在研究生理系統時，生理訊號的分形維度(fractal dimension)和頻譜(frequency spectra)是很重要的兩項指標。前者可萃取出隱藏於雜訊中集體訊號的強度，後者可找出訊號的節律。在此論文中，首先我們以頻譜分配函數(spectral distribution function)來估測頻譜密度，此方式在尋找節律時擁有高的對比率和低的錯誤率。在論文中我們也和傳統的periodogram時頻分析方式和傅立葉方法(Fourier method) 作了比較，其模擬的結果優於periodogram和傅立葉方法。其次，將此方法結合分形維度，進行大白鼠儲尿期間膀胱之交感和副交感神經的交互作用研究。實驗共用了6隻完整的成年母鼠並完成了18次的實驗，分析了其外尿道括約肌的肌電圖和膀胱壓力圖，結果顯示儲尿期間外尿道括約肌並沒有任何頻率出現，此外，其分形維度(1.5918±0.0157) 顯示訊號強度並無可觀的變化。另一方面，膀胱訊號顯示出8Hz的副交感神經頻率擁有高之平均訊雜比(signal-to-noise ratio (SNR) = 19.9001 decibel (dB))，而19 Hz的交感神經頻率亦擁有高平均訊雜比SNR=22.8330 dB。此外，於儲尿期間膀胱的分形維度(1.4796±0.0092)比外尿道括約肌的(1.5918±0.0157)更具相對強的變化及統計意義(P < 0.01)。最後，我們可推斷：儲尿期間，外尿道括約肌並無收縮，膀胱的交感神經及副交感神經的功能是相互協和，而不是相互抑制。

The fractal dimension (FD) and spectral frequency of physiological signals are two important indices in the study of physiological functions. The former can extract the intensity and the latter the rhythm of signals embedded in random noise. In this dissertation, a new time-frequency spectral estimation method via spectral distribution function (SDF) that can detect rhythms with high contrast and low error-probability is invoked. This method is based on the fact that SDF includes the discrete components that are characteristic during rhythmic oscillation and, hence, is ideal for analyzing non-stationary signals whose spectral properties evolve over time. It is proved that the proposed method is unbiased and consistent. We also compare it with the conventional time-frequency approaches such as periodogram-based and Fourier methods. Simulation results indicate that the proposed method outperforms periodogram-based and traditional Fourier methods.
Secondly, both indices were used to study the involved muscles of bladder and external urethral sphincter (EUS) in the lower urinary tract during the urine storage phase. Eighteen experiments were performed on six intact adult female Wistar rats and then the electromyogram (EMG) of EUS and cystometrogram (CMG) of bladder were analyzed. Results indicated that the EUS did not contain any significant spectral frequencies in the storage phase. Furthermore, its FDs (1.5918±0.0157) indicated that no appreciable amount of signal intensities was observed in the EUS. On the other hand, the bladder exhibited parasympathetic frequency of 8 Hz with signal-to-noise ratio (SNR) = 19.9001 decibel (dB) for group mean, and sympathetic frequency of 19 Hz with SNR = 22.8330 dB for group mean. In addition, its FDs (1.4796±0.0092) indicated relatively persistent intensities during storage as compared to that of EUS (1.5918±0.0157) with statistical significance (P < 0.01). Consequently, we conclude that the EUS is not activated during the phase of storage. It is the bladder that is under the cooperative, not antagonistic, innervations of sympathetic and parasympathetic nervous systems with discernible rhythmic frequencies and intensities.

[1] H. Akaike, “Power spectrum estimation through autoregressive model fitting,” Ann. Inst. Statist. Math., vol. 21, pp. 407-419, 1969.
[2] F.J.F. Barrington, The component reflexes of micturition in the cat, Brain, vol 54, pp. 177–188, 1931.
[3] V. A. Billock, G. C. de Guzman and J. A. Scott Kelso, "Fractal time and 1/f spectra in dynamic images and human vision, " Physica D, vol 148, pp.136–146, 2001.
[4] R B. Blackman and J. W. Tukey, "The measurement of power spectra from the point of view of communications engineering". Bell Sysf. Tech. J., vol. 33, pp. 185-282, 485-569, 1958.
[5] P. Bloomfield, Fourier Analysis of Time Series – An Introduction. New York: Wiley, 1976.
[6] R.N. Bracewell, The Fourier Transform and Its Applications, 3d edition. New York: McGraw Hill, 2000.
[7] M.F. Campbell, P.C. Walsh, and A.B. Retik, "A.B. Campbell's Urology," Philadelphia, PA: Saunders, 2002.
[8] K. W. Chan and H. C. So, "An eigenvalue residuum-based criterion for detection of the number of sinusoids in white Gaussian noise," Digitial Signal Process., vol. 13, pp.275-83, 2003.
[9] H. L. Chan, H. H. Huang, and J. L. Lin. "Time-frequency analysis of heart rate variability during transient segments," Annals of Biomedical Engineering, vol. 29, pp. 983-996, 2001.
