WK in scrambler thing REVD
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Short Word-document section, marked by its author as obsolete but kept for a while, from Phil's scrambler folder. It recalls the continuum Wiener-Khintchine theorem from his Fourier transform notes, using the convolution theorem and the autocorrelation. It then derives a Z-transform analogue for an infinite sequence of repeated P-length subsequences, relating the pulse-train power spectrum to the Z transform of the autocorrelation sequence. The text has some garbled equations.
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This is obsolete but keep for a while.
(j) The Z-Transform Wiener-Khintchine Theorem
In FT Section 32 (c) we derive the continuum version of the Wiener-Khintchine theorem as follows, making use of the Convolution Theorem FT (3.6),
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 FT (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 theorem, although strictly the theorem is that Rx(ω) = |X(ω)|2.
We shall now obtain a similar Wiener-Khintchine theorem for an infinite sequence which contains repeated P-length subsequences. We define the autocorrelation sequence as
rs ≡ !Syntax Error, Iyn yn+s = !Syntax Error, Iy-n y-n+s
where in the last expression we took n→-n. We can then construct a Z Transform version of the Wiener-Khintchine theorem as follows:
rs = !Syntax Error, I ys-n y-n
as = !Syntax Error, I∆t bs-n cn 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 ↔ 1
Thus, the diagonalized z-domain form A"(z) = ∆t B"(z) C"(z) is
R"(z) =Y"(z) Y"(z)* = | Y"(z) |2 .
It is shown in FT (34.14) that
P(ω) = T1 Ppulse(ω) (1/T) !Syntax Error, I !Syntax Error, I ym* yn eiω(m-n)T FT (34.14)
which we easily rewrite as
P(ω) = Ppulse(ω) (T1/T) | Y"(z) |2 (2.5.43)
Using (2.5.42) we get,
P(ω) = Ppulse(ω) R"(z) . (2.5.44)
This says that the spectral power density of repeated P-subsequence infinite pulse train can be obtained by computing the Z Transform of the autocorrelation sequence and multiplying by Ppulse(ω). This then is our Z Transform version of the Wiener-Khintchine theorem.