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Automated Estimation of the Phase Between Thoracic and Abdominal Movement Signals Alexis L. Motto*, Member, IEEE, Henrietta L. Galiana, Fellow, IEEE, Karen A. Brown, and Robert E. Kearney, Fellow, IEEE
Abstract—This paper presents a new procedure for the automated estimation of the phase relation between thoracic and abdominal breathing signals measured by inductance plethysmography (RIP). This estimation is achieved using linear filters, binary converters and an exclusive-or gate. The filters are designed offline from prior knowledge of the spectrum of subjects’ respiration, reducing computational complexity and providing on-line processing capabilities. Some numerical results based on simulated time series and infant respiration data are provided, showing that the new method is less biased than the Pearson correlation method, commonly used for assessment of thoracoabdominal asynchrony. Our method offers further advantages: 1) it works with uncalibrated measurements; 2) it provides quantitative phase estimates with no need to estimate the underlying frequency of the breathing signals; 3) it does not require nonconvex optimization search algorithms; and 4) it is easy to implement and to automate. Index Terms—Biosignals, correlation, filtering, infants, noninvasive monitoring, phase estimation, respiratory plethysmograph signals, sleep, smoothing.
I. INTRODUCTION
T
HERE have been previous attempts to automate the analysis of the phase relation between thoracic and abdominal signals measured by noninvasive respiratory inductance plethysmography (RIP) [2]–[4], [11], [17]. Prisk et al. [11] reported that the maximum linear (or Pearson) correlation method was the best; i.e., least biased, of six commonly used time-domain procedures for analyzing thoracoabdominal asynchrony. However, that conclusion was based on results obtained under favorable experimental conditions using sinusoidal and triangular waveforms and breathing signals from lightly anesthetized rhesus monkeys showing breathing periods of qusai-constant magnitudes, frequencies, and baseline drifts (or trends). In reality, RIP signals, from infants in particular, have time-varying amplitudes and frequencies and are corrupted by correlated noise processes with unknown statistical properties Manuscript received July 19, 2003; revised August 29, 2004. This work was supported by a grant from the Natural Sciences and Engineering Research Council of Canada (NSERC). The work of A. L. Motto was supported in part by an NSERC Postdoctoral Fellowship Award. Asterisk indicates corresponding author. *A. L. Motto is with the Department of Biomedical Engineering, McGill University, Montreal, QC H3A 2T5, Canada (e-mail:
[email protected]). H. L. Galiana and R. E. Kearney are with the Department of Biomedical Engineering, McGill University, Montreal, QC H3A 2T5, Canada (e-mail:
[email protected];
[email protected]). K. A. Brown is with the Department of Anaesthesia, McGill University Health Center/Montreal Children’s Hospital, Montreal, QC H3H 1P3, Canada (e-mail:
[email protected]). Digital Object Identifier 10.1109/TBME.2005.844026
such as movement artifacts and cardiogenic oscillations. Thus, to obtain reliable results under realistic assumptions by means of the Pearson correlation method, the respiratory signals must first be partitioned into segments of similar statistics. A breath-by-breath detection algorithm is also required to determine the breathing frequency for each segment. Further, a nonconvex optimization sub-problem must be solved for each segment in order to determine the time-lag or time-shift of the peak of the correlation function. The optimal time-shift and the average frequency of each segment can then be used to estimate the phase difference between two corresponding signal segments. These requirements render the Pearson correlation method computationally intensive, difficult to implement and to automate, let alone to use for online processing. Therefore, it is relevant to develop a new signal processing procedure that is less restrictive, less biased, and easier to implement and to automate than the Pearson correlation method. The remaining sections are organized as follows. Section II outlines a phase estimation procedure based on Pearson correlation. Section III presents a new procedure for the automated phase estimation of thoracic and abdominal breathing signals. Section IV illustrates the application of our method to simulated time series and infant respiration signals and compares the results to those obtained using the Pearson correlation methods. It shows that our method yielded less biased estimates. Section V provides some concluding remarks. II. PHASE ESTIMATION USING PEARSON CORRELATION The Pearson correlation method (also known as maximum linear correlation method) consists of determining the Pearson correlation coefficients of subsets of two series; for example and . Let and be two subsets of and , respectively, where denotes a shift , denotes the maxindex such that imum shift allowed, and denotes the number of data points ). The Pearson correlation between at zero shift (i.e., and is defined as (1a) where
