Multirate filter banks are a powerful tool in advanced signal processing, allowing for efficient analysis and manipulation of signals at different sampling rates. They decompose signals into frequency subbands, enabling various applications in compression, communication, and adaptive filtering.

Understanding multirate filter banks involves key concepts like decimation, interpolation, and polyphase representation. These techniques form the foundation for designing analysis and synthesis filter banks, which are crucial for applications such as subband coding, wavelet transforms, and multicarrier modulation systems.

Multirate filter bank fundamentals

• Multirate filter banks are a key concept in advanced signal processing that involve processing signals at different sampling rates
• They enable efficient analysis, processing, and synthesis of signals by decomposing them into frequency subbands
• Understanding the fundamentals of multirate filter banks is essential for designing and implementing various signal processing applications

Decimation and interpolation

• Decimation reduces the sampling rate of a signal by an integer factor $M$, effectively discarding $M-1$ samples for every $M$ samples
  • Involves an anti-aliasing lowpass filter followed by downsampling to prevent aliasing in the decimated signal
• Interpolation increases the sampling rate of a signal by an integer factor $L$, inserting $L-1$ zero-valued samples between each original sample
  • Requires a post-filtering operation to remove the imaging components introduced by the upsampling process
• Decimation and interpolation are fundamental building blocks in multirate signal processing and are used extensively in filter bank design

Analysis and synthesis filter banks

• An analysis filter bank decomposes an input signal into multiple frequency subbands using a set of bandpass filters
  • Each subband signal is typically decimated to maintain the overall sampling rate
• A synthesis filter bank reconstructs the original signal from the subband signals by interpolating and filtering each subband, then summing the results
  • The synthesis filters are designed to cancel out the aliasing introduced during the decimation process in the analysis stage
• The combination of analysis and synthesis filter banks allows for efficient processing and manipulation of signals in the subband domain

Polyphase representation

• Polyphase representation is a mathematical tool that simplifies the analysis and implementation of multirate systems
• It decomposes a filter into multiple branches, each operating at a lower sampling rate, which reduces computational complexity
• The polyphase components of a filter $h[n]$ are defined as $e_k[n] = h[nM+k]$, where $M$ is the decimation/interpolation factor and $k = 0, 1, ..., M-1$
• Polyphase representation allows for efficient implementation of decimation and interpolation operations using the noble identities

Noble identities

• The noble identities are a set of rules that allow for the interchange of decimation/interpolation operations with filtering operations in a multirate system
• The first noble identity states that decimation by a factor $M$ followed by filtering with $H(z)$ is equivalent to filtering with $H(z^M)$ followed by decimation by $M$
• The second noble identity states that filtering with $G(z)$ followed by interpolation by a factor $L$ is equivalent to interpolation by $L$ followed by filtering with $G(z^L)$
• Applying the noble identities in conjunction with polyphase representation simplifies the design and implementation of multirate filter banks

Filter bank design techniques

• Various design techniques have been developed to construct multirate filter banks with desirable properties such as perfect reconstruction, aliasing cancellation, and good frequency selectivity
• The choice of design technique depends on the specific requirements of the application, such as the number of subbands, the desired filter characteristics, and the allowable complexity

Cosine modulated filter banks

• Cosine modulated filter banks (CMFBs) are a class of filter banks where the analysis and synthesis filters are obtained by cosine modulation of a single prototype filter
• The prototype filter is typically a lowpass filter with good stopband attenuation and a smooth transition band
• CMFBs offer several advantages, such as perfect reconstruction, good frequency selectivity, and efficient implementation using fast cosine transforms
• Examples of CMFBs include the modified discrete cosine transform (MDCT) and the extended lapped transform (ELT) used in audio coding applications

Quadrature mirror filter banks

• Quadrature mirror filter banks (QMFBs) are a special case of two-channel filter banks where the analysis and synthesis filters are related by a quadrature mirror symmetry
• The analysis filters are designed such that their magnitude responses are mirror images of each other around the quadrature frequency $\omega = \pi/2$
• QMFBs can achieve perfect reconstruction and good frequency selectivity with relatively low complexity
• They are commonly used in subband coding applications, such as audio compression (MPEG audio) and image compression (JPEG 2000)

Paraunitary filter banks

• Paraunitary filter banks are a class of filter banks where the polyphase matrix of the analysis filters is unitary, i.e., its inverse is equal to its Hermitian transpose
• This property ensures perfect reconstruction and orthogonality between the subband signals, which is desirable for energy compaction and decorrelation
• Paraunitary filter banks can be designed using lattice structures or by factorizing the polyphase matrix into a product of elementary matrices
• Examples of paraunitary filter banks include the discrete wavelet transform (DWT) and the lapped orthogonal transform (LOT)

