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Third International Workshop on Advanced Computational Intelligence August 25-27,2010 - Suzhou, Jiangsu, China A New LMS Algorithm with Application to Fixed Satellite Communications Haibin Li, Yanguang Han, and Xinping Guan Abs-A distinctive new LMS algorithm is proposed in this paper. Both fast convergence rate and small steady state error can be obtained through non-linear weighting N errors of the current moment and past moments in this algorithm. The algorithm analyzed that the ideal step size is achieved by increasing the error information, building new nonlinear relations between step size factor and error function and eliminating the effect of correlated noise sequences. Then the new algorithm is applied to the fixed satellite communications system in which the channel is affected by multipath effect and weather conditions attenuation. Simulation results show that the new algorithm converges faster than usual algorithms under approximate steady state performance. The signal BER performance is greatly improved with the new adaptive algorithm equalizer in the fixed satellite communications system. I. INTRODUCTION T RANSMITTING signal reaches the terminal through more than one path in the high speed broadband fixed satellite digital communications systems, which is known as multipath propagation. Each path will have different amplitude attenuation and phase delay, so that e signal is expanded in the time domain at the receiver, which results in inter-symbol interference (lSI). lSI and channel fading affect the efficiency and quality of communication. Therefore, it is necessa to use equalization algorithm to mitigate inter-symbol interference and signal distortion caused by the fixed satellite channel. The least mean square (LMS) algorithm is one of the most popular filter algorithm in adaptive signal processing [1], [2]. It has been extensively applied in many areas duo to its simplicity, robustness and implementing easily. Within the LMS algorithm the step size is an important parameter that influences the algorithm's convergence performance. For fixed step size LMS algorithm, the choice of the step size reflects a tradeoff between steady state error and the speed convergence. A small step size gives small steady state eor but also a longer convergence time constant. In order to solve the contradictions between convergence speed and steady Manuscript received April 3, 2010. This work was supported by the Hebei province Postdoctoral Fellow ofchina under Grant A1550. H. B. Li is with the Key Lab ofIndustrial Computer Control Engineering of Hebei Province, Qinhuangdao, CO 066004 China (+86 13785934001; e-mail: hbli@ ysu.edu.cn). y. G. Han is with the Key Lab of Industrial Computer Control Engineering of Hebei Province, Qinhuangdao, CO 066004 China (e-mail: hguang@yahoo. com). X. P. Guan is with the Key Lab of Industrial Computer Control Engineering of Hebei Province, Qinhuangdao, CO 066004 China (e-mail: [email protected]). 978-1-4244-6337-4/10/$26.00 @2010 IEEE 72 state error of the fixed step size LMS algorithm, reference [3] proposed a variable step size LMS algorim. However, reference [4] analysis points out that this algorithm in [3] is vulnerable to the effect of independent noise, and proposes the MVSS _ LMS algorithm which the adaptive errors is not correlative when it is close to the optimal parameters and the correlation is also small in the process of convergence, leading to that step size reduces too fast and converges too slow. II. FID SATELLITE COMMUNICATIONS CHNEL The weather conditions will cause changes of the signal amplitude, phase, polarization and downlink beam incidence angle, which lead to the decline of the signal transmission quality and the increase of bit error rate. Among all weather conditions, rain causes the most serious propagation loss in fixed satellite communications at Ka-band [6]. Based on the satellite chnel propagation characteristics measurements statistical model [5], multipath effect fading can be expressed as a complex Gaussian random process, in which the envelope is Rayleigh distribution and phase is uniformly distributed between 0 and 2 . The satellite communications channel total attenuation is blessed with non-equency selective, which is slow enough and is only affected by the weather conditions [7]. TABLE I FIXED SATELLITE CHANNEL ENVELOPE AND PHASE MODEL PARAMETERS Weather Envelope Envelope Phase Phase Conditions Mean Variance Mean Variance Clear sky 0.413 0.00087 0.0072 0.00357 Cloudy 0.498 0.00025 0.0086 0.00405 Cumulus 0.346 0.00272 0.0154 0.00864 cloudy Overcast 0.440 0.00041 0.0274 0.00414 Intermittent 0.483 0.00003 0.0088 0.00546 light rain Thunder 0.436 0.01386 0.0068 0.00414 shower Light snow 0.488 0.00034 0.0088 0.00442 Blowing 0.500 0.00021 0.0089 0.00435 snow Rain 0.662 0.0200 -0.0089 0.03077 When the signal passes through the channel, the equivalent low-pass received signal (not including the Gaussian white noise) is simplified as follows: x(t) = Kexp(j)u I (t) 0 t T , (1) where T is modulation symbol width, u I (t) signal source outputs in time domain expression, K, are signal

