Performance Analysis of Multisplit Time Varying LMS Algorithm

Performance Analysis of Multisplit Time Varying LMS Algorithm

Abdullah M. K.

Department of Electronics & Telecommunication MPGI School of Engineering, Nanded Maharashtra, India

A. I. Tamboli

Department of Electronics & Telecommunication SGGS Institute of Engineering & Technology,Nanded Maharashtra, Nanded

AbstractIn this paper multisplit time varying LMS algorithm (MS TVLMS) is proposed and its performance is compared with multisplit LMS (MS LMS) and affine combination of two LMS adaptive filters. Two performance criteria are utilized: minimum mean square error (MSE) and convergence rate.MS TVLMS utilizes the idea of split filtering with linearly constrained optimization scheme. Then multi split adaptive filter is obtained by introducing continuous split procedure that results into a Hadamard domain adaptive filter. Simulation results with proposed algorithm are presented and compared with MS LMS and affine combination. Moreover proposed algorithm has minimum mean square error and better convergence rate as compared with MS LMS and affine combination.

KeywordsTV LMS,MS LMS,affine combination

1. INTRODUCTION

   LMS algorithm is one of the most widely used algorithms for adaptive signal processing because of its simplicity and robustness. But its performance in terms of convergence rate and tracking ability depends on eigen value of input signal correlation matrix. Split adaptive filter has evolved as a better solution to improve convergence rate and reduce computational burden. Fundamental principles were proposed in [1] for real Toeplitz matrices. Subsequently the same technique was extended to classical algorithms in linear prediction theory[2].K.C.Ho and P.C.Ching proposed split LMS adaptive filter for AR modeling[3] and then P.C.Ching and K.F.Wan generalized it to a so called unified approach [4],[5] by introduction of continuous splitting and corresponding application to a general transversal filtering problem.

   This paper is organized as follows: Section II recalls principles of transversal filter and its connection with hadamard transform.Multisplit adaptive filter is considered and is updated with TV LMS.Simulation results are presented in Section IV.Conclusions are reported in Section V.

2. MULTISPLIT AND HADMARD TRANSFORM

   Consider the classical scheme of an adaptive transversal filter as shown in Figure 1, in which the N-by-1 tap-weight vector of the filter w(n)=[w0(n), …, wN-1(n)]t has been split into its symmetric and antisymmetric parts:

   w(n) = ws(n)+ wa(n) (1)

   where ws(n) = ½ [w(n) + Jw(n)], wa(n)= ½ [w(n) Jw(n)] and J is the reflection matrix.

   Fig.1.Split Adaptive transversal filtering

   GSC structure [6] with symmetry and antisymmetry constraints the split transversal filtering scheme in Figure turns into the form represented by Figure 2.

   Figure 2.GSC implementation of split filter

   Let N=2M, where M is an integer number greater than one. Now, if each branch in Figure 1 is considered separately, the transversal filters ws(n) and wa(n) can also be split into their symmetric and antisymmetric parts[6].

   Figure 3.Multisplit adaptive filtering

   The above multi-split scheme can be viewed as a linear transformation of x(n) denoted by

   x (n) Ttx(n) (2)

   e(n)=d(n)-y(n) (6)

   W(n)=W(n-1)+ne(n)x(n) (7)

   (8)

   n= 0 *n (9)

3. SIMULATION RESULTS

Simulation flowchart

Start

Initialize the filter order & Convergence factor

where

Consider input & Desired Signal

Apply Linear Transform to input array and filter coefficients

(3)

Compute Error

Compute filter output

Above transform results into Hadamard transform[6] as shown below

Update Coefficients

Figure 4.Hadamrd transform of input x(n)

Time varying LMS can be applied for updating the parameter With no increase in computational complexity.

Hadamard matrix of order 2M is constructed as follows:

Yes

Iterations < Max

Find MSE

MS TV LMS algorithm:

Initialization:

For i=0,1,..N-1,.set wi(0)=0 Updating:

1)T transform on input x(n)

2) TV LMS algorithm:

(4)

No

Simulation Parameters

Input signal parameters:

y(n)=W(n-1)xT(n) (5)

Amplitude: 1

Frequency: 500Hz

Sampling Frequency: 10000Hz Initial Phase: 0

Noise Parameters:

Amplitude: 0.15 Type: Gaussian Mean: 0

Variance: 1

Initial Seed: 10

Filter Parameters

Filter Type: FIR Order: 32

Structure: Direct form-I Window: Rectangular

No. Of Iterations: variable Convergence Factor: time varying

Figure.5.Convergence behavior of MS LMS

Desired signal parameters:

Sinusoidal signal of 500Hz frequency with

amplitude 1.

Table 1: MSE for different algorithms

From above table it can be observed that MSE of MS TVLMS is less than MS LMS and affine combination.

Figure 6.Convergence behavior of MSTV LMS

Figure 7.Convergence behavior of MSTV LMS and MS LMS

Figure 8.Convergence behavior of Affine Stochastic

Figure 9.Convergence behavior of Affine Error

Figure 10.Comparision of MS LMS, MS TVLMS and affine combination

From above graphs it can be observed that MS TV LMS has minimum MSE and better convergence rate as compared to MS LMS and affine combination of two LMS adaptive filters.

IV CONCLUSIONS AND FUTURE SCOPE

This paper studied the performance of MS TV LMS algorithm. Here, input vectors as well as filter coefficients are split as symmetric and asymmetric parts using Hadamard transform. Hadamard transform is a linear transform, which operates on time domain samples of input and impulse response of filter. Simulation results show the better performance of proposed algorithm over MS LMS and affine combination in terms of MSE and convergence rate.

The same procedure can also be repeated in frequency domain. The input vector is split as low frequency part and high frequency part, each part is separately applied adaptive filtering algorithm, which leads to sub band adaptive algorithm. As no transformation is required and only requires filter banks, it is less computationally brden.

V. REFERENCES

1. P. Delsarte and Y.V.Genin, The Split Levinson Algorithm,IEEE Trans. On Acoust. ,Speech and Signal Processing,vol.ASSP- 34;no.3;pp.470-478,June 1986.

2. P. Delsarte and Y.V.Genin, On the splitting of Classical Algorithms in Linear Prediction Theory,IEEE Trans. On Acoust. ,Speech and Signal Processing,vol.ASSP-35;no.5;pp.645-653,May 1987.

3. K.C.Huo and P.C.Ching, Performance Analysis of a Split-Path LMS Adaptive filter for AR Modelling,IEEE Trans. On Signal Processing,vol.40;no.6;pp.1375-1382;June 1992.

4. P.C.Ching and K.F.Wan, A Unified Approach to Split Structure Adaptive Filtering,Proceedings of the IEEE- ICASSP95,Detroit,USA;May 1995.

5. L.S.Resende,J.M.T.Romano and M.G.Bellanger, Adaptive Split Transversal Filtering:A Linearly Constrained Approach,.Proceedings of IEEE 2000 AS-SPCC,Lake Louise,Canada,Oct.2000.

6. S.Haykin, Adaptive Filter Theory ,3rd edition;Prentice-Hall;New Jersy;1996.