Frequenrate

Frequenrate is a metric used in signal analysis to describe how quickly the frequency content of a signal changes over time. Denoted FR(t), it is the time derivative of the instantaneous frequency f_inst(t). The concept combines the idea of frequency with a rate of change and is typically expressed in units of hertz per second (Hz/s).

For a signal s(t) with an analytic representation s_a(t) = A(t) e^{i φ(t)} obtained via the Hilbert transform, the instantaneous frequency is f_inst(t) = (1/2π) dφ/dt, and the frequenrate is FR(t) = df_inst/dt = (1/2π) d^2φ/dt^2. In practice FR can be estimated by computing the analytic signal, extracting φ(t), differentiating to obtain f_inst, and differentiating again to obtain FR; smoothing is commonly applied to reduce noise.

Interpretation and use: A signal with a constant frequency has FR = 0. If the frequency changes linearly with time, FR is constant and equal to the slope of f_inst(t). More complex signals with multiple components require time–frequency decomposition to estimate FR for each component. Applications include radar and sonar chirp-rate tracking, FM demodulation, audio pitch drift analysis, and seismic frequency drift studies. FR provides a dynamic complement to traditional spectral measures.

Limitations: Accurate estimation of FR depends on reliable phase unwrapping and noise suppression; numerical differentiation can amplify noise and bias results. It relates to, but is distinct from, instantaneous frequency and from broader time–frequency representations such as spectrograms or wavelet scalograms. The term frequenrate is used in theoretical discussions to emphasize the rate of frequency variation rather than the frequency content itself.