This paper shows how to make gravitational‑wave templates for the Laser Interferometer Space Antenna (LISA) without ever building a full, high‑cadence waveform. The authors exploit the fact that signals from binary systems occupy a sparse set of lines in a time‑frequency view. By generating only the local pieces around those lines, they avoid computing and storing the full oscillating waveform across an entire multi‑year observation.

The core tools are a particular time‑frequency transform called the Wilson‑Daubechies‑Meyer (WDM) transform and a fast way of handling the instrument response called time‑delay interferometry (TDI) on the fly. The WDM transform has a symmetry that makes it possible to remove the rapidly changing carrier phase (a process called heterodyning) in narrow bands. TDI on the fly evaluates the slowly varying detector response on a sparse solar‑system‑barycenter (SSB) grid. Together these ideas let the code assemble templates from sparse carrier tracks and local packets rather than from every second of data.

The paper describes two practical ways to turn a source model into a template. One is a time‑domain partition of unity with heterodyned fast Fourier transforms. The other is a lookup table of overlaps between short chirp fragments and the WDM filters. The authors also introduce an alternative called “TDI Tapestry,” in which the two gravitational‑wave polarizations are first combined and transformed to the WDM domain and the detector response is applied afterward. These choices let the same machinery work for a wide range of sources, including nearly monochromatic galactic binaries, massive black‑hole binaries that sweep quickly through frequency at merger, and extreme mass‑ratio inspirals (EMRIs) with many harmonics.

Why this matters: LISA signals can last months or years and the instrument noise can change over time and have gaps. Traditional template calculations would evaluate the waveform at every delayed time entering every TDI sample, which is very costly. The methods here greatly reduce that cost and the memory needed to hold templates. The paper reports algorithmic improvements such as a general grid placement scheme and a Cartesian treatment of slow modulations that avoids tricky phase unwrapping. These points make the approach highly parallel and practical for long, complex signals.