Refining the ambiguity domain characteristics of non-stationary signals for improved time-frequency analysis: Test case of multidirectional and multicomponent piecewise LFM and HFM signals

Abstract This paper aims at providing a more accurate description of the ambiguity domain characteristics of a piecewise multicomponent non-stationary signals with focus on piece-wise linear frequency modulated (LFM) (PW-LFM) signal and a mixed LFM and hyperbolic FM (HFM). The main motivation comes from the observed PW-LFM nature of several real life signals. It is essential that the characteristics of these types of signals be taken into account for the design of high resolution Time–Frequency Distributions (TFDs) and therefore to improve the application and interpretation of time–frequency signal analysis and processing. In this paper the ambiguity function (AF) of a general PW-LFM signal is derived exactly and then analyzed to deduce important properties. The precise location and behavior of both auto-terms and cross-terms of a general piecewise LFM signal can be deduced from its AF. Numerical simulations using different types of test signals confirm the analytical derivations. An extension of the PW-LFM test signal is also presented by using HFM signals. For such signals, the exact analytical expression of auto-terms is given as well as the expression of cross-terms between HFM and LFM. The results obtained can be used in future studies to design more advanced quadratic time–frequency distributions (QTFDs) that exhibit improved properties in terms of resolution and accuracy.

[1]  Boualem Boashash,et al.  Time-Frequency Distributions Based on Compact Support Kernels: Properties and Performance Evaluation , 2012, IEEE Transactions on Signal Processing.

[2]  Boualem Boashash,et al.  Estimating and interpreting the instantaneous frequency of a signal. I. Fundamentals , 1992, Proc. IEEE.

[3]  E. Kummer.  De integralibus quibusdam definitis et seriebus infinitis. , 1837 .

[4]  Massimiliano Martone,et al.  A multicarrier system based on the fractional Fourier transform for time-frequency-selective channels , 2001, IEEE Trans. Commun..

[5]  Boualem Boashash,et al.  Time-Frequency Signal Analysis and Processing: A Comprehensive Reference , 2015 .

[6]  Andrew K. Chan,et al.  Linear frequency-modulated signal detection using Radon-ambiguity transform , 1998, IEEE Trans. Signal Process..

[7]  Y. Katznelson.  An Introduction to Harmonic Analysis: Interpolation of Linear Operators , 1968 .

[8]  A. Zayed.  A convolution and product theorem for the fractional Fourier transform , 1998, IEEE Signal Process. Lett..

[9]  John M. O'Toole,et al.  Time-Frequency Processing of Nonstationary Signals: Advanced TFD Design to Aid Diagnosis with Highlights from Medical Applications , 2013, IEEE Signal Processing Magazine.

[10]  F. Perez-Martinez,et al.  Helicopter Classification with a High Resolution LFMCW Radar , 2009, IEEE Transactions on Aerospace and Electronic Systems.

[11]  Boualem Boashash,et al.  Robust multisensor time-frequency signal processing: A tutorial review with illustrations of performance enhancement in selected application areas , 2017, Digit. Signal Process..

[12]  Fulei Chu,et al.  Recent advances in time–frequency analysis methods for machinery fault diagnosis: A review with application examples , 2013 .

[13]  Yang Yang,et al.  Application of Parameterized Time-Frequency Analysis on Multicomponent Frequency Modulated Signals , 2014, IEEE Transactions on Instrumentation and Measurement.

[14]  Boualem Boashash,et al.  A robust high-resolution time-frequency representation based on the local optimization of the short-time fractional Fourier transform , 2017, Digit. Signal Process..

[15]  Boualem Boashash,et al.  An improved time-frequency noise reduction method using a psycho-acoustic Mel model , 2018, Digit. Signal Process..

[16]  Samir Ouelha,et al.  Efficient software platform TFSAP 7.1 and Matlab package to compute Time-Frequency Distributions and related Time-Scale methods with extraction of signal characteristics , 2017, SoftwareX.

[17]  A. Enis Çetin,et al.  Cross-term-free time–frequency distribution reconstruction via lifted projections , 2015, IEEE Transactions on Aerospace and Electronic Systems.

[18]  Qiong Wu,et al.  Coprime sampling for nonstationary signal in radar signal processing , 2013, EURASIP J. Wirel. Commun. Netw..

[19]  Boualem Boashash,et al.  Performance evaluation of time-frequency image feature sets for improved classification and analysis of non-stationary signals: Application to newborn EEG seizure detection , 2017, Knowl. Based Syst..

[20]  Amine Naït-Ali,et al.  Generalized multi-directional discrete Radon transform , 2013, Signal Process..

[21]  Bengt Hansson,et al.  Automated analysis of song structure in complex birdsongs , 2016, Animal Behaviour.

[22]  Ljubisa Stankovic,et al.  A method for time-frequency analysis , 1994, IEEE Trans. Signal Process..

[23]  John M. O'Toole,et al.  Time-Frequency Detection of Slowly Varying Periodic Signals with Harmonics: Methods and Performance Evaluation , 2011, EURASIP J. Adv. Signal Process..

[24]  Abdul Rahim Abdullah,et al.  Power Quality Analysis Using Bilinear Time-Frequency Distributions , 2011 .

[25]  Jin Jiang,et al.  Time-frequency feature representation using energy concentration: An overview of recent advances , 2009, Digit. Signal Process..

[26]  B. Armstrong.  Spectrum line profiles: The Voigt function , 1967 .

[27]  K. Roach.  Meijer G function representations , 1997, ISSAC.

[28]  Yang Yang,et al.  Chirplet Path Fusion for the Analysis of Time-Varying Frequency-Modulated Signals , 2017, IEEE Transactions on Industrial Electronics.

[29]  Boualem Boashash,et al.  Designing high-resolution time-frequency and time-scale distributions for the analysis and classification of non-stationary signals: a tutorial review with a comparison of features performance , 2017, Digit. Signal Process..

[30]  Boualem Boashash,et al.  An Improved Design of High-Resolution Quadratic Time–Frequency Distributions for the Analysis of Nonstationary Multicomponent Signals Using Directional Compact Kernels , 2017, IEEE Transactions on Signal Processing.

[31]  Ivan Volaric,et al.  A compressive sensing based method for cross-terms suppression in the time-frequency plane , 2015, 2015 IEEE 15th International Conference on Bioinformatics and Bioengineering (BIBE).

[32]  Zhang Zhen-song,et al.  Radar signal processing based on a new cross-term reduced Wigner-Ville distribution , 2013 .

[33]  K.G. Foote,et al.  Transmit-Signal Design and Processing Strategies for Sonar Target Phase Measurement , 2007, IEEE Journal of Selected Topics in Signal Processing.

[34]  Douglas L. Jones,et al.  A signal-dependent time-frequency representation: fast algorithm for optimal kernel design , 1994, IEEE Trans. Signal Process..

[35]  Boualem Boashash,et al.  Automatic signal abnormality detection using time-frequency features and machine learning: A newborn EEG seizure case study , 2016, Knowl. Based Syst..

[36]  Matthias Patzold,et al.  The Wigner Distribution of Sum-of-Cissoids and Sum-of-Chirps Processes for the Modelling of Stationary and Non-Stationary Mobile Channels , 2016, 2016 IEEE 83rd Vehicular Technology Conference (VTC Spring).