Digital Signal Processing Reference
In-Depth Information
of the same unit circle. This illustrates that the FFT is least suitable for:
(i) processing FIDs with poor SNR, and for (ii) solving the quantification
problem via postprocessing total shape spectra from the FFT in an attempt
to extract peak parameters, as done in the usual fitting algorithms in MRS,
e.g., VARPRO, AMARES, LCModel, etc.
5.5 Model reduction problem via Pade canonical spectra
Employing (5.12), the canonical representations from (5.9) can be rewritten
in the following forms
K
T
(z
±1
−z
±
r,P
)
P
±
K
T
(z
±1
)
Q
±
K
T
(z
±1
)
=
p
±
K
q
±
K
r=1
K
T
(z
±1
−z
±
s,Q
)
s=1
8
<
:
9
=
;
r
′
,s
′
∈K
F
K
G
+K
F
K
G
(z
±1
−z
±
r
′
,P
)
(z
±1
−z
±
r,P
)
=
p
±
K
q
±
K
r
′
=K
G
+1
r=1
. (5.17)
K
G
K
G
+K
F
(z
±1
−z
±
s,Q
)
(z
±1
−z
±
s
′
,Q
)
s=1
s
′
=K
G
+1
The canonical quotient in the curly brackets of (5.17) is equal to unity. This
stems from the exact cancellation of the numerator and denominator polyno
mials, as a result of the coincidence of the invoked poles and zeros via (5.11)
in Froissart doublets. Thus, we can simplify (5.17) as
K
G
+K
F
K
G
(z
±1
−z
±
r,P
)
(z
±1
−z
±
r,P
)
P
±
K
T
(z
±1
)
Q
±
K
T
(z
±1
)
=
p
±
K
q
±
K
=
p
±
K
q
±
K
r=1
r=1
. (5.18)
K
G
+K
F
K
G
(z
±1
−z
±
s,Q
)
(z
±1
−z
±
s,Q
)
s=1
s=1
Upon convergence in the FPT
(±)
, it follows
p
±
K
G
+K
F
= p
±
K
G
(5.19)
q
±
K
G
+K
F
= q
±
K
G
. (5.20)
Using (5.19) and (5.20), we can reduce (5.18) to the sought relationship
P
±
K
G
+K
F
(z
±1
)
Q
±
K
G
+K
F
(z
±1
)
=
P
±
K
G
(z
±1
)
Q
±
K
G
(z
±1
)
(K
F
= 1, 2, 3,...).
(5.21)
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