Digital Signal Processing Reference
In-Depth Information
|∼10
−4
.
The main emphasis in
section 6.5
is on separation of stable (genuine) from
unstable (spurious) resonances. Spurious spectral structures are extremely
sensitive to even the slightest changes in the input data. Therefore, for the
outlined reasons of being able to clearly see what is caused by noise alone,
when comparing the results of quantifications in the noiseless and noisy cases
in
section 6.5
,
we shall use time signals{c
n
when{r
n
}is of a level which might also affect the 4th digit, |r
n
}and{c
n
+ r
n
}, respectively, with
the common part{c
n
}built from the same 4digit accurate input parameters.
Importantly, the envisaged comparisons will always be performed for the
same total (N) as well as partial (N
P
) signal lengths and the identical sam
pling time τ (i.e., the same acquisition time T = Nτ) for the noiseless{c
n
}
and noisy{c
n
+ r
n
}data, respectively. To cohere with chapter 3, the com
mon noiseless{c
n
}from
section 6.5
is built from the same 4digit accurate
input parameters given in Table 3.1. Furthermore, for both the 12 and
4digit accuracies from
sections 6.4
and
6.5
,
the deterministic time signals
{c
n
}(0≤n≤N−1) have the full signal length N = 1024 and are sampled
at the dwell time τ =1 ms. This same N and τ were also employed for the
noiseless FID from chapter 3. To such input data{c
n
}(0≤n≤N−1), ran
dom numbers as stochastic signal points{r
n
}(0≤n≤N−1) are added and
taken to be comprised of 25 genuine fundamental harmonics{z
k
}. More pre
cisely, these c
n
's are built from the corresponding sum of 25 complex damped
exponentials{z
k
}with constant amplitudes{d
k
}. Once sampling of such a
deterministic time signal is completed, the only known quantities that remain
available are τ , N and this tabulated FID. In other words, the previously
given input data set{z
k
,d
k
}and the number K of the elements of this set are
to be considered as unknown. This is where spectral analysis comes into play.
Specifically, the process of solving the quantification problem for this FID
amounts to reconstruction of the exact 100 complexvalued spectral parame
ters{z
k
,d
k
}(1≤k≤25), after the true number K = 25 has been unequivo
cally retrieved. To reemphasize, these deterministic time signal points{c
n
}
for the noisefree FID, that are repeated in their noisy counterparts{c
n
+r
n
},
K
k=1
d
k
z
k
(0≤n≤N−1), as per (3.1),
with the exact data for the input values of the fundamental spectral param
eters{ν
k
,d
k
}. Here, the elements{z
k
}are the signal poles, z
k
= e
iω
k
τ
are sampled from the formula c
n
=
where
ω
k
= 2πν
k
and Im(ω
k
) > 0 (1≤k≤K).
In the 12digit input data from
section 6.4
,
an unprecedented challenge is
placed on the subsequent usage of the Padeguided reconstruction by introduc
ing the minuscule difference of only 1×10
−11
ppm in the last digit of chemical
shifts between the 11th and 12th fundamental harmonics. Further, yet another
level of remarkable challenge is designed for the Padeoptimized quantification
by entering all the 25 phases φ
k
of the input amplitudes d
k
(1≤k≤25) as
precisely 12digit zeros that are extremely di
cult to reconstruct with the in
put accuracy. Likewise, along the lines of chapter 3, in the present
section 6.5
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