Performance Analysis and Comparison of Adaptive Filters for Wireless Channel Equalization

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International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459, Volume 2, Issue 5, May 2012)

341

Performance Analysis and Comparison of Adaptive

Filters for Wireless Channel Equalization

M.S.Chavan

1

, R.H.Chile

2

, S.R.Sawant

3

1_{Department of Electronics and Telecommunication Engineering, P.V.P.I.T.Budhgaon, Sangli}

2

Department of Instrumentation engineering, S.G.G.S.College of Engineering Nanded

3_{Department of Science and Technology Shivaji University Kolhapur}

1_{mschavan67@gmail.com}

Abstract: In this paper we have applied radial basis function neural networks for equalization of wireless channel in the presence of additive noise. We have been tested the adaptive filter algorithms of simple Kalman filter, extended Kalman filter, rbfn Kalman filter, rbfn extended Kalman filter, fuzzy Kalman filter and fuzzy extended Kalman filter which might be considered applicable for wireless channel equalization in presence of noise. The Kalman filter is one of the most widely used due to its simplicity, optimality, tractability and robustness. However, the application of the KF to nonlinear systems can be difficult. The most common approach is to use the extended Kalman filter. We have shown that performance of the fuzzy extended Kalman filter is superior. More specifically, this paper highlights the error rate performance of the adaptive filters for channel equalization. Simulation results demonstrate that the adaptive fuzzy Kalman filter and fuzzy extended Kalman filter works successfully, efficiently.

Keywords:Adaptive Filter, Channel Equalization, Kalman Filter, Extended Kalman Filter, RBFN, fuzzy.

I. INTRODUCTION

In digital communications, a considerable effort has been devoted to the study of data transmission system that utilizes the available channel bandwidth efficiently. The objective here is to design a system that accommodates highest possible rate of data transmission, subject to a specified reliability that is usually measured in terms of the error rate or average probability of symbol error. The radio channels in mobile radio systems are extremely random and time-variant. An adaptive filtering algorithm requires the knowledge of the “desired” response so as to form the error signal needed for adaptive process to function. In theory, the transmitted sequence (originating at the transmitter output) is the “desired” response for the adaptive equalization [16].

In practice, however, with the adaptive equalizer located at the receiver, the equalizer is physically separated from the origin of its ideal desired response. One method to generate the desired response locally in the receiver is the use of training sequence, in which a replica of the desired response is stored in the receiver. Naturally the generator of this stored reference has to be electronically synchronized with the known transmitted sequence. With a known training sequence, the adaptive filtering algorithm used to

adjust the equalizer coefficients corresponds

mathematically to searching for the unique minimum of quadratic error-performance. Second method is decision directed method, in which, a good facsimile of the transmitted sequence is being produced at the output of the decision device in the receiver. Accordingly, if this output were the correct transmitted sequence, it may be used as the “desired” response for the purpose of Adaptive Equalization. In the present paper we extend the use of Kalman filters to the training of general input, multi-output RBF networks [1], [10]. For linear dynamic systems the Kalman filter is known to be an optimal estimator. For nonlinear systems with noise, the Kalman filter can be extended by linearizing the system around the current parameter estimates [2], [10].

II. KALMANFILTER

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International Journal of Emerging Technology and Advanced Engineering

342 xk = Axk – 1 + Buk – 1 + wk – 1 . (1)

With a measurement:

zk = Hxk + vk (2)

p(w) ~ N(0, Q) (3)

p(v) ~N(0, R) (4)

The random variables in equations (1) and (2) represent the process and measurement noise respectively. They are assumed to be independent of each other or in other words they are uncorrelated. The noise is assumed to be white and with normal probability distributions. The process noise covariance matrix Q or measurement noise covariance matrix R may change with each time step or measurement, however we assume here they are constant matrices and in the difference equation which relates the states at previous time step to the state at current step.

