Biomedical Engineering Reference
In-Depth Information
T
t
w
t
f
t
r
Fig. 5.5
Trapezoidal pulse train
5.2.2.1 Effect of Pulse Width on UWB Transmit Spectrum
The trapezoidal waveform generated by the pulse generator can be characterised
by properties, such as rise and fall times (t
r
,t
f
), pulse width (t
w
) and pulse period
(T), as shown in Fig.
5.5
.
If equal rise and fall times are assumed for UWB pulses (i.e. t
r
= t
f
), the basic
UWB pulse (p (t)) can be expressed in following manner:
8
<
0
t\
t
r
t
w
=
2
A
ð
t
þ
t
r
þ
t
w
=
2
Þ=
t
r
t
r
t
w
=
2
t
t
r
=
2
p
ð
t
Þ¼
A
t
w
=
2\t\t
w
=
2
ð
5
:
3
Þ
:
A
ð
t
r
þ
t
w
=
2
t
Þ=
t
r
t
w
=
2
t
t
w
=
2
þ
t
r
0
t [ t
w
=
2
þ
t
r
where A is the maximum pulse amplitude, t
r
is the rise and fall times of the pulse
and t
w
is the pulse width. The Fourier series expansion of the pulse train can be
expressed as below:
X
n2
n
¼
n1
x
ð
t
Þ¼
2A
ð
t
w
þ
t
r
Þ
T
sin pf
n
t
r
pf
n
t
r
sin pf
n
ð
t
w
þ
t
r
Þ
pf
n
ð
t
w
þ
t
r
Þ
cos
ð
2pnft
Þ
ð
5
:
4
Þ
where f
n
= n/T, T is the time period of the pulse train, and n is an integer, n
1
= f
1
/f
and n
2
= f
2
/f where f
1
and f
2
are upper and lower cut off frequencies for the BPF.
The sensor nodes described in this chapter are intended to operate in the frequency
range of 3.5-4.5 GHz. Hence f
1
= 3.5 GHz and f
2
= 4.5 GHz for this particular
design. This frequency band is chosen in order to avoid possible interference that
can be generated due to the operation of other equipment in RF bands, such as
5 GHz WLAN. The first sinc function in (
5.4
) is determined by the rise time of the
pulses, while the second sinc function depends on both rise time and pulse width.
This results in an output waveform that is a function of both rise time and pulse
width.
Transmit spectrums of simulated UWB pulse streams with pulse widths of 1,
0.5 and 2 ns are shown in Fig.
5.6
. These UWB transmit spectrums are obtained
using simulations conducted in Advanced Design Systems, which is a commercial
