Use of the Discrete Fourier Transform in the Measurement of Frequencies and Levels of Tones

TLDR

The study applies digital computers and discrete Fourier transform techniques to measure single‑frequency tones. The authors develop three classes of weighting functions and derive efficient formulas that minimize memory usage and computation time for estimating tone frequency and level from DFT coefficients. Simulations show that with 512 samples at 7040 Hz, the DFT system achieves a maximum frequency error of about 0.03 Hz and a level error of roughly 0.03 dB for two tones near 1000 Hz separated by ~50 Hz.

Abstract

This paper considers the application of a digital computer and discrete Fourier transform (DFT) techniques to the measurement of signals known to comprise only single-frequency tones. We discuss the use of weighting functions to improve the effective selectivity of a measurement system that estimates the frequencies and levels of tones from the coefficients of their DFT. We present three classes of weighting functions which may be used to improve the inherent accuracy of such a system. The form of the weighting functions was chosen to minimize the amount of computer memory required, without using too much computer time. Several formulas are derived for estimating the frequency and level of a tone from its DFT coefficients. We chose the formulas to minimize computation time. Simulation results indicate that, through the use of a proper weighting function, a DFT measurement system that uses 512 samples taken at a sampling frequency of 7040 Hz can be designed so that the maximum error in the frequency estimates of two tones near 1000 Hz and separated by approximately 50 Hz is about 0.03 Hz. The corresponding maximum error in the level estimate is on the order of 0.03 dB.