Novel Algorithms for Optimal Waveforms Designin Multicarrier Systems

Mohamed SialaUniversity of Carthage,

MEDIATRON Laboratory,

High School of Communications,

2083 Ariana, Tunis, Tunisia

Fatma AbdelkefiUniversity of Carthage,

CoSIM Laboratory,

High School of Communications,

2083 Ariana, Tunis, Tunisia

Zineb HraiechUniversity of Carthage,

MEDIATRON Laboratory,

High School of Communications,

2083 Ariana, Tunis, Tunisia

Abstract—In this paper we propose novel efficient algorithmsthat allow optimal waveform design for multicarrier systems,specifically for the Orthogonal Frequency Division Multiplexing(OFDM) based systems. We start from the rapidly time-varyingand strongly delay-spread characteristics of the prorogationchannel that are used for such transmission systems and includethem in the closed-form expression of the Signal to Interfer-ence plus Noise Ratio (SINR) of the considered multicarriersystem. Using our algorithms for maximizing the derived SINRexpression leads to obtain the optimal waveforms at both theTransmitter (Tx) and the Receiver (Rx) sides, that significantlyreduce the system Inter-Carrier Interference (ICI)/Inter-SymbolInterference (ISI) and guarantee maximal SINRs for realisticmobile radio channels. Our proposed algorithms allow to designwaveforms at the Tx/Rx sides that are adapted to the channelpropagation conditions and hence result in spectacular perfor-mance enhancement compared to multicarrier systems usingconventional waveforms. Simulation results are given to supportour claims.

I. INTRODUCTION

The Maximization of the overall data rate transmission in

communication systems represents actually the main challenge

and objective of many research works. This objective generally

faces a high interference level that exists in the received

signal and that dramatically disturbs and degrades its quality.

Interesting solutions were proposed in order to overcome

this problem, including the Orthogonal Frequency Division

Multiplexing (OFDM) on which latest cellular technologies

rely on, such as LTE-A and that is used as a broadband

transport modulation in the physical layer of many advanced

communication systems [1]. This modulation offers several

advantages over conventional single carrier approaches such

as an enhanced capacity of the OFDM based system, a

high spectral efficiency for it and immunity to Inter-Symbol

Interference (ISI). In addition to these benefits, equalization

on a narrow band subcarrier is less complex in terms of

processing than other broadband schemes that don’t use the

OFDM transmission [2], [3].

Although there are many advantages of the OFDM ap-

proach, several research studies highlighted a set of drawbacks

of the OFDM transmission, including mainly the spectral leak-

age of Digital Fourier Transform (DFT)-based OFDM systems

that can result in an important interference level through the

OFDM subcarriers, awareness to carrier frequency offsets and

also constrained bandwidth efficiency due to the adjunction

of Cyclic Prefix (CP) used for the channel equalization in the

frequency domain. However, the OFDM approach efficiently

solves the problem of the frequency selective fading channel

by the mean of low-complexity equalizers and this does

represent a crucial aspect in the case of highly frequency-

selective channels. Contrary to high mobility situations that

are commonly envisaged for the next Fifth Generation (5G)

of mobile communication systems, the wireless propagation

channel is time-frequency variant where the time disper-

sion emerges from the multipath characteristic and the time-

selectivity arises from the Doppler spread that damages the

orthogonality induced in the OFDM signal and consequently

results in oppressive Inter-Carrier Interference (ICI) [4]. Thus

a non-orthogonal future wireless multi-carrier scheme with

malleable waveforms would represent an interesting solution

that potentially reduces the ISI and ICI and also minimizes

the energy spreading. This could also be a powerful candidate

to be used for the 5G systems. rithm This paper focuses on

minimizing the ICI and ISI through an adequate choice and

design of the OFDM used waveforms, even in cases where

the OFDM signal induced orthogonality is destroyed. We note

that in this same context, several solutions were proposed in

the literature to mitigate the ICI and ISI when the propagation

channel is doubly dispersive. One of the envisaged solution

is the one proposed in [4], that banks on waveform based on

Hermite-Gaussian function whose time-frequency density is

equal to 2 and that leads to a better time-frequency localization

compared to the OFDM basic rectangular pulse. This solution

reduces the ICI/ISI levels but at the same time results in

a spectral efficiency reduction and remains more complex

from a processing point of view than the CP-OFDM scheme.

