Elsevier

Signal Processing

Volume 91, Issue 4, April 2011, Pages 740-749
Signal Processing

Adaptive time–frequency analysis based on autoregressive modeling

https://doi.org/10.1016/j.sigpro.2010.07.020Get rights and content

Abstract

A new adaptive method for discrete time–frequency analysis based on autoregressive (AR) modeling is introduced. The performance of AR modeling often depends upon a good selection of the model order. The predictive least squares (PLS) principle of Rissanen was found to be a good criterion for model order estimation in stationary processes. This paper presents a modified formulation of the PLS criterion suitable for non-stationary processes. Efficient lattice filters based on the covariance assumption are used to estimate the model parameters of all model orders less than some maximum order M. It is shown that the resulting complexity is no larger than M. The modified PLS criterion allows the model order to adapt to non-stationary processes and, in turn, compute adaptive AR based time–frequency representations (TFRs). Examples of time–frequency analyses for synthetic and bio-acoustical signals are provided as well as comparisons to classical time–frequency representations.

Graphical abstract

Research highlights

►Autoregressive method to estimate time-varying spectra of non-stationary signals. ►Adaptation of autoregressive model order using fixed forgetting factor. ►Implementation of efficient recursive algorithm for time-frequency analysis. ►Experiments show high dynamic range while preserving time-frequency concentration.

Introduction

In recent years, there has been increasing interest in developing adaptive schemes for time–frequency analysis [1], [2], [3], [4], [5], [6], [7] and, in particular, time–frequency representations (TFRs) with an underlying parametric model [8], [9], [10], [11], [12]. For stationary signals, the parametric AR modeling approach yields high-resolution power spectral density (PSD) estimates even for short data records (see [13]). Furthermore, researchers have developed very efficient adaptive time- and order-recursive algorithms to compute the AR coefficients of non-stationary signals [8], [9], [10], [11].

Based on an AR model, Griffiths [14] derived an adaptive gradient descent algorithm for computing the instantaneous frequency of non-stationary signals. Hodgkiss and Presley [15] investigated the behavior of the gradient transversal filter, the gradient lattice and the least squares lattice when used to track time-varying sinusoidal components. The authors demonstrated that the recursive least squares approach with a forgetting factor performs at least as well as the gradient methods. In [16], a recursive least squares algorithm with a sliding window is presented to estimate the PSD of non-stationary signals. However, the author did not implement criteria to estimate the order, ‘because they considerably increase the computation time’. Cho et al. [17] introduced a variable forgetting factor, which is adapted to the time-varying signal by an extended prediction error criterion accounting for the non-stationarity of the signal. Using a Kalman filter to obtain the time-varying AR coefficients, the authors of [18] propose an equivalent method to the least squares lattice in [15] for computing an instantaneous power spectral density.

Jones and Baraniuk [19] presented an adaptive algorithm for TFR computation of non-stationary signals that is based on a signal-dependent radially Gaussian kernel. The kernel of their adaptive optimal-kernel (AOK) TFR adapts to the signal over time utilizing a short-time ambiguity function. Another adaptive algorithm for TFR computation of non-stationary signals, which is not based on AR modeling, can be found in [20] and in the review [21] of such algorithms for the quantitative analysis of electroencephalograms (EEGs).

All of the AR based methods cited above do not provide an estimate for the model order. Despite this, the performance of a TFR with an underlying AR model often depends upon a good selection of the model order. In this paper, we further discuss the modified predictive least squares criterion, which we introduced in [22], to determine a good estimate for the model order. We then compute the TFR of non-stationary signals using lattice filters based on the covariance assumption.

The paper is organized as follows. In Section 2, we review the time-varying autoregressive model and explain the importance of the forgetting factor λ. Section 3 addresses the model order selection and presents the modified PLS principle. In Section 4, we discuss the implementation of the underlying recursive algorithm and present a flowchart of the overall adaptive time–frequency analysis based on autoregressive modeling. The simulation results presented in Section 5 for synthetic and real-world signals confirm the fitness of the adaptive AR based algorithm for TFR computation. The paper concludes in Section 6 with a summary of the main results and directions for future research.

