The distribution of values of trigonometric sums with linearly independent frequencies: Kac's problem revisited |
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| Authors: | Richard Barakat |
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| Institution: | Division of Applied Sciences, Harvard University, Cambridge, MA 02138, U.S.A.;Bolt Beranek and Newman Inc., Cambridge, MA 02138, U.S.A. |
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| Abstract: | The average frequency with which a trigonometric sum having linearly independent frequencies achieves a given value in a specified time interval was solved by Kac in the form of a double integral. In the present paper, Kac's solution is generalized by allowing the amplitudes of the sinusoidal terms to be statistically independent random variables possessing compact probability density functions. The average frequency is expressed as a Fourier cosine series. Representative numerical calculations are shown. The asymptotic form of the average frequency is also discussed. |
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| Keywords: | average frequency trigonometric sums level crossing |
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