2 Averaging noise to offsets on circles

As shown in Fig. 1, the typical noise spectrum expected for the PLANCK High Frequency Instrument (HFI) features a drastic increase of noise power at low frequencies ${f\leq 0.01 ~ \rm {Hz}}$. We model this noise spectrum as:

$$\!\!\!\!\!\!\!\!\! S(f) = \sigma^2 \times \left( 1 + \sum_i \left( \frac{f_i}{f} \right) ^{\alpha_i} \right).$$ (1)

The knee frequency ${f_{\rm {knee}}}$ is defined as the frequency at which the power spectrum due to low frequency contributions equals that of the white noise. The noise behaves as pure white noise with variance ${\sigma^2}$ at high frequencies. The spectral index of each component of the low-frequency noise, ${\alpha_i}$, is typically between 1 and 2, depending on the physical process generating the noise.

The Fourier spectrum of the noise on the circle obtained by combining N consecutive scans depends on the exact method used. The simplest method, setting the circle equal to the average of all its scans, efficiently filters out all frequencies save the harmonics of the spinning frequency [Delabrouille et al.1998b]. Since the noise power mainly resides at low frequencies (see Fig. 1), the averaging transforms - to first order - low frequency drifts into constant offsets different for each circle and for each polarimeter. This is illustrated in the comparison between Figs. 4 and 5. More sophisticated methods for recombining the data streams into circles can be used, as $\chi^2$ minimization, Wiener filtering, or any map-making method projecting about ${6\ 10^5}$ samples onto a circle of about ${5\ 10^3}$ points. For simplicity, we will work in the following with the circles obtained by simple averaging of all its consecutive scans.

Figure 4: Typical 1/f2 low frequency noise stream. Here, ${f_{\rm knee}=f_{\rm spin}=0.016}$ Hz, ${\alpha =2}$ and ${\sigma=21 ~\mu\mathrm{K}}$ (see Eq. 1). This noise stream corresponds to 180 scans or 3 circles (60 scans per circle) or a duration of 3 hours

Figure 5: The residual noise on the 3 circles after averaging. To first approximation, low frequency drifts are transformed into offsets, different for each circle and each polarimeter. Note the expanded scale on the y-axis as compared to that of Fig. 4

We thus model the effect of low frequency drifts as a constant offset for each polarimeter and each circle. This approximation is excellent for ${f_{\rm {knee}}\le f_{\rm {spin}}}$. The remaining white noise of the h polarimeters is described by one constant ${h \times h}$ matrix.