IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 48, NO. 9, SEPTEMBER 2001

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A Short-Time Multifractal Approach for Arrhythmia Detection Based on Fuzzy Neural Network Yang Wang*, Yi-Sheng Zhu, Senior Member, IEEE, Nitish V. Thakor, Fellow, IEEE, and Yu-Hong Xu

Abstract—We have proposed the notion of short-time multifractality and used it to develop a novel approach for arrhythmia detection. Cardiac rhythms are characterized by short-time generalized dimensions (STGDs), and different kinds of arrhythmias are discriminated using a neural network. To advance the accuracy of classification, a new fuzzy Kohonen network, which overcomes the shortcomings of the classical algorithm, is presented. In our paper, the potential of our method for clinical uses and real-time detection was examined using 180 electrocardiogram records [60 atrial fibrillation, 60 ventricular fibrillation, and 60 ventricular tachycardia]. The proposed algorithm has achieved high accuracy (more than 97%) and is computationally fast in detection. Index Terms—Arrhythmia detection, fuzzy neural network, generalized dimension, multifractal.

I. INTRODUCTION

S

TUDY on the detection of abnormal cardiac rhythms is of important clinical significance. Ventricular fibrillation (VF) and ventricular tachycardia (VT) are life-threatening arrhythmias. In order to effectively offer high-energy defibrillation to VF and low-energy cardioversion to VT, automatic external defibrillators (AEDs) and implantable cardioverter defibrillators (ICDs) require arrhythmia classification algorithms that can distinguish shockable cardiac rhythms from nonshockable cardiac rhythms as accurately and as rapidly as possible [1], [2]. Various detection algorithms have been reported, such as VF-filter method [3], sequential hypothesis testing algorithm [4]–[6], fast template matching algorithms [7], time-frequency analysis [8], complexity measure [9] and wavelet analysis [10]–[12]. All these methods exhibit advantages and disadvantages, some being too difficult to implement and compute for AEDs and ICDs, some having low specificity in differentiating shockable and nonshockable arrhythmias [13]. Hence, we are exploring more sophisticated techniques to fully describe different cardiac arrhythmias. Cardiac rhythms are generated by dynamic systems that manifest nonlinear properties [14]–[16]. As effective analytic tools for nonlinearity, fractal geometry-based methods have been applied to cardiology recently [17]. In particular, multifractal ap-

proaches offer a new and potentially promising avenue for quantifying features of a range of physiologic signals that differ in health and disease. Many physiologic processes are in fact inhomogeneous, suggesting that different parts of the signal have different scaling properties. However, robust demonstration of multifractality for physiologic processes has been hampered due to their nonstationary properties [18]. Neural network computing has also been applied to rhythm detection, and is recognized as a powerful and promising technique for arrhythmia discrimination [19]–[22]. Evans et al. used this technique to distinguish beats of ventricular origin from normal sinus beats in electrocardiogram (ECG) signals, and test its accuracy (94% sensitivity and specificity) and noise robustness through human experiments [21]. Minami et al. made a Fourier-transform neural network by observing QRS complexes [22]. The performance of these networks is not satisfactory enough for clinical uses. It is necessary to develop new detection schemes with high level of accuracy, or equivalently, low false-positive and false-negative statistics. In this paper, we extend the multifractal approach after taking the nonstationary aspects of signals into consideration. The new method named short-time multifractality is proposed to describe arrhythmias. Then an advanced fuzzy Kohonen network (AFKN), which solves the intrinsic problems of the original one, is presented to increase classification accuracy. Thus, we have developed a detection algorithm which characterizes cardiac rhythms by short-time generalized dimensions (STGDs), and then inputs them to the detection network. The new method is essentially different from current approaches, since it deals with nonlinear and nonstationary physiologic processes from the viewpoint of multifractality. Using the short-time multifractal approach, the discrimination accuracy for each of VT, VF, and AF from 180 ECG records is more than 97%. The new algorithm is computationally efficient and well suited for real-time processing. With such capability, the short-time multifractal method is expected to be a powerful tool in biosignal processing. II. THEORY AND METHODS

Manuscript received October 26, 2000; revised June 2, 2001. This work was supported by the National Natural Science Foundation of China under Grant 69871019. Asterisk indicates corresponding author. *Y. Wang is with the Department of Biomedical Engineering, Shanghai Jiao Tong University, Shanghai 200030, China (e-mail: [email protected]). Y.-S. Zhu and Y.-H. Xu are with the Department of Biomedical Engineering, Shanghai Jiao Tong University, Shanghai 200030, China. N. V. Thakor is with the Department of Biomedical Engineering, Johns Hopkins School of Medicine, Baltimore, MD 21205 USA. Publisher Item Identifier S 0018-9294(01)07441-9.