0018-9294/$20.00 © 2005 IEEE
(1b)
(1c)
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(1d) and denote the mean of the samples and , respectively, given , and . We assume that all terms in (1) are well-defined. In the Pearson correlation method, the time shift for maximum correand , here denoted , is obtained lation between by solving the following optimization problem: (2) Let nals
be the average fundamental frequency of the sigand restricted to the time window , and define , where is an appropriate scaling factor. Authors reporting the Pearson correlation method have used as an estimate of the phase between thoracic and abdominal breathing signals. For an easier comparative analysis, in the numerical results (see Section IV), we will use the following wrapped phase variables in radians based on the Pearson correlation method: (3)
where the symbol denotes the absolute value of the element enclosed, and mod denotes the “modulo” function (i.e., where if and if , denoting the rounding of the element enclosed toward the ). nearest integer toward into the phase domain requires Note that the mapping of . knowledge of the frequency III. A NEW AUTOMATED PHASE ESTIMATION PROCEDURE We have developed a new method for the automated phase estimation of thoracic and abdominal breathing signals that has two main components. 1) The signal-to-noise ratio (SNR) is increased prior to processing, by means of nonrecursive, linear-phase, finite-duration, impulse response (FIR) bandpass filtering (e.g., see [8] and [9]). Linear phase filtering is used since the objective is to increase the SNR without distorting the phase relation between the thoracic and abdominal breathing signals. 2) The phase relationship between the filtered signals is estimated using an Exclusive-OR gate, similar to that used in phase-lock loops (for example, see [5], [12], [18]). Fig. 1 shows a block diagram of the new procedure for the automated analysis of thoracoabdominal asynchrony. Next, we describe the main components of the procedure. A. Linear-Phase bandpass Filter Linear-phase finite-impulse response (LPFIR) filtering is used because it does not distort the group-delay of the input signals. LPFIR filters can be designed using least-squares, the Remez exchange algorithm, the maximally flat method, or other algorithms [9], [13], [14], [16]. Thus, designing the
Fig. 1. Procedure for automated estimation of phase difference between thoracic and abdominal movement signals. The observed (noise-corrupted) breathing signals y n s n n n and y n s n n n are processed to produce an estimate of the absolute value of the phase, n , between the actual thoracic and abdominal breathing movements, denoted by s n and s n , respectively. Note that once the filter coefficients are specified, the new method is implemented using only multipliers, adders, comparators, and simple delay elements, which are numerically efficient and robust operators.
[ ] = [ ]+ [ ]
[]
[ ] = [ ]+ [ ]
[]
[]
filter involves choosing an optimality criterion, a filter order, and design method that is most suitable for the automated phase estimation of respiratory signals measured by inductive plethysmography. Remark 1: We performed discrete Fourier transforms on quiet breathing segments in 22 infants, previously reported by Brown et al. [4], and found that the power-spectral density was negligibly small for frequencies outside [0.4, 4] Hz. Hence we hypothesize that quiet thoracic and abdominal breathing signals are bandpass signals with spectrum [0.4, 4.0] Hz. We also observed that the range of fundamental frequencies of segments of infant quiet breathing is [0.4, 2.0] Hz. be the unit sample response and the frequency Let its sample response of the desired LPFIR filter, and denotes the filter order. The impulse response, where linear-phase FIR band-pass filter was designed using weighted least-squares error minimization [9]. For a type I FIR filter (i.e., even order and even symmetric sample impulse response), the filter coefficients were obtained by solving the following optimization:
where
is a weighting function (
,
(4) ), and (5)
(6)
Given the order , the frequency parameters in (6) and the of the desired LPFIR, any suitable opweighting function timization tool can be used to solve (4) for the optimal filter . We elected to use the Signal Processing coefficients Toolbox of Matlab [6]. In choosing the weighting function, we gave more weight to achieving a good low-frequency stop band ), because low-frequency movement artifacts (i.e., are prominent in inductance plethysmography.
