Pulse Height Analysis - Details

We assume that the pulses all have the same shape; that each pulse has the form A * S(t), where A is the amplitude, and S(t) the pulse shape. Note that significant amounts of non-linearity can give rise to variations in pulse shape with the energy of the event. (This is to be distinguished from a non-linear gain (voltage of the signal per incident energy) which could be substantial without causing significant variations in pulse shape). The optimal pulse height estimate, H, is the one which minimizes (in the least square sense) the difference between the noisy data, D(t), and the model of the pulse shape. Here we transfer the expression to be minimized into the frequency domain;

where N²(f) is the power spectrum of the noise. Setting the derivative of this expression to zero yields the optimal estimator of the pulse height as

This expression can be transferred back into the time domain as

where, F(t), the optimal filtering template, is constructed by taking the inverse Fourier transform of S(f)/N²(f). Thus to construct the optimal filter we must accumulate the average pulse shape and the power spectrum of the noise.

If the power spectrum of the noise were "white" (constant), the optimal filter would be identical to the average pulse shape. Since our noise increases as we approach DC, our optimal filters typically have a region with negative weights before the pulse arrives. In effect, this region is measuring the height of the variable baseline, and subtracting it from the region under the pulse. Note that this means that we need information before the pulse arrives. If there are some peaks in the noise spectrum, due to pickup from the power line or from vibrations, the S/N ratio will be small at those frequencies, and the filter will effectively "notch out" the noise, regardless of its phase.

A more complete description of optimal filtering can be found in section 13.3 of Numerical Recipes in C, 2nd ed. The relevant section can be downloaded in Postscript or EPS format here.

In the XRS instrument, Hi-res analysis is done on a fixed-length block of 2048 samples. This requires that only one pulse exist in that record (which takes about 167 mS). If there is more than one pulse within that time period ("pulse pileup"), we use a lower resolution method.