Digital Signal Processing Reference
In-Depth Information
v
R
(
t
)
=
s
(
t
)
c
(
t
)
+
n
(
t
)
2
P
S
d
(
t
)
c
(
t
) cos
(7.2)
⇒
v
R
(
t
)
=
ω
c
t
+
n
(
t
)
After multiplying the received signal by PN chip code c(t) we get,
2
P
S
d
(
t
)
c
2
(
t
) cos
v
m
(
t
)
=
ω
c
t
+
n
(
t
)
c
(
t
)
2
P
S
d
(
t
) cos
(7.3)
⇒
v
m
(
t
)
=
ω
c
t
+
n
(
t
)
c
(
t
)
The 2nd term of the Eq. (
7.3
) indicates the noise is chopped, i.e. (1) noise is spread
whereas wanted data is de-spread and hence (2) power spectral density lowers as
shown in the Fig.
7.7
.
Next, v
m
(t) is again passed through another multiplier, integrator, i.e. BPSK
detector. The probed signals are as follows.
=
P
S
(2 cos
2
v
o
(
t
)
ω
c
t
)
d
(
t
)
+
n
(
t
)
c
(
t
) cos
ω
c
t
(7.4)
The signal v
0
(t)isthensampledatt
=
kT
B
to get the final output v
L
(t).
T
B
v
L
(
t
)
=
v
o
(
t
)
dt
;
k
=
1
0
(7.5)
T
B
P
S
(2 cos
2
ω
c
t
dt
⇒
v
L
(
t
)
=
ω
c
t
)
d
(
t
)
+
n
(
t
)
c
(
t
) cos
=
d
(
t
)
0
By this way the data signal d(t) has been correctly detected at the end output.
7.4.3 Probability of Error Calculation
From the discussions of
Chaps 3
and
6
, it has been shown that, the probability of
error of BPSK for matched filter reception without employing spread spectrum is
2
erfc
E
S
1
/
2
1
P
e
=
(7.6)
η
η
2 is the two sided power spectral density
where,
E
S
is the symbol energy,
(variance) of the noise.
To consider the case of direct sequence spread spectrum incorporated with BPSK
(Eq.
7.2
), within bit interval 0
≤
t
≤
T
B
, the transmitted signal is
s
t
(
t
)
=
d
(
t
)
g
(
t
)
c
(
t
) cos 2
π
f
c
t
;0
≤
t
≤
T
B
(7.7)
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