We present theory and algorithms to perform an all-sky coherent search forperiodic signals of gravitational waves in narrow-band data of a detector. Oursearch is based on a statistic, commonly called the $\mathcal{F}$-statistic,derived from the maximum-likelihood principle in Paper I of this series. Webriefly review the response of a ground-based detector to thegravitational-wave signal from a rotating neuron star and the derivation of the$\mathcal{F}$-statistic. We present several algorithms to calculate efficientlythis statistic. In particular our algorithms are such that one can takeadvantage of the speed of fast Fourier transform (FFT) in calculation of the$\mathcal{F}$-statistic. We construct a grid in the parameter space such thatthe nodes of the grid coincide with the Fourier frequencies. We presentinterpolation methods that approximately convert the two integrals in the$\mathcal{F}$-statistic into Fourier transforms so that the FFT algorithm canbe applied in their evaluation. We have implemented our methods and algorithmsinto computer codes and we present results of the Monte Carlo simulationsperformed to test these codes.
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