Adaptive time–frequency analysis based on autoregressive modeling
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 . 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]where n=0, 1, 2…denotes the discrete-time sampling and u[n] is a stationary white noise process with zero mean and variance . The time-varying AR parameters 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)
An AR spectral analysis of non-stationary signals
Signal Processing
(1986)- 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...
- 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) Instantaneous mean frequency estimation using adaptive time–frequency distributions
2001 Canadian Conference on Electrical and Computer Engineering
(2001)- 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) Time-dependent ARMA modeling of nonstationary signals
IEEE Transactions on Acoustics, Speech and Signal Processing
(1983)- et al.
Joint time–frequency analysis
IEEE Signal Processing Magazine
(1999) Lattice filters for adaptive processing
Proceedings of the IEEE
(1982)Lattice methods for spectral estimation
Proceedings of the IEEE
(1982)