Polyphase Representation Research Articles

Combined adaptive antenna array and adaptive channel equalizer, based on multichannel fast recursive least squares algorithms

An adaptive antenna array (AAA) is a sort of directional antennas, which is able to suppress the signals of the interference sources. The AAA works well if there is no multipath propagation of the information signal. A device, which removes the multipath effects, is an adaptive equalizer. However, the independent AAA and equalizer do not work well if the interferences and multipath exist simultaneously. The purpose of this paper is to present a new efficient solution of the task by using of the AAA, whose weights are shared with the weights of the fractionally-spaced feed-forward (FF) part of the equalizer. The array output signal is combined with that of the feed-backward (FB) part of the equalizer. The array and such distributed equalizer operate as an indivisible adaptive filter, providing good performance and ability to receive the multipath information signals in the presence of the signals of the external interference sources. Because the FF part of the equalizer operates at the up-sampled rate, which is a few times higher than the information symbols rate, but the FB filter operates at the symbol rate, the computationally efficient fast recursive least squares (RLS) algorithms cannot be used directly in the considered adaptive unit. To solve this problem, it is proposed to use polyphase representation of the FF part of the equalizer. In this case, the architecture of the adaptive unit is viewed as a multichannel adaptive filter with unequal number of weights in channels. The paper presents the architecture of the adaptive unit and the details of its equalizer part. The mathematical description of a multichannel fast adaptive filtering algorithm, called stabilized fast a posteriori sequential technique (FAEST), fitted for the weights calculation of the considered unit is also presented. The verification of the proposed adaptive unit has been carried out via simulation, using a linear array with sixteen omnidirectional antennas, which operates in the conditions of –30 dB signal-to-interference ratio (SIR), 10…30 dB signal-to-noise ratio (SNR), receives the phase shift keying (8-PSK) desired signal, passed through a two-rays communication channel with about –65 dB gaps in the channel amplitude-frequency response. In the steady-state, the proposed adaptive unit demonstrates about –100 dB radiation pattern gaps towards the interference sources and the transient response with less than 500 symbols duration. The simulation demonstrates the ability of the proposed unit simultaneously to steer the main lobe of antenna array radiation pattern towards the desired signal source without a priori search of its angular location and to remove the signals of the external interferences and intersymbol interference in the array output signal. Due to these properties, the proposed AAA/equalizer can be used in equipment of modern mobile communication systems.

Discrete Wavelet Design for Bearing Fault Diagnosis Using Particle Swarm Optimization

Rolling bearings are widely used in a large variety of industrial applications. Therefore, it is necessary to provide an efficient fault detection and diagnosis mechanism to prevent component failure and poor performance during operation. This paper proposes a novel classification scheme based on the design of discrete wavelets best adapted to vibration signal analysis in order to identify and properly classify rolling bearing defects. Through polyphase representation of the wavelet filter bank, and using the particle swarm optimization (PSO) algorithm, the appropriate discrete wavelet associated filters are optimized to achieve the best fault classification accuracy. Simulation results show that the proposed wavelet design approach outperforms the well-known standard wavelets regardless the employed classifier and leads to an average fault classification improvement of about 2%.

Bivariate two-band wavelets demystified

There are several well known constructions of bivariate, compactly supported wavelets based on orthogonal refinable functions with dilation matrices of determinant 2 or -2. The corresponding filterbank consists of only two subbands: low-pass and high-pass. We unify these constructions and express their intrinsic structure via normal forms of the corresponding bivariate polyphase representations. A normal form is sparse and is obtained from the polyphase representation via a suitable unitary transformation. We characterize certain normal forms of bi-degree (1,1) and (2,2) and show that their non-zero elements correspond to the solutions of the univariate Quadrature Mirror Filter conditions. The unitary transformations are chosen to ensure sum rule conditions of certain order. We illustrate our results on some examples and address the quest for characterizing bivariate wavelet constructions of higher bi-degree.

Implementation of Cosine Modulated Digital Filter Bank on Processor with ARM Architecture

This paper presents a class of multichannel cosine-modulated filter banks (CMFB) of analysis based on the modulation effect with a fast discrete-cosine transformation of the fourth type (DCT-IV), which is calculated using the fast Fourier transform. As a prototype filter, a low-frequency filter with a finite pulse characteristic was used, frequency-shifted copies of which were made using an effective technology for polyphase representation of the filter Bank. The comparison of the number of arithmetic operations performed by digital down converter (DDC) based on cascade integral-comb (CIC) and CMFB based on different number of channels is given. A software description of the CMFB algorithm is presented in the form of block diagrams describing the capabilities of the Opencl and clfft software libraries for implementing the DCT-IV filter Bank and modulation algorithm on a GPU. The obtained algorithm was tested on an ARM family processor and a mali GPU with a table with sample rate for different number of channels with the maximum load of the graphics processor (GPU) and the minimum load of the Central processor (CPU).

