Accurate Identification of Periodic Oscillations Buried in White or Colored Noise Using Fast Orthogonal Search

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1 622 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 48, NO. 6, JUNE 2001 Accurate Identification of Periodic Oscillations Buried in White or Colored Noise Using Fast Orthogonal Search Ki H. Chon, Member, IEEE Abstract We use a previously introduced fast orthogonal search algorithm to detect sinusoidal frequency components buried in either white or colored noise. We show that the method outperforms the correlogram, modified covariance autoregressive (MODCOVAR) and multiple-signal classification (MUSIC) methods. Fast orthogonal search method achieves accurate detection of sinusoids even with signal-to-noise ratios as low as 10 db, and is superior at detecting sinusoids buried in 1 noise. Since the utilized method accurately detects sinusoids even under colored noise, it can be used to extract a 1 noise process observed in physiological signals such as heart rate and renal blood pressure and flow data. Index Terms 1 noise, fast orthogonal search, heart rate variability, MUSIC, power spectral density. I. INTRODUCTION IF ONE computes the power spectral density of heart rate, a so-called spectral component is prevalent and is one of the most dominant features among the periodic frequency components identified. One of the first observations of a spectrum in heart rate was reported by Kobayashi and Musha [4], who found that the power spectrum amplitudes are inversely proportional to their frequency [4]. A process is usually computed from long-term data recordings ( 1 h), is observed in very low-frequency ranges ( Hz), and is often the dominant frequency content in the computed spectrum. A similar process is also observed in renal blood pressure [10] and myocardial blood flow [3] data. In addition, processes are found in many other types of biological and nonbiological systems [12], [14]. Although the exact mechanism underlying, and the functional need for, a process is unclear, it is currently speculated that it may represent control of humoral and temperature regulations. This spectral process is also widely considered a noise source since it tampers with identification of the low ( Hz) and high-frequency ( 0.25 Hz) components of hear rate variability (HRV). The former are linked to both sympathetic and parasympathetic nervous control, while the latter are linked solely to parasympathetic nervous control. High spectral power associated with noise tends to mask the smaller spectral powers of the low and high frequencies Manuscript received December 23, 1999; revised February 23, This work was supported in part by a grant from the Whitaker Foundation. The author is with the Department of Electrical Engineering, City College of New York, Steinman Hall, Room 677, Convent@138th Street, New York, NY USA. Publisher Item Identifier S (01) and often impedes identification of low- and high-frequency regions of interest. noise also hampers identification of the tubuloglomerular feedback HZ) and myogenic ( Hz) mechanisms in renal blood pressure. A noise process is not the only noise source impeding the identification of frequency peaks in a spectrum, as other colored and also white noise sources may introduce additional problems. A noise process is not only possible in long-term data recordings but can also be observed in some short-term data [11], [15], [16]. Investigation of autonomic control of HRV data as well as renal autoregulation often involves analysis of short-term data, in which noise creates the same problems as it does in long-term data. Many novel techniques have been proposed for removing a noise process or obtaining only the periodic oscillatory components in a signal containing noise [3], [11], [14] [16]. Some of the more noteworthy methods applied are coarse-grain spectral analysis [15], a higher order cumulant-based approach [11], as well as wavelet transforms [14], [16]. In this paper, we used the fast orthogonal search (FOS) algorithm, originally proposed by Korenberg [5] [8] which is also robust in removing a noise process. In addition, the method presented is effective in identifying sinusoids buried in noise of various forms (e.g., white or colored noise, including noise). We provide simulation examples to compare the effectiveness of the FOS to that of the correlogram spectral method, the modified covariance autoregressive (MODCOVAR) method, and the multiple-signal classification (MUSIC) algorithm in identifying sinusoids buried in various types of noise. The MUSIC approach is chosen since it is one of the most powerful techniques