5 Advanced analysis of biomedical signals This chapter would look into the advanced analysis of biomedical signals by taking advantage of the nonstat ...
5 Advanced analysis of biomedical signals This chapter would look into the advanced analysis of biomedical signals by taking advantage of the nonstationary behavior of the long-term wearable signals. In the previous chapter, we saw that the nonstationarity of the signal could be tracked by using adap- tivelter algorithms, and adaptive segment boundaries could be obtained. Once eitherxed or adaptive segmented boundaries are obtained, then that particular segment could be windowed and will be treated as a (quasi) stationary segment for further analysis. Time-series modeling of the stationary segments provided interesting features for further machine learning appli- cations. The richness of feature extraction and removal of over- lapping signal structures could be extended if some advanced analysis of biomedical signals is performed. As shown in Fig. 5.1, this chapter on advanced biomedical signal analysis, along withChapters 3 and 4, forms the biggest circle in the connected healthcare diagram. Figure 5.1The significance of this chapter on biomedical signal analysis as it relates to the book is shown here. Biomedical Signal Analysis for Connected Healthcare.https://doi.org/10.1016/B978-0-12-813086-5.00003-7 Copyright2021 Elsevier Inc. All rights reserved.157 1. Introduction Biomedical wearablessignal analysis involves identifying signal behavior, extracting linear and nonlinear properties, compres- sion or expansion into higher or lower dimensions, and recog- nizing patterns. Signals are omnipresent. This statement certainly holds true when we are able to represent most station- ary and nonstationary phenomenon in mathematical expres- sions. These representations are able to give us keen insights into those phenomena and help us in identifying characteristic patterns of interest. As seen in previous chapters, signal process- ing involves analyzing analog/digital signals with the intention of measurement, reconstruction, quality improvement, compres- sion, feature extraction, and pattern recognition. Advancements in wearable sensor technologies have come a long way by making signal data acquisition, storage, and analysis easier, as well as opening doors for further improvisation considering unstruc- tured nature of long-term big data from IoT sensors. Someone might askIf signal analysis should be easier to achieve with developments in signal processing algorithms, then how come we have to deal with increasingly complex mathemat- ical representations and optimization problems?A suitable answer to this would be that modern day information theory relies extensively on big data signals being churned out by sensors from our natural and digital environments, and treating these signals requires algorithms that are highly efcient in storage and computation. Although the underlying algorithms constitute complex mathematical operations, theow of code is designed with the intention of processing maximum amount of signal data and discovering characteristic patterns in shortest time possible. This would be conducive only if we are able to seam- lessly stream and process data. This in turn motivates us to design better tools for capturing useful information from signals at the source and discarding unwanted signal artifacts, which would also lead to wearable sensors and hardware optimization. As a quick remark, we would like to highlight that this concept has been successfully implemented in state-of-the-art compressive sensing (CS) techniques for medical image acquisition, analysis, and reconstruction. Over the last few decades, signal processing has taken notable evolutionary leaps in terms of measurement from being simple techniques for analyzing short-term analog or digital signals in time, frequency, or joint timeefrequency (TF) domain to being complex techniques for continuous 158Chapter 5Advanced analysis of biomedical signals long-term analysis and interpretation in a higher dimensional domain. The intention behind this is simple, robust, and efcient feature extraction, i.e., to identify specic signal markers or prop- erties exhibited in one event and use them to distinguish from characteristics exhibited in another event. The objective of this chapter is to give the reader a systematic perspective of feature extraction methodologies which form the basis of machine learning, and hence intelligent systems for informed decision making. We delve into the vast world of feature extraction going across the evolutionary chain starting with basic time-domain operations, to frequency/TF domain transformations, to sparse signal representations and compressive sensing. It should be noted that in this chapter we have attempted to explain key biomedical signal feature extraction methods in general terms without detailing over mathematical derivations and specic application scenarios. Additionally, we have briey touched upon the aspects of curse and blessings of signal dimensionality which would help in building generalizable models and in deter- mining appropriate feature extraction for a given application. In other words, similar to how the laws of science behind some com- mon engineering techniques could be explained, in this chapter, we have attempted to postulate an approach toward a meaningful explanation behind those methods in developing a systematic reasoning as to which type of feature analysis method is suitable for a given biomedical application and its associated hardware and software resource constraints. 