1. Kalman Filtering
Presented by
Muhammad Irfan Anjum

2. Outline
 Introduction
 Dynamical Signal Models
 Scalar Kalman Filter
 Vector Kalman Filter
 Extended Kalman Filter
 Simulation results

3. Introduction
• Uses a series of measurements over time, and produces estimates of
unknown variables that tend to be more precise than those based on a
single measurement alone.
• Operates recursively on streams of noisy input data to produce a
statistically optimal estimate of the underlying system state.
• Two step process
– Estimates of current state variables with their uncertainties.
– Estimates are updated using weighted average after observing
output.
• Operates on real time data, no additional past information is required.

4. Dynamical Signal Models
x[ n ]
x[ n ]
ˆ
A[ n ]
A
w[ n ]
A[ n ]
w[ n ]
x[ n ]

5. Gauss Markov Process
1st order Gauss Markov Process:
s[ n ]
as [ n
1]
u[ n ]
u [n ]
n
s[ n ]
a
n 1
k
s [ 1]
a u [ n 1]
s[n ]
B
k 0
E ( s [ n ])
a
n
1
s
Az
Vector Gauss-Markov Model:
s[ n ]
As [ n
1]
Bu [ n ], n
0
n
s[ n ]
A
n 1
k
s [ 1]
A Bu [ n
k 0
1]
1

6. Scalar Kalman Filter
s[n ]
u [n ]
az
1
(a) Dynamical Model
x[n ]
s[n ]
ˆ
u[ n ]
~[ n ]
x
ˆ
s[ n | n ]
K [n]
w[n ]
az
ˆ
s[ n | n 1]
(b) Kalman Filter
1

7. Scalar Kalman Filter
Transmitted Signal:
s[ n ]
as [ n
1]
Received Signal:
x[ n ]
Prediction:
ˆ
s[ n | n
1]
ˆ
a s[ n
M [n | n
1]
a M [n
s[ n ]
u[ n ]
w[ n ]
1| n
1]
Minimum Prediction MMSE:
Kalman Gain:
2
1| n
1]
M [n | n
K [n]
2
ˆ
s[ n | n ]
Minimum MSE:
M [n | n]
ˆ
s[ n | n
(1
1]
M [n | n
w
Correction:
2
u
1]
K [ n ]( x[ n ]
K [ n ]) M [ n | n
1]
ˆ
s[ n | n
1]
1])

8. Vector Kalman Filter
u [n ]
s[n ]
B
Az
x[n ]
s[n ]
1
ˆ
u[ n ]
~[ n ]
x
ˆ
s[ n | n ]
K [n ]
h[n ]
w[n ]
h[n ]
Az
ˆ
s[ n | n 1]
1

9. Scalar state Vector Kalman Filter
Transmitted Signal:
s[ n ]
As [ n
Received Signal:
x[ n ]
h [ n ] s[ n ]
Prediction:
ˆ
s[ n | n
Minimum Prediction MMSE:
Kalman Gain:
1]
Bu [ n ]
T
ˆ
A s[ n
1]
M [n | n
1]
2
n
ˆ
s[ n | n ]
Minimum MSE:
M [n | n]
1| n
AM [ n
M [n | n
K [n]
Correction:
w[ n ]
1| n
T
1]
1]
BQB
T
1] h [ n ]
h [ n ]M [ n | n
ˆ
s[ n | n
(I
1]
1] h [ n ]
K [ n ]( x [ n ]
T
ˆ
h [ n ] s[ n | n
K [ n ] h [ n ]) M [ n | n
T
1]
1])

10. Vector state Vector Kalman Filter
Transmitted Signal:
Received Signal:
Prediction:
s[ n ]
As [ n
x[ n ]
H [ n ] s[ n ]
ˆ
s[ n | n
Minimum Prediction MMSE:
Kalman Gain:
1]
w[ n ]
ˆ
A s[ n
1]
M [n | n
Bu [ n ]
1]
1| n
AM [ n
M [n | n
K [n]
C [n]
Correction:
ˆ
s[ n | n ]
Minimum MSE:
M [n | n]
1| n
1] H
H [ n ]M [ n | n
ˆ
s[ n | n
(I
1]
1]
T
1]
BQB
T
[n]
1] H
T
K [ n ]( x [ n ]
K [ n ] H [ n ]) M [ n | n
[n]
ˆ
H [ n ] s[ n | n
1]
1])

11. Extended Kalman Filter
s[ n ]
Extended Kalman Filter
As [ n
x[ n ]
Vector Kalman Filter
1]
Bu [ n ]
H [ n ] s[ n ]
w[ n ]
a ( s[ n
1])
x[ n ]
a ( s[ n
s[ n ]
h ( s[ n ])
w[ n ]
ˆ
a ( s[ n
1])
1| n
Bu [ n ]
a
1])
s[ n
ˆ
h ( s[ n | n
h ( s [ n ])
1])
h
s[ n ]
A[ n
1]
a
s[ n
1]
|s[ n
ˆ
1 ] s [ n 1| n 1 ]
H [n]
1]
|s[ n ]
h
s[ n ]
|s[ n
ˆ
1 ] s [ n 1| n 1 ]
ˆ
s [ n |n 1]
|s[ n ]
ˆ
s [ n 1| n 1 ]

12. Extended Kalman Filter
ˆ
s[ n | n
1]
ˆ
a ( s[ n
M [n | n
1]
A[ n
1| n
1] M [ n
1])
1| n
M [n | n
K [n]
C [n]
T
1] A [ n
1]
1] H
H [ n ]M [ n | n
ˆ
s[ n | n ]
ˆ
s[ n | n
M [n | n]
(I
1]
T
T
BQB
[n]
1] H
T
K [ n ]( x[ n ]
K [ n ] H [ n ]) M [ n | n
1]
[n]
ˆ
h ( s[ n | n
1]))

13. Particle Tracking using Scalar Kalman filter

14. MMSE in Scalar Kalman filter particle tracking

15. Particle Tracking using Vector Kalman filter

16. Particle Tracking using Extended Kalman filter