difference between kalman filter & extended kalman filter

[Lekshmi B S]

extended kalman filter estimation

Hi all,,

What is the difference between kalman filtering and extended kalman filtering channel estimation...can nyone tel how they differ in their equations..i would also like to know the differences in formulating matlab codes for them...for sinusoidal wave as input, is an extended kalman filter must?

Thank you....

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Actually,,i am doing channel estimation using extended kalman filter for a sine wave input..this is the code i obtained..can anyone suggest whether any modifications or corrections are required....
I got the parameters from internet and so i am not familiar about why these values are used...can u please explain why these values............

Thank you..

clc;
clear all;
close all;
N = 350;
W = [ 1 0 ; 0 1 ]; % State-error
V = [ 1 0 ; 1 0 ]; % Measurement-error
H = [ 1 0 ; 0 0 ]; % Measurement-state
Q = [ 0.001 0 ; 0 0 ]; % Process noise covariance
R = [ 0.1 0 ; 0 0.01 ]; % Measurement noise covariance

x = zeros(2,N);
apriori_state = zeros(2,N);
aposteriori_state = zeros(2,N);
apriori_error = zeros(2,2,N);
aposteriori_error = zeros(2,2,N);
z = zeros(2,N);
K = zeros(2,2,N);

x(:,1) = [ 1 ; 3*pi/500 ];
aposteriori_state(:,1) = [ 1 ; 1*pi/500 ];
aposteriori_error(:,:,1) = [ 1 0 ; 0 1 ];
error2(:,1)=aposteriori_state(:,1)-x(:,1);
randn('state',2);
n1=sqrt(Q)*[randn;randn];
n2=sqrt(R)*[randn;randn];

for i = 2:N    
    x(:,i) = [sin(x(2,i-1)*(i-1));x(2,i-1)] + n1;
    z(:,i) = x(:,i) + n2;
    apriori_state(:,i) = [sin(aposteriori_state(2,i-1)*(i-1));aposteriori_state(2,i-1)];
    Fi = [0 (i-1)*cos(aposteriori_state(2,i-1)*(i-1)) ; 0 1 ];
    Qi = Q;
    Ri = R;
    apriori_error(:,:,i) = Fi*aposteriori_error(:,:,i-1)*Fi' + Qi;
    K(:,:,i) = apriori_error(:,:,i)*H' / (H*apriori_error(:,:,i)*H'+Ri);
    aposteriori_state(:,i) = apriori_state(:,i) + K(:,:,i) * (z(:,i) - apriori_state(:,i));
    aposteriori_error(:,:,i) = (eye(2) - K(:,:,i)*H) * apriori_error(:,:,i);
    error1(:,i)=apriori_state(:,i)-x(:,i);
    error2(:,i)=aposteriori_state(:,i)-x(:,i);
 end

figure(1);
t1=2:N;
subplot(3,1,1);
plot(t1,x(1,2:N));
ylim([-2 +2]);
title('Actual state');
xlabel('Time');
ylabel('Position');
grid on;
subplot(3,1,2);
plot(t1,apriori_state(1,2:N));
ylim([-2 +2]);
title('Predicted state');
xlabel('Time');
ylabel('Position');
grid on;
subplot(3,1,3);
plot(t1,aposteriori_state(1,2:N));
ylim([-2 +2]);
title('Updated state');
xlabel('Time');
ylabel('Position');
grid on;

figure(2);
t2=2:N;
subplot(2,1,1);
plot(t2,error1(1,2:N));
title('Error between actual value and predicted value');
xlabel('Time');
ylabel('Position');
grid on;
subplot(2,1,2);
plot(t2,error2(1,2:N));
title('Error between actual value and updated value');
xlabel('Time');
ylabel('Position');
grid on;