SPEX User's Manual - SRON

5.3 Data binning 93
Table 5.6: Coefficients for the approximations for λ k and c α
i a i b i
0 0.1342 1.3596
1 -0.3388 4.0609
2 0.7994 -4.3522
3 -1.1697 5.2225
4 0.9053 -3.7881
5 -0.2811 1.1491
f(x) =
a
π(a 2 + x 2 ) , (5.30)
where a is the width parameter. The FWHM equals 2a. Next we consider the Gaussian distribution,
given by
f(x) = 1 √
2πσ
e −x2 /2σ 2 , (5.31)
with σ the standard deviation. The FWHM is given by √ ln 256σ, which is approximately 2.3548σ.
We have computed the value for δ ≡ λ k / √ N numerically for both distributions. We have taken into
account also the effect of possible phase shifts, i.e. that the center of the distribution does not coincide
with a grid point but with a certain phase of the grid. The maximum bin width ∆/FWHM as a function
of the maximum absolute difference δ between the true and approximated cumulative density distribution
is plotted for both distributions in fig. 5.3.
Figure 5.3: Required bin width ∆ as a function of the accuracy parameter δ for a Gaussian
distribution (solid line) and a Cauchy distribution (dashed line).
We can approximate both curves with sufficient accuracy by a polynomial in x ≡ 10 log δ as
10 log(∆/FWHM) =
10 log(∆/FWHM) =
4∑ 10 log(δ)
c i ( ) i , (5.32)
10
i=0
4∑ 10 log(δ)
g i ( ) i , (5.33)
10
i=0