The concept of instantaneous frequency (IF), its definitions, and the correspondence between the various mathematical models formulated for representation of IF are discussed. The extent to which the IF corresponds to the intuitive expectation of reality is also considered. A historical review of the successive attempts to define the IF is presented. The relationships between the IF and the group-delay, analytic signal, and bandwidth-time (BT) product are explored, as well as the relationship with time-frequency distributions. The notions of monocomponent and multicomponent signals and instantaneous bandwidth are discussed. It is shown that these notions are well described in the context of the theory presented. >

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Estimating and interpreting the instantaneous frequency of a signal. I. Fundamentals

Hilbert transform in vibration analysis

This paper compares the Teager-Kaiser algorithm (TKA) and other similar local methods to the analytic signal (AS) procedure. The general concepts of the instantaneous amplitude and frequency are discussed. It is shown that only AS meets certain physical conditions for the amplitude, phase, and frequency (APF). The advantage of accuracy and simplicity of the AS is also demonstrated.

On the analytic signal, the Teager-Kaiser energy algorithm, and other methods for defining amplitude and frequency

An exact and general expression for the analytic wavelet transform of a real-valued signal is constructed, resolving the time-dependent effects of nonnegligible amplitude and frequency modulation. The analytic signal is first locally represented as a modulated oscillation, demodulated by its own instantaneous frequency, and then Taylor-expanded at each point in time. The terms in this expansion, called the instantaneous modulation functions, are time-varying functions which quantify, at increasingly higher orders, the local departures of the signal from a uniform sinusoidal oscillation. Closed-form expressions for these functions are found in terms of Bell polynomials and derivatives of the signal's instantaneous frequency and bandwidth. The analytic wavelet transform is shown to depend upon the interaction between the signal's instantaneous modulation functions and frequency-domain derivatives of the wavelet, inducing a hierarchy of departures of the transform away from a perfect representation of the signal. The form of these deviation terms suggests a set of conditions for matching the wavelet properties to suit the variability of the signal, in which case our expressions simplify considerably. One may then quantify the time-varying bias associated with signal estimation via wavelet ridge analysis, and choose wavelets to minimize this bias.

On the Analytic Wavelet Transform

Instantaneous frequency, taken as the derivative of the phase of the signal, is interpreted in the time-frequency literature as the average frequency of the signal at each time. We point out some difficulties with this interpretation, and show that for a generic two-component AM-FM signal, the interpretation holds only when the components are of equal strength. We conclude that instantaneous frequency and the average frequency at each time are generally two different quantities. One possible interpretation of the difference between these two quantities is suggested.

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Comments on the interpretation of instantaneous frequency

This review is concerned with applying the analytic signal in oscillation theory, where the concept of the analytic signal was hardly ever applied until recently. We treat the mathematical properties of the Hilbert transform and of the analytic signal, which allow one to determine the amplitude, phase, and frequency of any oscillation at any instant of time. For narrow-band oscillations and for broad-band oscillations that arise under slow frequency modulation, this definition agrees with the intuitive meaning of amplitude, phase, and frequency and with the quasistationary approximation, while allowing one to estimate the limits of applicability of the latter. We show that a number of radiotechnical devices (mixers, frequency modulators, detectors, frequency discriminators, etc.) transform the parameters of an oscillation as defined by the analytic signal. We establish the relationship between the adiabatic invariant and the equation of the oscillations for the analytic signal. This relationship allows one to construct a complete theory of the triode oscillator having a cubic characteristic, in which the capacity of the circuit and the transconductance of the tube slowly fluctuate. Here we get a new result in the second approximation, namely: we calculate the influence of the flicker effect on the instantaneous frequency of the oscillator; the corresponding spectral line width is substantial in practice. In conclusion, we treat some paradoxes and supplementary examples that illustrate the technical and physical significance of the introduced concepts.

https://iopscience.iop.org/article/10.1070/PU1977v020n12ABEH005479/pdf

D E Vakman

Papers

Amplitude, phase, frequency—fundamental concepts of oscillation theory