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Remark 2: For the numerical studies, we used the following , values for the design of the filter coefficients: , , . The weight function was set to 1000 and to 10 in the ) and the pass band (or first stop band (or ), respectively, and to 1 otherwise. The magnitude response of the desired linear-phase FIR filter is a piecewise linear function of (e.g., see [8] and [9]). B. Binary Converter Next, the binary converter operates pointwise on its input , as follows: signal, if if
(7)
portion of the total time (where denotes . The problem is then the sampling frequency) when . to choose or From Remark 1 (see Section III-A), we know that the spec, . It is trum of breathing is reasonable to choose such that (10) Thus, for infant respiration where , one can choose . , the function of the low-pass filter Given the window (see FIR2 in Fig. 1) is then to compute as the fraction of when the or . It is evident that an is the unbiased estimator of the fraction of time when mean estimator. Hence the impulse response of FIR2 is given by
C. Phase Estimation Using an Exclusive-OR Gate and a Low-Pass Filter (FIR2)
(11)
when The exclusive-OR (XOR) gate outputs and outputs when . Assume that and is constant over some the phase relation between specified time window. Then the proportion of time that is different from over the specified time window provides and in an indication of the phase relation between the said time window.1 This will be further clarified next. Consequently, the XOR gate is commonly used as a phase detector for digital signals [5]. and are periodic with (time-invariant) period If and if the part of the time when they are different over the period is denoted by , then their phase relation is given by (8) Note that if in (8) is substituted with an arbitrary nonnegative number (not necessarily an integer multiple of the fundamental period of and , here denoted ) such that , , then the phase between and can be estimated as (9) where denotes the amount of time over when . Thus, there is no need to estimate the underlying frequency of the signals studied. Since, as mentioned above, the frequency of spontaneous breathing and the phase between thoracic and abdominal breathing are in general time-varying, the corresponding vari, , and , ables , , and in (9) must be denoted respectively, where indicates the time-dependence. Hence, given the set of data points and , we want to determine using (9) where and are substiand , respectively. Here, represents the tuted with 1Note that thoracoabdominal asynchrony detection does not require the sign of the phase relation between thoracic and abdominal signals. If it is necessary to obtain the sign, one can substitute the XOR gate with a flip-flop [15]. In this paper, we use the XOR gate.
where denotes the number of sample points in the sliding window of length . We have (12) From (12), since condition
, for all ,
satisfies the
(13) where both inequalities can be multiplied by radians (or 180 in the range of degrees) to produce scaled values of radians (or [0, 180] degrees). In the comparative studies to be to denote the phase presented in Section IV, we will use estimates in radians obtained by our method where (14) A mathematical analysis of the performance of the new procedure for thoracoabdominal asynchrony is beyond the scope of this paper. Rather, we will illustrate our method by means of numerical examples and compare its performance with that of the Pearson correlation method. D. Phase Comparator The role of the phase comparator is to distinguish between the (asynchrony absent) and (asynchrony two hypotheses: present). Thus, given a threshold , the phase comif , and decides if . parator decides If appropriate statistical models of respiration signals and the noise corruption were available, the determination of an optimal threshold could be achieved using the Bayesian, the Minimax or the Neyman-Pearson detection framework (e.g., see [10]). This detection problem however is out of the scope of the present paper whose focus is on the development of the concept of a procedure for the automated estimation of the phase relation between thoracic and abdominal RIP signals.
