Abstract
When searching for new gravitational-wave or electromagnetic sources, the $n$ signal parameters (masses, sky location, frequencies,...) are unknown. In practice, one hunts for signals at a discrete set of points in parameter space, with a computational cost that is proportional to the number of these points. If that is fixed, the question arises, where should the points be placed in parameter space? The current literature advocates selecting the set of points (called a "template bank") whose Wigner-Seitz (also called Vorono\"i) cells have the smallest covering radius ($\equiv$ smallest maximal mismatch). Mathematically, such a template bank is said to have "minimum thickness". Here, for realistic populations of signal sources, we compute the fraction of potential detections which are "lost" because the template bank is discrete. We show that at fixed computational cost, the minimum thickness template bank does not maximize the expected number of detections. Instead, the most detections are obtained for a bank which minimizes a particular functional of the mismatch. For closely spaced templates, the fraction of lost detections is proportional to a scale-invariant "quantizer constant" G, which measures the average squared distance from the nearest template, i.e., the average expected mismatch. This provides a straightforward way to characterize and compare the effectiveness of different template banks. The template bank which minimizes G is mathematically called the "optimal quantizer", and maximizes the expected number of detections. We review optimal quantizer and minimum thickness template banks that are built as n-dimensional lattices, showing that even the best of these offer only a marginal advantage over template banks based on the humble cubic lattice.
Highlights
Many searches for gravitational-wave and electromagnetic signals are carried out using matched filtering, which compares instrumental data to waveform templates [1,2,3]
II we briefly review matched filtering, templates and template banks, the overlap between templates, and the mismatch function m on parameter space
IV we review the conventional wisdom for template placement, which is to select the template grid points so as to minimize the covering radius R for a given average WS cell volume
Summary
Many searches for gravitational-wave and electromagnetic signals are carried out using matched filtering, which compares instrumental data to waveform templates [1,2,3]. Provided that the grid points are not too widely separated, the best choice is the grid that minimizes the (normalized) second moment, which is the mean value of the squared distance (mismatch) to the nearest grid point. IV we review the conventional wisdom for template placement, which is to select the template grid points so as to minimize the covering radius R for a given average WS cell volume This minimizes a quantity known as the thickness of the lattice. If other adjustable parameters are kept fixed, choosing an optimal quantizer for the template grid in any stage will minimize the number of lost signals in that stage
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