Using the fast Fourier transform to determine the period of a physical oscillator with precision

Abstract

Measuring the period of torsion pendulums with precision has long been a formidable challenge in gravitation experiments, particularly those measuring the Newtonian gravitational constant G. An alternative method to fitting the position signal of the pendulum to a sine wave is the use of the power spectrum generated by the fast Fourier transform (FFT) as the source of information from which the period of oscillation can be determined. There are, however, known limitations to the use of a FFT to measure the period of a physical oscillator with precision. These limitations include two effects due to the finiteness of the duration of the sinusoidal data record and one effect due to the uncertainty of the starting phase of the oscillator relative to the window imposed by this duration. We have done a phenomenological study of the FFT using a desktop computer to imitate a precision oscillator having the physical characteristics of a finite damping constant and drift in the zero potential-energy position. Also, we have taken extensive data with a torsion pendulum, and analyzed them in this way. These studies show that for a real oscillator, such as the classical torsion pendulum, the FFT is a useful tool for determining the period of oscillation with the precision usually associated with larger, more complex, fitting algorithms. With good signal-to-noise ratio and under conditions appropriate to a torsion pendulum, the FFT method can measure the frequency or period to five parts in 106 or better.

The Newtonian gravitational constant: recent measurements and related studies

“…A research bibliography of all known measurements relating directly to G, including those searching for variations in it with respect to the temperature of the test bodies, their electromagnetic state, the orientation of their crystalline axes, etc was assembled by Gillies (1987). The experiments that made up these different categories of measurements were subsequently discussed in greater detail in articles published thereafter (Gillies 1988). Representative papers on a variety of these topics were selected and issued as a reprint book, too (Gillies 1992).…”

“…In spite of these modern motivations for the possible existence of temperature-dependent gravity, there has been very little laboratory activity aimed at studying such phenomenon since 1925. Virtually all of the experiments relevant to this topic were carried out before then (Gillies 1987(Gillies , 1988. Table 8.…”

Linguistic summaries of locally periodic time series

“…Among the various information conveyed in a time series, periodicity is an important one, frequently used in fields as diverse as astronomy [4], physics [5], energy production [6], speech analysis [7], zoology [8] or biology [9]. Moreover, the question of local periodicity occurs in many applicative contexts [10,11,12,13].…”

Determination of fundamental frequency of a physical oscillator by the period fitting method

“…In this case, we only need to determine the fundamental frequency with high precision from the experimental data. Traditional methods such as the fast Fourier transform ͑FFT͒ method, 8 the ''all-poles'' method, 9 and the nonlinear least-square fitting method 10 can provide more information about the signal, but they fail to determine the fundamental frequency optimally in the experiment of measuring G. In the FFT method, a power spectrum associated with the sampled data is calculated by the FFT. 8 Then the period is determined by calculating the centroid of the fundamental peak of the spectrum, and detailed information about different modes of the signal can be obtained from the spectrum.…”

“…For the second, the ''window'' effect and the ''phase'' effect lead to a serious deviation of the period in the FFT method. Although Goldblum and Ritter have done some beneficial work on this problem, 8 the two effects cannot be eliminated.…”