Biomedical Engineering Reference
In-Depth Information
8
<
n
B
ðÞ
if there is no pulse present
in a time slot
s
B
ðÞþ
n
B
ðÞ
if there is a pulse present
in a time slot
r
0
ðÞ¼
ð
3
:
10
Þ
:
where n
B
ðÞ
is the band-limited noise signal and s
B
ðÞ
is the filtered received
signal. With the assumption that the receiver node detects the presence of a pulse
by comparing the signal energies of the two BPPM slots (TS1 and TS2), proba-
bility of error for single pulse detection of the receiver with BPPM modulation
scheme can be derived from [
20
]:
t
E
p
N
0
0
1
2
@
A
P
e
¼
Q
ð
3
:
11
Þ
E
p
N
0
þ
T
s
B
2
where P
e
is the probability of error, B is the signal bandwidth (B = 1 GHz for the
receiver model used in the simulations), T
s
is the integration period which is equal
to the pulse width of 2 ns for the simulations, E
p
is the received signal energy
during the 2 ns integration period (T
s
) and Q ( ) represents the Q function. It should
be noted that the system described in [
20
] uses the detection of multiple pulses
during a single integration period, whereas the system presented in this chapter
detects only a single pulse within the integration period. For the system in [
20
], the
noise-by-noise product term increases due to the detection of multiple pulses
within the same integration window, significantly reducing the BER performance
of the system. Due to the independent detection of individual pulses and using
those individual detections to determine the presence of a bit, the noise by noise
product term does not affect the system presented in this chapter. Hence, the BER
equation presented in [
20
] reduces to (
3.11
) in this chapter for a single pulse
detection system.
When multiple PPB is sent, it is assumed that a bit is erroneous when more than
half the pulses sent per that bit are erroneous. If N pulses are sent per bit, prob-
ability that a bit being erroneous can be obtained by:
p
i
P
e
bit
¼
1
X
bc
i
¼
1
N
i
Þ
N
i
ð
1
p
ð
3
:
12
Þ
¼
N
!
i! N
i
N
i
where p = P
e
;
Þ
!
and
bc
is the inferior integer part of x. Modulation
ð
curves showing BER for different number of PPB are obtained based on (
3.12
) and
presented in Fig.
3.9
. It should be noted that the BER is plotted against pulse E
p
/N
0
in this figure. Bit energy can be obtained by the summation of pulse energies that
represent the bit. The results in Fig.
3.9
show that for the same E
p
/N
0
, sending
more number of PPB results in lower BER.
