Noise in spectra and map domain

healpy
astrophysics
Published

April 8, 2013

Spectra

NET or $$\sigma$$ is the standard deviation of the noise, measured in mK/sqrt(Hz), typical values for microwave amplifiers are 0.2-5.
This is the natural unit of the amplitude spectra (ASD), therefore the high frequency tail of the ASD should get to the expected value of the NET.
NET can also be expressed in mKsqrt(s), which is NOT the same unit.
mK/sqrt(Hz) refers to an integration bandwidth of 1 Hz that assumes a 6dB/octave rolloff, its integration time is only about 0.5 seconds.
mK/sqrt(s) instead refers to integration time of 1 second, therefore assumes a top hat bandpass.
Therefore there is a factor of sqrt(2) difference between the two conventions, therefore mK/sqrt(Hz) = sqrt(2) * mK sqrt(s)
See appendix B of Noise Properties of the Planck-LFI Receivers
http://arxiv.org/abs/1001.4608

Maps

To estimate the map domain noise instead we need to integrate the sigma over the time per pixel; in this case it is easier to convert the noise to sigma/sample, therefore we need to multiply by the square root of the sampling frequency:

sigma_per_sample = NET * sqrt(sampling_freq)

Then the variance per pixel is sigma_per_sample**2/number_of_hits

Angular power spectra

$$C_\ell$$ of the variance map is just the variance map multiplied by the pixel area divided by the integration time.

$C_\ell = \Omega_{\rm pix} \langle \frac{\sigma^2}{\tau} \rangle = \Omega_{\rm pix} \langle \frac{\sigma^2 f_{\rm samp}}{hits} \rangle$