Chapter 5. Frequency Domain Analysis

Transcription

1 Chapter 5 Frequency Domain Analysis

2 CHAPTER 5 FREQUENCY DOMAIN ANALYSIS By using the HRV data and implementing the algorithm developed for Spectral Entropy (SE), SE analysis has been carried out for healthy, cardiac-diseased and non-cardiac diseased subjects. This chapter presents the details of power spectral distributions and implementation of SE technique using MATLAB and discusses the results obtained for healthy and diseased subjects. 5.1 Introduction The study of oscillations of precise frequencies is the advantage of spectral analysis. The spectral disintegration of time series HRV (RR interval) data into addition of sinusoidal functions of a range of amplitudes and frequencies is performed by spectral analysis. A power spectrum drawn for magnitude verses frequency of the disintegration can be displayed. Power spectrum of HRV data represents the amplitude of heart rate fluctuations as a function of different frequencies. The information about the variability along with the oscillation frequency can be described by frequency domain analysis. For the power spectrum analysis various methods like Fast Fourier Transforms (FFT) and Auto Regressive (AR) techniques are commonly used to convert the time domain signal into frequency domain signal [69]. Frequency domain analysis for a HRV represents the spectrum analysis for heart rate signal. Methods of power spectral density (PSD) may be classified as parametric and nonparametric. Nonparametric methods are based on FFT. The advantage of nonparametric methods is that the processing speed is high compared to others, but they suffer from limited frequency resolution due to finite signal length. 67

3 The parametric methods permit the power spectral estimation by choosing an appropriate model, and then substituting the estimated values into the theoretical PSD expression. The higher frequency resolution is the advantage of parametric methods, but the drawback of the parametric methods is that the model order should be selected appropriately. In frequency domain analysis, very low frequency (VLF), low frequency (LF) and high frequency (HF) regions are generally considered. The specific frequency regions are defined as per the guidelines of the Task force. The distribution of power in each frequency band indicates the physiological state. Power (ms 2 ) is measured in absolute units in all the three regions of frequency. Powers of LF and HF are also measured in normalised units (n.u) which show the relative value of each power element in proportion to the total power, after the subtraction of VLF component. The ratio of LF/HF is evaluated to indicate the balance of sympathetic and parasympathetic nervous system. The comprehensive analysis for HRV in frequency domain is summarised as total power of entire RR interval (ms 2 ), VLF power in very low frequency range (ms 2 ), LF power in low frequency range (ms 2 ), HF power in high frequency range (ms 2 ), LF power in normalised units (nu), HF power in normalised units (nu)[95]. Heart rate is controlled by the sympathetic and parasympathetic branches of the Autonomic Nervous System (ANS). The variability of the heart rate is the result of a balance between the sympathetic and parasympathetic nerves. During periods of rest, the parasympathetic response dominates and the heart rate lowers. During activity, the sympathetic response dominates and the heart rate increases. The beat-to-beat variation in heart rate is termed the HRV. Since the heart rate is controlled primarily by the ANS, assessment of the HRV provides important information about sympathetic and parasympathetic cardiovascular control. In humans, the power spectrum of the HRV signal is divided into several frequency bands such as very low frequency (VLF), the low frequency (LF) and High frequency (HF). The frequency bands are not fixed but change in relation to change in autonomic modulations of the heart rate. 68

4 he Task force of European Society of Cardiology and North American Society of Pacing and Electrophysiology defined the frequency bands of VLF (0Hz Hz), LF(0.04Hz Hz) and HF (0.15Hz Hz). A parasympathetic blockade diminishes LF power, especially when the patient is in the supine position. The high frequency (HF) band is defined from 0.15Hz Hz and is associated only with parasympathetic activity. The power distribution within this band varies with the respiration rate and is more pronounced in the supine than the standing position. In case of dysfunction, compensatory changes occur in the ANS control system in order to keep the entire circulatory system within normal operating limits. In particular, the presence of Diabetic Mellitus is known to reduce HRV and disrupt the sympathetic and parasympathetic nervous systems in humans [117,118] FFT for Estimation of Power Spectrum Power Spectral Density (PSD) indicates the strength (power) of signal as a function of frequency. It reflects the frequencies where the power in energy is high or low. Computation of PSD is achieved by method called FFT directly. DFT converts signal from time domain to frequency domain.the PSD distributions are derived from this by plotting it as a function of frequency along horizontal axis and intensity along vertical axis.fft is an algorithm to compute the DFT more quickly. FFT algorithm characterizes the magnitude and phase of the signal.fft results noise which is removed by subtracting the mean from the data. Power spectral distributions show the frequencies which have signal power. Average of the signal is considered to be the power. This is equivalent to the square of the FFT s magnitude in frequency domain. The HRV data is irregular and hence to find the spectrum of irregular sample data is to resample it. The resampling of the data results uniform samples[119]. The resampling frequency selected for the HRV data is 4Hz.The reason for selecting this is that considering average heart beat as 120 beats per minute.(bpm) results 120/60=2samples/sec.or 2Hz. According to Nyquist the sampling rate must be twice to that resulting 4Hz. So the uniformly spaced data is converted into frequency domain using DFT. FFT algorithm of DFT which is fast is implemented in the thesis. So the 69

