Digital Signal Processing Reference
In-Depth Information
numerical values of the corresponding harmonic variables
8
<
:
{z
±
k,Q
z
±
k,Q
= z
±
k,P
}
:
genuine
poles
k∈K
G
{z
±
k,Q
}
⊇
(5.14)
k∈K
T
{z
±
k,Q
z
±
k,Q
= z
±
k,P
}
:
Froissart
poles.
k∈K
F
The associated genuine fundamental frequencies{ν
±
k,Q
}
k∈K
G
are extracted
from the whole set{ν
±
k,Q
}
k∈K
T
via
8
<
:
{ν
±
k,Q
ν
±
k,Q
= ν
±
k,P
}
k∈K
G
:
genuine
poles
{ν
±
k,Q
}
k∈K
T
⊇
(5.15)
{ν
±
k,Q
ν
±
k,Q
= ν
±
k,P
}
k∈K
F
:
Froissart
poles.
In these expressions, the following definitions of ν
±
k,P
and ν
±
k,Q
are employed
in terms of z
±
k,P
and z
±
k,Q
, respectively
=∓
i
ν
±
k,Q
=∓
i
ν
±
k,P
2πτ
ln(z
±
k,P
)
2πτ
ln(z
±
k,Q
).
(5.16)
This disentangling of the genuine from spurious (noise and noiselike) infor
mation is the signature of the concept of signalnoiseseparation, i.e., the SNS.
As noted, even noiseless FIDs have spurious information which is noiselike,
since it behaves similarly to noise. Namely, Froissart doublets are found in
spectral analysis by the FPT for any noisefree time signal and they accu
mulate at the circumference (|z|= 1) of the unit circle in the plane of the
complex harmonic variable. The borderline|z|= 1 of the essential singularity
points represents the natural limit (in the Weierstrass sense) with the maxi
mal probability of 1 (except at most for a nearly zeroarea set [58, 59, 61]) for
finding the locations of all the poles of the response function for time signals
generated from random numbers [44, 52, 58, 59].
In this context, it is important to recall that in the FFT, all frequencies
from the Fourier grid ω
FFT
k
≡2πk/T (0≤k≤N−1) are purely real and,
therefore, located exactly at the circumference|z|= 1 of the unit circle. As
such, the Fourier “fundamental” frequencies ω
FF
k
are entirely embedded in
noise throughout the circumference|z|= 1 of the unit circle with no proce
dure available for separation of physical signal from noise within the Fourier
analysis.
Overall, as stated before, due to its linearity, the FFT imports noise as
intact from the time to the frequency domain. Even worse, this situation is
further aggravated in the complex plane of the harmonic variable z by the
distribution of the corresponding Fourier grid points z
FFT
k
= exp (iω
FFT
k
τ).
Namely, all the elements of the Fourier set{z
FFT
k
}(0≤k≤N−1) exhibit
the feature|z
FFT
k
|= 1 which places them precisely at the unit distance from
the origin of the unit circle. Consequently, all the z
FF
k
's are maximally
mixed with noise, which predominantly populates the circumference|z|= 1
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