Xiumei Li

Local polynomial Fourier transform: A review on recent developments and applications

The local polynomial Fourier transform (LPFT), as a high-order generalization of the short-time Fourier transform (STFT), has been developed and used for many different applications in recent years. This paper attempts to review previous research work on the following issues of the LPFT. Firstly, the definition, the properties of the LPFT and its relationships with other transforms are reviewed. The LPFT for multicomponent signal is then presented. The polynomial time frequency transform (PTFT), which is the maximum likelihood estimator to estimate the parameters in the LPFT, as well as its properties and fast algorithms are discussed. By comparing with the Fourier transform (FT), the STFT and the Wigner-Ville distribution (WVD), the LPFT has its superiority in obtaining improved SNRs, which can be supported by theoretical analysis and computer simulations. Furthermore, the reassignment method is combined with the LPFT and the robust LPFT to improve the concentration of the signal representation in the time-frequency domain. Performances obtained by using various LPP-related methods are compared for signals in different noise environments, such as the additive white Gaussian noise (AGWN), impulsive noise, and the mixture of AGWN and impulsive noise.

LFM signal detection using LPP-Hough transform

This paper presents a new method to detect linear frequency modulated (LFM) signals by jointly using the local polynomial periodogram (LPP) and the Hough transform. Theoretical comparison is made on the 3dB signal-to-noise ratios (SNRs), achieved by the pseudo-Wigner-Ville distribution (PWVD) and the LPP, to show that the latter is important to achieve significant increase of noise margins in the Hough transform domain. The results of computer simulations are presented for the detection of mono- and multi-component LFM signals corrupted by additive white Gaussian noise and impulsive noise. It is also found that by using the time-frequency filtering, the computational complexity of the detection can be substantially reduced. Both the theoretical analysis and the simulation results show that the proposed method achieves significant performance improvement on detecting the LFM signals in very low signal-to-noise ratio environments.

The reassigned local polynomial periodogram and its properties

This paper defines a reassignment method based on the second-order local polynomial periodogram and investigates its properties with some mathematical proofs. Based on simulation results with various signals, such as parallel and crossed chirps, and parabolic and sinusoidal frequency modulated signals, comparisons with the reassigned spectrogram and smoothed pseudo-Wigner-Ville distribution are made to show the desirable ability of the reassigned local polynomial periodogram for improvement on the signal concentration in the time-frequency domain.

Compressive Sensing in Signal Processing: Algorithms and Transform Domain Formulations

Compressive sensing has emerged as an area that opens new perspectives in signal acquisition and processing. It appears as an alternative to the traditional sampling theory, endeavoring to reduce the required number of samples for successful signal reconstruction. In practice, compressive sensing aims to provide saving in sensing resources, transmission, and storage capacities and to facilitate signal processing in the circumstances when certain data are unavailable. To that end, compressive sensing relies on the mathematical algorithms solving the problem of data reconstruction from a greatly reduced number of measurements by exploring the properties of sparsity and incoherence. Therefore, this concept includes the optimization procedures aiming to provide the sparsest solution in a suitable representation domain. This work, therefore, offers a survey of the compressive sensing idea and prerequisites, together with the commonly used reconstruction methods. Moreover, the compressive sensing problem formulation is considered in signal processing applications assuming some of the commonly used transformation domains, namely, the Fourier transform domain, the polynomial Fourier transform domain, Hermite transform domain, and combined time-frequency domain.

Nuclear norm minimization framework for DOA estimation in MIMO radar

In this paper, the direction of arrival (DOA) estimation for noncircular sources in multiple-input multiple-output (MIMO) radar is dealt with by a novel nuclear norm minimization (NNM) framework. The proposed method exploits the noncircular property of signals to extend the data model for doubling the array aperture. Then a block sparse model of the extended data is formulated without the influence of the unknown noncircularity phase, and a novel signal reconstruction algorithm based on nuclear norm minimization is proposed to recover the block-sparse matrix. In addition, a weight matrix based on the reduced dimensional noncircular Capon (RD NC-Capon) spectrum is designed to reweight the nuclear norm minimization for enhancing the sparsity of solution. Finally, the DOA is estimated from the non-zero blocks of the reconstructed matrix. Due to exploiting the extended array aperture and block-sparse information, the proposed method provides superior DOA estimation performance and higher angular resolution. Furthermore, the proposed method has a low sensitivity to the priori information on the number of sources. Simulation results are presented to verify the effectiveness and advantages of the proposed method. HighlightsThe DOA estimation problem for noncircular sources in colocated MIMO radar is considered.A RD NC-Capon spectrum is formulation to design the weight matrix via solving a quadratic optimization problem.A reweighted nuclear norm minimization framework is proposed for DOA estimation.The proposed method provides superior DOA estimation performance and higher angular resolution due to exploiting the extended array aperture and block-sparse information.The proposed method has a low sensitivity to the priori information of the number of sources.

