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Here, the simulation of the
reconstruction from the complex impulse response is discussed. The dynamic
range and noise effect are also considered.
a) Impulse response dynamic range
The impact of the available dynamic
range is presented in Fig. 3-31 for 20, 40, 60 and 80 dB of dynamic
range, respectively. In this case, the dynamic range is defined from the dB
representation of the impulse response amplitude with an illumination light
source centered at 1300 nm with 40 nm bandwidth. Some oscillations
are observed for the smaller dynamic range examples, especially at the last
third of the grating reconstruction. Nevertheless, the impact is less important
as what has been observed in Fig. 3-29 for the reconstruction case from
the spectral response.
Fig. 3-31 Reconstructed coupling coefficient amplitude (top) and local Bragg
wavelength (bottom) for the FBG1 performed with layers thickness of 5 mm, a ratio
M/N = 10 for a impulse amplitude range of 80 dB (solid lines),
60 dB (dashed lines), 40 dB (dashed-dotted lines) and 20 dB
(dotted lines); the curves are translated for clarity
b)
Impulse response noise
The starting functions are the impulse
response amplitude ha and the phase difference hp. The
noisy impulse response is calculated in a similar manner as it has been
performed for spectral data in equation (3-22)
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(3-23) |
where Ao and As are
the noise amplitude offset and scale factor, respectively; Po and Ps
the noise phase offset and scale factor, respectively; and finally
"rand" a random number between ± 0.5. The results for 5, 10, 20 and 30 % scale noise and
phase offset of p/100, p/50, p/20 and p/10, respectively, are presented in Fig. 3-32. The local Bragg
wavelength, calculated from the derivative of the coupling coefficient phase,
has been necessarily performed on several reconstruction points to limit the
high variations of the noise. This procedure was not necessary for the
reconstruction from the noisy spectral response as the noise effect is spread
over the whole grating reconstruction. The coupling coefficient amplitude shows
only localized noise, but not an overall shape change, except from some
oscillations for very noisy cases. This indicates that the impulse noise impact
in the reconstruction is mainly restricted to the corresponding grating
position. Other experiments with single defect measurement impulse response
points have shown that the reconstruction presents also a single defect point
at the corresponding grating location. This explains that even for small phase
noise, the derivative noise is very important but the underlying local Bragg
wavelength is conserved if averaged on several points. Compared to the
reconstruction from a noisy spectral response, the reconstruction from a noisy
impulse response exhibits better results.
Fig. 3-32 Reconstructed coupling coefficient amplitude (top) and local Bragg
wavelength (bottom) for the FBG1 performed with layers thickness of 5 mm, a ratio
M/N = 10 for a different noisy impulse responses; solid lines :
Ao = 10-7, As = Ps = 5%,
Po = p/100; dashed lines : Ao = 10-6, As = Ps = 10%,
Po = p/50; dashed dotted lines : Ao = 10-5, As = Ps = 20%,
Po = p/20; dotted lines : Ao = 10-4, As = Ps = 30%,
Po = p/10; the curves are translated for clarity
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