Elsevier

Signal Processing

Volume 89, Issue 5, May 2009, Pages 791-806
Signal Processing

Keystone transformation of the Wigner–Ville distribution for analysis of multicomponent LFM signals

https://doi.org/10.1016/j.sigpro.2008.10.029Get rights and content

Abstract

Signal detection and parameter estimation for mono- and multicomponent linear frequency modulation (LFM) signals are studied by using the keystone transform of the Wigner–Ville distribution (WVD). The keystone-Wigner transform (KWT) introduces a weight factor containing a range of chirp rate into the time-lag instantaneous autocorrelation function and uses a one-dimensional (1-D) interpolation of the phase which we call keystone formatting. The proposed processing eliminates the effects of linear frequency migration (i.e., the frequency linearly varies along the time axis) to all the signal components even if their chirp rates are unknown. The Fourier transform (FFT) over the time variable to results of the KWT make the power of multicomponent LFM signal concentrated as the locations corresponding to their parameters. Furthermore, the KWT can be efficiently implemented using only complex multiplications and FFT based on the scaling principle instead of interpolating. The computational complexity of KWT is O(4N2log2N). Performance analysis is presented by using the perturbation method and verified by simulation results. Finally, the effectiveness of the KWT is validated by a real application example.

Introduction

Linear frequency modulation (LFM) signals are widely used in information system such as sonar, radar, and communications. To detect and estimate LFM signals is an important problem in these systems. For a long time, various methods based on maximum likelihood (ML) estimator are the predominant solutions to this problem. Most of these methods can be ascribed to multivariable optimization algorithm and usually computationally demanding in implementation [1], [2]. Though the algorithms in [3], [4] are much more efficient in computation, not applicable to multicomponent signals.

In the last two decades, there has been an intensive research effort to study time–frequency distributions (TFDs) in order to characterize and analyze nonstationary signals. Signal detection and parameter estimation are implemented using the linear (e.g., short-time Fourier transform) and quadratic transforms (e.g., the Wigner–Ville distribution). The Wigner–Ville distribution (WVD) is optimal, in the sense of maximum energy concentration about the instantaneous frequency (IF), for LFM signals [5]. In addition, it provides a high resolution representation in time and in frequency. However, due to interactions among signal components, representations of multicomponent signals contain interference (cross terms) in different regions of the time-frequency (TF) plane. Cross terms are sometimes a hindrance to interpretation since they carry redundant information and may obscure primary features of the signals. In order to suppress the cross terms, some multicomponent LFM signal analysis methods based on the Wigner–Hough transform (WHT) or the Radon–Wigner transform (RWT) are proposed in [6], [7]. Another TF analysis method, the fractional Fourier transform (FRFT), was developed in processing of LFM signals [8], which is implemented with the following operations, rotation of the signal coordinates around the origin and then integration along a straight line in TF plane.

In this paper, a new algorithm is proposed for multicomponent LFM signal analysis based on the WVD. It introduces a weight factor containing a range of chirp rate into the time-lag instantaneous autocorrelation function and interpolates from a hexagonal grid to a keystone-hexagonal grid. Then the effects of linear frequency migration, i.e., the frequency linearly varies along the time axis, for all LFM components in the TF plane can be eliminated through the keystone remapping. The LFM signals are transformed to horizontal lines in a new TF plane, where the time axis represents the new time variable after interpolation and the frequency axis still represents the frequency variable. This processing is called the keystone-Wigner transform (KWT) which is different from the rotation operators of FRFT. With a FFT over time variable, sharp peaks appear at the locations corresponding to the centroid frequencies and chirp rates (CFCRs) of the LFM signals. The multicomponent LFM signal can be detected with these peaks, and its parameters can be estimated the peak locations.

The paper is organized as follows. In Section 2, we derive the keystone transform of both the continuous-time WVD and the discrete-time WVD. In Section 3, we present the details of implementation including the choice of weight factors, keystone mapping and resolution. In Section 4, we describe the detection and parameter estimation problems based on the CFCR plane. In Section 5, performance analysis is provided and verified by simulation results. In Section 6, an application example is used to demonstrate the effectiveness of the KWT. Section 7 concludes the paper.

