〰️Signal Processing Unit 15 – Wavelet Applications in Signal Processing

Wavelet Basics

• Wavelets are mathematical functions used to analyze and represent signals or data
• Provide a way to decompose complex signals into simpler, more manageable components
• Allow for analysis at different scales or resolutions (multiresolution analysis)
• Wavelet transforms convert signals from time domain to time-frequency domain
• Basis functions are localized in both time and frequency
  • Enables capturing transient or non-stationary features in signals
• Characterized by their shape, scale, and position
• Offer advantages over traditional Fourier analysis for certain types of signals
  • Particularly useful for signals with discontinuities or sharp changes

Time-Frequency Analysis

• Combines time domain and frequency domain analysis techniques
• Provides information about how the frequency content of a signal changes over time
• Traditional Fourier analysis assumes signal is stationary and loses temporal information
• Short-Time Fourier Transform (STFT) introduced to address this limitation
  • Divides signal into fixed-size windows and applies Fourier transform to each window
  • Offers a fixed time-frequency resolution determined by the window size
• Wavelet transforms overcome the fixed resolution limitation of STFT
  • Provide variable time-frequency resolution adapted to the signal characteristics
• Continuous Wavelet Transform (CWT) uses scaled and shifted versions of a mother wavelet
• Discrete Wavelet Transform (DWT) uses discrete scales and positions for efficient computation

Wavelet Transforms

• Mathematical tools for representing signals using wavelets as basis functions
• Decompose signals into a set of wavelet coefficients
• Continuous Wavelet Transform (CWT)
  • Computes inner products between the signal and continuously scaled and shifted wavelets
  • Provides a highly redundant representation of the signal
  • Useful for signal analysis and feature extraction
• Discrete Wavelet Transform (DWT)
  • Uses discrete scales and shifts, typically powers of two (dyadic scales)
  • Provides a non-redundant representation of the signal
  • Implemented using filter banks with low-pass and high-pass filters
  • Decomposes signal into approximation and detail coefficients at each level
• Inverse Wavelet Transform reconstructs the original signal from the wavelet coefficients
• Wavelet Packet Transform extends DWT by allowing further decomposition of detail coefficients

Multiresolution Analysis

• Framework for constructing and analyzing wavelets and their properties
• Decomposes signals into a hierarchy of approximations and details at different scales
• Approximations represent the low-frequency or coarse information of the signal
• Details capture the high-frequency or fine information at each scale
• Wavelet functions are derived from a scaling function through dilation and translation
• Scaling function satisfies a two-scale relation, linking it to the wavelet function
• Multiresolution analysis ensures perfect reconstruction of the original signal
  • Approximations and details at each level can be combined to recover the signal
• Allows for efficient computation and storage of wavelet coefficients
• Provides a natural way to analyze signals at different levels of detail

Wavelet Families

• Different classes of wavelets with distinct properties and characteristics
• Haar wavelet
  • Simplest wavelet, resembling a step function
  • Discontinuous and non-differentiable
  • Useful for signals with sudden changes or discontinuities
• Daubechies wavelets
  • Family of orthogonal wavelets with compact support
  • Characterized by the number of vanishing moments
  • Higher vanishing moments provide better approximation of smooth signals
• Symlets
  • Nearly symmetrical wavelets derived from Daubechies wavelets
  • Designed to have minimal asymmetry while retaining other desirable properties
• Coiflets
  • Wavelets with additional vanishing moments for both the wavelet and scaling functions
  • Provide better approximation of polynomial signals
• Biorthogonal wavelets
  • Offer greater flexibility in design by relaxing orthogonality constraint
  • Allow for symmetric wavelet functions, which can be advantageous in certain applications
• Choice of wavelet family depends on the specific signal characteristics and application requirements

Signal Denoising

• Process of removing noise from signals while preserving important features
• Wavelet-based denoising exploits the sparsity of wavelet coefficients
• Noisy signal is transformed into the wavelet domain using DWT
• Wavelet coefficients are thresholded to suppress noise
  • Soft thresholding shrinks coefficients towards zero
  • Hard thresholding sets small coefficients to zero
• Threshold value is chosen based on noise level and desired signal-to-noise ratio
• Inverse DWT is applied to the thresholded coefficients to obtain the denoised signal
• Wavelet denoising is particularly effective for signals with non-stationary noise
• Outperforms traditional denoising methods (Wiener filtering) in many scenarios
• Allows for adaptive denoising based on the signal characteristics at different scales

Compression Techniques

• Wavelets enable efficient compression of signals and images
• Exploit the sparsity and decorrelation properties of wavelet coefficients
• Wavelet transform concentrates signal energy into a few large coefficients
• Small coefficients can be discarded or quantized coarsely without significant loss
• Thresholding and quantization of wavelet coefficients reduce data size
• Entropy coding (Huffman, arithmetic coding) further compresses the quantized coefficients
• Embedded coding schemes (EZW, SPIHT) allow progressive transmission and reconstruction
  • Transmit most significant coefficients first, followed by refinements
  • Enable scalable compression and quality control
• Wavelet-based compression standards (JPEG 2000) outperform traditional methods (DCT-based JPEG)
  • Provide better compression ratios and visual quality at low bitrates
• Wavelet packets offer more flexible decomposition for adaptive compression

Practical Applications

• Wavelet-based methods find applications in various fields
• Signal processing
  • Denoising, compression, and analysis of audio, speech, and biomedical signals
  • Feature extraction and pattern recognition
  • Transient detection and characterization
• Image processing
  • Image denoising, compression, and enhancement
  • Edge detection and texture analysis
  • Image fusion and super-resolution
• Communication systems
  • Channel coding and modulation using wavelet packets
  • Multicarrier modulation schemes (OFDM) with wavelet bases
• Biomedical applications
  • Analysis of EEG, ECG, and other physiological signals
  • Detection of abnormalities and diagnostic features
• Geophysical applications
  • Analysis of seismic and well log data
  • Characterization of subsurface structures and reservoirs
• Financial applications
  • Analysis of financial time series and market data
  • Detection of trends, patterns, and anomalies
• Wavelet-based methods continue to evolve and find new applications across disciplines