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IV. ILLUSTRATIONS WITH SIMULATED TIME SERIES AND INFANTS’ DATA From the foregoing, it is evident that (12) has a much lower computational cost and is easier to implement and to automate than (1)–(3). Thus, to complete the proof of the usefulness of our method, we must show that it yields phase estimates with bias and variance that are as good as or better than that produced by the Pearson correlation method when applied to RIP signals under a setting that is more general than the qusai-constant-frequency, -amplitude and -baseline-drift signals studied in [11]. To demonstrate this, we considered data drawn from simulated time series as well as clinical data from post-operative infants previously reported by Brown et al. ([4]). The numerical results obtained with our method were compared with those obtained with the Pearson correlation method.
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is a fixed phase whose value was specified for Note that each example to be described in the following subsections. 2) Simulated Additive Noise Processes: The noise present in noninvasive RIP signals, such as movement artifacts, electronic and sensor noises are nondeterministic and, therefore, must be modeled by some nontrivial stochastic processes. In this paper, we assume that all noise corruptions are additive, and we used white Gaussian processes to model electronic or sensor noise. To represent the noise processes with prominent low-frequency components that are typical of movement artifacts or gross body movements, we used a model of stochastic diffusion processes, so-called mean-reverting Itô processes (e.g., see [1] and [7]). and be the additive noise processes corrupting Let the thoracic and abdominal breathing signals, respectively, and define (16a)
A. Method All numerical analyzes in this paper were done using Matlab 6.5 (The MathWorks, Inc, Natick, MA). Data were sampled at ; the order of the linear-phase bandpass FIR filter was set to 200, corresponding to a 2-s time delay; the sample impulse response of the low-pass filter FIR2 was set as , for , where (or ). Thus, for example, an asynchrony lasting 4 s would produce a (i.e., ). phase estimate The phase estimates obtained by the Pearson correlation method was computed from (1)–(3) over the same sliding used for computing the phase estiwindow of length obtained by our method [see (14)]. mates Remark 3: Note that in solving the optimization problems (2) required by the Pearson correlation method, we constrained to within half the inverse of the the feasible values of lowest frequency of infant quiet breathing; that is, we set in (2). This is sufficient to cover the range of [0, 180] degrees given two signals with fundamental frequencies greater than or equal to 0.4 Hz such as infant breathing (see Remark 1 in Section III-A). required by the The average fundamental frequency Pearson correlation method [see (3)] were computed from the and . bandpass LPFIR filtered signals B. Simulated Data 1) Simulated Breathing Signals: Thoracic and abdominal breathing signals, here denoted and , respectively, were modeled as piecewise-linear frequency-modulated sinusoidal signals (15a) (15b) where (15c) if if if if
(15d)
(16b) (16c) (16d) and are independent white Gaussian prowhere, and are standard Wiener processes, cesses; and are mean-reverting Itô processes. , given by the stochastic differential equation A process , is a mean-reverting process if the is continuous, monotone decreasing in and function (16e) (16f) In the results shown in this paper, for the sake of simplicity, mean reversion was achieved by means of the drift functions (16g) (16h) , , and positive constants and . for real constants and fluctuate randomly, but Hence, the processes and , respectively. The betend to revert to some levels havior of this reversion depends on both the short term variance and and the speed of reversion parameters parameters and . In particular, we set the initial values of and , and the constants (16g) and (16h) as follows:
(16i) For the numerical simulation, the first terms in the right-hand side of (16a) and (16b) were approximated using the forward rectangular rule. To obtain the second terms on the right-hand side of (16a) and (16b), we used normal random numbers and added them cumulatively at the discrete time instants. All random numbers were generated using the pseudonormally distributed random number generator in Matlab 6.5 (MathWorks, Inc, Natick, MA). The state of the random generator was set to , , , and , respectively. 5, 2, 12, and 7 for
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TABLE I QUANTITATIVE RESULTS FROM SIMULATED DATA
E [1] denotes the sample mean of the vector enclosed; [1] denotes the sample standard deviation of the vector enclosed; and b = 0 , b = 0 , where denotes the true value of the phase relation between the simulated thoracic and abdominal signals).
