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Subspace Channel Identification for Multiuser MIMO STBC-OFDM Systems

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Abstract

This paper investigates the subspace channel identification for multiple-input multiple-output zero-padded orthogonal frequency-division multiplexing systems with space–time block code and virtual carriers (VCs). We first develop a new subspace channel identification model when the VCs exist. Then, two schemes, the forward–backward method and the repetition index scheme (RIS), are developed to generate many times equivalent symbols and then to enhance the performance of the subspace channel identification. With these methods, more accurate subspace channel identification can be obtained by using only a few received blocks. Further, the noise pre-whitening technique is investigated to decrease the nonwhite noise effect caused by the RIS scheme. We also provide complexity analyses of the proposed methods. With the acceptable computational cost, the proposed methods significantly improve the performances of the channel identification and the equalization. Simulations are carried out to demonstrate the effectiveness of the proposed methods.

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Acknowledgements

This work was supported in part by the Special Foundation for Young Scientists of Quanzhou Normal University of China under Grant No. 201330, in part by Fujian Province Education Department under Grants JAT170470, in part by the National Natural Science Foundation of China under Grant 61501041, in part by the Open Foundation of State Key Laboratory under Grant ISN19-19, in part by the Ministry of Science and Technology, Taiwan under Grants MOST 107-2221-E-030-003 and in part by Fu-Jen Catholic University of Taiwan under Grant A0106014.

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Appendix

Appendix

1.1 A. Proof of Lemma 1

Let \( {\mathbf{F}}_{D}^{H} = [{\mathbf{a}}_{{c_{0} }} , \ldots ,{\mathbf{a}}_{{c_{D - 1} }} ] = [{\mathbf{b}}_{0}^{H} , \ldots ,{\mathbf{b}}_{N - 1}^{H} ]^{H} \) where \( {\mathbf{a}}_{k} = (1/\sqrt N )[1 ,e^{j2\pi k/N} , \ldots ,e^{j2\pi (N - 1)k/N} ]^{{\text{T}}} \) and \( {\mathbf{b}}_{n} = (1/\sqrt N )[e^{{j2\pi nc_{0} /N}} , \ldots ,e^{{j2\pi nc_{D - 1} /N}} ] \), and define \( \alpha_{g} = {\mathbf{u}}_{g}^{H} {\mathcal{T}}_{L + 1,N} ({\bar{\mathbf{H}}}) = [\alpha_{g} (0), \ldots ,\alpha_{g} (N - 1)] \) where \( {\varvec{\upalpha}}_{g} (n) \in C^{1 \times 2K} \). The homogeneous system \( {\mathbf{u}}_{g}^{H} {\mathcal{T}}_{L + 1,N} ({\bar{\mathbf{H}}})({\mathbf{F}}_{D}^{H} \otimes {\mathbf{I}}_{2K} ) = {\mathbf{0}} \) is simplified as:

$$ \begin{aligned} \alpha_{g} \left( {{\mathbf{F}}_{D}^{H} \otimes {\mathbf{I}}_{2K} } \right){ = } & \sum\limits_{n = 0}^{N - 1} {\alpha_{g} (n)({\mathbf{b}}_{n} \otimes {\mathbf{I}}_{2K} )} = \sum\limits_{n = 0}^{N - 1} {\left( {1/\sqrt N } \right)\left[ {e^{{j2\pi nc_{0} /N}} \alpha_{g} (n), \ldots ,e^{{j2\pi nc_{D - 1} /N}} \alpha_{g} (n)} \right]} \\ = & \left[ {{\mathbf{a}}_{{c_{0} }}^{{\text{T}}} \beta_{g} ,\ldots ,{\mathbf{a}}_{{c_{D - 1} }}^{{\text{T}}} \beta_{g} } \right] = {\mathbf{0}} \\ \end{aligned} $$
(42)

where \( {\varvec{\upbeta}}_{g} { = }\left( {\begin{array}{*{20}c} {{\varvec{\upalpha}}_{g} (0)} \\ \vdots \\ {{\varvec{\upalpha}}_{g} (N - 1)} \\ \end{array} } \right) \) is a N × P matrix. The homogeneous system in (42) is equivalent to:

$$ \left( {\begin{array}{*{20}c} {{\mathbf{a}}_{{c_{0} }}^{{\text{T}}} {\varvec{\upbeta}}_{g} } \\ \vdots \\ {{\mathbf{a}}_{{c_{D - 1} }}^{{\text{T}}} {\varvec{\upbeta}}_{g} } \\ \end{array} } \right) = {\mathbf{F}}_{D}^{*} {\varvec{\upbeta}}_{g} = {\mathbf{0}} $$
(43)

