FT 34 (b) rewrite REVD
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Short working note dated 7.22.13 by Phil, part of the August 2013 update to his spectral theory book. It derives |X(ω)|², the energy spectrum E(ω) and power spectrum P(ω) for a general pulse train x(t) = Σ yn xpulse(t−tn), ending in a double sum over ym* yn e^{iω(m−n)T}. Phil notes his digital W-K R"(z) usage ended up in the main FT document and this file may be deleted. Some summation symbols are garbled in the text.
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Rewrite of FT Section 34 (b) PhL 7.22.13
Summary: I am looking for some place to use my digital W-K R"(z) thing. The statement of this digital W-K gets inserted earlier in Sec 32. One page only below. Maybe we can delete this file. I did end up wedging the W-K R"(z) usage into this section 34 (b) which I think is the logical place it should appear, see FT doc for how it turned out.
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(b) Spectral power density for a General Pulse Train
In Section 33 (d) we dealt with simple pulse trains. Here we consider the more general amplitude modulated pulse train. In all equations, one can replace !Syntax Error, I by !Syntax Error, Ito adapt the equation to a finite pulse train instead of an infinite one. We start then with
x(t) = !Syntax Error, I yn xpulse(t -tn) (33.1) (34.5)
X(ω) = (1/T1)Xpulse(ω) X'ω) = Xpulse(ω) X"(z) (25.3) (34.6)
X"(z) = X'ω)/T1 = !Syntax Error, Iyn e-iωnT = Y"(z) (24.2) (34.7)
Comment:
To find the frequency-domain power spectrum of a signal x(t), our first task is to compute | X(ω) |2. From (34.6) we get
|X(ω)|2 = |Xpulse(ω)|2 (1/T1)2 |X'ω)|2 = |Xpulse(ω)|2 | X"(z) |2 z = eiωT (34.8)
We therefore must deal with the following object, using (34.7),
| X"(z) |2 = (1/T1)2 |X'ω)|2 = | !Syntax Error, Iyn e-iωnT | 2 = | Y"(z) |2
= !Syntax Error, Iyn e-iωnT !Syntax Error, Iym* e+iωmT = !Syntax Error, I !Syntax Error, I ym* yn eiω(m-n)T (34.9)
so that
| X(ω) |2 = | Xpulse(ω) |2 !Syntax Error, I !Syntax Error, I ym* yn eiω(m-n)T . (34.10)
From box (34.4) we then have
E(ω) ≡ |X(ω)|2/2π = (1/2π) | Xpulse(ω) |2 !Syntax Error, I !Syntax Error, I ym* yn eiω(m-n)T (34.11)
P(ω) ≡ = (1/2πT) | Xpulse(ω) |2!Syntax Error, I !Syntax Error, I ym* yn eiω(m-n)T (34.12)
which we can write as (again using box (34.4) results)
E(ω) = T1 Ppulse(ω) !Syntax Error, I !Syntax Error, I ym* yn eiω(m-n)T (34.13)
P(ω) = T1 Ppulse(ω) (1/T) !Syntax Error, I !Syntax Error, I ym* yn eiω(m-n)T (34.14)
P = !Syntax Error, Idω P(ω) . (33.31) (34.15)
Not knowing details of the yn there is not much else we can do in these expressions.