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Adaptive Filtering Techniques using Cyclic Prefix in OFDM Systems for Multipath Fading Channel Prediction

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Abstract

This paper presents adaptive channel prediction techniques for wireless orthogonal frequency division multiplexing (OFDM) systems using cyclic prefix (CP). The CP not only combats intersymbol interference, but also precludes requirement of additional training symbols. The proposed adaptive algorithms exploit the channel state information contained in CP of received OFDM symbol, under the time-invariant and time-variant wireless multipath Rayleigh fading channels. For channel prediction, the convergence and tracking characteristics of conventional recursive least squares (RLS) algorithm, numeric variable forgetting factor RLS (NVFF-RLS) algorithm, Kalman filtering (KF) algorithm and reduced Kalman least mean squares (RK-LMS) algorithm are compared. The simulation results are presented to demonstrate that KF algorithm is the best available technique as compared to RK-LMS, RLS and NVFF-RLS algorithms by providing low mean square channel prediction error. But RK-LMS and NVFF-RLS algorithms exhibit lower computational complexity than KF algorithm. Under typical conditions, the tracking performance of RK-LMS is comparable to RLS algorithm. However, RK-LMS algorithm fails to perform well in convergence mode. For time-variant multipath fading channel prediction, the presented NVFF-RLS algorithm supersedes RLS algorithm in the channel tracking mode under moderately high fade rate conditions. However, under appropriate parameter setting in \(2\times 1\) space–time block-coded OFDM system, NVFF-RLS algorithm bestows enhanced channel tracking performance than RLS algorithm under static as well as dynamic environment, which leads to significant reduction in symbol error rate.

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Appendix

Appendix

1.1 Brief Description of \({2\times 1}\) STBC-OFDM System Model

Let us consider \(2\times 1\) STBC-OFDM system with two transmitters (TX1 and TX2) and one receiver (RX1) antennas (as in [44]), in which the kth and \((k+1)\)th OFDM symbol blocks are denoted as \({\vec {\mathbf{X }}}_{\mathbf{o }} =\left[ {{\begin{array}{cccc} {X_{0,k} }&{}\; {X_{1,k} }&{}\; \ldots &{}\; {X_{N-1,k} } \\ \end{array} }} \right] _{N\times 1}^T \) and \({\vec {\mathbf{X }}}_{\mathbf{e }} =\left[ {{\begin{array}{cccc} {X_{0,k+1} }&{}\; {X_{1,k+1} }&{}\; \ldots &{}\; {X_{N-1,k+1} } \\ \end{array} }} \right] _{N\times 1}^T \), respectively. By performing IFFT operation, the \({\vec {\mathbf{X }}}_{\mathbf{o }}\) and \({\vec {\mathbf{X }}}_{\mathbf{e }} \) can be expressed in time domain as \({\vec {\mathbf{x }}}_{\mathbf{o }} =\left[ {{\begin{array}{cccc} {x_{0,k} }&{}\; {x_{1,k} }&{}\; \ldots &{}\; {x_{N-1,k} } \\ \end{array} }} \right] _{N\times 1}^T \) and \({\vec {\mathbf{x }}}_{\mathbf{e }} =\left[ {{\begin{array}{cccc} {x_{0,k+1} }&{}\; {x_{1,k+1} }&{}\; \ldots &{}\; {x_{N-1,k+1} } \\ \end{array} }} \right] _{N\times 1}^T \), respectively. Subsequently by incorporating CP, these two OFDM symbol blocks are represented as

$$\begin{aligned} {\vec {\mathbf{x }}}_{{\mathbf{co,o }}} =\left[ {{\begin{array}{ccccc} {{\vec {\mathbf{x }}}_{{\mathbf{cp,o }}}^T }&{}\quad {{\vec {\mathbf{x }}}_{\mathbf{o }}^T } \\ \end{array} }} \right] _{(N+G)\times 1}^T \quad \hbox {and}\quad {\vec {\mathbf{x }}}_{{\mathbf{co,e }}} =\left[ {{\begin{array}{cccc} {{\vec {\mathbf{x }}}_{{\mathbf{cp,e }}}^T }&{}\quad {{\vec {\mathbf{x }}}_{\mathbf{e }}^T } \\ \end{array} }} \right] _{(N+G)\times 1}^T \end{aligned}$$
(45)

Applying the space–time block-coding scheme using these two OFDM symbol blocks with CP [44], it can be shown that

$$\begin{aligned} {\vec {\mathbf{x }}}_{{\mathbf{STBC }}} =\left[ {\begin{array}{cc} +{\vec {\mathbf{x }}}_{{\mathbf{co,o }}}&{}\quad +{\vec {\mathbf{x }}}_{{\mathbf{co,e }}} \\ -{\vec {\mathbf{x }}}_{{\mathbf{co,e }}}^*&{}\quad +{\vec {\mathbf{x }}}_{{\mathbf{co,o }}}^*\\ \end{array}} \right] \end{aligned}$$
(46)

