Perfect Reconstruction Filter Banks Research Articles

Designing optimal prototype filters for maximally decimated Cosine Modulated filter banks with rapid convergence

An analytic design of a prototype filter for M-channel maximally decimated cosine-modulated Near Perfect Reconstruction (NPR) filter banks is proposed in this work. The prototype filter is created using the least-square (CLS) method with weighted constraints, which is one-dimensional and requires single-parameter optimization. Compared to existing approaches, this suggested method achieves rapid convergence by analytically determining the optimal step size, ensuring the 3 dB cutoff frequency at π/2 M. The simulation results for design examples outperform the techniques in the available literature in terms of amplitude and aliasing distortion, reaching distortion around 2.4489 × 10−4 and 3.4907 × 10−9, respectively. This optimization algorithm's usefulness is further demonstrated with the sub-band coding of ECG signals. Implementing optimal prototype filters has tangible real-world effects, especially in critical sectors like healthcare and communications, improving diagnostics accuracy, data transmission efficiency, and overall performance.

A fast-convergence, tree-structured, fully-parallel digital background timing mismatch calibration method for four-channel TIADC based on perfect reconstruction filter bank

This paper presents a fully-digital background timing mismatch calibration method for four-channel time-interleaved analog-to-digital converter (TIADC). Based on three improved dual-channel perfect reconstruction filter banks (PRFBs), the proposed method for correcting timing mismatch adopts the tree-structured four-channel PRFB. Unlike the direct four-channel PRFB, the improved structure utilizes fewer filter taps to achieve higher filter performance, making it suitable for real-time background calibration. Furthermore, timing mismatch is estimated by performing correlation calculations on the outputs of sub-channels with the variable step-size least mean squares (LMS) algorithm, being approximately eight times faster than the fixed step-size LMS algorithm. Our method verified by a 12-bit 3 GS/s TIADC test platform indicates that the spurs caused by the timing mismatch for single-tone and multi-tone signals can be effectively suppressed, close to 35 and 40 dB, respectively. The evaluation results on the Xilinx Vertix-7 VC707 platform indicate that the proposed method reduces FPGA hardware resources consumption by nearly 40% compared to the foreground pre-calibration method.

An improved wide-range, low-complexity, fully-parallel digital background timing mismatch calibration method for dual-channel TIADC based on perfect reconstruction filter bank

This paper presents a fully-digital background calibration method for calibrating timing mismatch in dual-channel time-interleaved analog-to-digital converter (TIADC). The proposed method estimates timing mismatch by performing correlation calculations on the outputs of sub-channels and corrects this mismatch using the improved perfect reconstruction filter bank (PRFB). The PRFB, which is improved by employing a polynomial fitting method, offers superior performance with fewer filter taps and extends the timing mismatch correction capability to almost the entire sampling clock cycle. The improved PRFB uses only one twentieth of the taps used in the original PRFB, and its dual-channel FIR filters have anti-symmetric taps. To enhance the real-time calibration capability of the system, this paper gives a fully-parallel hardware structure for calibrating offset, gain, bandwidth, and timing mismatches. The effectiveness of the proposed hardware structure is verified through a 10-bit 3GS/s dual-channel TIADC test platform. Simulation and measurement results show that the proposed calibration method significantly improves the signal-to-noise-and-distortion (SNDR) and spurious-free dynamic range (SFDR) of both single-tone and multi-tone input signals, which makes the proposed method a viable solution for practical TIADC applications.