[10] M.B. Chancellor and N. Yoshimura, "Neurophysiology of stress urinary incontinence," Rev. Urol. 6 (Suppl 3) pp.S19-S28, 2004.
[11] S. Chang, M. C. Hsyu, H. Y. Cheng, S. H. Hsieh, "Synergic co-activation of muscles in elbow flexion via fractional Brownian motion,” Chinese J. Physiol., vol. 51, pp. 360-368, 2008.
[12] S. Chang, S. J. Hu and W. C. Lin, "Fractal dynamics and synchronization of rhythms in urodynamics of female Wistar rats," J. Neurosci. Methods, vol. 139, pp. 271-9, 2004.
[13] S. Chang, S. J. Li, M. J. Chiang, S. J. Hu, and M. C. Hsyu, "Fractal dimension estimation via spectral distribution function and its application to physiological signals," IEEE Trans. Biomed. Eng., vol. 54, pp. 1895-8, 2007.
[14] S. Chang, S. T. Mao, T. P. Kuo, S. J. Hu, W. C. Lin, and C. L. Cheng. "Fractal geometry in urodynamics of lower urinary tract, " Chin. J. Physiol., vol 42 pp. 25-31, 1999.
[15] S. Chang, S. T. Mao, S. J. Hu, W. C. Lin, and C. L. Cheng. " Studies of detrusor-sphincter synergia and dyssynergia during micturition in rats via fractional Brownian motion. IEEE Trans. Biomed. Eng., vol. 47 pp. 1066-73, 2000.
[16] S. Chang, M. C. Hsyu, H. Y. Cheng, S. H. Hsieh, C. C. Lin, "Synergic co-activation in forearm pronation," Annals of Biomed. Eng., vol. 36, pp. 2002-18, 2008.
[17] R. M. Chen, and S. A. Chang. "State and Parameter-Estimation for Dynamic-Systems with Colored Noise - an Accuracy Approach Based on the Minimum Discrepancy Measure, " IEEE Trans. Auto. Control, vol. 36 pp. 813-823, 1991.
[18] C.L. Cheng, C.Y.Chai and W.C. de Groat, "Detrusor-sphincter dyssynergia induced by cold stimulation of the urinary bladder of rats," Am. J. Physiol, vol. 272, pp.1271–1282, 1997.
[19] D. G. Childers, Modern Spectrum Analysis. New York: IEEE Press, 1978.
[20] K. H. Chon, "Accurate identification of periodic oscillations buried in white or colored noise using fast orthogonal search, " IEEE Trans. Biomed. Eng., vol. 48, pp. 622-9, 2001.
[21] J. W. Cooley and J. W. Tukey, " An algorithm for the machine calculation of Fourier series, " Math. Comput., vol. 19, pp. 297-301, 1965.
[22] J. W. Cooley, P. A. W. Lewis, and P. D. Welch, "Historical notes on the fast Fourier transform, " IEEE Trans. Audio Electro-acoust., vol. AU-15, pp. 76-79, 1967.
[23] W.C. De Groat, Urinary Incontinence. Chapter 10, Mosby-Year book, 1997.
[24] W.C. De Groat, J.W. Downie, R.M. Levin, A.T. Long Lin, J.F.B. Morrison, O. Nishizawa, W.D. Steers and K.D. Thor, "Basic neurophysiology and neuropharmacology, " In: Incontinence, edited by Abrams, P., Khoury, S. and Wein, A. Plymouth, UK: Plymouth Distributors Ltd, pp. 105–155, 1999.
[25] P.M. Djuric, "A model selection rule for sinusoids in white Gaussian noise," IEEE Trans. Signal Process., vol. 44, pp. 1744-51, 1996.
[26] J. L. Doob, Stochastic Processes. New York: Wiley, 1953.
[27] P. Embrechts and M. Maejima, Selfsimilar Processes. Princeton: Princeton University Press, 2002.
[28] J. J. Fuchs, "Estimating the Number of Sinusoids in Additive White Noise," IEEE Trans. Signal Process., vol. 36, pp.1846-53, 1988.
[29] W.H. Gaskell, The Involuntary Nervous System, London, Longmans: Green and Co.,1916.
[30] P. T. Gough, "A Fast Spectral Estimation Algorithm-Based on the Fft," IEEE Trans Signal Processing, vol. 42, pp. 1317-1322, 1994.
[31] A. C. Guyton and J. E. Hall, Textbook of Medical Physiology, Philadelphia, W.B. Saunders, 2001
[32] B.L. Hao, H.C. Lee and S.Y. Zhang, "Fractals related to long DNA sequences and complete genomes," Chaos, Solitons & Fractals, vol. 11, pp. 825-836, 2000.
[33] M. H. Hayes III, Statistical Digital Signal Processing and Modeling. New York: Wiley, 1996.
[34] S. Haykin (ed.), Advances in Spectrum Analysis and Array Processing, vol. 3. Englewood Cliffs, NJ: Prentice Hall, 1995
[35] C. Karanikas, "The Hausdorff dimension of very weak self-similar fractals described by the Haar wavelet system," Chaos, Solitons & Fractals , vol. 11 pp. 275-280, 2000.
