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 } \mathrel { \hbox to 0.0 pt { \lower 3.0 pt \hbox { $ \mathchar 536 $ } \hss } \raise 2.0 % pt \hbox { $ \mathchar 318 $ } } 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 ( detC ) ^ { 1 / 2 N } 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 ) .