Qiuhui Chen

A B-spline approach for empirical mode decompositions

We propose an alternative B-spline approach for empirical mode decompositions for nonlinear and nonstationary signals. Motivated by this new approach, we derive recursive formulas of the Hilbert transform of B-splines and discuss Euler splines as spline intrinsic mode functions in the decomposition. We also develop the Bedrosian identity for signals having vanishing moments. We present numerical implementations of the B-spline algorithm for an earthquake signal and compare the numerical performance of this approach with that given by the standard empirical mode decomposition. Finally, we discuss several open mathematical problems related to the empirical mode decomposition.

Multivariate Filter Banks Having Matrix Factorizations

In this paper we design vector-valued multivariate filter banks with a polyphase matrix built by a matrix factorization. These filter banks are suitable for the construction of multivariate multiwavelets with a general dilation matrix. We show that block central symmetric orthogonal matrices provide filter banks having a uniform linear phase. Several examples are included to illustrate our construction.

STABILITY OF FRAMES GENERATED BY NONLINEAR FOURIER ATOMS

In this paper, we study the stability of two kinds of frames generated by nonlinear Fourier atoms. The first result is the Kadec type ¼-theorem. The second states that the nonlinear windowed Fourier atoms form a frame of L2(ℝ).

FUNCTIONS WITH SPLINE SPECTRA AND THEIR APPLICATIONS

In this paper, we introduce a family of real-valued functions which have spline spectra. They extend the well-known Sinc function and generally are the restrictions to the real line of analytic functions in a strip containing the real axis. We investigate various properties of these functions including those related to interpolation, orthogonality, and stability. Moreover, a sampling formula is provided for their construction and some applications for signal analysis are given.

Rational orthogonal systems are Schauder bases

It is well known that orthogonal rational systems (Takenaka-Malmquist or TM systems) are bases of the closures of their spans in all Hardy spaces, . In this paper, we further prove that they are, in fact, Schauder bases in those spaces. We simultaneously treat both the contexts in the unit disc and the upper-half space

GENERALIZED PALEY–WIENER THEOREMS

Non-harmonic Fourier transform is useful for the analysis of transient signals, where the integral kernel is from the boundary value of Mobius transform. In this note, we study the Paley–Wiener type extension theorems for the non-harmonic Fourier transform. Two extension theorems are established by using real variable techniques.

On the matrix completion problem for multivariate filter bank construction

Abstract We survey the main techniques for the construction of multivariate filter banks and present new results about special matrices of order four and eight suitable for their construction.

Time-Frequency Aspects of Nonlinear Fourier Atoms

In the standard Fourier analysis one uses the linear Fourier atoms e int : n ∈ ℤ. With only the linear phases nt Fourier analysis can not expose the essence of time-varying frequencies of nonlinear and non-stationary signals. In this note we study time-frequency properties of a new family of atoms e inθa (t) : n ∈ ℤ, non-linear Fourier atoms, where a is any but fixed complex number with |a| < 1, and dθa (t) a harmonic measure on the unit circle parameterized by t. The nonlinear Fourier atoms e inθ a (t) : n ∈ ℤ were first noted in [12] with some examples and theoretically studied in [8]. In this note we show that the real parts cos θ a (t), |a| < 1, form a family of intrinsic mode functions introduced in the HHT theory [5]. We prove that for a fixed a the set e inθa (t) : n ∈ ℤ, constitutes a Riesz basis in the space L 2([0, 2π]). Some miscellaneous results including Shannon type sampling theorems are obtained.

System identification by discrete rational atoms

A novel adaptive frequency-domain system identification method for linear time-invariant systems is proposed in this paper. It finds poles for discrete rational atoms with discrete frequency responses. The theoretical foundation, including adaptive decomposition principle and decomposition convergence rate, is established. The algorithm of the adaptive pursuit is also provided in this paper.

Amplitudes of mono-component signals and the generalized sampling functions

There is a recent trend to use mono-components to represent nonlinear and non-stationary signals rather than the usual Fourier basis with linear phase, such as the intrinsic mode functions used in Norden Huangs empirical mode decomposition [12]. A mono-component is a real-valued signal of finite energy that has non-negative instantaneous frequencies, which may be defined as the derivative of the phase function of the given real-valued signal through the approach of canonical amplitude-phase modulation. We study in this paper how the amplitude is determined by its phase for a class of signals, of which the instantaneous frequency is periodic and described by the Poisson kernel. Our finding is that such an amplitude can be perfectly represented by a sampling formula using the so-called generalized sampling functions that are related to the phase. The regularity of such an amplitude is identified to be at least continuous. Such characterization of mono-components provides the theory to adaptively decompose non-stationary signals. Meanwhile, we also make a very interesting and new characterization of the band-limited functions.