IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 55, NO. 2, FEBRUARY 2007313
Spectrally Efficient UWB Pulse Shaping
With Application in Orthogonal PSM
João A. Ney da Silva and Marcello L. R. de Campos
Abstract—In this paper, we present a method to obtain a set of
orthogonal pulses to be used in pulse-shape modulation (PSM)
for ultra-wideband communications. The pulses are built as
linear combinations of Hermite functions, which are shown to
have unique advantageous features. Mathematical restrictions of
orthogonality and spectral eficiency are introduced as guidelines
to a fully explained search procedure to find the best set of pulses.
Additionally, this procedure is adapted and used to find a single
FCC-compliant pulse shape. A quaternary PSM scheme is imple-
mented with orthogonal pulses obtained by the proposed method,
and the results of a simulation are shown.
Index Terms—Hermite functions, orthogonal functions, orthog-
onal modulation, pulse-shape modulation (PSM), pulse shaping,
spectral compliance, ultra-wide bandwidth (UWB), wireless.
important developments for this kind of system. In these sys-
tems, signals are impulsive, i.e., they are extremely short in
time, and therefore, have a highly spread spectrum. In order to
convey data information, pulses are directly controlled by the
chosen pulse-modulationscheme. Thechoice ofthemodulation
scheme is of great importance to the overall system perfor-
mance. The most common schemes are based on pulse-position
modulation (PPM) , or on pulse-amplitude modulation
Recently a new modulation technique was proposed by the
authors, making use of the shape of the pulse: pulse-shape mod-
ulation(PSM).It wasoriginally basedontwowaveformsor-
thogonal in time commonly used for modeling the pulse shape.
In the same work, the concept was developed to include
thogonal pulses, with a spectral restriction, in order to achieve
-ary orthogonal PSM, and a set of four orthogonal pulses was
The use of Hermite functions in UWB communications has
been addressed in several other recent works, as in  and ,
whichusea modifiedHermitefunction,and ,whichbringsto
discussion the effect of the antennae on the transmitted pulses.
The work of  shows that UWB antennae are undergoing a
LTRA-WIDE bandwidth (UWB) communication is re-
ceiving growing interest from researchers addressing
plications of the IEEE Communications Society. Manuscript received October
17, 2003; revised January 5, 2005 and July 8, 2006.
The authors are with the Programa de Engenharia Elétrica, COPPE/Univer-
sidade Federal do Rio de Janeiro, RJ, Brazil 21945-970 (e-mail: campos@lps.
ufrj.br). J. A. N. da Silva is also with the Departamento de Engenharia Elétrica
(DE3), Instituto Militar de Engenharia, Praça Gal. Tibúrcio, 80, 21290-270,
Brazil (e-mail: email@example.com).
Digital Object Identifier 10.1109/TCOMM.2006.887493
manifold evolution. As long as the effect of the channel on the
transmitted signal varies with the antenna type, independently
developingtheUWB waveforms is still a good approach.More-
over, after choosing an advantageous pulse waveform, one can
choose finding an adequate antenna or previously modify the
signal to be sent, in a prewarping approach.
higher order sets of orthogonal pulses for PSM. It is also shown
how this flexible method can be adapted to obtain pulses with
specific spectral criteria, through the important example of de-
signing a pulse compliant with the Federal Communications
Commission (FCC) spectral mask. The obtained PSM schemes
proving to be competitive against PAM and PPM schemes. Due
work well in more elaborate channels, e.g., multipath channels.
In the next section, we review the main types of UWB mod-
ulation, outlining their differences in terms of their expressions
for the modulated pulse streams and theoretical performances.
In Section III, the Hermite functions are introduced, and
their utility is explained using the concept of Hermite spaces.
Section IV describes in detail the method for obtaining or-
thogonal pulse shapes with some desired properties, based on
Hermite functions. In Section V, as an example of how the
method can be adapted to obtain pulses in conformity with
very specific spectral requirements, we describe the design
of a pulse complying with the FCC requirements for indoor
systems. In the following section, we show the performance of
some schemes based on the types of modulation approached in
the paper, including PSM using the proposed pulses. Finally,
some conclusions are presented in Section VII.
II. MODULATION TECHNIQUES
In this section, we give a brief review of the main types of
modulation to be used with UWB.