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Page 1: [IEEE 2010 Third International Workshop on Advanced Computational Intelligence (IWACI) - Suzhou, China (2010.08.25-2010.08.27)] Third International Workshop on Advanced Computational

Third International Workshop on Advanced Computational Intelligence August 25-27,2010 - Suzhou, Jiangsu, China

A New LMS Algorithm with Application to Fixed Satellite Communications

Haibin Li, Yanguang Han, and Xinping Guan

Abstract-A distinctive new LMS algorithm is proposed in

this paper. Both fast convergence rate and small steady state

error can be obtained through non-linear weighting N errors of

the current moment and past moments in this algorithm. The

algorithm analyzed that the ideal step size is achieved by

increasing the error information, building new nonlinear

relations between step size factor and error function and

eliminating the effect of correlated noise sequences. Then the

new algorithm is applied to the fixed satellite communications

system in which the channel is affected by multipath effect and

weather conditions attenuation. Simulation results show that

the new algorithm converges faster than usual algorithms under

approximate steady state performance. The signal BER

performance is greatly improved with the new adaptive

algorithm equalizer in the fixed satellite communications

system.

I. INTRODUCTION

TRANSMITTING signal reaches the terminal through more than one path in the high speed broadband fixed satellite digital communications systems, which is known

as multipath propagation. Each path will have different amplitude attenuation and phase delay, so that the signal is expanded in the time domain at the receiver, which results in inter-symbol interference (lSI). lSI and channel fading affect the efficiency and quality of communication. Therefore, it is necessary to use equalization algorithm to mitigate inter-symbol interference and signal distortion caused by the fixed satellite channel.

The least mean square (LMS) algorithm is one of the most popular filter algorithm in adaptive signal processing [1], [2]. It has been extensively applied in many areas duo to its simplicity, robustness and implementing easily. Within the LMS algorithm the step size is an important parameter that influences the algorithm's convergence performance. For fixed step size LMS algorithm, the choice of the step size reflects a tradeoff between steady state error and the speed convergence. A small step size gives small steady state error but also a longer convergence time constant. In order to solve the contradictions between convergence speed and steady

Manuscript received April 3, 2010. This work was supported by the Hebei province Postdoctoral Fellow of china under Grant A1550.

H. B. Li is with the Key Lab ofIndustrial Computer Control Engineering of Hebei Province, Qinhuangdao, CO 066004 China (+86 13785934001; e-mail: hbli@ ysu.edu.cn).

y. G. Han is with the Key Lab of Industrial Computer Control Engineering of Hebei Province, Qinhuangdao, CO 066004 China (e-mail: h....Yguang@yahoo. com).

X. P. Guan is with the Key Lab of Industrial Computer Control Engineering of Hebei Province, Qinhuangdao, CO 066004 China (e-mail: [email protected]).

978-1-4244-6337-4/10/$26.00 @2010 IEEE 72

state error of the fixed step size LMS algorithm, reference [3] proposed a variable step size LMS algorithm. However, reference [4] analysis points out that this algorithm in [3] is vulnerable to the effect of independent noise, and proposes the MVSS _ LMS algorithm which the adaptive errors is not correlative when it is close to the optimal parameters and the correlation is also small in the process of convergence, leading to that step size reduces too fast and converges too slow.