Biorthogonal filter banks

• Biorthogonal filter banks relax the orthogonality constraint of paraunitary filter banks, allowing for more design flexibility
• The analysis and synthesis filters are designed to be biorthogonal, i.e., their impulse responses satisfy a biorthogonality condition
• Biorthogonal filter banks can achieve perfect reconstruction and good frequency selectivity, while having more degrees of freedom in the filter design compared to paraunitary filter banks
• The 9/7 and 5/3 biorthogonal wavelet filter banks are widely used in image compression standards such as JPEG 2000

M-channel filter banks

• M-channel filter banks are a generalization of two-channel filter banks, where the input signal is split into $M$ subbands using $M$ analysis filters
• The subbands are decimated by a factor of $M$ and processed independently, then interpolated and recombined using $M$ synthesis filters
• M-channel filter banks offer greater frequency resolution and flexibility compared to two-channel filter banks, but at the cost of increased complexity
• The design of M-channel filter banks often involves optimization techniques to achieve perfect reconstruction, aliasing cancellation, and desired filter characteristics

Filter bank properties

• Filter banks are characterized by various properties that determine their performance and suitability for different applications
• These properties include perfect reconstruction, aliasing cancellation, coding gain, frequency selectivity, and computational complexity

Perfect reconstruction

• Perfect reconstruction (PR) is a desirable property of filter banks, where the output signal is a delayed version of the input signal without any distortion or aliasing
• PR is achieved when the analysis and synthesis filters are designed to cancel out the aliasing and amplitude distortions introduced by the decimation and interpolation operations
• PR filter banks are essential for lossless signal processing applications, such as audio and image compression, where the original signal needs to be perfectly reconstructed

Aliasing cancellation

• Aliasing is a distortion that occurs in multirate systems when the sampling rate is changed, causing the frequency components to overlap and interfere with each other
• Aliasing cancellation is a property of filter banks where the aliasing components introduced by the decimation operations in the analysis stage are canceled out by the synthesis filters
• Aliasing cancellation is a necessary condition for perfect reconstruction and is achieved by carefully designing the analysis and synthesis filters to satisfy certain conditions

Coding gain

• Coding gain is a measure of the energy compaction and decorrelation achieved by a filter bank, which determines its effectiveness in signal compression applications
• It is defined as the ratio of the arithmetic mean to the geometric mean of the subband variances, expressed in decibels (dB)
• Higher coding gain indicates better energy compaction and decorrelation, leading to more efficient signal compression
• Paraunitary and biorthogonal filter banks are often designed to maximize the coding gain for a given signal statistics

Frequency selectivity

• Frequency selectivity refers to the ability of a filter bank to separate the input signal into distinct frequency subbands with minimal overlap and leakage
• Good frequency selectivity is important for applications such as subband coding, where the subbands need to be processed independently without interference
• The frequency selectivity of a filter bank is determined by the stopband attenuation and the transition bandwidth of the analysis and synthesis filters
• Techniques such as cosine modulation and optimization-based design can be used to improve the frequency selectivity of filter banks

Computational complexity

• Computational complexity is an important consideration in the design and implementation of filter banks, especially for real-time and resource-constrained applications
• The complexity of a filter bank depends on the number of subbands, the filter lengths, and the implementation structure (direct form, lattice, polyphase)
• Efficient implementations, such as polyphase structures and fast transforms (DCT, FFT), can significantly reduce the computational complexity of filter banks
• Trade-offs between complexity and performance (PR, aliasing cancellation, frequency selectivity) need to be considered in the design process

Applications of multirate filter banks

• Multirate filter banks have found numerous applications in various fields of signal processing, including audio and image compression, communication systems, and adaptive filtering
• The ability to efficiently process and manipulate signals in the subband domain has made filter banks a powerful tool for a wide range of applications

Subband coding

• Subband coding is a compression technique that involves decomposing a signal into frequency subbands, quantizing and encoding each subband separately, and then reconstructing the signal from the encoded subbands
• Filter banks are used to perform the subband decomposition and reconstruction, exploiting the energy compaction and decorrelation properties to achieve efficient compression
• Examples of subband coding include MPEG audio compression (Layer III, known as MP3) and JPEG 2000 image compression, which use cosine modulated and biorthogonal wavelet filter banks, respectively

Wavelet transforms

• Wavelet transforms are a class of multiresolution signal representations that provide time-frequency localization and multiscale analysis
• Wavelet transforms can be implemented using iterated two-channel filter banks, where the lowpass subband is recursively decomposed to obtain a dyadic frequency decomposition
• The discrete wavelet transform (DWT) and its variants (lifting scheme, wavelet packets) have found applications in image compression (JPEG 2000), denoising, and feature extraction
• Filter banks designed using wavelet bases, such as Daubechies and biorthogonal wavelets, have desirable properties for wavelet transform applications

