optimal pulses robust to insufficient synchronization for

Optimal Pulses Robust to Insufficient Synchronization for

OFDM/OQAM Systems

Yu Zhao1, Xihong Chen

1, and Qun Zhang

2

1Air and Missile Defense College of Air Force Engineering University, Xi’an 710051, China

2Information and Navigation College of Air Force Engineering University, Xi’an 710077, China

Email: {sunzy54321, xhchen0315217, pepper24}@163.com

Abstract—In this paper, a novel pulse shaping method is

presented for Orthogonal frequency division multiplexing with

offset quadrature amplitude modulation (OFDM/OQAM)

systems. Designed as a linear combination of normalized

Hermite functions, the proposed pulse is optimized with respect

to interference caused by insufficient synchronization such as

carrier frequency offset (CFO) and timing offset (TO). The

optimization problem, based on signal-to-interference ratio (SIR)

maximization criterion, is carried out using an efficient iterative

algorithm. Simulation results show that this optimal pulse

outperforms traditional prototype pulses in the presence of

insufficient synchronization. Index Terms—OFDM/OQAM, pulse shaping, SIR, carrier

frequency offset, timing offset

I. INTRODUCTION

OFDM is employed in various systems due to its

robustness against time dispersion caused by multipath

propagation. These systems are lattice structures formed

by time-frequency shifts of a rectangular prototype pulse.

However, the cyclic prefix or guard time interval used to

prevent inter-symbol interference (ISI) will cost a loss of

spectral efficiency. Moreover, the rectangular pulse

which exhibits a sinc-shaped frequency response will

result in serious inter-carrier interference (ICI) in

frequency dispersion channels. In order to counteract the

above drawbacks, some pulse-shaping OFDM systems

(PS-OFDM) were proposed [1]-[3]. With lower sidelobe

and faster spectral decay pulse shape, PS-OFDM systems

are more robust to frequency dispersion caused by

Doppler effect and carrier frequency offset (CFO).

However, the underlying requirement in PS-OFDM is

that the pulses must be orthogonal or biorthogonal to

each other. However, a well localized orthogonal pulses

set cannot be achieved for unit density structures due to

Balian-Low theorem. Thus, many structures employ half

lattice density which results in a loss of bandwidth

efficiency.

Filter Bank Multicarrier (FBMC) is intensively studied

now as an alternative to OFDM in order to increase

bandwidth efficiency and to improve the robustness

Manuscript received December 10, 2014; revised May 12, 2015.

This work was supported by National Natural Science Fund of China

under Grant no. 60971100.

Corresponding author email: [email protected].

against various dispersive channels [4]. The most popular

FBMC technology is the OFDM/OQAM modulation

which has already been introduced in the TIA’s Digital

Radio Technical Standards and considered in WRAN

(IEEE802.22) [5]. In OFDM/OQAM systems, the

modulation used for each subcarrier is a staggered offset

QAM denoted as OQAM. The OQAM method introduces

a half symbol duration time offset between the real and

imaginary components of QAM (quadrature amplitude

modulation) constellation. With this approach, the

orthogonality constraint only holds in real field.

Therefore, a well time-frequency localized pulse can be

achieved in systems with unit lattice density.

In this paper, a novel OFDM/OQAM pulse shape

modeled as a linear combination of normalized Hermite

Journal of Communications Vol. 10, No. 5, May 2015

301©2015 Journal of Communications

doi:10.12720/jcm.10.5.301-307

OFDM/OQAM systems suffer from high sensitivity to

some desynchronization like CFO and timing offset (TO)