In PAM, a given waveform is sent with different amplitudes
correspondingtodifferent data beingtransmitted. There are two
waveforms often used, which are obtained from the derivatives
of the Gaussian function, therefore being generically called
Gaussian monopulses. The first derivative of the Gaussian
function gives rise to
0090-6778/$25.00 © 2007 IEEE
314 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 55, NO. 2, FEBRUARY 2007
Fig. 1. Gaussian and Rayleigh pulses.
which is also called Rayleigh pulse , where
factor. From the second derivative of the Gaussian function, we
is a time-scale
The Gaussian pulses are plotted in Fig. 1.
The basic PAM signal, composed of a stream of modulated
pulses, is given by
amplitude of the th pulse corresponding to the symbol repre-
sented. In binary modulation, we may have the optimum an-
tipodal case by making, for instance,
representing bit 1.
In the antipodal case, detection may be performed by a single
correlator using as template signal a normalized-energy pulse
, which will result in correlation values equal to 1 or -1. Its
theoretical performance over additive white Gaussian (AWG)
channels is the standard performance for antipodal signals 
is usually eitheror , and is the
is the frame duration or
is the complementary error function.
-ary modulation is achieved using
. In this case, the waveforms are no longer an-
tipodal, and the bit-error probability will depend on the cross-
correlation values between each pair of symbol waveforms.
different levels of
in the transmitted pulse. The modulated pulse stream becomes
symbol represented. In a binary modulation, we have typically
representing bit 0, and
is a time delay.
Reception of a PPM binary signal can be made using a single
correlator  with a special template waveform obtained from
the sum of the two possible waveforms, one of them inverted
and delayed by
is the delay of the th pulse corresponding to the
representing bit 1, where
between the two waveforms, we achieve positive values at the
-delayed pulses. This maximizes the distance between signals
for binary PPM with this kind of pulse. The theoretical perfor-
mance of generic nonorthogonal signals with cross-correlation
in the presence of white Gaussian noise is 
for minimum (most negative) cross-correlation
-ary PPM schemes use
symbols. The choice of these delays is critical, and must be ju-
diciously made. In , a quaternary PPM was tested, where the
four delays were uniformly spaced in time, being multiples of a
given . As the values of cross-correlation depend only on the
relative delay between the transmitted pulse and each template,
was chosen to minimize these values.
different delays to represent the
PSM was recently proposed in , and also independently in
. Both works make use of the orthogonality of Hermite func-
mite functions, generating pulses whose spectra are clearly dis-
tinct, which may be a problem when we have bandwidth restric-
tions. Although this can be easily solved by scaling the pulse in
time, enlarging its duration, this solution causes a reduction in
optimized combinations of Hermite functions in order to obtain
pulses occupying roughly the same spectra, therefore achieving
the rest of this paper, we refer only to the PSM modulation pro-
posed in .The original idea of PSM consistsof an orthogonal
binary modulation, where two orthogonal waveforms are used
theoretical performance of binary orthogonal PSM follows that
for any orthogonal signals in AWG channels
-ary orthogonal PSM schemes, we need to use sets of
pulse waveforms which are all orthogonal to each other. This
DA SILVA AND DE CAMPOS: SPECTRALLY EFFICIENT UWB PULSE SHAPING 315
can be obtained from sets of orthogonal functions. The modula-
tion proposed in  uses pulses obtained from the linear combi-
nation of a given number of orthogonal Hermite functions using
the concept of linear signal spaces. Among manypossibilities, a
search procedure selected the optimum combination that results
in a set of pulses occupying the most similar frequency bands.
