Planck Pre-Launch Status Papers - APC - Université Paris Diderot ...

D.1. Step 1-extrapolate uncalibrated noise to nominal frontend unit temperatureThis is a non-trivial step to be performed if we wich to considerall the elements in the extrapolation. Here we focus on a zeroorderapproximation based on the following assumptions:1. the radiometer noise temperature is dominated by the frontendnoise temperature, such that T noise ∼ T FEnoise ;2. we neglect any effect on the noise temperature given by resistivelosses of the front-end passive components;3. we assume the variation in T FEnoise to be linear in T phys.Based on these assumptions, the receiver noise temperature atnominal front-end temperature can be written asT noise (T nom ) = T noise (T test ) + ∆T phys ,∂T physA. Mennella et al.: LFI calibration and expected performanceIf we refer to ρ as the ratio σ(T sky)σ(T in )ρ = T sky + T noiseFrom Eqs. (D.7)and(2), we obtainσ =FE∂Tnoise(D.2)previous equation infers that[1 + bG0 (T sky + T noise ) ] 2∂G FEG FE (T test ) ∂T phys∆T phys = σ cal =(D.3)=(T in + T noise (T test ))d(t) = d 1(t) + d 2 (t)2(D.5)√given by σ d(t) =∑ Nj=1d(t) =w jd j (t)∑ Nj=1 w j(D.7)σ −2d j (t)given by⎛ ⎞−1/2N∑σ d(t) = ⎜⎝ σ −2d j (t) ⎟⎠1 + bG 0 (T sky + T noise ) · (D.8) j=1where ∆T phys = T nom − T test .Asimilarbutslightlydifferentrelationship can be derived for the gain factor G 0 . Weconsider that G 0 = const × G FE G BE ,andthatwecanwriteG FE (T nom ) = G FE 1(T test )(1 + δ), where δ =ln(10)10∂G FE (dB)∂T phys∆T phys ,i.e.,G 0 (T nom ) = G 0 (T test )(1 + δ).From the radiometer equation we have that σ ∝ (T in + T noise ),from which we can writeσ(T nom ) ≡ σ nom = σ(T test ) (T in + T noise (T nominal ))whereη == σ(T test )(1 + η), (D.4)∂TFEnoise∂T phys[(T in + T noise (T test ))] −1 ∆T phys .D.2. Step 2 – extrapolate uncalibrated noise to T skyFrom this point, we consider quantities such as T noise ,whitenoise level, and G 0 ,extrapolatedtothenominalfront-endtemperatureusing Eqs. (D.2), (D.3), and (D.4). Therefore, we nowomit the superscript “nom” so that, for example, σ ≡ σ nom .We now start from the radiometer equation in which, for eachdetector, the white noise spectral density is given byδT rms = 2 T in + T noise√ β· (D.6)We now attempt to find a similar relationship for the uncalibratedwhite noise spectral density linking δV rms to V out .WebeginfromEq. (C.5) andcalculatethederivativeofV out using Eq. (2) andδT rms from Eq. (D.6). We obtainσ = V out√ β[1 + bG 0 (T in + T noise )] −1 ,where β is the bandwidth and V out is the DC voltage output of thereceiver. Considering the two input temperatures T in and T sky ,then the ratio isσ(T sky )σ(T in ) = V out(T sky )V out (T in ) × 1 + bG 0(T in + T noise )and use Eq. (2) toplaceinexplicit form the ratio of output voltages in Eq. (D.8) sothatσ(T sky ) = ρ × σ(T in ), we have[ 1 + bG0 (T in + T noise )×T in + T noise 1 + bG 0 (T sky + T noise )D.3. Step 4-calibrate extrapolated noise] 2· (D.9)G 0[1 + bG0 (T sky + T noise ) ] 2 × 2 T sky + T noise√ β· (D.10)If we call σ cal the calibrated noise extrapolated at the sky temperatureand consider that, by definition, σ cal = 2 T sky+T√ noiseβ,theG 0σ. (D.11)Appendix E: Weighted noise averagingAccording to the LFI receiver design, the output from each radiometeris produced by combinating signals from two correspondingdetector diodes. We consider two differenced and calibrateddatastreams coming from twodetectorsofaradiometerleg, d 1 (t)andd 2 (t). The simplest way to combine the two outputsis to take a straight average, i.e.,, (E.1)so that the white noise level of the differenced datastream isσ 2 d 1 (t) + σ2 d 2 (t) .Thisapproach,however,isnotoptimal in cases where the two noise levels are unbalanced, sothat the noise of the averaged datastream is dominated by thenoisier channel.An alternative to Eq. (E.1)isgivenbyaweightedaverageinwhich weights are represented by the inverse of the noise levelsof the two diode datasteams, i.e.,d(t) = w 1d 1 (t) + w 2 d 2 (t)w 1 + w 2, (E.2)or, more generally, in the case where we average more than twodatastreams,· (E.3)For noise-weighted averaging, we choose the weights w j =,sothatthewhitenoiseofthedifferenced datastream is. (E.4)Page 15 of 16