Fig. 2. Estimates of the phase difference between two simulated time series (Sim. Series) representing thoracic (s [n]) and abdominal (s [n]) breathing signals modeled as piecewise-linear frequency modulated sinusoidal signals, derived from (15), corrupted by additive uncorrelated noise (Example A1 or left panels) and additive correlated noise (Example A2 or right panels), derived from (16). Note that the noise processes in Example A2 are identical (thus, correlated). In both examples, the phase between (s [n]) and (s [n]) was set to 0:8 radians. Here, and in all subsequent figures, [n] and [n] in the bottom two panels are the phase estimates obtained by the Pearson correlation method and our method, respectively (solid lines). In the case of simulated time series, the dashed lines show the true phase values. The estimates obtained from the Pearson correlation method ( ) is very biased and unreliable.
Finally, the simulated thoracic and abdominal signals with noise were defined as (17) (18)
Fig. 3. Sample means and standard deviations of estimate biases. The data in (a) and (b) was obtained from the setting of Example A1 (see Fig. 2) for imposed values of ranging from 0 to 180 degrees. The data in (c) and (d) was obtained from the setting of Example A2 for fixed phase = =4 radian and a window length L ranging from 4 to 20 s.
C. Experimental Data . where 3) Analysis of Simulation Results: Recall that and denote the phase estimates obtained by the Pearson correlation method and our method, respectively. Fig. 2 shows that only our method obtained good estimates of the phase relation between two piecewise-linear frequencymodulated signals with uncorrelated additive noise (Example A1) and correlated additive noise (Example A2). This is quantitatively summarized in Table I where the estimates obtained have negligible biases and variances while by our method that obtained by the Pearson correlation method show high biases and variances. Fig. 3 shows that our method produced less biased estimates than the Pearson correlation methods for fixed values of the ranging from total synchrony to total asynchrony true phase (left panels). It also shows that our method remained less biased when the window length was increased (right panels) from 4 s to 20 s. In fact, it improved as the window length increased, which is consistent with the mathematical derivations in Section III-C. Note however that as the window length increases, the probability of misclassifying events of shorter duration would also increase, and this is inherent to all methods.
1) Description of Data: We now consider phase estimation in breathing periods from 4 infants aged 41, 42, 45, and 45 weeks, weighing 3.0, 2.5, 5.4, and 5.5 kg, respectively. These data were previously reported by Brown et al. [4] as part of another study with appropriate ethics approval. This provided a convenient initial database for the validation of asynchrony detection. In brief, all the clinical studies received informed parental consent. The data set was obtained as follows. The observed continuous-time ribcage and abdominal signals (NIMS, Respitrace Plus, North Bay Village, FL), here denoted and , respectively, were amplified and filtered with 15 Hz 8-pole Bessel filters (Frequency Devices, Haverhill, MA), and sampled at 50 Hz with a 12-bit analog-to-digital converter (Data Translation, Marlborough, MA). These data were stored on a computer using LABDAT data acquisition software (RHT-InfoDat, Montreal). No attempt was made to calibrate the signals in absolute terms. For conciseness’ sake and easy reference, define (19a) (19b)
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Fig. 4. Phase analysis of a 20-s period of ribcage (rc[n]) and abdomen (ab[n]) breathing excursions measured by inductance plethysmography from an infant (45 weeks post-conceptual, 5.4 kg) with qusai-constant offsets (Example B1 or left panels) and a linear drift on the ribcage signal (Example B2 or right panels). Observe that rc[n] and ab[n] are almost totally synchronous in Example B1, and somewhat asynchronous in Example B2. In Example B2, however, our method seems to yield less biased and more uniform estimates than the Pearson correlation method.
