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

- Muhammad O Bin Saeed
^{1}, - Azzedine Zerguine
^{1}Email author and - Salam A Zummo
^{1}

**2013**:135

https://doi.org/10.1186/1687-6180-2013-135

© Bin Saeed et al.; licensee Springer. 2013

**Received: **15 July 2013

**Accepted: **26 July 2013

**Published: **6 August 2013

## Abstract

A lot of work has been done recently to develop algorithms that utilize the distributed structure of an *ad hoc* wireless sensor network to estimate a certain parameter of interest. One such algorithm is called diffusion least-mean squares (DLMS). This algorithm estimates the parameter of interest using the cooperation between neighboring sensors within the network. The present work proposes an improvement on the DLMS algorithm by using a variable step-size LMS (VSSLMS) algorithm. In this work, first, the well-known variants of VSSLMS algorithms are compared with each other in order to select the most suitable algorithm which provides the best trade-off between performance and complexity. Second, the detailed convergence and steady-state analyses of the selected VSSLMS algorithm are performed. Finally, extensive simulations are carried out to test the robustness of the proposed algorithm under different scenarios. Moreover, the simulation results are found to corroborate the theoretical findings very well.

## Keywords

## 1 Introduction

*ad-hoc*wireless sensor networks has created renewed interest in distributed computing and opened up new avenues for research in the areas of estimation and tracking of parameters of interest where a robust, scalable, and low-cost solution is required. To illustrate this point clearly, consider a set of

*N*sensor nodes spread over a wide geographic area, as shown in Figure1. Each node takes sensor measurements at every sampling instant. The goal here is to estimate a certain unknown parameter of interest using these sensed measurements. In a centralized network, all nodes transmit their readings to a fusion center where the processing takes place. However, this system is prone to any failure of the fusion center. Furthermore, large amounts of energy and communication resources may be required for the complete signal processing task between the center and the entire network to be successfully carried out. These resources needed could become considerable, as the distance between the nodes and the center increases[1].

An *ad hoc* network, on the other hand, depends on distributed processing with the nodes communicating only with their neighbors and the processing taking place at the nodes themselves. As no hierarchical structure is involved, any node failure would not result in the failure of the entire network. Extensive research has been done in the field of consensus-based distributed signal processing and resulted in a variety of algorithms[2].

In this work, a brief overview of the virtues and limitations of these algorithms[3–10] is conducted, thus providing the background against which our contribution is justified. Incremental algorithms organize the network in a Hamiltonian cycle[3]. The estimation is completed by passing the estimate from node to node, improving the accuracy of the estimate with each new set of data per node. This algorithm is termed the incremental least mean squares (ILMS) algorithm as it uses the LMS algorithm[11] with incremental steps within each iteration[3]. A general incremental stochastic gradient descent algorithm was developed in[4], for which[3] can be considered a special case. These algorithms are heavily dependent on the Hamiltonian cycle and in case of a node failure, a new cycle has to be initiated. The problem of reconstructing the cycle is non-deterministic polynomial-time hard[12]. A quantized version of the ILMS algorithm was suggested in[5]. A fully distributed algorithm was proposed in[6], called diffusion least mean squares (DLMS) algorithm. In the DLMS algorithm, neighbor nodes share their estimates in order to improve the overall performance; see also[7]. Ultimately, this algorithm is robust to node failure as the network is not dependent on any single node. On the other hand, the authors in[8] introduced a scheme to adapt the combination weights at each iteration for each node instead of having as fixed weights for the shared data. In this case, the performance is improved if more weight is given to the estimates of the neighbor nodes that are providing more improvement per iteration.

All the previously mentioned algorithms are generally based on a non-constrained optimization technique, except in[8] which uses constrained optimization to adapt the weights only. However, the authors in[9] use the constraint that all nodes converge to the same steady-state estimate to derive the distributed LMS algorithm. This algorithm is unfortunately hierarchical, thus making it complex and not completely distributed. To remedy this situation, a fully distributed algorithm based on the same constraint was suggested in[10]. The algorithms in[9, 10] have been shown to be robust to inter-sensor noise, a property not possessed by the diffusion-based algorithms. However, it has been shown in[7] that diffusion-based algorithms perform better in general.

