11 Conclusions

The expected data rate from the PLANCK Low Frequency Instrument is $\approx 260$ kbits/sec. The bandwidth for the scientific data download currently allocated is just $\approx 60$ kbit/sec. Assuming an equal subdivision of the bandwidth between the two instruments on-board PLANCK, an overall compression rate of a factor 8.7 is required to download all the data.

In this work we perform a full analysis on realistically simulated data streams for the 30 GHz and 100 GHz channels in order to fix the maximum compression rate achievable by loss-less compression methods, without considering explicitly other constrains such as: the power of the on-board Data Processing Unit, or the requirements about packet length limits and independence, but taking in account all the instrumental features relevant to data acquisition, i.e.: the quantization process, the temperature/voltage conversion, number of quantization bits and signal composition.

As a complement to the experimental analysis we perform in parallel a theoretical analysis of the maximum compression rate. Such analysis is based on the statistical properties of the simulated signal and is able to explain quantitatively most of the experimental results.

Our conclusions about the statistical analysis of the quantized signal are: I) the nominally quantized signal has an entropy $h \approx 5.5$ bits at 30 GHz and $h \approx 5.9$ bits at 100 GHz, which allows a theoretical upper limit for the compression rate $\approx 2.9$ at 30 GHz and $\approx 2.7$ at 100 GHz. II) Quantization may introduce some distortion in the signal statistics but the subject requires a deepest analysis.

Our conclusions about the compression rate are summarized as follows: I) the compression rate $C_{{\rm r}}$ is affected by the quantization step, since greater is the quantization step higher is $C_{{\rm r}}$ (but worse is the measure accuracy). II) $C_{{\rm r}}$ is affected also by the stream length $L_{{\rm u}}$, i.e. more circles are compressed better then few circles. III) the dependencies on the quantization step and $L_{{\rm u}}$ for each compressor may be summarized by the empirical formula (12). A reduced compression rate $\mbox{$C_{{\rm r},1}$ }$ is correspondingly defined. IV) the $C_{{\rm r}}$ is affected by the signal composition, in particular, by the white noise rms and by the dipole contribution, the former being the dominant parameter and the latter influencing $C_{{\rm r}}$ for less than $\approx 6\%$. The inclusion of the dipole contribution reduces the overall compression rate. The other components (1/f noise, CMB fluctuations, the galaxy, extragalactic sources) have little or no effect on $C_{{\rm r}}$. In conclusion, for the sake of compression rate estimation, the signal may be safely represented by a sinusoidal signal plus white noise. V) since the noise rms increases with the frequency, the compression rate $C_{{\rm r}}$ decreases with the frequency, for the LFI $\Delta\mbox{$C_{{\rm r}}$ }/\mbox{$C_{{\rm r}}$ } \lesssim 10\%$. VI) the expected random rms in the overall compression rate is less than $1\%$. VII) we tested a large number of off-the-shelf compressors, with many combinations of control parameters so to cover every conceivable compression method. The best performing compressor is the arithmetic compression scheme of order 1: arith-n1, the final $C_{{\rm r},1}$ being 2.83 at 30 GHz and 2.61 at 100 GHz. This is significantly less than the bare theoretical compression rate (9) but when the quantization process is taken properly into account in the theoretical analysis, this discrepancy is largely reduced. VIII) taking into account the data flow distribution among different compressors the overall compression rate for arith-n1 is:

$$\overline{\mbox{$C_{{\rm r}}$ }}_{,{\rm arith-n1}} \approx 2.65 \pm 0.02.$$

This result is due to the nature of the signal which is noise dominated and clearly excludes the possibility to reach the required data flow reduction through loss-less compression only.

Possible solutions deal with the application of lossy compression methods such as: on-board averaging, data rebinning, or averaging of signals from duplicated detectors, in order to reach an overall lossy compression of about a factor 3.4, which coupled with the overall loss-less compression rate of about 2.65 should allow to reach the required final compression rate $\approx 8.7$. However each of these solutions will introduce heavy constrains and important reduction of performances in the final mission design, so that careful and deep studies will be required in order to choose the best one.

Another solution to the bandwidth problem would be to apply a coarser quantization step. This has however the drawback of reducing the signal resolution in terms of $\Delta T/T$.

Lastly the choice of a given compressor cannot be based only on its efficiency obtained from simulated data, but also on the on-board available CPU and on the official ESA space qualification: tests with this hardware platform and other compressors will be made during the project development. Moreover, we are confident that the experience which will be gained inside the CMB community developing ground, balloon and space based experiments, as the development of full prototypes of the on-board electronics, will provide us with a solid base to test and improve compression algorithms. In addition the final compression scheme will have to cope with requirements about packet length and packet independence. We discuss briefly this problems recalling two proposals (Maris 1999a, 1999b) which suggest solutions to cope with these constrains.

Acknowledgements

We warmly acknowledge a number of people which actively support this work with fruitful discussions, in particular F. Argüeso, M. Bersanelli, L. Danese, G. De Zotti, E. Gaztñaga, J. Herrera, N. Mandolesi, P. Platania, A. Romeo, M. Seiffert and L. Toffolatti and K. Gorski and all people involved in the construction of the Healpix pixelisation tools, largely employed in this work, G. Lombardi from Siemens - Bocholt and G. Maris from ETNOTEAM - Milano, for fruitful discussions about compression principles and their practical application, P. Guzzi and R. Silvestri from LABEN - Milano for explanations and suggestions about the PLANCK-LFI data acquisition electronics. At last we acknowledge the referee Miguel A. Albrecht, for the useful suggestions and corrections, which improve significantly the text readability and accuracy.