A gradient-based approach to optimization of compressed sensing systems

A new framework to incoherent dictionary design is proposed and a gradient descent-based algorithm is derived to obtain the optimal dictionary.Based on a parametric technique, a gradient descent-based algorithm is derived to design the robust sensing matrix.The expression of derivative for each of the two algorithms is explicitly derived.The validity of the proposed approaches is confirmed with experiments carried out using synthetic data and real images. This paper deals with a gradient-based approach to optimizing compressed sensing systems. An alternative measure is proposed for incoherent sparsifying dictionary design. An iterative procedure is developed for searching the optimal dictionary, in which the dictionary update is executed using a gradient descent-based algorithm. The optimal sensing matrix problem is investigated in terms of minimizing HGF2, where H is the target of Gram matrix of desired coherence property. Unlike the traditional approaches, G is taken as the Gram of the normalized equivalent dictionary of the system, ensuring that HGF2 has the designated physical meaning. A gradient descent-based algorithm is derived for solving the optimal sensing matrix problem. The validity of the proposed approaches is confirmed with experiments carried out using synthetic data and real images.

Uncertainty principle of the second-order LPFT

This paper studies the uncertainty principle of the second-order local polynomial Fourier transform (LPFT). It shows that the uncertainty product of the LPFT is time-independent when the Gaussian window is used to segment the signal. Meanwhile when the extra parameter is estimated correctly, the uncertainty product of the LPFT becomes a constant. Compared to the short-time Fourier transform and the Wigner-Ville distribution, it shows that the LPFT provides a better resolution of signal presentation in the time-frequency domain. Simulation for a speech signal is also given to confirm that the LPFT is capable of revealing more spectrum details when the frequency contents change dramatically.

An approach to classification and under-sampling of the interfering wireless signals

Abstract Classification of interfering signals that belong to different wireless standards is important topic in wireless communications. In this paper, we propose a procedure for separation and classification of wireless signals belonging to the Bluetooth and to the IEEE 802.11b standards. These signals operate in the same frequency band and may interfere with each other. The procedure is made of a few steps. In the first step, the separation of signal components is done using the eigenvalue decomposition approach. The second stage is based on the compressive sensing approach, used to reduce the number of transmitted samples. A suitable transform domain is chosen for each separated component using l 1 -norm as a measure of sparsity. Since the Bluetooth signals are less sparse compared to the IEEE 802.11b signals, after choosing sparse domain, additional sparsification needs to performed to further enhance the sparsity. In the last step of the procedure, the classification is performed by observing the time-frequency characteristics of the reconstructed separated components. The theory is proved by the experimental results.

Reconstruction and classification of wireless signals based on compressive sensing approach

The procedure for the classification and reconstruction of randomly under-sampled signals transmitted through the communication channel, is proposed in this paper. The focus of this work is on the wireless communication signals that operate in the same frequency band and may interfere with each other. In the first stage, the separation of signal components is done by applying the concept of eigenvalue decomposition. Next, the compressive sensing approach is used to reduce the number of transmitted samples and to provide accurate signal reconstruction upon transmission. In the last step, the classification is done by observing the time-frequency characteristics of reconstructed separated components. The theory is proved by the experimental results.

Systematic analysis of uncertainty principles of the local polynomial Fourier transform

In this paper, we show that there are a number of uncertainty principles for the local polynomial Fourier transform and local polynomial periodogram. Systematic analysis of uncertainty principles is given, explicit expressions of the uncertainty relations are derived, and an example using the chirp signal and the Gaussian window function is given to verify the expressions.