possible W-K for section 2_5 REVD
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Working note by Phil dated 3.26.05, kept in a 'Not needed anymore' scrambler folder because the material is already in his FT notes. It derives the continuum Wiener-Khintchine theorem from the autocorrelation, then a Z-transform version for discrete sequences of repeated P-symbol subsequences. It ends with the power spectrum as a sum of delta-function lines using Fourier sum identities. Equation text is partly garbled.
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This is the Title PhL 3.26.05
This is all installed now in FT so not longer needed in scrambler.
(j) The Wiener-Khintchine pathway to the power spectrum.
In FT Section 32 (c) we derive the continuum version of the W-K theorem as follows:
rx(t) !Syntax Error, Idt"x(t - t") x(-t") . // energy units FT (32.7)
a(t) = !Syntax Error, I dt" b(t-t") c(t") A(ω) = B(ω) C(ω) . FT (3.6)
b(t) = x(t) ↔ B(ω) = X(ω)
c(t) = x(-t) ↔ C(ω) = X(ω) = X(ω)* // from (7.1) and (7.2)
Thus, the diagonalized frequency domain form A(ω) = B(ω) C(ω) is
Rx(ω) = |X(ω)|2 . FT (32.8)
Since
P(ω) ≡ = FT (33.23)
where T is the duration of a pulse train, we find that
P(ω) = 2πT Rx(ω) . FT (34. 3)
This says that the spectral power density of a pulse train can be obtained by computing the Fourier Integral Transform of the autocorrelation function and multiplying by 2πT. We can regard this as the Wiener-Khintchine (W-K) theorem, although strictly the W-K theorem is that Rx(ω) = |X(ω)|2.
Although this was not addressed in FT, one can obtain a similar W-K theorem for a discrete sequence {yk}. In (2.5.33) we define the autocorrelation sequence as
rs ≡ (1/P) . s = any integer (2.5.33)
If we regulate our infinite sequence as being (2N+1) copies of the P-symbol subsequence (where N is very large), then we can write the above as
rs ≡ (1/P) ≈ !Syntax Error, Iym ym+s = !Syntax Error, Iym ym+s .
We continue to represent the infinite duration of the pulse train by the symbol T, knowing that it is going to cancel another T later on, so we write the limit as
rs = !Syntax Error, Iym ym+s = !Syntax Error, Iy-m y-m+s
where for the last expression we took m→-m. We can then construct a Z Transform version of the W-K theorem as follows:
rs = !Syntax Error, I ys-m y-m
as = !Syntax Error, I∆t bs-m cm A"(z) = ∆t B"(z) C"(z) FT (24.5)
as = rs ↔ A"(z) = R"(z)
bk = yk ↔ B"(z) = Y"(z)
ck = y-k ↔ C"(z) = Y"(z-1) = Y"(z)* // if the ck are real
Δt ↔ (T1/T)
Thus, the diagonalized z-domain form A"(z) = ∆t B"(z) C"(z) is
R"(z) = (T1/T) Y"(z) Y"(z)* = (T1/T) | Y"(z)|2
It is shown at the start of FT Appendix F that
P(ω) = Ppulse(ω) (T1/T) | Y"(z) |2
and thus we may write
P(ω) = Ppulse(ω) R"(z)
This is our Z-transform version of the Wiener-Khintchine theorem. We can now use this compute the power spectrum of a sequence composed of repeated P-length subsequences. Consider:
rs = <an2> s = NP = β
rs = <anan+s> s ≠ NP = α (2.5.35)
The Z transform of this autocorrelation sequence is:
R"(z) = Σs=-∞∞ rs z-s = β Σs=NP z-s + α Σs≠NP z-s
The first sum is over s = 0, ±P, ±2P, etc. Let s = rP so then, using z = eiωΔt,
Σs=NP z-s = Σr=-∞∞ z-Pr = !Syntax Error, I(eiωΔt)-Pn = !Syntax Error, Ie-PiωΔtn
Then using FT (13.2) with k = -PΔt
!Syntax Error, Ieink = !Syntax Error, I2πδ(k - 2πm) -∞ < k < ∞ . FT (13.2)
and setting k = -ωPΔt we get
!Syntax Error, Ie-PiωΔtn = !Syntax Error, I2πδ(ωPΔt - 2πm)
and so
Σs=NP z-s = !Syntax Error, I2πδ(ωPΔt - 2πm)
As for the second sum,
Σs≠NP z-s = !Syntax Error, Iz-n – !Syntax Error, I z-Pn = !Syntax Error, I2πδ(ωΔt - 2πm) – !Syntax Error, I2πδ(ωPΔt - 2πm)
We have then shown that
R"(z) = β Σs=NP z-s + α Σs≠NP z-s
= β [!Syntax Error, I2πδ(ωPΔt - 2πm) ] + α [!Syntax Error, I2πδ(ωΔt - 2πm) – !Syntax Error, I2πδ(ωPΔt - 2πm) ]
= (β-α) !Syntax Error, I2πδ(ωPΔt - 2πm) ] + α [!Syntax Error, I2πδ(ωΔt - 2πm)
Now from ** we know that P(ω) = Ppulse(ω) R"(z) so we find that
P(ω) = Ppulse(ω) { (β-α) !Syntax Error, I2πδ(ωPΔt - 2πm) ] + α [!Syntax Error, I2πδ(ωΔt - 2πm) }
Now Δt = T1 so
P(ω) = Ppulse(ω) { (β-α) !Syntax Error, I2πδ(ωPT1 - 2πm) ] + α [!Syntax Error, I2πδ(ωT1 - 2πm) }
= Ppulse(ω) { (β-α)(1/PT1) !Syntax Error, I2πδ(ω- 2πm/PT1) ] + α (1/T1) [!Syntax Error, I2πδ(ω - 2πm/T1) }
= Ppulse(ω)(1/T1) { (β-α)(1/P) !Syntax Error, I2πδ(ω- mω1/P) ] + α [!Syntax Error, I2πδ(ω - mω1) }
= Ppulse(ω)ω1 { (β-α)(1/P) !Syntax Error, Iδ(ω- mω1/P) ] + α [!Syntax Error, Iδ(ω - mω1) }