2.4 Algorithms and products
2.4.1 Level 0 products and algorithms Level 0 data is the data stream
received directly from the instrument without any
further processing. Since MIPAS performs some
onboard processing, this does not mean that
Level 0 data is free of processing. The
interferogram recorded by the detectors undergo a
series of modification before being downlinked
to the ground station. The main processing steps
are:
2.4.1.1 Decimation and filtering
In order to lower the size of the
signals to be transmitted, measured
interferograms are filtered and decimated.
This operation is part of the onboard
processing.
Neglecting the dispersion phenomenon inducing
a nonnull phase, an observed interferogram
is basically a real and symmetrical
function. The symmetry is about ZPD and, by extension
about every multiple of MPD. The Fourier
transform of such an interferogram is a real
and symmetrical spectrum with symmetry about
every multiple of the sampling
frequency. In other words, the full spectrum
will show on one half the true physical
spectrum and on the other half the image of
this spectrum. Depending on the
convention, this second half may be
displayed as negative frequencies or as
frequencies above the sampling frequency
divide by 2, as displayed in the figure
below.
For a given sampling rate of s_{s}
(equal to 7692 cm^{“1}
for MIPAS, corresponding to a
laser operating at 1300 nm), the Nyquist
sampling theorem states that this sampling
frequency defines a fixed spectral band
of maximum width s_{s}
/ 2. This spectral band is quite large
and can be reduced. The principle of
data compression is to sample at a lower
rate by decimating the interferogram (taking
one point out of n) already
sampled by the metrology laser system. The
result is a reduced number of interferogram
data points that permits a smaller data
throughput.

Figure 2.2 Filtering and decimation scheme 
When a spectrum is band limited between s_{0}
and s_{1}
, the sampling frequency can be
reduced up to 2s_{1}
without any information loss as stated
by the Nyquist sampling theorem. Reducing
the sampling frequency further can
produce spectral overlap that disturbs the
interesting information (aliasing effect).
However, since there is a useless spectral
region from 0 to s_{o}
, it is possible to sample at a lower
rate than 2s_{1}
and still keep all the
information. For a real filtered signal,
where both the desired physical band and its
image are present, the lowest possible
sampling frequency preserving the
information is twice the spectral
bandwidth . For MIPAS, complex filters
have been devised in such a way that it has
no image passband, by defining its
imaginary part antisymmetrical such that it
produces a compensating negative image.
After such a filtering, the only
undersampling condition is:
s_{s}
= s_{1}
 s_{o}
  eq 2.1 
Thus, the decimation factor can be two times
larger after complex filtering. The integer
ratio of the initial sampling frequency to
the new one is called the decimation
factor, noted DF. Since the folding
frequencies are not restricted to be out of
the band of interest, there is no
additional restriction on the decimation
factor. It is then possible to better
optimize the decimation factor. This is
where a gain can be made with respect to
data reduction. The shape of the
apodisation function applied to the
filtering impulse response (FIR) is
critical. It must produce sufficient
smoothing of the filter, but must avoid
widening it to the point of reducing too
much the effective bandwidth of the pass
bands and the possible data compression.
MIPAS FIR filters respect
these criteria and are defined over 256 taps
using 16bit coefficients. The isolation of
the various MIPAS filters range from
65 to 87 dB. The processing
needed for the proper recovery of the
wavenumber axis for each spectrum consists
of computing a Fourier transform of the
decimated signal, unfolding of the spectral
axis (for cases where spectral limits do not
exactly correspond to an integer factor of
the band width), followed by axis limit
identification. Further details about this
procedure can be found in the ATBD
Ref. [1.6 ]
.
2.4.1.2 Word length reduction During the formatting of the data
stream by the SPE, the word length (or bit
size) of the interferogram is reduced on a
fraction of the interferogram. Due to the
typical shape of an interferogram (see the
figure below), the full dynamic range (16
bits) is used only near the ZPD. Far from the ZPD, only a small fraction of
the ADC range is used. The
regions far from the ZPD, on both side of the
interferogram, can thus be coded using a
smaller number of bits without loosing any
information. The size of the data transmitted is
thus significantly reduced.
full size

Figure 2.3 Typical analogue (left) and digitalised (right) interferograms. 
2.4.1.3 Data compression
2.4.1.4 Formatting into instrument source packets