The method used for obtaining such pulses is detailed in this
paper. The resulting modulated pulse stream is given by
chosen, depending on the data content.
is the waveform associated with the th pulse
III. ORTHOGONAL HERMITE PULSES
A. Hermite Functions
Sets of orthogonal functions arise from the solution of some
partial differential equations. Particularly interesting are those
used as solutions to the Sturm–Liouville equation, such as Le-
gendre, Bessel, and Hermite functions , . These are in-
finite sets with an unbounded number of functions orthogonal
to each other. As it occurs with Fourier series, they can be used
to expand some functions in terms of the base of orthogonal
functions. For the above-mentioned sets, this is valid for all
square-integrable (or finite-energy) functions. This way, any of
these sets can be said to form a basis of the space of finite-en-
ergy functions, referred to as
of linear signal spaces , i.e., collections of signals
and belong to the collection, then any linear
also belongs toit.In
Observing the equations of the Gaussian pulses (1) and (2)
the same characteristics, which makes them particularly inter-
esting for the purpose of forming UWB pulses, for they can be
easily related to the Gaussian pulses. This also means that the
orthogonality of Hermite functions, constitute the motivation to
tions, given by
space. This is a particular case
has finite value.
by the formulas
are the Hermite polynomials, recursively obtained
Fig. 2. First three Hermite functions.
It is interesting to observe the Fourier transforms of the Her-
mite functions, which also have a recursive equation, presented
B. Hermite Spaces
As already mentioned, the Hermite functions form a basis of
. We can select finite subspaces from this infinite-dimension
space by selecting onlya finite number offunctions from the or-
thogonal set. Therefore, the
-dimensional Hermite space can
We can immediately identify both Gaussian pulses of (1) and
(2) as being contained in the Hermite space of dimension 3.
Indeed, the first one is also contained in the Hermite space of
An interesting fact, proved with the help of time–frequency
analysis theory , is that the Hermite spaces are maximally
concentrated in both time and frequency. This allows getting
the shortest effective duration waveforms without having a fre-
quency content higher than necessary. Besides avoiding further
difficulties for signal generation, this guarantees optimal use of
the communications resources measured by the duration-band-
C. Hermite Pulses
Hermite pulses of
waveform belonging to the
th order are defined  as any signal
th dimension Hermite space.
316 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 55, NO. 2, FEBRUARY 2007
These pulses can thus be expressed as
Hermite pulses can be expressed as
is an-degree polynomial. Alternatively,
relating the considered pulse to the
This way, each signal of the Hermite space
through its orthonormal basis. From (15), it can be easily seen
that the Gaussian and Rayleigh pulses are particular cases of
Hermite pulses. Using (16) and
is a vector composed of the first
is the projection vector, a real column vector
first Hermite functions.
, we have the following
The orthogonality between the pulses is made clear by the or-
thogonality of these projection vectors, which relate the wave-
forms of (1) and (2), with
, to the first three Hermite func-
IV. SEARCHING FOR ORTHOGONAL SETS
We wish to find a set of
, orthogonal to each other, with no DC component, and oc-
cupying the most possibly similar frequency bands. This set of
pulses can be represented by a matrix
from each pulse
th-order Hermite pulses,
formed by the
where the second index designates each particular pulse.
Notice that the requirement of absence of the DC component
eliminates thetrivial solutionof usingexactlytheHermitefunc-
tions as the set of desired pulses. Moreover, it poses a linear re-
striction to the vector components
lines of matrix
, leading to
, and therefore, to the
A. General Procedure to Obtain a Set of
At this point, we should search within the
vector space represented by the columns of
we are looking for an orthogonal basis of this space, where the
basis vectors represent the pulses we want, which are subjected
to the following restrictions:
• they are orthogonal;
• they have zero DC component;
• they occupy the most possibly similar frequency bands.
With this in mind, we developed the following procedure,
translating the above statements into mathematical restrictions.
, make [from (23)]
. In some sense,
2) Determine the generic form of the projection vector and
then build the equation expressing the pulse by the func-
tions of the basis
(For the purpose of clarity, whenever possible, we omit the
index referring to the pulse.)
3) Apply the zero-DC restriction given by
a) Express the algebraic relation between elements of
is a linear function resulting from (26).
b) Apply the same relation to
number of variables in
, according to (24)].
