Burst Transmission Symbol Synchronization in the Presence of Cylce Slip Arising from Different Clock Frequencies

Somaye Bazin

bazin.somayeh@gmail.com

Mahmoud Ferdosizade Naeiny

Electrical Engineering Department, Shahed Univeristy

Ferdosizadeh@shahed.ac.ir

Roya Khanzade

Electrical Engineering Department, Shahed Univeristy

royakhanzade@gmail.com

Abstract—In digital communication systems different clock frequencies of transmitter and receiver usually is translated into cycle slips. Receivers might experience different sampling frequencies from transmitter due to manufacturing imperfection, Doppler Effect introduced by channel or wrong estimation of symbol rate. Timing synchronization in presence of cycle slip for a burst sequence of received information, leads to severe degradation in system’s performance that represents as shortening or prolonging of bit stream. Therefor the necessity of prior detection and elimination of cycle slip is unavoidable. Accordingly, the main idea introduced in this paper is to employ the Gardner Detector (GAD) not only to recover a fixed timing offset, its output is also processed in a way such that timing drifts can be estimated and corrected. Deriving a two steps algorithm, eliminates the cycle slips arising from wrong estimation of symbol rate firstly, and then iteratively synchronize symbol’s timing of a burst received signal by applying GAD to a feed forward structure with the additional benefits that convergence and stability problems are avoided, as they are typical for feedback schemes normally used by GAD. The proposed algorithm is able to compensate considerable symbol rate offsets at the receiver side. Considerable results in terms of BER confirm the algorithm’s proficiency.

Keywords—cyclic slip-Gardner TED- Timing Recovery

1. INTRODUCTION

Timing recovery as a process of sampling at the right times is critical in digital communication receivers. In principle, the problem is formulated through Maximum Likelihood (ML). The direct computation of ML problem, performs the task of jointly detection and estimation of the message bits and the timing offset. However this solution conveys the exhaustive search methods that imposes a lot of computation and makes the solution impractical. To avoid the complexity of direct computation of ML problem, iterative solutions are introduced [1-3]. The general idea of iterative timing recovery scheme is to improve the timing estimation accuracy by multiple exploiting the timing information provided by a set of samples and application of this estimation to regenerate a

new set of samples that iteratively approaches to the local maximum of the likelihood function. ML based timing recovery methods usually ignore the time varying timing offsets and proceed under the assumption of fixed synchronization parameter estimation [4-6]. However in practice, the timing offset may vary with time, due to the different clock frequencies in transmitter and receiver caused by fractional error in baud rate estimation, manufacturing imperfection and etc. [7].

Different clock frequencies in transmitter and receiver leads to linear increasing of timing offset from symbol to symbol. While, timing offset increases linearly for successive symbols, Synchronizers may fail to track this time varying delay. Since getting far away from true value makes the estimator falls into the adjacent stable operating point and synchronizer starts to keep tracking this new stable operating point. Consequently one symbol inserted into or erased from the sequence. This is called Cycle Slipping (CS). There is also another source of CS which is the large phase variance of voltage controlled oscillators (VCOs) caused by low signal to noise ratio (SNR) that is not subjected in this paper. As long as cycle slips occur, system’s performance decreases dramatically due to the relative loss of synchronization caused by symbol insertion or omission in the sequence. In order to alleviate the adverse effect of cycle slip it has to be eliminated before applying timing synchronization.

Although, several studies have considered the problem of cycle slipping in synchronizers [8-10], few authors have proposed the solution [11-13]. Typically, solutions concern about error tracking synchronizers in low SNRs which are based on closed feedback loop. While the good tracking performance of feedback schemes is not deniable, they require, in counterpart relatively long acquisition time that makes them unsuitable for burst transmission schemes .In this sense, a feed forward structure based on extracting timing delay estimation from the statistics of received samples, and then adjusting the time by some sort of interpolation is more suitable. In this work, in order to utilize the band width efficiency of Non-Data-Aided (NDA) estimators and effective flexibility of interpolation, Gardner TED [14] and Farrow filter [15] are used in a feed forward structure as timing delay estimator and interpolation filter respectively.

In accordance with the above statements, this work is motivated by the objective of deriving a novel algorithm which employs GAD in a non-conventional manner so that not only the fixed timing offset is recovered, GAD’s output is also processed in a way such that considerable timing drifts can be estimated and corrected, which is not addressed in the literatures. CS is eliminated at the first step and then Gardner timing delay estimation in cooperation with Farrow based interpolation filter Makes the algorithm suitable for a more agile and efficient iterative receiver.

