5.2 Polyphase decomposition

Polyphase decomposition is a powerful technique in signal processing that breaks down signals into smaller components. It allows for efficient implementation of multirate systems, reducing computational complexity and enabling parallel processing.

This method is crucial for filter banks, wavelet transforms, and sampling rate conversion. By decomposing signals and filters into polyphase components, engineers can design more efficient and effective digital signal processing systems.

Polyphase decomposition fundamentals

Definition of polyphase decomposition

• Polyphase decomposition is a mathematical technique used to represent a discrete-time signal or system as a sum of polyphase components
• Involves decomposing a signal or system into a set of subsystems with reduced sampling rates
• Enables efficient implementation of multirate signal processing algorithms

Key properties of polyphase components

• Each polyphase component operates at a lower sampling rate compared to the original signal
• Polyphase components are obtained by downsampling the original signal by a factor of M
• Polyphase components can be recombined to reconstruct the original signal
• Polyphase decomposition preserves the overall frequency response of the system

Mathematical notation for polyphase representation

• A signal $x[n]$ can be decomposed into M polyphase components $x_k[n]$, where $k = 0, 1, ..., M-1$
• The polyphase components are defined as: $x_k[n] = x[nM + k]$
• The original signal can be reconstructed as: $x[n] = \sum_{k=0}^{M-1} x_k[n/M]$, where $n/M$ represents upsampling by a factor of M

Applications of polyphase decomposition

Role in multirate signal processing

• Polyphase decomposition is a fundamental tool in multirate signal processing
• Enables efficient implementation of decimation and interpolation operations
• Allows for the design of filter banks with perfect reconstruction properties

Efficient implementation of filter banks

• Polyphase decomposition can be used to implement filter banks with reduced computational complexity
• By decomposing filters into polyphase components, the number of required multiplications and additions can be significantly reduced
• Polyphase structures enable parallel processing of subband signals

Polyphase decomposition in wavelet transforms

• Wavelet transforms can be implemented using polyphase decomposition
• Polyphase decomposition allows for efficient computation of wavelet coefficients
• Enables the design of critically sampled wavelet filter banks with perfect reconstruction

Polyphase decomposition of FIR filters

Type 1 vs Type 2 polyphase decompositions

• Type 1 polyphase decomposition: The filter coefficients are divided into M polyphase components
• Type 2 polyphase decomposition: The filter coefficients are divided into M polyphase components with a delay of one sample between components
• Type 1 is commonly used for decimation filters, while Type 2 is used for interpolation filters

Noble identities for polyphase structures

• Noble identities are mathematical properties that allow for the simplification of polyphase structures
• The first noble identity states that decimation by M followed by filtering is equivalent to filtering followed by decimation by M
• The second noble identity states that filtering followed by interpolation by L is equivalent to interpolation by L followed by filtering

Polyphase implementation of decimation filters

• Decimation filters can be efficiently implemented using polyphase decomposition
• The input signal is first filtered by a set of polyphase components
• The filtered signals are then downsampled by a factor of M to obtain the decimated output
• Polyphase implementation reduces the computational complexity compared to direct form implementation

Polyphase implementation of interpolation filters

• Interpolation filters can also be implemented using polyphase decomposition
• The input signal is first upsampled by a factor of L by inserting zeros between samples
• The upsampled signal is then filtered by a set of polyphase components
• The filtered signals are combined to obtain the interpolated output

Polyphase decomposition of IIR filters

Challenges with IIR polyphase decomposition

• Polyphase decomposition of IIR filters is more challenging compared to FIR filters
• IIR filters have feedback paths, which introduce dependencies between polyphase components
• Direct polyphase decomposition of IIR filters may lead to instability or loss of perfect reconstruction property

Approximation methods for IIR polyphase decomposition

• Approximation methods can be used to overcome the challenges of IIR polyphase decomposition
• One approach is to approximate the IIR filter with a high-order FIR filter and then apply polyphase decomposition
• Another approach is to use allpass decomposition, which decomposes the IIR filter into a cascade of allpass filters