Parallel implementation of empirical mode decomposition for nearly bandlimited signals via polyphase representation

Nearly bandlimited signals play an important role in the biomedical signal processing community. The common method to analyze these signals is via the empirical mode decomposition approach which decomposes the non-stationary signals into the sums of the intrinsic mode functions. However, this method is computational demanding. A natural idea to reduce the computational cost is via the block processing. However, the severe boundary effect would happen due to the discontinuities between two consecutive blocks. In order to solve this problem, this paper proposes to realize the parallel implementation via polyphase representation. That is, the empirical mode decomposition is implemented on each polyphase component of the original signal. Then each sub-signals are combined after upsampling. The simulation results show that our proposed method achieves the approximate intrinsic mode functions both qualitatively and quantitatively very close to the true intrinsic mode functions. Besides, compared with the conventional block processing method which significantly suffered from the boundary effect problem, our proposed method does not have this issue.

Computer cryptography through performing chaotic modulation on intrinsic mode functions with non‐dyadic number of encrypted signals

This study proposes a computer cryptographic system through performing the chaotic modulation on the intrinsic mode functions with a non-dyadic number of the encrypted signals. First, the empirical mode decomposition is applied to an input signal to generate a set of intrinsic mode functions. Then, these intrinsic mode functions are categorised into two groups of signals. Next, a type 1 polyphase is employed to represent each group of signals. These polyphase components are combined to generate a non-dyadic number of polyphase components. Second, the chaotic modulation is applied to these combined polyphase components for performing the encryption in the time frequency domain. To reconstruct the original signal, first, the chaotic demodulation is applied to the encrypt components to reconstruct the combined polyphase components. Then, the original groups of intrinsic mode functions are reconstructed through the type 2 polyphase representation and the original signal is reconstructed. Compared with the chaotic filter bank system, the proposed approach enjoys the nonlinear and adaptive property of the empirical mode decomposition. Therefore, a better security performance can be achieved particularly for the non-stationary signals. Compared with the conventional chaotic modulation approach, the proposed system allows performing the cryptography in the time frequency domain.

Extending Classical Multirate Signal Processing Theory to Graphs—Part II: M-Channel Filter Banks

This paper builds upon the basic theory of multirate systems for graph signals developed in the companion paper (Part I) and studies $M$ -channel polynomial filter banks on graphs. The behavior of such graph filter banks differs from that of classical filter banks in many ways, the precise details depending on the eigenstructure of the adjacency matrix $\boldsymbol {A}$ . It is shown that graph filter banks represent, (linear and) periodically, shift-variant systems only when $\boldsymbol {A}$ satisfies the noble identity conditions developed in Part I. It is then shown that perfect reconstruction graph filter banks can always be developed when $\boldsymbol {A}$ satisfies the eigenvector structure satisfied by $M$ -block cyclic graphs and has distinct eigenvalues (further restrictions on eigenvalues being unnecessary for this). If $\boldsymbol {A}$ is actually $M$ -block cyclic then these PR filter banks indeed become practical, i.e., arbitrary filter polynomial orders are possible, and there are robustness advantages. In this case, the PR condition is identical to PR in classical filter banks—any classical PR example can be converted to a graph PR filter bank on an $M$ -block cyclic graph. It is shown that for $M$ -block cyclic graphs with all eigenvalues on the unit circle, the frequency responses of filters have meaningful correspondence with classical filter banks. Polyphase representations are then developed for graph filter banks and utilized to develop alternate conditions for alias cancellation and perfect reconstruction, again for graphs with specific eigenstructures. It is then shown that the eigenvector condition on the graph can be relaxed by using similarity transforms.

Minimum-Energy Wavelet Frames on Local Fields

To overcome the computational complexity, numerical instability and the complication of change of bases problem, we introduce the concept of minimum-energy wavelet frame on local fields of positive characteristic and present its equivalent characterizations in terms of their wavelet masks. Besides, in view of polyphase representation of the wavelet masks, we give a necessary and sufficient condition for the existence of minimum-energy wavelet frames on local fields. At last, we present the decomposition and reconstruction formulae of minimum-energy wavelet frame for \(L^2(K)\).

