Training sequence based frequency-domain channel estimation for indoor diffuse wireless optical communications
- Jun-Bo Wang^{1, 2}Email author,
- Xiu-Xiu Xie^{1},
- Yuan Jiao^{1} and
- Ming Chen^{3}
https://doi.org/10.1186/1687-1499-2012-326
© Wang et al; licensee Springer. 2012
Received: 2 September 2011
Accepted: 23 September 2012
Published: 30 October 2012
Abstract
Channel estimation is a key technology for wireless optical communication (WOC) systems. Based on the training sequence (TS), this article develops three frequency domain (FD) channel estimation approaches for diffuse wireless optical channels. Considering the propagation property of light and the characteristics of optical modulation, this article establishes a link model for the indoor WOC systems. Using the established link model, three FD channel estimation methods, such as the LS method, the minimum mean square error (MMSE) method and the ORL-MMSE method are proposed. The computational complexity analysis gives time complexities of the three channel estimation methods in FD. This article also evaluates the performance of the proposed three methods by computer simulation, measuring in terms of MSE and BER, respectively. Further more, based on the Cramer–Rao Bound theorem, optimal TSs are found and tabulated for different channel responses and TS lengths. By using the optimal TSs, channel estimation errors measured by BER performance are also investigated.
Keywords
1 Introduction
Motivated by the research[7–15], this article focuses on the TS-based channel estimation methods in FD for WOC systems. Considering the propagation property of light and the characteristics of optical modulation, this article establishes the mathematical channel model for indoor WOC systems. Further, based on the established link model, three FD channel estimation methods, such as the LS method, the minimum mean square error (MMSE) method and the modified MMSE method which can be called optimal low-rank MMSE (OLR-MMSE) are proposed. In addition, this article compares the complexity of the three methods. Furthermore, the performance of the proposed three methods are studied and compared, measuring in terms of Mean square error (MSE) and bit error ratio (BER), respectively. Based on the Cramer–Rao Bound (CRB) theorem, optimal TSs are found and tabulated for different channel responses and TS lengths. By using the optimal TSs, channel estimation errors measured by BER performance are also investigated.
The rest of this article is organized as follows: the mathematical model for WOC systems is described in Section 2. Three channel estimation methods in FD for WOC channels using a know sequence are proposed in Section 3. Section 4. analyzes the computational complexity of the proposed FD channel estimation algorithms. The simulation results of performance of the proposed three channel estimation methods are provided in Section 5. Based on the CRB theorem, Section 6. investigates a valid performance metric for determining optimal TSs. In Section 7., optimal TSs are obtained by enumerative search and the simulation results of BER performance are shown. Conclusions are drawn in Section 8.
2 Mathematical model in frequency domain (FD) for diffuse WOCs
where M = N + J − 1 and {n_{ i }} denotes the channel noise.
where H = [H_{0},H_{1},…,H_{M−1}]^{T},N = [N_{0},N_{1},…,N_{M−1}]^{T} is the Gaussian noise with variance σ^{2}.
3 TS based FD channel estimation method
In data aided channel estimation, the known information to the receiver is inserted in information symbols so that the current channel state can be estimated. By analyzing the relationship between the known TS and the received symbols, the instantaneous channel impulse response can be estimated[16].
3.1 Least-squares (LS) estimation
where H is the objective of the channel estimation,${\stackrel{\u0304}{\mathbf{A}}}^{-1}\mathbf{N}$ denotes the estimated error. It can be observed that, the estimated channel impulse response is free from the interference between taps. In other words, LS channel estimation method is independent of the correlation properties of TS. However, the estimation error variance is largely depending on the channel noise.
3.2 MMSE estimation
From the Equation (17), the MMSE approach can alleviate the effect of channel noise in some degree when compared with the LS approach. However, the MMSE channel estimation requires prior knowledge of the channel coefficient covariance matrix R_{H} and noise covariance matrix R_{H}, which means that the MMSE method is more complicated than the LS approach.
3.3 OLR-MMSE estimation
The MMSE estimator, as described in (17), is of considerable complexity since a matrix inversion is needed every time the data in$\stackrel{\u0304}{\mathbf{A}}$ changes. We can reduce the complexity of this estimator by averaging over the transmitted data[15].
where β = E{|A_{ k }|^{2}} E{|1/A_{ k }|^{2}}. To further reduce the complexity of the estimator, we proceed with the OLR approximations below.
where U is a unitary matrix containing the singular vectors and ⋀ is a diagonal matrix containing the singular values λ_{1} ≥ λ_{2} ≥ ⋯ ≥ λ_{ M } on its diagonal. The SVD also dramatically reduces the calculation complexity of matrices.
The OLR-MMSE estimator can be interpreted as first projecting the LS estimates onto a subspace and then performing the estimation. Because the subspace has a small dimension (as small as J + 1) and still describes the channel well, the complexity of OLR-MMSE estimator is much lower than MMSE estimator with a good performance.
4 Computational complexity analysis
Even when an algorithm is computationally feasible in principle, it may not be realizable in practice if the algorithm requires an inordinate amount of resources (such as running time, storage space, and communication bandwidth) to obtain the solution. Thus, the computational complexity of an algorithm is an important factor for the realization. Moreover, running time is of primary concern in most cases. Generally, the running time needed by an algorithm expressed as a function of the size of the input parameters is called the time complexity of the algorithm[19]. In this section, the computational complexities of the proposed three FD channel estimation algorithms are analyzed in terms of time complexity.