2. PROBLEM FORMULATION

2.1 Signal model Assume a traditional communication system, where the transmitted signal is corrupted

by passing through AWGN channel which also imposes a timing delay and carrier frequency and phase offset to the received signal as follows:

( ) = ∆ ℎ( − − ) + ( ) (1)

Where denotes the zero mean unit variance, independently and identically distributed (i.i.d.) symbols that might be taken from any linear modulation scheme. and ∆ are phase and carrier frequency offsets. ℎ( )is a square root raised cosine pulse and ( ) is a complex zero-mean additive white Gaussian noise with two sided power spectral density of /2. Moreover , and are unknown timing delay, pulse duration and the length of the transmitted signal respectively.

At the receiver side, the signal in (1) should be matched filtered and the transmitted symbols should be regenerated by sampling ( ) in − ̂ time instants, where ̂ is timing delay estimation provided by synchronizer. Even if receiver has exact information about symbol rate, there still may exist some fractional difference between transmitter and receiver clock frequencies due to the implementation imperfection. However, blind receivers have to estimate symbol rate that always conveys some estimation error. In this case the regenerated symbols are located in − ̂ time instants, where: = + (2)

and is the difference between transmitter and receiver symbol duration and can be either positive value or negative one. The discrete samples are:

( ) = ( ∆ ) ( + ) ( − ̂ − + )

= ( ∆ ) ( − + ( + ) ) (3)

Where is i.i.d. normally distributed variable with variance , ( ) is convolution of ℎ( ) with channel impulse response and the analog pre-filter response. represents + and stands for − ̂. Obviously, timing delay varies for different symbols of a received burst, due to the

variable timing delay part which is increased linearly by . Traditional approaches assume this variation is slow in comparison with symbol interval and they approximate the timing delay over a number of symbol periods that synchronization parameter can be considered as quasi-constant [16], however ignoring this variation would degrade the performance as it will be well illustrated in simulation results.

2.2 Cycle-Slips in synchronizer Typically, Gardner’s Timing Error Detection (TED) provides timing estimation to

synchronize the received symbols using the samples at twice rate of the symbol rate, according to the following equation [14]:

Herecomplex

( ) =

The timing destimati

carrier

synchro

As lovalue ofdelay toa quasithat

2.

For alternatsided basamplesequals

e ( )is thex conjugatio

( ∆∗ [+ 1)

obvious facdelay estimaion is negl

and phase

onized regar

ong as +f that +o the adjacei-periodic f= . The p

.2.1 Lemm

the sake otively changandwidth ofs involving + .

( ) = e timing eron. Pluggin

)( ∆ ( )) ) − (ct is that thation, likewligible, con

offset can

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ma

of simplicityges betweenf 1/ , as it for symbo

F

∗( + 1/2)rror of the g (3) into (4

∗ ∗)

∆ ) "e phase off

wise the impnsidering an

n be omitte

ior carrier an| TED is cas from this and CS hap

f ( ) whiof ( ) is pr

y, transmittn -1 and +1is depicted

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Figure .1 g(t)

)[ ( + 1)kth symbol

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ynchronizati

acking the tisynchroniznon-unifor

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(5)

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e that ouble essive value

Considering bandwidth of ( ) and doing some manipulation in order to discard

ineffective sentences in (5), a simplified extend is obtained, As long as | + | ≤

Let ( ) represents the timing offset estimation of th symbol. For any arbitrary positive

value of that 0 < < and + = − , ( ) is generated as follows: ( ) = [ ∗ ∗( − ) + ∗ ∗(− )] × [( − ) 2 − + ( − ) − 2 − ] (6)

Or: ( ) = [ ∗ ∗( − ) + ∗ ∗(− )]× [ 2 − − − 2 −+ − 2 − − 2 − ] (7)

Evidently: ( − ) ≤ ( ) −2 − ≤ 2 −

Noting to the fact that and are even +1 or -1, term − − −− can be ignored in (7) and consequently: ( ) = | ( )|

Which is always positive irrespective to what and are, and confirm the correctness of delay estimation.

