Search Results (16)

Discrete cosine transforms (DCTs) are widely used in intelligent electronic systems for storing, processing, and transmitting data. Their popularity stems, on the one hand, from their unique properties and, on the other hand, from the availability of fast algorithms that minimize the computational and hardware complexity of their implementation. Until recently, the Type VIII DCT had been one of the least studied variants, with virtually no publications addressing fast algorithms for its implementation. However, this situation has changed, making the development of efficient implementation methods for this transform a timely and important research problem. In this paper, several algorithmic solutions for implementing the Type VIII DCT are proposed. A set of Type VIII DCT algorithms for small lengths N = 3, 4, 5, 6, 7 is presented. The effectiveness of the proposed solutions is due to the possibility of successful factorization of small-sized DCT-VIII matrices, leading to a reduction in the computational complexity of implementing transforms of this type. Compared with direct matrix–vector computation, the proposed algorithms achieve an approximate 53% reduction in the number of multiplications, at the cost of an increase of about 21% in the number of additions. This work continues a series of previously published studies aimed at creating a library of small-sized algorithms for discrete trigonometric transforms. Full article

This paper presents new fast algorithms for the type VII discrete cosine transform (DCT-VII) applied to input data sequences of lengths ranging from 3 to 8. Fast algorithms for small-sized trigonometric transforms enable the processing of small data blocks in image and video coding with low computational complexity. To process the information in image and video coding standards, the fast DCT-VII algorithms can be used, taking into account the relationships between the DCT-VII and the type II discrete cosine transform (DCT-II). Additionally, such algorithms can be used in other digital signal processing tasks as components for constructing algorithms for large-sized transforms, leading to reduced system complexity. Existing fast odd DCT algorithms have been designed using relationships among discrete cosine transforms (DCTs), discrete sine transforms (DSTs), and the discrete Fourier transform (DFT); among different types of DCTs and DSTs; and between the coefficients of the transform matrix. However, these algorithms require a relatively large number of multiplications and additions. The process of obtaining such algorithms is difficult to understand and implement. To overcome these shortcomings, this paper applies a structural approach to develop new fast DCT-VII algorithms. The process begins by expressing the DCT-VII as a matrix-vector multiplication, then reshaping the block structure of the DCT-VII matrix to align with matrix patterns known from the basic papers in which the structural approach was introduced. If the matrix block structure does not match any known pattern, rows and columns are reordered, and sign changes are applied as needed. If this is insufficient, the matrix is decomposed into the sum of two or more matrices, each analyzed separately and transformed similarly if required. As a result, factorizations of DCT-VII matrices for different input sequence lengths are obtained. Based on these factorizations, fast DCT-VII algorithms with reduced arithmetic complexity are constructed and presented with pseudocode. To illustrate the computational flow of the resulting algorithms and their modular design, which is suitable for VLSI implementation, data-flow graphs are provided. The new DCT-VII algorithms reduce the number of multiplications by approximately 66% compared to direct matrix-vector multiplication, although the number of additions decreases by only about 6%. Full article

Accurate and efficient evaluation of the intersection area between two axis-aligned ellipses is essential in applications where the coordinate system or underlying geometry naturally imposes alignment. However, most existing numerical integration techniques are designed for arbitrarily oriented ellipses, and their generality typically requires adaptive refinement or solving higher-degree algebraic intersection formulations, leading to greater computational cost than necessary in the axis-aligned case. This study introduces two analytically derived, fixed-cost Gauss–Legendre quadrature formulations for computing the intersection area in the axis-aligned configuration. The first is a sine-mapped Gauss–Legendre quadrature, which applies a trigonometric transformation to improve conditioning near endpoint singularities while retaining constant-time evaluation. The second is an enhanced two-panel affine-normalized formulation, which splits the intersection domain into two sub-intervals to increase local accuracy while maintaining a fixed computational cost. Both methods are benchmarked against adaptive Simpson integration, polygonal discretization, and Monte Carlo sampling over 10,000 randomly generated ellipse pairs. The two-panel formulation achieves a mean relative error of 0.003% with runtimes more than twenty times faster than the adaptive reference and remains consistently more efficient than the polygonal and Monte Carlo approaches while exhibiting comparable or superior numerical behavior across all tested regimes. Full article

