SPEX User's Manual - SRON

94 Response matrices
for the Cauchy distribution (5.32) and Gaussian distribution (5.33) respectively, with the coefficents given
in table 5.7.
Table 5.7: Coefficients for the approximations for the Cauchy and Gauss distribution.
i c i g i
0 0.400 0.277
1 5.065 3.863
2 9.321 8.470
3 9.333 9.496
4 3.584 3.998
The rms deviation of these fits over the range 10 −9 < δ < 1 is 0.0086 for the Cauchy distribution and
0.0070 for the Gauss distribution. Outside of this range the approximation gets worse, however those
values for δ are of no practical use.
The bin size as a function of the number of resolution elements R and number of photons per resolution
element N r is now obtained by combining (5.32) or (5.33) with (5.28) and using
λ k = √ N r δ. (5.34)
Figure 5.4: Required bin width ∆ relative to the FWHM for a Gaussian distribution as a function
of N r for R=1, 10, 100, 1000 and 10000 from top to bottom, respectively.
In fig. 5.4 we show the required binning, expressed in FWHM units, for a Gaussian lsf as a function of R
and N r , and in fig. 5.5 for a Lorentzian.
For the Gaussian, the resolution depends only weakly upon the number of counts N r . However in the case
of a pure Lorentzian profile, the required bin width is somewhat smaller than for a Gaussian. This is due
to the fact that the Fourier transform of the Lorentzian has relatively more power at high frequencies than
a Gaussian (exp [−ωσ] versus exp [ −(ωσ) 2 /2 ] respectively). For low count rate parts of the spectrum,
the binning rule 1/3 FWHM usually is too conservative!
5.3.9 Final remarks
We have estimated conservative upper bounds for the required data bin size. In the case of multiple
resolution elements, we have determined the bounds for the worst case phase of the grid with respect to
the data. In practice, it is not likely that all resolution elements would have the worst possible alignment.
In fact, for the Gaussian lsf, the phase-averaged value for δ at a given bin size ∆ is always smaller than