In this paper, we present an assortment of both standard and advanced Fourier techniques that are useful in the analysis of astrophysical time series of very long duration -where the observation time is much greater than the time resolution of the individual data points. We begin by reviewing the operational characteristics of Fourier transforms (FTs) of time series data, including power spectral statistics, discussing some of the differences between analyses of binned data, sampled data, and event data, and briefly discuss algorithms for calculating discrete Fourier transforms (DFTs) of very long time series. We then discuss the response of DFTs to periodic signals, and present techniques to recover Fourier amplitude "lost" during simple traditional analyses if the periodicities change frequency during the observation. These techniques include Fourier interpolation which allows us to correct the response for signals that occur between Fourier frequency bins. We then present techniques for estimating additional signal properties such as the signal's centroid and duration in time, the first and second derivatives of the frequency, the pulsed fraction, and an overall estimate of the significance of a detection. Finally, we present a recipe for a basic but thorough Fourier analysis of a time series for well-behaved pulsations.

We report on more than 7 yr of monitoring of PSR J0537À6910, the 16 ms pulsar in the LMC, using data acquired with RXTE. During this campaign the pulsar experienced 23 sudden increases in frequency (''glitches'') amounting to a total gain of over 6 ppm of rotation frequency superposed on its gradual spin-down of ¼ À2 ; 10 À10 Hz s À1

We present the serendipitous discovery of 16 ms pulsed X-ray emission from the Crab-like supernova remnant N157B in the Large Magellanic Cloud. This is the fastest spinning pulsar associated with a supernova remnant (SNR). Observations with the Rossi X-ray Timing Explorer (RXTE), centered on the field containing SN1987A, reveal an X-ray pulsar with a narrow pulse profile. Archival ASCA X-ray data confirm this detection and locate the pulsar within 1 ′ of the supernova remnant N157B, 14 ′ from SN1987A. The pulsar manifests evidence for glitch(es) between the RXTE and ASCA observations which span 3.5 years; the mean linear spin-down rate is Ṗ = 5.126 × 10 −14 s s −1 . The background subtracted pulsed emission is similar to other Crab-like pulsars with a power law of photon index of ∼ 1.6. The characteristic spin-down age (∼ 5000 years) is consistent with the previous age estimate of the SNR. The inferred B-field for a rotationally powered pulsar is ∼ 1 × 10 12 Gauss. Our result confirms the Crab-like nature of N157B; the pulsar is likely associated with a compact X-ray source revealed by ROSAT HRI observations.

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The binary period of 4U1626-67 has been found from a careful analysis of its optical pusations. A single lower frequency sidelobe of the 2.4% amplitude 7.68s optical pulsations from this X-ray pulsar bee~ detected on at least 3 different nights in Fourier transforms of high speed photometry obtained with the CTIO 4m telescope. The 0.42% sidelobe pulsations have a frequency which is 0.4011(21) mHz lower than the frequency of the direct pulsations near 130.26 mHz. The weaker sidelobe pulsations are interpreted as arising from X-ray to optical reprocessing on the com~anion star and are shifted to the lower frequency by the rotation frequency of the binary orbit because the X-ray pulsar spins in the same sense as the orbital motion (direct, or prograde). The •orbital period is refined by connecting phases to be either