We have developed two new methods for finding binary pulsars that search complementary portions of parameter space. Both techniques examine small regions of the Fourier transform of a time series and identify the distinctive but very low signal-to-noise patterns produced by binary pulsars. The first technique is a Fourier-domain version of traditional "acceleration" searches which can coherently detect a binary pulsar if the orbital period is much longer than the observation time (Porb ≳ 10Tobs). Based on Fourier-domain matched filtering, it provides sensitivities similar to time domain versions, but with great computational advantages (i.e. greatly decreased memory usage, increased speed, and an inherently parallel nature) when the time series is very long ( ≳ 108 points). The second technique is a computationally "cheap" but incoherent search for binary pulsars where Porb ≲ ⅔Tobs. This search detects periodic sidebands created by orbital phase modulation of a binary pulsar's signal using a two-stage Fourier analysis. Sideband searches have the potential to detect very interesting exotic pulsar systems in ultra-compact binaries that would be impossible to detect using traditional techniques. We have implemented both searches along with many other advanced Fourier "tricks" into a suite of pulsar search software called PRESTO . Using this software we have discovered a very interesting 3 ms binary pulsar with a 102 m orbital period and a ∼10 MJupiter companion in the globular cluster NGC 6544, a most-likely isolated 4.71 ms pulsar in the globular cluster Terzan 5, and have confirmed the new 65.6 ms X-ray pulsar discovered in the supernova remnant 3C 58.
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