old AMI ensemble method section

This is the Title PhL 3.26.05 Note that page numbering is turned on in this template. (f) Power Spectral Density Calculation using the Ensemble Method The starting point is (35.5) which we repeat here, <P(ω)> = Ppulse(ω) !Syntax Error, I !Syntax Error, I <ymym+s> z-s . (35.5) In section (c) we determined these ensemble averages for random variables Yn and Ym <ymyn> = (-p2/a) a|m-n| m ≠ n // a ≡ (1-2p) <yn2> = p . (37.8) which we write now as <ymym+s> = (-p2/a) a|s| s ≠ 0 // a ≡ (1-2p) <yn2> = p . (37.8) Then (35.5) may be written as <P(ω)> = Ppulse(ω) !Syntax Error, I { p + (-p2/a)!Syntax Error, I a|s|z-s } But we calculated {.....} in (37.10) so we don't repeat all that work and just quote the result, { p + (-p2/a)!Syntax Error, I a|s|z-s } = z = eiωT Thus we have <P(ω)> = Ppulse(ω) !Syntax Error, I = Ppulse(ω) ( !Syntax Error, I[1] ) The remaining sum is either 2πδ(0) or (2N+1) in the limit N→∞. But (T/T1) is this same quantity, so we find that ( ) = 1 and our final result is <P(ω)> = Ppulse(ω) This is then the mean of the spectral power densities of all the pulse trains in the ensemble. Since we are in the N → ∞ limit, this is the same as P(ω) for any pulse train in the ensemble so we end up with P(ω) = Ppulse(ω) (37.15) which is the same as (37.14) as computed by the autocorrelation method. As usual, in the "ensemble method", there is no need to talk about <....>1 type averages, or autocorrelation sequences.