Fast evaluation of trigonometric polynomials from hyperbolic crosses

Interpolation lattices for hyperbolic cross trigonometric polynomials

Error analysis of the Wiener–Askey polynomial chaos with hyperbolic cross approximation and its application to differential equations with random input

Hardware implementation of the fast Fourier transform (FFT) function consists of multiple consecutive arithmetic operations over complex numbers. Applying floating-point arithmetic to FFT coprocessors leads to a wider dynamic range and allows the coprocessor to collaborate with general purpose processors via the standard floating-point arithmetic. This offloads compute-intensive tasks from the primary processor and overcomes floating-point concerns such as scaling and overflow/underflow detection. The downside, however, is that floating-point units are slower than the fixed-point counterparts. One of the popular ways to improve the speed of floating-point FFT units is to merge the arithmetic operations inside the butterfly units of a FFT architecture. This leads to a butterfly architecture based on multi-operand adders. Butterfly units are designed, in two of the most recent works, using three-operand and four-operand adders. However, the work reported here by the present authors goes further and a butterfly architecture based on a five-operand adder is proposed. Simulation results demonstrate that the proposed butterfly architecture is 50% smaller than the fastest previous work with about 17% latency overhead. Compared with the smallest previous work, the proposed design is 47% smaller and 8% faster.

Fast Fourier Transform (FFT) is computationally an efficient algorithm that transforms a function in time domain to function in frequency domain. All 3G and 4G wireless technologies use OFDM and FFT is an integral block of transceiver loop. In this paper three FFT architectures are designed that fall within the Cooley-Tukey class of algorithms. These FFT designs are targeted for OFDM applications especially in future generations of Software Defined Radio (SDR) systems. The FFT is performed on 64-point complex valued input samples with 16-bit precision. Each core is optimized for a combination of area, power, and speed. The design is targeted for a partially reconfigurable FPGA platform such as Xilinx Virtex-4 or above. The FFT cores have been simulated in Model Sim and synthesized using Xilinx ISE software. The cores can be readily downloaded on to target FPGA board to check for run time configurability. The performance evaluation of these cores show that they run at clock frequency of about 80MHz.

The stationary monochromatic radiative transfer equation (RTE) is a partial differential transport equation stated on a five-dimensional phase space, the Cartesian product of physical and angular domain. We solve the RTE with a Galerkin FEM in physical space and collocation in angle, corresponding to a discrete ordinates method (DOM). To reduce the complexity of the problem and to avoid the “curse of dimension”, we adapt the sparse grid combination technique to the solution space of the RTE and show that we obtain a sparse DOM which uses essentially only as many degrees of freedom as required for a purely spatial transport problem. For smooth solutions, the convergence rates deteriorate only by a logarithmic factor. We compare the sparse DOM to the standard full DOM and a sparse tensor product approach developed earlier with Galerkin FEM in physical space and a spectral method in angle. Numerical experiments confirm our findings.

We introduce an efficient way of performing polynomial multiplication in a class of finite fields GF(pm) in the frequency domain. The Fast Fourier Transform (FFT) based frequency domain multiplication technique, originally proposed for integer multiplication, provides an extremely efficient method for multiplication with the best known asymptotic complexity, i.e. O(n log n log log n). Unfortunately, the original FFT method bears significant overhead due to the conversions between the time and the frequency domains, which makes it impractical to perform multiplication of relatively short (160 - 1024 bits) integer operands as used in many applications. In this work, we introduce an efficient way of performing polynomial multiplication in finite fields using the FFT. We show that, with careful selection of parameters, all the multiplications required for the FFT computations can be avoided and polynomial multiplication in finite fields can be achieved with only O(m) multiplications in addition to O(m log m) simple shift, addition and subtraction operations. We show that, especially in constrained devices where multiplication is expensive, polynomial multiplication in the suggested finite fields using the FFT outperforms both the schoolbook and Karatsuba methods for practically small finite fields, e.g., relevant to elliptic curve cryptography.

This paper analyzes the complexity of the contraction fixed point problem: compute an e-approximation to the fixed point V * = Γ(V * ) of a contraction mapping r that maps a Banach space B d of continuous functions of d variables into itself. We focus on quasi linear contractions where Γ is a nonlinear functional of a finite number of conditional expectation operators. This class includes contractive Fredholm integral equations that arise in asset pricing applications and the contractive Bellman equation from dynamic programming. In the absence of further restrictions on the domain of Γ, the quasi linear fixed point problem is subject to the curse of dimensionality, i.e., in the worst case the minimal number of function evaluations and arithmetic operations required to compute an e-approximation to a fixed point V * e B d increases exponentially in d. We show that the curse of dimensionality disappears if the domain of Γ has additional special structure. We identify a particular type of special structure for which the problem is strongly tractable even in the worst case, i.e., the number of function evaluations and arithmetic operations needed to compute an e-approximation of V * is bounded by Ce -p where C and p are constants independent of d. We present examples of economic problems that have this type of special structure including a class of rational expectations asset pricing problems for which the optimal exponent p = 1 is nearly achieved.