Another attractive alternative technique is the Lattice-OFDM

[6] that consists in adapting the signal waveforms and the

lattice feature to the propagation channel conditions. Other

approaches to reduce ICI/ISI are based on further equalization

treatment at the receiver side such as the Least-Squares QR

(LSQR) algorithm [7]. Note also that muticarrier systems

based on filter-banks were investigated in [8], [9] and used

a Gabor frame for the analysis bank and its dual frame for the

synthesis bank, in order to mitigate the ICI/ISI effect.

This paper investigates two equivalent efficient algorithms

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that, through a maximization step of the Signal to Interference

plus Noise Ratio (SINR), deduce the OFDM optimized wave-

forms that should be used at the transmitter and the receiver

sides. These algorithms present the advantage of the flexibility

to non-orthogonal multicarrier schemes and the simplicity of

their implementation.

The work described in this paper is organized as follows: in

Section II we describe the multicarrier system model where we

specify its transmitter and receiver blocks. We also detail the

propagation channel models that will be used in these paper

investigations. In Section III we focus on the derivation of the

SINR expression in the case of our considered multicarrier

system. We dedicate Section IV to explain the derivation of our

two proposed equivalent optimization waveforms algorithms.

In Section V we illustrate the obtained optimization numerical

results and we highlight the efficiency of our optimized

waveforms algorithms. Finally, in Section VI we give the

conclusions of our work and its perspectives.

II. NOTATIONS

The boldface lower case letters denote vectors and the bold-

face upper case letters refer to matrices. The superscripts .T

and .∗, denote the transpose and the element-wise conjugation,

respectively. In addition, E and � refer to the expectation

operator and the component-wise product of two vectors or

matrices, respectively. I denotes the identity matrix with ones

in the main diagonal and zeros elsewhere.

III. SYSTEM MODEL

This section provides preliminary concepts and notations re-

lated to the considered multicarrier scheme and the channel as-

sumptions and model. Starting from the symbols, lattices, and

waveform inspired from the framework of Weyl-Heisenberg,

frames are discussed through this section. Furthermore, we

consider their discrete time version in order to simplify the

theoretical derivations that will be investigated.

A. OFDM Transmitter and receiver blocks

We denote by T the OFDM symbol duration and by F the

frequency separation between two adjacent subcarriers. Let

am,n, m,n ∈ Z, be the transmitted symbol at time nT using

subcarrier mF and assumed to be independent identically

distributed (i.i.d.) with zero mean and energy equal to E.

The baseband transmitted signal is given by the following

expression:

e(t) =∑m,n

amnϕmn(t),

where ϕmn(t) = ϕ(t − nT )ej2πmFt denotes the time and

frequency shifted version of the OFDM transmitter prototype

waveform ϕ(t) used to transmit the symbol amn and that is

assumed to have a unitary energy, means ‖ϕ(t)‖ = 1.

The transmitted signal is sampled at a sampling rate R =1Ts

, where Ts = TN is the sampling period such that N ∈

N, 1TsF

= Q ∈ N and N > Q. Let Q denotes the number

of the OFDM signal sub-carriers. The sub-carrier frequencies

correspond to mF = mQTs

, where m = 0, 1, . . . , Q − 1. Note

that the parameter (N−Q)Ts can be confused with the notion

of Cyclic Prefix (CP) in a conventional OFDM system and the

time-frequency plane density refers to δ = 1FT = N

Q .

Let the infinite vector e = [. . . , e−2, e−1, e0, e1, e2, . . .]T =

(eq)q where eq , also denoted [e]q, be the sampled version of

the transmitted signal at time qTs with q ∈ Z such that e =∑m,n amnϕmn. We denote by ϕmn = (ϕq−nN )q�(ej2π

mqQ )q

the vector that results from a time shift of nNTs = nTand a frequency shift of mF = m

QTsof the transmission

prototype vector ϕ = (ϕq)q . The waveform unitary energy

assumption is also maintained in the discrete version, i.e.,

‖ϕ‖2 =∑

q∈Z |ϕq|2 = 1.

