Parallel Channel Estimator and Equalizer for Mobile OFDM Systems

Abstract

Mobile OFDM refers to OFDM systems with fast moving transceivers, in contrast to traditional OFDM systems whose transceivers are stationary or have a low velocity. In this paper, we use the basis expansion model (BEM) to model time-varying OFDM channels. Using different BEM’s, we investigate various architectures to implement the least-squares (LS) channel estimation and its corresponding zero-forcing (ZF) channel equalization. The experimental results show that our implementation for mobile OFDM systems is capable of combatting the time variation of mobile OFDM channels, and more hardware resource utilization is necessary compared with a traditional OFDM design which fails in a time-varying scenario. For mobile OFDM systems, different BEM’s are available for the channel modeling. We observe that the so-called Critically sampled Complex-Exponential BEM (CCE-BEM) leads to the most efficient hardware architecture while still maintaining high modeling accuracy.

Appendix: Detailed Derivation of (9)

Let us start from the noiseless version of (8) as

$$\begin{aligned} {\bf r}^{(p)} =& \sum_{q=- Q}^Q {{{\bf{D}}^{(p)}_{q}} {{\bf \Delta}^{(p)}_q} {\bf{b}}^{(p)}}, \end{aligned}$$

where ${{\bf{D}}^{(p)}_{q}}$ is a submatrix obtained from ${{\bf{D}}_{q}}$ by only selecting rows (columns) corresponding to ${\bf r}^{(p)}$ in ${{\bf{r}}_{\rm{F}}}$ (${\bf b}^{(p)}$ in ${\bf b}$), and ${{\bf \Delta}^{(p)}_{q}}$ is obtained from ${{\bf \Delta}_{q}}$ by selecting the rows of ${\bf b}^{(p)}$ in ${\bf b}$.

We first notice that ${\bf \Delta} _{q} = \operatorname{diag} ( {\bf F}^{(L)} {\bf c}_{q} )$ as specified in (7), and thus we can specify ${{\bf \Delta}^{(p)}_{q}}$ as

$$\begin{aligned} {{\bf \Delta}^{(p)}_q} = \operatorname{diag} \bigl( {\bf F}^{(L,p)} {\bf c}_q \bigr), \end{aligned}$$

where ${\bf{F}}^{(L,p)}$ collects the rows of ${\bf{F}}^{(L)}$ corresponding to the positions of ${\bf b}^{(p)}$ in ${\bf b}$.

To this end, it is clear that

$$\begin{aligned} {{\bf \Delta}^{(p)}_q} {\bf{b}}^{(p)} =& \operatorname{diag} \bigl( {\bf F}^{(L,p)} {\bf c}_q \bigr) { \bf{b}}^{(p)} \\ =& \operatorname{diag} \bigl( {\bf{b}}^{(p)} \bigr) {\bf F}^{(L,p)} {\bf c}_q. \end{aligned}$$

(25)

Substituting (25) into ${\bf r}^{(p)}$, we obtain

$$\begin{aligned} {\bf r}^{(p)} =& \sum_{q=- Q}^Q {{\bf{D}}^{(p)}_{q}} \bigl( \operatorname{diag} \bigl({ \bf{b}}^{(p)} \bigr) {\bf F}^{(L,p)} \bigr) {\bf c}_q \\ =& \bigl[{{\bf{D}}^{(p)}_{-Q}}, \ldots, {{ \bf{D}}^{(p)}_{Q}} \bigr] \\ &{}\times {\bf I}_{2Q+1} \otimes \bigl( \operatorname{diag} \bigl( { \bf{b}}^{(p)} \bigr) {\bf F}^{(L,p)} \bigr) \\ &{} \times \bigl[{\bf c}_{-Q}^T, \ldots, {\bf c}_{Q}^T \bigr]^T \\ =& {{\bf{D}}^{(p)}} \bigl( {{\bf I}_{2Q+1} \otimes \bigl( \operatorname{diag} \bigl( {\bf{b}}^{(p)} \bigr) {\bf F}^{(L,p)} \bigr)} \bigr){\bf{c}} , \end{aligned}$$

(26)

where ⊗ stands for the Kronecker product, ${{\bf{D}}^{(p)}} =[{{\bf{D}}^{(p)}_{-Q}}, \ldots, {{\bf{D}}^{(p)}_{Q}}]$ and ${\bf c} = [{\bf c}_{-Q}^{T}, \ldots, {\bf c}_{Q}^{T} ]^{T}$.

Consequently, if we denote

$$\begin{aligned} {\bf A }^{(p)} = {{\bf{D}}^{(p)}} \bigl( {{{ \bf{I}}_{2Q + 1}} \otimes \bigl( {\operatorname{diag} \bigl( {{ \bf{b}}^{(p)}} \bigr) {\bf{F}}^{(L,p)}} \bigr)} \bigr) \end{aligned}$$

as defined in (10), we obtain

$$\begin{aligned} {\bf r}^{(p)}={\bf A}^{(p)}{\bf{c}} \end{aligned}$$

which is the noiseless version of (9).