1.1 Evolution of feature extraction methods In simple terms, feature extraction is the process of unveiling hidden characteristic information about the input signal and its behavior of its sources. That is, we are able to represent a given input signal by a set of features which represent a specic behavior or pattern depicted by the signal or a compact or useful representation of the signal. Feature extraction is usually a dimensionality reduction or data compression/reduction pro- cess, and helps in reducing the number of resources required to analyze a signal in a connected IoT context. In other words, given a large input signal with redundant components, perform- ing feature extraction on it would yield a smaller set of represen- tative data which could describe the original signal with sufcient accuracy and also help in building an efcient and robust pattern classier system. Chapter 5Advanced analysis of biomedical signals159 It is suggested that people-centered or application-dependent features are extracted rather than generic features, as they would better suit and depict signal behavior and underlying patterns. For example, when we are attempting to analyze and classify mu- sic signals, we do not need mean or variance of the signals or even its root mean squares, since the room or environment set- tings will be tuned accordingly, but we might need to use them when audio signals are taken from nonmodiable sources. Before we proceed with extracting information from signals, as described inChapter 3, we usually discretize the continuous analog signals into discrete digital signals using an A-to-D con- verter. This helps in identifying characteristic patterns over discrete time intervals which otherwise cannot be observed if the signal is processed in analog form. At grass roots level, the easiest way to analyze time-domain signals is byltering them, which helps in removing unwanted artifacts from the signals such as random noise, third party/source components or values, and unwanted signal patterns. The most appropriate method or signal preprocessing method will be the one that can produce an output most suited to feature extraction. This method can be devised through two possible approaches: (a) if the artifact char- acteristics (such as noise patterns) are known, we can design appropriate adaptivelters as explained inChapter 4, or (b) if the artifact properties are unknown and could be an issue with biomedical wearables, we need to preprocess the signal using signal decomposition methods which would be covered later in the chapter. Let us provide an overview of some common feature extraction methodologies applied to real-world biomedical sig- nals in the past few decades. To make it simpler for the reader, as depicted in Fig. 5.2, let us group all the available signal process- ing and feature extraction techniques into the followingfour gen- erations: (1)Time Domain (2)Frequency Domain (3)Joint TF Domain (4)Signal Decomposition and Sparse Domains The reader may note that the list of methods included in this chapter is by no means exhaustive and that we have studied some key feature extraction methods in biomedical signal pro- cessing, and have attempted tond out the most efcient method from each generation. This study will further dene the criteria to design an intelligent feature extractor adapted for biomedical signals. In order to better explain and demonstrate our views on various feature extraction techniques, we have as running examples in Sections 2e5. The classical view of 160Chapter 5Advanced analysis of biomedical signals biomedical signal properties could be captured in one or many of the following fourNmethods: (i)Non-stationarity (ii)Non-Linearity (iii)Normal/Gaussian distributions and (iv)Near-sparse representations There are many mathematical and statistical tools to test these properties. For example, Kwiatkowski, Phillips, Schmidt, and Shin test [ 1] could be used for testing stationarity. Brocke DecherteScheinkman [ 2] could be used for testing non-linearity, KolmogoroveSmirnov test [ 3] could be used for Gaussianity test, and the sparsity could be tested using Gini index [ 4]. The modern view of biomedical signal analysis from a data sci- ence perspective could be seen of features that need to handle the fourV's: (i)Velocity (ii)Variability (iii)Veracity and (iv)Variety Given the connected healthcare context and the streaming na- ture of signals from wearable devices, the data have velocity, the variability is a natural phenomenon associated with physiological signals and they could be further amplied due to the data acqui- sition in a continuous and open context, veracity is typically asso- ciated with the uncertainty in labeling the data as most of the biomedical signals captured from wearable devices may not Figure 5.2Evolution of biomedical signal analysis techniques captured in four generations. Chapter 5Advanced analysis of biomedical signals161 have the structure and properties associated with clinical grade signals, the possibility of multimodal sensing and measurement, and differences in device functionalities provide variety to the signals obtained. This chapter is organized as follows: Sections 2e6will describe key signal analysis methods that could be used for various applications most importantly for feature extraction and analysis. Further to this, inSection 7, we will highlight some areas concerning the curse and blessing of dimensionality and how it affects feature extraction and machine learning. Finally, we will conclude this chapter with some discussions and observa- tions inSection 8. It should be noted that some of the topics covered in prior chapters could also be seen as time domain (Sec- tion 2) and frequency domain (Section 3) methods. 