Assuming a linear time-varying multipath channel h(p, q)with q and p standing respectively for the normalized obser-

vation time and the time delay, the received signal has the

following expression:

rq =∑m,n

amn

∑p

h(p, q)[ϕmn]q−p + nq

=∑m,n

amn[ϕmn]q−p + nq,

where [ϕmn]q =∑

p h(p, q)[ϕmn]q is the channel distorted

version of ϕmn and n = (nq)q denotes the discrete complex

Additive White Gaussian Noise (AWGN), the samples of

which are centered, uncorrelated with common variance N0.

The decision variable

Λkl = 〈Ψkl, r〉 = ΨHkl r (1)

on the transmitted symbol amn is obtained by projecting the

received signal r on the receiver pulse Ψkl, where Ψkl =

(Ψq−lN )q�(ej2πkqQ )q is the time and frequency shift version

of the received vector dedicated to demodulate amn. Perfect

demodulation is achieved since the transmitter waveform and

the receiver one are bi-orthogonal. Note that this condition

could not be perfectly satisfied and we will consider a general

framework for it in this paper.

B. Channel model

For simplification sake, assume that the Linear Time Variant

(LTV) channel h(p, q) is Wide-Sense Stationary with Un-

correlated Scattering (WSSUS). Then, the channel discrete

scattering function has the following expression:

S(p, ν) =∑Δq

φh(p,Δq)e−j2πνTsΔq,

where φh(p1, p2;Δq) = E[h∗(p1, q)h(p2, q + Δq)] =φh(p1, Δq)δK(p2 − p1) defines the corresponding autocorre-

lation function with δK being the Kronecker symbol.

To simplify the derivations, we presume a channel with a

finite path number, K, with the following channel impulse

response:

h(p, q) =

K−1∑k=0

hkej2πνkTsqδK(p− pk),

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where hk, νk and pk are respectively the amplitude, Doppler

frequency and the time delay of the kth path. The paths ampli-

tudes hk are assumed to be i.i.d. complex Gaussian variables

with zero mean and average powers equal to πk = E[|hk|2]such that

∑K−1k=0 πk = 1. We choose an exponential truncated

decaying model. Let 0 < b < 1 be the decaying factor, such

that the paths powers can be expressed as π=1−b1−bK

bk.

In this paper, we consider a radio mobile channel where

the scattering function S(p, ν) has periodical classical Doppler

spectral density α(ν), with period 1Ts

. This scattering function

obeys to the Jakes model that is decoupled from the dispersion

in the time domain denoted β(p). This means that S(p, ν) =β(p)α(ν), such that β(p) =

∑K−1k=0 πkδK(p− pl) and

α(ν) =

⎧⎨⎩

πBd

1√1−( 2ν

Bd)2

if |ν| < Bd

2

0 if Bd

2 ≤ |ν| ≤ 12Ts

(2)

where Bd is the Doppler speard and∫α(ν)ej2π

νTs

k dν =J0(πBdTsk).

IV. SINR ANALYSIS

Without loss of generality, we will focus on the evaluation

of the SINR for the symbol a00. Referring to (1), the decision

variable on a00 has the following expression:

Λ00 = a00 〈Ψ00, ϕ〉+∑

(m,n) �=(0,0)

amn 〈Ψ00, ϕmn〉+ 〈Ψ00,n〉 .

This decision variable is the sum of three elements: a useful

element, an interference one and a noise one. The purpose

of our optimization approach is to minimize the mean power

of the interference element, resulting from symbols amn such

that (m,n) �= (0, 0), for a fixed value of the useful element

power.

A. Average Useful Power

The useful term corresponding to a00 is given by U00 =a00 〈Ψ00, ϕ00〉. For a given realization of the channel, the

average power of the useful terms is given by PhS =

E| 〈Ψ00, ϕ00〉 |2. Therefore, the average of the conditional

useful power over channel realizations is PS = E[PhS ]h,

where [ϕ00]q =∑K−1

k=0 hk[ϕ00]q−pkej2πνkTs(q−pk). Let σp(v)

denote the time shift operator by p sample durations of the

vector v = (νq)q , i.e. σp(v) = (νq−pk)q and Φν denote the

Hermitian matrix with (p, q)th entry ej2πνTs(p−q). Under these

notions, we deduce that:

PS = E ΨHKSϕS(p,ν)Ψ ,

where we define the useful signal Kernel matrix as

KSϕS(p,ν) =

∑K−1k=0 πkΦνk

� (σpk(ϕ00)σpk

(ϕ00)H). Since

PS is a positive entity, then the Kernel matrix is a positive

Hermitian matrix. Hence, given any choice of the transmitter

prototype ϕ, one can maximize the useful signal power by

choosing the receiver prototype vector Ψ as the eigenvector

of the KSϕS(p,ν) that corresponds to its maximum eigenvalue.

