}^*}} \right\} }$ and ${\text{E}}\left\{ {h_{k,n,{m^*}}^{({n_r},{n_t})}{{(h_{k,n',{m^*}}^{({n_r},{n_t})})}^*}} \right\}$ when ${m^*} = {\beta _i}{\text{{ mod}}}_{}^{}{N_C}$, respectively, where $h_{k,n,n'}^{({n_r},{n_t})} = {[\mathbf{{H}}_k^{({n_r},{n_t})}]_{n,n'}}$. The expectation of the unit is given by

$$\begin{aligned}&{\text{{E}}}\left\{ {h_{k,n,m}^{({n_r},{n_t})}{{\left( h_{k,n',m'}^{({n_r},{n_t})}\right) }^*}} \right\} \nonumber \\&\quad = {\text{{E}}}\left\{ {\frac{1}{{N_C^2}}\sum \limits _{l = 1}^{L} {\left( \begin{array}{l} {e^{ - j2\pi \frac{{m - m'}}{N_C}(l-1)}}\\ \sum \limits _{n''' = 1}^{{N_C}} {\sum \limits _{n'' = 1}^{{N_C}} {\alpha _{l,k,n'''}^{({n_r},{n_t})}{{\left( \alpha _{l,k,n''}^{({n_r},{n_t})}\right) }^*}{e^{j2\pi \left( \frac{{m - n}}{{{N_C}}}(n'''-1) - \frac{{m' - n'}}{{{N_C}}}(n''-1)\right) }}} } \end{array} \right) } } \right\} \end{aligned}$$

(46)

with $\alpha _{l,k,n'''}^{({n_r},{n_t})}{\left(\alpha _{l,k,n''}^{({n_r},{n_t})}\right)^*} = \left[\mathbf{{R}}_{\mathbf{{\alpha }}_{l,n}^{}}^{(0)}\right]_{n'',n'''}^{}$ given in (13).