A novel PRFB decomposition for non-stationary time-series and image analysis

This work presents a novel perfect reconstruction filterbank decomposition (PRFBD) method for nonlinear and non-stationary time-series and image data representation and analysis. The Fourier decomposition method (FDM), an adaptive approach based on Fourier representation (FR), is shown to be a special case of the proposed PRFBD. In addition, adaptive Fourier–Gauss decomposition (FGD) based on FR and Gaussian filters, and adaptive Fourier–Butterworth decomposition (FBD) based on Butterworth filters are developed as the other special cases of the proposed PRFBD method. The proposed theory of PRFBD can decompose any signal (time-series, image, or other data) into a set of desired number of Fourier intrinsic band functions (FIBFs) that follow the amplitude-modulation and frequency-modulation (AM-FM) representations. A generic filterbank representation, where perfect reconstruction can be ensured for any given set of lowpass or highpass filters, is also presented. We performed an extensive analysis on both simulated and real-life data (COVID-19 pandemic, Earthquake and Gravitational waves) to demonstrate the efficacy of the proposed method. The resolution results in the time-frequency representation demonstrate that the proposed method is more promising than the state-of-the-art approaches.

Measures for the Degree of Time Variance: Application to Filter Banks

Quantifying a system’s time-invariance deviation may reveal how it represents and processes the signals. The main objective of this paper is to quantify the departure of a time-varying system from being time-invariant. In particular, the paper quantifies the time variance of filter banks using kernel representation. We define the ℓ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> measure for the degree of time variance (MDTV <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ℓ₂</sup> ) based on the discrete-time domain form of the kernel representation and the ℓ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sup> measure (MDTV <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ℓ_∞</sup> ) based on the matrix representation of the LPTV kernel. We show that off-diagonal matrix elements can be used to deduce the time-varying behavior of an LPTV system, proving that time-invariant and time-varying components can be separated. The matrix form of the LTI system, which is closest to an LPTV system, has been derived. This research offers a new kernel-based definition for time variance for any time-varying system and provides explicit expressions for MDTV for three distinct scenarios, namely, a single channel (SC), a uniform filter bank (UFB), and a non-uniform filter bank (NUFB). For each scenario, we determine the LPTV kernel and impulse response of the nearest LTI system to plot the MDTV versus decimation factor. Also, MDTV generated by an SC and a UFB with identical low pass filters in each channel is identical. We also observe that two-channel perfect reconstruction (PR) filter banks result in zero MDTV. Finally, we have compared the proposed approach with earlier work and demonstrated its better efficiency in terms of computational complexity.

Frame multiresolution analysis of continuous piecewise linear functions

The Franklin wavelet is constructed using the multiresolution analysis (MRA) generated from a scaling function [Formula: see text] that is continuous on [Formula: see text], linear on [Formula: see text] and [Formula: see text] for every [Formula: see text]. For [Formula: see text] and [Formula: see text], it is shown that if a function [Formula: see text] is continuous on [Formula: see text], linear on [Formula: see text] and [Formula: see text], for [Formula: see text], and generates MRA with dilation factor [Formula: see text], then [Formula: see text]. Conversely, for [Formula: see text], it is shown that there exists a [Formula: see text], as satisfying the above conditions, that generates MRA with dilation factor [Formula: see text]. The frame MRA (FMRA) is useful in signal processing, since the perfect reconstruction filter banks associated with FMRA can be narrow-band. So it is natural to ask, whether the above results can be extended for the case of FMRA. In this paper, for [Formula: see text], we prove that if [Formula: see text] generates FMRA with dilation factor [Formula: see text], then [Formula: see text]. For [Formula: see text], we prove similar results when [Formula: see text]. In addition, for [Formula: see text] we prove that there exists a function [Formula: see text] as satisfying the above conditions, that generates FMRA. Also, we construct tight wavelet frame and wavelet frame for such scaling functions.

