Image Processing Reference
In-Depth Information
on linear prediction (LP) methods and autoregressive (AR) modeling of the FID,
whereas the latter are based on NLLS optimization to directly fit the damped
sinusoid model on the data.
The first example of the black-box approach applied to the analysis of FID
data is the so-called linear prediction singular value decomposition (LPSVD)
method due to Barkhuijsen [35], who applied the well-known approach by
Kumaresan and Tufts (KT) [36]. The aim is to estimate the amplitude and
frequency of K -complex damped sinusoids embedded in noise when N data
samples of the process are observed. v k = [ f k , a k , d k ,
φ k ] are unknown parameters.
The method is based on the observation that signal x ( n ) can be modeled by an
AR process according to the following backward equation system
K
1
xn
()
=
bxn k
(
+
)
+
wn
()
(12.12)
k
k
=
where the b k 's are the backward prediction coefficients. When Equation 12.12 is
used to describe the process generated by Equation 12.1, an interesting relation
does exist between b k 's and signal frequencies:
K
K
Bz
()
=+
1
bz
k
= −
(
1
e
−+ −
dj
2
π
f
z
1
)
(12.13)
k
k
k
k
=
1
k
=
1
Thus, the unknown frequencies in Equation 12.1 are easily obtained by
estimating the b k 's and by rooting the polynomial B ( z ). The identification of b k 's
is obtained by solving the following linear system derived from Equation 12.12
Ab
=−
h
(12.14)
where
x
*( )
1
x
*( )
2
x
*( )
L
x
*( )
2
x
*( )
3
x
*(
L
+
1
)
A
=
xN
*(
Lx
)
* (
NL
− +
1
)
x
* (
N
1
)
(12.15)
b
b
()
()
1
2
x
x
*( )
*( )
0
1
b
=
h
=
bL
()
xNL
*(
−−
1
)
Search WWH ::




Custom Search