new FT 32 (f) REVD
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Working document dated 3.26.05 from Phil's spectral theory book files, used to develop a new final subsection of Section 32 without changing existing equation numbers. It defines the autocorrelation sequence of pulse train amplitudes, shows its Z transform equals |Y(z)|^2, and compares this with the ordinary Wiener-Khintchine theorem. It then relates the power spectrum P(ω) to the pulse spectrum and the Z transform of the autocorrelation. Some equations are garbled in extraction.
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New FT Section 32 (f) PhL 3.26.05
This is a new section at the end of Section 32, so no existing equation numbers get altered. We just tack this on at the end. But as I read now, I think <...> should be <....>1. !!! Yes, in FT I have fixed that up. This is all good. So this doc is just where I developed this new section 32 (f).
(f) Z Transform Wiener-Khintchine theorem for a Pulse Train
The Wiener-Khintchine theorem was stated above in section (c) as
Rx(ω) = |X(ω)|2 (32.8)
where Rx(ω) is the Fourier Integral transform of the autocorrelation function rx(t)
rx(t) ≡ !Syntax Error, I dt' x(t') x(t' + t) . (32.1)
From the amplitudes yn of an infinite pulse train one can define an autocorrelation sequence in analogy with the autocorrelation function,
rs ≡ limN→∞ [!Syntax Error, I yn yn+s ] = <yn yn+s> . (32.16)
Although rx(t) (our particular definition) has no normalization factor, we have added to the definition of rs in order to obtain the finite result rs = <yn yn+s> .
It is convenient to use this shorthand notation for rs,
rs = !Syntax Error, I yn yn+s = !Syntax Error, I yn yn+s (32.17)
where T = (2N+1)T1 is the duration of the pulse train. We know that this infinite T is going to cancel another T in any "application" so we allow it to exist temporarily, as in (33.22) where T = [2πδ(0)]T1 .
The Z transform of rs is given by
R"(z) ≡ !Syntax Error, I rs z-s = !Syntax Error, I{ !Syntax Error, I yn yn+s } z-s = !Syntax Error, I !Syntax Error, I [yn zn ] [yn+s z-(n+s)]
= !Syntax Error, I !Syntax Error, I [yn zn ] [ym z-m] m ≡ n+2
= [ !Syntax Error, I yn zn] [!Syntax Error, Iym z-m ] = Y"(z)* Y"(z)
so we have obtained this Z Transform version of the Wiener-Khintchine theorem, which we compare to the regular version,
R"(z) = | Y"(z) |2 Z Transform Wiener-Khintchine (32.18)
Rx(ω) = |X(ω)|2 . regular Wiener-Khintchine (32.8)
Here X(ω) is the Fourier Integral Transform of the pulse train x(t),
x(t) = !Syntax Error, I yn xpulse(t -tn). (25.1)
while Y"(z) is the Z transform of the sequence of pulse train amplitudes.
We now jump ahead in our presentation in order to show a key use of R"(z). Below we will define the spectral power and energy density P(ω) and E(ω) of an infinite pulse train and its pulse as
P(ω) ≡ = (33.23)
Ppulse(ω) ≡ = . (33.24)
The regular Wiener-Khintchine theorem (32.8) that Rx(ω) = |X(ω)|2 then tell us
P(ω) = or E(ω) = . (34.3)
Later we will show that
P(ω) = T1 Ppulse(ω) (1/T) !Syntax Error, I !Syntax Error, I ym* yn eiω(m-n)T (34.14)
= Ppulse(ω) | Y"(z) |2 . (32.19)
Inserting the Z transform Wiener-Khintchine theorem (32.18) then gives
P(ω) = Ppulse(ω) R"(z) (32.20)
which provides a simple relation between P(ω) and R"(z).