On the Design of Two-Dimensional Quincunx Filterbanks with Directional Vanishing Moment Based on Eigenfilter Approach

This paper presents a two-dimensional (2-D) eigenfilter technique to design 2-D perfect reconstruction (PR) filterbanks with directional vanishing moment (DVM). In this paper, we first introduce the DVM constraint for 2-D directional filterbank design. The proposed DVM constraint is imposed in 2-D eigenfilter formulation to obtain maximally flat 2-D low-pass analysis filter. Then, we use this designed analysis filter in 2-D time-domain formulation of PR condition to obtain complementary low-pass synthesis filters. The PR and DVM constraints are imposed to obtain 2-D low-pass synthesis filter. It is shown that the proposed 2-D filters satisfy the PR criteria with prescribed DVM. The performance of 2-D PR filters is evaluated in image denoising application. The performance of designed filterbank is compared with well-known existing methods in terms of peak signal-to-noise ratio to validate the results.

Design of Two Channel Biorthogonal Filterbanks using Euler Frobenius Polynomial

This paper presents a technique to design a new class of biorthogonal perfect reconstruction (PR) filterbanks. In this technique, we use Euler Frobenius polynomial (EFP) to design maximally flat Euler Frobenius halfband polynomial (EFHP). This is obtained by imposing vanishing moments (VMs) and PR constraints on EFP. The resulted EFHP is used in three and four step lifting structure to determine analysis low-pass and high-pass filters. The lifting halfband kernels are designed using EFHP. It has been ensured that the proposed filters satisfy the linear phase and PR property. Also, the proposed filters have frame bound ratio very close to unity. Several design examples are presented and the properties of proposed filters are compared with existing filters. It has been ensured that the proposed filters give more regularity as compared to existing filterbanks.

ZapLine: A simple and effective method to remove power line artifacts

Power line artifacts are the bane of animal and human electrophysiology. A number of methods are available to help attenuate or eliminate them, but each has its own set of drawbacks. In this brief note I present a simple method that combines the advantages of spectral and spatial filtering, while minimizing their downsides. A perfect-reconstruction filterbank is used to split the data into two parts, one noise-free and the other contaminated by line artifact. The artifact-contaminated stream is processed by a spatial filter to project out line components, and added to the noise-free part to obtain clean data. This method is applicable to multichannel data such as electroencephalography (EEG), magnetoencephalography (MEG), or multichannel local field potentials (LFP). I briefly review past methods, pointing out their drawbacks, describe the new method, and evaluate the outcome using synthetic and real data.

Fast-convolution multicarrier based frequency division multiple access

Fast-convolution multicarrier (FCMC), the asynchronous waveform with ultra-low sidelobe, has appeared to be a promising waveform technique for future wireless communications. In this paper, we investigate the filter bank optimization as well as receiver design including low-complexity channel estimator and equalizer for FCMC based frequency division multiple access (FDMA). Starting from the conventional multi-carrier signals, we first derive a vectorized signal model as a framework to systematically design the FCMC transceiver. For nearly perfect reconstruction (NPR) filter banks design, an optimization criterion which consists of minimizing received signal segments mean square error (MSE) is proposed. From the fact that Toeplitz matrices can be asymptotically diagonalized by discrete Fourier transform (DFT) matrix, the channel equalizer can be simplified to one-tap frequency domain equalizer when DFT size is large enough. The minimum mean square error (MMSE) criterion is then applied to calculate the coefficients of one-tap frequency domain equalizer. In practice, due to the channel fading, the channel estimation has to be performed to obtain the channel state information (CSI) which is required by the channel equalization. To this end, we propose a combined cyclic prefix (CP) and cyclic suffix (CS) pilot structure which facilitates to estimate the frequency domain CSI directly in the receiver end. The proposed FCMC based FDMA features low-complexity receiver, adjustable users bandwidth and low peak-to-average power ratio. Simulation results confirm the performance of the proposed scheme.