[36] S. M. Kay, Modern Spectral Estimation, Theory and Application. Englewood Cliffs, NJ: Prentice Hall, 1988.
[37] S. B. Kesler, Modern Spectrum Analysis II. New York: IEEE Press, 1986.
[38] A.N. Kolmogorov, "Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum," C.R. (Doklady) Acad. Sci. USSR, vol. 16, pp.115–118, 1940.
[39] L. H. Koopmans, The Spectral Analysis of Time Series. New York: Academic Press, 1974.
[40] M.N. Kruse, A.L. Belton and W.C. de Groat, "Changes in bladder and external urethral sphincter function after spinal cord injury in the rat," Am. J. Physiol., vol. 264, pp.1157–1163, 1993.
[41] J.N. Langley, The Autonomic Nervous System. England, Cambridge, Heftier & Sons, 1921.
[42] O.R. Langworhty, L.C. Kolb and L.G. Lewis, Physiology of Micturition. Baltimore, The Williams & Wilkins Company, 1940.
[43] S.C. Liu and S. Chang, "Dimension estimation of discrete-time fractional Brownian motion with applications to image texture classification," IEEE Trans. Image Processing, vol. 6, pp.1176–1184, 1997.
[44] E. G. Lovett and J. B. Myklebust. "Approximate minimum bias multichannel spectral estimation for heart rate variability," Annals of Biomedical Engineering, vol. 25, pp.509-520, 1997.
[45] B.B. Mandelbrot, "The Fractal Geometry of Nature," San Francisco: Freeman, 1982.
[46] S. L. Marple, Digital Spectral Analysis with Applications, Englewood Cliffs, NJ: Prentice Hall, 1987.
[47] L. Nottale, "On the transition from the classical to the quantum regime in fractal space–time theory," Chaos, Solitons & Fractals, vol. 25, pp.797-803, 2005.
[48] E. Parzen, “Modern empirical spectral analysis,” Tech. Report N-12, Texas A&M Research Foundation, College Station, TX, 1980.
[49] D. B. Percival and A. T. Walden, Spectral Analysis for Physical Applications: Multitaper and Conventional. Univariate Techniques. Cambridge University Press, 1993.
[50] B. Porat, Digital Processing of Random Signals: Theory and Methods. Englewood Cliffs, NJ: Prentice Hall, 1994.
[51] M. B. Priestley, Spectral Analysis and Time Series. London ; New York: Academic Press, 1981.
[52] J. Proakis, C. M. Rader, F. Ling and C. L. Nikias, Advanced Digital Signal Processing. New York: Macmillan, 1992.
[53] E. A. Robinson, "A Historical-Perspective of Spectrum Estimation," Proceedings of the IEEE, vol. 70, pp. 885-907, 1982.
[54] X. Rode, "Musical sound signals analysis/synthesis: sinusoidal+residual and elementary waveform models", Applied Signal Processing , vol. 4, pp.131-141, 1997.
[55] L. L. Scharf, Statistical Signal Processing: Detection, Estimation and Time Series Analysis. Reading, MA:Addison Wesley, 1991.
[56] M. B. Siroky, and R. J. Krane, Urinary Incontinence. Chapter 14, Mosby-Year book, 1997.
[57] D. Sheehan, "Discovery of the autonomic nervous system," Arch. Neurol. Psychiat., vol. XXXV, pp.1081-1115, 1936.
[58] S.J. Shefchyk, "Sacral spinal interneurones and the control of urinary bladder and urethral striated sphincter muscle function," J. Physiol., vol. 533, pp.57–63, 2001.
[59] S. Standring, Gray's Anatomy. New York: Churchill Livingstone, 2005.
[60] P. Stoica and R. Moses, Spectral Analysis of Signals. Englewood Cliffs, NJ: Prentice Hall, 2005.
[61] C. W. Therrien, Discrete Random Signals and Statistical Signal Processing. Englewood Cliffs, NJ: Prentice Hall, 1992.
[62] D. J. Thomson, "Spectrum Estimation and Harmonic-Analysis," Proceedings of the IEEE, vol. 70, pp. 1055-1096, 1982.
[63] E. Tsakiroglou and A. T. Walden, "From Blackman-Tukey pilot estimators to wavelet packet estimators: a modern perspective on an old spectrum estimation idea," Signal Processing, vol. 82, pp. 1425-1441, 2002.
[64] R.F. Voss, "Fractals in nature: From characterization to simulation," In: The Science of Fractal Images, edited by Peitgen, H.O. and Saupe, D. New York: Springer-Verlag, pp. 21–70, 1988.
[65] J. C. Winters, and R. A. Appell, Urinary Incontinence. Chapter 24, Mosby-Year book, 1997.
[66] E. Wong and B. Hajek, Stochastic processes in engineering systems. New York: Springer-Verlag, 1985.
[67] K. Yoshida, Functional analysis, 6th ed. Berlin; New York: Springer-Verlag, 1980.