`TV' SUMMARY Homomorphic analysis of ECG signals are presented in this master thesis.Four typical ECG signals are considered and deconvolved into their minimum and maximum phase components through cepstral filtering. The complex cepstra of the signals are linearly filtered to extract the basic wavelet and the excitation function. The ECG signals are, in general, mixed phase and hence, exponantial weighting is done to aid deconvolution of the signals. The basic wavelet for normal ECG approximates the action potential of the muscle fiber of the heart. At the end of the attempt about heart procedure, there had been a succesfull result. If we are given a signal that is the sum of two components signal whose Fourier transforms occupy different frequency band, then it is possible to separate the two components with a linear filter. The fact that linear systems are relatively easy to analyze and are useful in separating signals combined by addition is a direct consequence of the property of superposition, which defines the class of linear systems.This observation leads to the consideration of classes of nonlinear systems that obey a generalised principle of superposition. Such systems are represented by algebraically linear transformations between input and output vector spaces and have thus been called homomorphic systems. In 1963 Bogert, Healy, and Tukey published a paper with a rather unusual title. In this paper they observed that the logarithm of the power spectrum of a signal containing an echo had an additive periodic component due to the echo, and thus the Fourier transform of the log-power spectrum should exhibit a peak at the echo delay. This function they termed the cepstrum, paraphrasing the word spectrum because: `In general, we find ourselves operating on the frequency side in ways customary on the time side and vice versa`.Bogert et al. went on to define a whole vocabulary to complement this new signal processing technique;however,only the term `cepstrum` has been widely used. Since the power spectrum is the Fourier transform of the autocovariance function and is always positive, we can think of the cepstrum as being the output of the characterictic system D* when the input is an autocorrelation. Since the power spectrum is always positive, only the real logarithm is required. In general, we must use the complex logarithm and complex Fourier transforms; so to emphasize the relationship while maintaining the distinction, we call the output of the characteristic system the complex cepstrum. We hasten to add that the complex cepstrum x(n) is, of course, real for real inputs x(n). We retain the term `cepstrum` for use when only the real logarithm used.Previous studies suggest that cepstrum analysis is suited to data which consist of wavelets. This is true even if shapes of the wavelets are not known prior to analysis. For instance, the power cepstrum was succesfully applied in radar analysis, where the arrival time of the main wavelet was determined by reducing interference, and in marine exploration where source depth was determined and ocean bottom was mapped. Considerable emphasis is given in this chapter to cepstrum applications in medicine, including diastolic heart sound analysis for the detection of coronary artery disease, ECG pattern classification, and speech signal decomposition for theoretical as well as bandwith compression application purposes. The cepstrum method serves as an alternative approach to linear prediction in that it does not make any assumption regarding the characteristics of the data sequences. Bogert et al developed the complex cepstrum approach to find echo arrival times in a composite signal by decomposing the non additive constituents. The term cepstrum represents the power spectrum; it is defined as a function of pseudo-time, t, the spectral ripple frequency or quefrency. The cepstrum terms defined by bogert et al, are summarized below: Throughout the thesis, both terms are used so as not to confuse readers. Cepstral analysis is applied mainly in cases where the signal contains echoes of some fundamental wavelet. By means of the power cepstrum, the times of the wavelet and the echoes can be determined. The complex cepstrum is used todetermine the shape of the wavelet. These techniques have been discussed in the literature with various applications. It has been applied to the analysis of EEG signals,to ECG signals,and to the speech signal. CEPSTRA Cepstrum analysis is concerned with the deconvolution of two signal types: the fundamental (basic) wavelet and a train of impulses (excitation function). The composite signal can be represented in terms of power, complex, or phase cepstra. The basic wavelet y(nT) can be recovered by lowpass filtering the complex cepstrum and taking the inverse z-transform of the resultant sequnce. Note that for effective wavelet recovery it is essential that the frequencies of the wavelet and the excitation function can also be recovered by first highpass filtering the complex spectra and then taking the inverse z-transform of the resultant signal. VIThe recovery process requires that the filtered complex cepstrum be z- transformed, exponentiated, and inverse z-transformed. The y(nT) and v(nT) sequences can be restored since the necessary phase information has been retained. Filters are defined in the pseudo-frequency domain and are analogus to lowpass, highpass filters in the frequency domain. In summary, the complex cepstrum contains the phase information and therefore allows reconstruction of the composite signal. The power cepstrum can be calculated from the