Periodically nonuniform sampling and averaging of signals in multiresolution subspaces associated with the fractional wavelet transform

Periodically nonuniform sampling is one of the most fundamental theories in the signal processing community. Several sampling theorems, which aid in the reconstruction of periodically nonuniform sampled signals in the fractional Fourier transform (FrFT) and the linear canonical transform (LCT) domain, have been proposed in the existing literature. However, we are given only finite number of samples in practice. Therefore, these theorems may not be suitable for the perfect reconstruction of a signal because of the slow decay of synthesizing functions. Focusing on this issue, we first investigate the reconstruction of a periodically nonuniform sampled signal in multiresolution subspaces associated with the recently-proposed fractional wavelet transform (FrWT). By means of perfect reconstruction (PR) filter banks associated with the FrFT, we derive the expressions of the synthesizing functions which can be guaranteed to be compactly supported, under a mild restriction that the number of channels is constrained by the support of scaling function. In order to relax this restriction, we further propose a periodically nonuniform averaging and reconstruction theorem of signals in multiresolution subspaces associated with the FrWT. Simulation results are presented to validate our algorithms.

Periodically Nonuniform Sampling and Reconstruction of Signals in Function Spaces Associated With the Linear Canonical Transform

In practice, we are given only a finite number of samples. Sampling theorems with slow-decayed synthesizing functions, therefore, may not be suitable for perfect reconstruction of a signal. Focusing on this issue, we investigate the reconstruction of a periodically nonuniform sampled signal in function spaces associated with the linear canonical transform (LCT). We propose a sampling theorem in which compactly supported synthesizing functions can be guaranteed by the existence of finite impulse response perfect reconstruction filter banks associated with the LCT. We also present the simulation results to validate our algorithm.

Wavelet operator for multiscale modeling of a nuclear reactor

Abstract This article introduces a methodology of designing a wavelet operator suitable for multiscale modeling. The operator matrix transforms states of a multivariable system onto projection space. In addition, it imposes a specific structure on the system matrix in a multiscale environment. To be specific, the article deals with a diagonalizing transform that is useful for decoupled control of a system. It establishes that there exists a definite relationship between the model in the measurement space and that in the projection space. Methodology for deriving the multirate perfect reconstruction filter bank, associated with the wavelet operator, is presented. The efficacy of the proposed technique is demonstrated by modeling the point kinetics nuclear reactor. The outcome of the multiscale modeling approach is compared with that in the single-scale approach to bring out the advantage of the proposed method.

Learning Filter Bank Sparsifying Transforms

Data is said to follow the transform (or analysis) sparsity model if it becomes sparse when acted on by a linear operator called a sparsifying transform. Several algorithms have been designed to learn such a transform directly from data, and data-adaptive sparsifying transforms have demonstrated excellent performance in signal restoration tasks. Sparsifying transforms are typically learned using small sub-regions of data called patches, but these algorithms often ignore redundant information shared between neighboring patches. We show that many existing transform and analysis sparse representations can be viewed as filter banks, thus linking the local properties of patch-based model to the global properties of a convolutional model. We propose a new transform learning framework where the sparsifying transform is an undecimated perfect reconstruction filter bank. Unlike previous transform learning algorithms, the filter length can be chosen independently of the number of filter bank channels. Numerical results indicate filter bank sparsifying transforms outperform existing patch-based transform learning for image denoising while benefiting from additional flexibility in the design process.

M-Channel Perfect Recovery of Coarsened Graphs and Graph Signals With Spectral Invariance and Topological Preservation

In this paper, an M -channel perfect reconstruction filter bank based on a coarsening algorithm is proposed. Compared with most of the designs of graph filter banks that do not consider the graph reconstruction, our proposed design can provide the perfect graph reconstruction as well as perfect graph signal reconstruction. In the analysis part, the proposed filter bank provides a coarse version of the input graph as well as coarsened graph signal spectral invariant to the input signal. In the synthesis part, the filter bank perfectly reconstructs the input graph as well as input signal from their coarse version. The spirit of the proposed design is to partition the spectral information of the input graph and input graph signal into every channel. The partition method is adjustable and can be nonuniform. Two intuitive schemes named sort-by-eigenvalue and sort-by-intensity for uniform partition are introduced. In the proposed design, the coarsening operators are obtained through an existing coarsening algorithm, while the recovery operators are defined by taking the conjugate transpose of the coarsening operators. Besides, we address the relation between the proposed design and a framework of sampling graph signals based on the discrete sampling theory. It is shown that the coarsening operator in the proposed design is actually a special case of the sampling operator in the framework based on the discrete sampling theory. Experimental results are presented to demonstrate the effectiveness of the proposed design of filter banks.

