It is in general a reasonably good assumption that the total noise (excluding systematics that might be scan-synchronous or correlated to the signal), be a Gaussian process that can be described by a Power Spectral Density (PSD, in Volts2 of electrical signal per Hz) typically of the form:
Here is a "knee frequency" at which low-frequency noise and white noise contributions to the power spectral density are statistically equal, and is a spectral index, typically between 1.0 and 2.5, depending on the dominant physical process which generates low frequency noise. It is a good approximation to assume that vanishes for and , with and .
In an experiment for which the scanning strategy consists in scanning repeatedly the same circle of the sky, the sky signal for any given pixel is calculated by averaging the "samples" corresponding to this pixel. When the white noise contribution dominates the total noise, i.e. when and are such that
which can be simplified to if , and , or to if and , then the rms of the final noise on one scan circle obtained by averaging N consecutive scans is just the rms of the original noise divided by . This is not the case when the noise is significantly coloured (i.e. not white), since averaging consecutive scans actually reduces the bandwidth of the signal in such a way that most of the low-frequency noise contribution at frequencies is filtered out. In Fourier space, the spinning and averaging process keeps all components of the signal that are harmonics of the spinning frequency, and cuts other components (this is only an approximation, but good enough for this discussion). In our case, the low-frequency contribution to the standard deviation will be much larger on a 2-hour data stream than on the corresponding circle of data obtained by averaging.
Figure 2: Example of 8 consecutive hours of simulated low-frequency noise, with and Hz. These values are typical for ground-based experiments which suffer from thermal fluctuations and atmospheric noise (for the PLANCK HFI, due to the extremely favourable observing conditions, we expect the low-frequency drifts to be dominated by electronics noise, i.e. and instead). The top panel shows drifts due to the low-frequency part of the noise only, with a spectrum of the form . The effect of adding the white noise contribution (which has a standard deviation of 1.00) would be a strong broadening of the line. The bottom panel shows the absolute value of the FFT of the total noise. The knee frequency of 0.10 Hz is clearly visible
The top plot of Fig. 2 (click here) is a plot of 8 hours of the low-frequency part of a noise with a spectrum of the form of Eq. (1 (click here)), with and Hz (which is extremely pessimistic for the PLANCK HFI, but illustrates our point better than more realistic noise: drifts on the scale of 8 hours are almost imperceptible for and Hz unless the noise is significantly smoothed to filter high frequency components). Such figures for the noise are quite typical for ground-based CMB experiments. Only the low-frequency component is represented in the top panel of Fig. 2 (click here) (which corresponds thus to a spectrum or PSD of ). Units for noise generation have been normalised so that the rms of the white part of the noise per sample is 1.00 (and the rms per resolution element on a data circle, obtained from averaging the 120 consecutive scans of one 2-hour period for one detector, would be if there were no low frequency noise). In this simulation, the sampling frequency is 2034 samples per minute, which for PLANCK corresponds to one sample per 10 arcminute pixel. The bottom plot of Fig. 2 (click here) is the absolute value of the Fast Fourier Transform of the noise realization with white noise included as well as the low-frequency part. The knee frequency of about 0.10 Hz is clearly visible.
Figure 3: Residual total noise on 4 consecutive circles, each obtained by averaging 120 consecutive 1 minute scans of the 8-hour sample of the simulated low-frequency noise plotted in Fig. 2 (click here). The data for each circle looks flat at this scale, but each circle has a very different offset
Low frequency drifts with an amplitude of a few (whereas the rms of the white part of the noise only is 1.00) are obvious in Fig. 2 (click here). The standard deviation of the total 8-hour noise (white + low-frequency) is about 2.49, that of the low-frequency part is 2.28, and that of the white part is 1.00. However, what actually happens on circles after 120 consecutive scans have been averaged is shown in Fig. 3 (click here), where the data corresponding to the four circles obtained from this 8-hour signal are plotted with the same units on the vertical axis. Here, the white noise component has been included. The variance on each individual circle is much smaller than that of the original signal, because each circle is obtained by averaging 120 scans as explained above. Also, it is impressive to notice that almost all the power of low frequency noise now appears in the form of a different "offset" Ai for each circle i. The average level of all the circles is not the same, but low frequency drifts on each circle are much smaller than on the original time sequence.
Figure 4: Here we show the centred signal for each of the circles of Fig. 3 (click here). Note the scale compared to that of Fig. 2 (click here) and Fig. 3 (click here). Even for the very pessimistic assumed values for and , low frequency drifts are barely visible in the white noise
It is important to check whether the low frequency noise will have any effect other than to add a different offset to the data corresponding to each circle. Figure 4 (click here) shows the centred noise for each of the above 4 circles. Even with the very pessimistic values of and used here, low-frequency drifts are almost too low to be distinguished in the dominating white noise. If however we remove the white part of the noise and look at the four circles (Fig. 5 (click here)), we see clearly that some low frequency drifts are still there, which would appear if a drastic smoothing of the data were performed.
A crude estimate of the increase of the standard deviation of the noise on a circle (compared to the standard deviation of white noise only) can be obtained by integrating the PSD between Hz and . This method underestimates slightly the rms increase because drifts at frequencies lower than the spinning frequency are not totally cut out by the spin-chopping. The steeper the noise spectrum, the less accurate this method is. A rigorous calculation can be found in (Janssen et al. 1996) for the special case of .
Although this additional noise power is quite low, it is at low frequency only, and the effect on the accuracy of the measurement of interesting cosmological quantities (e.g., the power spectrum of the fluctuations) should be investigated. For instance, it should be kept in mind that when the resolution of maps is degraded by smoothing, the standard deviation of pure white noise scales as the inverse of the size of the pixel, whereas the standard deviation of parallel stripes scales roughly as the square root of the size of the pixel. Quantifying the effect of striping on sophisticated statistical tests or pattern recognition methods is even harder, and may require the help of numerical simulations of two-dimensional noise maps.
Figure 5: Residual of low frequency component of the noise only for each of the circles of Fig. 4 (click here). These structures, which can barely be distinguished in Fig. 4 (click here), would appear more and more clearly if the data were smoothed