Int. J. Communications, Network and System Sciences, 2010, 3, 843-849
doi:10.4236/ijcns.2010.311114 Published O nline Novem ber 2010 (http:// www.SciRP.org/journal/ijcns)
Copyright © 2010 SciRes. IJCNS
Power Spectral Analysis of Orthogonal Pulse-Based
TH-UWB Signals
Sudhan Majhi, Paul Richardson
Electrical and Computer Engineering, University of Michigan, Dearborn, USA
E-mail: sudhanmajhi@gmail.com, richarpc@umd.umich.edu
Received April 28, 2010; revised July 8, 2010; accepted September 10, 2010
Abstract
The paper analyzes power spectral density (PSD) of orthogonal pulse-based signals for time hopping ultra
wideband (TH-UWB) systems. Our extensive studies show that the PSD of these signals depends not only on
the time dithering code and the modulation schemes, but also on the energy spectral density (ESD) of or-
thogonal pulses. The different order orthogonal pulses provide different ESD which changes the shape of
continuous spectral component with symbols. We show that orthogonal pulse-based signals reduce the dy-
namic range of amplitude of discrete spectral components. Further, we reduce the dynamic range by adopting
longer TH code over orthogonal pulse-based signals. As a result, UWB system performance improves with
average transmitted power. The theoretical analysis of PSD of orthogonal pulse-based TH-UWB signal is
provided in details and verified through simulation results.
Keywords: Time Hopping Ultra Wideband Radio, Orthogonal Pulse Modulation, Power Spectral Density
1. Introduction
Impulse radio Ultra Wideband (IR-UWB) radio is a
promising technology for short range wireless commu-
nications. The information is conveyed by short-duration
pulse, which has potential to deliver high data rates with
low power spectral densities (PSD). Federal Communi-
cation Commission (FCC) has limited the PSD of the
signal to –41.25 dBm/MHz in the band from 3.1 GHz to
10.6 GHz. However, the UWB signal interferes with
other radio signals due to its noise like signal character-
istic. The power distribution of UWB signal over a large
bandwidth can be id entified by analyzing the PSD of the
signal. The reduction of PSD of UWB signal reduces
UWB interference on narrowband (NB) systems and
keeps UWB signal under the FCC spectral mask [1].
Recently, orthogonal pulses have taken a lot of atten-
tion in time hopping UWB (TH-UWB) systems [2]. Sev-
eral modulation schemes such as pulse shape modulation
(PSM), combined on-off -keying (OOK) and PSM, called
OOK-PSM, combined PSM and pulse position modula-
tion (PPM), called PSM-PPM, combined orthogonal
PPM (OPPM) and bi-orthogonal PSM (BPSM), called
OPPM-BPSM, have been proposed to simplify the
TH-UWB systems [3-5]. The orthogonal pulse-based
modulation schemes are becoming more popular due to
the simplicity of newly developed pulse designed algo-
rithms [6,7]. Th e efficiency of these orthogonal pulses is
studied in [8,9]. The spectral characteristic of these or-
thogonal pulses is provided in [10]. However, only few
papers described the PSD of orthogonal pulse-based
modulation schemes.
In this paper, we analyze the power spectral de nsity of
orthogonal pulse-based TH-UWB signal for PSM, BPSM,
OPPM-PSM and OPPM-BPSM schemes. We show that
orthogonal pulse-based modulation schemes adjust the
continuous power spectrum component with FCC limita-
tion and reduce the dynamic range of the amplitude of
the discrete spectral components. Again we smooth the
PSD of the signal by employing longer TH code over the
orthogonal pulse-based modulation. The simulation re-
sults are provided for PSD of TH-UWB signal for OPPM
and OPPM-PSM schemes based on modified Hermite
pulses (MHPs).
The rest of the paper is organized as follows. Section 2
discusses a generalized orthogonal pulse-based TH-UWB
signal. Section 3 presents derivation of PSD of orthogo-
nal pulse-based TH-UWB signal. Section 4 characterizes
PSD of TH-UWB signal for PSM, BPSM and OPPM-
BPSM schemes. In Section 5, simulation results and
discussion are provided. Finally, conclusion is drawn in
Section 6.
844 S. MAJHI ET AL.
2. Generalized Orthogonal Pulse-Based
UWB Signal Model
In M-ary PSM scheme, different symbols are transmitted
by different order orthogonal pulses. It requires M or-
thogonal pulses at the transmitter and M correlators at the
receiver. In M-ary BPSM scheme, symbols are transmit-
ted by using positive or negative amplitude of the or-
thogonal pulses. It requires orthogonal pulses at
the transmitter and correlators at the receiver to
transmit and receive all M possible symbols. In OPPM-
BPSM scheme, M symbols are transmitted by using L
pulse positions and N biorthogonal pulses where
/2M
/2M
k
M
2= , and [5].
l
L2= 1
2lk
=N
OPPM-BPSM is a generalized modulation scheme of
PSM, BPSM, OPPM-PSM and OPPM-PSM scheme [5].
By changing the number of pulse positions and orthogo-
nal pulses, one can construct a wide variety of modulai-
tons. For example, M-ary PSM scheme can be designed
by using one pulse position and M orthogonal pulses
whereas an M-ary BPSM can be constructed by using
one pulse position and biorthogonal pulses. M-ary
OPPM-PSM can be constructed by using L positions and
2N orthogonal pulses.
/2M
Since PSM, BPSM and OPPM-PSM are particular
cases of OPPM-BPSM, only OPPM-BPSM signal model
is provided. The M-ary OPPM-BPSM signal for
user is given as thk
)(=)( )()()()( k
lc
k
jf
k
nmtx
j
k
mnl TcjTtwdEts