A. Short-Time Multifractal Characterization Given a point set on which a measure is defined, let be the cubes of the -coordinate mesh that is defined by cover . Within the th cube,

0018–9294/01$10.00 © 2001 IEEE

(1)

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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 48, NO. 9, SEPTEMBER 2001

For small positive

and , the number of cubes with has a scaling exponent a multifractal object [23]

if

is

for (2) is the number of cubes with . The where gives a description multifractal singularity spectrum of the set. On the other hand, multifractal generalized dimensions are defined by [24]

for

and

for

(3a)

ln  ln

Fig. 1. " some time instant.

( )

p t; " curves

(1w = 1.5 s) of a ECG record at

rectangular window is one of the suitable window functions. The STGDs are defined by

for

(7a)

(3b) for

and are dual In fact, the two descriptions and can be deduced one from the other in most general cases thanks to a Legendre transform [25] (4a) (4b) Up to now, the presence of nonstationarity in signals is a major problem in contemporary cardiology, which limits the application of multifractal analysis [18]. Given an ECG record with different rhythms, the multifractal method can tell whether there is arrhythmia in the time series, but cannot determine where (or when) an individual abnormal rhythm with , its happens. For a nonstationary signal characteristic is varying with time. Since the information of the signal near a time instant can not be obviously shown by the singularity spectrum or generalized dimensions, the short-time multifractality is proposed in this work to deal with the time-varying property of the signal. , the number of the cubes ( -interFor the point set . The and measure vals) required to cover is in (1) can be generalized to and that are relative to the time instant (5) with

, and (6a) (6b) (6c)

is the sign function. It can be known from (6) that where is a window function that centers on the origin, and the

(7b)

is weighted by in (5), gives After the signal prominence to the intervals near the time instant and, theredescribes the nonstationary fore, the spectrum property of the signal. Similarly, one could get the short-time by defining the multifractal singularity spectrum and measure . The spectrums are equivalent and can be deduced from each other using the Legendre transform at a specific time instant. affect the compuThe attributes of the window function . One can define the window length tation of (8) makes a notable impact on the analysis results The size of retrogrades to when is a (see part IV), and . rectangular window with For a specific window function, one can get the curve with fixed and . Fig. 1 curves of a ECG record for shows the and 1.5 s at some time instant . The is . However, slope of the curve for the sampling rate of a signal is definite, and it is hard to let . Since the curve tends to a straight the time interval by line when is small, we can estimate the value of calculating the slope of the linear part in the curve , let and Choose small ( when 1). Then one can approximate a line by points . the least squares method using the The slope of the line is the estimative value of

WANG et al.: A SHORT-TIME MULTIFRACTAL APPROACH FOR ARRHYTHMIA DETECTION BASED ON FUZZY NEURAL NETWORK

(a)

(b) Fig. 2. (a) Typical waveforms and (b) their t = 4 of VF (left) and VT (right).

q



D(t; q )

curves for q = 2 and

Since short-time multifractality reflects not only the nonlinearity, but also the nonstationarity of signals, it is expected to provide deeper description of arrhythmia than multifractality. curves Fig. 2 shows typical waveforms and their 2 and 4 of VT and VF, respectively. It is manifested that for the STGD of the signals is varying with time, and for different rhythm, STGD vibrates mainly at different domain, which enables us to discriminate different types of cardiac arrhythmias.

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In this paper, we present an AFKN based on the work of Tsao et al., the new classification algorithm is as follows. is the set of n input vectors in . Let denote the set of weight vectors with ( is the weight vector attached to the th output neuron, ). There are kinds of vectors to be classified, and is the set of positive integers related to the types of input vectors, in which represents the type of . is the Euclidean norm. some small positive constant, and the AFKN1. Fix , so that the domain of desired number of output neurons th, output for the th kind of vector is the th , and th nodes. . Choose a AFKN2. Initialize with , decreasing sequence is the iterate limit. and : AFKN3. For : a) For for all , and is the winner Compute node for (10) Compute learning rates

for

(11)

B. Rhythm Classification by AFKN We have developed a new fuzzy Kohonen network for arrhythmia detection. The Kohonen clustering network (KCN) is well known for cluster analysis. KCN is an unsupervised scheme that finds the “best” set of weights for hard clusters in an iterative, sequential manner. The structure of KCN consists of two layers: an input layer, and an output layer. Each output node has a weight vector attached to it, and it is this network weight vector that is adjusted during learning. Given an input vector, the neurons in the output layer compete among themselves and the winner (whose weight has the minimum distance from the input) updates its weights and those of some set of predefined neighbors. The process is continued until the weight factors “stabilize” [26]. However, KCN suffers from several major problems [27]. Termination is not based on optimizing any model of the process or its data. The final weight vectors usually depend on the sequence of the input data. Different initial conditions usually yield different results. Several parameters of the KCN algorithm, such as the learning rate, and the size of the neighborhood, must be varied from one data set to another to achieve “useful” results. These shortcomings will lower the accuracy of classification. To solve the intrinsic problems of KCN, Tsao et al. proposed a fuzzy Kohonen clustering network (FKCN) which integrates the Fuzzy -Means model into the learning rate and updating strategies of the Kohonen network [28]. However, the topological orderliness of weights organization after training, which is the property of KCN, is lost by FKCN [29].

for

(12a)

for

(12b)

for

(12c) (13)

b) Update all

weight vectors

with (14)

. . c) Compute , stop; else next . d) If Notes on the AFKN algorithm. is fixed, (11) shows that for each , 1) When is inversely proportional to the distance from to the . Thus, it can be known winning weight vector from (12) that the closer becomes the distance from to , the greater is the learning rate and effective update neighborhood, and vice versa. Therefore, both the learning rate and the update neighborhood in the AFKN scheme is automatically adjusted with and the during evolution. distance

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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 48, NO. 9, SEPTEMBER 2001

2) AFKN is nonsequential. Updates on all weight vectors are performed after each pass through . Hence, is independent of the data lathe iterate sequence bels. for is not in the 3) During learning, if the winner node , domain of desired output, namely, the center of the update neighborhood will move toward the desired output node according to (12). is fixed, (12) manifests that for each , the 4) When learning rate decreases when the distance from the updating node to the center of the neighborhood increases. This is similar to the KCN algorithm and, therefore, AFKN keeps the topologically orderliness in the organization of weight vectors. Comparing with the classical Kohonen network, AFKN is nonsequential, supervised, and automatically controls both the learning rate distribution and effective update neighborhood, which highly improve the ability of discrimination and make the detection algorithm more accurate. By inputting STGDs of a signal to the classification network, individual arrhythmia could be recognized. C. Comparative Analysis Since short-time multifractality is introduced as a novel approach for discriminating abnormal cardiac rhythms, it is necessary to test our technique using the method of surrogate data [30]. The surrogate signals are generated by randomizing point-to-point increments of each ECG record. Hence every new signal is a random walk [31]. The signal preserves the original distribution of increments but destroys the correlation among them. By the Student’s -test, STGDs of the surrogate data could be statistically compared with STGDs of the original records. Moreover, the new algorithm is compared with two other approaches currently used for arrhythmia detection. We select the VF-filter algorithm [3] and the classification algorithm based on wavelet transform and neural network [12] to perform on the same set of data used by our algorithm. The VF-filter approach relies on that different cardiac rhythms approximate sinusoidal waveforms to different extents. The mean period of each ECG record is calculated from the discrete FFT. The data are then combined with a copy of the data which has been shifted by half a period, If the data approximate a periodical signal, they will, therefore, cancel. This process is equivalent to applying a narrow bandstop filter centered on the mean signal frequency. The output of the process is, therefore, referred to as the VF-filter leakage. For a data segment with samples , the leakage is calculated as

