A variable step-size incremental LMS solution for low SNR applications

doi:10.1016/j.sigpro.2020.107730

Signal Processing

Volume 178, January 2021, 107730

https://doi.org/10.1016/j.sigpro.2020.107730 Get rights and content

Abstract

Many practical applications built around adaptive algorithms require solutions that work at low signal-to-noise ratio (SNR) values. In this situation, a variable step-size algorithm is a must to accommodate this scenario. A plethora of variable step algorithms exist in the literature for high SNR values. This paper presents a different variable step-size incremental least mean square strategy that provides excellent performance at low SNR values. A detailed theoretical analysis has been carried out for the proposed algorithm. Simulations have been performed for different scenarios to demonstrate the validity of the proposed strategy.

Introduction

In the literature, researchers have presented numerous distributed algorithms for estimation over wireless adaptive networks [1]. The first least mean square (LMS) algorithm based distributed algorithms were proposed in [2], [3], [4]. In situations where the LMS algorithm can not be used, for reasons like a time-varying scenario or highly correlated input, the variable-step version of the LMS algorithm is used instead. Until the work of [5], and later extended in [6], all the available studies used the fixed step-size strategy for the diffusion LMS algorithm [3]. Several other algorithms having a varying step-size strategy have since been presented [7], [8], [9], [10], [11], [12], [13].

Every algorithm has its advantages and disadvantages, however, nearly all the proposed algorithms use the diffusion scheme [5], [6], [7], [8], [9], [10], [11], [12]. The work in [5] used a variable step-size strategy with the incremental scheme. However, the first work to fully deal with a variable step-size incremental LMS (VSSILMS) was presented in [13]. This strategy, though, was well suited to high signal-to-noise ratio (SNR) scenarios. However, in a situation where there is a power failure, only then would a backup system be made available. Systems operated by this mechanism will not have enough power to fully sustain this situation. In this case, eventually they will operate in a low-power scenario and the SNR will be reduced consequently. The devised algorithms for a high SNR scenario will not be suitable for a low SNR environment.

This situation will also happen in a limited-power transmitter scenario or in a high interference region. In this case, constrained optimization problems must be devised for this purpose. Unless an exact characterization for the interference is obtained, it is difficult to consider such algorithms. The best way is to design low-power scenario algorithms.

This work then looks at remedying this situation and hence developing an algorithm for this purpose. The variable step-size strategy first introduced in [14] was shown to perform better in sparse environments. Due to the sparse nature of the setup, the SNR of the system is inherently lower than a non-sparse environment. Based on this reason and the better performance of the strategy in such a scenario, as shown in [14], we apply the said variable step-size strategy here in an incremental setup. More specifically, we make the following contributions:

1.
It is shown through simulations that the proposed incremental algorithm, using a variable step-size, provides a better steady-state error compared with the algorithm from [13] for low SNR scenarios.
2.
Complete theoretical analysis has been carried out for the proposed algorithm. This includes a bound on the expectation on the step-size as well as the mean-square analysis, which includes the learning behavior during the transient phase as well as the steady-state analysis.

The rest of the paper is divided as follows. The system model is presented in Section 2, followed by a brief description of the Incremental LMS (ILMS) and VSSILMS algorithms. The proposed strategy for varying the step-size in low SNR scenarios is presented in Section 3. Section 4 presents the theoretical analysis, which is followed by experimental results in Section 5. The work concludes in Section 6.

Section snippets

System model

A geographical area with N wireless sensors, connected through a Hamiltonian cycle, as presented in Fig. 1, is considered. The desired parameters of interest are collectively modeled using a vector, $w^{o} \in R^{M}$ . The input data for any node k ∈ N at a certain time instant, i, is modeled as a regressor row vector, $u_{k} (i) \in R^{M}$ . The output at node k is, thus, a scalar value, d_k(i), corrupted by additive noise, such that $d_{k} (i) = u_{k} (i) w^{o} + v_{k} (i),$ with v_k(i) being an additive noise with a zero mean.

The fixed

Proposed algorithm

The aim of this work is to present a variable step-size strategy using the incremental scheme for scenarios with low SNR. The step-size update equation proposed in this work is inspired from the work in [14] and is governed by the following recursion: $μ_{k} (i + 1) = α μ_{k} (i) + γ | e_{k} (i) |,$ where γ and α are positive control parameters as defined above. Note that recursion (5) was for the first time devised in [14].

Inserting (5) in the update of the step-size of the VSSILMS yields the proposed incremental

Theoretical analysis

In this section, various analyses are carried out to assess the behaviour of the proposed algorithm. We begin by introducing the weight-error vector, ${\tilde{w}}_{k} (i),$ given by ${\tilde{w}}_{k} (i) = w^{o} - w_{k} (i) .$

Inserting (6) in (4) and solving gives ${\tilde{w}}_{k} (i) = [I_{M} - μ_{k} (i) u_{k}^{T} (i) u_{k} (i)] {\tilde{w}}_{k - 1} (i) - μ_{k} (i) u_{k}^{T} (i) v_{k} (i) .$

Results and discussion

Experimental results are reported in this section for the proposed algorithm. The testing is done under four different scenarios. Firstly, a performance comparison is presented between the proposed algorithm and the algorithm from [13]. This comparison is done because the computational complexity for the two algorithms is similar. However, at low SNR values, the proposed algorithm is shown to perform better. In the next simulation, the transient theoretical analysis results using (21) are

Conclusion

This work proposes another variable step-size scheme for the ILMS algorithm for distributed estimation in low signal-to-noise ratio (SNR) scenarios. The proposed algorithm has been studied in comparison with the VSSILMS algorithm from [13]. The complexity of the two algorithms is similar but the proposed algorithm is shown to outperform the VSSILMS algorithm from [13] when the SNR is low. A detailed theoretical analysis for our algorithm has been presented. A comparative study between

Declaration of Competing Interest

None.

Acknowledgment

The authors acknowledge the support provided by the Deanship of Scientific Research at KFUPM under Research Grant SB181001.

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Short communicationA variable step-size incremental LMS solution for low SNR applications

Abstract

Introduction

Section snippets

System model

Proposed algorithm

Theoretical analysis

Results and discussion

Conclusion

Declaration of Competing Interest

Acknowledgment

Adaptive networks

Proc. IEEE

Incremental adaptive strategies over distributed networks

IEEE Trans. Signal Process.

Diffusion least-mean squares over adaptive networks: Formulation and performance analysis

IEEE Trans. Signal Process.

Distributed LMS for consensus-based in-network adaptive processing

IEEE Trans. Signal Process.

Variable step-size least mean square algorithms over adaptive networks

Proceedings of the 10th International Conference on Information Science, Signal Processing and their Applications (ISSPA 2010)

A variable step-size strategy for distributed estimation over adaptive networks

EURASIP J. Adv. Sig. Process.

A new variable step-size strategy for adaptive networks

Proceedings of the 2011 Conference Record of the Forty Fifth Asilomar Conference on Signals, Systems and Computers (ASILOMAR)

A noise-constrained algorithm for estimation over distributed networks

Int. J. Adap. Cont. Signal Process.

Optimal variable step-size diffusion LMS algorithms

Proceedings of the IEEE Workshop on Statistical Signal Processing (SSP)

Short communication
A variable step-size incremental LMS solution for low SNR applications