complex cepstrum. The analysis of ECG signals in the time and frequency domains has been utilized in clinical diagnosis and is well documented in the literature. In the time domain analysis, te amplitudes and duration of the P, QRS and T complexes have been used to construct the feature vector. Pattern recognition techniques were then applied to the feature vector for classification purposes. However, extraction of the feature vector is often complicated by the presence of background noise and other artifacts.interfering with the pattern. In the frequency domain, analysis of the ECG signal has often resulted in unsuccessful! discrimination among pathological states. In an effort to produce more conclusive spectral results, Amazeen et al. utilized the phase information for information for distinguishing normal from abnormal ECGs. Murthy et al. proposed homomorphic filtering and cepstrum analysis. The ECG signal was taken as the output of a system driven by excitation functions. Note that the ECG signal represented the convolution of the basic wavelet and excitatory functions. The complex cepstrum of the ECG signal was filtered to separate the system (basic wavelet) and the excitation functions. In this approach, the basic wavelet represented the action potential generated by the heart, while the excitation function represented the excitation pattern of the heart muscle during the cardiac cycle. In their thesis, analyzed ECG recordings that exhibited normal characteristics and recordings showing left bundle branch block beat, pacemaker and PVC (Premature Ventricular Contraction) beat. Finally, the basic wavelet of the ECG signal was recovered upon linear filtering of the complex cepstra. Prior to filtering, the complex cepstra were recalculated for an exponentially weighted input data sequence xa(n). The original input sequence x(n) was multiplied by an appropriate weighting factor a`, where alfa represents some constant. This exponential weighting was provided in order for the sequence x(n) to exhibit minimal phase characteristics and ultimately ameliorate the basic wavelet recovered from the reconstructed cepstra.Exponential weighting of a convolution yields a convolution of exponantially weighted sequences. In summary, Murthy et al. utilized homomorphic filtering and the complex cepstrum method to decompose the input data sequence x(n) into its wavelet and excitatory function components.The results of their analysis reveal that the basic wavelet components closely resemble action potentials in cardiac muscle fibers while Vllthe excitatory functions follow the excitation pattern evident in the heart muscle during the cardiac cycle. The power cepstrum does not contain information about the phase. If the phase is of interest, we must consider the complex cepstrum of the sequence defined as the inverse z-transform of the complex logarithm of X(Z). If the sequence x(nT) is the convolution of two sequences u(nT) and h(nT), namely, x(nT)=u(nT)*h(nT), then X(Z)=U(Z) H(Z) logX(ZHogU(Z)+logH(Z) and since the inverse transform is a linear operation, the complex cepstrum is x(nT)=//(nT)+Â(nT) Hence the complex cepstrum of the convolution of two sequences equals the sum of their cepstra, the complex cepstrum is thus an operator converting convolution into summation. Its application to deconvolution problems becomes apparent. Assume that x(nT), u(nT), and h(nT) are the output, input, and impulse response sequences of a discrete linear system, respectively. If w(nT) and /?(nT) occupy different quefrency ranges, then the complex cepstrum can be liftered (filtered) to remove one. In the complex cepstrum phase information is retained therefore it can be inverted, to yield the deconvolved h(nT) or u(nT). The computation of the complex cepstrum has to be carefully considered since the complex logarithm is not singled valued. The imaginery part of the complex logarithm is the phase. If it is presented in module 2% form (principal value), then discontinuities will appear in the phase term. This will occur due to the jump from 2% to zero, when the phase is being increased over 2%. Phase un warping algorithms must be employed to overcome this problem. A simple solution is to compute the relative phase between adjacent samples, add them together in order to get a cumulative, unwarped phase. The complex cepstrum can be implemented by means of the DFT replacing the Z transform. This is true since the sequences are of finite length. The region of convergence for the Z transform includes the unit circle allowing the Z transform and its inverse to be evaluated for Z=exp(jw); therefore : x(nT)=IDFT{log(DFT{x(nT)})}. This equation is of great computational importance since the DFT and IDFT can be very effectively calculated by the FFT algorithm. HOMOMORPHIC FILTERING Let us consider again the example given above. Here the sequence tx(nT)} can be samples of a speech signal, the sequence (h(nT)} the weighting sequence of the vocal tract, and (u(nT)} the samples of the pressure wave exciting the vocal tract during voiced utterance, when the vocal cords are vibrating. The pressure (u(nT)} can be modeled as a train of very narrow pulses appearing at a frequency known as the fundamental frequency or the pitch. We are interested both in the vmsequence (h(nT)} in order to learn about the vocal tract characteristics, and in the sequence (u(nT)} in order to estimate the pitch. Eqation above gives the complex cepstrura, as the sum of the cepstra of the input and the vocal tract responses. Assume that in the quefrency range we have : Â(nT)=0 for«>»0 and M(nT)=0 forn<no