Section snippets

Time-varying autoregressive modeling

A discrete-time non-stationary process can be modeled by a time-varying autoregressive (TVAR) process of model order m represented by the linear difference equation [23]x[n]=k=1mamk[n]x[nk]+u[n],where n=0, 1, 2…denotes the discrete-time sampling and u[n] is a stationary white noise process with zero mean and variance σu2. The time-varying AR parameters amk[n],k=1,2,,m are commonly found as the least squares solution of predefined, accumulated prediction errors. Assuming the AR parameters

Selection of model order

In addition to the estimation of the AR model parameters, the selection of an appropriate model order m in (1) is an important factor in time series modeling. The model order determines the amount of memory required to represent the process. Higher model orders may capture more details of the process but essentially require more data to accurately estimate its parameters.

A convincing criterion for model order estimation has been proposed by Rissanen [26], which is referred to as the predictive

Algorithmic implementation

The application of the MPLS criterion requires the computation of the a priori prediction error of all model orders up to a predefined maximum order M. Although a large class of least squares (LS) algorithms can be used to solve this task, recursive least squares (RLS) algorithms based on lattice structures offer a very efficient implementation. One RLS algorithm from [10] is considered in this paper, namely the growing memory covariance least squares (GMCLS) lattice algorithm. To adapt to

Experimental results

To demonstrate the desirability of the GMCLS algorithm for improving the time–frequency analysis of time-varying signals, we conduct simulations on three different synthetic signals and one bio-acoustical signal. The resulting TFRs are then compared with the Wigner distribution (WD), the spectrogram (squared magnitude of the STFT), and the adaptive optimal-kernel (AOK) TFR introduced by Jones and Baraniuk [19].

Conclusion

In this paper, an AR based method to estimate time-varying spectra of non-stationary signals is presented utilizing a modified PLS criterion to estimate the time-varying model order. It has been shown through simulations that the MPLS criterion is a valid means to select the model order for non-stationary signals. We implemented the very efficient GMCLS lattice algorithm, enabling us to compute TFRs even for long signals. The objective of these time-varying AR TFRs is to identify the properties

Acknowledgment

The authors would like to thank Stephan Sand for his contribution to the research presented in this paper.

References (31)

  • N. Martin

    An AR spectral analysis of non-stationary signals

    Signal Processing

    (1986)
  • J. Gosme et al.

    Adaptive diffusion as a versatile tool for time–frequency and time-scale representations processing: a review

    IEEE Transactions on Signal Processing

    (2005)
  • S.-C. Chan, Z. Zhang, Adaptive window selection and smoothing of lomb periodogram for time–frequency analysis of time...
  • S. Pola et al.

    Estimation of the power spectral density in nonstationary cardiovascular time series: assessing the role of the time–frequency representations (TFR)

    IEEE Transactions on Biomedical Engineering

    (1996)
  • S. Krishnan

    Instantaneous mean frequency estimation using adaptive time–frequency distributions

    2001 Canadian Conference on Electrical and Computer Engineering

    (2001)
  • R. McDonald et al.

    Prediction of the quality ratings of tracheoespohageal speech using adaptive time–frequency representations

    2008 Canadian Conference on Electrical and Computer Engineering (CCECE 2008)

    (2008)
  • Y. Grenier

    Time-dependent ARMA modeling of nonstationary signals

    IEEE Transactions on Acoustics, Speech and Signal Processing

    (1983)
  • S. Qian et al.

    Joint time–frequency analysis

    IEEE Signal Processing Magazine

    (1999)
  • B. Friedlander

    Lattice filters for adaptive processing

    Proceedings of the IEEE

    (1982)
  • B. Friedlander

    Lattice methods for spectral estimation

    Proceedings of the IEEE

    (1982)
  • M. Honig et al.

    Adaptive Filters: Structures, Algorithms and Applications

    (1984)
  • J. Proakis et al.

    Advanced Digital Signal Processing

    (1992)
  • Y. Abramovich et al.

    Time-varying autoregressive (TVAR) adaptive order and spectrum estimation

    Conference Record of the 39th Asilomar Conference on Signals, Systems and Computers

    (2005)
  • S. Kay

    Modern Spectral Estimation: Theory and Application

    (1999)
  • L. Griffiths

    Rapid measurement of digital instantaneous frequency

    IEEE Transactions on Acoustics, Speech and Signal Processing

    (1975)
  • Cited by (0)

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