A self-synchronizing method for asynchronous code acquisition in band-limited spread spectrum...

2410 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 8, AUGUST 2009

A Self-Synchronizing Method forAsynchronous Code Acquisition in

Band-Limited Spread Spectrum CommunicationsFrancesco Benedetto, Member, IEEE, and Gaetano Giunta, Member, IEEE

Abstract—This paper addresses the problem of initial syn-chronization of pseudo-noise code. A new code acquisitiontechnique for spread-spectrum communication systems usingband-limited chip waveforms is presented. Unlike conventionalpower detector based on testing the estimated maximum ofthe ambiguity function, the devised detector exploits a fastparabolic interpolation, running on three estimated ambiguitysamples in the neighborhood of the coarse estimate. Performanceanalysis is carried out in comparison with conventional detector.Mathematical expressions for the probability of false alarm anddetection are derived. They are numerically evaluated, underoperating settings, by a reduced Taylor’s expansion up to thesecond order. The theoretical results, substantiated by computersimulations, have evidenced that the devised method is wellsuited for asynchronous spread-spectrum communications. Inparticular, the acquisition performance depends on the actualoffset between the received and the reference code waveforms,which are randomly distributed (in chip-asynchronous systems)within one sampling period. In fact, the parabolic interpolationtechnique outperforms the conventional detector for a wide rangeof code offsets because it is able to self-synchronize the testingvariable around the true offset.

Index Terms—Code acquisition, fast methods, hypothesis test-ing, performance analysis, spread-spectrum communications.

I. INTRODUCTION

SPREAD Spectrum (SS) techniques have raised a sig-nificant importance in mobile communications in recent

years [1]. A common form of SS is Direct-Sequence Spread-Spectrum (DS/SS) in which the transmitter multiplies a binarydata sequence by a higher rate pseudo-random or pseudo-noise(PN) binary sequence. In particular, DS/SS Code-DivisionMultiple-Access (CDMA) has emerged as the incoming stan-dard for third-generation radio-mobile transmission technol-ogy [2], since it offers significant advantages in terms ofchannel capacity, mobile power consumption, link qualityand resilience to multi-path propagation [3]. However, inorder to exploit the advantages of a SS signal in a CDMAsystem, receivers must synchronize the local PN code withthe incoming PN code [4]-[5]. Thus, quickly achieving andthen maintaining PN code synchronization is a critical issueto be satisfied because even a small misalignment can causedramatic signal-to-noise ratio (SNR) degradation. Usually, this

Paper approved by S. N. Batalama, the Editor for Spread Spectrum andEstimation of the IEEE Communications Society. Manuscript received January22, 2007; revised July 11, 2007.

The authors are with the Digital Signal Processing and MultimediaCommunications Lab, Dept. of Applied Electronics, University of ROMATRE, via della Vasca Navale 84, 00146 Rome, Italy (e-mail: {fbenedetto,giunta}@ieee.org).

Digital Object Identifier 10.1109/TCOMM.2008.08.070030

Fig. 1. Structure of PN code-acquisition systems.

task is solved via a two-step approach: first an initial codeacquisition, which synchronizes transmitters and receivers towithin an uncertainty of a chip period, is realized; second,a code tracking phase, which performs and maintains finesynchronization is carried out [4].

Apart from decoding approaches (not addressed herein) thatre-formulate code acquisition as a decoding problem from thereceived signal [6], the most widely used and studied methodsfor acquiring PN codes doing correlations for all sequencesare: (maximum likelihood) full parallel search, serial search[7], and hybrid search [8]. In each of these methods, corre-lations between the incoming and the locally generated PNsequence are realized. In the first case (full parallel search),correlations are formed for all possible PN code offsets. Insimple serial search, only one of the correlations used in thefull parallel search is formed, while hybrid search tests a smallset of possible alignments in parallel and then repeats this teston another set of observations until the correct code offsetis discovered. In all cases, a threshold test is performed toaccept or reject the presence of useful signal characterizedby a given code in the search region under investigation. Thetest sequentially searches for the most likely codes and theiroptimum timing shift as reliable candidates for code (and codeoffset) acquisition [2], [9]. Conventional testing methods forthe presence of a pilot synchronization signal (with a givenspreading code offset) rely on the power detector [10]. Thistest distinguishes between two different hypotheses (see Fig.1): the in-sync condition (hypothesis H1), which correspondsto the case of presence of the tested code with the offsetdetected by the receiver’s systematic timing offset; and theout-of-sync case (hypothesis H0) which conversely states theabsence of that code with the considered offset [9]. Thesetwo cases differ because the output of a matched filter isideally constant in the former condition, while it randomlyvaries in the latter one. In fact, it is well known that theuser codes employed are orthogonal only if the users arechip-synchronized with each other. In practice, any pair of