5 procedure of estimating PSD involved converting the unevenly spaced samples into evenly spaced samples by linear interpolation. The Power spectral density is estimated equation (5.1) using MATLAB on HRV data and the plotted the power spectral density distributions. The power spectral distributions are derived by applying FFT[120] to HRV data and using MATLAB tool for FFT, plots of PSD distribution are obtained as shown in figures[5.1,5.2,5.3 and5.4] are obtained. The signal is freed from noise by subtracting the mean of the data from the actual data before applying FFT tool. For corresponding power to various frequencies of the PSD distribution plots is noted and all the noted powers are added resulting the PSD. The parametric method of estimating PSD is having algorithmic approach hence easy to implement and fast to compute. The parameter selection required for this method is N the data length which is the length of the record. Sequence of operations to be performed for Computation of spectral entropy(se) is described below 1. Transformation of HRV data into power spectrum by applying FFT 2. Computation of the power spectral density(psd) 3. Normalization of the PSD 4. Computation of spectral entropy(spen) using Shanon s entropy as given below H = - Σ P f ln P f (5.1) Where P f is the PDF at frequency f described Step1 is achieved by finding DFT using equation (5.2) [120] Where X (w) is the Discrete Fourier Transform (DFT) of x[n]. Where x[n] is the HRV data. Step2 is performed by estimating the power using equation (5.3) [120] (5.3) Step 4 is performed by using equation (5.4) [120] 70

6 Where S (W) is the Power Spectral Density (PSD) and X F (W) is the FFT [120]. The Power Spectral Density (PSD) distribution as a function of frequency is determined by the method described in using FFT for healthy subjects is plotted and shown in figure 5.1 Figure 5.1: Power Spectral Density of healthy subject Figure 5.1 shows that the PSD distribution of HRV for healthy subjects. It is seen from the figure the power density is spread widely over the frequency range from 0Hz Hz covering both lower and higher frequency range. It may be noted that for healthy subjects there are no significant high amplitude power density peaks occurring at any particular frequency range in the FFT spectrum. The Power Spectral Density (PSD) distribution as a function of frequency for Hypothyroid subjects is plotted and shown in figure 5.2 Figure 5.2: Power Spectral Density of Hypothyroid subject 71

7 Figure 5.2 shows that the PSD distribution of HRV for Hypothyroid subjects is spread over a narrow frequency range from 0Hz Hz. The PSD indicates that the PSD is confined to the lower frequency range compared to healthy subjects. The figure clearly shows for Hypothyroid subjects, there is a strong occurrence of high amplitude power density peaks in lower frequency region between 0.025Hz Hz. The Power Spectral Density (PSD) distribution as a function of frequency for Depression subjects is plotted and shown in figure 5.3. Figure 5.3: Power Spectral Density of Depression subject Figure 5.3 shows that the PSD distribution of HRV for Depression subjects is very much confined to a very narrow frequency region from 0Hz - 0.1Hz compared to healthy subjects. The figure shows for Depression subjects the occurrence of strong high amplitude power density peaks at very low frequency region between 0Hz Hz. The Power Spectral Density (PSD) distribution as a function of frequency for CHF subjects is plotted and shown in figure 5.4. Figure 5.4: Power Spectral Density of CHF subject 72