available for harmonic analysis [1], [9]. Autoregressive (AR) model-based spectral methods, including the MUSIC approach, work well if the noise is white, but they suffer when the noise is colored (including noise). Nonetheless, they are currently among the best methods available to detect harmonics [9]. Experimental data collected is often contaminated by colored noise rather than purely white noise. The simulation examples will concentrate on short-term data recordings rather than long-term data, even though the algorithm proposed is equally effective for both cases. II. DESCRIPTION OF FOS ALGORITHM The algorithm utilized is the FOS, a method first introduced by Korenberg [5], [6]. FOS is a general-purpose modeling technique which has been applied to a number of practical problems, /01$ IEEE

2 CHON: ACCURATE IDENTIFICATION ON PERIODIC OSCILLATIONS BURIED IN WHITE OR COLORED NOISE 623 such as determination of the structure and coefficients of nonlinear difference equation models [5], [6]. For example, FOS has been used to estimate a sum of sinusoids buried in white or colored noise [1], [6] [8]. In the present paper, we use the previously developed FOS to obtain sinusoidal series. An output can be described by a polynomial or power series of the form where are the basis functions with and is the total number of basis functions. The is the noise source terms. The basis functions are the sinusoidal series such that More generally, for 1, 2, Substituting (3) into (1) provides (1) and (2) and (3) The and are the cosine and sine amplitudes, respectively. Thus, the task of the FOS algorithm is to determine (also and ) successively by searching through a set of candidate frequencies as specified according to the desired frequency resolution. The candidate frequency which reduces the mean-square error (mse) by the greatest amount is chosen. In addition, a threshold level is set so that the program no longer searches for candidate frequencies once a pre-specified number of frequencies (as determined by those frequencies which reduce the mse by the greatest amount) is reached. A description of the implementation of the FOS algorithm is provided in the Appendix as well as [5] and [6]. III. SIMULATION RESULTS We will compare the performance of the FOS approach in detecting periodic frequencies to other spectral methods: the correlogram, MODCOVAR and MUSIC spectral methods. We have selected MODCOVAR since this method is one of the most widely utilized AR spectral methods, especially in the biomedical engineering community. The MUSIC method was also chosen for comparison since this eigenvalue analysis method is one of the most powerful techniques available for detecting sinusoids buried in noise. With the MUSIC approach, selection of the correct initial model order is not as crucial as it is when using MODCOVAR, provided that an initial guess of the model order is larger than the true model order. However, based on an initial guess of the model order, proper judgment is needed to select only those significant eigenvalues from the pool of eigenvalues estimated using the MUSIC method. For further detailed implementation of the MODCOVAR and MUSIC methods see [9]. A nonmodel-based spectral analysis (4) method of Blackman and Tukey s correlogram approach was chosen since it is one of the most widely used methods in detecting periodic components in data. All of the methods to be compared to FOS rely on the use of discrete Fourier transform (DFT). Some of the well known disadvantages associated with DFT are that the fundamental frequency of the signal analyzed must be known a priori and that all frequencies are harmonically related to the fundamental frequency. In addition, if the number of data points is not equal to an integer multiple of the fundamental period, then the estimated spectrum will smear the adjacent frequency content. The FOS method avoids all of these problems associated with using DFT. In addition, the frequency resolution obtainable with FOS is not limited by the sampling rate and DFT segment size, but is limited only by the additional computational time with finer frequency resolution. Generalized harmonic analysis (GHA), largely the work of N. Wiener [13], differs from the FOS in that GHA does not search a set of candidate frequencies and has lower frequency resolution than the FOS. The first simulation example consists of the sum of two cosine components with frequencies of 0.15 and 0.4 Hz, and with unit amplitude values for both frequencies. Noise sources of any type were not added to this signal. The signal was sampled at 1 Hz for a total of 1024 samples. The autocorrelation lags was set to 512 to 512 (frequency