B. Average interference power

The interference term within the decision variable Λ00 is

given by I00 =∑

(m,n) �=(0,0) amn 〈Ψ00, ϕmn〉 that results

from the contribution of amn such that (m,n) �= (0, 0). The

mean power of PhI over channel realizations is given by

PI = E[PhI ]h = E

∑(m,n) �=(0,0)

E[| 〈Ψ00, ϕmn〉 |2]h.

By re-iterating the same derivation as the one in Section IV-A,

we find that:

PI = E ΨHKIϕS(p,ν)Ψ .

where the interference Kernel matrix is ex-

pressed as KIϕS(p,ν) =∑K−1

k=0 πkΦνk�

(∑

(m,n) �=(0,0) σpk(ϕmn)σpk

(ϕmn)H). Since PI is always

positive, then KIϕS(p,ν) is also a positive semidefinite matrix.

We deduce that for any choice of the transmitter prototype

vector ϕ, one can minimize the interference power by

choosing the receiver prototype vector Ψ as the eigenvector

of KIϕS(p,ν) that corresponds to its smallest eigenvalue. It

is important to highlight an important property that both

KIϕS(p,ν) and KIΨS(−p,−ν) do verify, namely

ΨHKIϕS(p,ν)Ψ = ϕHKIΨS(−p,−ν)ϕ. (3)

This property is very important for the following part of this

paper: given any arbitrary choice of the receiver prototype

vector Ψ , we can optimize the choice of the transmitter

prototype vector ϕ through a minimization of the quadratic

form with Kernel matrix KIΨS(−p,−ν). Note that the same

relationship remains true between KSϕS(p,ν) and KSΨ

S(−p,−ν).

C. Average noise power

The noise average power is given by:

PN = E[| 〈Ψ00,n〉 |2] (4)

= ΨH00E[nn

H ]ΨH00. (5)

Since the noise is assumed to be white, therefore its covariance

matrix is equal to Rnn = E[nnH ] = N0I. Consequently, as

the prototypes are of unitary energy, then PN = N0.

D. SINR expression

Using the obtained expressions of the useful power PS , the

interference power PI and the noise power N0, the SINR has

the following expression:

SINR =PS

PI + PN=

ΨHKSϕS(p,ν)Ψ

ΨHKIϕS(p,ν)Ψ + N0

E

. (6)

This expression is valid for normalized transmitted energy and

Tx/Rx prototype functions ϕ and Ψ . A possible approach to

minimize the SINR consists in minimizing the denominator of

(6), ΨHKIϕS(p,ν)Ψ , subject to a fixed useful power and this

can be performed through the Lagrange multiplier method.

This approach is useful to minimize the normalized interfer-

ence power for a fixed N0

E value. In this case our denominator

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minimization problem becomes equivalent to consider the

following Lagrangian function:

Qλ,SNRS(p,ν) (ϕ,Ψ ) = ΨH(KIϕS(p,ν) − λKSϕ

S(p,ν))Ψ (7)

where λ is the Lagrange multiplier that depends on the Signal

to Noise Ratio (SNR) value. For such a problem, the common

solutions proposed in the literature are based on presenting

the SINR as a function of ambiguity functions [10]. In this

paper we investigate an approach based on Kernel functions

depending on Ψ and ϕ, that we will detail in the following

sections.

V. OPTIMIZATION TECHNIQUE

Our main objective consists in determining the couple

(ϕ,Ψ ) that maximizes the SINR for a given SNR value.

As mentioned previously, this is equivalent to optimize the

auxiliary function Qλ,SNRS(p,ν) (ϕ,Ψ ) where the parameter λ must

be evaluated. For this purpose, we investigate two equivalent

approaches that we will explain in the sequel.