Partitioning \( {\mathbf{u}}_{g} = [{\mathbf{u}}_{g}^{{\text{T}}} (0), \ldots ,{\mathbf{u}}_{g}^{{\text{T}}} (P - 1)]^{\text{T}} \) and using \( {\varvec{\upalpha}}_{g} = {\mathbf{u}}_{g}^{H} {\mathcal{T}}_{L + 1,N} ({\bar{\mathbf{H}}}) = [{\varvec{\upalpha}}_{g} (0), \ldots ,{\varvec{\upalpha}}_{g} (N - 1)] \), we have \( {\varvec{\upalpha}}_{g} (n) = [{\mathbf{u}}_{g}^{H} (n), \ldots ,{\mathbf{u}}_{g}^{H} (n + L)]{\bar{\mathbf{H}}} \), and \( {\varvec{\upbeta}}_{g} \) is calculated by

$$ {\varvec{\upbeta}}_{g} { = }\left( {\begin{array}{*{20}c} {{\mathbf{u}}_{g} (0)} & \cdots & {{\mathbf{u}}_{g} (P - 1 - L)} \\ \vdots & \ddots & \vdots \\ {{\mathbf{u}}_{g} (L)} & \cdots & {{\mathbf{u}}_{g} (P - 1)} \\ \end{array} } \right)^{H} {\bar{\mathbf{H}}} = {\mathcal{H}}_{P,L + 1}^{H} ({\mathbf{u}}_{g} ){\bar{\mathbf{H}}} $$
(44)

We further reallocate the row order of \( {\bar{\mathbf{H}}} \) by

$$ {\bar{\mathbf{H}}} = \left( {\begin{array}{*{20}c} {{\mathbf{h}}_{1} (0)} & {{\mathbf{h}}_{2} (0)} \\ {{\mathbf{h}}_{2}^{*} (L)} & { - {\mathbf{h}}_{1}^{*} (L)} \\ {{\mathbf{h}}_{1} (1)} & {{\mathbf{h}}_{2} (1)} \\ {{\mathbf{h}}_{2}^{*} (L - 1)} & { - {\mathbf{h}}_{1}^{*} (L - 1)} \\ \vdots & \vdots \\ {{\mathbf{h}}_{1} (L)} & {{\mathbf{h}}_{2} (L)} \\ {{\mathbf{h}}_{2}^{*} (0)} & { - {\mathbf{h}}_{1}^{*} (0)} \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {{\mathbf{e}}_{1}^{{\text{T}}} \otimes {\mathbf{I}}_{J} } \\ {{\mathbf{e}}_{2L + 2}^{{\text{T}}} \otimes {\mathbf{I}}_{J} } \\ {{\mathbf{e}}_{2}^{{\text{T}}} \otimes {\mathbf{I}}_{J} } \\ {{\mathbf{e}}_{2L + 1}^{{\text{T}}} \otimes {\mathbf{I}}_{J} } \\ \vdots \\ {{\mathbf{e}}_{L + 1}^{{\text{T}}} \otimes {\mathbf{I}}_{J} } \\ {{\mathbf{e}}_{L + 2}^{{\text{T}}} \otimes {\mathbf{I}}_{J} } \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {{\mathbf{h}}_{1} (0)} & {{\mathbf{h}}_{2} (0)} \\ {{\mathbf{h}}_{1} (1)} & {{\mathbf{h}}_{2} (1)} \\ \vdots & \vdots \\ {{\mathbf{h}}_{1} (L)} & {{\mathbf{h}}_{2} (L)} \\ {{\mathbf{h}}_{2}^{*} (0)} & { - {\mathbf{h}}_{1}^{*} (0)} \\ \vdots & \vdots \\ {{\mathbf{h}}_{2}^{*} (L)} & { - {\mathbf{h}}_{1}^{*} (L)} \\ \end{array} } \right) = {\mathbf{E\bar{G}}} $$
(45)

Substituting (44) and (45) into (43), the homogeneous system becomes \( {\mathbf{F}}_{D}^{*} {\mathcal{H}}_{P,L + 1}^{H} ({\mathbf{u}}_{g}^{{}} ){\mathbf{E\bar{G}}} = {\mathbf{0}} \), which completes the proof of Lemma 1.\( \square \)

1.2 B. Proof of Lemma 2

We rewrite the mapping matrix P as \( {\mathbf{P}} = ({\mathbf{e}}_{{c_{0} }} , \ldots ,{\mathbf{e}}_{{c_{D - 1} }} ) \) where \( {\mathbf{e}}_{i} \) is the ith column vector of \( {\mathbf{I}}_{N} \). Then we have

$$ \begin{aligned} {\varvec{\Phi}}_{N} {\mathbf{P}} & = {\text{diag}} (\phi_{0} , \ldots ,\phi_{N - 1} )({\mathbf{e}}_{{c_{0} }} , \ldots ,{\mathbf{e}}_{{c_{D - 1} }} ) \\ & = (\phi_{{c_{0} }} {\mathbf{e}}_{{c_{0} }} , \ldots ,\phi_{{c_{D - 1} }} {\mathbf{e}}_{{c_{D - 1} }} ) = ({\mathbf{e}}_{{c_{0} }} , \ldots ,{\mathbf{e}}_{{c_{D - 1} }} )\text{diag} (\phi_{{c_{0} }} , \ldots ,\phi_{{c_{D - 1} }} ) \\ & = {\mathbf{P}}\varPsi_{D} \\ \end{aligned} $$
(46)

where \( \varPsi_{D} = {\text{diag}} (\phi_{{c_{0} }} , \ldots ,\phi_{{c_{D - 1} }} ) \).\( \square \)

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Yuan, Y., Zhang, B. & Yu, JL. Subspace Channel Identification for Multiuser MIMO STBC-OFDM Systems. Circuits Syst Signal Process 38, 4115–4140 (2019). https://doi.org/10.1007/s00034-019-01047-8

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