MATLAB Implementation of Cepstrum Analysis with Code Examples

Cepstrum is a signal processing technique primarily used for analyzing periodic structures in frequency spectra. It is implemented by applying logarithmic transformation to the signal's spectrum followed by inverse Fourier transform, widely applied in speech signal processing, fault diagnosis, and related fields.

### Fundamental Principles of Cepstrum Signal Preprocessing: Perform Short-Time Fourier Transform (STFT) on the input signal to obtain spectral information. Logarithmic Transformation: Apply logarithmic operation (using absolute values) to the spectrum to amplify periodic features. Inverse Transformation: Execute inverse Fourier transform on the logarithmic spectrum to obtain the cepstral sequence.

### MATLAB Implementation Approach Spectral Calculation: Use the `fft` function to compute Fourier transform of the input signal. Logarithmic Processing: Apply `log10` and `abs` functions to convert spectral magnitude to logarithmic scale. Inverse Transformation: Perform inverse transform using `ifft` on the logarithmic spectrum to obtain the final cepstrum.

### Implementation Example and Extensions Given a speech signal `y` with sampling frequency `fs`, the cepstrum can be calculated as follows: Compute the signal's spectrum using Fourier transform. Apply logarithmic transformation and execute inverse Fourier transform. Adjust the display range of cepstrum, typically focusing on the first half of data due to symmetry.

Furthermore, cepstrum can be utilized for fundamental frequency detection and harmonic analysis tasks, where signal characteristics are extracted by observing peak positions in the cepstral domain. The implementation involves careful parameter selection for STFT window size and overlap to optimize periodic pattern detection.