We consider here that the photometers in the fiber interferometer are detector noise limited. Each signal measurement is then affected by an additive, stationary noise, uncorrelated with the data. For each interferogram, a realization of the four noise signals is recorded in the background current sequences. These noise signals have to be taken into account when estimating the photometric signals and the squared coherence factor.
The measurement 
of the signal produced by
photometric detector 
is the sum of 
and the additive noise
:
The estimator 
of 
that minimizes the mean quadratic
error with the actual signal is obtained by optimal filtering (Press et
al. 1988):
where 
is the Wiener filter, whose expression is, when the
signal and the noise are uncorrelated:
The power spectrum of the background current can be used to estimate
.
The quality of the deconvolution signal 
is critical to achieve a good interferogram correction.
What matters is not the average of 
but rather its smallest
value: if at the minimum the local signal to noise ratio is too low, then
the division is numerically unstable and the interferogram correction
process is useless. Therefore it is necessary to adopt a selection scheme:
if the minimum value of 
or 
is below a
certain rejection threshold, then the interferogram is discarded and does
not lead to a fringe visibility measurement. For a photometric signal
, the rejection threshold can be expressed as a multiple of
the standard deviation 
of the
corresponding background sequence, filtered by 
.
There is a trade-off in the choice of the rejection threshold, between
rejecting only a few interferograms and having a better correction. For
sufficiently good data this choice is not critical, as the correction
quality does not significantly depend on the minimum value of the
as long as this value is greater than
10
. For faint objects or in bad seeing
conditions, one may have to use rejection thresholds as low as
3
.
We should now seek an estimator of the coherence factor that is not biased
by detector noise. The signal M(x) actually measured at the output of the
interferometric detector contains an additive noise 
:
If the data processing described in the preceding sections is applied to
M, it leads in the wave number space to
and, since the signal and the noise are uncorrelated, their moduli add
quadratically:
which yields, in the integral form that is assumed to be valid when there
is differential piston:
An estimator 
of the noise integral is obtained by applying
to the current signal the same treatment that was used to process M. A
better estimator is obtained by processing many realizations of the
interferometric background current (for example, all the background
sequences of a complete batch), and averaging the resulting power spectra.
Eq. (58 (click here)) then provides an estimator of S:
from which Eq. (45 (click here)) enables us to establish an unbiased
estimator of the squared modulus of the coherence factor, averaged over the
optical bandpass:
The quantity 
is the final result of the data
reduction process on a single interferogram.
Figure 7: Result of the data reduction on a batch of 122 interferograms of
Boo. The adopted shape factor was 
cm.
Because of mediocre seeing conditions, the rejection threshold was set to
3: only 47 interferograms met this level and led to a measurement of the
squared coherence factor. For that batch the final estimate of the squared
coherence factor is 
A batch of interferograms leads to a collection of n measurements of the
fringe visibility. An example is shown in Fig. 7 (click here).
The final estimate of the squared coherence factor is the average of the
, and if 
is the standard deviation
of the estimator of 
, then the standard error is
The final error estimate for 
should include the
uncertainties on 
and 
.
Fluctuations in the measurement of 
can be
attributed to three main noise sources:
is not exactly
the noise integral 
;
that differs from the true deconvolution signal
P. An adequate rejection threshold should ensure that the deconvolution
noise is low (or even negligible);
, bandpass
limited by the atmospheric transmission window) its effects are always
negligible.
Although we have so far considered only one interferometric output, each
scan provides two measurements of 
. Coherence factor
measurements on each output must be treated separately because each channel
has its own instrumental transfer function. But the correlation between the
two channels can tell us about the relative importance of the noise
sources.
Detector noise is fully uncorrelated between the two interferometric outputs. Conversely, the differential piston perturbations are exactly the same on each output. Deconvolution noise is very strongly correlated since the deconvolution signal is the same for both channels, but not fully correlated because the deconvolution is applied to two separate measurements of the interferometric signal, each one affected by independent noise.
Let 
be the estimator of 
for
Channel 1 (Eq. 61 (click here)) and 
for Channel
2. The following holds without making any assumption on the nature of the
noise sources:
where 
is the correlated part of the noise for
Channel i, and 
is the uncorrelated
part. For each interferogram, it is possible to measure the difference
between the two estimators, i.e.
and for a batch of scans the variance of 
is the sum of
the variances of the uncorrelated noises:
If two photometers of the same type have been employed to measure the
interferometric outputs, the statistics of the uncorrelated noise is the
same on both channels. It is then possible to determine the variance of the
uncorrelated noise:
The variance of the correlated noise follows immediately from Eq. (63 (click here)):
From what was said above, we can with a good approximation identify
with the detector noise, and
with the joint contribution of piston and
deconvolution noise. Thus simultaneous measurements of both interferometric
outputs make it possible to tell the relative contribution of detector
noise to the total noise.
This is useful to work out the optimum fringe speed v, which is the
result of a compromise. On the one hand detector noise increases with v,
because for a given optical bandpass the fringe signal is spread over a
wider frequency bandpass. On the other hand the faster the interferogram is
scanned, the more frozen is the seeing for the duration of the scan, and
the piston perturbations are smaller. In the 
example, 44%
of the noise variance is uncorrelated, which means that no single source
dominates the noise.