To avoid the effect of running speed, the running time of an algorithm is usually measured by the number of operations needed by the algorithm to obtain the solution. However, in most cases, it is very difficult to determine the exact number of operations of an algorithm. Therefore, using the big-O notation, the time complexity of an algorithm is usually expressed in an asymptotic upper bound on the number of operations[20].
Complexity comparison
Channel estimation method | Multiplication | Addition |
---|---|---|
FD-LS | O(M) | O(1) |
FD-MMSE | O(M^{3}) | O(M^{3}) |
FD-OLR-MMSE | O(r M^{2}) | O(r M^{2}) |
5 Quality measure of FD channel estimate
To evaluate the performance of the proposed FD channel estimation schemes, the proposed three methods are studied and compared by computer simulation. Here, two comparisons are considered. Firstly, the proposed three methods are compared in terms of MSE, which denotes the accuracy of the channel estimation. Secondly, the three methods are compared, measuring by BER, which denotes the system performance influenced by channel estimation.
where h and$\widehat{\mathbf{h}}$ are the IDFT of H and$\widehat{\mathbf{H}}$, respectively. Without lose of generality, we assume that the length of TS used in channel estimation N = 26 and the number of channel taps J = 3.
6 TS optimization
The CRB provides a lower bound on the statistical variance of any unbiased estimator, which means that the lower bound on CRB is the minimum variance for an estimator can obtain. The CRB for random channels has been wildly used as a performance measure for the design of optimal TSs[17, 18]. In this section, we first derive a closed-form expression for the CRB for the estimation of H.
where x^{H} is the Hermitian transposition of x, p(Y,H) is the joint probability density function of received signal sequences and channel coefficients, the derivatives are evaluated at the true value of H and the expectation is taken with respect to p(Y,H).
Based on (35) optimal binary sequences are found by exhaustive computer search. The resulting sequences offer the best performance of error covariance at the output of the channel estimator.
Complexity comparison
Optimal TS search method | Multiplication | Addition |
---|---|---|
FD method | O(M log_{2}M) | O(M log_{2}M) |
TD method | O((M + 1)J^{2}) | O((M + 1)J^{2}) |
7 Enumerative computer search for optimal TSs
As the placement of N clusters of pilot symbols is shown in Figure2, a computer search has been performed to find the FD optimal TSs satisfying the Equation (35) for a given number of channel order J and TS length N.
Hexadecimal value of optimal TS
Channel | TS length (N) | ||||
---|---|---|---|---|---|
orderJ | 2J | 2J+ 1 | 2J+ 2 | 2J+ 3 | 2J+ 4 |
J = 2 | F | 13 | 27 | 3D | BC |
J = 3 | 3D | 5E | 97 | 0B7 | 35F |
J = 4 | ED | 08B | 20B | 3DD | F59 |
J = 5 | 3DD | 765 | DAB | 1677 | 2A6F |
J = 6 | EB7 | 0F35 | 36A3 | 6D77 | B3AF |
J = 7 | 32A6 | 2B67 | F36A | 15B67 | 23597 |
To evaluate the performance, it is desired to compare our proposed optimal TSs in FD with the optimal TSs in TD for FD-LS channel estimation in terms of the BER via computer simulations, respectively. Although no TSs in TD have been proposed in published articles for WOC system, in our former research, we have investigated the optimal TSs for TD channel estimation for WOC systems[14].
8 Conclusions
This article investigates the TS-based channel estimation in FD for indoor WOC systems. Firstly, the mathematical model is established for indoor WOC systems. Based on the established link model, three channel estimation methods, such as the LS method, the MMSE method and the ORL-MMSE method are proposed. Moreover, this article evaluates the performance of the proposed three methods by computer simulation, measuring in terms of MSE and BER, respectively. The mathematical analysis and the simulation results show that the LS estimator performs the worst of all methods and it has a lowest computational complexity, however. The MMSE estimator yields the best performance with its relatively higher complexity. The OLR-MMSE estimator gives the best trade off between performance and complexity. Based on the CRB theorem, a performance measure has been proposed to assess the quality of pilot symbols for FD channel estimation. Then, optimal TSs are found by a computer search. It has been demonstrated that the proposed method for searching the optimal TSs based on FD techniques invites a lower complexity search strategy compared with TD techniques. It should be noted that the optimal TSs in FD have a better BER performance compared to the TSs in TD. The minimum value is obtained by the channel estimation method with a lower channel taps when the TS length hold fixed. Moreover, with the number of channel taps held fixed, the longer the TS length is, the better the channel estimation performance can be obtained.
Declarations
Acknowledgements
This work is supported by National Nature Science Foundation of China (No. 61102068, No.61172077 and No. 60972023), China Postdoctoral Science Foundation (No. 20110490389), Research Fund of National Mobile Communications Research Laboratory, Southeast University (No. 2012A06), the Research Fund for the Doctoral Program of Higher Education (No. 20113218120017), the open research fund of the State Key Laboratory of Integrated Services Networks, Xidian University (ISN12-11), NUAA Research Funding (NS2011013) and the Fundation of Graduate Innovation Center in NUAA (No.kfjj20110213 and No.kfjj20110129).
Authors’ Affiliations
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