Theory, Design and Tests on a Prototype Module of a Compact ...

2. THE PERTURBED TRANSMISSION MATRIX 55
<strong>and</strong> Pi = I+B, I being the identity matrix <strong>and</strong> if one defines b = 2jωM
then
⎛
0
B = ⎝
0
⎞
b
⎠ ,
0
⎛
1
Qi = ⎝
0
⎞
0
⎠
−1
(4.42)
2.2. The perturbed transmission matrix of the whole chain.
The behavior of the whole chain is represented by the product of the
transmission matrix for each cell; in the case of errors, this product is
not given from T N because the T is different for each cell. It is given
by the product of terms like T + Pp + Qp instead.
Ttot = ( ˙
T + P1 + Q1)( ˙
T + P2 + Q2) . . . ( ˙
T + PN + QN)
= ˙ T N + P1 T ˙ . . . ˙
T
+
(N−1)times
˙ T P2 T ˙ . . . ˙
T
+ . . . +
(N−2)times
˙ T . . . ˙
T
PN−1
(N−2)times
˙ T
+ ˙ T . . . ˙
T
PN + Q1 T ˙ . . .
(N−1)times
˙
T
+
(N−1)times
˙ T Q2 T ˙ . . . ˙
T
(N−2)times
+ . . . + ˙ T . . . ˙
T
QN−1
(N−2)times
˙ T + ˙ T . . . ˙
T
QN + high order terms
(N−1)times
(4.43)
where we neglected the product of two or more perturbed matrices.
Then, in a short form we can write
Ttot ∼ = ˙
T N +
N
T˙ p−1 Pp ˙ T N−p +
p=1
The first term of the previous sum is
N
T˙ p−1 Qp ˙ T N−p
p=1
T˙ p−1 Pp ˙ T N−p = UΛ p−1 U −1 Pp UΛ N−p U −1
= ε + p UΛ p−1 U −1 (I + B)UΛ N−p U −1 =
T˙ N−1 + ε + p UΛ p−1 U −1 BUΛ N−p U −1
= ε + p
(4.44)
<strong>and</strong> since T can be expressed as the product of a diagonal matrix by
the eigenfunctions matrices, one obtains
Ttot = ˙
T N + ˙
T N−1
+
N
p=1
N
ε + p +
p=1
N
p=1
ε − p UΛ p−1 U −1 QpUΛ N−p U −1
ε + p UΛ p−1 U −1 BUΛ N−p U −1 +
(4.45)
The expressions for the third <strong>and</strong> fourth terms of previous formula are
obtained in a similar way. Then, the expression for the transmission