Allpass decomposition for IIR filters

• Allpass decomposition is a technique used to decompose IIR filters into a cascade of allpass filters
• Allpass filters have a flat magnitude response and introduce only phase delay
• By decomposing an IIR filter into allpass filters, the stability and perfect reconstruction properties can be preserved
• The allpass filters can then be implemented using polyphase structures for efficient computation

Polyphase filter banks

Uniform vs non-uniform filter banks

• Filter banks can be classified as uniform or non-uniform based on the sampling rates of the subbands
• Uniform filter banks have equal bandwidth and sampling rates for all subbands
• Non-uniform filter banks have varying bandwidth and sampling rates for different subbands
• Polyphase decomposition can be applied to both uniform and non-uniform filter banks

Perfect reconstruction conditions

• Perfect reconstruction is a desirable property of filter banks, ensuring that the original signal can be perfectly reconstructed from the subband signals
• For perfect reconstruction, the analysis and synthesis filters must satisfy certain conditions
• The polyphase components of the analysis and synthesis filters should form a paraunitary matrix

Cosine modulated filter banks

• Cosine modulated filter banks are a class of filter banks that use cosine modulation to generate the analysis and synthesis filters
• The prototype filter is designed in the frequency domain and then modulated to obtain the subband filters
• Cosine modulated filter banks have efficient implementation using polyphase structures
• They provide good frequency selectivity and perfect reconstruction properties

Paraunitary filter banks

• Paraunitary filter banks are a special case of perfect reconstruction filter banks
• The polyphase matrix of a paraunitary filter bank is unitary, meaning that its inverse is equal to its conjugate transpose
• Paraunitary filter banks have orthogonal subband signals and provide energy preservation
• They can be efficiently implemented using lattice structures or factorization methods

Computational efficiency of polyphase structures

Reduced complexity compared to direct form

• Polyphase structures offer reduced computational complexity compared to direct form implementations
• By decomposing filters into polyphase components, the number of multiplications and additions can be significantly reduced
• Polyphase structures exploit the redundancy in the computations and avoid unnecessary operations

Exploiting parallelism in polyphase implementations

• Polyphase structures are well-suited for parallel implementation
• The polyphase components can be processed independently, allowing for parallel computation
• Parallel processing can further improve the computational efficiency of polyphase implementations
• Multicore processors or hardware accelerators can be utilized to exploit parallelism

Polyphase decomposition in hardware implementations

• Polyphase decomposition is widely used in hardware implementations of signal processing systems
• FPGAs and ASICs can benefit from the reduced complexity and parallelism offered by polyphase structures
• Hardware implementations can take advantage of the regular structure and modular nature of polyphase decomposition
• Polyphase structures can be optimized for hardware resources and power efficiency

Advanced topics in polyphase decomposition

Multidimensional polyphase decomposition

• Polyphase decomposition can be extended to multidimensional signals and systems
• Multidimensional polyphase decomposition allows for efficient processing of images, videos, and other multidimensional data
• The concepts of polyphase components and perfect reconstruction can be generalized to higher dimensions
• Multidimensional polyphase decomposition finds applications in image compression, video coding, and computer vision

Generalized polyphase representation

• Generalized polyphase representation is a framework that extends the concept of polyphase decomposition
• It allows for the representation of signals and systems with arbitrary sampling rates and delays
• Generalized polyphase representation provides a unified framework for analyzing and designing multirate systems
• It enables the development of advanced signal processing algorithms and techniques

Polyphase decomposition in rational sampling rate systems

• Polyphase decomposition can be applied to systems with rational sampling rate changes
• Rational sampling rate systems involve both decimation and interpolation operations
• Polyphase decomposition allows for the efficient implementation of rational sampling rate converters
• It enables the design of filter banks with perfect reconstruction properties for rational sampling rates