finite pulse trains
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Short working note by Phil dated 8.3.13, part of the Spectral Theory Book update on dropping ensembles. It writes the power spectrum as the single-pulse spectrum times the pulse-train autocorrelation, expressed through the z-transform with z = e^{iωT}. It works out N = 0 (one pulse) and N = 1 (three pulses) explicitly, then notes that <y_n* y_{n+s}> cannot equal <y_n>^2 since it varies with s. Some summation symbols were garbled in extraction.
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Finite Pulse Trains PhL 8.3.13
I am trying now to "treat" finite pulse trains in the new "no ensemble" environment. I have these facts:
rs ≡ !Syntax Error, I yn* yn+s ≡ <yn* yn+s>
R"(z) = (!Syntax Error, I yn* zn) (!Syntax Error, I ym z-m) = !Syntax Error, I !Syntax Error, I yn* ym z-(m-n)
P(ω) = Ppulse(ω) | Y"(z) |2 = Ppulse(ω) R"(z)
I think each of these equations is exact for finite pulse trains.
Case 0: Finite Pulse train with N = 0 has just a single pulse.
rs ≡ !Syntax Error, I yn* yn+s only exists from -2N to 2N which here is just s = 0
r0 = |y0|2
R"(z) = |y0|2
P(ω) = Ppulse(ω) |y0|2
This is all I think perfectly accurate.
Case 1: Finite Pulse train with N = 1 has three pulses
rs ≡ !Syntax Error, I yn* yn+s ≡ <yn* yn+s> only exists from -2 to 2
r-2 = !Syntax Error, I yn* yn-2 = [ y-1*y-3 + y0*y-2 + y1*y-1] = y1*y-1
r-1 = !Syntax Error, I yn* yn-1 = [ y-1*y-2 + y0*y-1 + y1*y0] = [ y0*y-1 + y1*y0]
r0 = !Syntax Error, I yn* yn = [ y-1*y-1 + y0*y0 + y1*y1]
and so on.
R"(z) = (!Syntax Error, I yn* zn) (!Syntax Error, I ym z-m)
= (!Syntax Error, I yn* zn) (!Syntax Error, I ym z-m)
= [ y-1* z-1 + y0* + y1* z] [ y-1 z1 + y0 + y1 z-1] z = eiωT
P(ω) = Ppulse(ω) R"(z)
Again, I think this is all exact. Now is this true? :
<yn* yn+s> = <yn>2 ?
Obviously it is not true because if it were true, <yn* yn+s> could not vary with s.