8 Figure 5.4 shows that the Power Spectral Density (PSD) distribution of HRV for CHF subjects. It is seen from the figure that the PSD is confined to a very narrow region of frequency range 0Hz Hz compared to healthy subjects. The figure shows that for CHF subjects the occurrence of high amplitude power density peak at very low frequency range between 0Hz Hz. The PSD derived from the FFT analysis of the HRV clearly illustrates that, for the cardiac and non-cardiac diseased subjects the power spectral energy is concentrated in a very small low frequency range exhibiting maximum spectral power concentration at lower frequencies compared to healthy subjects where the spectral power is distributed over both high and low frequency range. The results of power spectral analysis also show that for cardiac and non-cardiac diseased subjects, the high frequency componets in the power spectrum are totally absent and the spectral energy is mainly concentrated in lower frequency region. The absence of high frequency components in the power spectrum of cardiac diseased (CHF) subjects may be attributed to the absorption of high frequency components by the blocks occurring in the circulatory stream and in non- cardiac diseased subjects to ANS dysfunctio.[123,124] The Power Spectral Density (PSD) as a function of frequency for AF subjects is plotted and shown in figure 5.5. Figure 5.5: Power Spectral Density of AF subject Figure 5.5 shows that the PSD distribution of HRV for AF subjects. It is seen from the figure that the power density is spread over wide range of frequency covering both high and low frequency region from 0Hz - 0.5Hz. 73

9 The figure also indicates the occurrence of high amplitude power density peaks at both higher and lower frequency regions. The frequency range for power spectral distributions extends beyond the frequency range of healthy subjects. The power spectrum distribution for AF subjects indicates hyper dynamic activity of the heart and generation of additional spectral components in both lower and higher frequency regions. Principle behind AR method: It is assumed that the HRV data sequence RR(n) as the output of the system driven by white noise w(n) for Autoregressive (AR) power estimation. Typical transfer function of the AR system is an all pole function as described by the equation given below. [120,121] p RR( z) 1 p (5.5) i 1 a z i i 1 Difference equation (5.6)is used to describe the p th order AR process generated by RR(n). RR n) a RR( n 1) a RR( n 2) a prr( n ) (5.6) ( 1 2 p Hence the expression for the Power spectrum is 2 jw S X ( e ) (5.7) jw [ a e ] i From equation (5.7) it is clear that to find power spectrum the AR(p) model parameters a 1,a 2..a p and variance ψ 2 of the noise need to be estimated..we estimated the model parameters also known as AR coefficients using Yule-Walker method available as a direct tool in the MATLAB. In this procedure we used W (n) is taken as an estimation error. Using Equation (5.5) estimates are found using MATLAB. The error e( n) RR( n) RR'( n) (5.8) Where RR' ( n) ai RR( n i) where i ranges from 1 to p ( 5.9) 74

10 Least square predictor is used to find the minimum squared error E t by taking the square of equation (5.8). minimising Et with coefficients leads to Yule-Walker equations.. a g (5.10) where τ is the autocorrelation matrix of RR(n) with entries τ(i,j)= τ xx abs(i-j),g=[ transpose of [τ xx(1) τ xx(2) τ xx(3) ] and a=transpose of[a 1 a 2 a p ] Variance ψ 2 is expressed as 2 xx ( 0) a1 xx (1)... a p xx ( p) (5.11) since all the parameters required have been defined AR spectrum psd can be estimated. The selection of model order is the challenge involved in the PSD estimation. The model order is estimated by using Akaike criterion (AIC) [121,122]. 2 2 AIC( p) log[ ( p)] P (5.12) N Where ψ 2 (p) is the estimated variance at order p. Number of samples of the signal is N. The true order is the one at which AIC is minimum. Selection of model order AR Power spectral density estimation model order is not known. The challenge is to select the appropriate model order. The model order is estimated by using Akaike criterion (AIC)EQ(5.12). In the proposed work model parameters a 1,a 2..a p are estimated first. Then minimum squared error is found using least square predictor. using eq(5.10) known as Yule walker equation. Then variance is estimated using q(5.11).then AIC(p) is estimated using equation(5.12) for various orders and minimum value of AIC is noted,corresponding order is the model order selected. All the estimations corresponding to equation (5.7)- (5.12)are done using MATLAB tool. It is found that order 16 the AIC is least that is 0.5. For values of p from 2 to 16 AIC is noted. From the obtained AIC has a maximum of 1.8. Since AIC criterion showed minimum value at order 16. The model order is fixed as 16.Also it was observed that the model order was found lower with lower sampling frequencies. For frequencies less than 500Hz.But the sampling frequency is 500Hz the order selected satisfying AIC is 16 in the proposed work. Using MATLAB tool power spectram is plotted and PSD is estimated from the PSD distributions thus 75

