Program for Signal Transformation Using Hilbert Transform

The Hilbert transform is a fundamental signal processing technique primarily used for analyzing instantaneous frequency and phase characteristics of non-stationary signals. Its core principle involves converting real-valued signals into analytic signals, enabling extraction of instantaneous amplitude and phase information. In MATLAB implementation, this is typically achieved using the hilbert() function which returns the analytic signal from the input data.

In signal processing applications, the Hilbert transform finds utility in several key scenarios: Instantaneous Frequency Measurement: By computing the derivative of the phase angle from the analytic signal, one can obtain instantaneous frequency variations. This is particularly valuable for non-stationary signal analysis such as speech processing and vibration analysis. The implementation typically involves using the angle() and diff() functions to extract and differentiate the phase component. Phase Extraction: The Hilbert transform effectively separates the phase component of signals, making it crucial in communications and radar systems. The algorithm works by creating a 90-degree phase-shifted version of the original signal through convolution with the Hilbert kernel. Amplitude Envelope Analysis: The magnitude of the transformed analytic signal represents the envelope of the original signal, useful for detecting modulation characteristics or abnormal fluctuations. This can be computed using the abs() function on the analytic signal output.

The fundamental implementation approach involves frequency domain processing through Fourier transform, where negative frequency components are removed and phase adjustments are made to generate the analytic signal. This method offers high computational efficiency, making it suitable for real-time signal analysis applications. The algorithm can be implemented using FFT operations with proper frequency domain filtering.

To further enhance computational efficiency or adapt to specific application requirements, one can consider combining the Hilbert transform with Short-Time Fourier Transform (STFT) or wavelet transforms for joint time-frequency analysis. This hybrid approach provides more accurate instantaneous frequency estimation, particularly for signals with rapidly varying characteristics. Implementation would involve windowing techniques for STFT integration or wavelet filter banks for multi-resolution analysis.