A. First approach

To assess the value around which we should choose the

Lagrange multiplier, we perform a derivation of the SINR

involving the transmitted or receiver prototype vectors. Then

we do the same for the auxiliary functions and we make an

identification of the obtained terms. Hence, the gradient of the

SINR with respect to Ψ is given by:

∂SINR

∂Ψ=

−2ΨHKSϕS(p,ν)Ψ

(ΨHKIϕS(p,ν)Ψ + N0

E )2(KIϕS(p,ν)Ψ (8)

− 1

SINRKSϕ

S(p,ν)Ψ ). (9)

The vector Ψ leading to the optimum SINR corresponds to a

null value of the gradient, i.e.

KIϕS(p,ν)Ψ −1

SINRKSϕ

S(p,ν)Ψ = 0. (10)

Similarly, the vector Ψ that cancels the auxiliary gradient

function (7) must verify the following equality:

∂

∂ΨQλ,SNR

S(p,ν) (ϕ,Ψ ) = 2(KIϕS(p,ν) − λKSϕS(p,ν))Ψ = 0.

(11)

Referring to expression (10), the Lagrange multiplier λ should

be equal to the inverse of the SINR. The optimization problem

could be solved by the general eigenvalue problem (GEP),

since we have to solve (11). As our object is to maximize

the SINR and as 1λ is an eigenvalue of (KIϕS(p,ν),KSϕ

S(p,ν))

in expression (10), the optimum value λopt = 1SINRmax

corresponds to its maximum eigenvalue in order to meet our

expectations. It results that Ψopt is the eigenvector associated

to the eigenvalue λopt of the GEP (KSϕS(p,ν), KIϕS(p,ν)) (see

Algorithm 1). We are coming up with a Lagrange multiplier

evaluation that can be characterized as a direct, simple and

streamlined approach.

Algorithm 1: First approach

Require: channel parameters (K, pk, νk, hk, Ts), ϕ(0), ε,Ψ (0) = 0, e(Ψ) = e(ϕ) = 2, k = 0, SNR

Compute KSϕ(0)

S(p,ν) and KIϕ(0)

S(p,ν)

while e(Ψ) > ε or e(ϕ) > ε doλ = eig(KSϕ(k)

S(p,ν),KIϕ(k)

S(p,ν))k ← k + 1Ψ (k) eigenvector associated to λmax

Evaluate KIΨ(k)

S(−p,−ν) and KSΨ (k)

S(−p,−ν)

ν = eig(KSΨ (k)

S(−p,−ν),KIΨ(k)

S(−p,−ν))

ϕ(k) eigenvector associated to νmax

Evaluate errors: e(Ψ) = ‖Ψ (k+1) − Ψ (k)‖ and

e(ϕ) = ‖ϕ(k+1) −ϕ(k)‖end while

B. Second approach

Another direct optimization method consists in diagonlizing

the SINR denominator of expression (6) and then perform

a basis change that will simplify the expression of this

denominator, so that our optimization problem becomes a

maximization one that implies finding the eigenvector of the

SINR numerator that corrresponds to its maximum eigen-

value. More precisely, we first introduce the Kernel function

KINϕS(p,ν) = KIϕS(p,ν) + (N0

E )I. The eigendecomposition

of KINϕS(p,ν) is KINϕ

S(p,ν) = UΛUH , where U is a

unitary matrix, Λ is a diagonal one with nonnegative real

numbers on the diagonal. Then, the SINR denominator can

be written as ΨHKINϕS(p,ν)Ψ = ΨHUΛUHΨ = uHu

where u = Λ12UHΨ . Since KIϕS(p,ν) is a positive semidefinite

matrix, then all the entries of Λ are positive and greater thanN0

E > 0. Therefore, Ψ = UΛ−12u and the SINR expression

becomes the following:

SINR =uHΦu

uHu,

where Φ = Λ−12UHKSϕ

S(p,ν)UΛ−12 is a positive matrix.

Hence, maximizing the SINR is equivalent determining the

maximum eigenvalue of Φ and its associated eigenvector

umax. Therefore, Ψopt = UΛ−12 umax

||UΛ−12 umax||

(see Algorithm 2).

It is important to note that this second approach is slower

than the first one in terms of necessary compilation resources

and leads approximately to the same performances but more

stable numerically.