resolution ) and Hanning window used on the autocorrelation sequence prior to performing Fourier transform. The MODCOVAR model order was set to five based on the use of the Akaike information criterion [2]. For the MUSIC method, we selected an initial guess for the model order of 30. We used only the first four largest eigenvalues to estimate the MUSIC spectrum, however, as there was a distinct separation between the first four and the remaining 26 eigenvalues. For the FOS method, the frequency resolution was set to 0.01 Hz. As the Nyquist frequency was 0.5 Hz, we set the number of candidate frequencies to be searched to 50. With the initial threshold level of terms to be selected set to five among 50 candidate frequencies, and with the preset 95% confidence interval model order search criterion, the FOS algorithm terminated after selecting only two frequencies and their respective amplitude values. Fig. 1 shows the frequencies obtained from the four methods. All four methods correctly estimated the sinusoidal frequencies. Note that MUSIC, MODCOVAR, and the correlograms power spectral density values were normalized by each respective maximal amplitude values. So far so good! To determine the robustness of the four methods, the next simulation we ran consisted of a sum of cosines of three frequencies, two of which were very close ( 0.03, 0.04, and 0.2Hz). Short data segments of 128 and 512 points were generated with Gaussian white noise added so that the signal-to-noise ratios (SNRs) were 10, 0, and 10 db for each data segment. The SNR was computed as (variance of the signal/variance of the noise). The results based on the four methods compared are shown in Tables I (512 points) and II (128 points). YES in Tables I and II indicates that all three frequencies were detected without also detecting spurious frequency peaks. Thus, if all three frequencies were detected but if other spurious peaks with magnitude as large as any of the three true frequency peaks were present,

3 624 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 48, NO. 6, JUNE 2001 Fig. 1. Comparison of MUSIC, MODCOVAR, correlogram, and FOS methods for identification of two sinusoids (f = 0.15 and f = 0.4). TABLE I COMPARISON OF THE FOUR METHODS WITH N = 512 POINTS. YES INDICATES THAT ALL THREE FREQUENCIES WERE DETECTED WITHOUT ALSO DETECTING SPURIOUS FREQUENCY PEAKS TABLE II COMPARISON OF THE FOUR METHODS WITH N = 128 POINTS. YES INDICATES THAT ALL THREE FREQUENCIES WERE DETECTED WITHOUT ALSO DETECTING SPURIOUS FREQUENCY PEAKS then a NO mark was indicated in either Table I or II. With 512 data points, MUSIC, MODCOVAR, and the correlogram correctly identified the three frequencies for SNR up to 0 db. When the data was reduced to 128 points, the three methods correctly identified the three frequencies for SNR only up to 10 db. However, we observe that for FOS, for both 128 and 512 data points, the three frequencies were correctly identified for SNR as low as 10 db. We have found that 11 db is the lowest SNR at which FOS can unequivocally provide correct detection of the three frequencies. With the initial threshold level of terms to be selected set to 10, and a 95% confidence interval model order selection criteria, the FOS initially selected five frequencies. However, since the model residual (i.e., the prediction error) did not decrease significantly after the first three frequencies were estimated, only the first three frequencies were selected. Note that the FOS method, having this criteria, is at a definite advantage when selecting the correct model order over the other three methods. The model order selected for MODCOVAR was quite high, 70, in order to resolve the two closely spaced frequency peaks ( 0.03, 0.04) when SNR 10 db for 512 and SNR 0dB for 128 data points. One consequence of the high model order used for MODCOVAR is that it produced many spurious peaks in the spectrum. Note that the frequency resolution for both MODCOVAR and MUSIC was set to Hz, and the correlogram s frequency resolution was set to Hz. We chose a lower frequency resolution for the correlogram than for the other two methods, since above Hz the correlogram detected further spurious peaks. Thus, the inability of the MODCOVAR, MUSIC, and correlogram methods to resolve closely spaced peaks is not due to insufficient frequency resolution in the spectrum plot. The estimated spectrum for 512 data points with SNR 10-dB is shown in Fig. 2. The computational time for all four methods compared took less than a second for this simulation example. The FOS is as efficient as the other methods because it employs a fast algorithm which rapidly