0090-6778/09$25.00 c© 2009 IEEE

BENEDETTO and GIUNTA: A SELF-SYNCHRONIZING METHOD FOR ASYNCHRONOUS CODE ACQUISITION IN BAND-LIMITED SPREAD SPECTRUM . . . 2411

codes may present a relevant cross-correlation for non-zerochip offset. Such a residual correlation acts as a randomvariable (the codes are usually modulated by independent datastreams), characterized by a noise-plus-interference variancedepending on the effective time synchronization.

The constant false alarm rate (CFAR) criterion, often em-ployed to perform effective tests [11], is adopted to determinethe threshold value. Power detector is chosen to limit thecomputational costs of the decision device, in order to allowfaster sequential hypothesis rejection. That is because ofrejection is much more likely than acceptation in code serialsearch devices for spread spectrum communications [12]-[14].Most of the aforementioned code-timing estimation schemesimplicitly assume rectangular chip waveforms which are notband-limited and seldom used in practice. Real widebandCDMA systems utilize band-limited chip waveforms, suchas the raised-cosine pulse [15], and in contrast with thenumerous research works on code-timing estimation withrectangular chip waveforms, limited studies are available thattake into account band-limited chip waveforms (e.g. [16],[25]-[27]). Recently, new interesting solutions for the self-synchronization of linear receivers for asynchronous direct-sequence code-division multiple-access have been proposedin the seminal works by Psaromiligkos and Batalama in [28]-[29]. The same fundamental ideas can be applied also for theacquisition phase and initial cell search.

Addressing some of these issues, this paper proposes a codeacquisition technique based on a parabolic interpolation for SScommunication systems with band-limited chip waveforms.Our goal is to obtain via the proposed method a finer syn-chronization in the initial cell search versus the conventionalapproach (i.e. power detector). In fact, a parabolic estimatorprovides a finer estimate by interpolating only three samplesof the ambiguity function in the neighborhood of the coarseestimate [17]. The remainder of this paper is organized asfollows. In Section II, the system model is depicted and thebasic frameworks of the conventional procedure (i.e. the powerdetector approach) are showed. In Section III, the new codeacquisition technique based on the parabolic interpolation isillustrated while in Section IV a theoretical analysis of theperformance of the new method is carried out. Our numericalresults and comparisons are outlined in Section V, and finally,our conclusions are depicted in Section VI.

II. SYSTEM MODEL

The block scheme of the conventional acquisition receiverbased on a matched filter is depicted in Fig. 2. In thepresence of additive white Gaussian noise, it is well knownthat optimum receiver performs matched filtering operationsfollowed by active PN correlator (or chip matched filter) [1].The correlated output is then passed through the detector.Each sample output from the detector is compared with agiven threshold. Depending upon the comparator output, thereceiver either accepts the corresponding PN code phase andgoes into tracking mode or continues to search. Followingthe simple model used in [24], we denote with r(t) thecomplex representation of the received signal at the front-endacquisition receiver, with Tc the chip duration, and with τ the

time delay normalized to Tc:

r(t) =√

2Ecd (t − τTc) c (t − τTc) cos (ωct + φ) + n (t)(1)

where Ec is the received chip energy; d(t) is the data sequencewaveform; Tc is the chip duration; τ is the time delaynormalized to Tc; ωc is the carrier angular frequency; φ isthe phase distributed uniformly over [0, 2π). Then, we havec (t) =

∑∞i=−∞ cipTc (t − iTc), where ci ∈ {−1, +1} is

the i-th chip of the pseudo-noise (PN) code sequence andpTc (t) is the PN code waveform assumed to be band-limited(such as the pulse waveform with square root raised-cosinespectrum [16]). Moreover, we denote with n(t) a complexwhite Gaussian process with two sided power spectral densityequal to N0/2, accounting for both thermal noise and multi-user interference effects. At the output of the MF, we havea chip-waveform, with raised-cosine spectrum and roll-off γ,and defined as:

hTc (t) =sin

(πTc

t)

cos(γ π

Tct)