Suppose that for all th symbols that + >| |, ( ) represents the relevant timing

offset estimation, Similarly assume is a positive value,0 < < , that make the delay

exceeds from , so that + = + . Relatively ( ) can be achieved through

following equation: ( ) = [ ∗ ∗( ) + ∗ ∗( − )] × [( − ) + 2 + ( − ) − 2 ] (8)

Regenerating of ( ) for + = − + corresponds to: ( ) = [ ∗ ∗( ) + ∗ ∗( − )] [( − ) + 2 + ( − ) − 2 ] (9)

It is con

Or equ): −exceedsthat, dealternatperiodicConvennumberat least either lo

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ncluded from

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ong burst w

Cycle-Slips and d

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∈ [−1, 1] is can be expr

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s long as t

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Figu

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able to track

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ure.2. illustrati

nd correctioymbol rate o where: = +lized symbo

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This non-unipped for paffset is assumhis contributg the receiverst with sign

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on of the transm

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preted as a purst signal ough the m

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he DFT of dtion of CS h

onstant timinther words: =me cases it e zero due o extract roximated birst zero pand finally th

number of sion the follo= 1 +period of (with the le

mathematicaion of a perdata randomerm of ( )ed by usin( )] ans for Discr

=depicted sighappening, ng delay w = 0 0

might be mto the noisby: =

by -point adding of he index tha

symbols thaowing equa+

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and the aded, it can b

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ength of (FT for effin approxim

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rom the mafrom Gardn

= 0.1, SNR =tion.

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aximum likener’s TED is

= 10db. CS lea

= 0.1, SNR =

ulation type: B.

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s, CS correovery, sincea DC compo

and the ge

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ads to

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(12)

term ain to ection e it is onent eneral

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to adjusFarrow

Supp

And

In thsymboltiming a

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st samplinginterpolatio

pose (0) i would be

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hose cases tere is no commial interpoation of Farrsidered in ore when then for interpo

g time. Hereon structure

s the DC co

e approxima

ords, averagt provides ain interpola

that the fracmputationalolation is prow structur

order to gene polynomiaolation’s ou

e this methoin order to

omponent of

(0) =ated as:

= 1ging the evaan estimatioator.

Figure.4.

ctional delayl need to upreferred dure can be fi

nerate the oal order, forutput.

od is used iimprove the

f clearly:

( ) (0)

luated timinon of consta

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ue to the imind in [19].outputs, appr example,

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ng error by ant timing d

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ion with diginstants.

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l the interpoficients, Faron efficiencerent polynoange polynos to 3 resul

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TED over acan be use

olator’s strurrow structucy. The detomial orderomial in Falt in a follo

ng by

(13)

(14)

all the ed for

ucture ure of tailed rs can arrow owing

Figuroutput asamplesby internew appand thedelay isalgorithposition

To vSimulatroot raisAWGNthe DFT

In figsignal mWhile recoversymbol performimprovecorrectetypical in which

(

re. 5 illustrand . Notes are generarpolation, thproximation

e selected ss modified hm leads tons.

verify the ations are cased cosine f

N channel. TT length for

g 6 system’modulation.timing reco

ry”. The burate offset

mance, due ed dramaticed timing rapproach thh, timing de

) = −+ −

rates the ree that in ordated for eachhey are usen of . If amples are using (18)

o the timin

algorithm’s arried out wfilter with roThe interpolr CS correct

’s performan The propoovery with

urst length t is set tto the CS ecally by prrecovery plohat it synchrelay can be

16 + 12+ −12+ −espective sader to exploh symbol tim

ed to evalua≅ 0, the geused to deand this pr

ng recovery

Figure.5 ill

4 SIMUL

proficiencywith BPSK oll-off factolation filter ion is 5000.

nce is evalused algorithout CS coof the receto 0.1. As existence inrior eliminaot. In fig 7ronizes the tassumed pi

− 12 + 16+ 2 −3 +amples invo

oit the timinming interv

ate ( ), thenerated sametermine throcedure is and it loc

lustration of in

LATION RES

y, simulationand QPSK

or of 0.5 in torder in the.

