Digital Signal Processing Reference
In-Depth Information
p
A
y
Finding a best-fit straight line involves estimating
from the given
and
.
t
, we get
Premultiplying (6.15) by
A
t
t
A
y ¼ðA
AÞp:
ð
6
:
16
Þ
t
Here
ðA
AÞ
is a 2
2 matrix and if its inverse exists, we can write
p ¼ A
þ
y
ð
6
:
17
Þ
1
t
t
¼ðA
AÞ
A
y;
ð
:
Þ
6
18
A
þ
is known as the Moore-Penrose pseudo-inverse [1]. Equation (6.18) can be
rewritten as
where
p
is the estimate of
p
corresponding to the best-fit straight line and
!
!
1
X
N
i
¼
1
x
i
x
X
N
i
¼
1
x
i
y
i
t
i
p ¼
ð
6
:
19
Þ
1
;
S
xx
S
x
S
xy
S
y
¼
ð
6
:
20
Þ
S
x
N
where
x
i
is defined by (6.12) and the terms S
xx
;
S
x
;
S
y
;
S
xy
are defined in (6.22) and
(6.23).
But the above solution is computationally intensive as it involves matrix multi-
plications and a matrix inversion. A recursive solution is available for (6.19) but
even though it is not as computationally intensive, it still requires considerable time
[2, 3].
6.3.3.1 Closed-Form Solution
To avoid such time-consuming calculations, a closed-form solution for (6.20) is
used [4]. The expressions are
p
1
¼
ð
NS
xy
S
x
S
y
Þ
ð
p
2
¼
ð
S
y
p
1
S
x
Þ
N
^
and
^
;
ð
6
:
21
Þ
S
x
2
NS
xx
Þ
where
X
X
N
N
S
x
¼
x
i
and
S
y
¼
y
i
;
ð
6
:
22
Þ
i
¼
1
i
¼
1
X
X
N
N
x
i
S
xx
¼
and
S
xy
¼
x
i
y
i
:
ð
6
:
23
Þ
i
¼
1
i
¼
1
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