πTc

t

[1 − 4γ2

(t

Tc

)2] (2)

where obviously hTc (t) = pTc (t) ⊗ pTc (t). The receivedsignal is then sampled with a sampling interval Tc/Qc wherethe Qc ≥ 1 is referred to as over-sampling factor. In fact,practical acquisition and signal processing requires more thanone sample per chip; hence, using only one sample couldresult in a significant performance loss [18]. A typical choicefor the over-sampling factor is Qc = 2, which renders thealiasing-induced estimation error negligible [19]. Moreover,as already noted by [20], such choice matches the Nyquistcriterion. Hence in the following of this paper, we consideran over-sampling factor Qc = 2. Let us now consider Nsamples of the complex envelope of the received signal afterthe matched filter {r(iTc/2 − τTc), i = 1, ..N}, discretizedwith the normalized timing offset τ , being τ = 0 (without lossof generality) in the chip-synchronous case while (randomlydistributed) −0.5 ≤ τ ≤ 0.5 in chip-asynchronous commu-nications, being independent of the sampling times (eitherinteger or half integer multiples of the chip period Tc) [9].The power detector first estimates the cross-correlation ρw(τ)(despreading the generic w-th data block made of N chips)between r(t) and the (known) code candidate c(t), shifted by(iTc/2+kTc), where k is the normalized local PN code delayand is assumed to be an integer without loss of generality[20]1:

ρw(τ) =1N

N∑i=1

r

(iTc

2− τTc

)· c∗

(iTc

2+ kTc

)(3)

Then, the power detector simply accumulates W blocksof the square magnitude of the estimated cross-correlationρw(τ), in order to avoid the effect of the frequency and phaseuncertainty [2]:

Zk (τ) =1W

W∑w=1

|ρw(τ)|2 (4)

1Where, for sake of compactness, we have omitted the dependence ofρw(τ) from k, i.e. ρw(τ) = ρw(k, τ).


Fig. 2. Block scheme of both the conventional power detector and the parabolic interpolation scheme (oversampling factor Qc = 2).

and compares the (k-th) currently examined decision variableto a pre-selected threshold ν:

Zk (τ) >H1 ν or Zk (τ) <H0 ν (5)

The testing variable Zk (τ) is asymptotically (W → ∞)Gaussian because of the central limit theorem, assumingthe statistical independence (or at least time-limited corre-lation) between the W data-blocks. The threshold ν can beasymptotically tuned from a straightforward evaluation ofthe Gaussian integral for a fixed probability of false alarm.Under such condition, the mean E[Zk(τ)|H0] and the variancevar[Zk(τ)|H0] are evaluated under the null hypothesis tocompute the threshold ν, while the mean E[Zk(τ)|H1] andthe variance var[Zk(τ)|H1] conversely determine the asymp-totic testing power. The Constant False Alarm Rate (CFAR)procedure is employed to perform effective tests [21]. Inparticular, the CFAR test is accomplished in two successiveparts: first, a threshold is determined to limit the false-alarmprobability Pfa = P [Zk(τ) ≥ ν|H0] at a given reducedvalue (size of the test); second, the probability of detectionPD = P [Zk(τ) ≥ ν|H1] (power of the test) is evaluatedfor the threshold previously determined. The probability offalse alarm must be tuned to guarantee a very low numberof possible false alarms, which eventually imply a relevantpenalty time to the acquisition device. Large probabilities ofdetection (up to 100%) are typical of well-performing testingvariables [1], [11].

III. FAST ACQUISITION METHOD

In this section, we discuss the structure of the interpolation-based acquisition approach, which forms the basis for allthe mathematical derivations to be introduced in Section IV.Referring again to Fig. 2 and using the same notation of theprevious section, let us introduce the following quantities:

x = Zk

(τ − 1

2

), y = Zk

(τ +

12

), z = Zk (τ) (6)

where z is the same as expressed by (4), while y and x arethe detector’s outputs evaluated at Tc/2 before and after z,respectively. The new detector now searches for the apex ofthe parabolic function g(τ) = aτ2+bτ +c fitted over the threedetector’s outputs x, y, and z expressed by (6). The parameters

of the parabola fitting the measured correlation are evaluatedas follows:

a =2 (x − 2z + y)

T 2c

, b =(y − x)

Tc, c = z (7)

