SCIENCE CHINA Information Sciences, Volume 61, Issue 2: 022304(2018) https://doi.org/10.1007/s11432-016-9053-3

Partial intersection sphere decoding with weighted voting for sparse rotated V-OFDM systems

• AcceptedFeb 15, 2017
• PublishedAug 24, 2017
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Abstract

Vector orthogonal frequency division multiplexing (V-OFDM) is a general system that builds a bridge between OFDM and single-carrier frequency domain equalization in terms of intersymbol interference level and receiver complexity. In this paper, we focus on a rotated V-OFDM system over a broadband sparse channel owing to its large time delay spread and fewer nonzero taps. In order to collect the multipath diversity, a simple rotation matrix is designed, independent of the number of subchannels at the transmitter. For the rotated V-OFDM receiver, a partial intersection sphere decoding with weighted voting method is proposed by exploiting the sparse nature of the multipath channel. The proposed receiver chooses the transmitted vector from the set with the maximum likelihood estimation generated using the partial intersection sphere decoding method. For an extreme case, such as when a candidate set is empty, which usually occurs at a low signal-to-noise ratio (SNR), an efficient weighted voting system is used to estimate the transmitted vectors symbol-by-symbol. Simulation results indicate that the proposed receiver improves the symbol error rate performance with reduced complexity, especially for low SNR scenarios.

Acknowledgment

This work was supported by National Natural Science Foundation of China (Grant No. 61401194), China Scholarship Council (Grant No. 201406190083), and Program B for Outstanding Ph.D. Candidate of Nanjing University (Grant No. 201501B013).

References

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• Figure 1

The block diagram of a rotated V-OFDM modulation system.

• Figure 2

(Color online) Example of the updating process of the partial intersection sphere decoding method.

• Figure 3

(Color online) Example of the weighted voting system for strong noise scenarios.

• Figure 4

(Color online) Comparison of probability mass function of $\kappa$ with $K=16$ and $M=8,16,32,64$, respectively.

• Figure 5

(Color online) Comparison of probability mass function of $\kappa$ with $M=16$ and $K=2,4,8,16$, respectively.

• Figure 6

Time flow of PIS-WV decoding for rotated V-OFDM system. The $l$th subchannel is taken as an example.

• Figure 7

(Color online) Comparison of V-OFDM system with different rotation angles.

• Figure 8

(Color online) Diversity orders of PIS-WV decoding for power delay profile.

• Figure 9

(Color online) Symbol error rates of different receivers over sparse rotated V-OFDM system.

• Figure 10

(Color online) Decoding complexities of different receivers over sparse rotated V-OFDM system.

•

Algorithm 1 Partial intersection sphere decoding with weighted voting

Require:$\boldsymbol~Y,~\boldsymbol~h,~L,~M,~P,~r$

Output:$\widehat{\boldsymbol~X}$

Initialization: $\mathcal~U^0\gets\varnothing,~\mathcal~X^0\gets\varnothing,~\boldsymbol\beta\gets\boldsymbol0$;

for $l\gets0,1,\ldots,L-1$

for $m\gets0,1,\ldots,M-1$

$Y_{lM+m}$ is the $m$th entry of column vector $\boldsymbol~Y_l$;

$\boldsymbol{\mathcal{H}}_l^m$ is the $1\times\kappa$ vector aligned as the $m$th row and the $(m-i_0)\boldsymbolod~M$th, $(m-i_1)\boldsymbolod~M$th$,\ldots,(m-i_{\kappa-1})\boldsymbolod~M$th columns of $\boldsymbol{\mathcal{H}}_l$;

Generate a set of $\kappa$-dimensional symbol sequences: $\mathcal~S^m\gets\{\boldsymbol~S\Big|\boldsymbol~S\in\mathbb~X^{\kappa},~\big|Y_{lM+m}-\boldsymbol{\mathcal{H}}_l^m\boldsymbol~S\big|\leqslant~r\}$;

Generate a set of current coordinates of nonzero entries: $\mathcal~V^m\gets\left\{v\big|\forall~i\in\mathcal~I,~v\gets(m-i)\boldsymbolod~M\right\}$;

Generate a intersection of the existing and current nonzero coordinates sets: $\mathcal~W^m\gets\mathcal~U^m\bigcap\mathcal~V^m$;

Construct an injective mapping of coordinates: $f:~\ell\rightarrow(m-i_\ell)\boldsymbolod~M,~\ell\in\{0,1,\dots,\kappa-1\}$;

for all $\boldsymbol~S\in\mathcal~S^m$

Generate a set of symbol sequences for a given current $\boldsymbol~S\in\mathcal~S^m$:$\mathcal~X_{\boldsymbol~S}^{m+1}\gets\big\{\boldsymbol~X^{m+1}\big|\boldsymbol~X^m\in\mathcal~X^m,~\forall~w\in\mathcal~W^m,~X_w^{m+1}=S_{f^{-1}(w)};~\forall~\nu\in\complement_{\mathcal~V^m}\mathcal~W^m,~X_\nu^{m+1}\gets~S_{f^{-1}(\nu)}\big\}$;

The candidate receives one vote: $\beta_{lM+v}^m(S_{f^{-1}(v)})\gets\beta_{lM+v}^m(S_{f^{-1}(v)})+1,~v\in\mathcal~V^m$;

end for

Generate a set of entire symbol sequences for all $\boldsymbol~S\in\mathcal~S^m$: $\mathcal~X^{m+1}\gets\bigcup_{\boldsymbol~S\in\mathcal~S^m}\mathcal~X_{\boldsymbol~S}^{m+1}$;

Update the existing nonzero coordinates set: $\mathcal~U^{m+1}\gets\mathcal~U^m\bigcup\mathcal~V^m$;

end for

if $\mathcal~X^M=\varnothing$ then

Weighted voting decision symbol-by-symbol: $\widehat~X_{lM+k}\gets\mathop{\arg\max}_{X\in\mathbb~X}\sum_{m=0}^{M-1}\frac{\beta_{lM+k}^m(X)}{\left|\mathcal~S^m\right|},~k\in\{0,1,\ldots,\linebreak~M-1\}$;

else

Maximum likelihood estimation from candidate set: $\widehat{\boldsymbol~X}_l\gets\mathop{\arg\min}_{\boldsymbol~X_l^M\in\mathcal~X^M}\big\|\boldsymbol~Y_l-\boldsymbol{\mathcal{H}}_l\boldsymbol~X_l^M\big\|_2$;

end if

end for

return $\widehat{\boldsymbol~X}$.

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