chapter 6 channel estimation using kalman filter...
TRANSCRIPT
126
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
127
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.
128
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
129
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)
130
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)
131
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)
132
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)
133
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)
134
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
135
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
136
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)
137
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)
138
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
139
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)
141
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)
142
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.
143
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
144
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.