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Register a new userData compression on board the PLANCK Satellite Low Frequency Instrument: optimal compression rate
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Data on board the future PLANCK Low Frequency Instrument (LFI), to measure the Cosmic Microwave Background (CMB) anisotropies, consist of $N$ differential temperature measurements, expanding a range of values we shall call $R$. Preliminary studies and telemetry allocation indicate the need of compressing these data by a ratio of $c_r simgt 10$. Here we present a study of entropy for (correlated multi-Gaussian discrete) noise, showing how the optimal compression $c_{r,opt}$, for a linearly discretized data set with $N_{bits}=log_2{N_{max}}$ bits is given by: $c_r simeq {N_{bits}/log_2(sqrt{2pi e} ~sigma_e/Delta)}$, where $sigma_eequiv (det C)^{1/2N}$ is some effective noise rms given by the covariance matrix $C$ and $Delta equiv R / N_{max}$ is the digital resolution. This $Delta$ only needs to be as small as the instrumental white noise RMS: $Delta simeq sigma_T simeq 2 mK$ (the nominal $mu K$ pixel sensitivity will only be achieved after averaging). Within the currently proposed $N_{bits}=16$ representation, a linear analogue to digital converter (ADC) will allow the digital storage of a large dynamic range of differential temperature $R= N_{max} Delta $ accounting for possible instrument drifts and instabilities (which could be reduced by proper on-board calibration). A well calibrated signal will be dominated by thermal (white) noise in the instrument: $sigma_e simeq sigma_T$, which could yield large compression rates $c_{r,opt} simeq 8$. This is the maximum lossless compression possible. In practice, point sources and $1/f$ noise will produce $sigma_e > sigma_T$ and $c_{r,opt} < 8$. This strategy seems safer than non-linear ADC or data reduction schemes (which could also be used at some stage).
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