2. Time-domain analysis Starting with the basics, we explore simple time-domain signal pro- cessing techniques, wherein analog/digital signals are analyzed over time. The visualization would tell us how the signal values change over time, and how to use those values for predictions and regression analysis. Signal processing in time domain usually involves extracting characteristic properties or features from a specic time window con- taining sayNdiscrete samples. The time window can be randomly selected, considering that most biomedical signals we encounter are nonlinear and nonstationary, but the underlying patterns and properties could remain the same for a specicphenomenon exhibited by the signals source. One of the efcient time-domain methods is the autocorrelation function (ACF), and it provides a high-level understanding of the periodicity and rhythm in a signal [ 5]. The ACF is computed in a way that the signal is compared with its own time-shifted version, and is provided as follows: R xxsE[x(t)x(ts)] or in the discrete domain as R xxkExnxnk PN n1xnxnkpn wherep(n) is the discrete probability density function (PDF), and in certain cases, it is assumed to be an uniform PDF, and thereby making the calculation of the ACF much more straight forward. At the ground level, the most basic features which could be extracted from the signal would be its statistical properties such as mean and standard deviation (variance). These generic fea- tures are generally applicable when we are trying to classify or recognize commonly occurring patterns in signals. This being said, there are various application-specic feature extraction 162Chapter 5Advanced analysis of biomedical signals techniques in time domain such as envelope analysis [6], autore- gressive (AR) modeling/linear predictive coding (LPC) [ 7], and cepstral analysis [ 8]. Of these, AR modeling and LPC revolve around the same idea: the future values of a discretized signal are calculated as a func- tion of current and previous values,while cepstrum analysis propagates the idea of rate of change of different frequency spec- trum bands of the signal. It could also be interpreted as homo- morphicltering, wherein the signals have been transformed by joint addition and multiplication operations [ 9]. AR modeling helps in enhancing data compaction, signal resolution, deconvo- luting overlapping signal peaks, and also reduces signal noise. But, this method has a downside as well: the model order cannot be determined a priori and needs to be optimized for a given biomedical signal. Sometimes, if the model order is too small, the main statistical properties of the original signal might get ignored, and if the order is too big, it might result in including additional noise along with overtting of features. Also one must note that AR modeling is applicable only to a stationary win- dow selected from a nonstationary signal which means that contin- uous feature extraction is possible only if we adaptively orxedly segment the nonstationary signal into specic sized windows. AR modeling has been extensively applied in biomedical signal anal- ysis including ECG analysis [ 13] bioacoustics signals [10], EEG analysis [ 16], cell and tissue characterization [14], EMG signal analysis [ 11], and characterizing human activity signals [15]. Cepstral analysis was initially applied to analyze seismic echoes from earthquakes and other geophysical signals [ 17]. The technique was then extended to processing radar signals and human speech analysis [ 18] and has proven to be effective in discriminating between human sounds. It may be noted here that the cepstral coefcients calculation does not require explicit computation of Fourier Transform, and hence, it could be consid- ered as time-domain feature. Given that the cepstral coefcients are computed by taking logarithm of the transfer function or spec- trum of the signal, they could be considered as simple nonlinear features. Also cepstral modeling requires that the signal to be analyzed must be stationary over a given time interval. But, as we know that real-world signals cannot be stationary, a work- around to use these time-domain techniques is to perform adap- tive orxed segmentation on the signal before feature extraction. For representing human speech signals, we mainly use the power cepstrum as a feature vector which leads us to an improvised set of features known as Mel-Frequency Cepstrum Coefcients or MFCCs [ 19]. These features are calculated by transforming the spectrum into a Mel scale, thus creating Mel-frequency Cepstrum. Chapter 5Advanced analysis of biomedical signals163 These MFCCs efciently capture spectral energy measurements over short time windows and are modeled after the human audi- tory system. These features are highly useful in current state-of- the-art applications in voice recognition, pitch detection, speaker recognition, speech-based emotion recognition, and pathological voice analysis [ 20,21]. In time-domain signal processing, linear predictive coding (LPC) has given good results in audio and speech signal compres- sion [ 22]. In terms of feature extraction, the features extracted from LPC process are the linear predictionlter coefcients (as seen inChapter 4) which help in efcient realization for hardware implementations. Depending on a specic signal processing application, a linear predicting coder/lter can be customized to generate suitable coefcients or parameters which can be