Abstract
There are several practical problems associated with the use of the Laplace transform to study input–output relationships in the laboratory. In particular, it is extremely difficult to obtain the Laplace integral transform for measured signals, and even if the transform is known, obtaining the inverse transform can be problematic. At the root of these problems is the lack of efficient numerical methods to calculate the Laplace transform and its inverse [146].
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Notes
- 1.
Jean Baptiste Joseph Fourier (1768–1830), French mathematician and physicist.
- 2.
James William Cooley (b. 1926), American mathematician; John Wilder Tukey (1915–2000), American mathematician.
- 3.
Michael Faraday (1791–1867), English physicist and chemist.
- 4.
Leopold Kronecker (1823–1891), German mathematician.
- 5.
Leon Ong Chua (b. 1936), American engineer.
- 6.
For a very accessible account of this derivation, see [629].
- 7.
Norbert Wiener (1894–1964), American mathematician; Aleksandr Yakovlevich Khinchin (1894–1959), Soviet mathematician.
- 8.
Max Planck (1858–1947), German theoretical physicist.
- 9.
Werner Karl Heisenberg (1901–1976), German theoretical physicist.
- 10.
This phenomenon was in fact discovered by Henry Wilbraham (1825–1883), English mathematician, and rediscovered by Josiah Willard Gibbs (1839–1903), American physicist, chemist, and mathematician.
- 11.
Marc-Antoine Parseval des Chênes (1755–1836), French mathematician.
- 12.
Ary L. Goldberger (b. 1949), American physician–scientist and fractal physiologist.
- 13.
Jan Evangelista Purkyně, or the Latinized Johannes Evangelist Purkinje (1787–1869), Czech anatomist and physiologist.
- 14.
Michael R. Guevara (PhD 1984), Canadian physiologist; Tim J. Lewis (PhD 1998), Canadian biomathematician.
- 15.
Claude Elwood Shannon (1916–2001), American mathematician, engineer, and cryptographer; Harry Nyquist (1889–1976), American electronic engineer.
- 16.
Note that we are just using one of our favorite tricks.
- 17.
This definition of Sinc(f) is the one most used in the signal-processing literature. In the mathematical literature, this function is written as sinc(t) = sin(t)∕t. However, (8.39) is the definition that will be encountered most often by laboratory investigators.
- 18.
We do not consider the important topic of digital filters. We have kept our focus on analog filters because we wish to draw analogies to the filtering properties of biological dynamical systems.
- 19.
John D. Hunter (1968–2012), American neuroscientist.
- 20.
Do not use PyLab’s version of psd(). The function matplolib.mlab.psd() produces the same power spectrum as that obtained using the corresponding programs in Matlab.
- 21.
Julius Ferdinand von Hann (1839–1921), Austrian meteorologist.
References
J. S. Bendat and A. G. Piersol. Random data: Analysis and measurement procedures, 2nd ed. John Wiley & Sons, New York, 1986.
W. E. Boyce and R. C. DiPrima. Elementary differential equations and boundary value problems, 9th ed. John Wiley & Sons, New York, 2005.
R. N. Bracewell. The Fourier transform and its applications, 2nd ed., revised. McGraw–Hill, San Francisco, 1986.
G. A. Bryant. Animal signals and emotion in music: coordinating affect across group. Frontiers in Psychology, 4:1–12, 2013.
B. A. Cohen and A. Sances. Stationarity of the human electroencephalogram. Med. Biol. Eng. Comput., 15:513–518, 1977.
J. W. Cooley and J. W. Tukey. An algorithm for machine calculation of complex Fourier series. Math. Computation, 19:297–301, 1965.
C. E. Epstein and J. Shotland. The bad truth about Laplace’s transform. SIAM Rev., 50:504–520, 2008.
A. S. French and A. V. Holden. Alias-free sampling of neuronal spike trains. Kybernetik, 5:165–171, 1971.
V. Fugère and R. Krahe. Electric signals and species recognition in the wave-type gymnotiform fish Apteronotus leptorhynchus. J. Exp. Biol., 213:225–236, 2010.
W. B. Gearhart and H. S. Shultz. The function sinx∕x. The College Mathematics Journal, 21:90–99, 1990.
A. L. Goldberger, V. Bharagava, B. J. West, and A. J. Mandell. On the mechanism of cardiac electrical stability: the fractal hypothesis. Biophys. J., 48:525–528, 1985.
J. D. Hunter, J. G. Milton, P. J. Thomas, and J. D. Cowan. Resonance effect for neural spike time reliability. J. Neurophysiol., 80:1427–1438, 1998.
F. M. Klis, A. Boorsma, and P. W. J. DeGroot. Cell wall construction in Saccharomycetes cerevisiae. Yeast, 23:185–202, 2006.
T. J. Lewis and M. R. Guevara. 1∕f power spectrum of the QRS complex does not imply fractal activation of the ventricles. Biophys. J., 60:1297–1300, 1991.
G. Meyer-Kress, I. Choi, N. Weber, R. Bargas, and A. Hübler. Musical signals from Chua’s circuit. IEEE Trans. Circuits Systems II Analog Digital Signal Processing, 40:688–695, 1993.
P. Moller. Electric fishes: History and behavior. Chapman & Hall, London, 1995.
E. Niedermeyer and F. Lopes da Silva. Electro-encephalography: Basic principles, clinical applications and related fields. Urban & Schwarzenberg, Baltimore, 1987.
K. Omori, H. Kojima, R. Kakani, D. H. Stavit, and S. M. Blaugrund. Acoustic characteristics of rough voice: subharmonics. J. Voice, 11:40–47, 1997.
R. J. Peterka, A. C. Sanderson, and D. P. O’Leary. Practical considerations in the implementation of the French–Holden algorithm for sampling neuronal spike trains. IEEE Trans. Biomed. Engng., 25:192–195, 1978.
W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. Numerical recipes: The art of scientific computing, 3rd ed. Cambridge University Press, New York, 2007.
X. Rodet. Sound and music from Chua’s circuit. J. Circuits Systems Computers, 3:49–61, 1993.
J. A. Simmons. Resolution of target range by echolocating bats. J. Acosut. Soc. Amer., 54:157–173, 1973.
A. D. Straw, B. Branson, T. R. Neumann, and M. H. Dickinson. Multi-camera real-time three-dimensional tracking of multiple flying animals. J. Roy. Soc. Interface, 8:6900–6914, 2010.
W. van Drongelen. Signal processing for neuroscientists: Introduction to the analysis of physiological signals. Academic Press, New York, 2007.
H. Zakon, J. Oestreich, S. Tallarovic, and F. Triefenbach. EOD modulations of brown ghost electric fish: JARs, chirps, rises, and dips. J. Physiol. Paris, 96:451–458, 2002.
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Milton, J., Ohira, T. (2014). Frequency Domain II: Fourier Analysis and Power Spectra. In: Mathematics as a Laboratory Tool. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-9096-8_8
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