The transmitted wireless OFDM signals encounter multipath fading with L channel tap coefficients, which leads to \(G\ge 2L-2\) in the present scenario. The two independent wireless channel tap coefficient vectors are \({\vec {\mathbf{h }}}_{{\mathbf{1,k }}} \) (from TX1 to RX1) and \({\vec {\mathbf{h }}}_{{\mathbf{2,k }}} \) (from TX2 to RX1), respectively. The wireless channel tap coefficient vectors are assumed to be constant for two OFDM symbol blocks, such that

$$\begin{aligned} {\vec {\mathbf{h }}}_{{\mathbf{1,k }}} =\vec {\mathbf{h }}_{{\mathbf{1,k+1 }}} \hbox { and } {\vec {\mathbf{h }}}_{{\mathbf{2,k }}} =\vec {{\mathbf{h }}}_{{\mathbf{2,k+1 }}}\quad \hbox {for period } T_\mathrm{STBC} =2\left( {N+G} \right) T_s \end{aligned}$$
(47)

The received composite wireless signals at the receiver are

$$\begin{aligned} {\vec {\mathbf{y }}}_{{\mathbf{co,k }}} =\left[ {{\begin{array}{ccccc} {{\vec {\mathbf{y }}}_{{\mathbf{cp,k }}}^T }&{}\quad {{\vec {\mathbf{y }}}_{\mathbf{k }}^T } \\ \end{array} }} \right] _{(N+G)\times 1}^T \quad \hbox {and}\quad {\vec {\mathbf{y }}}_{{\mathbf{co,k }}+1} =\left[ {{\begin{array}{ccccc} {{\vec {\mathbf{y }}}_{{\mathbf{cp,k }}+1}^T }&{}\quad {{\vec {\mathbf{y }}}_{{\mathbf{k }}+1}^T } \\ \end{array} }} \right] _{(N+G)\times 1}^T \end{aligned}$$
(48)

The cyclic prefix part is extracted at the receiver considering \(G=2L-2\) (as in Sect. 2.2) to provide

$$\begin{aligned} {\vec {\mathbf{y }}}_{{\mathbf{cp,k }}}= & {} \vec {{\mathbf{d }}}_{\mathbf{o }} {\vec {\mathbf{h }}}_{{\mathbf{1,k }}} +\vec {{\mathbf{d }}}_{\mathbf{e }} {\vec {\mathbf{h }}}_{{\mathbf{2,k }}} +\vec {{\mathbf{n }}}_{{\mathbf{cp,k }}} \quad \hbox {in } k\mathrm{th} \hbox { OFDM symbol block period} \end{aligned}$$
(49)
$$\begin{aligned} {\vec {\mathbf{y }}}_{{\mathbf{cp,k+1 }}}= & {} -\vec {{\mathbf{d }}}^{{{*}}}_{\mathbf{e }} {\vec {\mathbf{h }}}_{{\mathbf{1,k+1 }}} +\vec {{\mathbf{d }}}^{{{*}}}_{\mathbf{o }} {\vec {\mathbf{h }}}_{{\mathbf{2,k+1 }}} +\vec {{\mathbf{n }}}_{{\mathbf{cp,k+1 }}} \quad \hbox {in } \left( {k+1} \right) \!\mathrm{th} \hbox { OFDM symbol block period}\nonumber \\ \end{aligned}$$
(50)

where

$$\begin{aligned} {\vec {\mathbf{d }}}_{\mathbf{o }}= & {} \left[ {{\begin{array}{ccccc} {x_{-L+1,k} }&{}\quad {x_{-L,k} }&{}\quad \ldots &{}\quad {x_{-2L+2,k} } \\ \vdots &{}\quad \ddots &{}\quad &{}\quad \vdots \\ \vdots &{}\quad &{}\quad \ddots &{}\quad \vdots \\ {x_{-1,k} }&{}\quad {x_{-2,k} }&{}\quad \ldots &{}\quad {x_{-L,k} } \\ \end{array} }} \right] _{(L-1)\times L} \end{aligned}$$
(51)
$$\begin{aligned} {\vec {\mathbf{d }}}_{\mathbf{e }}= & {} \left[ {{\begin{array}{ccccc} {x_{-L+1,k+1} }&{}\quad {x_{-L,k+1} }&{}\quad \ldots &{}\quad {x_{-2L+2,k+1} } \\ \vdots &{}\quad \ddots &{}\quad &{}\quad \vdots \\ \vdots &{}\quad &{}\quad \ddots &{}\quad \vdots \\ {x_{-1,k+1} }&{}\quad {x_{-2,k+1} }&{}\quad \ldots &{}\quad {x_{-L,k+1} } \\ \end{array} }} \right] _{(L-1)\times L} \end{aligned}$$
(52)