The Lomb–Scargle method (LSM) constitutes a robust method for frequency and amplitude estimation in cases where data exhibit irregular or sparse sampling. Conventional spectral analysis techniques, such as the discrete Fourier transform (FT) and wavelet transform, rely on orthogonal mode decomposition and are inherently constrained when applied to non-equidistant or fragmented datasets, leading to significant estimation biases. The classical LSM, originally formulated for univariate time series, provides a statistical estimator that does not assume a Fourier series representation. In this work, we extend the LSM to multivariate datasets by redefining the shifting parameter $\tau$ to preserve the orthogonality of trigonometric basis functions in ${\mathbb{R}}^n$. This generalization enables simultaneous estimation of the frequency, phase, and amplitude vectors while maintaining the statistical advantages of the LSM, including consistency and noise robustness. We demonstrate its application to solar activity data, where sunspots serve as intrinsic markers of the solar dynamo process. These observations constitute a randomly sampled two-dimensional binary dataset, whose characteristic frequencies are identified and compared with the results of solar research. Additionally, the proposed method is applied to an ultrasound velocity profile measurement setup, yielding a three-dimensional velocity dataset with correlated missing values and significant temporal jitter. We derive confidence intervals for parameter estimation and conduct a comparative analysis with FT-based approaches. Full article

Discrete sine transform (DST) has numerous applications across various fields, including signal processing, image compression and coding, adaptive digital filtering, mathematics (such as partial differential equations or numerical solutions of differential equations), image reconstruction, and classification, among others. The primary disadvantage of DST class algorithms (DST-I, DST-II, DST-III, and DST-IV) is their substantial computational complexity (O (N log N)) during implementation. This paper proposes an innovative decomposition and real-time implementation for the DST-IV. This decomposition facilitates the execution of the algorithm in four or eight sections operating concurrently. These algorithms, which encompass 4 and 8 sections, are primarily developed using a matrix factorization technique to decompose the DST-IV matrices. Consequently, the computational complexity and execution time of the developed algorithms are markedly reduced compared to the traditional implementation of DST-IV, resulting in significant time efficiency. The performance analysis conducted on three distinct Graphics Processing Unit (GPU) architectures indicates that a substantial speedup can be achieved. An average speedup ranging from 22.42 to 65.25 was observed, depending on the GPU architecture employed and the DST-IV implementation (with 4 or 8 sections). Full article

This paper presents new fast algorithms for the fourth type discrete Hartley transform (DHT-IV) for input data sequences of lengths from 3 to 8. Fast algorithms for small-sized trigonometric transforms can be used as building blocks for synthesizing algorithms for large-sized transforms. Additionally, they can be utilized to process small data blocks in various digital signal processing applications, thereby reducing overall system latency and computational complexity. The existing polynomial algebraic approach and the approach based on decomposing the transform matrix into cyclic convolution submatrices involve rather complicated housekeeping and a large number of additions. To avoid the noted drawback, this paper uses a structural approach to synthesize new algorithms. The starting point for constructing fast algorithms was to represent DHT-IV as a matrix–vector product. The next step was to bring the block structure of the DHT-IV matrix to one of the matrix patterns following the structural approach. In this case, if the block structure of the DHT-IV matrix did not match one of the existing patterns, its rows and columns were reordered, and, if necessary, the signs of some entries were changed. If this did not help, the DHT-IV matrix was represented as the sum of two or more matrices, and then each matrix was analyzed separately, if necessary, subjecting the matrices obtained by decomposition to the above transformations. As a result, the factorizations of matrix components were obtained, which led to a reduction in the arithmetic complexity of the developed algorithms. To illustrate the space–time structures of computational processes described by the developed algorithms, their data flow graphs are presented, which, if necessary, can be directly mapped onto the VLSI structure. The obtained DHT-IV algorithms can reduce the number of multiplications by an average of 75% compared with the direct calculation of matrix–vector products. However, the number of additions has increased by an average of 4%. Full article

In this paper, algorithms for calculating different types of generalized trigonometric and hyperbolic functions are developed and presented. The main attention is focused on the Ateb-functions, which are the inverse functions to incomplete Beta-functions. The Ateb-functions can generalize every kind of implementation where trigonometric and hyperbolic functions are used. They have been successfully applied to vibration motion modeling, data protection, signal processing, and others. In this paper, the Fourier transform’s generalization for periodic Ateb-functions in the form of Ateb-transform is determined. Continuous and discrete Ateb-transforms are constructed. Algorithms for their calculation are created. Also, Ateb-transforms with one and two parameters are considered, and algorithms for their realization are built. The quantum calculus generalization for hyperbolic Ateb-functions is constructed. Directions for future research are highlighted. Full article

We discuss determining a finite sample linear time-invariant (FS-LTI) system’s impulse response function, $h\left[n\right],$ in the frequency domain when the input testing function is a uniform window function with a width of L and the output is limited to a finite number of effective samples, M. Assuming that the samples beyond M are all zeros, the corresponding infinite sample LTI (IS-LTI) system is a marginally stable system. The ratio of the discrete Fourier transforms (DFT) of the output to input of such an FS-LTI system, $H_0\left[k\right]$, cannot be directly used to find $h\left[n\right]$ via inverse DFT (IDFT). Nevertheless, $H_0\left[k\right]$ contains sufficient information to determine the system’s impulse response function (IRF). In the frequency-domain approach, we zero-pad the output array to a length of N. We present methods to recover $h\left[n\right]$ from $H_0\left[k\right]$ for two scenarios: (1) $N\ge \max (L,M+1)$ and N is a coprime of L, and (2) $N\ge L+M+1$. The marginal stable system discussed here is an artifact due to the zero-value assumption on unavailable samples. The IRF obtained applies to any LTI system up to the number of effective data samples, M. In demonstrating the equivalence of $H_0\left[k\right]$ and $h\left[n\right]$, we derive two interesting DFT pairs. These DFT pairs can be used to find trigonometric sums that are otherwise difficult to prove. The frequency-domain approach makes mitigating the effects of interferences and random noise easier. In an example application in radar remote sensing, we show that the frequency-domain processing method can be used to obtain finer details than the range resolution provided by the radar system’s transmitter. Full article

This study proposes a novel factorization method for the DCT IV algorithm that allows for breaking it into four or eight sections that can be run in parallel. Moreover, the arithmetic complexity has been significantly reduced. Based on the proposed new algorithm for DCT IV, the speed performance has been improved substantially. The performance of this algorithm was verified using two different GPU systems produced by the NVIDIA company. The experimental results show that the novel proposed DCT algorithm achieves an impressive reduction in the total processing time. The proposed method is very efficient, improving the algorithm speed by more than 4-times—that was expected by segmenting the DCT algorithm into four sections running in parallel. The speed improvements are about five-times higher—at least 5.41 on Jetson AGX Xavier, and 10.11 on Jetson Orin Nano—if we compare with the classical implementation (based on a sequential approach) of DCT IV. Using a parallel formulation with eight sections running in parallel, the improvement in speed performance is even higher, at least 8.08-times on Jetson AGX Xavier and 11.81-times on Jetson Orin Nano. Full article

A basic tenet of linear invariant systems is that they are sufficiently described by either the impulse response function or the frequency transfer function. This implies that we can always obtain one from the other. However, when the transfer function contains uncanceled poles, the impulse function cannot be obtained by the standard inverse Fourier transform method. Specifically, when the input consists of a uniform train of pulses and the output sequence has a finite duration, the transfer function contains multiple poles on the unit cycle. We show how the impulse function can be obtained from the frequency transfer function for such marginally stable systems. We discuss three interesting discrete Fourier transform pairs that are used in demonstrating the equivalence of the impulse response and transfer functions for such systems. The Fourier transform pairs can be used to yield various trigonometric sums involving $\frac{\mathrm{s}\mathrm{i}\mathrm{n}\left(\pi k/N\right)}{\mathrm{s}\mathrm{i}\mathrm{n}\left(\pi Lk/N\right)}$, where k is the integer summing variable and N is a multiple of integer L. Full article

Fractional Galilei invariant advection–diffusion (GIADE) equation, along with its more general version that is the GIADE equation with nonlinear source term, is discretized by coupling weighted and shifted Grünwald difference approximation formulae and Crank–Nicolson technique. The new version of the backward substitution method, a well-established class of meshfree methods, is proposed for a numerical approximation of the consequent equation. In the present approach, the final approximation is given by the summation of the radial basis functions, the primary approximation, and the related correcting functions. Then, the approximation is substituted back to the governing equations where the unknown parameters can be determined. The polynomials, trigonometric functions, multiquadric, or the Gaussian radial basis functions are used in the approximation of the GIADE. Moreover, a quasilinearization technique is employed to transform a nonlinear source term into a linear source term. Finally, three numerical experiments in one and two dimensions are presented to support the method. Full article

