Digital Signal Processing Reference
In-Depth Information
By computing the Cauchy residues of (2.191), we are led to the following
expressions for the Pade amplitudes
=
p
±
K−1
q
±
K
K−1
k
′
=1
(z
±
k
−z
±
k
′
)
d
±
k
(2.192)
K
k
′
=1,k
′
=k
(z
±
k
−z
±
k
′
)
that are equivalent to (2.183). Such a procedure is valid only for the purely
Lorentzian lineshapes encountered in a nondegenerate spectrum with non
coincident fundamental frequencies. By this algorithm we can also reconstruct
the corresponding FID as a sum of K attenuated complex harmonics with
timeindependent amplitudes d
±
k
K
e
inω
±
k
τ
c
±
n
=
d
±
k
c
±
n
Im(ω
±
k
) > 0.
≈c
n
(2.193)
k=1
The expounded concept ought to be modified if one or more confluent (coin
cident) fundamental frequencies are present in the input FID. In such a case,
more than one amplitude corresponds to the same given frequency. This is
properly described by inclusion of multiple roots of the characteristic polyno
mials Q
±
K
(z
±1
). The corresponding lineshapes are nonLorentzians and they
are associated with the degenerate part of the entire spectrum. Let us assume
that there are J≤K degenerate fundamental frequencies in the input FID.
Further, let the k th degenerate frequency have the multiplicity m
k
whose
maximal value is denoted by M
k
M
k
= max{m
k
} M
1
+ M
2
++ M
J
= K.
(2.194)
The inequality 1≤m
k
≤M
k
with m
k
> 1 implies that the k th root of
Q
±
K
is repeated m
k
times. The special case m
k
= 1 is associated with non
degeneracy. When all the multiplicities m
k
of the k th root of Q
±
K
(z
±1
) are
included, the Cauchy residues of the quotient
P
±
K
(z
±1
)/Q
±
K
(z
±1
) become
P
±
K−1
(z
±
k
)
[(d/dz
±1
)
m
k
Q
±
K
(z
±1
)]
z
±1
=z
±
k
D
±
k,m
k
=
.
(2.195)
These expressions extend (2.183) to the case when there are fundamental
frequencies that coincide exactly with each other. The associated expressions
that generalize (2.188) to the case of simultaneous presence of simple and mul
tiple roots of Q
±
K
(z
±1
) in the FPT
(±)
are represented by the mixed complex
spectra
P
±
K−1
(z
±1
)
Q
±
K
(z
±1
)
J
M
k
D
±
k,m
k
(z
±1
−z
±
k
)
m
k
.
G
±
K
(z
±1
) =
=
(2.196)
m
k
=1
k=1
These results permit the exact reconstruction of the general form of the FID
for degenerate and nondegenerate fundamental frequencies. The degenerate
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