Fourier Analysis From Networked Measurements Using Time Synchronization

Can frequency-domain analysis be made from time-domain measurements taken over networks with poor temporal characteristics, that is, poor or no guarantees about when commands or data will be delivered, such as the public Internet? Our answer to this question is ldquoyes.rdquo Provided that the times that samples are taken are measured accurately, Fourier analysis can be performed, even when the samples are taken at nonuniform intervals. Due to the tendency of packet delays in networks to have long-tailed probability distributions, it is also desirable not to make assumptions concerning the probability distribution of the time intervals between samples. We show that the combination of network time synchronization and stabilization of measurement apparatus using the IEEE 1588 Protocol and the methods for Fourier analysis of nonuniformly spaced data from the literature are sufficient to provide spectra from measurements taken with networked sensors. The key steps of our method are an accurate timestamping of the measurements followed by a Fourier transform. The Fourier transform may be accomplished by either resampling the measurements into a constant rate so that fast Fourier transforms may be used or performing a least squares fit of a sine/cosine basis to the data. Both approaches were tested using measurements taken over the public Internet using measurement apparatus synchronized using IEEE 1588.