Transmultiplexers

• Transmultiplexers are systems that convert between time-division multiplexed (TDM) and frequency-division multiplexed (FDM) signals using multirate filter banks
• In a transmultiplexer, a set of input signals are first interpolated and filtered by a synthesis filter bank to generate an FDM signal, which is then transmitted over a channel
• At the receiver, an analysis filter bank is used to demultiplex the FDM signal into the original input signals, which are then decimated to obtain the TDM output
• Transmultiplexers are used in communication systems for efficient multiplexing and demultiplexing of multiple signals over a shared channel

Multicarrier modulation

• Multicarrier modulation is a technique used in wireless communication systems to transmit data over multiple frequency subcarriers in parallel
• Filter banks are used to implement multicarrier modulation schemes, such as orthogonal frequency-division multiplexing (OFDM) and filtered multitone (FMT)
• In OFDM, the synthesis filter bank is implemented using an inverse FFT, while the analysis filter bank uses an FFT, enabling efficient modulation and demodulation of the subcarriers
• Filter banks designed with good frequency selectivity and low intercarrier interference are essential for the performance of multicarrier modulation systems

Adaptive filtering

• Adaptive filtering is a technique used to adjust the coefficients of a filter in real-time based on the characteristics of the input signal and a desired output or error signal
• Multirate filter banks can be used in adaptive filtering applications to perform subband adaptive filtering, where each subband is processed by a separate adaptive filter
• Subband adaptive filtering offers several advantages, such as reduced computational complexity, improved convergence speed, and the ability to handle non-stationary signals
• Examples of subband adaptive filtering include acoustic echo cancellation, noise reduction, and system identification in subbands

Advanced topics in multirate filter banks

• As the field of multirate signal processing continues to evolve, several advanced topics have emerged that extend the capabilities and applications of filter banks
• These topics include nonuniform filter banks, multidimensional filter banks, the lifting scheme, filter bank frames, and oversampled filter banks

Nonuniform filter banks

• Nonuniform filter banks are a generalization of uniform filter banks, where the decimation and interpolation factors can vary across the subbands
• This allows for more flexible frequency decompositions and better adaptation to the signal characteristics and application requirements
• Nonuniform filter banks can be designed using tree structures, such as octave-band filter banks and wavelet packets, or by combining uniform filter banks with resampling operations
• Applications of nonuniform filter banks include audio and image compression, time-frequency analysis, and feature extraction

Multidimensional filter banks

• Multidimensional filter banks extend the concepts of multirate signal processing to higher-dimensional signals, such as images and video
• In a multidimensional filter bank, the input signal is decomposed into subbands along multiple dimensions (e.g., rows and columns for images) using separable or non-separable filters
• Multidimensional filter banks can achieve higher compression ratios and better energy compaction compared to one-dimensional filter banks
• Examples of multidimensional filter banks include separable wavelet transforms, quincunx filter banks, and directional filter banks used in image and video compression

Lifting scheme

• The lifting scheme is a flexible and efficient framework for constructing biorthogonal wavelet filter banks and performing wavelet transforms
• It decomposes the wavelet transform into a series of simple lifting steps, which involve splitting, predicting, and updating the input signal
• The lifting scheme allows for in-place computation, reduced memory requirements, and integer-to-integer wavelet transforms, making it suitable for hardware implementations
• Lifting-based filter banks have been used in various applications, such as image compression (JPEG 2000), progressive coding, and lossless compression

Filter bank frames

• Filter bank frames are a generalization of filter banks that allow for overcomplete signal representations and provide robustness to signal degradations
• In a filter bank frame, the number of subbands is greater than the decimation factor, resulting in a redundant representation of the input signal
• Filter bank frames can be designed to have desirable properties, such as tight frames, dual frames, and frame bounds, which ensure stable and invertible signal representations
• Applications of filter bank frames include signal denoising, sparse signal representation, and robust signal transmission

Oversampled filter banks

• Oversampled filter banks are a class of filter banks where the total decimation factor is less than the number of subbands, resulting in an overcomplete signal representation
• Oversampling introduces redundancy in the subband signals, which can be exploited for improved noise reduction, robustness to channel errors, and reduced aliasing
• Oversampled filter banks can be designed using the same techniques as critically sampled filter banks, such as cosine modulation and paraunitary factorization
• Applications of oversampled filter banks include subband adaptive filtering, signal denoising, and robust signal transmission in the presence of channel impairments