4 CHON: ACCURATE IDENTIFICATION ON PERIODIC OSCILLATIONS BURIED IN WHITE OR COLORED NOISE 625 is effective in identifying only periodic components, ignoring broadband spectral content including noise, noise can subsequently be extracted from a time series. We generated a noise source, shown in Fig. 4(a), with its power spectrum displaying an inverse fractional power law (6) where need not be an integer value. The noise process of 2048 data points was generated using the spectral synthesis method where the spectrum obeying (6) undergoes an inverse Fourier transform with phases randomly drawn from a uniform distribution [12]. Fig. 4(b) shows the addition of two sinusoids (7) Fig. 2. Comparison of MUSIC, MODCOVAR, correlogram, and FOS methods for identification of three sinusoids (f = 0.03, f = 0.04, and f = 0.2) with white noise added so that SNR = 010 db. finds the optimal solution. The MUSIC method, as shown in Fig. 2, provided better results than either the correlogram or the MODCOVAR methods. However, it resulted in an additional frequency peak at 0.43 Hz. For the MUSIC method, the 15 largest eigenvalues were used from the initial model order selection of 80. Using more than six eigenvalues did not further improve detection of the three frequencies, but as with MODCOVAR (AR model order 70), MUSIC detected many spurious peaks. Clearly, the FOS method provided far more accurate results than did the other three methods. Adeney and Korenberg have found similar superior performance of the FOS to that of MUSIC [1]. To further explore the capabilities of FOS in frequency detection, the following 251 data point sinusoid was generated where, are random phases, with 0.06, 0.065, and To the above sinusoids GWN was added to achieve SNR of 10, 0, and 10 db. Similar to the previous example, FOS provided the best detection of the three frequencies corrupted with significant noise. Only FOS was able to detect correctly the three frequencies for a SNR as low as 10 db, as shown in Fig. 3. Fig. 3(b) show a finer frequency scale for the plots in Fig. 3(a). Model orders of 90 and 60 were needed to identify 0.06 and 0.07 Hz for MODCOVAR and MUSIC, respectively. For SNR higher than 0 db, MUSIC, MODCOVAR, and the correlogram all correctly detected the three closely spaced frequencies. IV. APPLICATION OF FOS FOR EXTRACTION OF COMPONENTS In this section we will show the efficacy of the FOS method in extracting only the periodic signals buried in noise. As FOS (5) where is random phase with 0.03 and 0.25 added to a noise process. The SNR, as computed by taking the ratio of the variances of the noise and the sinusoids, was 20 db. We applied all four methods (correlogram, MODCOVAR, MUSIC, and FOS) to examine if any of these techniques can extract only the periodic signal and ignore the fractal components. The power spectra of the time series shown in Fig. 4(b) are shown in Fig. 5 as calculated by the four methods. Due to the low SNR, the correlogram shows small peaks especially at 0.03 and 0.25 along with a noise process. MODCOVAR and MUSIC also show a noise process with 0.03 and FOS, however, is able to distinguish correctly the two frequency components ( 0.03 and 0.25) but it does not distinguish the noise process, as shown in Fig. 5. Since FOS only estimates periodic signals (even under significant noise, either white or colored), a noise process can subsequently be extracted by subtracting the estimated sinusoidal process from the signal containing both noise and the sinusoids. The noise derived (subtraction of the estimated sinusoids from the signal containing both noise and the sinusoids) by FOS is shown in Fig. 6(b) and the actual noise process is shown in Fig. 6(a). Note that the estimated and the actual noise processes are nearly identical. This is to be expected since the FOS correctly extracts only the sinusoidal components, leaving a residual nearly identical to the actual noise. Since increasing the in (6) produces more correlated noise, and we wished to examine the performance of the four methods in a more correlated noise setting, we set to 1.5. As in the previous example (when 1), we also set the SNR to 20 db. The three conventional methods (MUSIC, MODCOVAR, and correlogram) again all detected the noise process and only one of the two sinusoidal frequencies (not shown). The inability of the AR methods to distinguish colored noise, including noise, has been previously reported [9], [11], which has been confirmed in this study via simulation examples. FOS, however, correctly only detected the two frequencies. Thus, the FOS method is a very robust algorithm because it is undeterred by the adverse effects of colored, white and noise present in the data.