11 obtained. Also it was observed that, the model order was found lower with lower sampling frequencies. For frequencies less than 500hz.But the sampling frequency is 500hz the order selected satisfying AIC is 16 in the work. Lower sampling frequency.with the optimum order selected the data is applied to Kubios software[137 ] where power is estimated for the corresponding frequency components and the PSD is estimated for one set of data. The PSD estimated using FFT and AR method are same values, AR method has the advantage of showing the frequency components smoothened. But in the thesis the implementation and verification is done only using FFT based method. This is because AR method involves tedious process of finding suitable order also AR computationally complex as the order is higher, where as the FFT method is simple and computationally fast to implement. Selection of parameters for PSD estimation Length of the data=length of the record=n Resampling frequency = 4Hz Model order p = Spectral Entropy Spectral Entropy (SE) represents the spectral complexity of the time series data. The SE describes the spread of power spectrum. Low value of SE indicates the concentration of the energy in few frequencies. SE quantifies the distribution width of the power spectrum [123,124]. From equation (3.21) SE is expressed as given below. SE P f ln P f (5.13) f Where Pf is the Probability Density Function (PDF) at frequency f Spectral Entropy(SE) is estimated from the power spectral density distributions derived from the procedure described in

12 Further the HRV data of the healthy and diseased subjects are investigated by estimating the Spectral Entropy (SE). The SE has been calculated using HRV data of both the healthy, cardiac and non-cardiac diseased subjects are shown in figure 5.6. Figure.5.6 SE values of healthy and diseased subjects It is seen from the figure 5.6 that the Spectral Entropy (SE) indicates higher value for healthy subjects (SE ~ 2.10) and lower SE values for diseased subjects. The SE values for Hypothyroid subjects ~0.40 and for Depression subjects SE~0.60. The SE values for non-cardiac diseased subjects are found to be the lowest compared to cardiac diseased subjects such as CHF (SE~ 0.75) and AF (SE~1.20) [123,124]. The average values of SE determined for healthy and diseased subjectsare tabulated in the table 5.1 and plotted and shown in the figure 5.7. Table 5.1: Average values of SE of disesed subjects and healthy subjects SUBJECT AVERAGE SE Healthy 2.00 Hypothyroid 0.45 Depression 0.70 CHF 0.84 AF

13 It is seen from the table that the average Spectral Entropy (SE) indicate higher value for healthy subjects (~2.0) and lower SE values for diseased subjects[123,124]. It is evident from the table 5.1 and figure 5.7 that for healthy subjects the average SE ~ 2.0, whereas for Hypothyroid subjects, the average SE ~0.45, for Depression subjects average SE~ 0.70 and for CHF subjects the average SE ~ Figure 5.7: Average values of SE of healthy and diseased subjects The results also indicate that Hypothyroid subjects show the lowest value for SE compared to other diseased subjects. The AF subjects indicates SE~1.20 which is lower than healthy subjects. Table 5.2: SE values (%) of diseased subjects with respect to healthy subjects SUBJECT SE (%) WRT TO HEALTHY SUBJECTS Healthy 100 Hypothyroid 23 Depression 35 CHF 42 AF 60 The table 5.2 displays the SE values for cardiac and non-cardiac diseased subjects in comparison to healthy subjects. The table indicates that for Hypothyroid subjects the SE~ 78

14 23%, Depression subjects ~ 35%, CHF~42% and AF ~60% of healthy subjects. The Lower Values of SE for cardiac diseased subjects (CHF, AF) agrees with the findings of Veera Reddy and Anuradha (2005) [11]. From the table 5.2 it is clearly evident that SE values clearly indicate low entropy values for cardiac (CHF ~42%) diseased and non-cardiac (Hypothyroid ~23% and Depression ~35%) diseased subjects. The results indicate that the spectral content of HRV for cardiac and non-cardiac diseased subjects are much lower compared to that of healthy subjects and suggests lower dynamic cardiac activity for the diseased subjects[124]. Further it is seen from the table that the SE values for non-cardiac diseased (Hypothyroid ~23% and Depression ~35%) subjects are much lower compared to cardiac (CHF~42%) diseased subjects. The table also indicates that among non-cardiac subjects the Hypothyroid subjects display lowest averaged entropy values ~23% compared to Depression subjects ~35%. For AF subjects all SE values show entropy values ~60% of healthy subjects indicating lesser spectral content compared to healthy subjects. 79