  The Fast Folding Algorithm (FFA) is a phase-coherent search technique for
periodic signals. It has rarely been used in radio pulsar searches, having been
historically supplanted by the less computationally expensive Fast Fourier
Transform (FFT) with incoherent harmonic summing (IHS). Here we derive from
first principles that an FFA search closely approaches the theoretical optimum
sensitivity to all periodic signals; it is analytically shown to be
significantly more sensitive than the standard FFT+IHS method, regardless of
pulse period and duty cycle. A portion of the pulsar phase space has thus been
systematically under-explored for decades; pulsar surveys aiming to fully
sample the pulsar population should include an FFA search as part of their data
analysis. We have developed an FFA software package, riptide, fast enough to
process radio observations on a large scale; riptide has already discovered
sources undetectable using existing FFT+IHS implementations. Our sensitivity
comparison between search techniques also shows that a more realistic
radiometer equation is needed, which includes an additional term: the search
efficiency. We derive the theoretical efficiencies of both the FFA and the
FFT+IHS methods and discuss how excluding this term has consequences for pulsar
population synthesis studies.
