Frequency-Domain Analysis of Linear Time-Periodic Systems

In this paper, we study convergence of truncated representations of the frequency-response operator of a linear time-periodic system. The frequency-response operator is frequently called the harmonic transfer function. We introduce the concepts of input, output, and skew roll-off. These concepts are related to the decay rates of elements in the harmonic transfer function. A system with high input and output roll-off may be well approximated by a low-dimensional matrix function. A system with high skew roll-off may be represented by an operator with only few diagonals. Furthermore, the roll-off rates are shown to be determined by certain properties of Taylor and Fourier expansions of the periodic systems. Finally, we clarify the connections between the different methods for computing the harmonic transfer function that are suggested in the literature

Unwrapping phase fluctuations in one dimension

Correlation functions in one-dimensional complex scalar field theory provide a toy model for phase fluctuations, sign problems, and signal-to-noise problems in lattice field theory. Phase unwrapping techniques from signal processing are applied to lattice field theory in order to map compact random phases to noncompact random variables that can be numerically sampled without sign or signal-to-noise problems. A cumulant expansion can be used to reconstruct average correlation functions from moments of unwrapped phases, but points where the field magnitude fluctuates close to zero lead to ambiguities in the definition of the unwrapped phase and significant noise at higher orders in the cumulant expansion. Phase unwrapping algorithms that average fluctuations over physical length scales improve, but do not completely resolve, these issues in one dimension. Similar issues are seen in other applications of phase unwrapping, where they are found to be more tractable in higher dimensions.Comment: 14 pages, 7 figures. arXiv admin note: text overlap with arXiv:1806.0183

Data analysis of gravitational-wave signals from spinning neutron stars. V. A narrow-band all-sky search

We present theory and algorithms to perform an all-sky coherent search for periodic signals of gravitational waves in narrow-band data of a detector. Our search is based on a statistic, commonly called the $\mathcal{F}$-statistic, derived from the maximum-likelihood principle in Paper I of this series. We briefly review the response of a ground-based detector to the gravitational-wave signal from a rotating neuron star and the derivation of the $\mathcal{F}$-statistic. We present several algorithms to calculate efficiently this statistic. In particular our algorithms are such that one can take advantage of the speed of fast Fourier transform (FFT) in calculation of the $\mathcal{F}$-statistic. We construct a grid in the parameter space such that the nodes of the grid coincide with the Fourier frequencies. We present interpolation methods that approximately convert the two integrals in the $\mathcal{F}$-statistic into Fourier transforms so that the FFT algorithm can be applied in their evaluation. We have implemented our methods and algorithms into computer codes and we present results of the Monte Carlo simulations performed to test these codes.Comment: REVTeX, 20 pages, 8 figure

Efficient Image Evidence Analysis of CNN Classification Results

Convolutional neural networks (CNNs) define the current state-of-the-art for image recognition. With their emerging popularity, especially for critical applications like medical image analysis or self-driving cars, confirmability is becoming an issue. The black-box nature of trained predictors make it difficult to trace failure cases or to understand the internal reasoning processes leading to results. In this paper we introduce a novel efficient method to visualise evidence that lead to decisions in CNNs. In contrast to network fixation or saliency map methods, our method is able to illustrate the evidence for or against a classifier's decision in input pixel space approximately 10 times faster than previous methods. We also show that our approach is less prone to noise and can focus on the most relevant input regions, thus making it more accurate and interpretable. Moreover, by making simplifications we link our method with other visualisation methods, providing a general explanation for gradient-based visualisation techniques. We believe that our work makes network introspection more feasible for debugging and understanding deep convolutional networks. This will increase trust between humans and deep learning models.Comment: 14 pages, 19 figure

Closed-Form Critical Conditions of Subharmonic Oscillations for Buck Converters

A general critical condition of subharmonic oscillation in terms of the loop gain is derived. Many closed-form critical conditions for various control schemes in terms of converter parameters are also derived. Some previously known critical conditions become special cases in the generalized framework. Given an arbitrary control scheme, a systematic procedure is proposed to derive the critical condition for that control scheme. Different control schemes share similar forms of critical conditions. For example, both V2 control and voltage mode control have the same form of critical condition. A peculiar phenomenon in average current mode control where subharmonic oscillation occurs in a window value of pole can be explained by the derived critical condition. A ripple amplitude index to predict subharmonic oscillation proposed in the past research has limited application and is shown invalid for a converter with a large pole.Comment: Submitted to an IEEE Journal on Dec. 23, 2011, and resubmitted to IEEE Transactions on Circuits and Systems-I on Feb. 14, 2012. My current six papers in arXiv have a common reviewe

The exponentially convergent trapezoidal rule

It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators