European Journal of Scientific Research - EuroJournals

832 Taba Mohamed Tahar, S. Femmame and D. Mossadeg
L
n = ∑
k = 0
k
L
n−k
a~
n−k
n−k
+ n = ∑ [ hk j ] an−
k + ω n
k = 0
x h j ω
As a result, we can obtain a new (derotated) sequence:
~ x = x
−n
j
n
=
n
L
∑
k = 0
−k
n
[ hk
j ] a~
−
n−k
+ j ω n
~ (19)
Derotation not only changes the GMSK detection into a simpler BPSK detection problem, it
can also create channel diversity useful in blind equalization [1]
Since { }
n
1
n
a ~ is real-valued sequence, we can induce two sub-channel outputs from (20):
L
{ ~
−k
−n
x } = Re[
h j ] a~
+ Re[
j ]
n
∑
k=
0
x = Re ω
x
2
n
k
n−k
L
{ ~
−k
−n
x } = Im[
h j ] a~
+ Im[
j ]
∑
k=
0
n−k
n
= Im ω (21)
n
k
n
Where the common input is a BPSK signal, i.e.
~ = = ± 1.
−
a n
n
j an
From the BPSK input data sequence, two sub-channels can be generated without over-sampling
and extra antenna,
{ } { } { } { } k
1
−k
hk
= Re( hk
j ) and
2
−
hk
= Im( hk
j
We hence arrive at the familiar equation in SIMO [7] (Single Input / Multiple Output) models:
X[k] = Hã[k] + n[k]
Where:
{ }
[ ]
{ [ ] } ,
⎡Re
x[
k]
⎤ { }
x k = ⎢
Im
⎥
⎣ x k ⎦ { } ,
⎡Re
H ⎤ { }
H = ⎢
Im
⎥ []
⎣ H ⎦ { [] } .
⎡Re
n[
k]
⎤
n k = ⎢
Im
⎥
⎣ n k ⎦
(22)
Re(
k
)
k
Im( h
− share no common zeros [2].
H will have full column rank if { } h − and { }
4. Cross-Relation Method
Consider a SIMO system of q outputs given by:
= ∑
=
M
k 0
k j
y(
l)
h(
k)
s(
l − k)
+ n(
l)
(23)
The noise free outputs yi(k), 1≤ i≤ q are given by:
yi(k) = hi(k) * s(k), 1≤ i≤ q (24)
where “*” denotes convolution. Using commutativity of convolution, it follows:
hj(k) * yi(k) = hi(k) * yj(k), 1≤ i