(1)
where j is the index of time frame, Etx is the transmitted
energy of the signal and {1}
m
d
0n
f}{)(k
j
c
<0 )(
k
jNc
1=
m
d
1/2, M
, is the
amplitude of pulse , n represents the order of the
orthogonal pulse. The pulses have finite energy and are
normalized to ensure equal energy per transmission, that
is where . The pulse
repetition interval Tf is divided into N
h time slots of
length Tc, where , is a pseudorandom
TH code sequence, and . is the
additional time shift from nominal position. For PSM
scheme, , and . For
BPSM scheme, ,
1,2)=(m
1M
1
)(k
l
0=
)(k
l
1=
)(
)( tw k
n
1
chTTN
1, M
0,1,=n
=|)(| 2dttwn


0,1,=n
h
m
d
n0,1,=
and
. For OPPM-PSM scheme, ,
and
0=
)k
1=
m
(
l
d
N,
12
<<<<
L
f
T
where 0=
1
. For
OPPM-BPSM scheme, 1=
m
d and other terms are
same as OPPM-PSM.
3. Power Spectral Density Analysis
In orthogonal pulse based signal, different symbols are
transmitted by different order orthogonal pulses. The
continuous spectrum, energy spectral density (ESD),
changes with symbol. The discrete spectral component
changes with orthogonality of the pulses and TH code.
Therefore, a mathematical frame work is essential to
understand the orthogonal pulse based PSD in the pres-
ence of deterministic TH code. We assume that the
analysis is only for 1 user. For simplicity, the super-
script/subscript terms in (1) are omitted/modified. After
some modification, sum of M symbol can be written
from (1) as
)(=)(,
1
0=
1
0= lchlffpll
s
N
h
M
l
pTchTTlNtwats


(2)
where l is the amplitude and l
a
is the pulse position.
The terms l, l
a
and l are independent and sta-
tionary process. The index is related to TH code,
hl , and TH period, p. To simplify the analysis of the
PSD of TH-UWB signal, it is assumed that the number
of time frames for a symbol is s and it is equal to
p. Since (2) depends on the time dithering, it can be
written in continuous form as
wp
c,
N
N
N
).(=)( fpp
l
TlNtsty
(3)
The PSD is computed by evaluating the Fourier trans-
form (FT) of the autocorrelation function of , i.e., )(ty

