Digital Filters - Digital Signal Processing: A Practitioner's Approach - page 74

1.5

Moving window

1

0.5

y - axis

0

-0.5

-1

- axis

x

-1.5

0 1 2 3 4 5 6 7

Time

Figure 3.32 Linear regression over a rectangular moving window

Now the solution for such overdetermined equations in a least-squares sense is
p ¼ A

Þ 1

t t

ð A A , where our interest is p ð 1 Þ¼ m. We can derive a closed-form

solution as

0 1

6
N ð N 1 Þ

12
N ð N 2

¼

@ A A

m
c

1 Þ t

t

: ð 3 : 56 Þ

6
N ð N 1 Þ

2 ð 2N þ 1 Þ
N ð N 1 Þ t

Even though (3.56) looks very complicated, it is only a pair of FIR filters with their
impulse responses given as h k ¼f 2 ; ...g and h k ¼f 1

; 4 ; 6 ; 8 ; 1 ; 1 ; 1 ; ...g . For a six-

point moving linear regressor, the rate is given as

( " # " # ) 1

6 2

: ð 6 þ 1 Þ X X

6 1
i ¼ 0 k i

6 1

1

_

k ¼ 2 ð i þ 1 Þ k i ð 3 : 57 Þ

t 1

i ¼ 0

This is shown in Figure 3.33, which is a SIMULINK block diagram. In fact, this is a
Kalman filter or a state estimator.

3.12.2 Fitting a Sine Curve
There are many situations where you know the frequency, but unfortunately, by the
time you get the signal, it is corrupted and of the form y k ¼ sin ð 2

fk Þþ n k , where
n k is a zero-mean uncorrelated noise. The problem is to get back the original signal.
Immediately what strikes our mind is a high-Q filter. Unfortunately, stability is an
issue for these filters. In situations of this type, we use model-based filtering.
Consider an AR process representing an oscillator (poles on the unit circle):

x k ¼ px k 1 x k 2 where 2 p 2 : ð 3 : 58 Þ