TABLE I THE STATISTICAL RESULTS OF STGDS FOR DIFFERENT WINDOWS LENGTHS

The classifier is developed by using five-scale Daubechies 4 wavelet transform for extracting features and then using a neural network to classify the cardiac rhythms. Six descriptors are derived by computing the energy of each scale over beats of the ECG record. A two-layer radial basis function neural network (RBFNN) [32] to detect arrhythmias is then used to classify the feature vector. III. DATA COLLECTION The test set used in our work includes 60 VT, 60 VF, and 60 atrial fibrillation (AF). The ventricular tachyarrhythmia data were recorded from patients being evaluated for automatic cardioverter defibrillator implantation. During electrophysiologic procedures, malignant ventricular arrhythmias were induced by altering current shocks with the intent to determine parameters for defibrillation (such as defibrillation energy threshold). This procedure helps confirm the successful operation of the defibrillator to be implanted. Recordings were obtained from an external chest lead, a transcardiac signal from a patch electrode placed on the apex, and a catheter inserted in the superior vena cava. The data were recorded on a battery-operated FM instrumentation tape recorder (TEAC model R-61). 120 records were visually selected as being sufficiently distinctive examples of VF and VT. The chest lead signal was digitized at a rate of 200 Hz using a Metrabyte Model Dash-8 data acquisition system. AF records were extracted from ECG recordings of MIT-BIH Arrhythmia Database according to the beat and rhythm annotations. Each record of the three rhythms was selected to be 6 s long. In addition, an 8-s ECG record consisting of AF and premature ventricular contraction (PVC) was selected from the database. (VT could be viewed as continuous episodes of PVC.) Since the sampling rate of this database is 360 Hz, the extracted records were down-sampled to 200 Hz. All the ECG signals 1 Hz) to remove baseline drift, were highpass filtered ( 60 Hz) to remove power line and then bandstop filtered ( interface. IV. RESULTS

Leakage

(15)

where is the number of data points in one mean period. Al-Fahoum and Howitt proposed a classification algorithm based on wavelet transform (WT) and neural network in 1999.

A. Short-Time Generalized Dimensions Computation The rectangular window function is applied in our work. For 0.3, 0.6, and 1.2 s, i.e., 60, 120, a specific window length ( 2, 4), ( 0.6 s, and 240 data points) and ( .) of 180 ECG records are calculated. For each , the mean and standard deviation of STGDs for AF, VF, and VT are estimated by statistical approach (see Table I),

WANG et al.: A SHORT-TIME MULTIFRACTAL APPROACH FOR ARRHYTHMIA DETECTION BASED ON FUZZY NEURAL NETWORK

respectively. Fig. 3 illustrates the AF/PVC signal and its curves for 1.2, 1.8, and 2.4 s, and 2, 4. When the window length is very short, for the definite sampling rate of the records, the points used to calculate STGD are too few to make the algorithm stable, so that the estimated STGDs vibrate intensely, which makes different rhythms hard to discriminate. In Table I, the standard deviation is large and STGDs of the three rhythms overlap with each other greatly 0.3, 0.6 s). With when the window size is too small ( the increase of the window length, the overlapping areas decrease. On the other hand, when the window size is excessively large, the signal length used to compute STGDs is so long that the property of an individual time instant cannot be fully reflected. In Fig. 3, AF and PVC cannot be differentiated in terms 2.4 s, while the difference of the two of STGDs with rhythms becomes clear with the decrease of the window length. Hence, the choice of suitable window length is important to the arrhythmia analysis. In this paper, the window length is taken as 1.2 s, i.e., 240 data points.

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*P: PVC

(a)

(b)

(c)

B. Rhythm Detection 1.2 s and time interval Let window length 0.6 s. For each record, calculate , , then choose the maximum, minimum, and mean of the two sets, respectively, as the characteristic , parameters of the signal. The parameters, written as , , , , and , are input to the classification neural network. There are three kinds of 3), arrhythmias (VT, VF, and AF) to be distinguished ( and the number of desired output nodes for each rhythm is 6. Fig. 4 shows the structure of the neural network used for detection. For the training of the neural network, 50% ECG data of each rhythm are selected randomly. After no more than 500 training iterations, the network is evaluated with the rest of the data. Sensitivity, specificity and accuracy for each class are calculated in order to measure the performance of rhythm classification. The detection results for the 180 ECG records are listed in Table II. Sensitivities and specificities of more than 95% are obtained. For discrimination of different rhythms, accuracies have achieved more than 97%. and impleThe algorithm is developed using Borland C mented on a PC. The time required to compute STGDs and detect individual rhythm by network for each signal is 0.03 s. The CPU type of our PC is Pentium II 233-Hz.