Techniques for Constructing Biorthogonal Bipartite Graph Filter Banks

The processing of data defined on irregular discrete domains, i.e., graph signals, is becoming an emerging area with great application potential. Using spectral graph theory, Narang and Ortega (2013) laid the framework for two channel filter banks with critical sampling for bipartite graph signals. The bipartite graph filter bank can be extended to any arbitrary graph using the notion of separable filtering. The design of the biorthogonal filter banks by Narang and Ortega (2013) is based on the factorization of a maximally flat polynomial. The factorization technique does not allow much control of the spectral response of the graph filters, resulting in response asymmetry. In this paper, we present a generic framework for constructing biorthogonal graph filter banks that does not require factorization. We introduce the notion of polyphase representation and ladder structures for graph filter banks. We show that filters having virtual spectral symmetry and almost energy preservation can be constructed without any sophisticated optimization. Fine control of the spectral response can also be achieved with ease.

Scaling Laplacian Pyramids

Laplacian pyramid--based Laurent polynomial (LP$^2$) matrices are generated by Laurent polynomial column vectors and have long been studied in connection with Laplacian pyramidal algorithms in signal processing. In this paper, we investigate when such matrices are scalable, that is, when right multiplication by Laurent polynomial diagonal matrices results in paraunitary matrices. The notion of scalability has recently been introduced in the context of finite frame theory and can be considered as a preconditioning method for frames. This paper significantly extends the current research on scalable frames to the setting of polyphase representations of filter banks. Furthermore, as applications of our main results we propose new construction methods for tight wavelet filter banks and tight wavelet frames.

Multi-channel filter banks associated with linear canonical transform

The linear canonical transform (LCT) has been shown to be a powerful tool for optics and signal processing. This paper investigates multi-channel filter banks associated with the LCT. First, the perfect reconstruction (PR) conditions are analyzed and design method of PR filter banks for the LCT is proposed, which demonstrates that the LCT based filter banks can inherit conventional design methods of filter banks in the Fourier domain. Then polyphase decompositions in the LCT domain are defined and polyphase realization of the LCT based filter banks is derived in terms of polyphase matrices. Furthermore, multi-channel cyclic filter banks associated with the LCT are proposed by defining circular convolution in the LCT domain. The PR design method and polyphase representation of cyclic filter banks for the LCT are derived similarly. Finally, simulations validate the proposed design methods of the LCT based filter banks and also demonstrate potential application of the LCT based cyclic filter banks in image subband decomposition.

Decompositions of trigonometric polynomials with applications to multivariate subdivision schemes

We study multivariate trigonometric polynomials satisfying the sum-rule conditions of a certain order. Based on the polyphase representation of these polynomials relative to a general dilation matrix, we develop a simple constructive method for a special type of decomposition of such polynomials. These decompositions are of interest in the analysis of convergence and smoothness of multivariate subdivision schemes associated with general dilation matrices. The approach presented in this paper leads directly to constructive algorithms, and is an alternative to the analysis of multivariate subdivision schemes in terms of the joint spectral radius of certain operators. Our convergence results apply to arbitrary dilation matrices, while the smoothness results are limited to two classes of dilation matrices.

A wavelet optimization approach for ECG signal classification

Wavelets have proved particularly effective for extracting discriminative features in ECG signal classification. In this paper, we show that wavelet performances in terms of classification accuracy can be pushed further by customizing them for the considered classification task. A novel approach for generating the wavelet that best represents the ECG beats in terms of discrimination capability is proposed. It makes use of the polyphase representation of the wavelet filter bank and formulates the design problem within a particle swarm optimization (PSO) framework. Experimental results conducted on the benchmark MIT/BIH arrhythmia database with the state-of-the-art support vector machine (SVM) classifier confirm the superiority in terms of classification accuracy and stability of the proposed method over standard wavelets (i.e., Daubechies and Symlet wavelets).

Two Dimensional Non-separable Adaptive Directional Lifting Structure of Discrete Wavelet Transform

In this paper, we propose a two dimensional (2D) non-separable adaptive directional lifting (ADL) structure for discrete wavelet transform (DWT) and its image coding application. Although a 2D non-separable lifting structure of 9/7 DWT has been proposed by interchanging some lifting, we generalize a polyphase representation of 2D non-separable lifting structure of DWT. Furthermore, by introducing the adaptive directional filteringingto the generalized structure, the 2D non-separable ADL structure is realized and applied into image coding. Our proposed method is simpler than the 1D ADL, and can select the different transforming direction with 1D ADL. Through the simulations, the proposed method is shown to be efficient for the lossy and lossless image coding performance.