char- acterized as features for a particular signal. For example, when we investigate into analyzing structural changes in signals, we apply a lattice-lter-based prediction discussed inChapter 4to generate reection coefcients as our features. Lattice predictionlters have proven to beexible signal processing tools as they are easy to optimize and have a modular structure [ 10]. When analyzing trends and patterns in biomedical signals, we must al- ways consider two important aspects: (a) structure and dynamics of the physiological system emanating the signal, and (b) signal properties reecting the physiological systems effect on the signal morphology. These aspects help us understand some of the key characteristics of physiological or biomedical signals, which could be further extracted as morphology features for pattern classica- tion, disease identication, and severity categorization. It should be noted that in most cases rather than being some sort of a spe- cic method, morphological feature extraction is simply an offshoot of time-domain signal analysis techniques, built using a combination of mathematical/geometrical functions and algo- rithms. Some of the other popular time-domain feature extraction techniques include computing form factor (FF) of a signal which is expressed as a function of activity and mobility features and also the zero crossing rate (ZCR) [ 23]. In computing ZCR, the number of times the signal crosses the baseline is calculated, and this gives an approximate idea about the instantaneous frequency varia- tions of a signal. For example, morphological feature extraction has been applied in signals such as Electrocardiogram (ECG), Phonocardiogram (PCG), which have some specic morpholog- ical shapes. Similarly, periodicity index of limb movements in sleep is computed as a movement-based morphological feature from accelerometry signals for identifying the severity of Periodic Limb Movements [ 25]. FF is another important measure of 164Chapter 5Advanced analysis of biomedical signals complexity of the signals. It is based upon the notion of variance as a measure of signal activity [ 24]. Mobility (Mx) is the square root of the ratio of the activity (vari- ance) of therst derivative of the signal to the activity of the orig- inal signal. M xs2 x0 s2 x 1 2 sx0 sx Complexity or FF is the ratio of the mobility of therst deriv- ative of the signal to the mobility of the signal itself. FF Mx0 Mxsx00= sx0 sx0= sx Deriving linear relationships within signal components is a desirable approach for developing machine learning tools, so that we can extract suitable features from the input data for the learning algorithm. However, difculty arises when the wearable signal possess properties of nonlinearity and nonseparation of classes or patterns exhibited by a group of similar signals emanated from similar sources. For example, neuromuscular ac- tivities detected using accelerometry techniques are highly likely to exhibit random variations and differences when captured from different test subjects. This nullies the possibility of applying xed signal processing tools for pattern classication. One solu- tion to address this problem is the application of signal depen- dant kernel methods to the input signal, which would transform it into a higher dimensional space, with a compact rep- resentation and where it would be easier to distinguish between signal classes. In kernel-based time-series modeling methodologies, complex real-world signals undergo dimension transformation, usually from a single-dimension time-series signal to a collection of mul- tidimension mapped points in anN-dimensional space [ 26]. The aim of a kernel function is to reduce the complex signal to simple set of features or representative values in (usually) higher dimen- sional space, so that identifying patterns (mostly through visuali- zation) becomes easier. Although the kernel mapping might be nonlinear in nature, the interclass separation itself would be linear. Kernel-based signal modeling methods have been exten- sively used in analyzingnancial time series [ 26], biomedical time series such as gait signals [ 26], and speech and speaker Chapter 5Advanced analysis of biomedical signals165 recognition [27]. In order to better explain time-domain signal analysis, we have taken an example of a vibroarthrographic abnormal knee signal acquired from a test subject [ 10] and extracted a small set of features or coefcients using AR modeling and cepstral analysis, as illustrated in Fig. 5.3. Although time-domain analysis of physiological signals may yield us signicant statistical and morphological features, it sometimes would not be able to represent a signal accurately due to presence of certain artifacts or complex behavior, which could be addressed only through the application of complex methodologies such as mathematical transformations to another domain. Nevertheless, we still need to understand the evolu- tionary chain and hence the root methods of feature extraction before we delve into advanced methodologies. It may be noted here that time-domain techniques will always reveal core infor- mation about signal properties which would always prove helpful in developing or deriving higher dimensional trasnformations for pattern analysis. Of the various methods discussed in this section, AR models, cepstral analysis, and kernel methods have proved to be widely applied time-domain based biomedical signal feature extraction methods. Table 5.1summarizes some common time- domain signal analysis methods and application scenarios. 