In matrix form, (49) and (50) can be rearranged by invoking (47) to give

$$\begin{aligned} {\vec {\mathbf{y }}}_{{\mathbf{CP }}} =\vec {{\mathbf{d }}}_{{\mathbf{CP }}} {\vec {\mathbf{h }}}_{{\mathbf{CP }}} +\vec {{\mathbf{n }}}_{{\mathbf{CP }}} \end{aligned}$$
(53)

where

$$\begin{aligned} {\vec {\mathbf{y }}}_{{\mathbf{CP }}}= & {} \left[ {{\begin{array}{l} {{\vec {\mathbf{y }}}_{{\mathbf{cp,k }}} } \\ {{\vec {\mathbf{y }}}_{{\mathbf{cp,k+1 }}} } \\ \end{array} }} \right] _{2(L-1)\times 1} , \quad {\vec {\mathbf{d }}}_{{\mathbf{CP }}} =\left[ {{\begin{array}{cc} {+{\vec {\mathbf{d }}}_{\mathbf{o }} }&{} {+{\vec {\mathbf{d }}}_{\mathbf{e }} } \\ {-{\vec {\mathbf{d }}}^{*}_{\mathbf{e }} }&{} {+{\vec {\mathbf{d }}}^{*}_{\mathbf{o }} } \\ \end{array} }} \right] _{2(L-1)\times 2L} ,\\ {\vec {\mathbf{h }}}_{{\mathbf{CP }}}= & {} \left[ {{\begin{array}{cc} {{\vec {\mathbf{h }}}_{{\mathbf{1,k }}} } \\ {{\vec {\mathbf{h }}}_{{\mathbf{2,k }}} } \\ \end{array} }} \right] _{2L\times 1} =\left[ {{\begin{array}{cc} {{\vec {\mathbf{h }}}_{{\mathbf{1,k }}+1} } \\ {{\vec {\mathbf{h }}}_{{\mathbf{2,k }}+1} } \\ \end{array} }} \right] _{2L\times 1} \quad \hbox {and}\quad {\vec {\mathbf{n }}}_{{\mathbf{CP }}} =\left[ {{\begin{array}{cc} {{\vec {\mathbf{n }}}_{{\mathbf{cp,k }}} } \\ {{\vec {\mathbf{n }}}_{{\mathbf{cp,k+1 }}} } \\ \end{array} }} \right] _{2(L-1)\times 1} \end{aligned}$$

Analogous to (19), if the channel is assumed to be constant for two consecutive OFDM symbol blocks, then

$$\begin{aligned} \left[ {\begin{array}{l} {\vec {\mathbf{h }}}_{{\mathbf{1,2q }}} \\ {\vec {\mathbf{h }}}_{{\mathbf{2,2q }}} \\ \end{array}} \right] = \vec {{\mathbf{A }}}_{\mathbf{2 }} \left[ {\begin{array}{l} {\vec {\mathbf{h }}}_{{\mathbf{1,2q -\mathbf{2}}}} \\ {\vec {\mathbf{h }}}_{{\mathbf{2,2q -\mathbf{2}}}} \\ \end{array}} \right] {{{-}}}\left[ {\begin{array}{l} {\vec {\mathbf{w }}}_{{\mathbf{1,2q }}} \\ {\vec {\mathbf{w }}}_{{\mathbf{2,2q }}} \\ \end{array}} \right] \quad \hbox {for } q=1,2,3,\ldots \end{aligned}$$
(54)

where \({\vec {\mathbf{A }}}_{\mathbf{2 }} =\alpha {\vec {\mathbf{I }}}_{2L\times 2L} \) is the channel state transition matrix with \(2L\times 2L\) dimensional identity matrix \({\vec {\mathbf{I }}}\). Using the aforementioned Eq. (53), the presented channel estimation schemes are used to obtain \({\hat{{\mathbf{h }}}}_{\mathbf{1,k }} =\hat{{{\mathbf{h }}}}_{{\mathbf{1,k }}+1} \) and \({\hat{\mathbf{h }}}_{{\mathbf{2,k }}} =\hat{{{\mathbf{h }}}}_{{\mathbf{2,k+1 }}} \). Further, this estimated/predicted channel state information is used to decode the STBC-OFDM symbols, as in [1, 2, 44].

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Kohli, A.K., Kapoor, D.S. Adaptive Filtering Techniques using Cyclic Prefix in OFDM Systems for Multipath Fading Channel Prediction. Circuits Syst Signal Process 35, 3595–3618 (2016). https://doi.org/10.1007/s00034-015-0214-2

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