Bipartite Graph Filter Banks: Polyphase Analysis and Generalization

The work by Narang and Ortega [“Perfect reconstruction two-channel wavelet filter banks for graph structured data,” IEEE Trans. Signal Process. , vol. 60, no. 6, pp. 2786–2799, Jun. 2012], [“Compact support biorthogonal wavelet filterbanks for arbitrary undirected graphs,” IEEE Trans. Signal Process. , vol. 61, no. 19, pp. 4673–4685, Oct. 2013] laid the foundations for the two-channel critically sampled perfect reconstruction filter bank for signals defined on undirected graphs. This basic filter bank is applicable only to bipartite graphs but using the notion of separable filtering, the basic filter bank can be applied to any arbitrary undirected graphs. In this paper, several new theoretical results are presented. In particular, the proposed polyphase analysis yields filtering structures in the downsampled domain that are equivalent to those before downsampling and, thus, can be exploited for efficient implementation. These theoretical results also provide new insights that can be exploited in the design of these systems. These insights allow us to generalize these filter banks to directed graphs and to using a variety of graph base matrices, while also providing a link to the $\text{DSP}_G$ framework of Sandryhaila and Moura [“Discrete signal processing on graphs,” IEEE Trans. Signal Process. , vol. 61, no. 7, pp. 1644–1636, Apr. 2013], [“Discrete signal processing on graphs: Frequency analysis,” IEEE Trans. Signal Process. , vol. 62, no. 12, pp. 3042–3054, Jun. 2014]. Experiments show evidence that better nonlinear approximation and denoising results may be obtained by a better selection of these base matrices.

A novel approach for time–frequency localization of scaling functions and design of three-band biorthogonal linear phase wavelet filter banks

Design of time–frequency localized filters and functions is a classical subject in the field of signal processing. Gabor's uncertainty principle states that a function cannot be localized in time and frequency domain simultaneously and there exists a nonzero lower bound of 0.25 on the product of its time variance and frequency variance called time–frequency product (TFP). Using arithmetic mean (AM)– geometric mean (GM) inequality, product of variances and sum of variances can be related and it can be shown that sum of variances has lower bound of one. In this paper, we compute the frequency variance of the filter from its discrete Fourier transform (DFT) and propose an equivalent summation based discrete-time uncertainty principle which has the lower bound of one. We evaluate the performance of the proposed discrete-time time–frequency uncertainty measure in multiresolution setting and show that the proposed DFT based concentration measure generate sequences which are even more localized in time and frequency domain than that obtained from the Slepian, Ishii and Furukawa's concentration measures. The proposed design approach provides the flexibility in which the TFP can be made arbitrarily close to the lowest possible lower bound of 0.25 by increasing the length of the filter. In the other proposed approach, the sum of the time variance and frequency variance is used to formulate a positive definite matrix to measure the time–frequency joint localization of a bandlimited function from its samples. We design the time–frequency localized bandlimited low pass scaling and band pass wavelet functions using the eigenvectors of the formulated positive definite matrix. The samples of the time–frequency localized bandlimited function are obtained from the eigenvector of the positive definite matrix corresponding to its minimum eigenvalue. The TFP of the designed bandlimited scaling and wavelet functions are close to the lowest possible lower bound of 0.25 and 2.25 respectively. We propose a design method for time–frequency localized three-band biorthogonal linear phase (BOLP) wavelet perfect reconstruction filter bank (PRFB) wherein the free parameters can be optimized for time–frequency localization of the synthesis basis functions for the specified frequency variance of the analysis scaling function. The performance of the designed filter bank is evaluated in classification of seizure and seizure-free electroencephalogram (EEG) signals. It is found that the proposed filter bank outperforms other existing methods for the classification of seizure and seizure-free EEG signals.

