The role of quantum mechanics in signal processing - Signal Processing in MRS with Biomedical Applications - page 74

part of such an FID has the confluent harmonics given by a linear combination
of attenuated complex exponentials with coe cients that are nonstationary
(timedependent) amplitudes d ± k,n . Such a time dependence of the amplitudes
is given by a polynomial of degree M k so that

J M k

d ± k,n e inω ± k

d ± k,n D ± k,m k (nτ) m k −1

c ± n

τ

= = (2.197)

m k =1

k=1

where Im(ω ± k ) > 0. Hence, the simple analytical formulae also exist in the
FPT (±) for degenerate amplitudes in the case of nonLorentzian spectral
lineshapes with coincident resonances having exactly the same real parts of
the corresponding complex fundamental frequencies. This outlined analysis
demonstrates that the two GreenPade versions, the FPT (+) and FPT (−) , are
able to treat both Lorentzian and nonLorentzian spectra on the same foot
ing. Nondegenerate and degenerate spectra correspond to FIDs represented
by linear combinations of K damped complex exponentials with stationary
and nonstationary amplitudes, respectively.

2.8 Determination of the exact number K of resonances
Maximally accurate retrieval of the unknown true number K of physical reso
nances in a given FID is of primary importance. In the FPT, this number K
is determined exactly. Moreover, K can be reconstructed in either the time or
frequency domain analysis, by employing the Shanks transform [5] and Frois
sart doublets [44], respectively. The main outlines of the mathematical basis
upon which this is achieved will be presented in this section (for a number of
complementary aspects, see chapter 5).

2.8.1 Exact Shank's filter for finding K, including the funda-
mental frequencies and amplitudes: the use of Wynn's
recursion
In the case of (2.193) or (2.197), the Shanks transform, which is denoted by
e K (c n ), is defined as [5]

e k (c n )≡ H k+1 (c n )
H k ( 2 c n )

1
W k

H k+1 (c n )
H k (c n )

= (2.198)

k

W k = (z k ′ −1) = 0 (2.199)

k ′ =1