〰️Signal Processing Unit 13 – Filter Banks and Wavelets

Fundamentals of Filter Banks

• Filter banks decompose a signal into multiple frequency bands or subbands, enabling efficient processing and analysis
• Consist of a set of analysis filters that split the input signal and a set of synthesis filters that reconstruct the original signal
• Analysis filters divide the signal into low and high frequency components, while synthesis filters combine these components to recover the original signal
• Perfect reconstruction is achieved when the output signal is identical to the input signal, except for a possible delay and scaling factor
  • Requires the analysis and synthesis filters to satisfy specific conditions, such as orthogonality or biorthogonality
• Subband coding is a common application of filter banks, where each subband is processed or coded independently
• Polyphase representation simplifies the implementation of filter banks by decomposing filters into polyphase components
• Alias cancellation is crucial in filter bank design to prevent aliasing artifacts caused by downsampling and upsampling operations

Introduction to Wavelets

• Wavelets are mathematical functions that analyze signals at different scales and resolutions, providing both time and frequency localization
• Wavelet transforms decompose a signal into a set of basis functions called wavelets, which are scaled and shifted versions of a mother wavelet
• Mother wavelet is a prototype function that is dilated (scaled) and translated (shifted) to generate a family of wavelets
• Continuous Wavelet Transform (CWT) operates on a continuous-time signal and produces a continuous-scale representation
  • Provides high redundancy and is computationally intensive, mainly used for signal analysis and feature extraction
• Discrete Wavelet Transform (DWT) operates on a discrete-time signal and produces a discrete-scale representation
  • More efficient and practical for signal processing applications, as it reduces redundancy and computational complexity
• Wavelet transforms have good time-frequency resolution, making them suitable for analyzing non-stationary signals and detecting local features
• Multiresolution analysis is a key concept in wavelet theory, where a signal is decomposed into a set of approximation and detail coefficients at different scales

Types of Filter Banks

• Two-channel filter banks are the simplest form, consisting of two analysis filters (lowpass and highpass) and two synthesis filters
  • Used in subband coding and wavelet transforms, where the signal is divided into two frequency bands
• Multichannel filter banks extend the concept to more than two channels, allowing for finer frequency resolution and more flexible signal decomposition
• Uniform filter banks have equal bandwidth for all subbands, resulting in a uniform frequency division
  • Commonly used in audio and speech processing applications
• Non-uniform filter banks have varying bandwidths for different subbands, adapting to the signal characteristics or application requirements
  • Used in perceptual audio coding and image compression, where the human perceptual system is taken into account
• Orthogonal filter banks have analysis and synthesis filters that are orthogonal to each other, ensuring perfect reconstruction and energy preservation
• Biorthogonal filter banks relax the orthogonality constraint, allowing for more design flexibility and better frequency selectivity
• Cosine Modulated filter banks use cosine modulation to generate a set of analysis and synthesis filters from a single prototype filter
  • Computationally efficient and widely used in audio and speech coding applications

Wavelet Transforms

• Continuous Wavelet Transform (CWT) provides a continuous-scale representation of a signal, using a continuous set of scales and translations
  • Computed by correlating the signal with scaled and shifted versions of the mother wavelet
• Discrete Wavelet Transform (DWT) discretizes the scale and translation parameters, resulting in a discrete set of wavelet coefficients
  • Implemented using a filter bank structure, where the signal is passed through a series of lowpass and highpass filters followed by downsampling
• Wavelet decomposition refers to the process of applying the DWT recursively to the lowpass subband, creating a multi-level decomposition
  • Each level represents a different frequency resolution, with the lowpass subband capturing the approximation and the highpass subband capturing the details
• Wavelet reconstruction is the inverse process, where the wavelet coefficients are upsampled and passed through a set of synthesis filters to recover the original signal
• Wavelet packets generalize the wavelet transform by allowing decomposition of both lowpass and highpass subbands, creating a full binary tree structure
  • Offers more flexibility in adapting to signal characteristics and designing optimal bases for specific applications
• Lifting scheme is an efficient implementation of the DWT that replaces the filter convolutions with a series of simple lifting steps
  • Reduces computational complexity and enables in-place computation, making it suitable for hardware implementations

Filter Bank Design Techniques

• Frequency response masking (FRM) is a technique for designing sharp transition band filters by combining a prototype filter with a set of masking filters
  • Allows for the design of high-order filters with reduced computational complexity
• Interpolated FIR (IFIR) filters use interpolation to increase the filter order and improve the frequency response without increasing the number of multiplications
• Halfband filters have a frequency response that is symmetric around half the Nyquist frequency, with a transition band that is half the passband width
  • Used in two-channel filter banks and wavelet transforms for their efficiency and perfect reconstruction properties
• Complementary filters have frequency responses that sum up to a constant value (usually 1) across the entire frequency range
  • Essential for perfect reconstruction in filter banks and wavelet transforms
• Lattice structures represent filters as a cascade of lattice stages, each characterized by a single coefficient
  • Provide improved numerical stability, quantization noise performance, and efficient hardware implementation
• Optimization techniques, such as least squares and minimax methods, are used to design filters with desired frequency response characteristics
  • Involve minimizing an error function that measures the deviation between the desired and actual frequency responses
• Multirate filter bank design involves designing analysis and synthesis filters that operate at different sampling rates
  • Requires careful consideration of aliasing cancellation and perfect reconstruction conditions

Applications in Signal Processing

• Audio compression uses filter banks and wavelets to decompose audio signals into subbands, allowing for efficient coding and perceptual modeling
  • Examples include MP3, AAC, and WMA formats
• Image compression applies wavelet transforms to exploit spatial and frequency locality, achieving high compression ratios while preserving perceptual quality
  • JPEG 2000 standard uses the DWT for improved compression performance and scalability
• Denoising and signal enhancement rely on wavelet transforms to separate signal and noise components based on their different time-frequency characteristics
  • Wavelet shrinkage and thresholding techniques are commonly used to suppress noise while preserving signal details
• Feature extraction and pattern recognition benefit from the multiscale and time-frequency localization properties of wavelets
  • Wavelet coefficients serve as discriminative features for classification and recognition tasks
• Biomedical signal processing, such as ECG and EEG analysis, employs wavelet transforms to detect and characterize transient events and abnormalities
• Wireless communication systems use filter banks for channelization, equalization, and multicarrier modulation schemes like OFDM
• Radar and sonar signal processing apply wavelet transforms for target detection, classification, and imaging applications

Implementation and Algorithms

• Polyphase implementation of filter banks decomposes the filters into polyphase components, enabling efficient computation and reduced memory requirements
• Fast algorithms for the DWT, such as the Mallat algorithm and the lifting scheme, exploit the recursive structure of the transform to reduce computational complexity
  • Mallat algorithm uses a tree-structured filter bank approach, while the lifting scheme relies on a series of simple lifting steps
• Lattice filter structures provide modular and parameterized implementations, facilitating the design and realization of perfect reconstruction filter banks
• Boundary handling techniques, such as periodic extension and symmetric extension, address the issue of finite-length signals in wavelet transforms
  • Ensure proper signal extension at the boundaries to avoid artifacts and maintain perfect reconstruction
• Quantization and encoding of wavelet coefficients are crucial for practical applications, especially in compression and data transmission
  • Scalar quantization, vector quantization, and entropy coding techniques are commonly used to compress wavelet coefficients
• Hardware implementations of filter banks and wavelet transforms aim to optimize speed, power consumption, and resource utilization
  • DSP processors, FPGAs, and ASICs are popular platforms for realizing efficient hardware designs
• Software libraries and toolboxes, such as MATLAB's Wavelet Toolbox and Python's PyWavelets, provide high-level functions and utilities for filter bank and wavelet analysis

Advanced Topics and Current Research

• Multidimensional filter banks and wavelets extend the concepts to higher dimensions, enabling processing of images, videos, and volumetric data
  • Separable and non-separable approaches are used to design multidimensional filter banks
• Directional filter banks and wavelets, such as the contourlet and shearlet transforms, aim to capture directional features and edges in images more effectively
• Adaptive filter banks and wavelets allow for dynamic adaptation of the transform based on signal characteristics or application requirements
  • Examples include adaptive wavelet packets and best basis selection algorithms
• Compressed sensing and sparse representations leverage the sparsity of signals in the wavelet domain for efficient acquisition and reconstruction
  • Wavelet bases are commonly used as sparsifying transforms in compressed sensing frameworks
• Graph signal processing extends the concepts of filter banks and wavelets to signals defined on graphs, enabling analysis of network-structured data
• Multiresolution analysis on manifolds and non-Euclidean domains generalizes wavelet transforms to handle signals defined on complex geometries and topologies
• Deep learning and neural networks have been combined with wavelet transforms for various tasks, such as image super-resolution and signal denoising
  • Wavelet-based neural network architectures have shown promising results in capturing multiscale features and dependencies
• Quantum wavelet transforms and filter banks explore the application of these concepts in quantum computing and quantum signal processing domains