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SeeH. W. Bode: “Network Analysis and Feedback Amplifier Design”.
The reason that I have put\(z = \frac{{\omega _m }}{\omega }\) instead of\(\frac{{\omega _m }}{\omega }\) is that I prefer to have the upper limit of the integral ofR c equal to 1 instead of equal to ∞.
We have here assumed that there is no dependance between μ c and\(\left\{ {\mu _m^{(v)} } \right\}\).
See “Regelungstheorie” byJost Hänny.
Ifg[x(t)] is not a linear function with constant coefficients, we can considerg[x(t)]=g(t) as an input function.
According to a commonly used notationx normal (a,b) means thatx is normally distributed with the mean valuea and the standard deviationb.
With linear systems are here meant such systems which can be described by linear differential equations with constant coefficients.
Of course, it need not be a “message” in the ordinary sense. The word is used here only for the sake of convenience.
The symbol 0 () means as usual “small of the same order as”.
As before we assumex(t)≡0 andv λ(t)≡0 fort≤0.
It is easily seen that the use of different α-values αλ for differentk λ(τ) implies no further generality in this case.
Observe that σ and τ belong to the same time interval.
As shown in the theory of functional series this is possible under very genral conditions.
At least within a certain interval.
See for instance the treatise byR. S. Phillips in “Radiation Laboratory Series 25”. Many of Phillips' notations are used in this chapter.
It is easily shown that\(\mathop {\lim }\limits_{\tau \to \infty } R(\tau ) = [My]^2 \).
For the sake of simplicity I writeY(f) instead ofY (2πjf).
Shannon:A mathematical theory of communication. Bell Syst. Tech. J. July 1948, October 1948.—Shannon:Communication in the presence of noise. Proc. IRE. Jan. 1949. —Gabor:Theory of communication. J. IEE. Nov. 1946.—Tuller:Theoretical limitations on the rate of transmissions of information. Proc. IRE. May 1949.
For the sake of simplicityV has not been made dependent on the amplitude, but it is easy to extend the reasoning to this more general case.
To speak about informationlines would, of course, be inadequate.
There exist, of course, many other ways to introduce the influence of the noise on the quantity of information.
c is here a calibration factor.
Further we assume thatP{ε (f)=0} be practically zero.
See for exampleR. S. Phillips; Radiation Laboratory Series 25.
μ=number of periods.
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Sundström, M. Some statistical problems in the theory of servomechanisms. Ark. Mat. 2, 139–246 (1952). https://doi.org/10.1007/BF02590880
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DOI: https://doi.org/10.1007/BF02590880