Quaternionic one-dimensional linear canonical transform

The linear canonical transform (LCT), which is a generalized form of the classical Fourier transform (FT), the fractional Fourier transform (FRFT), and other transforms, has been shown to be a powerful tool in optics and signal processing. Many results of this transform are already known, including its convolution theorem. However, the formulation of the convolution theorem for the LCT has been developed differently and is still not having a widely accepted closed-form expression. In this paper, we first propose a generalized convolution theorem for the LCT and then derive a corresponding product theorem associated with the LCT. The ordinary convolution theorem for the FT, the fractional convolution theorem for the FRFT, and some existing convolution theorems for the LCT are shown to be special cases of the derived results. Moreover, some applications of the derived results are presented.

A counterexample is presented to show that the solution to the stochastic realization problem given in the above-titled letter (ibid., vol.75, no.9, p.1333-4, Sept. 1987) does not apply, in general, to the infinite impulse response systems. The author expands on the comments.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A general framework for sampling and reconstruction in function spaces associated with fractional Fourier transform

EDITORIAL article Front. Physiol., 05 November 2012Sec. Computational Physiology and Medicine Volume 3 - 2012 | https://doi.org/10.3389/fphys.2012.00425

Diffraction and holography are fertile areas for application of signal theory and processing. Recent work on 3DTV displays has posed particularly challenging signal processing problems. Various procedures to compute Rayleigh-Sommerfeld, Fresnel and Fraunhofer diffraction exist in the literature. Diffraction between parallel planes and tilted planes can be efficiently computed. Discretization and quantization of diffraction fields yield interesting theoretical and practical results, and allow efficient schemes compared to commonly used Nyquist sampling. The literature on computer-generated holography provides a good resource for holographic 3DTV related issues. Fast algorithms to compute Fourier, Walsh-Hadamard, fractional Fourier, linear canonical, Fresnel, and wavelet transforms, as well as optimization-based techniques such as best orthogonal basis, matching pursuit, basis pursuit etc., are especially relevant signal processing techniques for wave propagation, diffraction, holography, and related problems. Atomic decompositions, multiresolution techniques, Gabor functions, and Wigner distributions are among the signal processing techniques which have or may be applied to problems in optics. Research aimed at solving such problems at the intersection of wave optics and signal processing promises not only to facilitate the development of 3DTV systems, but also to contribute to fundamental advances in optics and signal processing theory.

So far there has not been paid attention to frames that are balanced, i.e. those frames which sum is zero. In this paper we consider balanced frames, and in particular balanced unit norm tight frames, in finite dimensional Hilbert spaces. Here we discover various advantages of balanced unit norm tight frames in signal processing. They give an exact reconstruction in the presence of systematic errors in the transmitted coefficients, and are optimal when these coefficients are corrupted with noises that can have non-zero mean. Moreover, using balanced frames we can know that the transmitted coefficients were perturbed, and we also have an indication of the source of the error. We analyze several properties of these types of frames. We define an equivalence relation in the set of the dual frames of a balanced frame, and use it to show that we can obtain all the duals from the balanced ones. We study the problem of finding the nearest balanced frame to a given frame, characterizing completely its existence and giving its expression. We introduce and study a concept of complement for balanced frames. Finally, we present many examples and methods for constructing balanced unit norm tight frames.

The proposed paper deals with the signal processing using Invertible Rapid Transform (IRT). A software tool for signal and image processing using IRT is presented.

Reproducing kernel ilbert spaces (RKHSs) have been at the core of successful non-parametric tools in signal processing, statistics, and machine learning. Despite their success, the computational complexity of these models often hinders their use in practice. Indeed, fitting RKHS models typically relies on representer theorems to express the solution space as a combination of kernels evaluated at the training samples. Thus, the computational cost of evaluating these models is proportional to the number of training samples, which in many applications is prohibitively high. This issue is often addressed by sparsifying the coefficients of the kernel expansion, despite the fact that classical representer theorems no longer hold in the presence of sparsity penalties. In this work, we propose to directly tackle sparse learning over RKHSs by posing it as a functional problem. In other words, by formulating the RKHS model as a sparse, continuous combination of atoms from an overparametrized, continuous dictionary containing the value of the kernel evaluated at every point of the function domain. We show that despite the infinite dimensionality and non-convexity of the underlying optimization problem, these models can be fit exactly and efficiently using duality. We illustrate the performance of this technique in numerical experiments.

