Digital Filters - Digital Signal Processing: A Practitioner's Approach - page 72

To bring this into standard form, we put

t

u ¼½ cos m sin m k t cos m k t sin m

t

and b ¼½ x o y o v x v y . Representing the output as a scalar y k ¼ x w cos m y w

sin m , we get

t

u b ¼ y k : ð 3 :

54 Þ

3.11.3.2 Pseudolinear Solution

The above formulation is a pseudo MA filter. This is an inverse problem of finding b

given y k and . The solution can be of block type or recursive type. For simplicity

we show a block solution which is given as

u

i 1 P i ¼ 1 ðu i y i Þ ;itis
depicted in Figure 3.31(a). It shows the motion of the watcher or observer, the
motion of the target and the estimated motion of the target. Figure 3.31(b) shows
the angle ð m Þ measurements.

b ¼ P i ¼ 1 u i u

20 60

Calculated

50

15

40

10

Real 30

Watcher

5

20

0 10

-5 0 5 10 0 100 200 300 400 500

Target and watcher Seconds

(b)

(a)

Figure 3.31 Watcher and target in (3.54)

A careful look at Figure 3.31(a) reveals that the estimate is biased, even though
the direction is correct. This is because we have introduced a small amount of noise
in the measured angle m . In fact, it is a very small bias but still significant. A

considerable amount of published work is available in the area known as target
motion analysis (TMA). There are many methods to remove this bias, such as
instrumental variable (IV) methods, maximum likelihood (ML) methods and IV
approximate ML (IVAML) estimators. There are many associated problems still to
be researched.