[6], [7]. CFO is mainly caused by different local

oscillators of the transmitter and receiver. TO may occur

when the actual sampling instant at the receiver is

different from the optimal one. Normally, CFO and TO

estimation are done prior to the demodulation, but the

estimation is usually not perfect. Fortunately, the

prototype pulses used in the systems are able to provide

different immunity against CFO and TO [8]. It has been

shown that well frequency-localized pulse, such as square

root raised cosine (SRRC), is in favor for the TO

tolerance. If equally severe CFO and TO are present,

isotropically time-frequency localized pulses such as

IOTA and weighted Hermite pulse (WHP) [9], [10], are

more suited. However, none of them can remain the best

place for all the cases. Therefore, design of robust pulses

to insufficient synchronization is an attractive research

topic. In [11], pulse shapes that have minimum ICI power

for a given value of CFO in PS-OFDM systems are

derived. In [12], a pulse shape design procedure which is

optimal with respect to interference due to CFO in

OFDM/OQAM systems is introduced. Recently, a novel

optimized generalized Gaussian pulse (OGGP) is

proposed [13]. The pulse has lower side lobes and faster

decay in time domain compared to conventional pulse

shaping methods. In [14], the author formulated a

problem of direct optimization of the filter coefficients to

minimize the stopband energy. Therefore, the optimized

filter (DOFC) can achieve significantly good frequency

selectivity.

functions is proposed. The pulse is designed to be robust

to insufficient synchronization. Firstly, the exact SIR

expression of the demodulated symbol versus CFO and

TO is derived. Then, the coefficients of Hermite

functions are optimized based on SIR maximization

criterion. Finally, the optimization problem is simplified

to a series of quadratically constrained quadratic

programs (QCQP) using an iterative algorithm. By

semidefinite relaxation, the QCQP can be translated to

semidefinite programming (SDP) problems which are

easy to be solved. Theoretical analysis and simulation

results show that the optimal pulse outperforms

traditional pulses especially in the systems with large

CFO and TO.

The rest of this paper is organized as follows. The

system model of OFDM/OQAM in the presence of CFO

and TO is described in the next section. In Section 3,

design of robust pulse is formulated as a SIR

maximization problem and the optimization algorithm is

presented. Then performance of the optimal pulse is

tested by numerical simulations in Section 4. Finally

some conclusions are drawn in Section 5.

II. SYSTEM MODEL IN THE PRESENCE OF CFO AND TO

A. Baseband Model for OFDM/OQAM

The transmitted baseband signal in the OFDM/OQAM

system is generated as:

0

12

, 0

0

, ( )

( ) ( )mn

Mj j mF t

m n

m n

m n t

s t a e e g t n

g

(1)

where ( / 2)( )mn m n is the phase term resulting

from the staggering rule of OQAM; ,m na denotes the

real-valued symbol located on the mth subcarrier at

time instant 0n . ,2m ka and ,2 1m ka can be obtained

by taking the real and imaginary components of a

complex-valued symbol from QAM constellation

respectively. They are assumed to be independent and

identically distributed with zero mean and average energy2

a . 0F is the inter-carrier spacing,

0 is the symbol

duration time (0 0 1/ 2F ); M is the subcarrier number

which is supposed to be even. ( )g t denotes the

symmetrical real-valued prototype pulse in this paper. Assuming an ideal channel, complete reconstruction of

the symbol ,m na can be obtained according to the

following orthogonality condition:

( )( )

, ,

0 0

( ), ( ) {( )

(( ) , ( ) )}

( , )

n l m k n l m k

m n k l R

g

g t g t j

A k m l n F

m k n l

(2)

where ( , ) is the 2-D Kronecker delta function.

( , )gA is the ambiguity function expressed as:

* 2( , ) ( / 2) ( / 2) j t

gR

A g t g t e dt (3)

Hence, the orthogonality can be ensured by finding a

real-valued and even prototype pulse of which the

ambiguity function satisfies [5]:

0 0(2 ,2 ) ( , )gA u vF u v , ( , )u v (4)

B. Effects of CFO and TO

In our analysis, phase offset and channel noise is

disregarded. Considering the insufficient synchronization,

we introduce a carrier frequency offset f and a timing

offset t .