)()(=)(
tytyEfPy
(4)
where denotes the FT and denotes the
expectation operator. Therefore, the PSD can be ex-
pressed as [11].
{.}{.}E
 

)( )()(
)(
1
)()( |)(|
1
=)(
*
2
*2
fp
qp
k
fp
qpp
fp
y
TN
k
ffSfSE
TN
fSfSEfSE
TN
fP

(5)
where and are two independent random vari-
ables with the same probability distribution function.
is the FT of . It can be expressed as
p q
)(fSp)(tsp
l
fj
lll
M
l
peafTfWfS

2
1
0=
)()(=)(
(6)
where is the FT of the transmitted pulse .
The time domain representation of (2 order
MHPs can be expressed as
)( fWl)(twl
h)tl
)(1)2()(2=)( 12twlttwtnw lll 
 (7)
The FT of can be expressed as
)(
1fwl
)](2)(
4
1
[=)(
1fWffWjfW lll
(8)
where “ · ” stands for derivative with respect to fre-
Copyright © 2010 SciRes. IJCNS
S. MAJHI ET AL.
845
quency. For M HP, is defined as )(
0fW
22
4
02=)( f
efW
(9)
The time and frequency domain representation of
MHPs are given in Figure 1.
)(fTl
th is the FT of the TH code which transmits the
symbol
l
.=)( )(
,
2
1
0=
f
Th
p
lN
c
T
hl
cfj
s
N
h
lefT 
(10)
To find the closed form expression of in (5),
the expectation of is to be evaluated. It is
given as
)( fPy
2
|)(| fSp


}.)(
)()()( {=|)(|
2
*
*
1
0=
1
0=
2
nl
fj
nln
lnl
M
n
M
l
p
eaafT
fTfWfWEfSE



 (11)
Since l and n are independent random variables
derived from the same process and l
a a
and n
are in-
dependent random variables derived from different
processes. Therefore, (11) can be rewritten as

}}.{}{}{
)()()()(
}{|)(||)({|=|)(|
)(2
**
1
0=
222
1
0=
2
nl
fj
nl
nlnl
M
nln
lll
M
l
p
eEaEaE
fTfTfWfW
aEfTfWfSE


(12)
Similarly, the second expectation in (5) can be ex-
pressed as
Figure 1. Time and frequency (logarithmic plot) domains
representation of modified Hermite pulses (MHPs).



11
**
=0 =02
{()()}= () ()()()
}{ .
MM
pql nln
ln jfln
ln
ESfSfWfWfT fTf
EaEaE e




*
(13)
The waveforms and are generated by
two i.i.d processes. Therefore, the expectation in (13) is
independent of and and equal to the case
)(tsp
n
)(tsq
l nl
of (12), i.e.,
)()()()(
}{}{}{=)}()({
**
1
0=
1
0=
)(2
*
fTfTfWfW
eEaEaEfSfSE
nlnl
M
n
M
l
nl
fj
nlqp



(14)
Substituting (12) and (14) in (5), the final PSD can be
formulated as in (15).
)()()()()(
)}({}{}{
|)(||)(|
}{}{}{}{=)(
**
1
0=
1
0=
2
)(2
22
1
0=
)(2
2
fp
k
nlnl
M
n
M
l
fp
nl
fj
nl
ll
M
l
fp
nl
fj
nlly
TkNffTfTfWfW
TNeEaEaE
fTfW
TNeEaEaEaEfP






(15)
4. Characterization of PSD of TH-UWB
Signal
Although UWB signals are alike in the frequency do-
main, they are diverse in the time domain due to their
different characteristics of time domain parameters p,
f, l and l. We see that the PSD of orthogonal
pulse-based modulation signals consists of continuous
and discrete spectral components which change with the
order of pulse waveforms and modulation schemes. The
variations of PSD over different orthogonal pulse-based
signaling are given in the following subsection.
N
T aw
4.1. PSD of M-ary Pulse Shape Modulation
(PSM)
In PSM scheme, symbols are modulated only by the or-
der of orthogonal pulses. The generalized terms in (15)
are specified by l = 1 and a0=
l
. The expectations of
these variables are , nlnl and
respectively. The PSD of the PSM
signal can be written from (15) as
1 {aE=}{ 2
l
aE 0=}{} aE
1=}{ )(2 nl
fj
eE


