WK adder for 2_5 REVD
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Draft appendix dated 7.21.13 (WK Attempt #2) from Phil's Scrambler work, marked as not needed anymore since the Wiener-Khintchine material was moved to his Fourier Transform notes. It recalls the continuum theorem via the convolution theorem, then derives a Z-transform version for repeated P-length sequences. It applies this to the autocorrelation of a maximal length sequence, summing delta-function combs to get the spectral power density. Equations are partly garbled in the extraction.
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WK Attempt #2 PhL 7.21.13
All WK stuff got moved to FT where it belongs. Here I still had it in scrambler in trying to get the MLS spectrum.
Appendix G:
(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. In (2.5.33) we define the autocorrelation sequence as
rs ≡ (1/P) !Syntax Error, Iyn yn+s . 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, using T = (2N+1)PT1,
rs ≡ (1/P) !Syntax Error, Iyn yn+s ≈ !Syntax Error, Iyn yn+s = !Syntax Error, Iyn yn+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, Iyn yn+s = !Syntax Error, Iy-n y-n+s
where for 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 ↔ (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 . (2.5.42)
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.
(k) Using Wiener-Khintchine to compute the MLS Spectrum
We shall compute R"(z) and then use the Z transform Wiener-Khintchine theorem (2.5.44) to get P(ω). Recall that
rs = <an2> s = NP = β
rs = <anan+s> s ≠ NP = α . (2.5.35)
Therefore
R"(z) = !Syntax Error, Irn z-n = β !Syntax Error, Iz-n + α !Syntax Error, Iz-n
= β !Syntax Error, Iz-n + α ( !Syntax Error, I z-n – !Syntax Error, Iz-n )
= (β-α) !Syntax Error, Iz-n + α !Syntax Error, I z-n . (2.5.45)
The first sum is over n = 0, ±P, ±2P and so on. We can replace summation index n by index N,
!Syntax Error, Iz-n = !Syntax Error, I z-NP = !Syntax Error, I z-nP (2.5.46)
From FT (24.1) we know that z lies on the unit circle in the z-plane and is related to ω by
z = eiωT FT (24.1)
where T1 is the duration of a pulse of the pulse train. We then have
!Syntax Error, Iz-n = !Syntax Error, I (eiωT)-nP = !Syntax Error, I e-iωTnP . (2.5.47)
According to FT (13.2) we know how to compute this sum,
!Syntax Error, Ieink = !Syntax Error, I2πδ(k - 2πm) -∞ < k < ∞ . FT (13.2)
so setting k = -ωT1P we get
!Syntax Error, Iz-n = !Syntax Error, I2πδ(-ωT1P - 2πm) = !Syntax Error, I2πδ(ωT1P + 2πm) = !Syntax Error, I2πδ(ωT1P - 2πm) (2.5.48)
where we use δ(x) = δ(-x) and in the last step take m→ -m.
Meanwhile, our other sum of interest in (2.5.45) is this one,
!Syntax Error, I z-n = !Syntax Error, I (eiωT)-n = !Syntax Error, I e-iωTn
which is just the previous sum without the P. Thus,
!Syntax Error, I z-n = !Syntax Error, I2πδ(ωT1 - 2πm) . (2.5.49)
Inserting (2.5.49) and (2.5.48) into (2.5.45) gives
R"(z) = (β-α) !Syntax Error, Iz-n + α !Syntax Error, I z-n
= (β-α) !Syntax Error, I2πδ(ωT1P - 2πm) + α !Syntax Error, I2πδ(ωT1 - 2πm)
= (β-α) (T1P)-1 !Syntax Error, I2πδ(ωT1P - 2πm/[T1P]) + α(T1)-1 !Syntax Error, I2πδ(ωT1 - 2πm/T1)
= (2π/T1) { (β-α) (1/P) !Syntax Error, Iδ(ωT1P - 2πm/[T1P]) + α !Syntax Error, Iδ(ωT1 - 2πm/T1) }
= ω1 { (β-α) (1/P) !Syntax Error, Iδ(ωT1P - mω1/P) + α !Syntax Error, Iδ(ωT1 - mω1) } (2.5.50)
and this concludes our calculation of the Z Transform of the autocorrelation sequence rn. It only remains to install this into the Wiener-Khintchine theorem (2.5.44)
P(ω) = Ppulse(ω) R"(z)
= Ppulse(ω)ω1 { (β-α) (1/P) !Syntax Error, Iδ(ωT1P - mω1/P) + α !Syntax Error, Iδ(ωT1 - mω1) } (2.5.51)
This is in agreement with (2.5.21) which we stole from FT (F.22b) .