AB example
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A short Word working file dated 3.26.05, intended as an example to follow equation (34.11) in Phil's spectral theory book. It computes the Fourier transform and power spectrum of a pulse train with amplitudes alternating between A and B, using delta-function comb identities. It works the box-pulse case with A = B = 1, where only the m = 0 term survives, and ends with Phil's note questioning a stray factor of 2 and the 2πδ(0) squaring step.
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This is the Title PhL 3.26.05
Note that page numbering is turned on in this template.
Example to put after (34.11)
ym = A if m is even, = B if m is odd
X"(z) = X'ω)/T1 = !Syntax Error, Iyn e-iωnT = A!Syntax Error, I e-iωnT + B !Syntax Error, I e-iωnT
n = 2m n = 2m+1
= A !Syntax Error, I e-iω(2m)T + B !Syntax Error, I e-iω(2m+1)T
= (A + Be-iωT) !Syntax Error, I e-iω(2m)T
= (A + Be-iωT) !Syntax Error, I2πδ(2ωT1 - 2πm) // using (13.2)
= (1/2)(A + Be-iωT) !Syntax Error, I2πδ(ωT1 - πm) // using (13.2)
Thus
X"(z) = (A + Be-iωT) !Syntax Error, I2πδ(2ωT1 - 2πm)
|X"(z)|2 = |A + Be-iωT|2 [2πδ(0)] !Syntax Error, I2πδ(2ωT1 - 2πm)
|X"(z)|2 = (1/4)|A + Be-iωT|2 [2πδ(0)] !Syntax Error, I2πδ(ωT1 - πm)
= (1/2)|A + Be-iωT|2 [2πδ(0)] !Syntax Error, I2πδ(2ωT1 - 2πm)
|X(ω)|2 = |Xpulse(ω)|2 | X"(z) |2
= |Xpulse(ω)|2 |A + Be-iωT|2 !Syntax Error, I2πδ(2ωT1 - 2πm)
P(ω) = =
= |Xpulse(ω)|2 (1/T1) |A + Be-iωT|2!Syntax Error, Iδ(2ωT1 - 2πm)
|A + Be-iωT|2 = (A + Be-iωT)( A + Be+iωT) = A2 + B2 + 2ABcos(ωT1)
δ(2ωT1 - 2πm) = (1/2T1) δ(ω - m(ω1/2))
P(ω) = (1/2) |Xpulse(ω)|2 (1/T1)2[A2 + B2 + 2ABcos(ω1T1)] !Syntax Error, I δ(ω - m(ω1/2))
Special case class: Suppose xpulse(t) = box height 1 width T1.
Xpulse(ω) = X(ω) = T1 sinc(ωT1/2)
P(ω) = (1/2) sinc(ωT1/2) [A2 + B2 + 2ABcos(ω1T1)] !Syntax Error, I δ(ω - m(ω1/2))
Special SubCase #1. A = 1 and B = 1.
[1 + 1 + 2cos(ω1T)] = 4cos2(ωT1/2)
P(ω) = (1/2) sinc(ωT1/2) 4cos2(ωT1/2) !Syntax Error, I δ(ω - m(ω1/2))
= 2 !Syntax Error, I sinc(mπ/2) cos2(mπ/2)δ(ω - m(ω1/2))
The factor here contains for m ≠ 0
2 sin(mπ/2) cos(mπ/2) = sin(mπ) = 0
so only the m = 0 term survives and the result is
P(ω) = 2 δ(ω)
But why do I get 2 here? Does my δ squaring operation fail?
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I need more work on when you can do the 2πδ(0) thing with a clear explanation of what scaling is allowed and what is not! I have swept this under the rug until now. More work needed as usual.