Therefore these are seperable in the quefrency domain. We now want to transfer eî(nT) and A(nT) from the quefrency back into the time domain. We have to subject the sequences to the inverse operation. This involves first the DFT followed by complex exponentation. In this thesis homomorphic filtering has been applied to the automatic classification of ECG. Normal, pacemaker, left bundle branch beat, pvc (Premature Ventricular Contraction) have been considered. It has been found that feature selection for diagnostic purposes could be more efficient using homomorphic filtering than by conventional methods. It has been also demonstrated that the basic wavelet of normal ECG signal evaluated by the homomorphic filtering closely approximates the action potential spike in the cardiac muscle fibers. FOURIER TRANSFORMS Digital signal processing is concerned with the representation of signals by sequences of numbers or symbols and the processing of these sequences. The purpose of such processing may be to estimate characteristic parameters of a signal or to transform a signal into a form which is in some sense more desirable. The classical numerical analysis formulas, such as those designed for interpolation, integration, and differantation, are certainly digital signal processing algoratihms. On the other hand, the availability of high speed digital computers has fostered the development of increasingly complex and sophisticated signal processing algorithms, and recent advances in integrated circuit technology promise economical implementations of very complex digital signal processing systems. Signal processing, in general, has a rich history, and its importance is evident in such diverse fields as biomedical engineering, acoustics, sonar, radar,seismology, speech communication, data communication, nuclear science, and many others. In many applications, as, for example, in EEG and ECG analysis or in systems for speech transmission and speech recognition, we may wish to extract some characteristic parameters. Alternatively, we may wish to remove interference, such as noise, from the signal or to modify the signal to present it in a form which is more easily interpreted by an expert. As another example, a signal transmitted over a communications channel is generally perturbed in a variety of ways, including channel distortion, fading, and the insertion of background noise. One of the objectives at the receiver is to compensate for these disturbances. In each case, processing of the signal is required. IXWe consider the finite sequences and linear shift-invariant systems in terms of the fourier and z-transforms. For the special case in which the sequence to be represented is of finite duration, i.e., has only a finite number of nonzero values, it is possible to develop an alternative Fourier representation, referred to as the Discrete Fourier Transform (DFT). As we shall discuss in this thesis, the DFT is a Fourier representation of a finite-length sequence which is itself a sequence rather than a continious function, and it corresponds to samples equally spaced in frequency of the Fourier Transform of the signal. In addition to its importance theoretically as a Fourier representation of sequences, the DFT plays a central role in the implementation of a variety of digital signal processing algorithms, as a result of the existence of an efficient algorithm for the computation of the DFT. The direct computation of the DFT reqires approximately N2 complex multiplication and addition operations. In 1965, Cooley and Tukey, in their famous paper presented an efficient method for calculating the DFT. Their method, known as the Fast Fourier Transform (FFT), requires only N log2 N operations (where N is a power of 2). For N=1024, the number of operations required by the FFT is ten times less than the number required for direct computation. Many different FFT algorithms have been derived for software and hardware implementations. Two commonly used algorithms are known as the Decimation in time and Decimation in frequency algorithms. The interested reader is referred to the vast literature on this subject. There are a number of points of view that can be taken toward the derivation and interpretation of the DFT representation of a finite-duration sequence. We have chosen in this thesis finite-length and periodic sequences, so we shall consider first the Fourier series representation of periodic sequences. While the Fourier representation of periodic sequences is important in its own right, we shall also apply those results to the representation of finite-length sequences of this periodic sequence corresponds to the DFT of the finite-length sequence. This is done by constructing a periodic sequence for which each period is identical to the finite-length sequence. As we shall see, the Fourier series representation

ÖZET Bu yüksek lisans tezinde, EKG işaretlerinin, homomorfik analizi incelenmiştir.Dört tipik EKG işareti ele alınmış ve özelliklerinin belirlenmesi için cepstral filtreleme yolu ile minimum ve maximum faz bileşenlerine ayrılmıştır, işaretlerin kompleks kepstrumlan temel dalga şeklini ve uyarıcı fonksiyonu ortaya çıkarmak için, lineer olarak filtrelenir. Genelde, EKG işaretleri, karışık fazlı olduklarından işaretlerin ayrıştırılmasına yardımcı olmak için exponansiyel bölmelendirme yapılır. Normal EKG için temel dalga şekli, kalbin kas fiberlerinin aksiyon potansiyeline benzer. Böylece incelenen kalbin çalışması hakkında doğru bilgi elde etme çalışmaları oldukça başarılı bir sonuç vermiştir. IV

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