Bing-Zhao Li

New sampling formulae related to linear canonical transform

Linear canonical transform (LCT) is an integral transform with four parameters a, b, c, d and has been shown to be a powerful tool for optics, radar system analysis, filter design, phase retrieval, pattern recognition, and many other applications. Many well-known transforms such as the Fourier transform, the fractional Fourier transform, and the Fresnel transform can be seen as special cases of the linear canonical transform. In this paper, new sampling formulae for reconstructing signals that are band-limited or time-limited in the linear canonical transform sense have been proposed. Firstly, the sampling theorem representation of band-limited signals associated with linear canonical transform from the samples taken at Nyquist rate is derived in a simple way. Then, based on the relationship between the Fourier transform and the linear canonical transform, the other two new sampling formulae using samples taken at half the Nyquist rate from the signal and its first derivative or its generalized Hilbert transform are obtained. The well-known sampling theorems in Fourier domain or fractional Fourier domain are shown to be special cases of the achieved results. The experimental results are also proposed to verify the accuracy of the obtained results. Finally, discussions about these new results and future works related to the linear canonical transform are proposed.

On Sampling of Band-Limited Signals Associated With the Linear Canonical Transform

Sampling is one of the fundamental topics in the signal processing community. Theorems proposed under this topic form the bridge between the continuous-time signals and discrete-time signals. Several sampling theorems, which aid in the reconstruction of signals in the linear canonical transform (LCT) domain, have been proposed in the literature. However, two main practical issues associated with the sampling of the LCT still remain unresolved. The first one relates to the reconstruction of the original signal from nonuniform samples and the other issue relates to the fact that only a finite number of samples are available practically. Focusing on these issues, this paper seeks to address the above from the LCT point of view. First, we extend several previously developed theorems for signals band-limited in the Fourier domain to signals band-limited in the LCT domain, followed by the derivation of the reconstruction formulas for finite uniform or recurrent nonuniform sampling points associated with the LCT. Simulation results and the potential applications of the theorem are also proposed.

Spectral Analysis and Reconstruction for Periodic Nonuniformly Sampled Signals in Fractional Fourier Domain

Nonuniform sampling occurs in many applications due to imperfect timebase or random events. Periodic nonuniform sampling is a special type of nonuniform sampling. The properties and applications of periodic nonuniform sampling signals in traditional Fourier domain have been extensively studied, but so far none of the research papers covering the spectral analysis and reconstruction of periodic nonuniformly sampled signals in fractional Fourier domain have been published. This correspondence is to explore the spectral properties of periodic nonuniformly sampled signals associated with the fractional Fourier transform. First, based on the uniform fractional Fourier transform kernel, the general spectral representation of periodic nonuniformly sampled signals has been derived. Second, detailed analysis of periodic nonuniformly sampled chirp signals in the fractional Fourier domain have been performed. The results can be used to estimate the chirp rate and the sampling offsets. Finally, a more simple relationship between the discrete fractional spectrum of periodic nonuniformly sampled signals and the continuous spectrum has been obtained. Based on this relationship, the original continuous spectrum can be reconstructed from periodic nonuniformly sampled signals in fractional Fourier domain. In addition, the simulations are carried out to verify the correctness of the results.

Wigner-Ville Distribution Associated with the Linear Canonical Transform

The linear canonical transform is shown to be one of the most powerful tools for nonstationary signal processing. Based on the properties of the linear canonical transform and the classical Wigner-Ville transform, this paper investigates the Wigner-Ville distribution in the linear canonical transform domain. Firstly, unlike the classical Wigner-Ville transform, a new definition of Wigner-Ville distribution associated with the linear canonical transform is given. Then, the main properties of the newly defined Wigner-Ville transform are investigated in detail. Finally, the applications of the newly defined Wigner-Ville transform in the linear-frequency-modulated signal detection are proposed, and the simulation results are also given to verify the derived theory.

Spectral Analysis of Sampled Signals in the Linear Canonical Transform Domain

The spectral analysis of uniform or nonuniform sampling signal is one of the hot topics in digital signal processing community. Theories and applications of uniformly and nonuniformly sampled one-dimensional or two-dimensional signals in the traditional Fourier domain have been well studied. But so far, none of the research papers focusing on the spectral analysis of sampled signals in the linear canonical transform domain have been published. In this paper, we investigate the spectrum of sampled signals in the linear canonical transform domain. Firstly, based on the properties of the spectrum of uniformly sampled signals, the uniform sampling theorem of two dimensional signals has been derived. Secondly, the general spectral representation of periodic nonuniformly sampled one and two dimensional signals has been obtained. Thirdly, detailed analysis of periodic nonuniformly sampled chirp signals in the linear canonical transform domain has been performed.