II. FIXED SATELLITE COMMUNICATIONS CHANNEL

The weather conditions will cause changes of the signal amplitude, phase, polarization and downlink beam incidence angle, which lead to the decline of the signal transmission quality and the increase of bit error rate. Among all weather conditions, rain causes the most serious propagation loss in fixed satellite communications at Ka-band [6]. Based on the satellite channel propagation characteristics measurements statistical model [5], multipath effect fading can be expressed as a complex Gaussian random process, in which the envelope is Rayleigh distribution and phase is uniformly distributed between 0 and 27r . The satellite communications channel total attenuation is blessed with non-frequency selective, which is slow enough and is only affected by the weather conditions [7].

TABLE I FIXED SATELLITE CHANNEL ENVELOPE AND PHASE MODEL

PARAMETERS

Weather Envelope Envelope Phase Phase Conditions Mean Variance Mean Variance

Clear sky 0.413 0.00087 0.0072 0.00357 Cloudy 0.498 0.00025 0.0086 0.00405 Cumulus 0.346 0.00272 0.0154 0.00864 cloudy Overcast 0.440 0.00041 0.0274 0.00414 Intermittent 0.483 0.00003 0.0088 0.00546 light rain Thunder 0.436 0.01386 0.0068 0.00414 shower Light snow 0.488 0.00034 0.0088 0.00442 Blowing 0.500 0.00021 0.0089 0.00435 snow Rain 0.662 0.0200 -0.0089 0.03077

When the signal passes through the channel, the equivalent low-pass received signal (not including the Gaussian white noise) is simplified as follows:

x(t) = Kexp(jtjJ)uI (t) 05, t 5, T , (1)

where T is modulation symbol width, uI (t) signal source

outputs in time domain expression, K, tjJ are signal

Page 2: [IEEE 2010 Third International Workshop on Advanced Computational Intelligence (IWACI) - Suzhou, China (2010.08.25-2010.08.27)] Third International Workshop on Advanced Computational

envelope and phase real Gaussian random variables respectively for the equivalent low-pass communications channel. For fixed satellite channel at Ka-band under various weather conditions, the signal envelope and phase Gaussian model parameters are shown in Table 1 which given in [8].

III. THE NEW LMS ALGORITHM

The adaptive equalization problem is to try to adjust a set of weights so that the satellite communication system output tracks a desired signal. Least mean square algorithm is built upon minimum mean square error between expectation response and fIlter actual output. In an ideal variable step size LMS algorithm, the adaptive step size should converge fast with high accuracy and have the step size remain unchanged after the unknown step size is reached. Input signal sequence

x pass through m -1 delay units in turn, at the moment k constitutes a signal input vector X (k) , the input vector is

denoted by

X(k) =[x(k),x(k-l),··,x(k -m+l)t. (2)

The fIlter tap weight vector is given by

W(k) = [ww Wk2 ,··, Wkmt . (3)

As it is well known, the optimal solution for the weight

vector is WoPI(k) = R-1r , where R is input signal

autocorrelation matrix defmed as R = E[X(k)XT (k)] ,

and r = E[d(k)X(k)] . R can be represented as

R = QAQT, where A is the eigenvalues matrix, and Q is

the eigenvector matrix of R . The fIlter actual scalar output is

m

satisfy 0 < p < 1 / Amax ' where Amax is input signal X ( k ) maximum eigenvalue of autocorrelation matrix.