3. Frequency-domain analysis In interpreting wearable signals, frequency or spectral transfor- mation and signal analysis in the transform domain has become an important aspect of physiological signal characterization. Figure 5.3Autoregressive and cepstral coefficients of a knee vibration signal. 166Chapter 5Advanced analysis of biomedical signals In simple words, a time-domain signal tells us how the real-world signal varies with time, whereas a frequency-domain signal indi- cates the rate of change in signal values and its spectral compo- sition. The most common transformation used in biomedical signal analysis is the Fourier Transform [ 28,29] which converts any given practical signal (time limited) into a sum of innite Table 5.1 Summary of timeedomain feature extraction methods. Method Advantages Disadvantages Application Scenarios AR modeling Parametric feature representation Mathematical (all pole) model Spectral estimation Filter/hardware realization Explainable featuresDtermining the correct model order Lower order ignores statistical properties Higher model order estimates noise Only applicable to stationary segmentsECG analysis, Bioacoutical signal analysis Biometric modeling, Spectral estimation of biomedical signals EOG, EMG analysis Gait analysis Audio signal analysis Cepstrum analysisModels human auditory system Simple nonlinear feature analysis Deconvolution properties Explainable featuresComputational complexity Only applicable to stationary segmentsPathological voice analysis Emotion recognition Speaker recognition Linear predictive coding (LPC)Could be derived directly from (AR) model parameters Good for signal compression Filter/hardware realization Explainable featuresModel order estimation Applicable to stationary segmentsData compression and transmission Bioacoustics modeling Speech encoding Morphology feature extractionSignal-adapted features Account for signal structure and hidden information Reduce signal complexity and dimensions Explainable featuresRobustness of the features Only applicable to small signal windowECG analysis Human activity analysis Detect periodic limb movements Kernel-based modelingNonlinear feature extraction Identification of latent featuresLacks explainability Nonlinear transforms may be complexEEG analysis Neuromuscular Chapter 5Advanced analysis of biomedical signals167 number of sinusoidal waves. Most notable contribution of using frequency transforms is they provide aid in identifying the artifact, noise, and overlapping signal frequencies masking vital signal in- formation, which could then be used to develop suitablexed or adaptivelters for noise removal. Most of the feature analysis and machine leanring applica- tions require only the frequency information (magnitude spec- trum) for pattern recognition and classication, although some do use the phase spectrum. Most importantly, a Fourier Trans- form gives us information about what frequencies are present in our signal and in what proportions. Transformation from time domain to frequency domain is a lossless transform and is based on an important Fourier transform property that convolu- tion in time domain is equivalent of multiplication in frequency domain, and vice versa. Fourier Transforms are applicable to both discrete and continuous signals, as long as they are inte- grable which might also lead the reader to explore Laplace Trans- forms [ 30] for continuous-time signals and systems analysis. The Fourier analysis domain is a vertical comprising of numerous methods which are derived from the basic Fourier Transform. From the continuous Fourier Transform, one could move to discrete-time Fourier Transform in which we deal with sampled (and quantized) time-domain signals, and the frequency-domain variable is still continuous. The discretized version of both frequency and time-domain variables leads to discrete Fourier TransformdDFT (discrete in time and discrete in frequency) [ 31]. Anynite signal ofNsamplesx 0;x1;x2;.;xN1can be trans- formed into frequency domain samples using the following DFT computation: X kX N1 n0 xn$ei2pkn=N;kZ Each DFT coefcientX kcomponent includes amplitude and phase information of the correspondingx nsignal component, for a partcular discrete frequency (k). Since DFT deals withnitely sized data, it can easily be implemented on signal processing hard- ware. These systems usually implement FFT (Fast Fourier Trans- form) which is a faster implementation of DFT [ 28]. DFT is computed by applying the FFT algorithm, which rapidly computes Fourier Transforms by factorizing the Fourier Transform matrix into a product of sparse or small number of signicant factors which could be considered as features for machine learning and subsequent pattern classication (other features include Eigen values or Eigen vectors). This leads to applying DFT/FFT into 168Chapter 5Advanced analysis of biomedical signals numerous applications in biomedical signal processing such as ECG analysis [ 32], audio spectral analysis [33], data compression and multichannel Electroencephalogram (EEG)ltering [ 34], and also in wireless networks for biomedical applications [ 35]. In case of multichannel signals, retaining original signal is often recom- mended so that reverting to time domain for changing areas/sam- ples of interest could be done conveniently. This implies that DFT/FFT tends to discard the nonselected time-domain samples during transformation, thus leading to the risk of loss of signal values/information. A major disadvantage of DFT is that it does not work well with nonstationary signals because of the nonlocalization property of the basis functions (consisting of sine and