uated and cohm is referrrrection is

eived signalit is shown

n CS uncoration of CS7, “burst bytiming delayecewise con

16− 52 ++

olving to gng informatival. Once thhen applyingmples are de final symrepeated. I

cates the sa

nterpolation

SULTS

n results arsignals wh

transmitter ae timing rec

ompared in red as “CS c

referred al is 300 symn, despite

rrected timinS through iny burst timiy over the nnstant and th

generate theion by Garde new sampg (13) and

decimated bymbols, otherIterative apamples at t

re presentedhich are shapand they arecovery bloc

terms of BEcorrected ti

as “CS uncmbols, and the significng recoveryntroduced aing recovernumber of She variation

e interpolatdner’s TEDples are obta(14) resultsy the factorrwise the tiplication ofthe right ti

d in this secaped by a sqe passed thrk is set to 3

ER of the Bming recov

corrected tithe norma

cant decreay plot, BERalgorithm inry” imply toymbols’ int

n can be igno

(15)

tion’s , two ained s in a r of 2 ming f this ming

ction. quare rough 3 and

BPSK very”. ming

alized se in

R has n CS o the terval ored.

Fig. 7. tyypical approac

Fig. 6. BE

ch and introdu

ER performanc

uced algorithm

ce, = 300,

m BER perform

= 0.1, =

mance for ϵ=0

= 5000

.1 and ϵ=0.0055 , L=5000, N

N=300

Fig. 9. B

Fig

BER performa

g. 8. BER perfo

ance versus no

formance for d

ormalized sym

different burst

mbol rate offse= 5000

lengths, = 0

et, modulation

0.1, = 5000

type BPSK, b

0

burst length =

300,

Fig.10. B

It meanrecoversuch thaany segtake plaintroducdue to tby bursproposedifferen

Figursymbol result iprecise

In figBPSK acorrespodemonsmainly error de

BER performa

ns that the ry algorithmat the timingments. Cleace than ϵ=ced algoriththe short bust algorithmed CS corrnt ϵ.

re.8 displayrate offset

s achieved,symbol rate

gure.9 and and QPSK sonds to 30strated as thdue to this

etected by T

ance versus no

received sim is appliedng error can early, for ϵ==0.005, andhm widenedurst length. Hm is improvrected algo

ys BER of is set to .

, since the e modificati

10 the influsignal is dep00 symbolshe absolutes fact that lTED. Howev

ormalized sym

ignal is divd to each se

be assumed=0.1the shord as SNR id and the pHowever, f

ved, due toorithm achi

BPSK sign1 As it is plonger bur

ion in CS co

uence of difpicted, whes. Overalle value of arger offsetver, fluctuat

mbol rate offse= 5000

vided into segment. Thed to be fixerter burst leincreased thpoor performfor the case

the longer eves consi

nal for diffpredictable,rsts results orrection blo

fferent normere Eb/N0 as

decliningnormalizedts in symbotion of the p

et, modulation

smaller burse length of d and thus Cength will ghe gap betwmance eventhat ϵ=0.00possible bu

derably be

ferent burst as the bursin a betterock.

malized symssumed to btrend in th

d symbol raol rate reduplots’ deriv

n type QPSK, b

sts and indeeach segmeCS happeniguarantee thween typicantuates for t05, the perfourst length.tter perform

lengths wht length incDFT resol

mbol rate ofbe constanthe performate offset inuces the accative, when

burst length =

ependent tient is calcuing is avoidhat CS doeal approachtypical apprormance of Meanwhilmance for

here normacreases the blution and

ffsets on BEand burst le

mance is clncreases, thcuracy of tin system ten

= 300,

ming ulated ded in es not h and roach burst e the both

alized better more

ER of ength learly his is ming

nds to

achieve better performance despite increasing in (for example when corresponds to 0.3) can be interpreted through trade of between DFT resolution and TED’s accuracy. More precisely augmentation leads to the faster CS happening and better DFT resolution due to the more periodic terms that is provided by larger in a fix burst length. While, in the other hand it reduces the accuracy of timing error detected by TED. Note that in CS inexistence ( = 0) system performance does not approach to the perfect synchronization, since system had to recover and estimate constant timing delay.

5 CONCOLUSION

A new algorithm for timing recovery by prior elimination of cycle slip is proposed. It is shown that linearly increased timing delay causes alternate cycle slips in synchronizer. A burst sequence of timing error provided by Gardner TED is used to indicate and eliminate CS. After CS correction iterative bandwidth efficient timing recovery is applied to the sequence of burst samples. The satisfactory simulation results, evaluated in terms of BER and compared with theoretical and other approaches confirm the system’s appropriate performance.

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