In the power detection approach the testing variable isrepresented by (5), now the testing variable is the (ordinate ofthe) apex of the parabolic function fitted around the detector’soutputs. The coordinates of the apex are:

abscissa : Gk = − b

2a,

ordinate : g (Gk) = − b2

4a+ c (8)

or using (7), they can be expressed as functions of x, y, andz, as follows:

Gk = f (x, y, z) = −Tc

4· x − y

x − 2z + y

g (Gk) = g (x, y, z) = z − 18· (x − y)2

x − 2z + y(9)

In this work, we have used a parabolic interpolation toapproximate the ambiguity function because we are dealingwith band-limited systems. Unlike rectangular chips, band-limited chip waveforms (such as the ones with square-rootraised-cosine spectrum) have auto-correlations that are wellmatched by parabolic functions in the neighborhood of itsmaximum, according to their Taylor’s expansion truncatedup to the second order [22]. To evaluate the validity of theapproximation of the ambiguity function (2) with a parabolicinterpolation, we have derived the normalized amplitude misfiterror εm versus the actual delay τ , see (10) at the top of thenext page, where sinc(t) = sin(πt)/πt. The graph of εm isreported in Fig. 3 versus the timing offset (normalized to Tc)in the range [-0.25, 0.25]. We can note that the misfit erroris less than 3% in the worst case from both analytical andgraphical results. As evidenced by simulation, the effect ofthe numerical error due to the parabolic misfit is negligiblerespect to the estimation variance of the detector’s outputs(6).

The analytic procedure has been described under a theo-retical viewpoint, in order to completely evaluate the systemperformance through the analysis reported in the next section.


εm (τ) =sinc (τ) · cos (γπτ)

[1 − 4γ2τ2]+

[sinc(τ+0.5)·cos(γπ(τ+0.5))

[1−4γ2(τ+0.5)2] − sinc(τ−0.5)·cos(γπ(τ−0.5))

[1−4γ2(τ−0.5)2]

]2

8 ·[

sinc(τ+0.5)·cos(γπ(τ+0.5))

[1−4γ2(τ+0.5)2] + sinc(τ−0.5)·cos(γπ(τ−0.5))

[1−4γ2(τ−0.5)2] − 2 · sinc(τ)·cos(γπτ)[1−4γ2τ2]

] (10)

It has to be underlined that, in real cases, the algorithmconsiders as possible candidates the only testing variables forwhich z is greater than x and y simultaneously. As detailed inSection IV-B, this is in accordance with the operating mode,where the test is realized not for all the detector’s outputs butonly for the local maxima.

IV. PERFORMANCE ANALYSIS

In this section, we evaluate the performance of the newacquisition method based on the parabolic interpolation. Letus now define, for sake of compactness and as done in [22],x = A1 + ε1, y = A2 + ε2, and z = A3 + ε3, whereA1 = E[x], A2 = E[y], A3 = E[z], ε = [ε1ε2ε3]T are zeromean random variables and E[·] denotes the expectation valueoperator. Therefore, under the assumption of small errors forthe three detector’s outputs (x, y, z), the estimation errors canbe accordingly evaluated by a three-dimensional (3-D) reducedTaylor expansion up to the second order around the initialpoint ε = 0 as follows [17]:

E [g] =12

3∑i=1

ϑ2g

ϑA2i

∣∣∣∣g=0

· E [ε2

i

]+

+12

3∑i=1

3∑m

ϑ2g

ϑAiAm

∣∣∣∣g=0

· E [εiεm] + B (11)

V ar [g] =3∑

i=1

[ϑg

ϑAi

]2∣∣∣∣∣g=0

· E [ε2

i

]+

+3∑

i=1

3∑m

ϑg

ϑAi· ϑg

ϑAm

∣∣∣∣g=0

· E [εiεm] (12)

where the term B in (11) accounts for the systematic bias, rep-resenting the parabolic misfit of the mean ambiguity functiong(A1, A2, A3) [17]. Following the same guidelines of [22], theasymptotic Gaussianity of Gk and g(Gk) defined in (9) canbe easy proved by a small error analysis (i.e. |εi| << |Ai|)of the three (asymptotically Gaussian) random variables (x,y, and z), since the first order of the Taylor approximationovercomes the higher orders (note that A1 + A2 − 2A3 �= 0in a neighborhood of the maximum).

