Data compression on board the PLANCK Satellite Low Frequency Instrument: optimal compression rate

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: $c_r \simeq {N_{bits}/\log_2(\sqrt{2\pi e} ~\sigma_e/\Delta)}$, where $\sigma_e\equiv (det C)^{1/2N}$ 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).