njit-etd2003-081 - New Jersey Institute of Technology

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73 Let the wavelet function ψ(t) be centered at (t0, Ω0) in the time-frequency plane. 3.7.2 Parseval Relation of Wavelet Transforms (Energy Preservation Property) The CWT has an energy conservation property that is similar to Parseval's formula of the Fourier transform. Now the Parseval relation in the wavelet transform can be shown as: holds for any signal f (t) which is squared-integrable. Its proof requires the admissibility condition, which was defined in Eq. (3.66). By using the dual relation of the wavelet transform in the time-frequency domain
74 The measure dadb used in the transform domain is consistent with the discussion a 2 of the scaling property. Scaling by s while conserving the energy will spread the wavelet transform by s in both the dimensions a and b, and thus a renormalization by 1 a is necessary. It is worth noting that the wavelet transform energy between the different scales is also preserves such that 3.7.3 Discrete Wavelet Transform Although the admissibility condition assures the complete representation of f (t) with its wavelet transform coefficients W f (a ,b) , it requires the wavelet transform operation to be performed for all values of a and b which are continuous parameters. This transform representation is not practical. One would prefer to perform the wavelet transform operation as few times as possible. Therefore these scaling or dilation, and translation or shift parameters, a and b respectively, are discretized. This discretization provides a transform grid or frame on the time-scale plane for the representation of signal f (t) . It is intuitive that this grid or frame should be defined properly such that the complete representation of f (t) is still possible. This is called the Discrete Wavelet Transform (DWT). This version of the wavelet transform reduces the redundancies of the wavelet space W f (a,b) significantly. The mathematical reasoning on the choice of frames or grids is perfectly treated in the literature.
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Copyright Warning & Restrictions Th
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ABSTRACT TIME-FREQUENCY INVESTIGATI
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TIME-FREQUENCY INVESTIGATION OF HEA
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APPROVAL PAGE TIME-FREQUENCY INVEST
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BIOGRAPHICAL SKETCH (Continued) D.A
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ACKNOWLEDGMENT The author wishes to
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TABLE OF CONTENTS Chapter Page 1 IN
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TABLE OF CONTENTS (Continued) Chapt
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LIST OF FIGURES Figure Page 2.1 The
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LIST OF FIGURES (Continued) Figure
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LIST OF FIGURES (Continued) Figure
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ABBREVIATIONS A ABP Arterial Blood
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ABBREVIATIONS (Continued) P PID Pro
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2 mortality and sudden death. [2] B
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4 1) Patients with severe pulmonary
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6 examined for determining the inte
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8 identified using principal compon
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1 0 1. Present the application of t
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12 1.3 Outline of the Dissertation
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CHAPTER 2 PHYSIOLOGY BACKGROUND Bio
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16 Figure 2.2 The systemic and pulm
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18 illustrated in Figure 2.3. The i
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20 2.2 Blood Pressure The force tha
Page 51 and 52: 22 2.4 The Nervous System Human beh
Page 53 and 54: 24 The sympathetic nerve fibers lea
Page 55 and 56: 26 Without these sympathetic and pa
Page 57 and 58: 28 center in the medulla, which con
Page 59 and 60: 30 Figure 2.6 Autonomic innervation
Page 61 and 62: 32 average heart rate was measured
Page 63 and 64: 34 However, they do note that there
Page 65 and 66: Figure 2.9 The placement of the pos
Page 67 and 68: 38 female. While more men suffer fr
Page 69 and 70: 40 Stage II: Moderate COPD - Worsen
Page 71 and 72: CHAPTER 3 ENGINEERING BACKGROUND Th
Page 73 and 74: 44 Two common types of time-frequen
Page 75 and 76: 46 STFT: Short-Time Fourier Transfo
Page 77 and 78: 48 3.3 The Analytic Signal and Inst
Page 79 and 80: 50 The advantage of using equation
Page 81 and 82: 52 3.5 Covariance and Invariance Th
Page 83 and 84: where H(f), S(f) are Fourier transf
Page 85 and 86: 56 Another shortcoming of the spect
Page 87 and 88: 58 should take the kernel of the WD
Page 89 and 90: 60 called the cross Wigner distribu
Page 91 and 92: 62 3.6.3 The Choi-Williams (Exponen
Page 93 and 94: 64 Figure 3.3 Performance of the Ch
Page 95 and 96: 66 [-Ω,Ω ], then its STFT will be
Page 97 and 98: 68 This condition forces that the w
Page 99 and 100: 70 where c is a constant. Thus, the
Page 101: Figure 3.5 The time-frequency plane
Page 105 and 106: 76 and the wavelet transform repres
Page 107 and 108: 78 Figure 3.6 Figure depicting the
Page 109 and 110: 80 The final step to obtain the pow
Page 111 and 112: 82 It should be noted that if the w
Page 113 and 114: 84 The normal respiration rate can
Page 115 and 116: Figure 3.12 Power spectrum of BP II
Page 117 and 118: RR similar manner to give: When com
Page 119 and 120: 90 when there is significant correl
Page 121 and 122: 92 3.12 Partial Coherence Analysis
Page 123 and 124: 94 after removal of the effects of
Page 125 and 126: 96 The bulk of the theory and appli
Page 127 and 128: 98 technique is measurement time. T
Page 129 and 130: 100 usually attainable. The key poi
Page 131 and 132: 102 variability exists in the propa
Page 133 and 134: 104 eXogenous input (ARX) was used
Page 135 and 136: 106 The baroreflex, an autonomic re
Page 137 and 138: 108 the principal components are no
Page 139 and 140: 110 The mathematical solution for t
Page 141 and 142: 112 3.15 Cluster Analysis The term
Page 143 and 144: 114 formed) one can read off the cr
Page 145 and 146: 116 3.15.5 Squared Euclidian Distan
Page 147 and 148: 118 Alternatively, one may use the
Page 149 and 150: 120 Sneath and Sokal used the abbre
Page 151 and 152: 122 may seem a bit confusing at fir
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CHAPTER 4 METHODS The purpose of th
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126 4.1.2.1 Autonomic Testing. HR V
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128 of heart rate, blood pressure,
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130 The patients who underwent LVRS
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132 panel of the Correct.vi. It was
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134 4.2.3 Power Spectrum Analysis o