(d)

 ( ) 1 =

Fig. 3. (a) Signal consisting of AF and PVC and its t D t; q curves for 2 (left) and q 4 (right) with different window lengths: (b) w 1.2 s; (c) w 1.8 s; and (d) w 2.4 s (d).

q

= 1 =

=

1 =

Fig. 4. The structure of the detection network. TABLE II PERFORMANCE

OF THE SHORT-TIME MULTIFRACTAL ARRHYTHMIA DETECTION

METHOD

IN

*Sensitivity TP/(TP FN), Specificity TN/(TN FP), Accuracy (TP TN)/(TP FN TN FP), where TP true positive, FN false negative, TN true negative, and FP false positive.

C. Surrogate Analysis and Other Methods For the random walk surrogate signals, the mean and standard deviation of STGDs for different window lengths are estimated in Table III. By the -test, the STGD values of the surrogate data show significant difference from the STGD values of origfor all the ]. In inal VT, VF and AF signals [ addition, comparing with the original records, the STGD range of surrogate signals overlaps heavily, which means that the surrogate data lose the short-time multifractal difference between original cardiac rhythms. The mean and standard deviation of the VF-filter leakage for VT, VF, AF are estimated in Table IV(a) by statistical approach. Since the leakage range of the three rhythms overlaps with each

TABLE III THE STATISTICAL RESULTS OF STGDS FOR SURRAGATE DATA

other, it is difficult to completely discriminate the arrhythmias. By an empirical study, the threshold values are set as 0.467 s for VT-VF and 0.631 s for VF-AF. The performance of the VF-filter

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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 48, NO. 9, SEPTEMBER 2001

TABLE IV (a) THE STATISTICAL RESULTS OF VF-FILTER LEAKAGE FOR DIFFERENT CARDIAC RHYTHMS (b) PERFORMANCE OF THE VF-FILTER ALGORITHM IN ARRHYTHMIA DETECTION

(a)

(b)

PERFORMANCE

TABLE V ALGORITHM BASED ON WT ARRHYTHMIA DETECTION

OF THE

AND

RBFNN

IN

algorithm is listed in Table IV(b). Although VF-filter algorithm may be simple to implement, its detection accuracy is not high enough for clinical applications. After half of the data are used to train the RBFNN, the detection results of the classification algorithm based on five-scale Daubechies 4 wavelet transform and RBFNN are listed in Table V. Comparing with the short-time multifractal approach, the algorithm shows high accuracy in detecting VF and AF as well, while its sensitivity in discriminating VT is not satisfactory enough. Moreover, the RBFNN is sensitive to the input order of the training vectors, which sometimes biases the classification results [12]. Since the improved Kohonen network is sequence independent, the novel classification algorithm is more robust.

representation or of signals, and our method is computationally less intensive, which makes it more practical in clinical environment. The new fuzzy Kohonen network developed in this paper overcomes the shortcomings of the original network. Its properties, such as label independence, automatic adjustment of learning rates and neighborhood, and topologic orderliness of weights, help make the neural network more flexible and accurate in arrhythmia detection. With such advantages, it could be widely used for pattern classification in biomedical applications. One limitation of the current study is that the algorithm performance was tested by just three kinds of arrhythmias (VT, VF, and AF). A large database including other types of abnormal cardiac rhythms, such as idioventricular rhythms and asystole, is necessary to further test the achievement of our method. The ECG data we used are single-channel records, which unavoidably lose some amount of information about cardiac activities. Moreover, the multifractal approach disregards the energy information in the signals. Utilizing multichannel records or taking the energy of arrhythmia into consideration may raise the classification accuracy even further. The window function plays a key role in our short-time multifractal approach. We set a specific window after studying the impact of window length on the analysis of nonstationarity in ECG. How to automatically get a suitable shape of the window function, or more generally, how to associate the measure with time for different cases, is the problem for our further study. The multifractal or short-time multifractal analysis has proven to provide useful characterization of physiologic processes. The method can be utilized not only in ECG, but also in many other fields such as electroencephalography, electromyography. The multifractal (including short-time multifractal) concepts open possibilities to be a new fundamental approach in biosignal processing. REFERENCES