Demosaicking by Alternating Projections: Theory and Fast One-Step Implementation

Color image demosaicking is a key process in the digital imaging pipeline. In this paper, we study a well-known and influential demosaicking algorithm based upon alternating projections (AP), proposed by Gunturk, Altunbasak and Mersereau in 2002. Since its publication, the AP algorithm has been widely cited and compared against in a series of more recent papers in the demosaicking literature. Despite good performances, a limitation of the AP algorithm is its high computational complexity. We provide three main contributions in this paper. First, we present a rigorous analysis of the convergence property of the AP demosaicking algorithm, showing that it is a contraction mapping, with a unique fixed point. Second, we show that this fixed point is in fact the solution to a constrained quadratic minimization problem, thus, establishing the optimality of the AP algorithm. Finally, using the tool of polyphase representation, we show how to obtain the results of the AP algorithm in a single step, implemented as linear filtering in the polyphase domain. Replacing the original iterative procedure by the proposed one-step solution leads to substantial computational savings, by about an order of magnitude in our experiments.

Polyphase decompositions and shift-invariant discrete wavelet transforms in the frequency domain

Given a signal and its Fourier transform, we derive formulas for its polyphase decomposition in the frequency domain and for the reconstruction from the polyphase representation back to the Fourier representation. We present two frequency-domain implementations of the shift-invariant periodic discrete wavelet transform (SI-DWT) and its inverse: one that is based on frequency-domain polyphase decomposition and a more efficient ‘direct’ implementation, based on a reorganisation of the à trous algorithm. We analyse the computational complexities of both algorithms, and compare them to existing time-domain and frequency domain implementations of the SI-DWT. We experimentally demonstrate the reduction in computation time achieved by the direct frequency domain implementation of the SI-DWT for wavelet filters with non-compact support.

Swarm Intelligence Approach to Wavelet Design for Hyperspectral Image Classification

Wavelets are known to be a valuable tool for analyzing hyperspectral images. In this letter, we propose to further improve their performance by means of a novel classification-driven design scheme that aims at deriving a wavelet that best represents in terms of between-class discrimination capability the spectral signatures conveyed by a given hyperspectral image. This is achieved by adopting a polyphase representation of the wavelet filter bank and formulating the wavelet optimization problem within a particle-swarm-optimization (PSO) framework. Experimental results show that the proposed wavelet design method outperforms the popular Daubechies wavelets whatever the classifier type adopted in the classification process.

TIGHT FRAME CHARACTERIZATION OF MULTIWAVELET VECTOR FUNCTIONS IN TERMS OF THE POLYPHASE MATRIX

The extension principles play an important role in characterizing and constructing of wavelet frames. The common extension principles, the unitary extension principle (UEP) or the oblique extension principle (OEP), are based on the unitarity of the modulation matrix. In this paper, we state the UEP and OEP for refinable function vectors in the polyphase representation. Finally, we apply our results to directional wavelets on triangles which we have constructed in a previous work. We will show that the wavelet system generates a tight frame for L2(ℝ2).

An improved partial Haar dual adaptive filter for rapid identification of a sparse echo channel

Recently, a coupled echo canceller was proposed that uses two short adaptive filters for sparse echo cancellation. The first filter operates in the partial Haar domain and is used to locate the channel's dispersive region; the second filter is then centered around this location to cancel the echo in the time domain. In this paper, we propose feasible solutions to improve the performance of this partial Haar dual adaptive filter (PHDAF) in practical applications. These include: (1) alleviating the dependence of the PHDAFs performance on the echo-path impulse response's bulk delay; (2) improving the tracking performance of the PHDAF in response to abrupt changes in the echo path; and (3) extending the original PHDAF structure to support the cancellation of multiple echoes. The proposed algorithmic solutions exploit the Haar transform's polyphase representation and make use of a novel peak tendency estimator (PTE) based on Dezert–Smarandache theory (DSmT). The improved PHDAF is evaluated in terms of its mean-square error (MSE) curves and its mean time to properly locate a dispersive region for different SNRs. Results show that enhanced performance can be obtained using the proposed solutions at a minimal increase in computational cost.

Alias-Component Matrices of Nonuniform Transmultiplexers

Nonuniform transmultiplexers can be used for interconversion of signals with different sampling rates between the time-division multiplexing format and the frequency-division multiplexing format. Here, we review the polyphase representation and discuss the ℋ∞ model-matching design of finite impulse response nonuniform transmultiplexers by semidefinite programming. Then, we study the alias-component matrices. It will be shown that the alias-component matrices of nonuniform transmultiplexers are generalizations of those of linear periodically time-varying systems. In particular, it is shown that the output at any given frequency is only dependent on the input at a finite set of frequencies.