)()(=)( fpfpfP ky (16)
where
22
1
0=
|)(||)(|1=)( fTfWTNfp ll
M
l
fp
(17)
and
Copyright © 2010 SciRes. IJCNS
846 S. MAJHI ET AL.
)(
)()()()()1(=)(**
1
0=
1
0=
2
fp
k
nlnl
M
n
M
l
fpk
TkNf
fTfTfWfWTNfp


(18)
We see that is continuous spectrum component.
It depends on the TH code and the ESD of the or-
der orthogonal pulse. Since ESD of different order or-
thogonal pulses are not identical, the selectio n of order of
the orthogonal pulses plays an important role for con-
tinuous spectral component.
)( fp thl
)( fpk is the discrete spectral components which in-
duces UWB interference on the other narrow band sys-
tems [12]. The discrete components of the signal appear
based on the term . It shows that the
position of discrete component depends on the TH code
and its dynamic range of amplitude depends on the or-
thogonality of pu lses. Since pulses are or thogon al in time
and frequency domains, the value of is
approximately zero, as a result, the dynamic range of
amplitude of the discrete spectral components becomes
very small. This small dynamic range increases the av-
erage transmitted power in pulse and improves the UW B
system performance. It helps UWB signal to coexist with
other systems without any serious performance degrada-
tion. In addition, it facilitates UWB signal to keep its
spectrum under the FCC spectral mask without mini-
mizing the averag e transmitte d power in the signal.
)( fp
kTkNf
)()( *fWfW nl
4.2. PSD of M-ary Biorthogonal PSM (BPSM)
In BPSM scheme, symbols are modulated by order and
amplitude of the pulses, i.e., and
1}{
l
a0=
l
.
The expectation of these variables are ,
1=}
2
)
{l
aE
(2 fj


0=}{}{ nlnl aEaE and . 1=}{ nl
eE
The corresponding PSD of BPSM scheme can be ex-
pressed from (15) as
))()((2exp
|)(|1=)(
,,
2
1
0=
1
0=
1
0=
fcklhl
l
s
N
k
s
N
h
M
l
fpy
TkhTccfj
fWTNfP

 
(19)
The continuous PSD component of BPSM signal is
same as PSM scheme. However, the discrete spectral
components become zero due to the antipodal pulse. The
PSD of the TH-UWB signal for BPSM scheme is
smoothed. This allows the signal to coexist with other
NB signals. The extensive studies found that any antipo-
dal signal has only continuous spectral component [12].
The continuous component can be easily fitted to FCC
by using appropriate MHPs.
4.3. PSD of M-ary OPPM-BPSM
For OPPM-BPSM scheme, and 1}{
l
a
1)(=
l
l,
where
is the constant time shift length. This implies,
, and
1{ 2
l
aE
=} 0=}{nlaaE
2fmTj

))/2(2cos(1=}{
TmfeE
.
The corresponding PSD of OPPM-BPSM signal can be
expressed as

))()((2exp
|)(|1=)(
,,
2
1
0=
1
0=
1
0=
fcklhl
l
s
N
k
s
N
h
M
l
fpy
TkhTccfj
fWTNfP