4) Build the normalized orthogonality equations
, replacing its first line
which consist of
5) Obtain the generic expression for the mean central fre-
quency and the frequency spreading.
a) Take the Fourier transform of
are given by (12) to
DA SILVA AND DE CAMPOS: SPECTRALLY EFFICIENT UWB PULSE SHAPING 317
b) Find the expression of the energy spectral density of
a pulse given by
c) Write the expressions of mean central frequency and
frequency spreading defined as
respectively. Note that (29)–(32) are functions of
d) Apply (27) to eliminate
from (31) and (32)
and (32), but in terms of
6) Find the functional to be minimized.
andhave the same expressions of (31)
with a suggested value of
ically determined to fit best
(this was empir-
to the cutoff and
correspond to the pulses being consid-
7) Do an extensive search over all sets that are solutions to
(28), looking for the one which minimizes
The procedure described above was applied to construct a set
of three orthogonal Hermite pulses in . The pulses found are
shown in Fig. 3, and correspond to the following vectors:
The three pulses are given by
Fig. 3. Three optimal orthogonal pulses.
B. Obtaining a Set of Four Orthogonal Pulses
For implementing an
orthogonal waveforms. Due to the binary nature of the data,
only powers of two are usually chosen. Next, we show the pro-
cedure to obtain a set of four orthogonal pulses suitable for a
quaternary UWB PSM
-ary orthogonal modulation, we need
which consist of 10 equations with 16 unknowns
318 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 55, NO. 2, FEBRUARY 2007
5.c) and 5.d) Calculate the mean central frequency and fre-
7) Do a search over all sets that are solutions to (46), looking
for the one which minimizes (52).
in the case of three pulses becomes prohibitive in this case for
two reasons. First, the number of variables to be scanned is
twice as large in the case of four pulses when compared with
the case of three pulses; second, the equations involved are now
much more complex. Therefore, we developed a procedure for
the cases with
that uses orthogonal square matrices for
is based on the following reasoning.
be an matrix of the form
which is essentially matrix
It is possible to relate
reflecting the effect of the zero-DC restriction
without the first line.
andthrough a combination ma-
From (28) we have that
. Decomposingin its eigenvalues and
, we have
we have a square orthogonal matrix
Now we can state the following procedure to search through
the solutions of (28).
1) Find the (numerical) combination matrix
, such that (54)
4) Generate an orthogonal matrix
(We recall that
6) Use (54) to get
7) Using the elements of
Next we show the application of this procedure to our case,
andis simply a diagonal
, calculate[using (33)–(36)].
DA SILVA AND DE CAMPOS: SPECTRALLY EFFICIENT UWB PULSE SHAPING319
4) We obtained different instances of
gorithm that generates orthogonal
generic orthogonal full matrix, we applied the algorithm
twice, generating a lower and an upper Hessenberg ma-
was obtained by the product of these two
orthogonal matrices. In the present case, we generate two
4 4 Hessenberg matrices, with three rotations each. This
way, we control the six parameters
the space of possible solutions. With this procedure, we
were able to generate a sufficient number of matrices with
enough diversity to find a good solution.
using a known al-
Givens rotations. In order to have a
The best set of four pulses found is shown in Fig. 4, and cor-
responds to the following vectors:
The spectra of these pulses can be seen in Fig. 5. Two pairs of
pulses have identical spectra. In Fig. 4, it can be seen that each
Fig. 4. Four orthogonal Hermite pulses.
Fig. 5. Frequency spectra of the four orthogonal pulses.
pulse corresponds to the inverted version of another one; more
which explains the identical frequency contents.
V. DESIGNING PULSES UNDER SPECIFIC
In some situations, it is desirable for a UWB system to con-
ample, we show how the search procedure can be modified to
find a single pulse waveform respecting the FCC regulation for
indoor systems .
The FCC regulation allows a maximum spectral density
within a window ranging from 3.1 to 10.6 GHz, at which point,
the allowed density falls 10 dB. There is also a thin band to be
avoided, between 0.96 and 1.61 GHz, with a 34 dB lower limit
to protect GPS systems. Therefore, besides respecting these
limits, a UWB system should use efficiently the main spectral
window in order to maximize its total allowable power.
320 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 55, NO. 2, FEBRUARY 2007
With this in mind, we state that the Hermite space in which
we will search must permit obtaining pulse spectra whose -10
dB points keep a relation close to that of the FCC window
which can be viewed as an alternative measure of the pulse’s
As we are looking for a single waveform, we have
, which implies that. The value of
over third-order pulses, we realized that they never have -10 dB
relations greater than 0.201. So we must try
reducestoa 4 1columnmatrixwith
, leading to the normalization (74). After freely choosing an
adequate value for
, we can follow the procedure from
steps 1 to 4.
