Instantaneous frequencies of signals obtained by the analytic signal method

Abstract

In this paper, the instantaneous frequency is defined as the one obtained by converting a real time signal to a complex analytic signal and by differentiating the time-dependent phase with respect to time. Theoretical expressions of instantaneous frequencies for signals given as combinations of sinusoidal components are presented. Those are compared to the results obtained by numerical methods using the discrete Fourier transform in order to confirm the validity of the expressions and to check accuracies of the numerical methods. A reason for the existence of negative instantaneous frequencies is given by a vector representation of signal components. The instantaneous frequencies of frequency-and amplitude-modulated signals are also discussed. For a periodic sinusoidal burst signal, it was found that the instantaneous frequency stays zero during the period when the signal is zero and takes a value equal to twice that of the sinusoid at the onset of the signal.

The song of the singing rod

“…An analysis (not presented here), employing the socalled instantaneous frequency method, [19][20][21][22][23][24] of the beat frequency reveals that it slightly increases over time as the rod vibration decays. One can audibly hear the increase in the beat frequency as well (increase of 1-2 Hz).…”

Measurement of sampling jitter in analog-to-digital and digital-to-analog converters using analytic signals

“…A Blackman window is used in this case since its first half can be used for fade-in and its last half can be used for fade-out. In this regard, the window function wðtÞ is written as follows: where XðiÞ is the discrete spectrum of the windowed signal wðtÞxðtÞ, and yðtÞ is derived by converting YðiÞ into a time-domain variable using an inverse DFT [8]. The temporal center of yðtÞ (N=4 t < 3N=4) is supplied to the analytic signal to calculate the amplitude modulation waveform and phase modulation waveform.…”