IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 52, NO. 8, AUGUST 2005 1485 Communications______________________________________________________________________ Adaptive Modeling and Spectral Estimation of Nonstationary Biomedical Signals Based on Kalman Filtering Mateo Aboy*, Oscar W. Márquez, James McNames, Roberto Hornero, Tran Trong, and Brahm Goldstein Abstract—Wedescribeanalgorithmtoestimatetheinstantaneouspower spectraldensity(PSD)ofnonstationarysignals.Thealgorithmisbasedona dualKalmanfilterthatadaptivelygeneratesanestimateoftheautoregres- sivemodelparametersateachtimeinstant.Thealgorithmexhibitssuperior PSDtrackingperformanceinnonstationarysignalsthanclassicalnonpara- metric methodologies, and does not assume local stationarity of the data. Furthermore,itprovidesbettertime-frequencyresolution,andisrobustto model mismatches. We demonstrate its usefulness by a sample application involving PSD estimation of intracranial pressure signals (ICP) from pa- tients with traumatic brain injury (TBI). Index Terms—Intracranial pressure, Kalman filter, linear models, spec- tral estimation, traumatic brain injury. I. INTRODUCTION Currently, power spectral density (PSD) estimation of physiologic signals is performed predominantly using classical techniques based on the fast Fourier transform (FFT). Nonparametric methods such as the periodogram and its improvements (i.e., Barlett’s, Welch’s, and Blackman-Tukey’s methodologies [1]–[4]) are based on the idea of es- timating the autocorrelation sequence of a random process from mea- sured data, and then taking the FFT to obtain an estimate of the power spectrum. The main two advantages of these techniques are their com- putational efficiency, due to the numerical efficiency of the FFT algo- rithm, and that they do not make any assumptions about the process except for its stationarity. This makes them the methodology of choice, particularly in situations where long data records need to be analyzed and there is no clear model for the process. Furthermore, the availability of long data records enables one to improve their statistical properties by averaging or smoothing. However, these techniques have some lim- itations. They require stationarity of the segments studied, do not work Manuscript received on March 1, 2004; revised January 2, 2005. A previous version of this paper was presented at the IEEE-EMBS 2004 conference. This work was supported in part by the Northwest Health Foundation, in part by the Doernbecher Children’s Hospital Foundation, and in part by the Thrasher Research Fund. Asterisk indicates corresponding author. *M. Aboy is with the Department of Electronics Engineering Technology, Oregon Institute of Technology, Portland, OR 97201 USA. He is also with the Biomedical Signal Processing Laboratory, Department of Electrical and Com- puter Engineering, Portland State University, Portland, OR 97201 USA (e-mail: mateoaboy@ieee.org). O. W. Márquez is with the Signal Theory and Communications Department, ETSI-Telecomunicación, University of Vigo, 36310 Vigo, Spain, EU. J. McNames is with the Biomedical Signal Processing Laboratory, Depart- ment of Electrical and Computer Engineering, Portland State University, Port- land, OR 97201 USA. R. Hornero is with the Department of Signal Theory and Communications, ETSI-Telecomunicación, University of Valladolid, 47011Valladolid, Spain, EU. T. Trong is with the Department of Biomedical Engineering, OGI School of Science and Engineering, Oregon Health and Science University, Portland, OR 97206 USA. B. Goldstein is with the Complex Systems Laboratory, Department of Pedi- atrics, Oregon Health and Science University, Portland, OR 97201 USA. Digital Object Identifier 10.1109/TBME.2005.851465 well for short data records, and have limited frequency resolution. Since physiologic signals are nonstationary in nature, these techniques are applied following the methodology of the short-time Fourier transform (STFT), where nonparametric methods are applied to short overlapping segments which are assumed to be stationary. This approach has also its limitations. It imposes a piecewise stationary model on the data and, since local stationarity requires the analysis segments to be short in du- ration, they have limited time-frequency resolution. Time-frequency resolution can be improved by using parametric methods of PSD estimation. The parametric approach is based on modeling the signal under analysis as a realization of a particular stochastic process and estimating the model parameters from its samples. In the absence of a priori knowledge about how the process is generated, parametric PSD is generally performed assuming an autoregressive (AR) model [4]. This is a popular assumption for several reasons: 1) many natural signals such as speech, music or seismic signals have an underlying autoregressive structure; 2) in general, any signal—not necessarily AR in nature—can be modeled as an AR process if a sufficiently large model order is selected; 3) the all-pole structure of AR enables for good spectral peak matching, which makes it a good model candidate for situations where we are more interested in the spectral peaks than valleys; and 4) estimation of the model parameters involves the solution of a linear system of equations, which can be solved efficiently. Even though parametric PSD can improve the frequency resolution, the current techniques for PSD estimation based on AR models (i.e., autocorrelation, co- variance, modified convariance, and Burg’s methods [5], [6]) assume stationarity. To analyze nonstationary signals they must also assume the signal is locally piecewise stationary. We describe a methodology to estimate the time-varying AR model parameters of nonstationary signals using an adaptive Kalman filter. This methodology produces instantaneous estimates of PSD, improved time-frequency resolution, and enables for nonstationary spectral anal- ysis in situations where data records are too short and the local sta- tionary model does not work well. The reliability of the algorithm was tested with synthetic data generated from different models (AR, MA, ARMA, and harmonic), and with real data from physiologic pressure signals. Following the description of this methodology, we demonstrate its usefulness by a sample application involving PSD estimation of in- tracranial pressure signals (ICP) from patients with traumatic brain in- jury (TBI). II. METHODS The adaptive Kalman filter algorithm we propose for instantaneous PSD estimation assumes an underlying autoregressive structure of the data. We chose an underlying AR model structure because of its in- trinsic generality and peak matching capabilities. These are important properties for the analysis of physiologic signals, since we are usually more interested in estimating the frequency at which the formant fre- quencies (peaks) occur than the valleys. Starting from this assumption, we modeled a given physiologic signal with a recursion of the form (1) where is the physiologic signal under analysis at instant , are the model parameters, are delayed 0018-9294/$20.00 © 2005 IEEE