where and denote the desired thoracic and abdominal breathing components, respectively, and and denote the noise processes corrupting them, which are assumed to be additive. 2) Analysis of Experimental Results: The signal segments considered in this illustration are representative of breathing periods in post-operative infants [4]. Since there is no means of knowing the exact phase relation between thoracic and abdominal breathing signals in practice, we cannot argue on a mathematical basis that any method yielded less biased estimates—this was argued on the basis of validation with simulated time series (see Section IV-B). It is difficult to validate a phase estimation scheme on signals with unknown mathematical properties. However, we can conclude from visual inspection (the gold standard in apnea detection) that our method yielded less biased phase estimates with lower variances. a) Quiet breathing signals with a constant trend on the thoracic signal: Example B1 (Fig. 4, left panels) shows the phase relation estimates obtained from a segment of qusai-sinusoidal breathing signals with qusai-constant offset (i.e., and from (19) are time invariant). Observe that both methods yielded good estimates with low variances. This is consistent with the results reported by Prisk et al. [11]. b) Quiet breathing signals with a linear trend on the thoracic signal: Example B2 (Fig. 4, right panels) shows the phase relation estimates obtained from a segment of breathing signals with a linear trend (i.e., from (19) is qusai-linear in time). By visual inspection, our method seems to yield less biased and more uniform (low-variance) estimates than the Pearson correlation method.
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Fig. 5. Phase analysis of a 20-s segment of ribcage (rc[n]) and abdomen (ab[n]) breathing excursions measured by inductance plethysmography from an infant (42 weeks postconceptional age, 2.5 kg) with a nonlinear trend on the thoracic signal (Example B3 or left panels). The same data were used to obtain Example B4 (right panels), except that the Pearson correlation method was applied to the outputs of the LPFIR bandpass filters proposed in our method (see Fig. 1). The LPFIR bandpass filtered ribcage and abdomen signals are here denoted rc[n] and ab[n], respectively. Visual inspection shows that the useful thoracic (s [n]) and abdominal s [n] breathing signals are uncoordinated when s [n] is increasing in both examples. The Pearson correlation estimates are more biased than our method.
c) Quiet breathing with a nonlinear trend: Example B3 (Fig. 5, left panels) shows the phase relation estimates obtained from a segment of breathing signals in the presence of a nonlinear trend in one signal and a qusai-constant trend in the other (i.e., and from (19) are nonlinear and qusai-constant, respectively, in the time index ). By visual inspection, our method obtained less biased estimates with lower variances of the phase between the desired thoracic and abdominal signals. This can be compared to the validation test in Fig. 2. d) Quiet breathing after preprocessing by an LPFIR bandpass filter: Example B4 (or Fig. 5, right panels) illustrates that the phase estimates obtained by the Pearson correlation method were still erratic even after pre-processing the raw signals by the LPFIR proposed in Section III-A. e) Quiet breathing with time-varying frequency, amplitudes and phase relation and slowly varying nonlinear, correlated trends: Example B5 (or Fig. 6) shows the phase relation estimates obtained from a segment of breathing signals with time-varying frequency, amplitudes and phase relation. Visual inspection shows that both methods obtained good estimates of the phase relation, our method being more uniform. f) Quiet breathing with time-varying frequency, amplitudes, phase relation, slowly varying nonlinear, correlated trends with simulated, additive, uncorrelated noise processes: Example B6 (Fig. 7) shows the phase relation estimates obtained from the same segment of breathing signals shown in Fig. 6, except that simulated additive noise processes were added. Clearly, only our method remained consistent, with results compatible with Fig. 6.