Euclidean Algorithm for Extension of Symmetric Laurent Polynomial Matrix and Its Application in Construction of Multiband Symmetric Perfect Reconstruction Filter Bank

For a given pair of s-dimensional real Laurent polynomials ( a → ( z ) , b → ( z ) ) , which has a certain type of symmetry and satisfies the dual condition b → ( z ) T a → ( z ) = 1 , an s × s Laurent polynomial matrix A ( z ) (together with its inverse A - 1 ( z ) ) is called a symmetric Laurent polynomial matrix extension of the dual pair ( a → ( z ) , b → ( z ) ) if A ( z ) has similar symmetry, the inverse A - 1 ( Z ) also is a Laurent polynomial matrix, the first column of A ( z ) is a → ( z ) and the first row of A - 1 ( z ) is ( b → ( z ) ) T . In this paper, we introduce the Euclidean symmetric division and the symmetric elementary matrices in the Laurent polynomial ring and reveal their relation. Based on the Euclidean symmetric division algorithm in the Laurent polynomial ring, we develop a novel and effective algorithm for symmetric Laurent polynomial matrix extension. We also apply the algorithm in the construction of multi-band symmetric perfect reconstruction filter banks.

Multidimensional Nonseparable Oversampled Lapped Transforms: Theory and Design

This paper extends the theory of the one-dimensional oversampled linear-phase perfect reconstruction filter banks (OLPPRFBs) developed by Gan et al. to multidimensional (MD) cases and proposes MD nonseparable oversampled lapped transforms (NSOLTs). NSOLTs allow us to achieve an overcomplete analysis-synthesis system with nonseparable, symmetric, real-valued, overlapping, and compact-supported filters. The proposed systems are based on lattice structures and the redundancy is flexibly controlled by the number of channels and downsampling ratio. The proposed structure is shown to be capable of constructing Parseval tight frames in any number of dimensions. The number of design parameters are examined under the Parseval tight frame constraint. In order to design NSOLTs specified for sparse approximation of image or volume data, an example-based design procedure is introduced. The effectiveness of this method is verified by examining design samples and evaluating their sparse approximation performance using the iterative hard thresholding algorithm for a natural image and MRI volume data patch.

A Multi Directional Perfect Reconstruction Filter Bank Designed with 2-D Eigenfilter Approach: Application to Ultrasound Speckle Reduction.

B-Mode ultrasound images are degraded by inherent noise called Speckle, which creates a considerable impact on image quality. This noise reduces the accuracy of image analysis and interpretation. Therefore, reduction of speckle noise is an essential task which improves the accuracy of the clinical diagnostics. In this paper, a Multi-directional perfect-reconstruction (PR) filter bank is proposed based on 2-D eigenfilter approach. The proposed method used for the design of two-dimensional (2-D) two-channel linear-phase FIR perfect-reconstruction filter bank. In this method, the fan shaped, diamond shaped and checkerboard shaped filters are designed. The quadratic measure of the error function between the passband and stopband of the filter has been used an objective function. First, the low-pass analysis filter is designed and then the PR condition has been expressed as a set of linear constraints on the corresponding synthesis low-pass filter. Subsequently, the corresponding synthesis filter is designed using the eigenfilter design method with linear constraints. The newly designed 2-D filters are used in translation invariant pyramidal directional filter bank (TIPDFB) for reduction of speckle noise in ultrasound images. The proposed 2-D filters give better symmetry, regularity and frequency selectivity of the filters in comparison to existing design methods. The proposed method is validated on synthetic and real ultrasound data which ensures improvement in the quality of ultrasound images and efficiently suppresses the speckle noise compared to existing methods.