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# chapter1 - Chapter 1 Mathematical Methods In this chapter...

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Unformatted text preview: Chapter 1 Mathematical Methods In this chapter we will study basic mathematical methods for characterizing noise pro- cesses. The two important analytical methods, probability distribution functions and Fourier analysis, are introduced here. These two methods will be used frequently through- out this text not only for classical systems but also for quantum systems. We try to present the two mathematical methods in a compact and succinct way as much as possible. The readers may find more detailed discussions in excellent texts [1]-[6]. In particular, most of the discussions in this chapter follow the texts by M.J. Buckingham [1] and by A.W. Drake [2]. 1.1 Time Average vs. Ensemble Average Noise is a stochastic process consisting of a randomly varying function of time and space, and thus is only statistically characterized. One cannot argue a single event at a certain time or position; one can only discuss the averaged quantity of a single system over a certain time (or space) interval or the averaged quantity of many identical systems at a certain time instance (or spatial position). The former is called time (or space) average and the latter ensemble average. Let us consider N systems which produce noisy waveforms x ( i ) ( t ), as shown in Fig. 1.1. 1 Figure 1.1: Ensemble average vs. time average. One can define the following time-averaged quantities for the i-th member of the en- semble: x ( i ) ( t ) = lim T →∞ 1 T Z T 2- T 2 x ( i ) ( t ) dt , (mean = first-order time average) (1.1) x ( i ) ( t ) 2 = lim T →∞ 1 T Z T 2- T 2 h x ( i ) ( t ) i 2 dt , (mean square = second-order time average) (1.2) φ ( i ) x ( τ ) ≡ x ( i ) ( t ) x ( i ) ( t + τ ) = lim T →∞ 1 T Z T 2- T 2 x ( i ) ( t ) x ( i ) ( t + τ ) dt . (autocorrelation function) (1.3) One can also define the following ensemble-averaged quantities for all members of the ensemble at a certain time: h x ( t 1 ) i = lim N →∞ 1 N N X i =1 x ( i ) ( t 1 ) = Z ∞-∞ x 1 p 1 ( x 1 ,t 1 ) dx 1 , (mean = first-order ensemble average) (1.4) h x ( t 1 ) 2 i = lim N →∞ 1 N N X i =1 h x ( i ) ( t 1 ) i 2 = Z ∞-∞ x 2 1 p 1 ( x 1 ,t 1 ) dx 1 , (mean square = second-order ensemble average) (1.5) 2 h x ( t 1 ) x ( t 2 ) i = lim N →∞ 1 N N X i =1 x ( i ) ( t 1 ) x ( i ) ( t 2 ) (1.6) = Z ∞-∞ x 1 x 2 p 2 ( x 1 ,x 2 ; t 1 ,t 2 ) dx 1 dx 2 . (covariance ) Here, x 1 = x ( t 1 ), x 2 = x ( t 2 ), p 1 ( x 1 ,t 1 ) is the first-order probability density function (PDF), and p 2 ( x 1 ,x 2 ; t 1 ,t 2 ) is the second-order joint probability density function. p 1 ( x 1 ,t 1 ) dx 1 is the probability that x is found in the range between x 1 and x 1 + dx 1 at a time t 1 and p 2 ( x 1 ,x 2 ; t 1 ,t 2 ) dx 1 dx 2 is the probability that x is found in the range between x 1 and x 1 + dx 1 at a time t 1 and also in the range between x 2 and x 2 + dx 2 at a different time t 2 ....
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## This note was uploaded on 12/29/2011 for the course PHYSICS 731 taught by Professor Appelbaum during the Fall '11 term at Maryland.

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chapter1 - Chapter 1 Mathematical Methods In this chapter...

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