Simple Signal Simulation Techniques

1. Introduction

The content of this web page is related to the desire to make simple signal simulations to test the deglitching module in the SPIRE pipeline. For a fixed scan rate, the signal bandwidth is largest at the shortest band: 250 microns. This wavelength provides the most difficult test for a deglitching algorithm.

For the greatest fidelity, the signal simulations should be produced in two dimensions, since the transfer function due to diffraction of the telescope is two dimensional. In Sections 2 and 4 below, I produce simulated 2D images with point sources and "1/f" emission, respectively. These are costly in terms of disk space. However, they produce many potential simulated timestreams each, namely the rows separated by at least a beam diameter which are mostly independent (at the high spatial frequencies at least).

A cheaper way to produce the simulations (Section 3 and 5) is to produce one dimensional timestreams. This is an oversimplification compared to "real" signals; for example, the 1D point source simulations correspond to the case of the point sources falling exactly on the scan trajectory.

For both the 1D and 2D simulations, the procedure is to: 1) produce an initial timestream or map with infinitesimal angular resolution, dictated by the sampling; 2) Fourier transform the timetream or map; 3) apply the low-pass transfer function of diffraction; 4) inverse Fourier transform to get back to the time domain.

I adopted 1 arcsec sampling for the 2D images. Given the <15 Hz bandwidth of the photometer electronics and the further bandwidth limitation due to diffraction, sampling of the timestreams at 1-2 arcsec is about right for the expected telescope scan rates of 30 - 60 arcsec/sec.

2. Point Sources in Two Dimensions

I place point sources randomly in a square image. Each source was assigned a flux from a power law probability distribution. I chose a P(F) dF = P_0 F^-2 dF distribution, with an upper limit of 1000 mJy and lower limit of 0.1 mJy. (To implement this, I borrowed an idea from Numerical Recipes. I start with a uniform deviate x between F_min/F_max and 1. Then I use F = F_min / x.) The lambda = 250 micron simulation depicted below is 8192 arcsec on a side and contains 1,000,000 point sources:
full 2D point source simulation, 2.3 degrees x 2.3 degrees, displayed 0 - 100 mJy
center of 2D point source simulation, 0.15 degrees x 0.15 degrees, displayed 0 - 30 mJy

I assumed a uniformly illuminated telescope of diameter 3.5 m to estimate the diffraction transfer function.

Below are some sample timestreams for 1 arcsec sampling:
row 4097, ASCII format
detail in row 4097
row 6345, ASCII format
detail in row 6345

And some sample timestreams for 2 arcsec sampling:
detail in row 4097 with 2" sampling, ASCII format
detail in row 6345 with 2" sampling, ASCII format

3. Point Sources in One Dimension

(I'm only planning to work on this if the scans need to be much longer than 10,000 samples).

4. Galactic Emission in Two Dimensions

I modeled Galactic emission as '1/f' structure. The technique was to describe the structure in the Fourier plane, apply the telescope transfer function, and then inverse FFT to generate the sky image (using the magnitude and ignoring the phase).

I ran two cases: structure amplitude going as (f + f0)^-1, and structure going as (f + f0)^-2. f is the spatial frequency, and f0 is a small number to keep the amplitude finite at DC. The phase of each Fourier component was a random number. In both cases, the simulation is 8192 arcsec x 8192 arcsec, and the telescope transfer function was for lambda = 250 micron and a uniformly illuminated 3.5 m telescope.
2D f^-1 Galactic simulation, 2.3 degrees x 2.3 degrees, displayed 0 - 460 Jy
2D f^-2 Galactic simulation, 2.3 degrees x 2.3 degrees, displayed 0 - 460 Jy

Below are some sample timestreams for 1 arcsec sampling:
row 4140 of f^-1 simulation, ASCII format
row 5630 of f^-2 simulation, ASCII format

And some zooms at 2 arcsec sampling:
zoom of row 4140 with 2" samping, ASCII format
zoom of row 5630 with 2" samping, ASCII format

5. Galactic Emission in One Dimension

(I'm only planning to work on this if the scans need to be much longer than 10,000 samples).

6. The Next Steps

  1. Convert (m)Jy to ADU to simulate the data coming in to the deglitching module. (Simple scale factor plus offset.)
  2. Convert data format from ASCII or FITS to SPIRE pipeline data format.
  3. Add white noise.
  4. Add glitches.
I could also apply the electronics low pass filter to the data, but I suspect that will have a small effect on how the deglitching module handles these timestreams.
CDD, 2008 Oct 30