The following steps are modified to pursue our intended
5) Obtain the pulse’s spectral energy density, either deter-
mining its literal expression or by numerical computation
with a convenient frequency grid.
a) Find, on the spectral density curve, the points of
-10 dB (above and below the peak frequency) and
-34 dB (below the peak frequency). Namely, in as-
Fig. 6. FCC-compliant pulse.
where, similar to FCC
defined at (69), we have
To scan the solution space, defined by the vector
it is convenient to make
An optimum pulse was found with
, corresponding to
which results in the following time equation:
An inverted version of this pulse is plotted in Fig. 6, and its
spectral energy density, over the FCC mask, is shown in Fig. 7.
VI. SIMULATION RESULTS
In , several PSM schemes were tested and compared with
PPM and other schemes. Most of them are polarity-dependent,
i.e., schemes in which the receiver needs to know the original
polarity of the transmitted signal to properly demodulate the
data. One quaternary PSM scheme was intentionally made po-
larity-independent, which means that its performance does not
depend on the knowledge of the polarity of the received pulse.
This may be an important advantage, since the transmitted
signal, regardless of the scheme used, can always arrive at the
receiver with its original polarity or its polarity inverted due
to reflections. Polarity independence is easily accomplished
DA SILVA AND DE CAMPOS: SPECTRALLY EFFICIENT UWB PULSE SHAPING 321
Fig. 7. Frequency spectrum of the FCC-compliant pulse.
Fig. 8. Performance comparison of different UWB modulation schemes in
with PSM, but a polarity-dependent version was implemented
for comparison. The tests simulated the transmission of the
modulated pulse stream. The following modulation schemes
1) PPM-pd: the original (polarity-dependent) binary PPM
scheme, proposed in .
2) PSM-pd: polarity-dependent binary PSM scheme using
waveforms of (1) and (2) for the pulses.
3) QPSM-pi: polarity-independent quaternary orthogonal
PSM (QPSM) scheme using the four orthogonal pulses
given by (68).
4) QPSM-pd: polarity-dependent version of quaternary PSM,
using the same four orthogonal pulses.
5) PAM-pd: polarity-dependent binary PAM scheme, whose
theoretical performance was plotted for comparison.
Fig. 8 shows the performance of the quaternary PSM
schemes compared with binary schemes PPM-pd, PSM-pd,
and PAM-pd. It should be noted that both quaternary schemes
surpass PSM-pd (their best binary version) when
beyond 5 dB. The polarity-dependent PSM has a small ad-
vantage over the polarity-independent version with respect to
bit-error rate (BER) performance. It is interesting to observe
that the quaternary modulation curves are steeper than those
from binary schemes, causing them to approach the PPM-pd
grows, with QPSM-pd meeting PPM-pd at
8 dB. These results show the advantages of using
-ary modulation schemes for orthogonal PSM.
The antipodal PAM-pd scheme confirms its theoretical superi-
ority, already indicated by (4) compared with (7) and (8), being
the best scheme plotted in the figure.
In this paper, we presented a method to obtain PSM modula-
tion with orthogonal pulse waveforms occupying the same fre-
quency band, an important feature to make efficient use of the
available spectrum. The method was suitably adapted for pulse
shaping, originating a special pulse waveform fully compliant
with the FCC spectral restrictions.
The good performances of PPM and PAM, with the superi-
ority of the latter, is in accordance with the theoretical previ-
sions shown in Section II. On the other hand, PSM presents the
unique feature of orthogonality, paying a price for it, which is
a shorter distance between its symbols. In the simple environ-
other schemes, while the orthogonality advantage is little ex-
plored. However, the use of orthogonality to build higher order
modulation schemes has proven to be advantageous, as it can be
deduced from the performance improvement in QPSM schemes
when compared with the binary PSM scheme. The orthogo-
nality, the polarity independence, and the time–frequency con-
centration of the pulses in our PSM scheme are promising for
performing in environments where there is uncertainty with re-
spect to the state of the received signal, such as channels with
multipath and jitter.
 R. A. Scholtz, “Multiple access with time-hopping impulse modula-
tion,” in Proc. MILCOM, 1993, vol. 2, pp. 447–450.