V. DISCUSSION This paper presents a new algorithm using short-time multifractality and fuzzy Kohonen network for arrhythmia detection. The satisfactory performance (high sensitivity for shockable rhythms and high specificity for nonshockable rhythms) of our approach makes the method suitable for development of safe arrhythmia detectors requiring minimal skilled human interaction. Owing to the small size of STGDs computation and the neural network, it is possible for the algorithm to realize real-time detection. Recently, time-frequency analysis techniques such as the Wigner–Vile distribution [8] and the wavelet transform [10] have been applied to detect arrhythmias. These techniques continuously transform ECG signals into three-dimensional (3-D) representations by adding an extra axis of frequency or scale. An abnormal rhythm could be recognized visually in the representation [22]. Although this suggests significant potential in arrhythmia detection, continuous and high computation loads are required. Comparing to the time-frequency analysis tools, the short-time multifractal approach also gives a 3-D

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Yang Wang was born in Shanghai, China, on May 8, 1976. He received the B.E. degree in electronic engineering from Shanghai Jiao Tong University, China, in 1998, and the M.S. degree in biomedical engineering from the same university, in 2001. His current research interests are nonstationary signal processing, neuroengineering, and microprocessor-based bioinstrumentation. Mr. Wang is a recipient of National Excellence Scholarship of China.

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Yi-Sheng Zhu (M’88–SM’90) graduated from the Department of Electrical Engineering, University of Science and Technology of China (USTC), Beijing, China, in 1968. Since 1978, he served on the faculty of USTC. Between 1986 and 1988, he was a Visiting Scholar in the Department of Biomedical Engineering, Johns Hopkins University, Baltimore, MD. Since 1994, he has been a Professor and Chairman of Department of Biomedical Engineering, Shanghai Jiao Tong University. His current research interests are digital signal processing, microcomputer-based medical instrumentation, and new medicine. Prof. Zhu is a recipient of six awards from the National Foundation of China and another one from SNSF.

Nitish V. Thakor (S’78–M’81–SM’89–F’97) received B.Tech. degree in electrical engineering from Indian Institute of Technology, Bombay, India, in 1974 and the Ph.D. degree in electrical and computer engineering from the University of Wisconsin, Madison, in 1981. He served on the faculty of Electrical Engineering and Computer Science of the Northwestern University between 1981 and 1983, and since then he has been with the Johns Hopkins University, School of Medicine, Baltimore, MD, where he is currently serving as a Professor of Biomedical Engineering. He teaches and conducts research on cardiovascular and neurological instrumentation, medical microsystems, signal processing, and computer applications. He has authored more than 110 peer-reviewed publications on these subjects. He serves on the editorial boards of IEEE TRANSACTIONS ON INFORMATION TECHNOLOGY IN BIOMEDICINE and Annals of Biomedical Engineering. He has recently established a Center for Neuroengineering at the Johns Hopkins University with the aim of carrying out interdisciplinary and collaborative engineering research for basic and clinical neurosciences. He is actively interested in developing international scientific programs, collaborative exchanges, tutorials and conferences on Neuroengineering and Medical Microsystems. Dr. Thakor is a recipient of a Research Career Development Award from the National Institutes of Health and a Presidential Young Investigator Award from the National Science Foundation, and is a Fellow of the American Institute of Medical and Biological Engineering. He is also a recipient of the Centennial Medal from the University of Wisconsin School of Engineering and recognition from the students of the Alpha Eta Mu Beta Biomedical Engineering student Honor Society.

Yu-Hong Xu received the B.S. degree in electrical engineering from Peking University, Peking, China, in 1990, the M.S. degree in biophysics from the State University of New York (SUNY), Buffalo, in 1992, and the Ph.D. degree from University of California, San Francisco, in 1996. From 1996 to 1997, she was a Postdoctoral Scientist at Chiron Corp.. She was a Senior Research Scientist at Merck & Co., from 1997 to 1999. She was appointed as a Professor in the Department of Biomedical Engineering in Shanghai Jiao Tong University, Shanghai, China, in 1999. She is also a Professor in the School of Pharmacy at the same university. Her research interests are in the areas of bioengineering, drug formulation and delivery, bioinstrumentation, etc.

A short-time multifractal approach for arrhythmia ...

[7], time-frequency analysis [8], complexity measure [9] and wavelet ..... Selecting the appropriate time-frequency analysis tool for the applica- tion,” IEEE.

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