The xˆ-k ε R n defined as the a priori state estimate at time step k when the process prior to step k is known, and the xˆk ε R n is a posteriori state estimate at step k when the measurement is known. The a priori and a posteriori estimates errors can be defined as:

ek- ≡ xk -xˆ-k , and

ek ≡xk - xˆk

The a priori estimate error covariance is then

Pk –

= E [ ek

ek -T

]

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The a posteriori estimate error covariance is,

Pk = E [ ek ekT ]

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The next step involves finding an equation that computes an a posteriori state estimate as a linear combination of an a priori estimate and a weighted difference between an actual measurement and a measurement prediction.

xˆk=xˆ-k+ K ( zk- Hxˆ-k) (7)

The Kalman gain calculated from the equation:

Kk = P–k HT (HP-k HT + R)–1

= P-k H T / HP-k HT + R (8)

The difference (zk- Hxˆ-k) is the measurement

innovation or residual. We see that as the R, measurement error covariance approaches zero, the gain K weights the residual more heavily. Prediction Estimation cycle is shown in the fig.1.

Initial estimates for xˆk – 1 and Pk – 1

Fig.1. Prediction Estimation Cycle

III. EXTENDED KALMAN FILTER

The Kalman filter addresses the general problem of trying to estimate the state x € Rn of a discrete-time controlled

process that is governed by a linear stochastic difference equation. But what happens if the process to be estimated and (or) the measurement relationship to the process is non-linear? Some of the most interesting and successful applications of Kalman filtering have been such situations. A Kalman filter that linearizes about the current mean and covariance is referred to as an extended Kalman filter or EKF [3]. Let us assume that our process again has a state vector x € Rn, but the process is now governed by the non-linear stochastic difference equation

xk= f (xk – 1, uk – 1, wk – 1) (9)

With a measurement z € Rm

Zk= h (xk, vk) (10)

Where wk and vk are the random variables and again represent the process and measurement noise.

The state is predicted by dynamic model is

Predicted state: xˆ-k = f (xˆk – 1, uk – 1, 0) (11)

Predicted estimate covariance:

Pk- = AkPk– 1AkT + WkQk– 1WkT (12)

The predictedstate is corrected by dynamic model. Kalman gain:

Kk = P –

k Hk T

(HkP

-k Hk T

+ VkRkVk T

)–1 (13)

Updated state estimate: xˆk = xˆ-k+ Kk (zk-h( xˆ-k,0) (14)

Updated state estimate covariance: Pk= (I –KkH) P-k (15)

Prediction Estimation Cycle is shown in the fig.2.

Time Update (“Predict”) (1) Project the state ahead

Xˆ-k

=

Ax

ˆ

k – 1

+

Bu

k – 1

(2) Project the error covariance ahead

P

k -

=

AP

k – 1AT

+

Q

Measurement Update (“Correct”)

(1)Compute the Kalman gain

Kk = P –k HT (HP-k HT + R)–1 (2) Update estimate with

measurement zk

xˆk=xˆ-k+K (zk- Hxˆ-k)

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International Journal of Emerging Technology and Advanced Engineering

343 Initial estimates for xˆk – 1 and Pk – 1

Fig.2. Prediction Estimation Cycle

IV. RBF KALMAN FILTER

An RBF consists of the m-dimensional input x being passed directly to a hidden layer. Suppose there are c neurons in the hidden layer. Each of the c neurons in the hidden layer applies an activation function which is a function of the Euclidean distance between the input and an m-dimensional prototype vector. Each hidden neuron contains its own prototype vector as a parameter. The output of each hidden neuron is then weighted and passed to the output layer. The outputs of the network consist of sums of the weighted hidden layer neurons [5].

Mathematically, the output of a RBF equalizer is,

( ( )) ∑ ( ) (16)

Where the basis function F is a sigmoidal function,

( ) ( ‖ ‖ ) (17)

x is the input vector of the equalizer [x1 x2 … xm], h

indicates the total number of hidden neurons, μ and σ refer to the center and width of the l-th hidden neuron. ||.|| is the Euclidean norm. The coefficient wl is the weight of the l-th hidden neuron to the output neuron. Let us consider a time series described by a nonlinear autoregressive model affected by both process and observational noise:

x [n] = ƒ{x [n-1],…,x [n-N],W} + r [n-1] (18)

y [n] = x [n] + q [n] (19)

Where ƒ is a (generally nonlinear) function parameterized by W. The equations above can be rewritten as