5 626 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 48, NO. 6, JUNE 2001 Fig. 3. Comparison of MUSIC, MODCOVAR, correlogram, and FOS methods for identification of three sinusoids (f = 0.06, f = 0.065, and f = 0.07) with white noise added so that SNR = 010 db. (a) (b) Fig. 4. (a) Time series plots of a 1=f noise process and (b) a 1=f noise process superimposed with two sinusoids (f = 0.03 and f = 0.25). (SNR = 020 db). V. CONCLUSION Detection of periodic oscillations occurring in time series such as heart rate and blood pressure often require spectral analysis. The correlogram, which is a nonparametric way of computing a spectrum, remains one of the most popular techniques since it provides reasonably good results with easy implementation. However, it suffers when high frequency resolution is desired since the resolution is no better than the reciprocal of the data record duration, independent of the characteristics of the data. AR methods based on the description of the second-order statistics provide better resolution than the correlogram. As we

6 CHON: ACCURATE IDENTIFICATION ON PERIODIC OSCILLATIONS BURIED IN WHITE OR COLORED NOISE 627 Fig. 5. Comparison of MUSIC, MODCOVAR, correlogram and FOS methods for identification of two sinusoids (f = 0.03 and f = 0.25) with a 1=f noise process. Note that a linear scale has been used for the FOS. Fig. 6. (a) Comparison of (a) FOS derived 1=f noise process versus (b) true 1=f noise process. (b) show in this paper, the AR methods do provide better resolution than the correlogram, but their disadvantage is the selection of a proper AR model order. A method utilized in this paper, the FOS, we show to be far superior to either the correlogram or AR methods in detecting sinusoids buried in white or colored noise, including noise. Furthermore, the FOS remains effective with short data records and significantly low SNR. Thus, the method presented is useful in discriminating periodic components from the unwanted stochastic component (noise source). It can be very useful in detecting periodic signals in physio-

7 628 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 48, NO. 6, JUNE 2001 logical data such as heart rate variability and renal blood pressure and blood flow data. This is important since a noise process often precludes identification of desired periodic frequency components in both cardiovascular and renal signals. (16) APPENDIX The FOS algorithm is to rearrange the right-hand side of (2) into a sum of terms that are mutually orthogonal over the portion of the data record extending from to (8) with (17) (18) (19) where the are the orthogonal functions constructed from the in (2) and (3). The orthogonal expansion coefficients are then chosen to achieve a least-squares fit, for example mse (9) The term in (8) can be determined by Gram Schmidt orthogonalization as (10) The mse in (9) is equivalently mse (20) The overbar, for example in (19) and (20) denotes a time-average computed over the portion of data record from to. The in (1) can be determined by (21) where where and (22) (11) It has been shown that the reduction in the mse by adding the th term-pair on the right-hand side of (4) is [7] Then (8) and the orthogonality of in (10) yields (23) (12) The construction of the orthogonal functions in (10) is time and memory consuming. To avoid this, the FOS algorithm directly computes the orthogonal expansion coefficients (an algorithm based on a modified Cholesky decomposition technique) without explicitly creating the orthogonal functions. As a consequence, computing time is significantly reduced. The coefficients are obtained as follows: where (13) (14) (15) where is the [see (16)] corresponding to and corresponds to. To find the frequency, use (13) (19) and (23) to evaluate for each candidate frequency. The candidate frequency with the largest value is chosen, and so on for additional frequency terms. The process is stopped when the mse of (20) is acceptably small or a limit on the number of preset model terms has been reached. More detailed description of the FOS algorithm can be found in [5] [8]. REFERENCES [1] K. M. Adeney and M. J. Korenberg, Fast orthogonal search for array processing and spectrum estimation, Inst. Elect. Eng. Proc. Vision Image and Signal Processing, vol. 141, pp , [2] H. Akaike, Power spectrum estimation through autoregression model fitting, Ann. Inst. Stat. Math., vol. 21, pp , [3] J. B. Bassingthwaighte, R. B. King, and S. A. Roger, Fractal nature of regional myocardial blood flow heterogeneity, Circ. Res., vol. 65, pp , [4] M. Kobayashi and R. Musha, 1=f fluctuation of heartbeat period, IEEE Trans. Biomed. Eng., vol. BME 29, pp , [5] M. J. Korenberg and L. D. Paarman, Applications of fast orthogonal search: Time-series analysis and resolution of signals in noise, Ann. Biomed. Eng., vol. 17, pp , 1989.