Approximating bandlimited signals associated with the LCT domain from nonuniform samples at unknown locations

The sampling theory describes ways of reconstructing signals from their uniform or nonuniform samples associated with the traditional Fourier transform (FT). Most of the published papers about the sampling theory require signals to be bandlimited in the FT domain and assume that the sample locations and the band width are all known. However, the sample locations are not always known and most of the signals are non-stationary in practical applications. In order to overcome these shortcomings, this paper provides an algorithm for approximating signals from nonuniform samples at unknown locations. These signals are not necessarily bandlimited in the FT domain, however bandlimited in the LCT domain. The experimental results are given to verify the accuracy of the algorithm.

Using the multi-living agent concept to investigate complex information systems

In this paper, we propose the multi-living agent (MLA) concept based on the of analysis the characteristics of complex information systems, especially those systems that require multi-functional operations under strict restraint strong countermeasures (SRSC) environment. First, we investigate the representation of the livelihood of the system under the SRSC conditions, and obtain the basic dynamical presentation of the MLA from the profile of the system’s function. Next, we propose the concept of the living self-organization based on the self-organization profile of the system, and derive a Two-Set model of the living self-organization mechanism. Further, based on the above results, we present a basic construction model of MLA-based information systems in the field of information security and countermeasures. A three-level negotiation-coordination mechanism is also derived. Finally, we present two practical examples to show how to use the MLA concept to analyze and study real complex information systems. In our opinion, the proposal of the MLA will bridge the gap between the research of the application and the basic application level of science. It provides the principle research methods and the supporting theories for the construction and analysis of complex information systems in the information security and countermeasures field.

The Poisson sum formulae associated with the fractional Fourier transform

The theorem of sampling formulae has been deduced for band-limited or time-limited signals in the fractional Fourier domain by different authors. Even though the properties and applications of these formulae have been studied extensively in the literature, none of the research papers throw light on the Poisson sum formula and non-band-limited signals associated with the fractional Fourier transform (FrFT). This paper investigates the generalized pattern of Poisson sum formula from the FrFT point of view and derived several novel sum formulae associated with the FrFT. Firstly, the generalized Poisson sum formula is obtained based on the relationship of the FrFT and the Fourier transform; then some new results associated with this novel sum formula have been derived; the potential applications of these new results in estimating the bandwidth and the fractional spectrum shape of a signal in the fractional Fourier domain are also proposed. In addition, the results can be seen as the generalization of the classical results in the Fourier domain.

Speech recovery based on the linear canonical transform

As is well known, speech signal processing is one of the hottest signal processing directions. There are exist lots of speech signal models, such as speech sinusoidal model, straight speech model, AM-FM model, gaussian mixture model and so on. This paper investigates AM-FM speech model by the linear canonical transform (LCT). The LCT can be considered as a generalization of traditional Fourier transform and fractional Fourier transform, and proved to be one of the powerful tools for non-stationary signal processing. This has opened up the possibility of a new range of potentially promising and useful applications based on the LCT. Firstly, two novel recovery methods of speech based on the AM-FM model are presented in this paper: one depends on the LCT domain filtering; the other one is based on the chirp signal parameter estimation to restore the speech signal in LCT domain. Then, experiments results are presented to verify the performance of the proposed methods. Finally, the summarization and the conclusion of the paper is given.

Sampling in the Linear Canonical Transform Domain

This paper investigates the interpolation formulae and the sampling theorem for bandpass signals in the linear canonical transformLCTdomain. Firstly, one of the important relationships between the bandpass signals in the Fourier domain and the bandpass signals in the LCT domain is derived. Secondly, two interpolation formulae from uniformly sampled points at half of the sampling rate associated with the bandpass signals and their generalized Hilbert transform or the derivatives in the LCT domain are obtained. Thirdly, the interpolation formulae from nonuniform samples are investigated. The simulation results are also proposed to verify the correctness of the derived results.