Spectrum analysis can quickly extract and analyze frequency domain features of signal, and it has been widely applied in fields of image processing, noise processing and signal processing. Fast Fourier transform (FFT) is fast and efficient, because it can efficiently decrease complexity of discrete Fourier transform. As a consequence, FFT is a very good method for image frequency spectrum analysis. In this paper, we propose to consolidate frequency domain representation by sparse representation (SR) and collaborative representation classification (CRC) which has excellent performance in comparison with general sparse representation-associated classification algorithms. Our proposed novel method has three main phases. The first phase utilizes FFT to extract frequency domain features of original images, which are complementary with representations of the original images. The second phase of our proposed novel method exploits CRC or SR to obtain scores of original images and obtained features, respectively. The third phase integrates the scores of original images and obtained features and uses them to classify images. The major contribution of the proposed method is that it is usually more robust than methods using only FFT, CRC or state-of-art method CIRLRC for image classification. The experiments of image classification demonstrate that the simultaneous use of FFT and CRC or sparse representation classification has high accuracy on image recognition.

The number of real multiplications and additions for all the FFT algorithms is computed for selected data lengths. The arithmetic operations and the memory required are the basis for evaluating the hardware cost for each algorithm. The accuracy of the FFT/NTT algorithm is derived, and the results are compared to previously published accuracy analyses of other FFT algorithms. The FFT/NTT algorithm provides an accurate computation of a prime length discrete Fourier transform (DFT), with the only error sources being input and trigonometric coefficient quantization. These error sources are also found in other FFT implementations. However, due to the implementation of the FFT/NTT algorithm, the effect of these noise sources on the outputs is minimum for data lengths greater than 32. In the comparison of the arithmetic operations counts, the P-length FFT/NTT required more operations than a multiple-radix FFT of length P-1. However, in some cases, the number of operations as compared to a compatible radix-2 FFT favored the FFT/NTT. A speed advantage can be realized if the FFT/NTT is implemented in a pipeline configuration.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

We develop a new algorithm for the fast computation of discrete sums $f(y_j) := \sum_{k=1}^N \alpha_k K(y_j-x_k)$ (j =1, . . ., M) based on the recently developed fast Fourier transform (FFT) at nonequispaced knots. Our algorithm, in particular our regularization procedure, is simply structured and can be easily adapted to different kernels K. Our method utilizes the widely known FFT and can consequently incorporate advanced FFT implementations. In summary, it requires ${\cal O} (N \log N +M)$ arithmetic operations. We prove error estimates to obtain clues about the choice of the involved parameters and present numerical examples in one and two dimensions.

This work proposes and analyzes an anisotropic sparse grid stochastic collocation&#13;\nmethod for solving partial differential equations with random coefficients and forcing terms (input&#13;\ndata of the model). The method consists of a Galerkin approximation in the space variables and a&#13;\ncollocation, in probability space, on sparse tensor product grids utilizing either Clenshaw–Curtis or&#13;\nGaussian knots. Even in the presence of nonlinearities, the collocation approach leads to the solution&#13;\nof uncoupled deterministic problems, just as in the Monte Carlo method. This work includes a priori&#13;\nand a posteriori procedures to adapt the anisotropy of the sparse grids to each given problem.&#13;\nThese procedures seem to be very effective for the problems under study. The proposed method&#13;\ncombines the advantages of isotropic sparse collocation with those of anisotropic full tensor product&#13;\ncollocation: the first approach is effective for problems depending on random variables which weigh&#13;\napproximately equally in the solution, while the benefits of the latter approach become apparent when&#13;\nsolving highly anisotropic problems depending on a relatively small number of random variables, as&#13;\nin the case where input random variables are Karhunen–Lo`eve truncations of “smooth” random&#13;\nfields. This work also provides a rigorous convergence analysis of the fully discrete problem and&#13;\ndemonstrates (sub)exponential convergence in the asymptotic regime and algebraic convergence in&#13;\nthe preasymptotic regime, with respect to the total number of collocation points. It also shows&#13;\nthat the anisotropic approximation breaks the curse of dimensionality for a wide set of problems.&#13;\nNumerical examples illustrate the theoretical results and are used to compare this approach with&#13;\nseveral others, including the standard Monte Carlo. In particular, for moderately large-dimensional&#13;\nproblems, the sparse grid approach with a properly chosen anisotropy seems to be very efficient and&#13;\nsuperior to all examined methods.

