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## 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 non-null 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 ss (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 ss / 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.

When a spectrum is band limited between s0 and s1 , the sampling frequency can be reduced up to 2s1 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 so , it is possible to sample at a lower rate than 2s1 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 anti-symmetrical such that it produces a compensating negative image. After such a filtering, the only undersampling condition is:

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 16-bit 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.

2.4.1.3 Data compression

2.4.1.4 Formatting into instrument source packets

Keywords: ESA European Space Agency - Agence spatiale europeenne, observation de la terre, earth observation, satellite remote sensing, teledetection, geophysique, altimetrie, radar, chimique atmospherique, geophysics, altimetry, radar, atmospheric chemistry

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