Digital Signal Processing Reference
In-Depth Information
An attempt to simultaneously circumvent the mentioned fundamental re
strictions of the Fourier method could be made by first acknowledging the
fact that each signal has its own inner structure. Subsequently, it would be
necessary to unfold the hidden structure and try to parametrize it. Finally, a
quantitative analysis should be carried out to determine the spectral features
by retrieving the resonance parameters{u
k
,d
(s)
}from the input time signal.
Such a strategy would lead to parametric estimations of spectra as a major
advantage relative to a shape processing by the Fourier analysis. Specifically,
the fact that a given time signal possesses its intrinsic structure through,
e.g., a number of constituent harmonics{d
(s
k
u
k
k
}, offers a definite chance for
resolution improvement beyond the Fourier bound 2π/T. As discussed, in
parametric estimations, the total acquisition time T of the signal is not criti
cal for definition of the spectral resolution. As such, parametric processing of
spectra can lower the Fourier bound 2π/T, thus yielding improved resolution
beyond the prescription of the Fourier uncertainty principle. Moreover, the
Fourier uncertainty principle within parametric processing is replaced by a
milder limitation imposed by 'the informational principle'. According to this
latter principle, no more information could possibly be obtained from a spec
trum in the frequency domain than what has already been encoded originally
in the time domain. Such a conservation of information translates into an
algebraic condition, which requires a minimum of 2K signal points to retrieve
all the spectral parameters{u
k
,d
(s)
}(1≤k≤K). This condition is rooted
in the requirement that the underlying system of linear equations must be
at least determined by the number of equations equal to the number of un
known parameters. In practice, all experimentally measured time signals are
corrupted with noise and this obstacle is partially counterbalanced by solving
the associated overdetermined system in which the number of signal points
exceeds the number 2K of the sought parameters{u
k
,d
(s)
k
}.
k
}originates from a physical system
which has already evolved from t
0
= 0 to t
s
= 0 before beginning to count
the time. It could also happen that in some experimental measurements, e.g.,
via ICRMS, the first few hundred signal points might be of such poor quality
that they must be discarded. In such cases, the spectral analysis deals with
the delayed Hankel matrix H
n
(c
s
) ={c
i+j+s
}where c
s
= (Φ
0
|U
s
|Φ
0
) = 0
appears as the first element. Within the FFT, this effect of a delay in the
considered signal is taken into account merely by skipping the first s points
{c
r
}(0≤r≤s−1) from the whole signal{c
n
}(0≤n≤N−1). However,
this is not good, since the resulting FFT spectra are of unacceptably poor
quality, which cannot be amended due to information loss in some of the
skipped original data. Hence, it is essential to have a signal processor which
can properly analyze data matrices{c
i+j+s
By definition, the delayed signal{c
n+s
}associated with the evolution of
the examined system from a nonzero initial time t
s
= sτ = 0.
In many interdisciplinary applications using signal processing methods,
the problem of spectral analysis of data records with delayed time series is
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