Improving FFT frequency measurement resolution by parabolic and Gaussian spectrum interpolation

Discrete spectra can be used to measure frequencies of sinusoidal signal components. Such a measurement consists of digitizing a compound signal, performing windowing of the signal samples and computing their discrete magnitude spectrum, usually by means of the fast Fourier transform algorithm. Frequencies of individual components can be evaluated from their locations in the discrete spectrum with a resolution depending on the number of samples. However, the frequency of a sinusoidal component can be determined with improved resolution by fitting an interpolating parabola through the three largest consecutive spectrum bins corresponding to the component. The abscissa of its maximum constitutes a better frequency approximation. Such a method has been used for tune measurement systems in circular accelerators. This paper describes the efficiency of the method, depending on the windowing function applied to the signal samples. A typical interpolation gain is one order of magnitude. Better results are obtained with Gaussian interpolation, offering frequency resolution improvement by more than two orders of magnitude when used with windows having fast sidelobe decay. An improvement beyond three orders of magnitude is possible with steep Gaussian windows. These results are confirmed by laboratory measurements. Both methods assume the measured frequency to be constant during acquisition and the spectral peak corresponding to the measured component to constitute a local maximum in a given band of the input signal discrete spectrum.