All the above-mentioned algorithms use a fixed step-size. In general, the step-size is kept the same for all nodes. As a result, the nodes with better signal-to-noise ratio (SNR) may converge quicker and provide a reasonably good performance. However, the nodes with poor SNR will not provide similar performance despite cooperation from neighboring nodes. As a result, a distributed algorithm may provide improvement in average performance but individually, some nodes will still not be performing similarly to the other nodes. One solution for this problem was provided by[8]. The work in[8] provides a computationally complex method to improve the performance of the DLMS algorithm. For every iteration, an error correlation matrix is formed for each node. A decision is made based on this decision as to the weights of the neighbors. Thus, the combiner weights are adapted at every iteration according to the performance of the neighbors of each node. Simulation results from[8] have shown slight improvement in the performance, but this has been achieved at the cost of high computational complexity.

In comparison, a much simpler solution was suggested in[13], using a variable step-size LMS algorithm. This resulted in the variable step-size diffusion LMS (VSSDLMS) algorithm. Preliminary results showed remarkable improvement in performance at a reasonably low computational cost. The idea is to vary the step-size at each iteration based on the error performance. Each node will alter its step-size according to its own individual performance. As a result, not only the average performance improves remarkably but the individual performances of the nodes also get better.

A different diffusion-based algorithm was proposed in[14] using the recursive least squares (RLS) algorithm to obtain the diffusion RLS (DRLS) algorithm. This DRLS algorithm provided exceptional results in both speed and performance. Another RLS-based distributed estimation algorithm has been studied in[15, 16]. The latter algorithm is hierarchical in nature, which makes its complexity higher than that of the DRLS algorithm. The RLS algorithm is inherently far more complex compared with the LMS algorithm. In this work, it is shown that despite the LMS algorithm being inferior to the RLS algorithm, using a variable step-size allows the LMS algorithm to achieve performance very close to that of the RLS algorithm.

In order to achieve better performance, various other algorithms have been proposed in the literature, such as in[17–19]. The works in[17, 18] propose distributed Kalman filtering algorithms that provide efficient solutions for several applications. A survey of distributed particle filtering is provided in[19]. This work takes a look at several solutions proposed for nonlinear distributed estimation. However, the focus of this work is primarily on LMS-based algorithms, so these algorithms will not be included in any further discussion.

Our work extends that of[13] and investigates in detail the performance of the VSSDLMS algorithm. Here, the most popular variable step-size LMS (VSSLMS) algorithms are first investigated and compared with each other. Based on the best complexity-performance trade-off, one variant of the VSSLMS algorithms is chosen for the proposed algorithm. Next, the incorporation of the selected VSSLMS algorithm in the diffusion scheme is described, and complete convergence and steady-state analyses are carried out. The stability of the algorithm is also analyzed. Finally, some simulations are carried out to first determine which of the various selected VSSLMS algorithms provide the best trade-off between performance and complexity, and then to compare the proposed algorithm with similar existing algorithms. Note that the performance of the proposed algorithm is assessed under different conditions. Interestingly, the theoretical findings are found to corroborate the simulation results very well. Moreover, a sensitivity analysis is performed on the proposed algorithm.

The paper is organized as follows. Section 2 describes the problem statement and briefly introduces the DLMS algorithm. Section 3 incorporates the VSSLMS algorithm into the DLMS algorithm, and then complete convergence and stability analyses are carried out. Simulation results are given in Section 4 followed by a thorough discussion of the results. Finally, Section 5 concludes the work.

*Notation.* Boldface letters are used for vectors/matrices and normal font for scalar quantities. Matrices are defined by capital letters and small letters are used for vectors. The notation (.)^{T} stands for transposition operation for vectors and matrices and expectation operation is denoted by *E*[.]. Any other mathematical operators that have been used will be defined as they are introduced in the paper.

## 2 Problem statement

*N*sensor nodes deployed over a geographical area for estimating an unknown parameter vector