X [n] = F{X [n-1]} + Br [n-1] (20)

Y [n] = cX [n] + q [n] (21)

Where , - [ , ,

, ]

, -

* , - + [

* , - , - + ,

,

]

V. RBF EXTENDED KALMAN FILTER

An RBF consists of the m-dimensional input x being passed directly to a hidden layer. Suppose there are c neurons in the hidden layer. Each of the c neurons in the hidden layer applies an activation function which is a function of the Euclidean distance between the input and an m-dimensional prototype vector. Each hidden neuron contains its own prototype vector as a parameter. The output of each hidden neuron is then weighted and passed to the output layer. The outputs of the network consist of sums of the weighted hidden layer neurons.

The performance of an RBF network depends on the number and location (in the input space) of the centers, the shape of the RBF functions at the hidden units and the method used for determining the network weights. The researchers have trained RBF networks by selecting the centers randomly from the training data. The extended Kalman filter can be applied to RBF optimization by reformulating the radial basis functions [5].

There have been a number of popular choices for the g(fi) function at the hidden layer of RBFs. The most common choice is a Gaussian function of the form,

g(v) = exp(−v /β 2 ) (22) Where,

β

is a real constant.

Other hidden layer functions that have often been used are: The thin plate spline function,

g (v)= v log √ v (23) The multiquadric function,

g (v)= v2 + β 2 )1=2 (24) The inverse multiquadric function,

g (v)=(v2 + β 2 )−1=2 (25) The hidden layer functions should have the following properties:

1. The response at a hidden neuron is always positive; 2. The response at a hidden neuron becomes stronger as the

input approaches the prototype;

3. The response at a hidden neuron becomes more sensitive to the input as the input approaches the prototype.

Measurement Update (“Correct”)

(1)Project the state ahead

xˆ-k = f (xˆk – 1, uk – 1, 0)

(2) Project the error covariance ahead

P

k-

= A

k

P

k– 1

A

kT

+ W

k

Q

k– 1

W

kT

Measurement Update (“Correct”) (1) Compute the Kalman gain

K

k = P –k

H

kT (Hk

P

-k HkT

+ V

k

R

k

V

kT)–1

(2)

Update estimate with measurement zk

xˆ

k

=xˆ

-k

+ K

k

(zk-h (x

ˆ-k,

0)

(2) Update the error covariance

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International Journal of Emerging Technology and Advanced Engineering

344

VI. FUZZY KALMAN FILTER

By combining, the Kalman filtering and the fuzzy logic have shown their respective superiority in different applications [13]. The fuzzy theory provides a more effective representation of natural phenomena. Furthermore fuzzy membership function can be used for improving the characterization of system and measurement noise while the Kalman filter gives statistical characterization can provide an effective mean of capturing a priori information to provide modeling of an uncertain complex nonlinear system such as wireless channels [8].

The Pre-Processor and the Projectile Estimates: In fuzzy Kalman filter, the output of the prediction part of revised form is referred as the pre-processor estimate and the corrector output is referred as the projectile estimate. Let k be the Fuzzy-Kalman projectile estimate and the

measurement error eZk defined as:

eZk = Zk – H k (26)

eZk is used as the input to a membership function with the

corresponding element for center of gravity crisp output. The state feedback is calculated according to a membership function that is a function of the estimate error. So the projectile estimate becomes:

k+1 = A k + Kk (Zk H k) (27)

Kalman gain works according to the priori estimation error covariance matrix Pk and the measurement noise

covariance Rk, assuming that the output matrix H is an

identity matrix. Then the a posteriori estimate of the Kalman filter is obtained as:

k = X¯+Kk(Xk + vk X¯k)

Kalman gain: The Kalman gain multiplies the error and is stated as:

Kk = A k H T

/ (H k H

T

(29)

Where Rk, P¯k are the measurement noise and the a priori

error covariance matrices.