At the receiver, the demodulated signal for

time-frequency position 0 0( , )n m is performed by

projecting the received signal onto the set of functions

0 0,{ ( )}m ng t with a modified inner product. Then the

demodulated data symbol can be expressed as:

0 0 0 0

0 0

2

, ,

2

,

ˆ ( ), ( )

{ ( ) ( ) }

j ft

m n m nR

j ft

m nR

a e s t g t t

e s t g t t dt

(5)

where {} is the real part operator. Substituting (1) and

(3) in (5) and introducing 0m m k and

0n n l ,

after some steps of straightforward manipulations, one

obtains that:

31 2 4

0 0 0 0

1

, ,

0

0 0

ˆ {

( , )}

Mjj j j

m n m k n l

m n

g

a a e e e e

A l t kF f

(6)

where 1 0 0 0 0(2 2 )m F t n f t f ,

2 0kn ,

3 ( )2

k l kl

, 4 0 0( )kF t l f . Then, it is

easy to separate the demodulated data symbol into the

useful and interference part, i.e.:

0 0 0 0 0 0 0 0, , , ,ˆ { }m n m n m n m na a J (7)0

with

1

0 0, ( , )j

m n ge A t f (8)

and

31 2 4

0 0 0 0, ,

0 0

0 0( , )

jj j j

m n m k n l

k l

g

J e a e e e

A l t kF f

(9)

In this paper, the phase estimation of the desired signal

section is assumed to be perfect at the receiver. Then, a

simple one-tap zero-forcing (ZF) equalizer is used to

compensate the phase rotation. Finally, it leads to the

equalized signal expressed as:

0 0 0 0

0 0 0 0

0 0 0 0

0 0

, ,

, ,

, ,

,

ˆ { } { }

( , )

m n m nZF

m n m n

m n m n

g m n ZF

Ja a

H H

A t f a J

(10)



where 1

0 0

j

m nH e

is the ZF equalizer coefficient at the

0 0( , )m n subcarrier. Noting that as long as the prototype

pulse is an even real-valued function, gA is also real

and even. Concerning the first part of the second equality

of (10), the desired symbol is attenuated by ( , )gA t f

which is caused by CFO and TO. As 2 0kn ,

2 1j

e . Therefore, the interference power can be

written as:

3 4

2

( )

0 0

0 0

2 2

3 4 0 0

0 0

2

4 0 0

2

0(

2

2

2

0, ),

{ ( , )}

cos ( ) ( , )

(0.5 0.5cos 2 ) ( , )

J ZF

j

g

k l

g

k l

g

k lt f

a

a

a

k l

P E J

e A l t kF f

A l t kF f

A l t kF f

(11)

Then, the SIR expression writes:

2

2

, 0 0

0 0

( , )

(( ,

, ) ( ))

,

g

k l g

k l

SA t f

It f A l t kF f

R t f

(12)

III. DESIGN OF ROBUST PULSE

A. Pulse Shape Design

Equation (12) illustrates that the SIR depends on the

prototype pulse through its ambiguity function while

CFO and TO are confirmed. The pulses are usually

designed based on the following criterion: 1) energy

concentration, 2) rapid-decay, 3) spectrum-nulling and 4)

channel characteristics. In the wireless communication,

SIR is the predominant parameter that characterizes the

system performance. Therefore, we search for the robust

prototype pulse based on SIR maximization criterion, i.e.:

( )( ) argmaxopt g tg t SIR (13)

The optimal pulse is designed as a linear combination

of Hermite functions which are maximally localized both

in time and frequency. The nth order Hermite function

( )nh t is defined by:

21/4( ) 2 2 ! 2n t

n nh t n H t e (14)

where ( )nH t is the nth order Hermite polynomial with

0 ( ) 1H t , 1( ) 2H t t , and it can be computed

recursively by: 1 1( ) 2 ( ) 2 ( ) n n nH t tH t nH t . Noting

that Hermite function corresponding to 0n is the

Gaussian pulse and it provides the best time and

frequency localization. Hermite function is the

eigenvector of its Fourier transformation:

( ) ( )n n nF h t h t , 2 jn

n e is the corresponding

eigenvalue. For 2 , 0,1,2...n i i , both ( )nh t and its

Fourier transformation are real-valued even functions.

Then the proposed pulse is expressed as: 1

2

0

( ) ( )L

i i

i

g t h t

(15)

In (15), the L most concentrated Hermite functions are

chosen. The coefficients i are assumed to be real.

