Wavelet analysis and time-frequency representations are powerful tools for analyzing biomedical signals. They allow us to examine both the time and frequency aspects of complex signals like EEGs and ECGs, giving us a more complete picture of what's happening in the body.
These techniques are especially useful for non-stationary signals, where the frequency content changes over time. By using wavelets and time-frequency analysis, we can capture important details and patterns in biomedical data that might be missed with traditional signal processing methods.
Wavelet Transform Fundamentals
Wavelet Transform Overview
- Mathematical tool for analyzing signals in both time and frequency domains simultaneously
- Decomposes a signal into a set of basis functions called wavelets, which are localized in both time and frequency
- Allows for multi-resolution analysis of signals, capturing both high-frequency details and low-frequency trends
- Particularly useful for analyzing non-stationary signals, where the frequency content changes over time (biomedical signals, such as EEG, ECG, and EMG)
Continuous Wavelet Transform (CWT)
- Transforms a continuous-time signal into a two-dimensional representation, providing information about the signal's time-frequency characteristics
- Involves convolving the signal with a set of scaled and translated versions of a mother wavelet
- The resulting wavelet coefficients represent the similarity between the signal and the wavelet at various scales and positions
- Provides a high-resolution time-frequency representation, but is computationally intensive and redundant
Discrete Wavelet Transform (DWT)
- Discretized version of the continuous wavelet transform, using a discrete set of scales and translations
- Decomposes a signal into a set of wavelet coefficients using a hierarchical, multi-resolution approach
- Employs a pair of filters (low-pass and high-pass) to recursively decompose the signal into approximation and detail coefficients
- Computationally efficient and non-redundant, making it suitable for practical applications (signal compression, denoising, and feature extraction)
Mother Wavelet Selection
- The choice of mother wavelet significantly impacts the wavelet transform's performance and interpretation
- Mother wavelets are the basis functions used in the wavelet transform, and their properties determine the transform's time-frequency resolution and localization
- Common mother wavelets include Haar, Daubechies, Symlets, and Coiflets, each with unique characteristics (support size, number of vanishing moments, and regularity)
- The selection of an appropriate mother wavelet depends on the signal's properties and the desired analysis objectives (signal matching, feature extraction, or noise reduction)
Time-Frequency Analysis Techniques
Multi-resolution Analysis (MRA)
- Framework for analyzing signals at multiple scales or resolutions, allowing for the extraction of both coarse and fine-grained information
- Decomposes a signal into a set of approximation and detail coefficients using the discrete wavelet transform
- Approximation coefficients represent the low-frequency content and provide a coarse-scale representation of the signal
- Detail coefficients capture the high-frequency content and provide fine-scale information about the signal's local features
- MRA forms the basis for various wavelet-based signal processing techniques (denoising, compression, and feature extraction)
Short-time Fourier Transform (STFT)
- Time-frequency analysis technique that extends the classical Fourier transform to analyze non-stationary signals
- Divides a signal into short, overlapping segments using a sliding window function and applies the Fourier transform to each segment
- Provides a time-frequency representation of the signal, revealing how its frequency content changes over time
- The choice of window function (Hamming, Hann, or Gaussian) and window size determines the trade-off between time and frequency resolution
- Limitations include fixed time-frequency resolution and the assumption of local stationarity within each window
Spectrogram Representation
- Visual representation of a signal's time-frequency content, obtained by computing the squared magnitude of the short-time Fourier transform
- Displays the signal's energy distribution across time and frequency, with time on the x-axis, frequency on the y-axis, and energy represented by color or intensity
- Allows for the identification of time-varying spectral patterns, such as chirps, transients, and non-stationary components
- Commonly used in biomedical signal analysis (EEG, ECG, and speech) for detecting and characterizing time-localized events and abnormalities
- Interpretation of spectrograms requires consideration of the chosen window function, window size, and overlap, as these parameters affect the time-frequency resolution and visualization