Accurate estimation of 3-D near-field source parameters is essential for applications such as radar systems, underwater acoustics, and biomedical imaging. However, traditional methods often fall short due to limitations in handling noise, interference, and environmental variability. This study aimed to develop a multidimensional factor model that integrates high-precision sensor arrays with advanced optimization techniques to address these challenges. The proposed model utilizes compressive sensing for sparse signal recovery and hybrid optimization frameworks, including deep learning-based reconstruction algorithms and metaheuristic methods, to enhance accuracy and computational efficiency.The research employed high-precision sensor arrays equipped with adaptive beam forming capabilities to capture multidimensional data. A comprehensive test bed simulated diverse environmental conditions, including fluctuating noise levels and multi-source interference, to evaluate the model's robustness. Comparative analysis with traditional methods and benchmark models was conducted using performance metrics such as Mean Square Error (MSE), processing time, and robustness against accuracy degradation.Results showed that the proposed model achieved a significantly lower MSE (as low as 0.011±0.001) compared to traditional methods and benchmark models. The average processing time per estimation was 0.35 seconds, significantly faster than the alternatives. Additionally, the model demonstrated superior robustness, maintaining accuracy degradation below 7% under challenging conditions. Statistical analyses, including ANOVA and t-tests, validated these findings, underscoring the model’s effectiveness and reliability.In conclusion, the proposed model addresses key limitations in traditional approaches, offering a robust and efficient solution for 3-D near-field parameter estimation. Practical recommendations include optimizing hardware-software integration, validating with real-world datasets, and exploring modular designs for scalability. These advancements position the model as a transformative tool in signal processing, with broad applicability across defense, healthcare, and environmental monitoring.

Given a signal, whether it is a discrete vector or a continuous function, one desires to write it in terms of simpler components. Typically, these components or “building blocks” form what is called a basis. A basis is an optimal set, containing the minimal number of elements needed to uniquely represent any signal in a given space. A frame can be thought of as a redundant basis, having more elements than needed. In fact, in any finite dimensional vector space every finite spanning set is a frame. The redundancy of a frame leads to a non-unique representation, however, this makes signal representation resilient to noise and robust to transmission losses. Frames are now standard tools in signal processing and are of great interest to mathematicians and engineers alike. This chapter presents a brief introduction to frames in finite dimensional spaces, and in particular discusses a highly desirable class of frames called equiangular tight frames. Possible research ideas suitable for an undergraduate curriculum are also discussed.

Tensor Diagonalization, A Useful Tool in Signal Processing

The virtual analog and digital communications laboratory LaViCAD has been designed at the UPC and it results a useful tool to verify different communication systems performance and also signal processing techniques, subjects given in courses typically included in the curriculum of any electrical engineering degree. LaViCAD tool has become a flexible, sustainable and on-line freely offered educational platform and it can be updated whenever new content is required. At pedagogical level, the use of a virtual laboratory facilitates the learning of certain matters, acting as a connection between the theoretical contents of a communications system course and their practical understanding and experimentation. Furthermore, LAVICAD provides resources for professors to organize different teaching activities in their courses. These activities can be used in different environments as for instance in a classroom given in the context of a full attendance course or in a homework activity at a distance learning course. The aim of this paper is two-fold: On one hand to describe the main technical features that make LAVICAD an original fully reusable and reprogrammable tool in signal processing and communication systems courses. On the other hand to show some preliminary academic results obtained with the use of LAVICAD verifying how it improves the level of success.

Sparse signal approximations have become a new tool in signal processing with wide ranging applications from source separation to signal acquisition. Recently, many algorithms for sparse signal reconstruction have been developed, however, greedy algorithms, such as Orthogonal Matching Pursuit algorithm, can have better performance than the other algorithms. Approximate orthogonal matching pursuit algorithms, such as gradient pursuit algorithm and conjugate gradient pursuit algorithm, can lead to fast approximations to Orthogonal Matching Pursuit. In this paper Newton pursuit algorithm for sparse signal reconstruction is proposed. The algorithm is a kind of directional pursuit algorithms with the same computational complexity compared to orthogonal matching pursuit algorithm, but it can have better performance than approximate orthogonal matching pursuit algorithms.

Geophysics has played a significant and efficient role in studying geological structures over the past decades as the goal of geophysical data acquisition is to investigate underground phenomena with the highest possible level of accuracy. The ground penetrating radar (GPR) method is used as a nondestructive method to reveal shallow structures by beaming electromagnetic waves through the Earth and recording the received reflections, albeit inevitably, along with random noise. Various types of noise affect GPR data, among the most important of which are random noise resulting from arbitrary motions of particles during data acquisition. Random noise which exists always and at all frequencies, along with coherent noise, reduces the quality of GPR data and must be reduced as much as possible. Over the recent years, discrete wavelet transform has proved to be an efficient tool in signal processing, especially in image and signal compressing and noise suppression. It also allows for obtaining an accurate understanding of the signal properties. In this study, we have used the autoregression in both wavelet and f-x domains to suppress random noise in synthetic and real GPR data. Finally, we compare noise suppression in the two domains. Our results reveal that noise suppression is conducted more efficiently in the wavelet domain due to decomposing the signal into separate subbands and exclusively applying the method parameters in autoregression modeling for each subband.

Principal Components Analysis (PCA) is an invaluable statistical tool in signal processing. In many cases, an on-line algorithm to adapt the PCA network to determine the principal projections in the input space is desired. Algorithms proposed until now use the traditional deflation or the inflation procedure to determine the intermediate components sequentially, after the convergence of the principal or minor component is achieved. In this paper, we propose a constrained linear network and a robust cost function to determine any number of principal components simultaneously. The topology exploits the fact that the eigenvector matrix sought is orthonormal. A gradient-based algorithm named SIPEX-G is also presented.