**w**

^{o}of size (

*M*× 1), as shown in Figure1. Each node

*k*has access to a time realization of a known regressor vector

**u**

_{ k }(

*i*) of size (1 ×

*M*) and a scalar measurement

*d*

_{ k }(

*i*) that are related by

where *v*
_{
k
}(*i*) is a spatially uncorrelated zero-mean additive white Gaussian noise with variance${\sigma}_{{v}_{k}}^{2}$ and *i* denotes the time index. The measurements, *d*
_{
k
}(*i*) and **u**
_{
k
}(*i*), are used to generate an estimate, **w**
_{
k
}(*i*) of size (*M* × 1), of the unknown vector **w**
^{o}. Assuming that each node cooperates only with its neighbors, then each node *k* has access to updates **w**
_{
l
}(*i*), from its${\mathcal{N}}_{k}$ neighbor nodes at every time instant *i*, where$\underset{l\ne k}{l\in {\mathcal{N}}_{k}}$, in addition to its own estimate, **w**
_{
k
}(*i*). Two different schemes have been introduced in the literature for the diffusion algorithm. The adapt-then-combine (ATC) scheme[7] first updates the local estimate using the adaptive algorithm used and then the intermediate estimates from the neighbors are fused together. The second scheme, called combine-then-adapt (CTA)[6], reverses the order. It is found that the ATC scheme outperforms the CTA scheme[7] and therefore, this work uses the ATC scheme.

*d*

_{ k }and

**u**

_{ k },${\mathbf{R}}_{\mathbf{u},k}=\mathrm{E}\left[{\mathbf{u}}_{k}^{\mathrm{T}}{\mathbf{u}}_{k}\right]$ is the auto-correlation of

**u**

_{ k }, and

*μ*is the step-size. The recursion defined in (3) requires full knowledge of the statistics of the entire network. A more practical solution utilizes the distributive nature of the network. The work in[6] gives a fully distributed solution, which has been modified and improved in[7]. The update equation is divided into two steps. The first step performs the adaptation, while the second step combines the intermediate updates from neighboring nodes. The resulting scheme is called

*adapt-then-combine*or ATC. Using the ATC scheme, the diffusion LMS algorithm is given as

where **Ψ**
_{
k
}(*i* + 1) is the intermediate update, *μ*
_{
k
} is the step-size for node *k*, and *c*
_{
lk
} is the weight connecting node *k* to its neighboring node$l\in {\mathcal{N}}_{k}$, where${\mathcal{N}}_{k}$ includes node *k*, and$\sum {c}_{\mathit{\text{lk}}}=1$. Further details on the formation of the weights *c*
_{
lk
} can be found in[6, 7].

## 3 Variable step-size diffusion LMS algorithm

The VSSLMS algorithms show marked improvement over the LMS algorithm at a low computational complexity[20–25]. Therefore, this variation is inserted in the distributed algorithm to inherit the improved performance of the VSSLMS algorithm. Different variations have their own advantages and disadvantages. A complex step-size adaptation algorithm would not be suitable because of the physical limitations of the sensor node. As shown in[23], the algorithm proposed by[20] shows the best performance as well as having low complexity. Therefore, it is well suited for this application. A further comparison of performance of these variants in the present scenario confirm our choice of the VSSLMS algorithm.

*f*[

*μ*

_{ k }(

*i*)] is the step-size adaptation function that is defined using the VSSLMS adaptation given in[20] where the update equation is given by

where *e*
_{
k
}(*i*) = *d*
_{
k
}(*i*) − **u**
_{
k
}(*i*)**w**
_{
k
}(*i*), 0 < *α* < 1 and *γ* > 0.

**w**

^{(o)}=

**Q**

**w**

^{ o }, and

**Q**=

*c*

*o*

*l*{

**I**

_{ M },

**I**

_{ M },…,

**I**

_{ M }} is a

*M*

*N*×

*M*matrix. Similarly, the update equations can be remodeled to represent the entire network

**G**=

**C**⊗

**I**

_{ M };

**C**is an

*N*×

*N*weighting matrix, where {

**C**}

_{ lk }=

*c*

_{ lk }; ⊗ is the Kronecker product;

**D**(

*i*) is the diagonal step-size matrix; and the error energy matrix,

**E**(

*i*), is given by

*k*is given by

where **R**
_{
k
} is the autocorrelation matrix for node *k*.

### 3.1 Mean analysis

**D**(

*i*) is independent of the regressor matrix

**U**(

*i*)[20]. Accordingly, for small values of

*γ*in (6), the following relation holds true asymptotically

**U**

^{T}(

*i*)

**U**(

*i*)] =

**R**

_{ U }is the auto-correklation matrix of

**U**(

*i*), and taking the expectation on both sides of (14) gives

where the expectation of the second term of the right-hand side of (14) is 0 since the measurement noise is spatially uncorrelated with the regressor and zero-mean, as explained earlier.