Within the limits of convergence, for fixed step size LMS algorithm, the larger step size is, the faster equalization algorithm convergence speed and tracking capability is. However, if the initial step size is set too large, it will lead to large oscillation and steady state performance deterioration during the convergence process, and the mean square error will also become large. Reducing step size can improve steady state accuracy for the algorithm, but the convergence performance and tracking performance will be deteriorated. In order to avoiding the conflict between convergence rate and steady state error, the variable step size update equation is given by in [3]

p(k + 1)' = ap(k) + r e2(k) (8)

with 0 < a < 1 , r > 0 and { Pmax if p( k + 1) , > Pmax p(k + 1) = Pmin if p(k + 1)' < Pmin '

p(k + 1)' other wise where p is restricted ranging from Pmin to Pmax to

guarantee stability of the algorithm, and 0 < Pmin < Pmax . The initial step size is usually taken to be Pmax ' but results

show that the algorithm is not sensitive to the choice. The step size is controlled by the size of prediction error and the parameters a , r in [3]. It can be seen that error is large at the

early stage of algorithm, so a larger step size is taken advantage. Thereafter output signal gradually approaches to the ideal value, and the error decreases, then the step size is

Yk = L wn;x(k -i + 1). (4) also reduced ensuring a small detuning. The constant Pmax is i=1

The error between y( k ) and the desired signal d (k) is

e(k) = d(k) -y(k) = d(k) -W(kf X(k) , (5)

where e( k) minimized in the mean square sense is zero mean

Gaussian independent sequence. The LMS adaptive algorithm is a gradient search algorithm

which computes a set of weight W (k) for realizing Wiener

fIlter. WOPI (k) is optimal estimate value of W (k) which

makes the mean square value of estimate error is the smallest, least mean square error is expressed as

E{e2(k)}min = E{d2(k)} -rTWoPI(k). (6)

The usual weight update recursion is of the form

W(k + 1) = W(k) + p(k)e(k)X(k), (7)

where p( k) is the step size factor. If p( k) is a constant, the

algorithm is fixed step size LMS (FVSS _ LMS) fIltering algorithm. Otherwise, the algorithm is variable step size LMS (VSS _ LMS) algorithm. In order to ensuring algorithm convergence in the process of iteration, The step size must

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chosen to ensure that the mean square error of the algorithm

remains bounded, and Pmin is chosen to provide a minimum

level of tracking ability. This algorithm is nearly perfect. Unfortunately, the usage of

the instantaneous squared error as a measure of the optimum results is significantly degradation in the presence of noise.

The desired output signal becomes to

d(k) = W;I(k)X(k)+ N(k) , (9)

where N ( k) independent Gaussian white noise that is not

related to input signal X (k ) . Taking advantage of

� '(k) = QT �(k) and X '(k) = QT X(k) , where

�(k) = W(k) -WoPI(k) is the transformed weight vector.

Putting (5) and (9) into (8) yields

p(k+ 1) = ap(k) + r[d(k) -WT (k)X(k)]2

Then we have

= ap(k) + r�T (k)X(k)XT (k)�(k) +r N2(k)�T (k)X(k). (10)

E{p(k+ I)} = aE{p(k)}+ r(E{N2(k)}

Page 3: [IEEE 2010 Third International Workshop on Advanced Computational Intelligence (IWACI) - Suzhou, China (2010.08.25-2010.08.27)] Third International Workshop on Advanced Computational

+E{�'T(k)A�'(k)}), (11)

where it is assumed that the � '( k) and X '( k) are mutual

independence. Clearly, E{�'T(k)A�'(k)} influences the

closeness of adaptive system to the optimal solution. However,

duo to the presence of E { N 2 ( k)}, the step size update

deteriorates at all stages of adaptation but particularly near the optimum. The step size is no longer an accurate reflection of the adaptive state. The performance of algorithm is greatly reduced with a poor signal, so that weight vector is hard to close to optimal value. To avoid this sensitivity to noise, a new algorithm is proposed. The idea is based on the fact that the error is poor and correlation among successive samples is small near the optimum. Thus, we use an estimate of the autocorrelation between present moment error and past N errors rather than time-averaged of e(k)e(k -1) in [4] to

control step size updating. The new algorithm step size iterative formula is defmed as the following equation:

p(k) = P p(k -1) + (1 -p)e(k)e(k -1) N

+ Z:e-i+1e\k-i+l) (12) i=1

and

Ji(k+ 1) = aJi(k) + r p2(k). (13)

Defme N

Ak = Ie-i+1e\k -i+l), (14) i=1

where a (0 < a < 1), P (0 < P < 1 ) and r (r > 0) are

the same as those of the MVSS _ LMS algorithm. The autocorrelation of the present error and the past N errors not only rejecting the independent noise sequence effect, but also

is an efficient measure of the closeness to the optimum. � is

patching factor. Nonstationary environment and large error will cause the step size to increase to provide faster tracking, while stationary environment and small error will result in a decrease in the step size to yield smaller misadjustment. Through non-linear weighting L error values of the present moment e( k) and past moments e( k -1) , ... ,

e(k -N + 1), p(k) is close to 0 and Ji is close to 0 and

achieving smaller misadjustment values at the time of steady state. The error coefficient is related to length between the corresponding time and the current time. The longer the length is, the smaller the error coefficient is. Therefore the independence noise has a little influence on new algorithm. The new algorithm has the attractive property of achieving a small fmal steady state error while providing fast convergence at early stages of adaptation, and it allows the adaptive filter to track changes in the system.

IV. SIMULATION RESULTS

In this section, we describe simulations performed to verify the theory developed in the previous sections, and to compare

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experimentally the performance of the new LMS algorithm to that of the algorithm proposed in [4] and the fixed step size LMS algorithm. Next, we consider an application of the new LMS adaptive equalizer to fixed satellite communications system. First of all, the convergence performance of the new LMS algorithm and other algorithms is simulated and analyzed, and then the bit error rate(BER) performance of the satellite communications system based on the new LMS

algorithm is also researched. In the simulations presented here, the unknown system to be

modeled is a FIR filter which the dimension is 8, and defmed as W*=[0.8783 -0.5806 0.6537 -0.3223 0.6577 -0.0582 0.2895 -0.2710]. Both system and the adaptive filter are excited by a correlated signal x( k) generated by(4)

x(k) = 0.9x(k -l)+v(k), (15)

where v( n) is a zero-mean, uncorrelated Gaussian noise of

unity variance. Parameters of these algorithms are selected to produce a comparable level of steady state error. Results are obtained by averaging 150 independent runs.

-15

-- 1 Fix ed LMS

-- 2 Lit.141 LMS -- 3New LMS

tOO 200 300 400 500 600 700 800 900

Number ofiterations Fig. l. Convergence characteristic of three kinds of equalization

algorithm.

In Fig. I, Curve I describes the convergence rate of the fixed step size LMS algorithm when the step size is

Ji = 0.01 , it is not until iterative 300 times that the curve

becomes convergent. It is noticed that the misadjustment level of the fixed step size algorithm with a small step size is achieved, but the convergence rate of this algorithm with a large iterative times is also achieved. Curve 2 describes the convergence speed of the MVSS _ LMS algorithm, in which the parameters are a = 0.95 , P = 0.95 , r = 0_001 , Jimax = 0.2 and Jimin = 0.001 . As can be seen, the

algorithm converges fast in the initial stage, but ultimately convergent result is not ideal. It is not until 400 iterations that the algorithm is convergent. Curve 3 describes the convergence speed of the newly proposed algorithm simulation result, where the parameters are a = 0.95 , P = 0.95 , N = 5 , r = 0.001 , Jimax = 0.2 and

Jimin = 0.001 . It converges after 50 iterations, and the

steady-state performance is perfect.

