Edwin Jan Klein - Universiteit Twente

Appendix B. Crosstalk Derivation
The adjacent channel crosstalk is the relative power fraction of a signal that is present
on a channel adjacent to the channel on which this signal is transmitted and is defined
as:
⎛ PDrop
( λ = λc
+ ∆λcs
) ⎞
CT = 10log⎜
⎟
⎜
⎟
(B.1)
⎝ PDrop
( λ = λc
) ⎠
where λc is the wavelength of the channel for which the resonator is in full resonance
and ∆λcs is the separation between the channels.
Then with φ1 the roundtrip phase of a resonator in resonance:
( 2π
)
ϕ 1 =
λ
c
2
. R.
Neff
and φ2 the roundtrip phase of the adjacent channel:
2
( 2π
)
ϕ2
= . R.
Neff
λ + ∆λ
Combined with the drop response of the resonator:
P
Drop
c
cs
H
=
1+
F . sin
The expression for the crosstalk then becomes:
C
197
(
2 ϕ
⎛ H ⎞
⎜
2 ⎟
2
⎜ 1+
F
⎟ ⎛
C . sin ( ϕ 2 / 2)
1+
FC
. sin ( ϕ1
/
CT = 10log
= 10log
⎜
⎜
⎟
2
H
⎝1
+ FC
. sin 2
⎜
2 ⎟
⎝ 1+
FC
. sin ( ϕ1
/ 2)
⎠
/
2)
2)
( ϕ / 2)
The phase φ2 is equal to m.2π because λc is the resonance frequency. In addition FC
can be written as:
F C
1
= 2
sin ( π / 2F
)
Combining (B.5) with (B.6) then gives the expression:
⎛
⎞
⎜
⎟
2
⎜
1
⎟ ⎛ sin ( π / 2F
) ⎞
CT = 10log
= 10log
⎜
⎟
⎜
⎟
2
2
1
2
⎝ sin ( π / 2F
) + sin ( ϕ / 2)
⎜1
+
. sin ( ϕ / 2)
2
⎟
⎠
2
2
⎝ sin ( π / 2F
)
⎠
Then, since
⎞
⎟
⎠
(B.2)
(B.3)
(B.4)
(B.5)
(B.6)
(B.7)