Substituting (15) in (3), we obtain:

1 1

0 0

( , ) ( , )L L

g p q pq

p q

A A

(16)

where* 2

2 2( , ) ( / 2) ( / 2) j t

pq p qR

A h t h t e dt ,and

they can be calculated according to [15]. Substituting (16)

in (12) and introducing ' '( , ) ( , ) ( , )pq p qA v A v A v , we

obtain:

' '

' '

' '

' '

' '

0 , , , 1

' '

0 , , , 1

( , , , )

( , , , )

p q p qp q p q L

p q p qp q p q L

S p q p q

SIRI p q p q

(17)

' '( , , , ) ( , )AS p fq p tq (18)

, 0

0

0

' '

0

( , ) (( , , , ), )k l

k lI p q p t f A t kF fq l

(19)

Once well localized pulses are used, it is sufficient to

consider the energy perturbation only from neighboring

symbols with negligible performance loss. Thus, we only

consider 1k and 1l in (19). Finally, the

optimization task amounts to finding , 0,1,... 1i i L

so that the objective function ( , )SIR t f is maximized.

B. Optimization Problem Formulation

A new variable w a a is introduced to simply the

SIR expression. denotes matrix Kronecker product

and 0 2 1...T

L a . Then, as a function of w , the

SIR expression (17) can be reorganized in the following

matrix form:

T

TSIR

w Sw

w Iw (20)

where ( ) ijsS , ( )ijinI , 20 , 1i j L are symmetrical

and positive definite (SPD) matrix respectively defined

by: ' '( , , , )ijs S p q p q ,

' '( , , , )ijin p q p qI , i pL q ,

' 'j p L q .

The ambiguity function can also be reorganized in a

matrix form:

( , ) ( , ) T

gA a A a (21)

where ( , ) ( ) ijaA , 0 , 1 i j L is defined by

( , ) ij ija A . In [10], it was proposed that the

orthogonality constraints (4) are applied at the origin

( , ) 0u v and other L-1 nearest grid points due to the

good localization property of ( )g t . Then the coefficients

1 2 1... T

La can be determined by substituting (21)

in (4) and solving the resulting L equations.



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As ( , ) gA is real-valued, the orthogonality conditions

(4) can be transformed to:

2

0 0(2 ,2 ) ( , )gA u vF u v (22)

Substituting (16) in (22), we obtain:

( , ) ( , )T u vv uw B w (23)

where ( , ) ( ) iju v bB ,20 1, Lji , is defined by:

0 0(2 ,2 )ij ub A vF , i pL q , ' 'j p L q . Considering

the effect of insufficient synchronization, the constraints

on the nulls of the ambiguity function is not necessary

[16]. Moreover, for the sake of design freedom, we relax

the constraints (23) to: (0,0) 1Tw B w .

Besides the orthogonality condition constraints on w ,

there are some other inherent constraints due to

Kronecker product operator. In fact, two kinds of equality

constraints and one inequality constraint are introduced

(see Appendix): 0Cw ; 0T

ijw D w , 20 , 1 i j L ;

0Gw . Eventually the optimization problem can be

formulated as shown as follows:

(P): maxT

Tw

w Sw

w Iw

s.t. 1: (0,0 1)TC w B w , 2 : 0T

ijC w D w

3: 0C Cw , 4 : 0C Gw (24)

C. Iterative Algorithm for SIR Maximization

The optimization problem (P) is a fractional

programming problem, and the objective function is

non-convex in this case. In general, such problem is

NP-hard. To solve this problem, it will be convenient to

adopt Dinkelbach’s parametric approach as follows [17].

We define as the set of feasible solutions of problem

(P). Associated with this problem, the equivalent

parametric problem is:

( ) max[ ( , ) ]

T TF fw

w w Sw w Iw (25)

( ) arg max ( , )

fw

w w (26)

If there exists 0 for which ( ) 0 F ,

( ) w w is an optimal solution of (P) [17]. Then, an

effective iterative algorithm (known as Dinkelbach’s

method) is summarized as follows:

Input: outerL (maximum number of iterations)

0outer (convergence tolerance)

1: Calculate the largest eigenvalue and its

corresponding eigenvector of 1

I S . If ,

terminate( is an globally optimal solution);

Otherwise go to Step 2

2: 0i , 0 0

3: do whileouteri L and

1( , )

i i outerf w

4: 1i i

5: Solve 1( )iF

to obtain the optimal

solution

iw (inner loop)

6: T T

i i i i iw Sw w Iw

7: end do

8: return

It has been proved that convergence to the optimal

solution is guaranteed. Thus, problem (P) is transformed

to a series of QCQP problems:

1max ( )T

i w

w S I w , s.t. 1C , 2C , 3C , 4C (27)

Due to the variety of , optimization problem in (27)

is a non-convex quadratic program. Semidefinite

relaxation (SDR) method can be used to generate the

suboptimal solutions with quite encouraging results [18].

