Nonhomogeneous Wavelet Systems in High Dimensions

It is of interest to study a wavelet system with a minimum number of generators. It has been showed by X. Dai, D. R. Larson, and D. M. Speegle in [11] that for any $d\times d$ real-valued expansive matrix M, a homogeneous orthonormal M-wavelet basis can be generated by a single wavelet function. On the other hand, it has been demonstrated in [21] that nonhomogeneous wavelet systems, though much less studied in the literature, play a fundamental role in wavelet analysis and naturally link many aspects of wavelet analysis together. In this paper, we are interested in nonhomogeneous wavelet systems in high dimensions with a minimum number of generators. As we shall see in this paper, a nonhomogeneous wavelet system naturally leads to a homogeneous wavelet system with almost all properties preserved. We also show that a nonredundant nonhomogeneous wavelet system is naturally connected to refinable structures and has a fixed number of wavelet generators. Consequently, it is often impossible for a nonhomogeneous orthonormal wavelet basis to have a single wavelet generator. However, for redundant nonhomogeneous wavelet systems, we show that for any $d\times d$ real-valued expansive matrix M, we can always construct a nonhomogeneous smooth tight M-wavelet frame in $L_2(R^d)$ with a single wavelet generator whose Fourier transform is a compactly supported $C^\infty$ function. Moreover, such nonhomogeneous tight wavelet frames are associated with filter banks and can be modified to achieve directionality in high dimensions. Our analysis of nonhomogeneous wavelet systems employs a notion of frequency-based nonhomogeneous wavelet systems in the distribution space. Such a notion allows us to separate the perfect reconstruction property of a wavelet system from its stability in function spaces

Second Repeating FRB 180814.J0422+73: Ten-year Fermi-LAT Upper Limits and Implications

The second repeating fast radio burst source, FRB 180814.J0422+73, was detected recently by the CHIME collaboration. We use the ten-year Fermi Large Area Telescope archival data to place a flux upper limit in the energy range of 100 MeV−10 GeV at the position of the source, which is ~1.1 × 10−11 erg cm−2 s−1 for a six-month time bin on average, and ~2.4 × 10−12 erg cm−2 s−1 for the entire ten-year time span. For the maximum redshift of z = 0.11, the ten-year upper limit of luminosity is ~7.3 × 1043 erg s−1. We utilize these upper limits to constrain the fast radio burst (FRB) progenitor and central engine. For the rotation-powered young magnetar model, the upper limits can pose constraints on the allowed parameter space for the initial rotational period and surface magnetic field of the magnetar. We also place significant constraints on the kinetic energy of a relativistic external shock wave, ruling out the possibility that there existed a gamma-ray burst (GRB) beaming toward Earth during the past ten years as the progenitor of the repeater. The case of an off-beam GRB is also constrained if the viewing angle is not much greater than the jet opening angle. All of these constraints are more stringent if FRB 180814.J0422+73 is at a closer distance

A Fast Blind Impulse Detector for Bernoulli-Gaussian Noise in Underspread Channel

The Bernoulli-Gaussian (BG) model is practical to characterize impulsive noises that widely exist in various communication systems. To estimate the BG model parameters from noise measurements, a precise impulse detection is essential. In this paper, we propose a novel blind impulse detector, which is proven to be fast and accurate for BG noise in underspread communication channels.Comment: v2 to appear in IEEE ICC 2018, Kansas City, MO, USA, May 2018 Minor erratums added in v