Adaptive Filtering for Incident Plane Wave Estimation

Mohammad Shukri Salman Electrical and Electronic Engineering

Mevlana (Rumi) University Se1cuklu-Konya, Turkey

Email: mssalman@mevlana.edu.tr

Abstract-Plane wave propagation is an important field of interest in magnetic theory. In this paper, we propose an adaptive filtering noise canceller that enables us to estimate an incident plane wave from the total plane wave seen in a medium. In order to test the proposed canceller, a comparison between the performances of the least mean square (LMS), recursive least squares (RLS) and recursive inverse (RI) algorithms is investi­gated. Simulation results show that the RI algorithm outperforms the LMS algorithm in terms of convergence rate and the ability of estimating the incident wave. Also, it outperforms the RLS algorithm in terms of the number of mathematical operations required at each iteration and the ability of suppressing noise.

Index Terms-RI algorithm, LMS algorithm, RLS algorithm.

I. INTRODUCTION

In the last decades, adaptive filtering techniques have been an active topic because it is widely applicable in many signal processing [1], communications [2] and image processing applications [3]. Designing and adaptive filter requires several important factors to be taken into account; such as rate of convergence, computational complexity and tracking abilities and accuracy of steady-state solution [1], [4], [5]. Hence, designing such a filter would be an issue of interest.

Least mean square (LMS) algorithm was the first proposed algorithm that tries to solve those issues [1]. However, its performance is poor, in terms of mean-square-error (MSE) and

convergence rate, if the eigenvalue spread (�:;,:�:j) of the

autocorrelation matrix is relatively high. The recursive least squares (RLS) was proposed to overcome these problems. Nevertheless, the RLS algorithm has high computational com­plexity, stability and bad tracking ability drawbacks which are important considerations, especially, in real time applications. Recently, a recursive inverse (RI) adaptive filtering algorithm was proposed [6] in order to overcome all these problems. Even though the computational complexity of the RI algorithm is less than that of the RLS algorithm, it can be further reduced by applying the fast implementation techniques using Fourier transform [7].

Measuring incident plane waves from the total plane waves propagating in a medium is an important thing to be studied in magnetic theory. In [8], authors employ the sign-LMS algorithm for estimating ultrasonic straight beam pulse-echo inspections. It extracts only two of the interface echoes mul-

ISBN: 978-1-4673-5613-8©2013 IEEE

Mehmet Fatih Yilmaz Mechatronics Engineering

Mevlana (Rumi) University Se1cuklu-Konya, Turkey

Email: mfyilmaz@mevlana.edu.tr

J E;(t,z)+ E'(t, z)+ n(t,Z) Adaptive "

� Filter

/

d(k)

+ y(k) �/ '\

\.. ./ e( k)

Fig. 1. Block diagram of the adaptive noise cancellation model.

tiple reflections for enhanced resolution and quality-enriched presentation.

In this paper, we propose an adaptive filtering noise can­celler that enables us to estimate an incident plane wave from the total plane wave measured in any medium. The proposed technique is shown in Fig. 1 assuming that the reflected plane wave ET and the noise parts are the unwanted parts of the measured wave. A comparison among the performances of the LMS, RLS and RI algorithms is investigated for the numerical example given in Section IV. Simulation results show that the RI algorithm outperforms the LMS algorithm in terms of convergence rate and the ability of estimating the incident wave. Also, it outperforms the RLS algorithm in terms of the number of mathematical operations required at each iteration and the ability of suppressing noise.

This paper is organized as follows. In Section II, the RI algorithm is reviewed. In Section III the problem description is provided in detail. In Section IV, a numerical example and the simulation results are given and discussed. Finally, the conclusions are drawn in Section V.

II. THE RECURSIVE INVERSE AL GORITHM

The RI algorithm and its performance in different areas and different experimental settings had been shown in detail [6], [9], [3]. In this section, we briefly review the RI algorithm.

The update equation of the RI algorithm is given by

w(k) = [I - fL(k)R(k)]w(k - 1) + fL(k)p(k). (1)

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XY Plane

Fig. 2. Plane wave reflection and transmission.

z

where w( k) is the filter tap weight vector, R( k) is the estimate

assumed in the region z < 0 (where the incident and reflected plane waves propagate) and medium 2 is in the region z > 0 (where the transmitted plane wave propagates). The incident electric field intensity is given as:

(5)

where Eo is the electric wave magnitude, w is the angular frequency, kl is wave number in medium 1 (kl = W.JElf..ll) and z is the direction of propagation. The reflected electric wave can be expressed as

Er(z, t) = Eorej(wt+kIZ)iiy

where r is the reflection coefficient which is given by

r = Ti2 - Til Ti2 + Til

(6)

(7)

where Ti2 = !iii and Til = r;;; are the intrinsic impedances V £2 V £1 of medium 2 and 1, respectively. After a certain period of time, the total electric wave measured in medium 1 will be

of the tap-input vector, (x(k)) autocorrelation matrix, p(k) EI(z, t) = Ei(Z, t) + Er(z, t) + n(z, t) (8) is the estimate of the cross-correlation vector between the desired output signal d(k) and the tap-input vector w(k). The where n(z, t) is a noise process.

correlations are estimated recursively as [10], IV. NUMERICAL EX AMPLE

R(k) = ,8R(k -1) + x(k)xT(k), (2)

p(k) = ,8p(k - 1) + d(k)x(k), (3)

where ,8 is the forgetting factor which is usually very close to one.

In [6], it has been shown that the step-size f..l is advantageous to be variable,

f..l(k) f..lo

1 - ,8k where f..lo < f..lmax, (4)

2(1 - ,8) Amax (Rxx)

and f..lmax

where Amax (Rxx) is the maximum eigenvalue of Rxx and Rxx = E{x(k)xT(k)} and R(k) = E{R(k)}.

III. PROBLEM DESCRIPTION

Electromagnetic (EM) plane waves propagate through a medium are usually affected by the medium parameters; such as permittivity (E), permeability (f..l) and conductivity (a) . When electric o r magnetic wave travels across two different media with an angle 81 a portion of this wave is reflected from the plane of incidence back to medium 1 with an angel 82, while the rest of the plane wave is transmitted into medium 2 83[11] as shown in Fig. 2.

Now, let us consider the normal incidence case of an electric plane wave propagating in the +z direction. Assume both me­dia are lossless dielectrics with permeability and permittivity constants as f..ll, EI, f..l2, E2, respectively. Where medium 1 is

ISBN: 978-1-4673-5613-8©2013 IEEE

Consider a uniform plane wave in vacuum is normally incident on an infinite lossless dielectric with permeability constant f..l = f..lo and permittivity constant E = 9Eo. The incident wave is given as Ei(Z, t) = lOcos(wt -klz)iiy with f = 3GH z. And the reflected wave is estimated as Er(z, t) = 10rcos(wt + klz)iiy with r = -0.5. Hence, after sometime, the total plane wave seen in medium 1 will be EI (z, t) = [10cos(wt - klZ) - 5cos(wt + kIZ)] iiy + n(z, t). Where n(z, t) is an additive white Gaussian noise (AWGN) sequence with zero mean and variance a; .

In this section, we employ the LMS [1] (see Table I), RLS [4] (see Table II) and RI algorithms to estimate the incident plane wave (Ei(Z, t» from the total wave seen in medium 1 (EI (z, t». We use the noise cancellation setting shown in Fig. 1. All the experiments were implemented with the following parameters: Filter length (N = 50 taps), the noise variance a; = 0.0225 and the number of independent Monte-Carlo runs is 100.

For the LMS algorithm: The step-size f..l = 4 X 10-6. For the RI algorithm: The step-size f..lo = 4 X 10-6 and ,8 = 0.995. For the RLS algorithm: ,8 = 0.995 and <5 = 1. Figs. 3, 4 and 5 show the estimated plane waves using the LMS, RLS and RI algorithms, respectively. From Fig. 3, we see that the LMS algorithm needs sometime to start estimating the incident plane wave (low convergence rate) and with 0.087r phase shift between the original incident plane wave and the estimated one. In Fig. 4, the RLS algorithm successfully estimates the incident plane wave and immediately (high convergence rate) but it requires a high number of mathematical operations. However, Fig. 5 shows that the RI algorithm also successfully

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estimates the incident plane wave and immediately (the same convergence rate of the RLS) with less noise fluctuations and less number of mathematical operations (See Table III to compare the number of mul./div.s and add./sub.s).

TABLE I

SUMMARY OF THE LMS ALGORITHM.

1. Filter Output.

y(k) = wH (k)x(k) 2. Estimation Error.

e(k) = d(k) -y(k) 3. Tap - Weight Adaptation.

w(k + 1) = w(k) + fLe*(k)x(k) where x(k) is the tap - input vector and d(k) is the desired filter output.

TABLE II

SUMMARY OF THE RLS ALGORITHM.

Initialize the algorithm by setting,

w(O) = 0 prO) = 8-11

and

8 _{ small positive constant for high SNR -

large positive constant for low SNR. for each instant of time, k = 1,2, . compute

S(k) = P(k-l)x(k) ) _ S(k) k(k -

/3+xH (k)SJrk) �(k) = d(k) -w (k -1)x(k) w(k) = w(k -1) + k(k)e(k)

and

P(k) = V1p(k -1) -(3-1k(k)xH (k)P(k -1).

TABLE III COMPUTATIONAL COMPLEXITY OF LMS, RI AND RLS ALGORITHMS

Mult.lDiv. Add./Sub.

LMS 2N + 1 2N

RI

RLS 3N2 + lIN + 9 3N2 + 7 N + 4

V. CONCLUSIONS

In this paper, a noise cancellation setting for estimating incident plane waves using adaptive filters is proposed. A comparison between the performances of the LMS, RLS and RI algorithms have been shown. Results show that the RI algorithm outperforms the LMS algorithm in terms of convergence rate and the ability of estimating the incident wave. Also, it outperforms the RLS algorithm in terms of the number of mathematical operations required at each iteration and the ability of suppressing noise.

REFERENCES

[1] B. Widrow and S. D. Stearns, Adaptive Signal Processing, Prentice Hall,

Eaglewood Cliffs, NJ, 1985.

ISBN: 978-1-4673-5613-8©2013 IEEE

Normal Incidence Estimation Using LMS Algorithm

15 - - - Original Incident Wave

10

5 Q) '0 � 0 Q. E ..:

-5

-10

-15

-0.5 -0.4 -0.3 -0.2 -0.1 o

Fig. 3. Estimated incident plane wave using the LMS algorithm.

Normal Incidence Estimation Using RLS Algorithm

15 - - - Original Incident Wave

10

Q) '0 :0 '" Q. E ..:

-5

-10

-15

-0.5 -0.4 -0.3 -0.2 -0.1 o

Fig. 4. Estimated incident plane wave using the RLS algorithm.

Normal Incidence Estimation Using RI Algorithm

15 - - - Original Incident Wave

10

-10

-15

-0.5 -0.4 -0.3 -0.2 -0.1 o z

Fig. 5. Estimated incident plane wave using the RI algorithm.

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[2] A. H. Sayed, Fundamentals of Adaptive Filtering, Wiley, New York, NY, USA, 2003.

[3] M. S. Ahmad, O. Kukrer and A. Hocanin, "A 2-D Recursive Inverse

Adaptive Algorithm," Signal Image and Video Processing, Springer, vol.

7, no. 2, pp. 221-226, March 2013. [4] S. Haykin, Adaptive Filter Theory, Prentice Hall, Upper Saddle River,

NJ, 4th edn., 2002.

[5] G-O. Glentis, K. Berberidis and S. Theodoridis, "Efficient least squares adaptive filtering algorithms for FIR transversal filtering," IEEE Signal

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[6] M. S. Ahmad, O. Kukrer and A. Hocanin, "Recursive inverse adaptive filtering algorithm," Digital Signal Processing, Elsevier, vol. 21, no. 4,

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Quasi-Newton LMS algorithm using FFT," Second International Con­

ference on Digital Illformation and Communication Technology and its Applications, (DICTAP2012), pp. 510-513, 2012.

[8] M. S. Mohammed and K. Ki-Seong, "Sign least mean square-based

deconvolution technique for ultrasonic testing," Russian Journal of Non­destructive Testing, vol. 48, no. 10, pp. 609-613, 2012.

[9] M. S. Ahmad, O. Kukrer and A. Hocanin, "The effect of the forgetting

factor on the RI adaptive algorithm in system identification," 10th International Symposium on Signals, Circuits and Systems (ISSCS2011),

pp. 1-4, 2011.

[10] G. O. Glentis, K. Berberidis and S. Theodoridis, "efficient least squares

adaptive algorithms for FIR transversal filtering", IEEE Signal Processing Magazine, pp. 13-41, July 1999.

[II] T. L. Wilson, K. Rohfls, S. Huttemeister, Tools of Radio Astronomy,

Springer, 5th edn., 2009.

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