A crucial step for SDR is to observe that

Tr( ( ))T Tw Hw H ww , where Tr( ) stands for the

trace of a matrix. We introduce a variable TM ww ,

and note that it is equivalent to M being a rank one and

positive semidefinite (PSD) matrix. Thus, we obtain the

following equivalent formulation of (27):

1max Tr(( ) )i M

S I M

s.t. 1: Tr( (0,0) ) 1C B M

2 : Tr( ) 0ijC D M , 20 , 1i j L

3: Tr( ) 0TC C CM , 4 :C Gw 0

5: 0, ( ) 1C rank M M (28)

here, 0M indicates that

that except for the non-convex rank-one constraint

( ) 1rank M in 5C , the remaining problem is convex.

Thus, by dropping the rank-one constraint, we obtain a

relaxed semidefinite programming (SDP) problem which

can be efficiently solved using the general purpose CVX

package [19].

IV. NUMERICAL RESULTS

Among the prototype pulses that have been considered

for our studies in this paper, the IOTA and WHP design

have isotropic time-frequency localization, and OGGP

has the lowest time-domain side lobes. The DOFC design,

on the other hand, has the lowest side lobes in frequency

domain. In this section, the SIR performance of the

optimal pulse will be compared with that of the above

prototype pulses. The experiments are established

through a 20MHz channel where M=256 subcarriers and

4-QAM are employed. Rectangular time-frequency

lattice is set to 0 0 1 2F . f and t are

normalized to 0F and

0 , respectively.

In Fig. 1, it is showed that system performance in

terms of SIR is greatly improved when increasing the

number L of Hermite functions. Nevertheless, the SIR


304©2015 Journal of

is PSD it can be seen M

improvement decreases with the number L. Moreover,

increasing L will induce computational complexity and

even cause instability in the optimization process.

Therefore, we choose L=8 as a compromise according to

the simulation.

Fig. 1. SIR comparison versus the number of Hermite functions

Fig. 2. SIR comparison versus CFO with TO=0

Fig. 3. SIR comparison versus TO with CFO=0

The curves for SIR versus normalized CFO are

presented in Fig. 2 for the case where timing is perfect. In

the comparison, the optimization was redone for each set

of f . The figure shows that the optimal pulse

significantly outperforms other prototype pulses for f

larger than 0.04 at the cost of the SIR of nearly 25 dB at

0f . It can be also concluded that more SIR

performance gain can be obtained for larger CFO.

The curves for SIR versus normalized TO are

presented in Fig .

As is shown in the figure, the optimal pulse has the

highest SIR for t larger than 0.025. We also note that,

the optimal pulse performs slightly inferior to other

pulses in the small TO regions. However, in this case, the

SIR level is still higher than 20 dB. Similar results can be

concluded that more SIR performance gain can be

obtained for larger TO.

Fig. 4. SIR comparison versus CFO and TO

As is shown in Fig. 2 and Fig. 3, in small CFO or TO

regions, the optimal pulse performs slightly inferior to

other pulses. This is because the constraints on the nulls

of the ambiguity function of the proposed pulse are

abandoned. It can also be seen that in the presence of

CFO error, the SIR curves of OGGP and DOFC are

displayed in the reversed order compared to TO case. It

confirms that an improvement of the robustness against

CFO error can be addressed by using the pulse like

OGGP which has good time localization. On the other

hand, good time localization means bad frequency

localization, therefore, it is not preferable in the TO error.

