Drifting sub-pulse analysis using the two-dimensional Fourier transform

The basic form of drifting sub-pulses is that of a periodicity whose phase depends (approximately linearly) on both pulse longitude and pulse number. As such, we argue that the two-dimensional Fourier transform of the longitude-time data (called the Two-Dimensional Fluctuation Spectrum; 2DFS) presents an ideal basis for studies of this phenomenon. We examine the 2DFS of a pulsar signal synthesized using the parameters of an empirical model for sub-pulse behaviour. We show that the transform concentrates the modulation power to a relatively small area of phase space in the region corresponding to the characteristic frequency of sub-pulses in longitude and pulse number. This property enables the detection of the presence and parameters of drifting sub-pulses with great sensitivity even in data where the noise level far exceeds the instantaneous flux density of individual pulses. The amplitude of drifting sub-pulses is modulated in time by scintillation and pulse nulling and in longitude by the rotating viewing geometry (with an envelope similar to that of the mean pulse profile). In addition, sub-pulse phase as a function of longitude and pulse number can differ from that of a sinusoid due to variations in the drift rate (often associated with nulling) and through the varying rate of traverse of magnetic azimuth afforded by the sight line. These deviations from uniform sub-pulse drift manifest in the 2DFS as broadening of the otherwise delta-function response of a uniform sinusoid. We show how these phase and amplitude variations can be extracted from the complex spectrum.