cosine waveforms) and components and therefore does not give the temporal resolution needed for time-varying signal analysis. Thus, DFT also fails in capturing instantaneous frequency information in nonstationary signals which can be very well be captured by joint TF methods. Postu- lation and application of DFT has also led to developing other novel variants of Fourier Transform such as STFT (Short-time Fourier Transform) and its associated energy distribution formu- lation called spectrogram which has also proved to be effective in solving time-varying signal analysis problems [ 36]. Similar to DFT, the Discrete Cosine Transform (DCT) is another widely used frequency-domain feature extraction tech- nique, which converts a discrete time-domain signal into a sum of cosine functions (whose coefcients are our features) which oscillate at different frequencies in the domain [ 37]. DCT is very similar to DFT, except that the cosine transform is applied only on real numbers and DCT uses only cosine functions for transformations. X lX N1 n0 xncosp n n1 2 l k0;.;N1 DCT is widely used for signal compression applications (audio and images) because of its energy conversation/compaction property. This emanates from its fundamental property convert- ing data points into a sum of cosine functions at different oscilla- tions, thus making the transformation orthogonal in nature. A disadvantage of DCT is that although the input values could be integers, the output will always be real valued. In order to main- tain integrity, we need some quantization steps to make the output integer valued. A good way to assess the frequency con- tent of a signal is to monitor its power spectral density at different points of interest. This also helps us innding periodicities from Chapter 5Advanced analysis of biomedical signals169 a short window in an otherwise complicated signal. Commonly used methods include the following: Bartletts method, Welchs method, and periodogram. Power spectral density could also be estimated from signal parameters or model coefcients, thereby leading to the notion of spectral estimation [ 41,42]. Spectral Estimation techniques could be categorized into parametric and nonparametric. The parametric approach works under the assumption that the input signal exhibits a specic pattern described by its statistical features derived using methods such as AR modeling. On the other hand, the nonparametric approach explicitly calculates the spectrum of the signal without considering its structure or properties. Based on the spectral den- sity estimation, various methods such as Bartletts method and Welchs method have been developed which employ variations of the conventional periodogram method for estimating spectral density parameters of a frequency-domain signal [ 43]. Despite their robustness to noise and quantization effect, spectral estima- tion techniques fall short when it comes to estimating densities of instantaneous frequency components as they are applied directly to anite signal window with an averaging effect. Nevertheless, these techniques have been applied for ECG analysis [ 45], EEG analysis [ 46], EMG analysis [47], and audio [44,48]. In order to better highlight utility of frequency-domain methods, we have shown a sample ECG signal and its FFT spec- trim in Fig. 5.4. The reason behind using ECG as an example is that its periodic nature allows us to identify signicant frequency features easily. The frequency domain analysis could be extended to beat-to-beat analysis involved in spectral analysis of heart rate variability and arrhythmia detection applications. When it comes to analyzing phase information and extracting features in fre- quency domain, Hilbert Transform is widely applied as it not only gives us instantaneous components but also provides an an- alytic representation of the signal. Hilbert Transforms dene rela- tionships between real and imaginary parts of complex signals. Unlike spectral estimation techniques, Hilbert Transform per- forms really well when extracting instantaneous time and frequency-domain features such as amplitude, phase, and domi- nant frequency [ 49]. The transformation is a convolution between signalx(t) and the Cauchy kernelkt 1 ptin time domain. b xtxtkt In frequency domain,b Xw isgnwXw Hilbert Transform ofx(t) is also called analytical signal, 170Chapter 5Advanced analysis of biomedical signals xHTtxtib xt Instantaneous amplitude atjx HTtj x2tb x2t q Phase angle 4ttan 1b xt xt Instantaneous frequency ut d4t dt Since Hilbert Transform is not a bounded operation, it may need certain additional operations before implementing it on discrete signals. Despite this, Hilbert Transform has been exten- sively applied in calculating instantaneous features of ECG sig- nals [ 50], EEG signals [51], and EMG signals [52].Fig. 5.5 illustrates the application of the Hilbert Transform on an ECG signal. Figure 5.4Frequency-domain representation of a sample ECG signal. Chapter 5Advanced analysis of biomedical signals171 Transforming to the frequency domain not only helps in bet- ter signal visualization and interpretation but also gives a local framework for low-level feature extraction and signal classica- tion. These methods are the building blocks for developing methods which transform a nonstationary signal into higher di- mensions (such as joint TF plane) for extracting characteristic and hidden information. For example, the Fourier Transform is the main method for transforming a signal to a spectral domain; the DCT is used for building dictionary of matching functions, which further help in signal size reduction through sparse repre- sentation; techniques such as DFT and spectral estimation pro- vide us a good basis for biomedical image analysis and 2D signal analysis, and works best with windowed signals. We know that Fourier Transform is widely used in most signal processing applications, but its cosine counterpart, the DCT, as shown in Fig. 5.6provides signal energy compaction and feature retention. The DCT feature extraction leads to exploiting the property of data compression, which in turn helps in hardware optimization of transmitterereceiver networks, for improved signal analysis of real-world signals. A key highlight of any frequency-domain technique transformation is the resolution change it imposes on the signal, which brings out the hidden Figure 5.5Hilbert Transform of the ECG signal inFig. 