VI. SIMULATION RESULTS

The results of the proposed optimization techniques are car-

ried out using the MATLAB language and for a discrete lattice

δ = 1FT . The optimal Tx/Rx waveform couple (ϕopt,Ψopt),

maximizing the SINR, are evaluated through Algorithm 1 or

Algorithm 2 for a fixed ϕ(0). These simulations are realized

for a high time and frequency dispersive channel characterized

by the channel obeying scattering function (2) defined on the

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Algorithm 2: Second approach

Require: channel parameters (K, pk, νk, hk, Ts), ϕ(0), ε,Ψ (0) = 0, e(Ψ) = e(ϕ) = 2, k = 0, SNR

Compute KSϕ(0)

S(p,ν) and KIϕ(0)

S(p,ν)

while e(Ψ) > ε or e(ϕ) > ε doKINϕ(k)

S(p,ν) = KIϕ(k)

S(p,ν) +1

SNRI

Compute [U,Λ] = eig(KINϕ(k)

S(p,˚))

Compute Φ = Λ−12UHKSϕ(k)

S(p,ν)UΛ−12

Compute [umax, λmax] = eig(Φ)k ← k + 1

Ψ (k) = UΛ−12 umax

||UΛ−12 umax||

Evaluate KIΨ(k)

S(−p,−ν), KSΨ (k)

S(−p,−ν) and

KINΨ (k)

S(−p,−ν) = KIΨ(k)

S(−p,−ν) +1

SNRI

Compute [V,Σ] = eig(KINΨ (k)

S(−p,−ν))

Compute Θ = Σ− 12VHKSΨ (k)

S(−p,−ν)VΣ− 12

Compute [vmax, νmax] = eig(Θ)

ϕ(k) = VΣ−12 vmax

||VΣ−12 vmax||

Evaluate errors e(Ψ) = ‖Ψ (k+1) − Ψ (k)‖ and

e(ϕ) = ‖ϕ(k+1) −ϕ(k)‖end while

time-frequency plane [0, Tm] × [−Bd

2 ,−Bd

2 ], where Tm and

Bd are respectively the delay and Doppler spreads.

In the simulations, we assumed that ϕ(0) is a Gaussian

pulse. Such a choice is motivated by the fact that such a pulse

exhibits excellent immunity against channel distortions be-

cause it is the most time-frequency localized Hermite function

of index 0. Most of simulations are carried under the following

parameters: BdTm = 0.01, Bd

F = 0.1 and FT = 1.25.

In Fig.1, we present the evolution of the SINR versus the

normalized maximum Doppler frequency for a normalized

channel delay spread values to Bd

F where Q = 128 and

for a waveform support duration equal to 2T . In this figure,

we compare our optimized transmitter waveform design with

the conventional OFDM system that deploys cyclic prefixes

(CP) of 8 or 48 samples. This figure demonstrates that our

approach outperforms the conventional OFDM system and the

gain that can be achieved in SINR is 5dB. As we see, the

gain remains practically unchanged and increases when the

waveform support duration increases as it is illustrated in Fig.2

where it can achieve 8dB when the support duration is equal

to 4T .

Fig.3 shows the evolution of the SINR in dB with respect

to FT where we compare the conventional OFDM (denoted

by "CW") and the optimized waveforms (denoted by "OW").

As can be expected, the proposed system outperforms the

conventional OFDM for a large range of channel dispersions,

especially in the case of a highly frequency dispersive channel

and this is whatever the support duration of the waveform.

This figure reveals a significant increase that can reach 8dB

in the obtained SINR when the support duration increases.

Furthermore, it represents a mean to find the adequate couple

(T, F ) of an envisaged application to insure the desired

transmission quality. Then, we can note that for a lattice

density equal to δ = 0.8 (FT = 1.25), coinciding with a

conventional OFDM system with a CP having one quarter of

the time symbol duration, the SINR can be above 20dB. As

we can also see through this figure, if the density decreases,

the SINR increases and therefore, OFDM transmission is less

subject to ICI and ISI. As we see, the SINR is almost the

same when the support duration, D, increases, i.e. D = 2and D = 4 lead approximately to the same SINR. This

could reduce considerably the computation complexity. In this

example, rather considering large support duration, we can

take in D = 2 since it leads to the same SINR as larger value

of D.

