Computing Instantaneous Frequency Functions

Instantaneous frequency is a fundamental concept in signal processing that characterizes the frequency components of a signal at specific time instants. In MATLAB, several techniques exist for computing instantaneous frequency functions, with the Hilbert transform-based approach being the most prevalent.

The theoretical foundation begins with understanding the formal definition of instantaneous frequency. For any given signal, instantaneous frequency can be derived from the phase derivative of its analytic signal. The analytic signal, constructed through Hilbert transformation, contains only positive frequency components of the original signal.

The standard MATLAB implementation involves three key steps: First, apply the hilbert() function to obtain the analytic signal (this built-in function directly returns the complex analytic representation). Second, compute the instantaneous phase using angle() or unwrap(angle()) functions to extract phase information. Third, differentiate the phase using diff() or gradient() functions to obtain instantaneous frequency. The core implementation can be structured as: analytic_signal = hilbert(original_signal); instantaneous_phase = unwrap(angle(analytic_signal)); instantaneous_frequency = diff(instantaneous_phase)/(2*pi*dt); where dt represents the sampling interval.

Important implementation considerations include phase unwrapping to avoid discontinuities and noise reduction techniques. Direct phase differentiation may amplify high-frequency noise, necessitating smooth differentiation methods like Savitzky-Golay filtering or moving average filters. Additionally, instantaneous frequency calculations require signals to satisfy the narrow-band assumption. For multi-component signals, pre-processing with empirical mode decomposition (EMD) or wavelet-based filtering may be necessary to isolate dominant components.

In time-frequency analysis applications, instantaneous frequency functions enable investigation of non-stationary signal behaviors, finding extensive utility in mechanical vibration analysis, biomedical signal processing (e.g., EEG and ECG analysis), and communication system monitoring where frequency modulation characteristics require precise tracking.