This paper presents a methodology to obtain the Fourier coefficients (FCs) and the derivative Fourier coefficients (DFCs) from an input signal. Based on the Taylor series that approximates the input signal into a trigonometric signal model through the Kalman filter, consequently, the signal’s and successive derivatives’ coefficients are obtained with the state prediction and the state matrix inverse. Compared to discrete Fourier transform (DFT), the new class of filters provides noise reduction and sidelobe suppression advantages. Additionally, the proposed Taylor–Kalman–Fourier algorithm (TKFA) achieves a null-flat frequency response around the frequency operation. Moreover, with the proposed TKFA method, the decrement in the inter-harmonic amplitude is more significant than that obtained with the Kalman–Fourier algorithm (KFA), and the neighborhood of the null-flat frequency is expanded. Finally, the approximation of the input signal and its derivative can be performed with a sum of functions related to the estimated coefficients and their respective harmonics. Full article

Explicit links of the multivariate discrete (anti)symmetric cosine and sine transforms with the generalized dual-root lattice Fourier–Weyl transforms are constructed. Exact identities between the (anti)symmetric trigonometric functions and Weyl orbit functions of the crystallographic root systems $A_1$ and $C_n$ are utilized to connect the kernels of the discrete transforms. The point and label sets of the 32 discrete (anti)symmetric trigonometric transforms are expressed as fragments of the rescaled dual root and weight lattices inside the closures of Weyl alcoves. A case-by-case analysis of the inherent extended Coxeter–Dynkin diagrams specifically relates the weight and normalization functions of the discrete transforms. The resulting unique coupling of the transforms is achieved by detailing a common form of the associated unitary transform matrices. The direct evaluation of the corresponding unitary transform matrices is exemplified for several cases of the bivariate transforms. Full article

In this study, the temperature distribution of a pavement was predicted by developing an analytic algorithm. The Laplace and inverse Laplace transform techniques and Gaussian quadratic formula were applied to a pavement system of an asphalt overlay placed over an existing concrete pavement. The temperature distribution of the previous cement concrete pavement with an asphalt overlay can be estimated with the proposed analytical method regardless of the depth and time. To conduct the method, the layer thicknesses, material thermal properties and climatic factors (including air temperature, wind velocity and solar radiation) were firstly input. Then, a discrete least-squares approximation of the interpolatory trigonometric polynomials was used to fit some specific measured climatic factors considered in the surface boundary condition, i.e., the measured solar radiation intensity and air temperature. The pavement surface convection coefficient can be approximately calculated by the wind speed. The temperature solutions are validated with the measured pavement temperature data of two different periods of a whole year (summer and winter). The obtained results demonstrate the feasibility of the developed analytical approach to predict the temperature distribution of the existing cement concrete pavement with an asphalt overlay in different weather conditions with acceptable accuracy. Full article

This paper investigates the problem of an optimal sensor placement for better shape deformation sensing of a new antenna structure with embedded or attached Fiber Bragg grating (FBG) strain sensors. In this paper, the deformation shape of the antenna structure is reconstructed using a strain–displacement transformation, according to the measured discrete strain data from limited FBG strain sensors. Moreover, a two-stage sensor placement method is proposed using a derived relative reconstruction error equation. In this method, the initial sensor locations are determined using the principal component analysis based on orthogonal trigonometric (i.e., QR) decomposition, and then a new location is sequentially added into the initial sensor locations one by one by minimizing the relative reconstruction error considering information redundancy. The numerical simulations are conducted, and the comparisons show that the proposed method is advantageous in terms of the sensor distribution and computational cost. Experimental validation is performed using an antenna experimental platform equipped with an optimal FBG strain sensor configuration, and the reconstruction results show good agreements with those measured directly from displacement sensors. The proposed method has a large potential for the strain sensor placement of complex structures, and the proposed antenna structure with FBG strain sensors can be applied to the future wing-skin antenna or flexible space-based antenna. Full article