ABC equation
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Word-document derivation dated 3.26.05, kept in the support folder of the Spectral Theory Book files. It works with a pulse sequence of weights A, B and C at delays 0, T and 2T, and finds the Fourier transform as a comb of delta functions at multiples of ω1/3. It then derives the power spectrum as the single-pulse spectrum times (1/3)|A + B e^{-iωT} + C e^{-i2ωT}|^2. Equation symbols are partly garbled by extraction errors.
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This is the Title PhL 3.26.05
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General results for ABC
X"(z) = X'ω)/T1 = !Syntax Error, Iyn e-iωnT = [ A + B e-iωT + C e-i2ωT]!Syntax Error, I e-iω(3m)T .
!Syntax Error, I e-iω(2m)T = 2π δ5(3ωT1,N/3)
X"(z) = [ A + B e-iωT + C e-i2ωT] 2πδ5(3ωT1,N/3) . (34.16)
X"(z) = [ A + B e-iωT + C e-i2ωT] 2π !Syntax Error, Iδ(3ωT1-2πm)
= (1/3) [ A + B e-iωT + C e-i2ωT] ω1 !Syntax Error, I δ(ω - mω1/3)
X(ω) = Xpulse(ω) = (1/3) [ A + B e-iωT + C e-i2ωT] ω1 !Syntax Error, I δ(ω - mω1/3)
|X"(z)|2 = | A + B e-iωT + C e-i2ωT|2 [2π δ5(2ωT1,N/2)]2
δ6(k,N/3) ≡
= | A + B e-iωT + C e-i2ωT|2
= | A + B e-iωT + C e-i2ωT|2 δ6(k,N/3)
= 3 = 3 = | A + B e-iωT + C e-i2ωT|2 δ6(k,N/3)
= (1/3) | A + B e-iωT + C e-i2ωT|2 δ6(3ωT1,N/3)
P(ω) = |Xpulse(ω)|2 (1/T1)(1/3) |A + Be-iωT+ C e-i2ωT|2 δ6(3ωT1,N/3)
P(ω) = Ppulse(ω) (1/3) |A + Be-iωT+ C e-i2ωT|2 2πδ6(3ωT1,N/3)
P(ω) = Ppulse(ω) (1/3) |A + Be-iωT+ C e-i2ωT|2 !Syntax Error, I2π δ(3ωT1 - 2πm)
P(ω) = Ppulse(ω) (1/3)2 |A + Be-iωT+ C e-i2ωT|2 !Syntax Error, I2π δ δ(ω - mω1/3)