Digital Filters - Digital Signal Processing: A Practitioner's Approach - page 73

3.12 Kalman Filter
The output of a filter is completely characterised by its coefficients and the initial
conditions (Figure 3.31A).

Filter coefficients (known)

γ k Noise

y k (m easured)

x k

u k ( known)

Σ

Digital filter

State of the filter (to be estimated)

Figure 3.31A Kalman filter: the output is completely characterised by the filter coefficients
and the initial conditions

For a simple explantion, consider (3.16), where the infinite sequence is generated
by a 3-tuple f 1 0 g in which there are two components: coefficients, known as

parameters; and the other values at various places inside the filter before the delay
elements, known as the state of the filter.
The output is generally a linear combination of these states. The aim of the
Kalman filter is to accurately estimate these states by measuring the noisy output.
The following examples illustrate the approach and its application.

: 8 ; 1 ;

3.12.1 Estimating the Rate
Estimating the rate occurs in radar, tracking and control. Let us consider a sequence
of angular measurements f 1 ; 2 ; ...; N g over a relatively small window.

It is required to find the slope or the rate. The problem looks simple, but using
conventional Newton forward or backward differences produces noisy rates.
Consider the simple plot of

k in Figure 3.32.
Now we place the x and y axes at the desired point, as shown in Figure 3.32. Note
that we have moved only the y-axis to the desired place. Then we have
¼ mk t þ c. Here we need to estimate the value of m. This choice of coordinate

frame ensures that k ¼ 0 at all times. We can write

0
@

1
A

0
@

1
A

1
2
.
N

0 1

or

t 1

m
c

¼

. .

¼ Ap: ð 3 : 55 Þ

ð N 1 Þ t 1