5.4. 172Chapter 5Advanced analysis of biomedical signals information to extractable level, as well as enhances signal visual- ization, especially when analyzing sharp discontinuities in signal behavior. Table 5.2summarizes key aspects and applications of various frequency-domain feature extraction methods. Table 5.2 Summary of frequency-domain feature extraction methods. Method Advantages Disadvantages Application Scenarios DFT/FFT Easy to implement on hardware Applicable to periodic/rhythmic signals Applicable to finite windowed signalsCannot be applied for transient analysis Does not provide true analysis of Does not work with nonstationary signals Cannot capture instantaneous frequency variationsWidely applied in biomedical signal analysis to understand filtering and frequency characteristics Spectral-domain feature extraction DCT Fast implementations Energy conservation/compaction Orthogonal transformationDoes not model transient activities Needs stationary segmentsSignal compression Biometrics Multichannel biomedical signal analysis Pathological voice analysis Spectral estimationModel-based/parametric spectrum Fewer computations Smoother spectral envelopesModel order selection Only applicable to stationary signal segmentsModel-based biomedical signal analysis Spectral feature analysis Hilbert TransformUseful for constructing analytic/ complex signals Extracts instantaneous features Provides single-sided spectrum Defines relationship between real and imaginary parts of complex signalsMight be sensitive to amplitude variations Helps in noise removal by orthogonal transformation of signal componentsEnvelope analysis of a signal, ECG and heart rate variability analysis Figure 5.6DCT of the ECG signal inFig. 5.4. Chapter 5Advanced analysis of biomedical signals173 4. Joint time-frequency analysis Through time-domain and frequency-domain techniques, we can usually extract low-level stationary features from a windowed signal. But, these features do not represent the true non- stationarity of real-world signals and capture only the global in- formation which roughly classies the signals. Real-world signals are nonlinear and nonstationary, and in most cases, their charac- teristic information lies in transient and localized components which can be analyzed only by transforming them to suitable di- mensions. One way to do this is to apply joint TF transformation to a nonstationary signal. By applying suitable processing on the TF decomposition parameters, subtle signal characteristics can be revealed. In many real-world applications, identication of these subtle differences makes a signicant impact in signal anal- ysis. Particularly, in classication applications using TF ap- proaches, there may be situations where a localized highly discriminative signal structure is diluted due to the presence of other overlapping signal structures. When we apply TF represen- tations to signal classication, we observe that there are small re- gions where multiple signal components overlap with varying discriminative characteristics. The power of any overlapping area is usually determined by the high-energy signal components which mask the discriminative characteristics of the low-energy components. Majority of nonstationary biomedical signals are composed of a mixture of coherent and noncoherent signal structures with varying localized overlapping regions. Their overlapping regions can be separated in order for understanding of the signal for extracting high discriminative features. Coherent signal compo- nents have denite TF localization, and hence, modeling and correlating them with dictionary elements is easier through greedy search methods such as matching pursuit (MP) algorithms [ 39]]. On the other hand, the noncoherent components need to be broken into smaller structures till their information is diluted across the whole dictionary. Composition of coherent and nonco- herent structures in a signal decides the energy distribution pattern in the signal and hence the decomposition algorithms. Ideally, a joint TF distribution needs to have a high clarity, zero cross-terms, and low computational complexity. A variation of the conventional Fourier Transform in the joint TF domain is the STFT which is used for computing the fre- quency and phase content of a localized signal over time. STFT extracts multiple frames of the signal to be analyzed using a moving-time window. The windows width is kept narrow so 174Chapter 5Advanced analysis of biomedical signals that the frame is stationary for signal analysis. Depending on the windowing function, we can design the STFT to be a narrowband or wideband transform. Narrow windows do not offer a good localization in the frequency domain, but rather in the time domain. When window function is innitely long, STFT turns into Fourier Transform, giving an excellent frequency localiza- tion, but does not yield any time information. On the other hand, when window function is innitely short, we get good time localization. One must also note that the STFT provides us a local analysis framework for nonstationary signal analysis. Based on Heisenbergs uncertainty principle, there is always a trade-off between time and frequency resolutions with one compromising for the other [ 55]. 