Accurate Frequency Analysis of Measured Time-Dependent Signals Over Short Intervals,

Abstract : The resolution of the frequency utilizing the Fourier Transform on a time-dependent signal is generally considered to be restricted to the inverse of the length of the time interval. This suggests that a one second long signal will permit a frequency resolution of 1 Hz. A much more precise determination of the frequency is possible by a careful review of the phase when the source has an unknown narrow-band frequency. For purposes of the analysis presented herein, narrow-band frequency implies a bandwidth less than the normally interpreted frequency resolution. The phase of the cross-spectral density of successive time intervals indicates the difference between the Fast Fourier Transform (FFT) frequency resolution and the frequency of the input signal. Since this calculation is a trivial step after the calculation of the FFT, this method can be easily implemented on real-time systems using existing hardware for the FFT. This method has been applied to the acoustic data obtained from a helicopter. The improved analysis of the Doppler shift of the frequency for the moving aircraft permitted a good estimate of the velocity of the approaching helicopter and its range at the closest point of approach using one microphone on the ground.