Tx/Rx waveforms, mainly ϕopt and Ψopt, corresponding to

the maximal SINR, are illustrated in Fig.4 for FT = 1.25,

BdTm = 0.01 and D = 2. This figure provides a comparison

between the optimized waveforms. As we remark in this figure,

the receiver and transmitter pulses are different from those of

the conventional OFDM system. This confirms our claims in

the sense that the conventional OFDM system not usually lead

to the optimal SINR. Fig.5 shows that the obtained transmitter

pulse reduces exponentially of about 80dB, the out-of-band

(OOB) emissions contrary to a conventional OFDM system

that requires large guard bands to do so and it can be observed

that the optimal prototype waveform is more localized. We

notice also that when D increases the gain becomes less

pronounced starting from a value of D = 4. More importantly,

since the optimized obtained waveform reduces dramatically

the spectral leakage to neighbouring subcarriers, inter-user

interference will be minimized especially at the uplink of

OFDMA systems, where users arrive at the base station with

different powers.

VII. CONCLUSION

In this paper, we investigated an optimal waveform design

that is especially suited for multicarrier transmissions over

rapidly time-varying and strongly delay-spread channels. For

this purpose, potential optimizing algorithms for the transmit-

ter and receiver waveforms were proposed. Moreover, these

waveforms provide a neat reduction in ICI/ISI and guarantee

maximal SINRs for realistic mobile radio channels. Simulation

results demonstrated the excellent performance of the proposed

solutions and highlighted the property of the efficient reduc-

tion of the spectral leakage obtained through the optimized

waveforms. As such, our proposed solutions can be seen as

an attractive candidate for the optimization of the spectrum

allocation in 5G systems. A possible challenging research axis

consists in extending the optimization for the OFDM/OQAM

and multi-pulse OFDM systems.

REFERENCES

[1] K. Heng H. Kopka and P. W. Daly, Interference Coordination for OFDM-based Multihop LTE-advanced Networks , IEEE Wireless Communica-tions, pp. 54–63, vol.18, issue 1, Feb. 2011.

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(a) CP = N −Q = 48.

(b) CP = N −Q = 8.

Fig. 1. Optimized SINR as a Function ofBdF

(Q = 128, SNR = 30dB,

BdTm = 10−2 and D = 2).

Fig. 2. Optimized SINR as a Function ofBdF

(Q = 128, SNR = 30dB,

BdTm = 10−2 and D = 4).

[2] L. J. Cimini, Analysis and Simulation of Digital Mobile Channel UsingOrthogonal Frequency Division Multiplexing, IEEE Transactions onCommunications, vol.33, no. 7, pp. 665–675, 1985.

[3] W. Y. Zou and Y. Wu, COFDM : An Overview, IEEE Transactions onBroadcasting, vol. 41, no.1, pp. 1–8, Mar. 1995.

[4] R. Haas and J.C. Belfiore, A Time-Frequency Well localized Pulse forMultiple Carrier Transmission, Wireless Personnal Communication, pp.1–18, 2007.

[5] P. Schniter, A new Approach to Multicarrier Pulse Design for DoublyDispersive Channels, In Proc. Allerton Conf. Commun., Control, andComputing, pp. 1012–1021, 2003.

[6] T. Strohmer and S. Beaver, Optimal OFDM Design for Time-FrequencyDispersive Channels, IEEE Trans. Commun., vol. 51, pp. 1111–1122, Jul.2003.

[7] G. Taubock, M. Hampejs, P. Svac, G. Matz, F. Hlawatsch and K.Grochenig, Low-Complexity ICI/ISI Equalization in Doubly DispersiveMulticarrier Systems Using a Decision-Feedback LSQR Algorithm, IEEE

Fig. 3. Optimized SINR as a Function of FT (Q = 128, SNR = 30dBand BdTm = 10−2).

Fig. 4. Tx/Rx Waveforms Optimization Results (D = 3).

Fig. 5. Transmit Waveform Power Spectral Density Optimization.

Transactions on Signal Processing, pp. 2432–2436, 2011.[8] D. Roque, C. Siclet and P. Siohan, A Performance Comparison of FBMC

Modulation Schemes with Short Perfect Reconstruction Filters, ICT,Jounieh, Liban, 2012.

[9] D. Roque and C. Siclet, Performances of Weighted Cyclic Prefix OFDMwith Low-Complexity Equalization, IEEE Communications Letters, vol.17, pp. 439–442, 2013.

[10] W. Kozek and A. F. Molisch, Nonorthogonal Pulseshapes for Multicar-rier Communications in Doubly Dispersive Channels, IEEE Journal onSelected Areas in Communications, vol. 16, no. 8, Oct. 1998.

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