*λ*

_{max}(

**G**

**B**)| < 1, where

**B**= (

**I**

_{ MN }− E[

**D**(

*i*)]

**R**

_{ U }). Since

**G**comes from

**C**and we know that ∥

**G**

**B**∥

_{2}≤ ∥

**G**∥

_{2}.∥

**B**∥

_{2}, we can safely infer that

**C**∥

_{2}= 1 and for noncooperative schemes, we have (

**G**=

**I**

_{ MN }), we can safely conclude that

where *λ*
_{max}(**R**
_{
u,k
}) is the maximum eigenvalue of the auto-correlation matrix **R**
_{
u,k
}. This scenario is different from that of the fixed step-size as in this case where the system is stable only when the mean of the step-size is within the limits defined by (20).

### 3.2 Mean-square analysis

In this section, the mean-square analysis of the VSSDLMS algorithm is investigated. Here, the weighted norm has been used instead of the regular norm. The motivation behind using a weighted norm stems from the fact that even though the MSD does not require a weighted norm, the evaluation of the EMSE depends on a weighted norm. In order to accommodate both these measures, a general analysis is conducted using a weighted norm, where a weighting matrix is replaced by an identity matrix for the case of MSD, where a weighting matrix is not required[26].

For ease of notation, we denote$\mathrm{E}\left[\widehat{\mathbf{\Sigma}}\right]={\mathbf{\Sigma}}^{\prime}$ for the remaining analysis.

#### 3.2.1 Mean-square analysis for Gaussian data

**R**

_{ U }=

**T**

**Λ**

**T**

^{T}, where

**Λ**is a diagonal matrix containing the eigenvalues for the entire network and

**T**is a matrix containing the eigenvectors corresponding to these eigenvalues. Using this eigenvalue decomposition, we define the following relations

**D**(

*i*) is block-diagonal, so it does not transform since

**T**

^{T}

**T**=

**I**. Using these relations, (21) and (24) can be rewritten, respectively, as

where$\stackrel{\u0304}{\mathbf{Y}}(i)=\stackrel{\u0304}{\mathbf{G}}\mathbf{D}(i){\stackrel{\u0304}{\mathbf{U}}}^{\mathrm{T}}(i)$.

**R**

_{ v }=

**Λ**

_{ v }⊙

**I**

_{ M }denote the block diagonal noise covariance matrix for the entire network, where ⊙ denotes the block Kronecker product[27] and

**Λ**

_{ v }is a diagonal noise variance matrix for the network. Hence, the second term of the right-hand side of (25) is

where **b**(*i*) = bvec{**R**
_{
v
}E[**D**
^{2}(*i*)]**Λ**}.

**A**

_{ k }is given by

where **Λ**
_{
k
} defines the diagonal eigenvalue matrix and *λ*
_{
k
} is the eigenvalue vector for node *k*.

**D**(

*i*) ⊙

**D**(

*i*)] can be written as

which characterizes the transient behavior of the network. Although (34) does not explicitly show the effect of the variable step (VSS) algorithm on the network’s performance, this effect is in fact subsumed in the weighting matrix, **F**(*i*) which varies for each iteration, unlike in the fixed step-size LMS algorithm where the analysis shows that this weighting matrix remains fixed at all iterations. Also, (33) clearly shows the effect of the VSS algorithm on the performance of the algorithm through the presence of the diagonal step-size matrix **D**(*i*).

#### 3.2.2 Learning behavior of the proposed algorithm

**D**

_{0}=

*μ*

_{0}

**I**

_{ MN }, we have for iteration (

*i*+ 1)

*i*from those of iteration (

*i*+ 1) and simplifying them, we get

The relations in (46) and (47) govern the transient behavior of the MSD and EMSE of the proposed algorithm. These relations show how the effect on the proposed algorithm’s transient behavior of the weighting matrix varies from one iteration to the next as the weighting matrix itself varies at each iteration. This is not the case in the simple fixed step-size DLMS in[6] where the weighting matrix remains constant for all iterations. Since the weighting matrix depends on the step-size matrix, which becomes very small asymptotically, then both the norm and influence of the weighting matrix also become asymptotically small. From the above relations, it is seen that both the MSD and EMSE become very small at steady-state because the weighting matrix itself becomes small at steady-state and these relations will then depend only on the product of the weighting matrices at each iteration.

### 3.3 Steady-state analysis

**G**, does not permit the weighting matrix,

**F**

**(**

**i**

**)**, to be evaluated separately for each node, this is not the case for the determination of the step-size at any node. Here, we define the misadjustment as the ratio of the EMSE to the minimum mean square error. The misadjustment value is used in determining the steady-state performance of the algorithm[11]. Therefore, taking the approach of[20], we first find the misadjustment, as given by

where **D**
_{
ss
} = diag{*μ*
_{
ss,k
}
**I**
_{
M
}}.