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Phase analysis of a 60-s segment of ribcage (rc[n]) and abdomen ab[n]) breathing excursions measured by inductance plethysmography from an infant (45 weeks postconceptional age, 5.5 kg). Visual inspection shows that the useful thoracic s [n] and abdominal s [n] breathing signals have a phase relation varying from some degree of synchrony to some degree of asynchrony. Note the time-varying frequency and amplitudes of s and s . Fig. 6. (
only simple multipliers, adders, comparators and simple delay elements (see Fig. 1), which is easy to understand and to implement; 2) it works with uncalibrated measurements since it maps the breathing signals into a phase space independent of their amplitudes (see Section III-C); 3) it does not require a breath-detection algorithm since there is no need to estimate the underlying frequency of the breathing signals (see Section III-C); 4) it does not assume underlying qusai-constant-frequency, -amplitude and -baseline-drift signals; and 5) it provides quantitative phase estimates in the [0, 180]-degree range. Further, the response time to sudden changes in phase depends on the filter length, as justified by filtering theory. Thus, for example, since the response to a step change is linear in time (e.g., see [8]), the slope of phase estimates versus time could be used to predict phase values within a few sample points, instead of globally delayed by the window length. In summary, we have presented a new method for the automated estimation of the phase relation between two time series. The method is applicable to a larger class of signals than that considered in previous approaches, including respiratory signals measured by uncalibrated noninvasive inductance plethysmography. The new method is well suited for the study of long data records such as sleep disordered breathing. It also is well suited for both offline analysis and online monitoring. The new method was successfully applied to various data including sinusoidal and nonsinusoidal signals with constant or nonlinear trends resulting from uncorrelated or correlated additive noise processes. Current studies are exploring the integration of the new procedure into the broader framework of automated classification and event detection including both central and obstructive apneas. REFERENCES
rc n n n
Fig. 7. Phase analysis of a 60-s segment of ribcage ( [ ] + [ ]) and abdomen ( [ ]+ [ ]) breathing signals from Fig. 6, except that uncorrelated mean-reverting Itô processes [ ] and [ ] were added to the thoracic and abdominal signals, respectively. Here, [ ] and [ ] are identical to those used in Fig. 2, except that the state of the random generator was set to 37 [ ] (see paragraph 5 in Section IV-B2). Only our method maintains for consistent estimates with those in Fig. 6.
ab n n n
Wn
nn
nn nn nn
V. CONCLUDING REMARKS For all simulated time series studied, our method yielded phase estimates with bias and variance that are as good as or better than those produced by the Pearson correlation method. For all infant data studied, although it was impossible to know the exact phase values between thoracic and abdominal signals, our method seemed to yield less biased phase estimates in accordance with visual inspection. The main advantages of our method over commonly used methods are: 1) it uses
[1] W. L. Andrew, “Maximum likelihood estimation of generalized Ito processes with discretey sampled data,” Econometric Theory, vol. 4, pp. 231–247, 1988. [2] M. Benameur, M. D. Goldman, C. Ecoffey, and C. Gauthier, “Ventilation and thoracoabdominal asynchrony during halothane anesthesia in infants,” J. Appl. Physiol., vol. 74, pp. 1591–1596, 1993. [3] K. A. Brown, B. Bissonnette, H. Holtby, B. Shandling, and S. Ein, “Chest wall motion during halothane anaesthesia in infants and young children,” Can. J. Anaesthesia, vol. 39, pp. 21–26, 1992. [4] K. A. Brown, R. Platt, and J. H. T. Bates, “Automated analysis of paradoxical ribcage motion during sleep in infants,” Pediatric Pulmonol., vol. 33, pp. 38–46, 2002. [5] W. F. Egan, Frequency Synthesis by Phase Lock. New York: Wiley, 1990, ch. 5. [6] Signal Processing Toolbox Users Guide, The MathWorks, Inc., Natick, MA, 1998. [7] B. Oksendal, Stochastic Differential Equations, 6th ed. New-York: Springer-Verlag, 2003. [8] A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing, 2nd ed. Upper Saddle River, N.J.: Prentice Hall, 1999. [9] T. W. Parks and C. S. Burrus, Digital Filter Design. New York: Wiley, 1987. [10] H. V. Poor, An Introduction to Signal Detection and Estimation, 2nd ed. New York: Springer-Verlag, 1994. [11] G. K. Prisk, J. Hammer, and C. L. Newth, “Techniques for measurement of thoracoabdominal asynchrony,” Pediatric Pulmonol., vol. 34, pp. 462–472, 2002. [12] J. G. Proakis and K. Salehi, Communication Systems Engineering, 2nd ed. Upper Saddle River, N.J.: Prentice Hall, 2002. [13] L. R. Rabiner, J. H. McClellan, and T. W. Parks, “FIR digital filter design techniques using weighted Chebyshev approximation,” Proc. IEEE, vol. 63, pp. 595–610, Apr. 1975.
