Filters for High Rate Pulse Processing

We introduce a filter-construction method for pulse processing that differs in two respects from that in standard optimal filtering, in which the average pulse shape and noise-power spectral density are combined to create a convolution filter for estimating pulse heights. First, the proposed filters are computed in the time domain, to avoid periodicity artifacts of the discrete Fourier transform, and second, orthogonality constraints are imposed on the filters, to reduce the filtering procedure’s sensitivity to unknown baseline height and pulse tails. We analyze the proposed filters, predicting energy resolution under several scenarios, and apply the filters to high-rate pulse data from gamma-rays measured by a transition-edge-sensor microcalorimeter.

💡 Research Summary

The paper presents a novel filter‑construction technique for processing high‑rate pulse signals that departs from conventional optimal filtering in two fundamental ways. Traditional optimal filters are derived in the frequency domain by multiplying the average pulse shape by the inverse of the noise power spectral density, which implicitly assumes periodicity due to the discrete Fourier transform (DFT) and yields a filter that is sensitive to low‑frequency components such as baseline drift and the long tails of preceding pulses. The authors address these shortcomings by (1) designing the filter directly in the time domain, thereby eliminating DFT‑induced cyclic convolution artifacts, and (2) imposing orthogonality constraints on the filter coefficients so that the filter is mathematically orthogonal to both a constant baseline and a chosen tail‑shape vector representing the residual part of a previous pulse.

Mathematically, the measured signal is modeled as (x(t)=A,s(t)+b+n(t)), where (A) is the pulse amplitude, (s(t)) the normalized pulse shape, (b) a slowly varying baseline, and (n(t)) zero‑mean Gaussian noise with covariance matrix (C). The goal is to estimate (A) by a linear operation (\hat A = h^{!T}x). Minimizing the variance (\sigma^{2}=h^{!T}Ch) under the constraints (\langle h, \mathbf{1}\rangle =0) (baseline orthogonality) and (\langle h, s_{\text{tail}}\rangle =0) (tail orthogonality) leads to a Lagrangian formulation. Solving the resulting linear system yields a closed‑form filter
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