4.1 Short-time Fourier Transform Short-time Fourier transform or Short-term Fourier tranform (STFT) is a natural extension of Fourier transform in addressing signal non-stationarity by applying windows for segmented anal- ysis. In the continuous domain STFT could be represented as, STFT fxtg Xs;w Z N N xtwtsejwtdt The discrete version of STFT could be expressed as: Xn;wX N mN xmwnmejwm wherew(n) is the analysis window, which is assumed to be nonzero only in the interval0;N1 Xn;kXn;wj w2p Nk X N mN xmwnmej2p Nkm Altering view of the discrete STFT is Xn;w 0X N mN xmejw0m wnm xne jw0n ;wn Chapter 5Advanced analysis of biomedical signals175 The signalx(n)isrst modulated withejw0n, and then passed through alter with impulse responsew(n). Thelterbank anal- ysis and synthesis frameworks for STFT are shown in Figs. 5.7 and 5.8respectively. An equivalent representation would be Xn;w 0ejw0n xn;wnejw0n BandwidthDfof the window functionwtis Df 2 R f2jwfj2df R jwfj 2df The spread in timeDtcan be given as Dt 2 R t2jwtj2dt R jwtj 2dt Resolution in time and frequency cannot be arbitrarily small, because their product is lower bounded Dt$Df 1 4p Figure 5.7Block diagram of STFT Filterbank Analysis. 176Chapter 5Advanced analysis of biomedical signals This is referred to as the uncertainty principle or Heisenberg inequality [ 53]. It means that one can only trade time resolution for frequency resolution or vice versa. Gaussian windows meet the bound with equality. Spectrogram of a simple TF visualization tool is obtained by jXn;uj 2 where the energy is distributed in the two-dimensional TF plane. This two-dimensional TF plane can be treated as an image or as a matrix or as a 2D PDF. An example of a time-varying spectral analysis using a signal such as a linear frequency modulated wave (linear chirp) could be done as follows: xtAcos u 1tu2u1 2Mt2 where the frequency band of the chirp isu 1uu2 Time period is 0tM; the instantaneous frequency of the signal is obtained by differentiating the argument ofx(t) w.r.tt, Figure 5.8Block diagram of STFT Synthesis (Filterbank). Chapter 5Advanced analysis of biomedical signals177 and due to the quadratic nature of the phase of the signal,rst or- der differentiation provides a linear instantaneous frequency estimate. utu 1t Mu2u1 Note that the instantaneous frequency becomes a function of time. The extension of instantaneous frequency estimates to mul- tiple frequency components at a given time needs a joint time and frequency distribution to compute marginals and moments from these 2D distributions. STFT has been extensively applied in audio signal processing [ 56], especially music genre classication [57] and speech signal synthesis [ 58]. STFT is desirable in applying to unimodal, univar- iate signals, wherein multiple component complexities do not exist, and moreover the signal artifacts and noise are very low. STFT features might need a threshold level during extraction andtting, as thexed window limits the amount of nonsta- tionary characteristics to be extracted. To illustrate the implemen- tation of STFT, we have taken into consideration a smartphone audio signal consisting of a series of bioacoustical audio events, as shown in Fig. 5.9. The spectrogram of the audio signal is shown Figure 5.9Audio signal recorded through a smartphone. 178Chapter 5Advanced analysis of biomedical signals inFig. 5.10. In order to overcome the limitations of STFT, the Wavelet Transform could be used to improve TF representation and resolution [ 38]. The Wavelet Transform is obtained in a similar fashion as STFT, i.e., the input signal is multiplied by a function and the transform is calculated separately for different frames of the time-domain signal. Wavelet Transform computa- tion is unique in three aspects: (i) it can be used to represent tran- sient signal structures that are quite common in biomedical signal analysis, (ii) the window width is varied for every single spectral component for optimized resolution results, and (iii) the signal is decomposed into wavelets. Scaling can cause either dilation or compression of signal, thus generating opportunities for multi- ple applications of the Wavelet Transform. Dilated versions could be cascaded in order to develop newer concepts such as scattering transforms or some similar deep belief networks for signal decom- position, whereas compression versions could be used in sensing technologies. Unlike the STFT which exhibits constant resolution at all times and frequencies, the Wavelet Transform has a good frequency and poor time resolution at low frequencies and vice versa. Figure 5.10Spectrogram of the signal inFig. 5.9. Chapter 5Advanced analysis of biomedical signals179 4.2 Wavelet transform Wavelet transform provides a joint TF (or joint time and scale) representation of a signal with moreexibility as compared to STFT [ 38]. The Continuous Wavelet Transform (CWT) of a signalx(t)is given as Xa;b 1 a pZ xtjtb a dt b/translation variable (continuous) and a/scaling variable (continuous) jtis known asmotherwavelet. The desirable properties ofmotherwavelet are Z N N jjtjdt > :2Xfforf>0 Xfforf0 0 forf
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