Section snippets

Continuous KWT

The signal to be analyzed is modeled by a multicomponent LFM signal as given byx(t)=i=0K-1xi(t)=i=0K-1Aiej[2πfit+πγit2]where t(-,+), K is the number of components present in the signal, Ai is the constant real amplitude, fi represents the centroid frequency and γi is the chirp rate. The continuous-time symmetric instantaneous autocorrelation function (SIAF) of signal x(t) is defined asRxC(τ,t)=x(t+τ2)x*(t-τ2)Therefore, the SIAF of signal x(t) in Eq. (1) has the following expressionRxC(τ,t)=

Implementation

In this section we will present some practical details of the keystone-Wigner implementation including the choice of the weight factor, keystone transform and resolution.

Detection and parameter estimation

In this section, we provide the algorithm for detecting and estimating the parameters of multicomponent LFM signals embedded in additive white Gaussian noise (AWGN). For clarity, we start with one component. Later on, we will generalize the procedure to the multicomponent case.

Performance analysis

Since the estimation algorithm is iterative, it inevitably suffers from error propagation phenomena. Errors in the weight factor determined by the last estimate of chirp rate propagate to the next estimate of chirp rate. The estimate of the initial chirp rate based on the SKWT is thus the most critical step and the SNR threshold, below which the algorithm looses its reliability, is essentially determined by the threshold related to the correct estimate of the initial chirp rate. For this

Application

In this section, we will present a KWT application to the inverse synthetic aperture radar (ISAR) imaging for maneuvering target. The processing is performed with raw radar data. The ISAR imaging techniques of maneuvering targets were introduced in many references such as [23], [24]. The target in this section is a Yake-42 airplane with the size of 36.3 m/34.8 m/9.8 m in the length/width/height respectively. The radar works in the C band, with a bandwidth of 400 MHz and a PRF of 400 Hz. A total of

Conclusions

In this paper, we have proposed a method for the analysis of multicomponent LFM signals embedded in AWGN based on the KWT. In the implementation of the method, an iterative estimation of the chirp rate is carried out by the SKWT which is the simple variants of the KWT. In order to suppress the interference induced by strong signals on the detection and estimation of weak signals, the Clean method is used, which provides significant improvements in terms of detection performance and estimation

Acknowledgments

This work was supported by the National Natural Science Foundation of China under grant 60890072 and 60725103. The authors wish to thank Dr. Junhai Su and the anonymous reviewers for their very helpful suggestions.

References (24)

  • P.M. Djuric et al.

    Parameter estimation of chirp signal

    IEEE Trans. ASSP

    (December 1990)
  • B. Boashash

    Estimating and interpreting the instantaneous frequency of a signal

    Proc. IEEE

    (April 1992)
  • T.J. Abatzoglou, Fast maximum likelihood joint estimation of frequency and frequency rate, in: Proceedings of IEEE...
  • S. Peleg et al.

    Linear FM signal parameter estimation from discrete-time observations

    IEEE Trans. Aerosp. Electron. Syst.

    (July 1991)
  • P. Rao et al.

    Estimation of instantaneous frequency using the discrete Wigner distribution

    Electron. Lett.

    (February 1990)
  • S. Barbarossa

    Analysis of multicomponent LFM signals by a combined Wigner–Hough transform

    IEEE Trans. Signal Process.

    (June 1995)
  • J.C. Wood et al.

    Radon transformation of time–frequency distributions for analysis of multicomponent signals

    IEEE Trans. Signal Process.

    (November 1994)
  • L. Qi et al.

    Detection and parameter estimation of multicomponent LFM signal based on the fractional Fourier transform

    Sci. China (ser. F)

    (2004)
  • R.P. Perry et al.

    SAR Imaging of moving targets

    IEEE Trans. Aerosp. Electron. Syst.

    (January 1999)
  • M. Xing et al.

    High resolution ISAR imaging of high speed moving targets

    IEE Proc.—Radar Sonar Navig.

    (April 2005)
  • T.A.C.M. Claasen et al.

    The Wigner distribution—Part II: discrete-time signals

    Philips J. Res.

    (1980)
  • J. Jeong et al.

    Alias-free generalized discrete-time time–frequency distributions

    IEEE Trans. Signal Process.

    (November 1992)
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