mods

This is the Title PhL 3.26.05 This is just a temp working file, everything here has been taken care of. How would section 20 look for a finite pulse train? x(t) =!Syntax Error, I xpulse(t - tn) tn = nT1 pulse train (14.17) X(ω) = Xpulse(ω) 2πδ5(ωT1,N) = c(ω) 2π T1δ5(ωT1,N) (14.19) xpulse(t) = T1 δ(t) = δ(t/T1) // dimensionless Xpulse(ω) = T1 (8.3) c(ω) = 1 (14.12) item 2 x(t) = !Syntax Error, Iδ(t - nT1) (14.12) item 1 (20.1) X(ω) = 2π T1δ5(ωT1,N) (14.19) w(t) = y(t) x(t) =!Syntax Error, Iy(t)δ(t - nT1) =!Syntax Error, Iy(tn)δ(t - nT1) =!Syntax Error, Iynδ(t - nT1) (20.3) Equation (20.5) no longer works without real delta functions!!! So we no longer have W(ω) = Y(ω) + !Syntax Error, IY(ω - mω1) . not true for finite train (20.7) So I really don't want to "add" anything to Section 20. Maybe the discrete FT can help here??? I see no further changes needed before Section 25. Section 25: I see no need to add anything here! So I think now I can jump to Section 32 and start worrying about things there. Section 32 is OK for any x(t) really, no need to add anything about finite pulse trains here. Section 33 definitely needs some finite pulse train addition! x(t) = !Syntax Error, I xpulse(t - nT1) (14.17) (33.xx) X(ω) = Xpulse(ω) 2πδ5(ωT1,N) (14.10) (33.xx) | X(ω) |2 = | Xpulse(ω)|2 [2πδ5(ωT1,N)]2 From (A.21) we now replace [2πδ5(ωT1,N)]2 = (2N+1) 2π δ6(k,N) so that | X(ω) |2 = (2N+1) | Xpulse(ω)|2 2π δ6(k,N) P(ω) = = (2N+1) 2π δ6(k,N) P1(ω) = = 2π δ6(k,N) Then we sort of stop for a while with this result which is exact for finite N. Section 34 what happens here? For a finite train we get down to here with no changes <rx(t)> = !Syntax Error, I !Syntax Error, I<aman> rpulse[t + (m-n)T1)] (34.7) <|X(ω)|2 > = |Xpulse(ω)|2 !Syntax Error, I !Syntax Error, I <aman> eiω(m-n)T (34.9) No changes at all here, just show finite sums is all! I added a trailer sectino showing the key results with finite sums. Section 35 what happens here? Now we have to do a separate finite N section. Finite Pulse train with Random Coefficients Using the above forms for the coefficient averages, (34.9) becomes <|X(ω)|2> = |Xpulse(ω)|2 { α!Syntax Error, I !Syntax Error, I [ eiω(m-n)T] +β!Syntax Error, I !Syntax Error, I [1] } (35.4) To evaluate the first double sum, we write it as !Syntax Error, I !Syntax Error, I [ eiω(m-n)T] = !Syntax Error, I { !Syntax Error, I [ eiω(m-n)T] – 1 } = ( !Syntax Error, I e+iωnT ) (!Syntax Error, Ie-iωmT ) – !Syntax Error, I1 = | !Syntax Error, I e+iωnT |2 - !Syntax Error, I1 We now quote two results from Appendix A !Syntax Error, I eink = 2π { } = 2πδ5(k,N) !Syntax Error, I 1 = (2N+1) (A.32) so the first double sum in (35.4) becomes !Syntax Error, I !Syntax Error, I [ eiω(m-n)T] = [2πδ5(ωT1,N)]2 - (2N+1) (35.5) The second double sum in (35.4) is just !Syntax Error, I !Syntax Error, I [1] = !Syntax Error, I [1] = (2N+1) (35.6) and therefore we may write (35.4) as <|X(ω)|2> = |Xpulse(ω)|2{ α [ [2πδ5(ωT1,N)]2 - (2N+1) ] + β (2N+1) } or = |Xpulse(ω)|2 { α [ - 1 ] + β } (35.7) = |Xpulse(ω)|2 { α + (β-α) } (35.7) = |Xpulse(ω)|2 { (1/4) + (1/4) } (35.7) Now we divide both sides by 2π. The left side then becomes <P(ω)>/(2N+1) = <P1(ω)> as in (33.6). And |Xpulse(ω)|2/2π = Ppulse(ω) so, <P1(ω)> = Ppulse(ω) { α + (β-α) } (35.8) At this point Appendix A says δ6(k,N) ≡ = (A.21) so I then have = |Xpulse(ω)|2 { α 2π δ6(k,N) + (β-α) } (35.7) Divide by 2π to get <P1(ω)> = Ppulse(ω) { (β-α) + α 2π δ6(k,N) } ********************************************* (a) Spectral Energy Density of an Finite Pulse Train We start with x(t) = !Syntax Error, I xpulse(t - nT1) (14.1) (33.1) X(ω) = !Syntax Error, I Xpulse(mω1) 2π δ(ωT1 - 2πm) (14.4) (33.2)