The assumption of small errors in the estimated ambi-guity function, substantiated and validated by the extensivecomputer simulations done in Section V, implies that ouranalysis is valid for a synchronization error within one chipperiod. This fundamental hypothesis is largely assumed inthe literature (see for instance [16], [26]-[27]) during theacquisition phase, otherwise it is not possible to effectivelyperform any initial synchronization (frequent false alarmsbring to large penalty times) and the following data reception.

Let us now evaluate the expressions for the mean valuesAi, the variances E[ε2

i ], and the covariances E[εi εm] of the

Fig. 3. Misfit error εm (normalized to 1) of the parabolic approximation ofthe ambiguity function.

estimators of the ambiguity function at the time lags (τ −1/2) · Tc, τ · Tc, and (τ + 1/2) · Tc. Recalling the definitionsof the system model presented in Section II, we obtain thefollowing expressions for the mean values, the variances andthe covariances, respectively:

Ai = E [Zk (τ)] = E[|ρw (τ)|2

](13)

E[ε2

i

]= var [Zk (τ)] =

1W

var[|ρw (τ)|2

](14)

E [εiεm] = cov [Zk (τ1) , Zk (τ2)]

=1W

cov[|ρw (τ1)|2 , |ρw (τ2)|2

](15)

For sake of compactness, let us now re-write the estimatedcross-correlation (3) as follows:

ρw (τ) = ejϕ(τ) · Ψ (τ) + N (τ) (16)

where Ψ (τ) represents the (ideal) despread correlation be-tween the noiseless and interference-free received signal andthe candidate code:

Ψ (τ) =1

2N

2N∑i=1

c

(iTc

2− τTc

)

·c∗(

iTc

2+ kTc

)(17)

with phase ϕ (τ) and a raised-cosine chip waveform suchexpressed by (2); while N(τ) is a zero mean complex randomstationary variable, with variance σ2

N and a square root raised-cosine waveform, and defined as:

N(τ) =1

2N

2N∑i=1

n

(iTc

2− τTc

)

·c∗(

iTc

2+ kTc

)(18)


Hence, the mean values and the variances can be nowexpressed as follows:

E[|ρw (τ)|2

]= Ψ2 (τ) + σ2

N (19)

E[|ρw (τ)|4

]= Ψ4 (τ) + 2σ4

N + 4Ψ2 (τ) · σ2N (20)

var[|ρw (τ)|2

]= E

[|ρw (τ)|4

]−

(E

[|ρw (τ)|2

])2

=

= 2Ψ2 (τ) · σ2N + σ4

N (21)

Let us now focus on the evaluation of the covariances. Sinceit is (22) (see top of next page), where, for sake of compactnesswe have N1 = N(τ1) and N2 = N(τ2). Then, assumingwithout loss of generality N(τ) stationary over the numberW of blocks, we have:

E [N1N2] = 0,

σ2N1

= σ2N2

= σ2N RN (τ) = E [N · N∗] (23)

E [N1N∗1 N2N

∗2 ] = E [N1N

∗1 ] · E [N2N

∗2 ] +

+E [N1N∗2 ] · E [N∗

1 N2] (24)

and using (23)-(24) in (22), we obtain (25) (see top ofnext page), where the cross-correlation RN (τ) has a raised-cosine waveform expressed by (2). Finally, the covariances areobtained as:

cov[|ρw (τ1)|2 , |ρw (τ2)|2

]= 2Ψ (τ1) · Ψ (τ2)RN (τ2 − τ1)

+ |RN (τ2 − τ1)|2 (26)

Using the expression of g(x, y, z) obtained from (9) in theprevious equations, after some algebra, (11) and (12) modifyas (27) and (28) (see next page).

The quantities expressed by the previous formulas, valid forany value of τ distributed in the range (−0.5 < τ < 0.5), arethen used to evaluate the searched threshold (under the out-of-sync hypothesis) and the asymptotic detection probability(under the in-sync hypothesis). In the following sub-sections,let us evaluate the performance of the system under the twohypotheses H1 and H0, respectively.

