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CHAPTER 6

CHANNEL ESTIMATION USING KALMAN FILTER

ALGORITHM

6.1 INTRODUCTION

Filtering is desirable in many situations in engineering and

embedded systems. For example, radio communication signals are corrupted

with noise. A good filtering algorithm can remove the noise from

electromagnetic signals while retaining the useful information. The Kalman

filter is a tool that can estimate the variables of a wide range of processes. In

mathematical terms Kalman filter estimates the states of a linear system. The

Kalman filter not only works well in practice, but it is theoretically attractive

because it can be shown that of all possible filters, it is the one that minimizes

the variance of the estimation error.

The Kalman filter is a set of mathematical equations that provides an

efficient computational (recursive) means to estimate the state of a process, in

a way that minimizes the mean of the squared error. The filter is very

powerful in several aspects:

i. It supports estimations of past, present, and even future states.

ii. It can do so even when the precise nature of the modeled

system is unknown.

The power of the Kalman Filter is that it operates online. This

implies that to compute the beat estimate of the state and its uncertainty, the

previous estimates by the new measurement. This implies that the previous

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data again is not considered, to compute the optimal estimates the previous

time step and the new measurement is considered.

The filter is very powerful in several aspects, it supports estimation

of past, present and future states and it can do even when the precise nature of

the modeled system is unknown. The Kalman filter can be thought of as

an estimator that provides three types of outputs given a noisy

measurement sequences and associated models.

The three different forms of Kalman filter are,

(i) State estimator

(ii) Measurement filter

(iii) Whitening filter

Also Kalman filter equation are fall into two groups. They are

time update equations and measurement update equation. Time update

equations are responsible for predicating the state and error covariance

estimates for the next time step. Measurement update equations are

responsible for filtering. Hence time update equations are called predictor

equations and measurement update equations are called corrector

equations. The time update projects the current state estimate ahead in

time whereas the measurement update adjusts the projected estimate by

an actual measurement at that time. The time update equations are

responsible for projecting forward (in time) the current state and error

covariance estimates to obtain the a priori estimates for the next time step.

The measurement update equations are responsible for the feedback for

incorporating a new measurement into the a priori estimate to obtain an

improved a posteriori estimate.

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6.2 AUTO REGRESSIVE (AR) PROCESS

AR models are mathematical models of the persistence, or

autocorrelation, in a time series. AR models are widely used in estimating the

coefficients of filters and other fields. There are several possible reasons for

fitting AR models to data. Modeling can contribute to understanding the

physical system by revealing something about the physical process that builds

persistence into the series.

AR models can also be used to predict behavior of a time series

from past values alone. This type of prediction can be used as a baseline to

evaluate possible importance of other variables to the system. AR models are

widely used for prediction of filter co efficient for rapid variations of channels

based on the knowledge of present and past samples. AR models can also be

used to remove persistence. Any stationary random process can be

represented as an infinite tap AR process.

An infinite tap AR process model is impractical and it is truncated

to an N-tap form. The truncated AR process model to represent the underlying

model driving the tap-gain process. Kalman filter could also be applied to

track the states of time varying channels. The state variable is defined as the

channel states, which can be modeled in an AR model. The state equation is

then built based on the AR model, which reflects the statistics of time varying

channel. A Kalman filter to track multi-path channel in a direct-sequence,

spread-spectrum communication system. In order to develop a model for the

tap-gain process, a complex gaussian random process can be represented by a

general AR model is shown in Figure 6.1

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Figure 6.1 Block diagram of auto regressive process

Any stationary random process can be represented as an infinite tap

AR process. An infinite tap AR process model is impractical. It is truncated to

an N-tap form. It will use the truncated AR process model to represent the

underlying model driving the tap-gain process. The AR process

mathematically can be expressed as,

N

mm=1

g (k) = (6.1)

where g (k) is the complex gain process, m parameters of the model, N is the

number of delays in the auto regressive model and w(k) is noise in which a

sequence of identically distributed zero-mean complex gaussian random

variables.

The block diagram for filter interpretation of AR process is shown

Figure 6.2.

Figure 6.2 Filter interpretation of AR process

w(k)

…..…..

…..

