From the corrected spectrum and interferogram calculated in Sect. 4.3 and
presented in Fig. 6, the modulus and the phase of the complex visibility are
computed in the range 
cm
(Fig. 11). The average measured
visibility is 1.006 with a standard deviation of 1.3% (Table 2). The
phase is almost perfectly zero, the error being less than 0.01 rad. The
oscillations in the visibility modulus account for most of the noise. The
closer the edges of the spectrum, the bigger the oscillations. The phase is
left unaffected by these oscillations. This is the Gibbs phenomenon
(Bracewell 1986). As shown in Fig. 7 the injected error signal is well
reconstructed by the reduction process on more than 
m on each side
around the zero opd position. The difference between the injected error
signal and the correction process output error signal is, as expected, of
the order of 
rms. It is a smooth and coherent noise
which explains the good reconstruction of the spectrum. A fast varying noise
with the same rms fluctuations would produce a spectrum with few
similarities with the original one. Before 
m and after
m the reconstructed piston signal is shifted by 
(or
the phase function is shifted by 
) with respect to the signal
reconstructed in the interval 
. The interpolations
necessary to compute the interferograms with fluctuating opds generate
errors in the interferograms that are similar to a noise. The 
-leap in
the output piston is due to this noise. As a consequence, although the
reduced interferogram is defined on a width 
m, the
actual spectral resolution is determined by a width 
m and is 
cm
. It is possible to
increase spectral resolution with an infinite S/N ratio if the method
explained at the end of Sect. 4.4.2 is extrapolated. The width of the
narrow band interferogram is the reverse of the bandwidth and the resolution
of the reconstructed wide band spectrum is then of the order of the
bandwidth of the narrow band spectrum.
More realistic simulations are necessary to determine how good the correction can
be when data are noisy and what should be expected with this method. Three S/N
ratios have been simulated: 100, 50, 20. The spectrum displayed in Fig. 8,
which is the result of the reduction and summation of 100 interferograms with
S/N=100, shows the limits of the method in its basic use. For such a S/N ratio
the spectrum reconstruction starts to be of poor quality and the inferred
visibility function gets less and less reliable. Spectra obtained with the
narrower filter method of Sect. 4.4.2 and displayed in Fig. 10 lead to the
visibility functions moduli and phases of Fig. 12. As for the case of the
infinite S/N ratio, the Gibbs phenomenon is visible. Spectral fluctuations
frequencies decrease with decreasing S/N ratios indicating that spectral
resolution is also decreasing, and fluctuations amplitudes increase with noise.
The statistics of the moduli of the measured visibility functions are presented in
Table 2. Visibility functions are measured with a quite good accuracy
for S/N ratios of 100 and 50 with average precisions of 
and 
respectively. Because of the Gibbs phenomenon, local errors can be larger than
on the edge of the spectrum. For S/N=20 the correction algorithm fails
to reconstruct workable visibility moduli. As far as phases are concerned, the
reconstruction is easier. The maximum error is 0.01rad for S/N=100 and 50
and it reaches 0.02rad for S/N=20.
From these results a few conclusions can be drawn on what can be expected from
the piston reduction process. First, in the simulations the fringes were scanned
at 
corresponding to a fringe frequency of 2274Hz at
2.2
m. For a scan length of 400
m it has been shown that the standard
deviation of piston is 10.5
m for average atmospherical conditions on the
IOTA interferometer and for this fringe speed. Each set of simulated data is
a record of 100 interferograms. After reduction the strength of piston is thus
attenuated by a factor of 10 and the standard deviation of optical path
fluctuations is 
at 2.2
m. The good quality of the
correction obtained with the proposed algorithm in realistic conditions of
observation shows that it is workable for real observations. Second, very good S/N
ratios are necessary if visibility measurements with precisions better than
are required. It turns out that the maximum level of noise that makes this
goal achievable corresponds to a S/N ratio close to 50. Third, this method
necessarily degrades the initial spectral resolution. After correction, the
estimated resolution for an infinite S/N ratio is about 
cm
and is of
the order of 
cm
or more for S/N ratios of 100 and 50. These
resolutions are poor relative to the resolutions needed for a fruitful analysis
of CO lines in the K band for example. Nevertheless, although it is
theoretically possible to indefinitely increase spectral resolution in
the case of an infinite S/N ratio as explained in Sect. 5.1, with these resolutions it is
possible to derive gradients and higher order terms from the computed visibilities
that should contain interesting informations.