 J. A. N. da Silva and M. L. R. de Campos, “Orthogonal pulse shape
modulation for impulse radio,” in Proc. ITS, 2002, pp. 916–921.
 M. Ghavami, L. B. Michael, and R. Kohno, “Hermite function based
orthogonal pulses for UWB communications,” in Proc. WPMC, Aal-
borg, Denmark, 2001, pp. 437–440.
 L. B. Michael, M. Ghavami, and R. Kohno, “Effect of timing jitter on
Hermite function based orthogonal pulses for ultra wideband commu-
nication,” in Proc. WPMC, Aalborg, Denmark, 2001, pp. 441–444.
 C. Mitchell, G. Abreu, and R. Kohno, “On the design of orthogonal
pulse-shape modulation for UWB systems using Hermite pulses,” J.
5, no. 4, pp. 328–343, Dec. 2003.
 H. G. Schantz, “Introduction to ultra-wideband antennas,” in Proc.
IEEE UWBST Conf., 2003, pp. 1–9.
using monopulse waveforms,” in Proc. MILCOM, 1999, vol. 2, pp.
 J. G. Proakis, Digital Communications, 4th ed.
 J.A.N.da SilvaandM.L. R.deCampos,“Performancecomparisonof
2003, vol. 2, pp. 789–793.
Supplements in Physics.New York: W. A. Benjamin, 1962.
New York: McGraw-
322IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 55, NO. 2, FEBRUARY 2007
 E. Butkov, Mathematical Physics.
 F. Hlawatsch, Time-Frequency Analysis and Synthesis of Linear Signal
Spaces, TheKluwer InternationalSeriesin EngineeringandComputer
Science.Norwell, MA: Kluwer, 1998.
 Federal Communications Commission (FCC), “Revision of Part 15
of the Commission’s Rules Regarding Ultra-Wideband Transmission
Systems,” 2002, ET Docket.
Reading, MA: Addison-Wesley,
João Abdalla Ney da Silva was born in Rio de
Janeiro, Brazil, in 1965. He received the Engineering
degree in electrical engineering from the Federal
University of Rio de Janeiro (UFRJ), Rio de Janeiro,
Brazil, in 1989, the M.Sc. degree in computer sci-
ences from Military Institute of Engineering (IME),
Rio de Janeiro, Brazil, in 1998, and is currently
working towards the Ph.D. degree from the Program
of Electrical Engineering at COPPE/UFRJ.
From January 1998 until 2000, he was a Develop-
ment Engineer at the Institute of Development and
Research, Technological Center of the Brazilian Army (IPD/CTEx), Rio de
Janeiro, Brazil, dealing with electronic warfare (EW) systems. Since 2004, he
has been with the Informations Technology Division (DTI) at CTEx developing
projects and studies in EW, software defined radios, and radar sensors. His re-
search interests include ultra-wideband communications, signal processing for
communications, and signal processing for radar signals.
Marcello L. R. de Campos was born in Niteroi,
Brazil, in 1968. He received the Engineering degree
(cum laude) from the Federal University of Rio de
Janeiro (UFRJ), Rio de Janeiro, Brazil, in 1990,
the M.Sc. degree from COPPE/UFRJ in 1991, and
the Ph.D. degree from the University of Victoria,
Victoria, BC, Canada, in 1995, all in electrical
In 1996, he was a Postdoctoral Fellow with the
Department of Electronics, School of Engineering,
UFRJ, and with the Program of Electrical En-
gineering, COPPE/UFRJ. From January 1997 until May 1998, he was an
Associate Professor with the Department of Electrical Engineering (DE/3),
Military Institute of Engineering (IME), Rio de Janeiro, Brazil. He is currently
an Associate Professor of Electrical Engineering, COPPE/UFRJ, where he
served as Department Vice-Chair and Chair in the years 2004 and 2005, respec-
tively. From September to December 1998, he was visiting the Laboratory for
Telecommunications Technology, Helsinki University of Technology, Espoo,
Finland. His research interests include adaptive signal processing, statistical
signal processing, and signal processing for communications.
Dr. de Campos served as IEEE Communications Society Regional Director
for Latin America in 2000 and 2001. In 2001, he received a Nokia Visiting Fel-
lowship to visit the Centre for Wireless Communications, University of Oulu,