Precision spectral peak frequency measurement using a window leakage ratio function

For power spectra of signals consisting of stationary sinusoids mixed with random noise, the frequency and amplitude of a spectral peak can be estimated with greater accuracy than the nearest frequency bin of the Fourier transform by exploiting the spectral leakage characteristics for the particular data window used. Techniques such as linear interpolation or an amplitude weighted average have inadequate precision due to the nonlinear leakage into adjacent bins and the dependence on data window type. This paper offers a new general algorithm presented using the Fourier coefficients ^ck of the input data window to produce a function which is the ratio of the side-bin amplitudes of the window in the frequency domain. The ratio function allows one to use the amplitudes of the adjacent bins of a spectral peak to precisely estimate the peak frequency and amplitude when the frequency does not lie exactly on a frequency bin (in between the discrete bins of a Fourier transform). Examples are provided for a number of popular data windows. The ratio function can be most easily implemented using a simplified log-ratio function for the window side bin magnitudes. A statistical analysis provides a useful frequency estimation error estimate given the signal-to-noise ratio of the spectral peak based on an approximation of the ratio of non-zero mean Gaussian variables. The benefits of this technique are not just improved estimation accuracy for amplitude and frequency, but also allow large spectral data files to be accurately reduced in size for remote monitoring of vibration spectra. An example is given of a methodology for reduction of spectral data file size without the loss of important signals for analysis where the file size is reduced by 88% with only a few percent error, which is mostly confined to the background noise in the reconstructed spectrum.