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Noise theory

A detailed discussion of noise theory is given in references 1 and 2.

For any quantity X(t) that exhibits noise, the noise power within a unit bandwidth or power spectral density SX(f) is defined as

  equation32

where X(f) is the Fourier transform of X(t), and tex2html_wrap_inline483 is an ensemble average. In the simplest case where the transitions that cause the noise are described by equation (2), where tex2html_wrap_inline485 is the lifetime of the fluctuation causing interaction, the spectral density is given by equation (3).

  equation45

  equation50

From simple statistical considerations <( tex2html_wrap_inline489 )2> can usually be found, for example in the case of number fluctuations it is given by Poisson statistics.

This equation is very general with the condition imposed that the interactions of the electrons are independent. For small fluctuations this is indeed true and thus the Lorentzian spectrum (equation (3)) appears often. At low frequencies ( tex2html_wrap_inline493  <<1) the spectrum is white, that is independent of frequency, while at high frequencies ( tex2html_wrap_inline493  >>1) it varies as 1/f2, and its half power point is at f=1/(2 tex2html_wrap_inline503 ).





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