**b**

_{ ss }=

**R**

_{ v }

**D**

*ss*2

**Λ**and${\mathbf{D}}_{\mathit{\text{ss}}}^{2}=\text{diag}\left\{{\mu}_{\mathit{\text{ss}},k}^{2}{\mathbf{I}}_{M}\right\}$. Now solving (52), we get

The above two steady-state relationships depend on the steady-state weighting matrix which becomes very small at steady-state, as explained before. As a result, the steady-state results for the proposed algorithms become very small compared to those for the fixed step-size DLMS algorithm.

## 4 Numerical results

In this section, several simulation scenarios are considered and discussed to assess the performance of the proposed VSSDLMS algorithm. Results have been conducted for different average SNR values. The performance measure used throughout these simulations is the MSD. The length of the unknown vector is taken as *M* = 4. The size of the network is *N* = 20 nodes. The sensors are randomly placed in an area of 1 unit square. The input regressor vector is assumed to be white Gaussian with auto-correlation matrix having the same variance for all nodes. Results averaged over 100 experiments are shown for the SNR value of 20 dB, a normalized communication range of 0.3, and a Gaussian environment.

*α*and

*γ*, these values are varied to check their effect on the performance of the algorithm. As can be seen from Figure3, the performance of the VSSDLMS algorithm degrades as

*α*gets larger. Similarly, the performance of the proposed algorithm improves as

*γ*increases as depicted in Figure4. This investigation therefore allows for a proper choice of

*α*and

*γ*to be made.

*M*= 2 and a communication range of 0.35. Two values for

*α*, namely

*α*= 0.995 and

*α*= 0.95, are chosen whereas

*γ*= 0.001.

## 5 Conclusions

The proposed variable step-size diffusion LMS (VSSDLMS) algorithm has been discussed in detail. Several popular VSSLMS algorithms are investigated and the algorithm providing the best trade-off between complexity and performance is chosen as the proposed VSSDLMS algorithm. Complete convergence and steady-state analyses have been carried out to assess the performance of the proposed algorithm. Simulations have been carried out under different scenarios and with different SNR values. A sensitivity analysis has been carried out through extensive simulations. Based on the results of this analysis, the values for the parameters of the VSSLMS algorithm were chosen. The proposed algorithm has been compared with existing algorithms of similar complexity and it has been shown that the proposed algorithm performed significantly better. Theoretical results were also compared with simulation results and the two were found to be in close agreement with each other. The proposed algorithm was then tested under different scenarios to assess its robustness. Finally, a steady-state comparison between theoretical and simulated results was carried out and tabulated and the results were also found to be in close agreement with each other.

## Declarations

### Acknowledgements

This research work is funded by King Fahd University of Petroleum and Minerals (KFUPM) under research grants FT100012 and RG1112-1&2.

