Spectrally Efficient UWB Pulse Shaping With Application in Orthogonal PSM
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 efficiency 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 implemented with orthogonal pulses obtained by the proposed method, and the results of a simulation are shown
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ABSTRACT: Choice of ultra wideband (UWB) pulse shapes has become an interesting challenge such that the spectral limits of the frequency band approved by the Federal Communications Commission (FCC) can be utilised efficiently. In this paper, we compare the efficiency of various UWB pulse shapes based upon their power spectral density (PSD) and autocorrelation functions (ACF). We propose a set of prolate spheroidal wave functions (PSWF) for a typical M-ary pulse shape modulation (PSM) scheme. In the process, we show closed form expressions for the ACF of PSWF-based UWB pulse shapes of different orders.Int. J. of Ultra Wideband Communications and Systems. 01/2014; 3(1):19 - 30.
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ABSTRACT: An optically tunable fractional order temporal differentiator implemented using a silicon-on-isolator microring resonator with a multimode interference (MMI) coupler is proposed and experimentally demonstrated. Through changing the input polarization state, the self coupling coefficient and the loss factor of the designed ring resonator with the MM! coupler are changed. Correspondingly, the coupling regime is changed. Through changing the coupling regime from over-coupled to under-coupled regime, the phase shift in the resonance wavelength is changed from <;π to >π. This tunable phase shift is used to implement a tunable fractional order photonic differentiator with an order tunable from <;1 to 1. The proposed fractional order differentiator is demonstrated experimentally. A Gaussian pulse with a bandwidth of 45 GHz is temporally differentiated with a tunable order of 0.37, 0.67, 1, 1.2, and 1.3.IEEE Photonics Technology Letters 01/2013; 25(15):1408-1411. · 2.04 Impact Factor
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ABSTRACT: In this paper, we propose a novel orthogonal bi-pulse ultra-wideband (UWB) system, which uses an even pulse and an odd pulse to convey information symbols in an alternating manner. Due to the orthogonality of these pulses, their corresponding received waveforms remain orthogonal after propagating through multipath channels. Then we consider two major challenges in the realization of our proposed UWB system: timing synchronization and symbol demodulation. In particular, the idea of timing with dirty template (TDT) in [L. Yang, G.B. Giannakis, Timing Ultra-Wideband signals with dirty templates, IEEE Trans. on Commun. 53 (11) (2005) 1952-1963] is employed for timing synchronization and the noncoherent scheme in [L. Yang, G.B. Giannakis, A. Swami, Noncoherent Ultra-Wideband (de)modulation, IEEE Trans. Commun. 55 (4) (2007) 810-819] is used to bypass channel estimation. Both of these techniques are characterized by correlating adjacent waveform segments. In the implementation of these techniques, we will gradually reveal the advantages of our proposed system. The correlation of adjacent waveform segments only contains the information of a single symbol. This enables a significant enhancement of the synchronization speed of TDT when no training sequence is transmitted. For the same reason, our demodulation approach completely mitigates the inter-symbol interference (ISI) in the second paper referred to, above, and entails a simple demodulator even in the presence of unknown timing errors. Simulations are also carried out to corroborate our analysis.Physical Communication 12/2008; 1(4):237-247.
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
314IEEE 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 , andis 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 theth 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 SHAPING315
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 theth 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 SHAPING317
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 cutoffand
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