Instantaneous Frequency, Group Delay, and Signal Generation Methods for Non-Stationary Signals

Non-stationary signals are characterized by statistical properties that vary over time, with frequency components that change throughout the signal duration. These signals are commonly encountered in fields such as radar, acoustics, and biomedical engineering.

Instantaneous Frequency Instantaneous frequency describes the signal's instantaneous rate of change at a specific time point, mathematically defined as the derivative of phase with respect to time. For analytic signals, the Hilbert transform can be employed to extract instantaneous frequency, revealing local frequency characteristics. In MATLAB implementation, the hilbert() function can be used to obtain the analytic signal, followed by differentiation of the unwrapped phase angle using the diff() function or angle() function with proper unwrapping.

Group Delay Group delay characterizes the propagation delay of different frequency components through a system, reflecting the system's phase response properties. Calculation typically involves analyzing the signal's phase spectrum using frequency domain methods. For non-stationary signal analysis, group delay can detect changes in propagation media or system response characteristics. Code implementation often involves using the grpdelay() function in MATLAB or calculating from the derivative of the phase response with respect to frequency.

Signal Generation Methods Common methods for generating non-stationary signals include: Linear Frequency Modulated signals (Chirp signals): Frequency varies linearly or non-linearly with time, widely used in radar and sonar systems. MATLAB's chirp() function allows generation with customizable sweep rates and frequency ranges. Multi-component signal synthesis: Simulating complex variations by superimposing multiple time-varying frequency signals, implementable using sinusoids with time-dependent frequency parameters. Time-frequency distribution based signal design: Reverse engineering signals using Wigner-Ville distribution or short-time Fourier transform coefficients. The Signal Processing Toolbox provides functions like spectrogram() for analysis and synthesis. Randomly modulated signals: Modulating frequency or amplitude through random processes to simulate real-world uncertainties, achievable using random number generators combined with modulation techniques.

When combined with time-frequency analysis tools such as wavelet transforms (using wavedec() or cwt() functions) and empirical mode decomposition (EMD), these methods can more accurately characterize the dynamic properties of non-stationary signals.