As the original definition on Hilbert spectrum was given in terms of total energy and amplitude, there is a mismatch between the Hilbert spectrum and the traditional Fourier spectrum, which is defined in terms of energy density. Rigorous definitions of Hilbert energy and amplitude spectra are given in terms of energy and amplitude density in the time-frequency space. Unlike Fourier spectral ana...

We estimate Fourier coefficients of a Boolean function which has recently been introduced in the study of read-once branching programs. Our bound implies that this function has an asymptotically “flat” Fourier spectrum and thus implies several lower bounds of its various complexity measures.

We present a detailed theoretical analysis for the determination of the total internal loss in Fabry-Perot resonators based on Fourier analysis of the emission or transmission spectrum. The observation of higher-order harmonics and their relative height in the Fourier-transformed spectrum allow us to quantify the total resonator loss. Because this new method considers both contrast and shape of...

When a periodic waveform with a discrete-harmonic spectrum is temporally windowed to make a signal, its spectrum becomes a continuous function of frequency. However, there are discrete-frequency representations for windowed signals such as the Fourier series representation of a periodically extended signal. This article introduces the concept of matching between the temporal window and the peri...

Most quantum algorithms that give an exponential speedup over classical algorithms exploit the Fourier transform in some way. In Shor’s algorithm, sampling from the quantum Fourier spectrum is used to discover periodicity of the modular exponentiation function. In a generalization of this idea, quantum Fourier sampling can be used to discover hidden subgroup structures of some functions much mo...

We advocate a compressed sensing strategy that consists of multiplying the signal of interest by a wide bandwidth modulation before projection onto randomly selected vectors of an orthonormal basis. First, in a digital setting with random modulation, considering a whole class of sensing bases including the Fourier basis, we prove that the technique is universal in the sense that the required nu...

In this paper, we study the result of applying a lowpass variant filtering using scaling-rotating kernels to both the spatial and spatial-frequency representations of a two-dimensional (2-D) signal (image). It is shown that if we apply this transformation to a Fourier pair, the two resulting signals can also form a Fourier pair when the filters used in each domain maintain a dual relationship. ...

Traditional linear signal processing is based on viewing signals as sequences or functions in time that flow in one direction, from past through present into future. Somewhat surprisingly, the assumption that the most basic operation that can be performed on a signal is a time shift, or “delay,” is sufficient to derive many relevant signal processing concepts, including spectrum, Fourier transf...

It has been previously shown that a radially periodic image having a symmetric radial period can be decomposed into a circular Fourier series of circular cosine functions with radial frequencies of f = 1/T , 2/T , . . . , and that its Fourier spectrum consists of a series of half-order derivative impulse rings with radii f = n/T (which are the Fourier transforms of the circular cosines in the s...

In this paper, we propose the Fourier frequency vector (FFV), inherently, associated with multidimensional Fourier transform. With the help of FFV, we are able to provide physical meaning of so called negative frequencies in multidimensional Fourier transform (MDFT), which in turn provide multidimensional spatial and space-time series analysis. The complex exponential representation of sinusoid...