Page 4: [IEEE 2010 Third International Workshop on Advanced Computational Intelligence (IWACI) - Suzhou, China (2010.08.25-2010.08.27)] Third International Workshop on Advanced Computational

Under ideal conditions, the fixed satellite communications channel declines slowly enough, and without the satellite data processing. Transmitter and repeater amplifiers don't have linear distortion [6]. In different weather conditions, we just simply need to change the two Gaussian process parameters and the real constant generator parameters. A complex Gaussian process simulates mUlti-path effects.

10°r-------�------��============� --+-- 1 Theory -----17- 2 After-Eql --e-- 3 8efore-Eql

3

10.5 '------'-------'-'--------B�--'--------' o 5 10 15

signal to noise ratio (dB) 20

Fig. 2. Bit error rate simulation comparison diagram of system.

In order to analyze the effects of new LMS filtering algorithm on BER performance of the fixed satellite communications system, Fig. 2 shows the behavior of the channel BER performance with the NVSS _ LMS filtering algorithm equalizer compared to A WGN channel and weather attenuation channel of fixed satellite. Sending signals is the Bernoulli random binary sequence. The number of data is the 105. The modulation method of signal is BPSK. Take thunder shower for example, setting the corresponding parameters according to the conclusion given by [6]. Equalizer weighted tap coefficients Mare 7. The value for a = 0.95 appears to be a good choice for all the

experiments, while the value for f3 = 0.95 and r = 0.001 which control algorithm convergence time is chosen arbitrarily. N = 3 is the past error numbers. Maximum and

minimum step size respectively are set Jlmax = 0.01 ,

Jlmin = 1 x 10-5 .For comparison easily, the ideal A WGN

channel BER performance curve is given. From Fig. 2 we can see that the coherent BPSK signal sent by modulator passes through the fixed satellite channel, inciting severe distortion and inter symbol interference. When the BER is

� = 1 x 104, signal corresponding loss of signal to noise

ratio(SNR) is up to IOdB if the NVSS_LMS algorithm equalizer is not applied in the communications system, otherwise the loss of SNR reduce to 2.8dB.

V. CONCLUSION

The adaptive new variable step size LMS filter algorithm has not only less computational complexity, but also has better convergence and steady state error performance than other algorithms. The analysis and simulation results indicate

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that the algorithm converges faster than the fixed step size LMS algorithm and MVSS _ LMS algorithms. The loss of signal to noise ratio is small under certain bit error rate conditions, and it has achieved the attractive result in the fixed satellite communications system.

REFERENCES

[I] T. S. Qiu and D. X. Wei, "Adaptive Signal Processing in Communication," Beijing: Publishing House of Electronics Industry, 2005, pp. 23-47.

[2] S. Haykin, "Adaptive Filter Theory", 4rd ed, B. Y. Zheng. Trans. Beijing: Publishing House of Electronics Industry, 2003, pp. 183-230.

[3] R. H. Kwong and E. W. Johnston, "A variable step size LMS algorithm," IEEE Transactions on Signal Processing, vol. 40, no.7, pp. 1636-1642, July. 1992.

[4] K. Mayyas and T. Aboulnasr, "A robust variable step-size LMS type algorithm: analysis and simulations," IEEE Transactions on Signal Processing, vol. 45, no.3, pp. 1408-1411, 1997.

[5] C. Loo, "Land mobile satellite channel measurement at Ka band using Olympus," IEEE Vehicular Technology, vol. 44, no.2, pp. 919-923, 1994.

[6] A. H. Wang and W. X. Luo, "Modelling of Ka-band satellite communication channel and system performance simulation," Journal of China Institute of Communications, vol. 22, no.9, pp. 61-69, 2001.

[7] C. Loo, "Impairment of digital transmission through a Ka band satellite channel due to weather conditions," International Journal Satellite Communication, vol. 16, no.3, pp. 137-145, 1998.

[8] C. Loo, "Statistical models for land mobile and fixed satellite communications at Ka-band," IEEE Vehicular Technology Conference, vol. 46, no.2, pp. 1023-1027, 1996.