Data 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 cᵣ 10. Here we present a study of entropy for (correlated multi-Gaussian discrete) noise, showing how the optimal compression cr,opt, for a linearly discretized data set with Nbits=₂Nmax bits is given by: cᵣ ≃ Nbits/₂(√2πe σₑ/Δ), where σₑ≡ (det C)¹/²ⁿ is some effective noise rms given by the covariance matrix C and Δ≡ R / Nmax is the digital resolution. This Δ only needs to be as small as the instrumental white noise RMS: Δ≃ σₜ ≃ 2 mK (the nominal μK pixel sensitivity will only be achieved after averaging). Within the currently proposed Nbits=16 representation, a linear analogue to digital converter (ADC) will allow the digital storage of a large dynamic range of differential temperature R= Nmax 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: σₑ ≃ σₜ, which could yield large compression rates cr,opt ≃ 8. This is the maximum lossless compression possible. In practice, point sources and 1/f noise will produce σₑ > σₜ and cr,opt < 8. This strategy seems safer than non-linear ADC or data reduction schemes (which could also be used at some stage).