Machine accurate quantification and illustrated signal-noise separation - Signal Processing in MRS with Biomedical Applications - page 205

6.5 Signal-noise separation via Froissart doublets with

pole-zero coincidences

Figures 6.1 -6.14 reveal how the concept of Froissart doublets within the
FPT (+) and FPT (−) can be successfully applied to synthesized noisefree and
noisecorrupted FIDs. Here, as mentioned, for noisefree{c n }and noise
corrupted{c n + r n }time signals, the 4digit input data for all the 100 spec
tral parameters taken directly from Table 3.1 are used to carry out spectral
analysis in this section. The specifics of the present model for noise run as
follows. We add random numbers{r n }to the noiseless time signal{c n }to
generate the noisy input FID data{c n + r n }(0≤n≤N−1). More pre
cisely, this additive noise r n is a set{r n }(0≤n≤N−1) of N random

Gaussdistributed zero mean numbers (orthogonal in the real and imaginary
parts) with the standard deviation σ = λ×RMS. Here, λ is the selected
noise level and the acronym RMS stands for rootmeansquare (or equiva
lently, the quadratic mean) of the noiseless FID. For the given noiseless set
{|c n |}generated with the 4digit spectral parameters from Table 3.1, RMS is

defined by the arithmetic mean (average) value RMS = (

N
n=0

|c n | 2 /N) 1/2 .

According to our noise model, adding λ % noise{r n }to noiseless data{c n }of

RMS noise−free would produce noisy data{c n + r n }whose RMS noise−corrupted

is λ % of RMS noise−free , so that RMS noise−corrupted = λ RMS noise−free . Here
λ is a fixed number expressed in percent. For example, adding 10% noise
would yield a new RMS (noisy), which is 10% of the old RMS (noiseless),
σ = 0.01 RMS noise−free . In the present computations, we shall fix the noise
level λ to be a constant number equal to 0.00289, so that σ = 0.00289 RMS
where, as stated, RMS≡RMS noise−free . The value 0.00289 in the standard
deviation σ of noise is chosen to approximately match 1.5% of the height of
the weakest resonance (n ◦ k = 13) in the spectrum. This noise level is su
cient to illustrate the main principles of Froissart doublets. The FPT can also
successfully handle FIDs with much higher noise levels for synthesized and
encoded data, as shown in Refs. [6, 9, 35].

6.5.1 Converged Pade genuine resonances and lack of con-

vergence of Froissart doublets in FPT (±) with a quarter
of full signal length

In practice, for the same level of accuracy in quantification, the FPT (+) needs
more signal points than the FPT (−) . However, in this subsection, to simplify
our comparative analyses of these two variants of the Pade methodology, we
shall choose the same partial signal length to show the fully converged re
constructions in the FPT (±) for the FFTstructured signal lengths (integer
powers of 2). This convention need not be followed when displaying certain