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

Abstract

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: $c_r \simeq {N_{bits}/\log_2(\sqrt{2πe} ~σ_e/Δ)}$, where $σ_e\equiv (det C)^{1/2N}$ is some effective noise rms given by the covariance matrix $C$ and $Δ\equiv R / N_{max}$ is the digital resolution. This $Δ$ only needs to be as small as the instrumental white noise RMS: $Δ\simeq σ_T \simeq 2 mK$ (the nominal $μK$ 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 $R= N_{max} Δ$ 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: $σ_e \simeq σ_T$, 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 $σ_e > σ_T$ 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).

LaTeX file, To appear in Proceedings of the UIMP98 workshop "The CMB and the Planck Mission" in Santander