scrambler WK REVD

This is the Title PhL 3.26.05 This was something I was going to put inside Scrambler. Not sure how it ended up, but at least lets rename this to be scrambler_WK and file it in the scrambler folder. // Actually, all this stuff got put in a new release of FT, so I didn't need to put it into scrambler. (j) Using the Z-transform Wiener-Khintchine to compute the MLS Spectrum Here we show an alternate method for computation of the spectral power density P(ω) of an MLS sequence. Equation FT (32.20) states what is essentially the Z Transform Weiner-Khintchine theorem, P(ω) = Ppulse(ω) R"(z) , FT (32.20) (2.5.42) which allows one to compute P(ω) from the Z transform R"(z) of the autocorrelation sequence rs. We are interested in an infinite pulse train which is periodic with period P and consists of a repeating subsequence of length P. In (2.5.33) we defined the autocorrelation sequence for this subsequence as rs ≡ (1/P) !Syntax Error, Iyn yn+s = !Syntax Error, Iyn yn+s = <yn yn+s> (2.5.33) where <yn yn+s> is an expectation value over the P-length subsequence. We can rewrite rs by averaging over 2M+1 copies of the repeated subsequence, rs = !Syntax Error, I yn yn+s ≈ !Syntax Error, I yn yn+s (2.5.xx) where on the right we assume M is very large. Defining large N = MP this can be written rs ≈ !Syntax Error, I yn yn+s ≈ !Syntax Error, I yn yn+s . (2.5.xx) For an infinite sequence we take N→∞ so that rs = limN→∞ [!Syntax Error, I yn yn+s ] = <yn yn+s> (2.5.xx) This last equation is the definition of rs that appears in FT (32.16) and which applies to any infinite sequence, not just a periodic one. Our task now is to compute R"(z) for our MLS sequence and then to insert it into (2.5.xx) above to obtain P(ω). The staring point is, rs = <an2> s = NP = β rs = <anan+s> s ≠ NP = α . (2.5.35) Therefore R"(z) = !Syntax Error, Irn z-n = β !Syntax Error, Iz-n + α !Syntax Error, Iz-n = β !Syntax Error, Iz-n + α ( !Syntax Error, I z-n – !Syntax Error, Iz-n ) = (β-α) !Syntax Error, Iz-n + α !Syntax Error, I z-n . (2.5.45) The first sum is over n = 0, ±P, ±2P and so on. We can replace summation index n by index N, !Syntax Error, Iz-n = !Syntax Error, I z-NP = !Syntax Error, I z-nP . (2.5.46) From FT (24.1) we know that z lies on the unit circle in the z-plane and is related to ω by z = eiωT FT (24.1) where T1 is the duration of a pulse of the pulse train. We then have !Syntax Error, Iz-n = !Syntax Error, I (eiωT)-nP = !Syntax Error, I e-iωTnP . (2.5.47) According to FT (13.2) we know how to compute this sum, !Syntax Error, Ieink = !Syntax Error, I2πδ(k - 2πm) -∞ < k < ∞ . FT (13.2) so setting k = -ωT1P we get !Syntax Error, Iz-n = !Syntax Error, I2πδ(-ωT1P - 2πm) = !Syntax Error, I2πδ(ωT1P + 2πm) = !Syntax Error, I2πδ(ωT1P - 2πm) (2.5.48) where we use δ(x) = δ(-x) and in the last step take m→ -m. Meanwhile, our other sum of interest in (2.5.45) is this one, !Syntax Error, I z-n = !Syntax Error, I (eiωT)-n = !Syntax Error, I e-iωTn which is just the previous sum without the P. Thus, !Syntax Error, I z-n = !Syntax Error, I2πδ(ωT1 - 2πm) . (2.5.49) Inserting (2.5.49) and (2.5.48) into (2.5.45) gives R"(z) = (β-α) !Syntax Error, Iz-n + α !Syntax Error, I z-n = (β-α) !Syntax Error, I2πδ(ωT1P - 2πm) + α !Syntax Error, I2πδ(ωT1 - 2πm) = (β-α) (T1P)-1 !Syntax Error, I2πδ(ω - 2πm/[T1P]) + α(T1)-1 !Syntax Error, I2πδ(ω - 2πm/T1) = (2π/T1) { (β-α) (1/P) !Syntax Error, Iδ(ω - 2πm/[T1P]) + α !Syntax Error, Iδ(ω - 2πm/T1) } = ω1 { (β-α) (1/P) !Syntax Error, Iδ(ω - mω1/P) + α !Syntax Error, Iδ(ω - mω1) } (2.5.50) and this concludes our calculation of the Z Transform of the autocorrelation sequence rn. It only remains to install this into the Z Transform Wiener-Khintchine theorem (2.5.44) P(ω) = Ppulse(ω) R"(z) = Ppulse(ω) ω1 { (β-α) (1/P) !Syntax Error, Iδ(ω - mω1/P) + α !Syntax Error, Iδ(ω - mω1) } . (2.5.51) This is in agreement with (2.5.21) which we stole from FT (F.22b) and upon which we based all our MLS spectral results in sections (h) and (i). Comment: Our initial computation of the MLS spectrum including the work of FT Appendix F made no use whatsoever of the autocorrelation sequence, much less its Z transform or the Wiener-Khintchine theorem. For an MLS spectrum these concepts just provide an alternate pathway to computing the spectrum, and of course it is gratifying to see the result come out the same by either method. Deep down, and perhaps not really that deep down, both methods are the same method with a different order of calculation, and the intermediate objects have different names or in some cases no names at all.