Variable-step-size based sparse adaptive filtering algorithm for channel estimation in broadband wireless communication systems
- Guan Gui^{1}Email author,
- Wei Peng^{2},
- Li Xu^{1},
- Beiyi Liu^{1} and
- Fumiyuki Adachi^{3}
https://doi.org/10.1186/1687-1499-2014-195
© Gui et al.; licensee Springer. 2014
Received: 17 December 2013
Accepted: 6 November 2014
Published: 22 November 2014
Abstract
Sparse channels exist in many broadband wireless communication systems. To exploit the channel sparsity, invariable step-size zero-attracting normalized least mean square (ISS-ZA-NLMS) algorithm was applied in adaptive sparse channel estimation (ASCE). However, ISS-ZA-NLMS cannot achieve a good trade-off between the convergence rate, the computational cost, and the performance. In this paper, we propose a variable step-size ZA-NLMS (VSS-ZA-NLMS) algorithm to improve the ASCE. The performance of the proposed method is theoretically analyzed and verified by numerical simulations in terms of mean square deviation (MSD) and bit error rate (BER) metrics.
Keywords
1 Introduction
Channel structures in different mobile communication systems
Generations of mobile communication systems | 2G cellular (IS-95) | 3G cellular (WCDMA) | 4G/5G cellular (LTE-Advanced~) |
---|---|---|---|
Transmission bandwidth | 1.23 MHz | 10 MHz | 20 ~ 100 MHz |
Time delay spread (assume) | 0.4 μs | 0.4 μs | 0.4 μs |
Sampling channel length | 1 | 8 | 16 ~ 80 |
Number of nonzero taps | 1 | 4 | 6 |
Channel model | Dense | Approximate sparse | Sparse |
It is well known that step size is a critical parameter which determines the estimation performance, convergence rate, and computational cost. However, ISS-NLMS and ZA-ISS-NLMS adopt a fixed step size, and as a result, they are unable to achieve a good balance between steady-state estimation performance and convergence speed. Different from ISS-NLMS [4], variable step-size NLMS (VSS-NLMS) was first proposed to improve the estimation performance [11] without sacrificing the convergence speed. Variable step size is controlled by the instantaneous square error of each iteration, i.e., lower error will decrease the step size and vice versa. To the best of our knowledge, the application of sparse VSS-NLMS to simultaneously exploit the channel sparsity and control the step size has not been reported in the literature.
In this paper, we propose a zero-attracting VSS-NLMS (ZA-VSS-NLMS) algorithm for sparse channel estimation. The main contribution of this paper is to propose the ZA-VSS-NLMS using VSS rather than ISS for estimating spare channels. In addition, the step size of the proposed algorithm is updated in each iteration according to the error information. In the following, conventional ZA-ISS-NLMS is introduced and its drawback is analyzed at first. ZA-VSS-NLMS is then proposed using an adaptive step size to achieve a lower steady-state estimation error. To derive the adaptive step size, different from the traditional VSS-NLMS algorithm in [11], two practical problems are considered: sparse channel model and tractable independent assumptions [12]. At last, numerical simulations are carried out to evaluate the proposed algorithm in terms of two metrics: mean square deviation (MSD) and bit error rate (BER).
The remainder of this paper is organized as follows. A system model is described and ZA-ISS-NLMS algorithm is introduced in Section 2. In Section 3, ZA-VSS-NLMS algorithm is proposed. Numerical results are presented in Section 4 to evaluate the performance of the proposed ASCE method. Finally, we conclude the paper in Section 5.