8 CHON: ACCURATE IDENTIFICATION ON PERIODIC OSCILLATIONS BURIED IN WHITE OR COLORED NOISE 629 [6] M. J. Korenberg, Fast orthogonal algorithms for nonlinear system identification and time-series analysis, in Advanced Methods of Physiological System Modeling, V. Z. Marmarelis, Ed. New York, NY: Plenum, 1989, vol. II, pp [7], Fast orthogonal identification of nonlinear difference equation and functional expansion models, in Proc. Midwest Symp. Circuit Sys., vol. 1, 1987, pp [8], A robust orthogonal algorithm for system identification and timeseries analysis, Biol. Cyber., vol. 60, pp , [9] S. L. Marple, Digital Spectral Analysis with Applications. Englewood Cliffs, NJ: Prentice-Hall, [10] D. J. Marsh, J. L. Osborne, and A. W. Cowley, 1=f fluctuations in arterial pressure and the regulation of renal blood flow in dogs, Amer. J. Physiol., vol. 258, pp. F1394 F1400, [11] B. Pilgram, P. Castiglioni, G. Parati, and M. Di Rienzo, Dynamic detection of rhythmic oscillations in heart-rate tracings: A state-space approach based on fourth-order cumulants, Biol. Cyber., vol. 76, pp , [12] R. F. Voss, Fractals in Nature: From Characterization to Simulation,H. O. Peitgen and D. Saupe, Eds. New York: Springer-Verlag, 1988, pp [13] N. Wiener, The Fourier Integral and Certain of Its Applications. New York: Dover, [14] G. W. Wornell, Wavelet-based representations for the 1=f family of fractal processes, Proc. IEEE, vol. 81, pp , [15] Y. Yamamoto and R. L. Hughson, Extracting fractal components from time series, Physica D, vol. 68, pp , [16] F. Yang and L. Wang, Modeling and decomposition of HRV signals with wavelet transforms, IEEE Eng. Med. Biol. Mag., vol. 16, pp , Ki H. Chon (M 96) received the B.S. degree in electrical engineering from the University of Connecticut, Storrs, the M.S. degree in biomedical engineering from the University of Iowa, Iowa City, and the M.S. degree in electrical engineering and the Ph.D. degree in biomedical engineering from the University of Southern California, Los Angeles, CA. He spent three years as a National Institutes of Health (NIH) Post-Doctoral Fellow at the Harvard MIT Division of Health Science and Technology and one year as a Research Assistant Professor in the Department of Molecular Pharmacology, Physiology, and Biotechnology at Brown University, Providence, RI. Currently, he is an Assistant Professor in the Department of Electrical Engineering at the City College of the City University of New York. His current research interests include biomedical signal processing and identification and modeling of physiological systems.