## Authors’ Affiliations

## References

- Estrin D, Girod L, Pottie G, Srivastava M: Instrumenting the world with wireless sensor networks. In
*Proceedings of the ICASSP*. Salt Lake City; 07–11 May 2001:2033-2036.Google Scholar - Olfati-Saber R, Fax JA, Murray RM: Consensus and cooperation in networked multi-agent systems.
*Proc. IEEE*2007, 95(1):215-233.View ArticleGoogle Scholar - Lopes CG, Sayed AH: Incremental adaptive strategies over distributed networks.
*IEEE Trans. Signal Process*2007, 55(8):4064-4077.MathSciNetView ArticleGoogle Scholar - Ram SS, Nedic A, Veeravalli VV: Stochastic incremental gradient descent for estimation in sensor networks. In
*Proceedings of the 41st Asilomar Conference on Signals, Systems and Computers*. Pacific Grove; 4–7 November 2007:582-586.Google Scholar - Rastegarnia A, Tinati MA, Khalili A: Performance analysis of quantized incremental LMS algorithm for distributed adaptive estimation.
*Signal Process*2010, 90(8):2621-2627. 10.1016/j.sigpro.2010.02.019View ArticleGoogle Scholar - Lopes CG, Sayed AH: Diffusion least-mean squares over adaptive networks: formulation and performance analysis.
*IEEE Trans. Signal Process*2008, 56(7):3122-3136.MathSciNetView ArticleGoogle Scholar - Cattivelli F, Sayed AH: Diffusion LMS strategies for distributed estimation.
*IEEE Trans. Signal Process*2010, 58(3):1035-1048.MathSciNetView ArticleGoogle Scholar - Takahashi N, Yamada I, Sayed AH: Diffusion least mean squares with adaptive combiners: formulation and performance analysis.
*IEEE Trans. Signal Process*2010, 58(9):4795-4810.MathSciNetView ArticleGoogle Scholar - Schizas ID, Mateos G, Giannakis GB: Distributed LMS for consensus-based in-network adaptive processing.
*IEEE Trans. Signal Process*2009, 57(6):2365-2382.MathSciNetView ArticleGoogle Scholar - Mateos G, Schizas ID, Giannakis GB: Performance analysis of the consensus-based distributed LMS algorithm.
*EURASIP J. Adv. Signal Process*2009. doi: 10.1155/2009/981030Google Scholar - Haykin S:
*Adaptive Filter Theory*. Englewood Cliffs: Prentice-Hall; 2000.Google Scholar - Papadimitriou CH:
*Computational Complexity*. MA: Addison-Wesley, Reading; 1993.Google Scholar - BinSaeed MO, Zerguine A, Zummo SA: Variable step size least mean square algorithms over adaptive networks. In
*Proceedings of ISSPA 2010*. Kuala Lampur; 10–13 May 2010:381-384.Google Scholar - Cattivelli FS, Lopes CG, Sayed AH: Diffusion recursive least-squares for distributed estimation over adaptive networks.
*IEEE Trans. Signal Process*2008, 56(5):1865-1877.MathSciNetView ArticleGoogle Scholar - Mateos G, Schizas ID, Giannakis GB: Distributed recursive least-squares for consensus-based in-network adaptive estimation.
*IEEE Trans. Signal Process*2009, 57(11):4583-4588.MathSciNetView ArticleGoogle Scholar - Mateos G, Giannakis GB: Distributed recursive least-squares: stability and performance analysis.
*IEEE Trans. Signal Process*2012, 60(7):3740-3754.MathSciNetView ArticleGoogle Scholar - Cattivelli F, Sayed AH: Diffusion strategies for distributed Kalman filtering and smoothing.
*IEEE Trans. Automatic Control*2010, 55(9):2069-2084.MathSciNetView ArticleGoogle Scholar - Olfati-Saber R: Distributed Kalman filtering for sensor networks. In
*Proceedings of IEEE CDC 2007*. New Orleans; 12–14 December 2007:5492-5498.Google Scholar - Hlinka O, Hlawatsch F, Djuric PM: Distributed particle filtering in agent networks: a survey, classification, and comparison.
*IEEE Signal Proc. Mag.*2013, 30(1):61-81.View ArticleGoogle Scholar - Kwong RH, Johnston EW: A variable step size LMS algorithm.
*IEEE Trans. Signal Process*1992, 40(7):1633-1642. 10.1109/78.143435View ArticleGoogle Scholar - Aboulnasr T, Mayyas K: A robust variable step-size LMS-type algorithm: analysis and simulations.
*IEEE Trans. Signal Process*1997, 45(3):631-639. 10.1109/78.558478View ArticleGoogle Scholar - Costa MH, Bermudez JCM: A robust variable step-size algorithm for LMS adaptive filters. In
*Proceedings of the ICASSP*. Toulouse; 14–19 May 2006:93-96.Google Scholar - Lopes CG, Bermudez JCM: Evaluation and design of variable step size adaptive algorithms. In
*Proceedings of the ICASSP*. Salt Lake City; 7–11 May 2001:3845-3848.Google Scholar - Mathews VJ, Xie Z: A stochastic gradient adaptive filter with gradient adaptive step size.
*IEEE Trans. Signal Process*1993, 41(6):2075-2087. 10.1109/78.218137View ArticleGoogle Scholar - Sulyman AI, Zerguine A: Convergence and steady-state analysis of a variable step-size NLMS algorithm.
*Signal Process*2003, 83(6):1255-1273. 10.1016/S0165-1684(03)00044-6View ArticleGoogle Scholar - Sayed AH:
*Fundamentals of Adaptive Filtering*. New York: Wiley; 2003.Google Scholar - Koning RH, Neudecker H, Wansbeek T: Block Kronecker products and the vecb operator. Economics Department, Institute of Economics Research, University of Groningen, Groningen, The Netherlands, Research Memo no. 351, 1990Google Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License(http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.