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[14] H. W. Schüssler and P. Steffen, “Some advanced topics in filter design,” in Advanced Topics in Signal Processing, J. S. Lim and A. V. Oppenheim, Eds. Engelwood Cliffs, NJ: Prentice-Hall, 1988, ch. 8, pp. 416–491. [15] A. S. Sedra and K. C. Smith, Microelectronic Circuits, 4th ed. New York: Oxford Univ. Press, 2004, pp. 1009–1013. [16] I. W. Selesnick, “Maximally flat low-pass digital differentiators,” IEEE Trans. Circuits Syst. II: Analog Digit. Signal Process., vol. 49, no. 3, pp. 219–223, Mar. 2002. [17] Y. Sivan, W. S. Davidson, T. Deakers, T. G. Keens, and J. L. Newth, “Ribcage to abdominal asynchrony in children undergoing polygraphic sleep studies,” Pediatric Pulmonol., vol. 11, pp. 141–146, 1991. [18] H. L. Van Trees, Detection, Estimation and Modulation Theory. Part II, Nonlinear Modulation Theory. New York: Wiley, 2001.
Alexis L. Motto (S’99–M’02) received the Dipl.Ing. degree in electrical engineering from École Supérieure Interafricaine de l’Électricité, Côte d’Ivoire, in 1993, the M.S. degree in applied computer science from École Polytechnique, Montreal, QC, Canada, in 1996, and the Ph.D. in electrical engineering from McGill University, Montreal, QC, Canada, in 2001. Currently, he is a Postdoctoral Fellow in the Department of Biomedical Engineering at McGill University where his research interests center on biomedical signal processing and applied probability.
Henrietta L. Galiana (M’87–SM’93–F’02) received the Bachelor’s degree in electrical engineering (Honors) from McGill University, Montreal, QC, Canada, in 1966, followed by the Master’s Elect. Eng. degree (biomedical) in 1968. After a few years working with L. Young at Massachusetts Institute of Technology’s (MIT’s) Man-Vehicle Lab, and a 7–year sabbatical, she returned to doctoral studies and received the Ph.D. degree in biomedical engineering in 1981. Following a Post-Doc at McGill’s Aerospace Medical Res. Unit, with Geoffrey Melvill Jones, she accepted a staff position in the Department of Biomedical Eng., where she now holds the position of Full Professor. Her research interests focus on signal processing and the modeling of control strategies for the orientation of eyes and head, and related issues of platform coordination and sensory fusion. Theoretical predictions are tested in the vestibular clinic for patient evaluation, and by porting to biomimetic robot systems. She is a past President (2002) of the IEEE Engineering in Medicine and Biology Society, and currently serves on the IEEE TAB Strategic Planning and Review Committee.
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Karen A. Brown received the MD and the FRCP(c) from University of Toronto, Toronto, ON, Canada, in 1980 and 1986, respectively. She is now an Associate Professor at McGill University and is Anesthetist-inChief at the Montreal Children’s Hospital.
Robert E. Kearney (M’76–SM’92–F’01) received the Ph.D. degree in mechanical engineering from McGill University, Montreal, QC, Canada, in 1976. He is a Professor and Chair of the Department of Biomedical Engineering at McGill University. He maintains an active research program that focuses on using quantitative engineering techniques to address important biomedical problems. Specific areas of research include: The development of algorithms and tools for biomedical system identification; the application of system identification to understand the role played by stretch reflexes and joint mechanics in the control of posture and movement; and the development of bioinformatics tools and techniques for proteomics. Dr. Kearney is a Fellow of the Engineering Institute of Canada and the American Institute of Medical and Biological Engineering. He is a recipient of the IEEE Millennium medal.