Measurement Error Covariance Matrix: Authors have used the projectile error covariance of the Fuzzy Kalman filter equation for constructing the membership function of ψ. The measurement error covariance is obtained from the projectile estimation error as:

Pk = H k H T _{+ R}

k (30)

Thus, the filtered state estimate is obtained from the predicted estimate as, Filtered state estimate = predicted state estimate + (Gain x Error)

VII. FUZZY EXTENDED KALMAN FILTER

The performance of a fuzzy system depends, on both its rule base and its membership functions. Given arule base, the membership functions can be optimized in order to obtain the best performance from the fuzzy system. The extended Kalman filter algorithm also resembles that of a predictor-corrector algorithm for solving numerical problems. It makes an approximation of the system states called the a priori estimate, which is used to predict the measurement that is about to arrive. This estimate is adjusted by the actual measurement, and thus obtains the posteriori estimate. Then the a posterioriestimate is used to predict the measurement at the next time step, and becomes the a priori estimate for the next time step. A Kalman filter that linearizes about the current mean and covariance is referred to as an extended Kalman filter (EKF). Our present an extended Kalman filter approach to optimize the membership functions of the inputs and outputs of the fuzzy controller. The performances of the fuzzy system before and after the optimization are compared, as well as the membership functions [8], [13].

The optimization of fuzzy membership functions can be viewed as a weighted least-squares minimization problem, where the error is the difference between the fuzzy system outputs and the target values for those outputs. We use z to denote the target vector for the fuzzy system outputs, and h (k) to denote the actual outputs at the kth iteration of the training.

Z (k) = [z1 (k)… zL (k)]T (31)

h (k) = [h1 (k)… hL(k)]T (32)

where L denotes the number of outputs of a fuzzy system. In order to cast the membership function optimization problem in a form suitable for Fuzzy Kalman filtering, we let the membership function parameters constitute the state of a nonlinear system, and let the output of the fuzzy system constitute the output of the nonlinear system to which the extended Kalman filter is applied. We denote the centroid, lower half-width and upper half-width of the ith fuzzy membership function of the jth input by cij,

bij-, bij+, respectively and we denote the centroid and the

half-width of the ith fuzzy membership function of the output by γi and βi respectively. The state of the nonlinear

system can be adapted as x = [ b11- b11+ c11 ….. bµ1-

bµ1+b11+ c µ1 b12- b12+ c12 …bV2- bv2+ cv2 β1 γi…..βk γk ]T

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International Journal of Emerging Technology and Advanced Engineering

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In our approach, after casting the fuzzy system to a nonlinear system, to which the extended Kalman filter will be applied, the derivative-based EKF approach is then carried out to optimize the membership functions. In fuzzy extended Kalman filter, except that fuzzy implementation all computation is similar to as above extended Kalman filter. Basic algorithm for fuzzy extended Kalman filters which is implemented in our software as follows:

Let, X (k) is the nonlinear input vector at time k, X_ (k) is the predicted state and it might be change, P defines the state error covariance matrix, Sigma v and sigma w they are measurements noise and process noise random variables respectively.

Q = σv * σv is process noise covariance and R = σw * σw is a measurement noise covariance.

Prediction: The predict state uses the approximate judgment from the previous time state to procedure an approximate of current stage.

x_( i ) = [sin(x_(2,i-1)*(i-1));x_(2,i-1)] (33) z_(i) = x_(1,i) (34)

Fuzzy implementation: Fuzzy Extended Kalman filter has main approach to optimize the membership functions of the inputs and outputs for satisfactory result. Here we introduce membership function to implement fuzzy in extended Kalman filter.

E = (z (i) - z_(i)); E = newfis('tipper');

E = addmf(E,'input',1,'fuzzy','gaussmf', [0 1]; (35) Jacobian function:

F = [0i*cos (x_(2,i)*i);01]; (36) The function used to compute the predicted filter stage

and predicted measurement can‟t be applied to the covariance directly. Instead a matrix of partial derivatives, so called the Jacobian is computed. At each time step, the Jacobian is evaluated with current predicted states. This process essentially linearizes the nonlinear function around the current approximation.

Prediction error covariance:

P =F*P*F '+ G*Q*G' (37) Where Q is process noise covariance, G is the Jacobian of the transfer functions due to the error.

The time update equations are responsible for projecting forward the current state and error covariance estimates to obtain the a priori estimates for next step.

