Environmental Engineering Reference
In-Depth Information
3.4.1
Preserving Measurement Accuracy in the TDR/FDR
Transformation
There are some crucial aspects that need to be considered to perform an accurate
TD/FD transformation, thus reducing errors on the assessment of the spectral re-
sponse of the system and in the extraction of
S
11
(
)
[29].
First, the time duration of the acquisition window,
T
w
, should be sufficiently long,
so as to include all multiple reflections due to the signal traveling back and forth
from the probe-end toward the generator. However, an excessively-long
T
w
would
lower the sampling frequency (since the maximum number of acquisition points for
a given instrument is fixed), thus limiting the possible frequency range of analysis.
Additionally, to reduce the spectral leakage caused by the rectangular truncation
window, two main strategies can be adopted. One is the application of the Nicolson
algorithm: a linear ramp is subtracted from the original TDR waveform, and each
of the acquired
N
points is scaled according to the
N
th
final point; in this way, the
value of the TDR waveform at the truncation point is zero. The resulting scaled
signal,
w
∗
(
f
nT
c
)
, can be written as [33]
w
(
NT
c
)
n
w
∗
(
nT
c
)=
w
(
nT
c
)
−
(3.12)
N
where
T
c
is the sampling period,
w
(
nT
c
)
is the rectangular-windowed signal, and
n
N
refers to each sampling point. The important feature of this algorithm
is that the FFT of the Nicolson-modified signal produces the same response as the
FFT of the original TDR signal [29]. Alternatively to the Nicolson algorithm, it
is possible to use the derivatives of the time domain signals calculated with the
backward difference method; however, generally, noise is slightly enhanced when
the derivative of two similar quantities is taken in digital systems [24, 29].
Moreover, the addition of a number of zeros to the Nicolson-modified waveform
(or to the derived signal) artificially enhances the frequency resolution of the ob-
tained FD data, thus corresponding to an ideal interpolation in FD [29].
The steps at the basis of the TD/FD approach are summarized in Fig. 3.9: this proce-
dure can be implemented, for example, in a MATLAB-based algorithm that, starting
from the TDR waveforms, extrapolates the
S
11
=
1
,
2
, ...,
(
)
.
In practical applications, the specification of the input signal,
v
0
f
(
)
, is crucial
for obtaining accurate results. As aforementioned, the input function should be the
unaltered signal generated by the TDR instrument and propagating (along the cable)
up to the section where the scattering parameter is being measured [29]: in fact, the
S
11
(
t
may be acquired by removing
one of the electrodes from the probe [24]; however, this can be unpractical for most
commercial probes, like the probe shown in Fig. 3.4, in which the central conductor
is attached to the molded probe-head.
A possible alternative is the use an artificial TDR waveform modeled as an ana-
lytical function [23]:
f
)
should be evaluated at this very section.
v
0
(
t
)
1
+
er f
[
α
t
(
t
−
t
0
)]
(
)=
v
0
,
art
t
(3.13)
2
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