Abstract
At the beginning of Chap. 1 we defined a system as a mapping of an input signal into an output signal, as expressed by (3.3).
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Notes
- 1.
This proof is believed to have been first presented by Norbert Wiener in “The Fourier Integral and Some of its Applications,” Cambridge University Press, New York, 1933, where the LTI operator is referred to as “the operator of the closed cycle.”
- 2.
Note that the zero input response (3.77) can also be obtained by exciting the system with \(f(t) = y(t_{0})\delta (t - t_{0})\). This is just a particular illustration of the general equivalence principle applicable to linear differential equations permitting the replacement of a homogeneous system with specified nonzero initial conditions by an inhomogeneous system with zero initial conditions using as inputs singularity functions whose coefficients incorporate the initial conditions of the original problem.
- 3.
These Fourier coefficients can be expressed in terms of the modified Bessel functions I n as follows: \(q_{n}^{\pm } = {(\pm i)}^{n}I_{n}( \frac{\alpha }{4\omega _{ 0}} )\).
- 4.
Here we adhere to the convention and define the pole in terms of the complex variable s = iω so that the roots of the denominator of the transfer function are solutions of \({s}^{2} - 2\alpha s +\omega _{ 0}^{2} = 0\).
- 5.
The problem has a long history. One of the early resolutions was on the thermal noise generated by the resistor. See, e.g., [11].
Bibliography
Lebensbaum MC(1963) Further study of the black box problem. In: Proceedings of the IEEE (Correspondence), vol 51, p. 864.
Papoulis A (1962) The fourier integral and its applications. McGraw-Hill Book Company, Inc., New York.
Slepian J (1949) Letters to the editor. Electr Eng 63:377.
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Wasylkiwskyj, W. (2013). Linear Systems. In: Signals and Transforms in Linear Systems Analysis. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3287-6_3
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