Z-1Z-1 Z-1

hk

x(k)

hk-1 h2 h1

h(k)

w(k) x(k)

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The mean value of AR process is,

N

g mm=1

(6.2)

The variance value of AR process is,

2g

N

mm=1

= E g(k)

N2

m gg nm=1

=

(6.3)

The autocorrelation value of AR process is,

gg

N

mm=1

R (q) = E g(k -q)g(k)

= E

N

m ggm=1

= (6.4)

The autocorrelation coefficient is,

gggg 2

x

N

m ggm=1

R (q)r (q) =

s

=

(6.5)

The AR-process x (k) can also be interpreted as a filtered version of

the driving noise w (k) . In the time domain, the filtering operation is equal to

the convolution and it is mathematically expressed as,

x(k) = h(k) * w(k) (6.6)

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6.2.1 Applications

i. Speech recognition and coding (storage)

ii. System identification

iii. Modelling and recognition of sonar, radar, geophysical signals

iv. Spectral analysis

6.2.2 Reasons for using AR Modeling

i. Each type of process (Moving Average (MA), AR, Auto

Regressive Moving Average (ARMA)) can be converted to

the other types.

ii. AR model can be found by solving linear set of equations,

unlike the others.

iii. AR spectrum can be calculated from a signal of length N at T,

which has better frequency resolution.

iv. AR model can maximize entropy.

v. AR model can have far fewer coefficients than the

corresponding MA model.

6.3 LEAST MEAN SQUARE (LMS) ALGORITHM

The filtering output can be written as,

N-1

ii=0

y(k) = w (k) x(k - i) (6.7)

The estimation of error can be defined as,

e(k)=d(k)-y(k) (6.8)

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Assume the signals are involved are real valued. The LMS

algorithm changes the filter taps weights so that e(k) is minimized in the mean

square sense. When the processes x(k) and d(k) are jointly stationary, this

algorithm converges to a set of tap-weights which on average are equal to the

Wiener-Hopf solution.

The LMS algorithm is a practical scheme for realizing wiener

filters without explicitly solving Wiener-Hopf solution. The conventional

LMS algorithm is a stochastic implementation of the steepest descent

algorithm. It simply replaces the cost function.

2 by its instantaneous coarse estimate 2ˆ

Substituting 2ˆ for in the steepest descent recursion and it is

obtained as,

2= -w(k +1) w (k) (6.9)

where

T0 1 N-1= w (k) w (k) .....w (k)w(k)

0 1 1

....T

Nw w w (6.10)

The ith element of the gradient vector 2e (k) is

2

i i

i

e (k) e(k)= 2e(k)w w

y(k)= - 2e(k)w

= -2 e(k) x(k - i)

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Then 2e (k) = - 2e(k) x (k)

where T= x(k) x(k -1) ...... x(k - N +1)x (k)

The tap-weighted vector adaptation is

= +2w (k +1) w(k) (6.11)

The above equation (6.11) referred as LMS recursion.

Advantages and disadvantages of LMS algorithm

i. Simplicity in implementation

ii. Stable and robust performance against different signal

conditions

iii. Slow convergence due to eignvalue spread.

6.3.1 Normalised LMS Algorithm

The principle characteristics of the normalized LMS algorithm is

based on the adaptation constant ˆ is dimensionless, whereas in LMS, the

adaptation has the dimensioning of a inverse power. The adaptation can be

calculated as,

2a + (6.12)

The normalized LMS algorithm with data dependent updation step size can be

expressed as,

2a +ME (u(k)) (6.13)

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The normalization is such that

i. The effect of large fluctuations in the power levels of the input

signal is compensated at the adaptation level.

ii. The effect of large input vector length is compensated by

reducing the step size algorithm.

The normalized LMS algorithm is convergent in mean square sense if 0 <

The performance analysis of normalized LMS algorithm is shown

in Figure 6.3. The three different level step size 0.075, 0.025 and 0.0075 has

been considered. The value of the step size is decreased the BER is

minimized. In step size 0.0075 the error rate is low at higher time steps

compared with other levels.

Figure 6.3 Performance analysis of LMS algorithm

Erro

r Rat

e

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The performance analysis of normalized LMS algorithm for various

step size is shown in Figure 6.4. For simulation, the step size is taken at four

different levels 1.5, 1.0, 0.5 and 0.1. In step size value is 0.1 the settling time

is high compared with other step size levels. It shows that, the value of the

step size is decreased the settling time is increased. The error rate is low at

µ=1.5 almost 0.05 is high compared with µ=1.0.