(20)
The PSDs of BPSM and OPPM-BPSM schemes are
identical. However, OPPM-BPSM can be used for higher
level modulation scheme for higher data rate systems.
Therefore, OPPM-BPSM modulation is an attractive
choice of TH-UWB signal from several aspects.
5. Simulation Results and Discussions
In this section, PSD is provided for orthogonal pulse-
based signaling and compared with conventional OPPM
scheme. In simulation, different orders of MHPs are used
with two different lengths of TH code 8 and 16. The
other simulation parameters are set to ns and
pulse width is 7 ns. 60=
f
T
Since BPSM and OPPM-BPSM have antipodal signal,
they have only continuous spectral component and shape
of their spectral is same as continuous component of non
antipodal signal. The only difference is that the spectral
of antipodal signal does not contain any discrete compo-
nent. The PSD in non antipodal modulation schemes is
more complicated. Since OPPM and OPPM-PSM are
special cases of OPPM-BPSM, OPPM and OPPM-PSM
have been chosen to compare the PSD of the signal. The
PSD of 8-ary OPPM is given in Figure 2 for order
pulse and in Figure 3 for and order pulses
with TH code of length 8 and . Since each time
only one pulse is used in OPPM scheme, orthogonality is
maintained by position not by pulse. The 3r order
pulse almost satisfies the FCC spectral mask except
some discrete components. However, and
order pulses do not satisf y the FCC spectral mask shown
in Figure 3. The dynamic range of the amplitude of dis-
crete components of OPPM scheme is about 8 dB which
is very high. The power of the signal is calculated based
on the line where the dynamic range is zero (4 dB bellow
from the pick point). As FCC rules, pick amplitude must
be bellow the –41.25 dBm limit. Therefore, the power of
the signal is calculated based on the line which is maxi-
mized up to –45.25 dBm. As a result, signal provides low
average transmitted power which degrades the system
3rd
d
4th
c
T5th
4t
7.5=
h5th
Copyright © 2010 SciRes. IJCNS
S. MAJHI ET AL.
Copyright © 2010 SciRes. IJCNS
847
Figure 2. PSD of 8-ary OPPM scheme with 3rd order MHP and TH code length is 8.
Figure 3. (a) PSD of 8-ary OPPM scheme with 4th order MHP; (b) PSD of 8-ary OPPM scheme with 5th order MHP and TH
code length is 8.
848 S. MAJHI ET AL.
Figure 4. PSD of 8-ary OPPM-PSM schemes for 4 positions and 5 pulses (0th and 3rd) w i th TH code of length 8.
Figure 5. PSD of 8-ary OPPM-PSM schemes for 4 positions and 2 pulses 0th and 3rd with TH code of length 16.
Copyright © 2010 SciRes. IJCNS
S. MAJHI ET AL.
Copyright © 2010 SciRes. IJCNS
849
performance. Not that if the dynamic range becomes zero,
the maximum limit becomes –41.25 dBm.
Figure 4 shows the PSD of 8-ary OPPM-PSM for 4
positions and 2 orthogonal pulses with TH code of length
8. We see that that dynamic range of the amplitude of the
discrete spectral component of OPPM-PSM scheme is 4
dB which is lesser than the OPPM scheme even the same
length of TH code is used. It is because of the orthogo-
nality of pulses. So by reducing dynamic range, we can
improve the UWB system performance by increasing the
average transmitted power in the signal pulse as well as
we can reduce the UWB interference over other radio
systems. Again by applying TH code over these or-
thogonal pulse-based modulations, dynamic range of
amplitude of discrete component further could be re-
duced. Figure 5 shows the PSD of 8-ary OPPM-PSM
with TH code of length 16 and . The dynamic
range is almost reduced to 1 dB. However, it can not be
reduced to zero whatever the length of TH code used.
We also see that the average transmitted power in Figure
5 is more than the previous cases. Therefore, orthogonal
pulse-based TH-UWB signaling has several advantages
than its complexity burden.
3.75=
c
T
6. Conclusion
PSD of orthogonal pulse-based TH-UWB signal has
been analyzed based on stochastic signal theory. This
shows that the continuous and discrete components
change with order of pulses and modulation schemes.
The discrete component can be removed by employing
an antipodal signal. If the signal is not antipodal, the dy-
namic range of amplitude of the discrete component can
be reduced by using orthogonal pulse-based modulation
and long TH code. Therefore, it has been proved that the
orthogonal pulse-based signaling not only reduce the
dynamic range of discrete spectral component also im-
prove the system performance by increasing the average
transmitted power in the signal.
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