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136 weighted-average value of the c
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138 For each given scale a within t
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140 frequency F to the wavelet func
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142 4.2.8 System Identification Ana
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144 In this study a simpler approac
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146 Table 4.2 Parameters That Make
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148 4.2.11 Cluster Analysis The sam
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150 viewing the time series of sequ
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Figure 5.2 BPV analysis of a COPD s
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Figure 5.3 HRV analysis of a normal
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Figure 5.4.1 Comparison of the HRV
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158 5.2 Time Frequency Analysis One
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Figure 5.5 Test signal with 3 sine
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162 Figure 5.6 (c) CWD of a signal
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164 Figure 5.7 (c) WT (dB4 wavelet)
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166 HRV more information about HRV
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168 Figure 5.9 (c) CWD plots of a n
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Figure 5.10 CWT (Morlet) HRV plot o
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172 The following figures show the
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174 Figure 5.15 CWT (Mexican Hat) H
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176 5.2.5 Best Wavelet Selection fo
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178 Table 5.1 Correlation Indices o
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180 5.2.6 Vagal Tone and Sympathova
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182 These figures basically show th
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184 Figure 5.20 Sympathetic and par
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188 5.2.7 Time-Frequency Analysis (
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190 Figure 5.29 3D and contour plot
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192 Figure 5.33 3D and contour plot
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Figure 5.44 Plot of raw respiration
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Figure 5.46 The LF partial coherenc
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Figure 5.48 HF partial coherence pl
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Table 5.2 Cross-Spectral Analysis o
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Table 5.3 Cross-Spectral Analysis o
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Figure 5.50 HF coherence of COPD (1
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212 For better presentation of the
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Figure 5.53 Coherence and partial c
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216 2. Interpretation of the transf
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218 covariances of the parameters,
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220 deviations are interpreted as A
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222 Figure 5.58 Bode plot of the HR
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224 In this section a simple model
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226 The data for all 47 COPD subjec
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228 Figure 5.60 Normal and COPD cla
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230 Figure 5.61 Normal and COPD cla
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232 Figure 5.62 Normal classificati
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234 5.7 Cluster Analysis The purpos
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236 Figure 5.64 Severity classifica
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238 both the normal and COPD subjec
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240 In summary, COPD subjects had h
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APPENDIX A EXERCISE PHYSIOLOGY A.1
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244 A.3 Figure Out Your Target Hear
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APPENDIX B ANALYSIS PROGRAM LISTING
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248 4) Click on file, close to exit
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250 • TN 11
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252 B.1.2 Partial Coherence Between
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254
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256 Block Diagram !rime of record K
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258
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260 B.2.2 Time — Frequency Analys
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262 This program provides the STFT
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264 G(:j+1)=G(:,j+1)/(2*sum(G(:j+1)
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266 T=(length(Signa)/sample)/(Times
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268 subplot(3, 1,3), plot(T,E); xla
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270 4. The program creates five out
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272 B.2.3.4 Program to Generate Sym
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274 ylabel('frequency'); title('Ins
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276 The program will run and output
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278 axis([0 1 0 2]); grid on; xlabe
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280 vagal=sum(TFDs(HFC,1:k)); symto
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282 plot(J,symtopar); %plot(A,symto
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284 4. Remove the constant levels a
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286 Make sure the agreement is quit
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288 B.2.6 Principal Components Anal
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290 Columns 12 through 15 'LF_pcoh_
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292 I= 1.0000 -0.0000 -0.0000 -0.00
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294 variances = 3.4083 1.2140 1.141
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296 B.2.7 Cluster Analysis Program
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298 end [R,C]=size(Data); if length
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300 B.2.8 Cross-correlation Program
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302 C.3 Partial coherence of HR and
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304 [13] Madwed, J., and R. Cohen.
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306 [41] Mallat, S. G., "A Theory f
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[70] Tazebay, M.V., R.T. Saliba and
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