2 ZA-ISS-NLMS algorithm
3 Proposed algorithm
where μ(n +1) is the VSS which is calculated from the estimation error and the variance of the additive noise. Comparing Equation 7 with Equation 4, it can be found that the step size is different, i.e., step size in Equation 4 is invariant while step size in Equation 7 is adaptively variant. There are two facts about μ(n) and ρ that should be noticed: 1) the variant step-size μ(n) is adopted to speed up the convergence speed in the case of large estimation error, while to ensure the stability in the case of small estimation error; 2) the parameter ρ, which depends on the initial step-size μ and regularization parameter λ, is utilized to exploit channel sparsity effectively. Otherwise, variant parameter ρ(n) = μ(n)λ may cause extracomputational complexity and ineffectiveness use of the channel sparsity.
- (A1):
Input vector x(t) and the additive noise z(t) are mutually independent at time t.
- (A2):
Input vector x(t) is a stationary sequence of independent zero mean Gaussian random variables with a finite variance${\mathrm{s}}_{\mathrm{x}}^{2}\text{.}$
- (A3):
z(t) is an independent zero mean random variables with variance${\mathrm{s}}_{\mathrm{z}}^{2}\text{.}$
- (A4):
$\tilde{\mathbf{h}}\left(n\right)$is independent of x(t).
4 Numerical simulations
Simulation parameters
Parameters | Values |
---|---|
Transmission bandwidth | W =40 MHz |
Delay spread | τ =0.8 μs |
Channel length | N =64 |
Number of nonzero coefficients | K =2 and 6 |
Distribution of nonzero coefficient | random Gaussian $\mathcal{C}\mathcal{N}\left(0,1\right)$ |
Threshold parameter for VSS-NLMS | $C=\left\{\begin{array}{c}\hfill 6.0\times {10}^{-6},\phantom{\rule{0.12em}{0ex}}\mathrm{\text{for}}\phantom{\rule{0.12em}{0ex}}\mathrm{\text{SNR}}=5\phantom{\rule{0.25em}{0ex}}\mathrm{dB}\hfill \\ \hfill 3.0\times {10}^{-6},\phantom{\rule{0.12em}{0ex}}\mathrm{\text{for}}\phantom{\rule{0.12em}{0ex}}\mathrm{\text{SNR}}=10\phantom{\rule{0.25em}{0ex}}\mathrm{dB}\hfill \\ \hfill 2.0\times {10}^{-6},\phantom{\rule{0.12em}{0ex}}\mathrm{\text{for}}\phantom{\rule{0.12em}{0ex}}\mathrm{\text{SNR}}=20\phantom{\rule{0.25em}{0ex}}\mathrm{dB}\hfill \end{array}\right.$ |
Received SNR E_{ s }/N_{0} | 0 ~ 40 dB |
Step size | μ = 0.5 and μ_{max} = 2 |
Regularization parameter | $\rho =0.0015\phantom{\rule{0.25em}{0ex}}{\sigma}_{n}^{2}$ |
Modulation schemes | 8 PSK, 16 PSK, 16 QAM, and 64 QAM |
Therefore, it has been confirmed that the proposed algorithm can achieve the advantages of good performance and fast convergence speed.
5 Conclusions
Step size is a key parameter for NLMS-based adaptive filtering algorithms to balance the steady-state estimation performance and convergence speed. Either ISS-NLMS or ZA-ISS-NLMS cannot update their step size in the process of adaptive error updating. In this paper, a ZA-VSS-NLMS filtering algorithm was proposed for channel estimation. Unlike the traditional algorithms, the proposed algorithm utilizes VSS which can update the step size adaptively according to the updating error. Therefore, the proposed method can achieve a better steady-state performance while keeping a comparable convergence speed when compared with the existing methods. Simulation results have been presented to confirm the effectiveness of the proposed method in terms of MSD and BER metrics.
Declarations
Acknowledgements
The authors would like to extend their appreciation to the anonymous reviewers for their constructive comments. This work was supported in part by the Japan Society for the Promotion of Science (JSPS) research activity start-up research grant (No. 26889050), Akita Prefectural University start-up research grant, as well as the National Natural Science Foundation of China under grants (Nos. 61401069, 61261048, and 61201273).
Authors’ Affiliations
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