Figure 6.4 Performance of various step size µ for normalizd LMS

algorithm

6.4 KALMAN FILTER EQUATIONS

Consider a discrete dynamic process x(k) which can be described

as,

ˆx(k) = (6.14)

Error Rate

Time Step n

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where x(k) is the state vector which includes the parameters to be

estimated, x(k) is the previous sample value of x(k) . is the state transition

matrix and w(k) is the process noise which is entering the system, it is

modeled as zero mean White Gaussian Noise (WGN).

The state prediction equation is,

x(k / k -1) = (6.15)

The error covariance prediction equation is,

Tcp(k / k -1) = (6.16)

where cQ is the variance of white gaussian noise.

The prior value of estimated error is,

e(k / k -1)= x(k) - x(k -1 / k -1) (6.17)

The posteriori value of estimated error is,

e(k)= x(k) - x(k / k -1) (6.18)

where x(k / k -1) is the priori state estimation.

The priori estimate error covariance is,

Tp(k / k -1) = E e(k / k -1) e(k / k -1) (6.19)

The posteriori estimate error covariance is,

Tp(k) = E e(k) e(k) (6.20)

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The measurement update equations in order to compute the Kalman

gain ( kK ) and it can be mathematically expressed as,

T T -1kK = p(k / k -1) H (H p(k / k -1) H + R)

T

T

p(k / k -1) H=Hp(k / k -1)H + R

(6.21)

where H is the predicted measurement and R is the measurement

of error covariance.

The state smoothening equation is,

kx(k / k -1) = x(k -1 / k -1)+ K e(k) (6.22)

where kK is the Kalman gain.

The vector differences between actual and predicted

measurements are,

e(k) = z(k) - H x(k -1 / k -1) (6.23)

The posterior estimation of the Kalman filter is,

kp(k) =(1 - K H)p(k -1 / k -1) (6.24)

The predicted error covariance matrix is defined in terms of

zero mean Gaussian estimation error vector.

TP(k / k -1) = E{[x(k) - x(k / k -1)][x(k) - x(k / k -1)] } (6.25)

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The filtered error covariance matrix is written as,

T}P(k -1 / k -1) = E{[x(k) - x(k -1 / k -1)][x(k) - x(k -1 / k -1)] (6.26)

After each time and measurement update pair, the process is

repeated with the previous a posteriori estimates used to predict the new a

priori estimates. This recursive nature is one of the very appealing features of

the Kalman filter. It makes practical implementation much more feasible than

an implementation of Wiener filter which is designed to operate on all of the

data directly for each estimate. Figure 6.5 shows a complete picture of the

operation of the filter.

Figure 6.5 Parameters updating for Kalman filter

The general block diagram of a Kalman filter is given in Figure 6.6.

The initial estimates for x(k) and p(k) depends on the state vector. For each

time period the vector z(k) is observed and error matrix e(k) are computed.

In some conditions, the error covariance are assumed as constants, both

the estimation error covariance and Kalman gain will stabilize quickly

Time update

(a) Predict the state matrix

of next state

(b) Predict the error

Measurement update

(a) Compute the Kalman gain

(b) Update the estimate

(c) Update the error

covariance matrix using

observed vector

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and then remain constant. Sometimes, these parameters can be precomputed

by either running filter off-line.

Figure 6.6 Block diagram of Kalman filter

6.4.1 Characteristics of Kalman Filter

i. It is optimal for adaptive process.

ii. It is recursive, the estimates are updated upon receipt of each

measurement.

iii. Kalman filter creates its own error analysis

iv. Kalman filters are easily reconfigured to handle wild data

points and model changes.

v. Kalman filter operates in vector/matrix format concepts and

operations are independent of number of states.

6.5 KALMAN FILTER BASED CHANNEL ESTIMATION FOR

OFDM SYSTEM

Kalman filter channel estimation algorithm is found to be best

suited for UWB OFDM systems for minimizing the errors in noisy

environments and to obtain an optimal estimate of desired quantities from

data provided by adaptive process. The Kalman filter is also used for

+ Z-1

H(k)

+Z(k) x(k+1) x(k)

y(k)

n(k) v(k)

140

predicting the future coefficients of an adaptive control system which will be

more useful for tuning the filter coefficients for fast fading channel.

Blind Kalman algorithm is proposed to blindly estimate the channel

parameters for an MB OFDM system. The channel parameters are considered

as states and the received signal is considered as measurements (Chen &

Wang 2000). The future value of the channel parameters are predicted by

using the past observations (Lawrence Kanfan 1971).

