Digital Signal Processing Reference
In-Depth Information
FAST PADE TRANSFORM INSIDE and OUTSIDE the UNIT CIRCLE, FPT
(+)
(Left) and FPT
(−)
(Right)
Distributions of Poles in a Complex Plane
Distributions of Poles in a Complex Plane
1.5
1.5
Harmonic variables (dots): Pade = Exact
Exact: z
k
= e
2i
πτ
ν
k
, Pade: z
+
Harmonic variables (dots): Pade = Exact
Exact: z
−1
+
k
−
k
k
= e
2i
πτ
ν
k
= e
−2i
πτ
ν
k
, Pade: z
−
k
= e
−2i
πτ
ν
1
1
Lip
Analytical
continuation
inside
the unit circle:
|z| < 1
0.5
FPT
(+)
0.5
N
P
= 256
0
Convergence
region:
|z| < 1
H
2
O
0
H
2
O
−0.5
−0.5
N
P
= 256
FPT
(−)
Convergence region: |z| > 1
−1
−1
Analytical continuation
outside the unit circle: |z| > 1
Lip
−1.5
−1.5
−1
0
1
−1
0
1
(i) Re(z
+
(iv) Re(z
−
k
)
k
)
Absorption Total Shape Spectrum
Absorption Total Shape Spectrum
18
18
PADE: FPT
(+)
PADE: FPT
(−)
NAA
B
0
=1.5T
NAA
B
0
=1.5T
N
P
= 256
N
P
= 256
14
14
Cr
Cr
PCr
NAAG
PCr
NAAG
10
10
Cho
PCho
Cho
PCho
Gaba
Gaba
Glu
Gln
Glu
Gln
Cr
Gln
Cr
Gln
Glu
Ins
Tau
Glu
Ins
Tau
PCr
PCr
NAA
Gaba
NAA
Gaba
6
6
Mobile
Lip (4x)
Mobile
Lip (4x)
Asp
Asp
Asp
Asp
H
2
O
H
2
O
PCho
PCho
2
2
0
0
5
4
3
2
1
5
4
3
2
1
(ii) Chemical shift (ppm)
(v) Chemical shift (ppm)
Absorption Component Shape Spectra
Absorption Component Shape Spectra
18
18
PADE: FPT
(+)
PADE: FPT
(−)
B
0
=1.5T
B
0
=1.5T
N
P
= 256
N
P
= 256
14
6
14
6
10
10
15
15
6
7
6
7
16
16
22
22
8
8
9
9
20
21
19
21
19
20
10
10
17
14
12
17
18
14
12
5
5
3
3
2
1
2
2
1
2
25
25
18
24
13
11
4
24
13
11
4
23
23
0
0
5
4
3
2
1
5
4
3
2
1
(iii) Chemical shift (ppm)
(vi) Chemical shift (ppm)
FIGURE 3.10
The initial convergence regions (top panels), absorption total shape spectra
(middle panels) and absorption component shape spectra (bottom panels)
in the FPT
(+)
(left) and FPT
(−)
(right). The middle panels display the
usual acronyms for the main MRdetectable metabolites in the healthy human
brain, whereas the bottom panels give the corresponding numbers of these
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