Signal Flow Graph Approach to Efficient DST I-IV Algorithms

In this paper, fast and efficient discrete sine transformation (DST) algorithms are presented based on the factorization of sparse, scaled orthogonal, rotation, rotation-reflection, and butterfly matrices. These algorithms are completely recursive and solely based on DST I-IV. The presented algorithms have low arithmetic cost compared to the known fast DST algorithms. Furthermore, the language of signal flow graph representation of digital structures is used to describe these efficient and recursive DST algorithms having $(n-1)$ points signal flow graph for DST-I and $n$ points signal flow graphs for DST II-IV

Signal Flow Graph Approach to Efficient DST I-IV Algorithms

In this paper, fast and efficient discrete sine transformation (DST) algorithms are presented based on the factorization of sparse, scaled orthogonal, rotation, rotation-reflection, and butterfly matrices. These algorithms are completely recursive and solely based on DST I-IV. The presented algorithms have low arithmetic cost compared to the known fast DST algorithms. Furthermore, the language of signal flow graph representation of digital structures is used to describe these efficient and recursive DST algorithms having (n�1) points signal flow graph for DST-I and n points signal flow graphs for DST II-IV

Reduced Multiplicative Complexity Discrete Cosine Transform (DCT) Circuitry

System and techniques for reduced multiplicative complex­ity discrete cosine transform (DCT) circuitry are described herein. An input data set can be received and, upon the input data set, a self-recursive DCT technique can be performed to produce a transformed data set. Here, the self-recursive DCT technique is based on a product of factors of a specified type of DCT technique. Recursive components of the technique are of the same DCT type as that of the DCT technique. The transformed data set can then be produced to a data con­sumer

A Fast Algorithm for the Inversion of Quasiseparable Vandermonde-like Matrices

The results on Vandermonde-like matrices were introduced as a generalization of polynomial Vandermonde matrices, and the displacement structure of these matrices was used to derive an inversion formula. In this paper we first present a fast Gaussian elimination algorithm for the polynomial Vandermonde-like matrices. Later we use the said algorithm to derive fast inversion algorithms for quasiseparable, semiseparable and well-free Vandermonde-like matrices having $\mathcal{O}(n^2)$ complexity. To do so we identify structures of displacement operators in terms of generators and the recurrence relations(2-term and 3-term) between the columns of the basis transformation matrices for quasiseparable, semiseparable and well-free polynomials. Finally we present an $\mathcal{O}(n^2)$ algorithm to compute the inversion of quasiseparable Vandermonde-like matrices

Fast Split-Radix and Radix-4 Discrete Cosine Transform Algorithms

The Discrete Fourier Transform (DFT) has a plethora of applications in applied mathematics and electrical engineering. Discrete Cosine Transform (DCT) is a real-arithmetic analogue of DFT. DCTs with orthogonal trigonometric transforms have been especially popular in recent decades due to their applications in digital video technology and high efficiency video coding. One can say that DCT is the key transform in image processing, signal processing, finger print enhancement, quick response code (QR code), multi-mode interface, etc. In this talk, we first introduce sparse and scaled orthogonal factorization for the DCT and inverse DCT. Afterwards, we present fast split-radix and radix-4 DCT and inverse DCT algorithms. We show that the proposed algorithms attain the lowest theoretical multiplication complexity and arithmetic complexity for 8-point DCT II/III matrices. We perform execution time of the proposed algorithms while verifying the connection to the order of the arithmetic complexity. Finally, the language of signal flow graph representation of digital structures is used to describe potential for real-world circuit implementation

A Fast Discrete Transform for Beam Digitization

Digital beamformers are popular due to the extensive usage in digital signal processing, including applications in radar, cellular networks, microwave imaging, and radio astronomy. When digital beamformers are considered, characteristics of the analog to digital converters e.g., dynamic range and instantaneous bandwidth, and the number of complex operations performed are of paramount importance in wireless communications. In here, we observe a hybrid of discrete transform matrices as the beam digitization transform matrix and present its sparse factorization. Next, the proposed factorization will be utilized to derive a fast algorithm while reducing the arithmetic complexity. Finally, the language of signal flow graphs will be utilized to connect the algebraic operations associated with the proposed algorithm to realize the system as an integrated circuit. This work is supported by the Faculty Innovative Research in Science and Technology, ERAU, Grant 13221

Vandermonde Neural Operators

Fourier Neural Operators (FNOs) have emerged as very popular machine learning architectures for learning operators, particularly those arising in PDEs. However, as FNOs rely on the fast Fourier transform for computational efficiency, the architecture can be limited to input data on equispaced Cartesian grids. Here, we generalize FNOs to handle input data on non-equispaced point distributions. Our proposed model, termed as Vandermonde Neural Operator (VNO), utilizes Vandermonde-structured matrices to efficiently compute forward and inverse Fourier transforms, even on arbitrarily distributed points. We present numerical experiments to demonstrate that VNOs can be significantly faster than FNOs, while retaining comparable accuracy, and improve upon accuracy of comparable non-equispaced methods such as the Geo-FNO.Comment: 21 pages, 10 figure

Integrating Communication and Sensor Arrays to Model and Navigate Autonomous Unmanned Aerial Systems

The emerging concept of drone swarms creates new opportunities with major societal implications. However, future drone swarm applications and services pose new communications and sensing challenges, particularly for collaborative tasks. To address these challenges, in this paper, we integrate sensor arrays and communication to propose a mathematical model to route a collection of autonomous unmanned aerial systems (AUAS), a so-called drone swarm or AUAS swarm, without having a base station of communication but communicating with each other using multiple spatio-temporal data. The theories of structured matrices, concepts in multi-beam beamforming, and sensor arrays are utilized to propose a swarm routing algorithm. We address the routing algorithm’s computational and arithmetic complexities, precision, and reliability. We measure bit-error-rate (BER) based on the number of elements in sensor arrays and beamformed output of the members of the swarm to authenticate and secure the routing for the decentralized AUAS networking. The proposed model has the potential to enable future drone swarm applications and services. Finally, we discuss future work on obtaining a machine-learning-based low-cost drone swarm routing algorithm