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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.

I. INTRODUCTION

U

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) [1], or on pulse-amplitude modulation

(PAM).

Recently a new modulation technique was proposed by the

authors, making use of the shape of the pulse: pulse-shape mod-

ulation(PSM)[2].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

found.

The use of Hermite functions in UWB communications has

been addressed in several other recent works, as in [3] and [4],

whichusea modifiedHermitefunction,and [5],whichbringsto

discussion the effect of the antennae on the transmitted pulses.

The work of [6] shows that UWB antennae are undergoing a

LTRA-WIDE bandwidth (UWB) communication is re-

ceiving growing interest from researchers addressing

or-

PaperapprovedbyR.Kohno,theEditorforSpreadSpectrumTheoryandAp-

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: abdalla@lps.ufrj.br).

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.

Inthispaper,wedescribeindetailthemethodusedforfinding

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

aretestedinanadditivewhiteGaussiannoise(AWGN)channel,

proving to be competitive against PAM and PPM schemes. Due

toitsgoodperformanceandspectralefficiency,weexpectitwill

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.

A. PAM

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

(1)

0090-6778/$25.00 © 2007 IEEE

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314 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 55, NO. 2, FEBRUARY 2007

Fig. 1. Gaussian and Rayleigh pulses.

which is also called Rayleigh pulse [7], where

factor. From the second derivative of the Gaussian function, we

have

is a time-scale

(2)

The Gaussian pulses are plotted in Fig. 1.

The basic PAM signal, composed of a stream of modulated

pulses, is given by

(3)

where

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,

0, and

representing bit 1.

pulse-repetition time.

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 [8]

is usually eitheror , and is the

representing bit

is the frame duration or

(4)

where

is the complementary error function.

-ary modulation is achieved using

amplitude for

. 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

B. PPM

InPPM,eachdatasymbolis representedbyaparticulardelay

in the transmitted pulse. The modulated pulse stream becomes

(5)

where

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 [1] 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

(6)

Optimizing

between the two waveforms, we achieve positive values at the

correlatoroutputforzero-delayedpulsesandnegativevaluesfor

-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

value

in the presence of white Gaussian noise is [2]

for minimum (most negative) cross-correlation

(7)

-ary PPM schemes use

symbols. The choice of these delays is critical, and must be ju-

diciously made. In [9], 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

C. PSM

PSM was recently proposed in [2], and also independently in

[3]. Both works make use of the orthogonality of Hermite func-

tions.Theworkdescribedin[3]usesamodulatedversionofHer-

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

themaximumachievablebitrate.Theworkdescribedin[2]uses

optimized combinations of Hermite functions in order to obtain

pulses occupying roughly the same spectra, therefore achieving

bestspectralefficiency.Intherestofthissectionandthroughout

the rest of this paper, we refer only to the PSM modulation pro-

posed in [2].The original idea of PSM consistsof an orthogonal

binary modulation, where two orthogonal waveforms are used

torepresentthedatabits.Thereception,basedonsignalcorrela-

tion,benefitsfromthetime-orthogonalitybetweenthewaveforms

used.Thewaveformsof(1)and(2)havesuchaproperty,related

tothefactofbeing,respectively,oddandeventimefunctions.The

theoretical performance of binary orthogonal PSM follows that

for any orthogonal signals in AWG channels

(8)

For

-ary orthogonal PSM schemes, we need to use sets of

pulse waveforms which are all orthogonal to each other. This

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DA SILVA AND DE CAMPOS: SPECTRALLY EFFICIENT UWB PULSE SHAPING 315

can be obtained from sets of orthogonal functions. The modula-

tion proposed in [2] 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

(9)

where

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 [10], [11]. 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 [12], i.e., collections of signals

that if

and belong to the collection, then any linear

combination

also belongs toit.In

have that

Observing the equations of the Gaussian pulses (1) and (2)

showninFig.1,onecannoticethattheyareformedbyaGaussian

curvemultipliedbyapolynomial.TheHermitefunctionspresent

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

technologytogenerateandreceiveHermite-basedpulsesmaynot

differmuchfromthatcurrentlyused.Furthermore,wemayexpect

thattheirfrequencycontentwillbenothigherthannecessary,asa

consequenceofthemaximalconcentrationintimeandfrequency

ofHermitespaces(cf.SectionIII-B).Theseissues,alongwiththe

orthogonality of Hermite functions, constitute the motivation to

usethesefunctions. Fig.2showssomeofthefirstHermitefunc-

tions, given by

space. This is a particular case

such

,we also

has finite value.

(10)

where

by the formulas

are the Hermite polynomials, recursively obtained

(11)

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

next

(12)

(13)

(14)

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

bedefinedasthespacespannedbythefirst

[12]

Hermitefunctions

span

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

dimension 2.

An interesting fact, proved with the help of time–frequency

analysis theory [12], 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-

width product.

C. Hermite Pulses

Hermite pulses of

waveform belonging to the

th order are defined [2] as any signal

th dimension Hermite space.

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316 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 55, NO. 2, FEBRUARY 2007

These pulses can thus be expressed as

(15)

where

Hermite pulses can be expressed as

is an-degree polynomial. Alternatively,

(16)

where

(17)

(18)

i.e.,

tions, and

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

projection vectors:

is a vector composed of the first

is the projection vector, a real column vector

Hermite func-

first Hermite functions.

is expressed

, we have the following

(19)

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

tions by

, to the first three Hermite func-

(20)

(21)

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

projection vectors

from each pulse

th-order Hermite pulses,

formed by the

(22)

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

(23)

A. General Procedure to Obtain a Set of

Orthogonal Pulses

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.

1) Given

, make [from (23)]

-dimensional

. In some sense,

(24)

2) Determine the generic form of the projection vector and

then build the equation expressing the pulse by the func-

tions of the basis

(25)

(For the purpose of clarity, whenever possible, we omit the

index referring to the pulse.)

3) Apply the zero-DC restriction given by

(26)

a) Express the algebraic relation between elements of

(27)

where

is a linear function resulting from (26).

b) Apply the same relation to

byalinearcombinationoftheothers.Thisreducesthe

number of variables in

to

, according to (24)].

4) Build the normalized orthogonality equations

, replacing its first line

fromto [or

(28)

which consist of

ables.

5) Obtain the generic expression for the mean central fre-

quency and the frequency spreading.

a) Take the Fourier transform of

equations withvari-

(29)

where

(14).

are given by (12) to

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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

(30)

c) Write the expressions of mean central frequency and

frequency spreading defined as

(31)

(32)

respectively. Note that (29)–(32) are functions of

.

d) Apply (27) to eliminate

from (31) and (32)

(33)

(34)

where

and (32), but in terms of

6) Find the functional to be minimized.

a) Make

andhave the same expressions of (31)

only.

and (35)

with a suggested value of

ically determined to fit best

frequencies of

b) Determine

(this was empir-

to the cutoff and

dB).

(36)

where and

ered.

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 [9]. The pulses found are

shown in Fig. 3, and correspond to the following vectors:

in (36).

(37)

The three pulses are given by

(38)

Fig. 3. Three optimal orthogonal pulses.

(39)

(40)

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

(41)

(42)

(43)

(44)

(45)

which consist of 10 equations with 16 unknowns

(46)