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Средняя пространственная частотная характеристика рефлектограммы с двухимпульсным зондирующим сигналом с частотным разнесением
The average spatial backscattered intensity spectrum of the OTDR with dual-pulse diverse frequency probe signal
Let us derive
the expression for the average power spectral characteristic of the
bandpass OTDR trace 34 or 35 with the carrier frequency
and random amplitude and phase modulation. Once again we calculate
the ACF of the random process
which in this case is the square of electric fields sum 30 and 31
i.e.
.
Like before we will use the Gaussian moment theorem [11] for the
complex scattering coefficients
,
and also consider the lack of correlation over the ensemble
between the fields, scattered by the different parts of the
dual-pulse
and
. The AKF
is found to be a sum of 16 terms, where due to the just mentioned
facts only 6 of them result in nonzero contribution, as a result we
have
got:
3636\* MERGEFORMAT ().
The
ACF 36 also depends only on the coordinate difference
and so the process
is wide-sense stationary. Applying Wiener–Khinchin theorem the
Fourier transform of 36 results in the PSD of the random process
:
3737\* MERGEFORMAT ().
The
linearly filtered processes
and
are statistically equal and the final PSD will take the form:
3838\* MERGEFORMAT (),
where
the convolution
was found in the second part of the paper for the probe pulses of
different amplitude shapes, the spatial frequency is defined as
.
Thus
the PSD of the random intensity fluctuation process
,
when the dual-pulse probe signal with different carrier frequencies
of the first and the second parts of the pair consists of several
components: the first term in 38 represents the constant part in
the received intensity, the second term correspond to the PSDs of the
intensity fluctuation processes, is
associated with the fields backscattered by the first
and the second
parts
of the dual-pulse, this PSD is concentrated near zero frequency and
defined previously 20, the third and the fourth terms correspond to
the PSDs of interference intensity fluctuations, associated with the
interference of the fields backscattered by the first
and the second
parts of the dual-pulse, they have the similar form as the PSD near
zero frequency, but due to harmonic component in the ACF 36 they
are concentrated near positive and negative frequency:
.
As before the
transition to the time domain could be made by multiplication of the
corresponding spatial frequencies by group velocity of light
,
so the PSD of the random process
is concentrated near zero frequency and near frequencies
,
where
.
It is now proved that using of dual-pulse with different carrier frequencies of the first and the second parts leads to appearance of three equally shaped PSDs where two PSDs are frequency shifted. The PSD that experience a positive frequency shift can be interpreted as power spectrum of the banpass quasi-harmonic signal, as mentioned before. One can also say that 38 shows the average spatial power spectrum of the OTDR intensity trace with dual-pulse diverse frequency probe signal.
The particular
form of the spectral characteristic 20 and 38 is defined by the
shape of the probe pulse
,
for the pulses of rectangular or Gaussian shape the spectral
characteristics could be derived using 22 и
26.