Fig. 4 gives the SIR comparison where the same

increment step for CFO and TO is set. WHP and IOTA

pulses lead to approximately the same performance, so

IOTA is not included in this comparison. As is shown,

the WHP design performances better than OGGP and

DOFC due to the fact that WHP provide identical

localization in time and frequency. Moreover, it can be

seen that the optimal pulse is superior to other pulses. It

confirms once again the efficiency of our method to

acquire a prototype pulse robust to insufficient

synchronization.

V. CONCLUSION

In this paper, we have proposed a novel approach to

design prototype pulses that lead to robust performance

2 4 6 8 10 120

5

10

15

20

L

SIR

(dB

)

CFO=TO=0.05

CFO=TO=0.1

CFO=TO=0.15

CFO=TO=0.2

0 0.05 0.1 0.15 0.2 0.25 0.30

5

10

15

20

25

30

normalized CFO

SIR

(dB

)

optimal pulse

OGGP

WHP

IOTA

DOFC

0 0.05 0.1 0.15 0.25

10

15

20

25

30

normalized TO

SIR

(dB

)

optimal pulse

OGGP

WHP

IOTA

DOFC

0.05 0.1 0.150

5

10

15

normalized CFO=TO

SIR

(dB

)

optimal pulse

OGGP

WHP

DOFC



. 3 for the case where CFO is fixed to zero

against insufficient synchronization for OFDM/OQAM

systems. The robust pulse was modeled as a linear

combination of normalized Hermite functions. Exact SIR

expression of demodulated symbol versus CFO and TO is

derived. The pulse shaping approach based on SIR

maximization criterion has been formulated as a

non-convex fractional programming problem. An

efficient iterative algorithm was proposed to transform

the NP-hard problem to a series of QCQP problems

which can be efficiently solved using SDR method.

Numerical simulations demonstrated that the optimal

pulse design method provides a significant gain over

traditional pulse shaping method. The advantage is

especially prominent in the case of systems with large

carrier frequency and timing offset.

APPENDIX

From the expression of w , we note that:

(i) ( 1) 1i L j jL iw w , 0 2,0i L j i ;

(ii)2

1 ( 1) 1iL j iL i j L jw w w , 0 2, 2i L i j L ;

(iii)2

2 0iL i iw , 0 1i L .

For (i), this equality constraint can be summarized by:

0Cw , where C is a 2 2L L matrix defined by

0 2

TT T T

L C C C 0 . [ ]iC is a

2( 1)i L matrix

given by:

1 ( 1)

1 1

0

i mn

n i L m

n mL i

otherwise

C (29)

For (ii), we deduce this kind of equality constraints:

0T

ij w D w , where ijD is a 2 2L L matrix defined by:

2 2( ( 1))

TT T T

ij ijiL L L i L L

D 0 E 0 , ijE is a 2L L matrix

given by:

1 1, 1

0.5 , ( 1) 1[ ]

0.5 ( 1) 1,

0

ij mn

m j n iL j

m i n j L j

m j L j n i

otherwise

E (30)

For (iii), we deduce this inequality constraint:

0Gw , where G is a 2 2L L matrix defined by:

1 ( 1),0 1

0mn

n m L m L

otherwise

G (31)

ACKNOWLEDGMENT

This work was supported by National Natural Science

Fund of China under Grant no. 60971100.

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Yu Zhao received the B.S. and M.S. degrees

in 2009 and 2011, respectively, from Air

Force Engineering University, Xi’an. He is

currently working toward the Ph.D. degree in

the Air and Missile Defense College. His

research interests include information theory

and multicarrier modulation techniques.

Xihong Chen received the M.S. degree in

communication engineering from Xidian

University, Xi’an, in 1992 and the Ph.D.

degree from Missile College of Air Force

Engineering University in 2010. He is

currently a professor with Air and Missile

Defense College, AFEU, Xi’an. His research

interests include information theory,

information security and signal processing.

Qun Zhang

received the M.S. degree in

mathematics from Shanxi Normal University,

Xi’an, in 1988, and the Ph.D. degree in

electrical engineering from Xidian University,

Xi’an,

in 2001.

He is currently a professor

with Information and Navigation College,

AFEU, Xi’an. His

research interests include

signal processing, clutter

suppression, and its

application in SAR and ISAR.



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