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