Let T0 1 N-1h(n) = [h (n),h (n), .....,h (n)] is the channel vector of size

N 1. Where N is number of subcarriers in an OFDM system and kh (n) is the

channel impulse response of kth subcarrier at nth instant. The proposed

method performs estimation in frequency domain. The choice of AR process

order ‘p’ represents the tradeoff between accuracy of the model and difficulty

in estimating channel parameters.

The state transition equations for the proposed method using

Kalman filter can be represented as

X(k)= X(k-1)+n(k)

X(k) is the channel state matrix which contains the channel vector

of order ‘p’. Hence the channel state matrix can be written as T T TX(k)=[h (k),........,h (k-p+1)] of size pN x 1. is the state transition matrix.

N N N

N N N

-A 1 -A 2 -A[p]I 0 0

=

0 I 0

(6.27)

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where NI and N0 are N×N identity matrix and zero matrix respectively.

A 1 ,A 2 ,......,A p are the state transition matrix for the respective state

vectors of each size N×N . The parameters in the matrix ‘A’ are computed by

using Yule-Walker equations from correlation of the channel gain.

According to input and output relation of an OFDM, the

measurement equation is

y(k)= B(k) X(k) + u(k) (6.28)

where T1 Ny(k) =[y (k),.....,y (k)] and iy (k) is the received signal value of the ith of

kth OFDM symbol at time instant ‘t’. TN NB(k)=[ (k),0 ,......,0 ]s and u(k) is the

measurement noise. (k)s is a N N diagonal matrix and it can be written as

OFDMˆS(k) = S (k)HNF . H

NF is the IFFT matrix and OFDMS (k) is the estimate of the

received symbol using previous state channel vector.

In the state space model, the above mentioned unknown parameters

channel state vector ‘h’ can be estimated through Kalman algorithm.

The Kalman gain K(k) of size pN pN can be expressed as

HK(k)= p(k/k-1)B (k)s(k) (6.29)

The innovation residual covariance matrix is N

1H 2S(k)= B(k)p(k/k-1)B (k)+ .

Where NI is the identity matrix of size N N and the covariance noise

assumed as uncorrelated.

The predicted state error covariance matrix p(k) is

Hp(k/k-1) = Cp (k-1)C + Q(k-1) (6.30)

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Q is the covariance matrix of the state noise and it is considered as identity

matrix multiplied by the variance 2 . The filtering error covariance matrix can

be written as

pNp(k)=[I -K(k)]p(k/k-1) (6.31)

The estimated channel matrix at time t is then expressed as

X(k) = CX(k-1) + K(k)e(k) (6.32)

where e(k) is the innovation vector and it is varied depends on the received

signal vector. The innovation vector can be written as

e(k) = y(k) - B(k) CX(k-1) (6.33)

y(k) is the received signal vector and B(k) is explained in the equation. The

channel estimate at instant time ‘t’ is

N N Nh(k)=[I ,0 ,......,0 ]X(k) (6.34)

Initial conditions for the above Kalman filter equation,

p(0)=I, Q(0)=I and X(0)=I (6.35)

The Performance of BER has been analyzed using Kalman Filter

based estimator is shown in Figure 6.7. It shows the BER analysis of the

proposed Kalman based estimator for different channel models. It can be seen

that the performance of CM 1 gives the lower error rate compared with other

channel models. In CM 4, the error rate is high compared to CM 2 at 50 dB

SNR levels.

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Figure 6.7 BER analysis of Kalman based estimator

0 5 10 15 20-45

-40

-35

-30

-25

-20

-15

-10

-5

SNR in dB

RLS Kalman

Figure 6.8 Comparison of RLS and Kalman based estimation

NM

SE in

dB

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Figure 6.8 shows the MSE comparison of RLS and Kaman filter

algorithms. It can be seen that the Kalman filter based channel estimation

gives almost 7 dB better performance in terms of MSE at 10 dB SNR level.

6.6 CONCLUSION

Kalman Filter algorithm has been developed for MB OFDM

channel models. State space equations are derived. In this estimation, the

knowledge of priori and posteriori channel datas are estimated. The

performance of the BER has been analyzed for various channel models CM 1

to CM 4. It shows that, when the distance is increased between the channels

the BER is increased . This means for CM 4 channel model, the BER is high

as compared to CM 1 channel model. The MSE comparison of RLS and

Kalman filter is done.