Innovation covariance:

S=H*P*H' + R; (38) Innovation covariance gives the difference between actual measurement and measurement prediction.

Extended Kalman gain:

P12=P*H' (39)

K=P12*inv(H*P12+R); (40) Kalman gain works accordingly to predicted error covariance and measurement noise covariance.

Corrected estimated state:

x_(,i) = x_(:,i)+K*(z(i)-z_(i)) (41) Here we update the corrected estimation of state with measurement z.

To update the corrected error covariance we have used matlab function:

P = (eye(2)-K*H)*P; (42)

Where H is the Jacobian of the transfer functions due to the variables involved

In the actual implementation of the filter, the measurement noise covariance is usually measured prior to operation of the filter. Measuring the measurement error covariance is possible because we need to be able to measure the process anyway (while operating the filter) so we should generally be able to take some off-line sample measurements in order to determine the variance of the measurement noise. The recursive nature is one of the very appealing features of the extended Kalman filter.

VIII. SIMULATIONRESULTS

Kalman Filter

Fig.3. Error performance of Kalman filter

0 10 20 30 40 50 60 70 80 90 100

0 1 2 3 4 5 6 7 8 9 10x 10

37 _{Mean square error}

M

S

E

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International Journal of Emerging Technology and Advanced Engineering

346 Extended Kalman Filter

Fig.4. Error performance of Extended Kalman filter

RBF Kalman Filter

Fig.5. Error performance of RBF Kalman filter

RBF Extended Kalman Filter

Fig.6. Error performance of RBF Extend Kalman filter

Fuzzy Kalman Filter

Fig.7. Error performance of Fuzzy Kalman filter

0 10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90

A

m

p

lit

u

d

e

time Mean square error

50 50.5 51 51.5 52 52.5 53 53.5 54 54.5 55 -0.1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

RBF Kalman output

E

de

0 5 10 15 20 25

0.9 1 1.1 1.2 1.3 1.4 1.5

Epochs

de

0 10 20 30 40 50 60 70 80 90 100

0 1 2 3 4 5 6

Fuzzy Kalman Filter

n(time)

M

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Fig.7. shows the mean square error plot of fuzzy Kalman filter with MSE vs time MSE is the performance parameter of filter we calculated as

MSE (t) = (Y (t) - X (t))2

Where X (t) = original signal at time„t‟ and the Y (t) = estimated signal at time„t‟.

[image:7.612.51.278.233.393.2]

Fuzzy Extended Kalman Filter

Fig.8. Error performance of RBF Extend Kalman filter

Fig.8. shows the mean square error plot of fuzzy extended Kalman filter. The performance parameter of filter is calculated as

MSE (t) = (Y (t) – X (t))2

Where X (t) = original signal at time„t‟ and the Y (t) = estimated signal at time„t‟.

[image:7.612.48.295.518.661.2]

IX. COMPARISION

Table I. MSE

Simple Kalman Filter

RBF Kalman Filter

Fuzzy Kalman Filter

Minimum 0.35e12 0.1421 0.080

Maximum 3.90e12 0.2083 3.350

Range of

MSE 3.55e12 0.0662 3.342

Table II.

MSE

Extended Kalman Filter

RBF Extended Kalman Filter

Fuzzy Extended Kalman Filter

Minimum 0.25 0.09 0.025

Maximum 2.70 0. 35 2.200

Range of

MSE 1.75 0.26 2.175

Table I and Table II shows the comparison of the error performance of the filters.

X. CONCLUSION

The success of neural network architecture depends heavily on the availability of effective learning algorithms. We have estimated performances and compared Kalman filter, RBF Kalman filter and fuzzy Kalman filters on the basis of the mean square errors. It has been observed that fuzzy Kalman filter has less mean square error rate than Kalman and RBF Kalman filters. We have also estimated performances and compared extended Kalman filter, RBF extended Kalman filter and fuzzy extended Kalman filters on the basis of the mean square errors. It has been observed that the fuzzy extended Kalman filter has less mean square error rate than extended Kalman and RBF extended Kalman filters. Simulation results demonstrate that the adaptive fuzzy Kalman filter adaptive fuzzy extended Kalman filter works successfully and efficiently.

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