Applications like fuel cells, batteries or food analysis are typical fields of interest for electrical impedance spectroscopy (EIS).Another field with promising results are EIS measurements in semiconductor (sc) gas sensor applications [1].Due to their high sensitivity and low cost, sc gas sensors are ideally suited for different applications, especially in the field of safety and security.The major drawbacks of sc gas sensors, i.e. poor selectivity and sensor baseline drift, can be overcome by using temperature cycled operation (TCO) with DC measurement [2] or EIS, both in combination with intelligent signal processing.Currently, EIS for sc gas sensors is only possible in laboratory environments using commercial impedance analyzers, which are characterized by high cost, poor mobility and long acquisition time.However, for efficient development of application specific sc gas sensor systems and/or a self-monitoring strategy [3] extensive field tests are necessary.Therefore, a new hardware concept for EIS of sc gas sensors based on FPGA-technology was developed.This concept also allows a combination of EIS with TCO for the first time.Additionally, state-of-the-art feature extraction methods are applied to sc gas sensor data and new features were tested [4]. MotivationDue to their high sensitivity and low cost, semiconductor gas sensors are ideally suited for safety and security applications like fire and leakage detection [5,6].The major drawbacks of this type of sensor are the poor selectivity, the sensor baseline drift and sensor poisoning.Especially for safety and security applications these drawbacks must be overcome to allow a proper function of the system.Temperature cycled operation (TCO) and electrical impedance spectroscopy (EIS) -both in combination with intelligent signal processing -are the main approaches for improving both selectivity and stability.In TCO mode the sensor temperature is changed periodically [2] and the DC resistance of the sensor is read-out continually during this cycle.For EIS the sensor temperature is usually kept constant and the sensor impedance is measured over a wide frequency range, e.g. during a frequency sweep [1,7,8].To detect a sensor malfunction due to sensor poisoning a self-monitoring strategy can be realized by combining TCO und EIS (fig. 1) [3].This strategy is based on redundant information from both modes which are evaluated separately and comparing both results.Additional information can be used for selectivity enhancement.

Intra-speaker variations due to perceptually induced stress or emotion adversely affect speech recognition system performance. In this paper, we combine auditory based (Mel frequency cepstral coefficients and linear predictive cepstral coefficients) features and articulatory based (voicedness) features for robust speech recognition. Voicedness features are derived using linear and teager energy operator (TEO) based nonlinear fast Fourier transform (FFT) spectra. Nonlinear properties are analyzed in both the time and frequency domains. In addition, we investigate the sensitivity of all these FFT spectra to stress and observe the performance of individual FFT spectra. The system is tested using stressed speech data from the speech under simulated and actual stress (SUSAS) database. The results show that articulatory based features help to improve the system performance. Furthermore, significant performance improvement has been observed when using the FFT spectrum which is less sensitive to stress

The recommended power quality (PQ) indices in electric power systems are defined based on the discrete Fourier transform (DFT). DFT is mostly evaluated applying fast Fourier transform (FFT) algorithm which requires signals to be periodic in nature. Hence, it evaluates PQ indices accurately for stationary PQ disturbances only. However, in case of nonstationary PQ disturbances signal is aperiodic and FFT results in erroneous assessment due to spectral leakage phenomena. Furthermore, FFT cannot provide time information of the PQ indices as computation is performed in frequency domain only. Which is significant shortcoming of the FFT when dealing with time-varying PQ disturbances to extract dynamic signature of the signal. Thereby, this paper presents a framework of redefining PQ indices for stationary and nonstationary PQ disturbances employing time-frequency distribution (TFD) method. Among all TFDs, reduced interference distribution (RID) represents the most suitable properties for analyzing time-varying PQ disturbances. Hence, in this work, it is employed to reformulate PQ indices typically used in electric power systems. The results of two synthetic examples considering stationary and nonstationary PQ disturbances imply that the RID-based redefined PQ indices are closer to the actual values. Also, they provide more accurate results than the existing transient PQ indices and traditional FFT-based method.

The Fast Fourier Transform (FFT) method for solving the 1-D heat diffusion equation offers an efficient approach for resolving partial differential equations (PDEs) with various time steps . FFT is used to transform the 1-D heat diffusion equation into the frequency domain and back to the time domain through inverse FFT. Using mathematical modeling with initial and Dirichlet boundary conditions, the numerical solutions produced by FFT are compared with analytical solutions. The accuracy of the method is validated using MAE and MSE calculated in Matlab. At several time intervals , the obtained MAE and MSE values indicate a good agreement between the numerical and analytical solutions, with very small errors. Numerical stability analysis confirms the reliability of the FFT method across various The variation in time step has a significant impact on the accuracy and stability of the solution. Smaller time steps improve accuracy and stability but require longer computation times. The optimal time step selected in this study is Increasing the number of discretization points also enhances accuracy but implies an increase in computational load and memory usage. The FFT method demonstrates good numerical consistency with increasing