A. In-Sync: Hypothesis H1

As pointed out in [9], the actual code offset is randomlylocated in one sampling period within the timing range(−0.5 < τ < 0.5) in a chip-asynchronous system, beingindependent of the sampling time instants (either integer orhalf integer multiples of the chip period Tc). The worst caseof erroneous synchronization with the conventional powerdetector will happen if the cross-correlation is computed for anoffset which is just in the middle between two chips. In such acase, an error of half chip will affect the estimates. Operatingat twice the chip rate, the worst offset for the power detectoris obviously located at Tc/4 instead of Tc/2, as in the case ofone sample per chip (see Fig. 4). The same happens for thenew detector because the parabola is fitted by three samplesaround the maximum given by the power detector. In fact, wehave verified by simulation (see Figs. 5–7) that the worst offsetis located at Tc/4. It should be noted that the correlation withoffset (τ = 1/4, i.e. τ ·Tc = ·Tc/4) could have a non-negligible

Fig. 4. Decision variables of both the power detector and the parabolicinterpolation scheme: a) Best case τ = 0, b)Worst case τ = 1/4, c) Middlecase τ = 1/8.

Fig. 5. Best Case τ = 0: Probability of detection (theory: asymptotic curve;simulation: 106 Monte-Carlo runs) of the conventional and the new methodsfor W = 15 and PF A = 10−3 versus SNR (per chip) of practical interest.

impact on the acquisition performance of the two detectors,as underlined in the following. However, it is in such a casethat the interpolation-based acquisition method dramaticallyoutperforms the conventional approach. While (27) and (28)are valid for every values of the timing offset τ , randomlydistributed, in the following let us focus on two special cases:τ = 0 (best case, see Fig. 4a) and τ = 1/4 (worst case,


E[|ρw (τ1)|2 , |ρw (τ2)|2

]= Ψ2 (τ1) · Ψ2 (τ2) + Ψ2 (τ1) · σ2

N2+

+Ψ2 (τ2) · σ2N1

+ E [N1N∗1 N2N

∗2 ]

+Ψ (τ1) · Ψ (τ2) ejϕ(τ1)e−jϕ(τ2)E [N∗1 N2] +

+Ψ (τ1) · Ψ (τ2) e−jϕ(τ1)ejϕ(τ2)E [N1N∗2 ] (22)

E[|ρw (τ1)|2 , |ρw (τ2)|2

]= Ψ2 (τ1) · σ2

N + Ψ2 (τ2) · σ2N +

+Ψ2 (τ1) · Ψ2 (τ2) + σ4N +

+ |RN (τ2 − τ1)|2 + 2Ψ (τ1) · Ψ (τ2) · RN (τ2 − τ1) (25)

E [g] =1

2 (E [x] − 2E [z] + E [y])3

·{

(E [y] − E [x])2 var [z] − (E [y] − E [z])2 var [x] +

− (E [x] − E [z])2 var [y] ++

(E2 [z] + E [x] E [y] − E [y] E [z] − E [x] E [z]

)cov [xy] +

+(E2 [y] − E [x] E [y] − E [y]E [z] + E [x] E [z]

)cov [xz] +

+(E2 [x] − E [x] E [y] + E [y] E [z] − E [x] E [z]

)cov [yz]

}+ B (27)

V ar [g] =1

128 (E [x] − 2E [z] + E [y])4

·{(

E2 [x] − 3E2 [y] + 2E [x] E [y] − 4E [x] E [z] + 4E [y] E [z])2

var [x] +

+(E2 [y] − 3E2 [x] + 2E [x] E [y] + 4E [x] E [z] − 4E [y] E [z]

)2var [y] +

+(6E2 [x] + 6E2 [y] + 20E [x] E [y] + 32E2 [z] − 32E [x] E [z]− 32E [y] E [z]

)2var [z] +

+4[(

E2 [x] − 3E2 [y] + 2E [x] E [y] − 4E [x] E [z] + 4E [y] E [z]) ·

· (E2 [y] − 3E2 [x] + 2E [x] E [y] + 4E [x] E [z] − 4E [y]E [z]

)]cov [xy] +

−8[(

E2 [x] − 3E2 [y] + 2E [x] E [y] − 4E [x] E [z] + 4E [y] E [z]) ·

· (3E2 [y] + 3E2 [y] + 10E [x] E [y] + 16E2 [z] − 16E [x] E [z] − 16E [y] E [z]

)]cov [xz] +

−8[(

E2 [y] − 3E2 [x] + 2E [x] E [y] + 4E [x] E [z] − 4E [y] E [z]) ·

· (3E2 [x] + 3E2 [y] + 10E [x] E [y] + 16E2 [z] − 16E [x] E [z] − 16E [y]E [z]

)]cov [yz]

}(28)

see Fig. 4b). We evaluate and compare the performance ofthe new estimator versus the conventional approach in thesetwo special cases (worst and best scenarios) because they arethe (lower and upper) bounds of the system performance.Indeed, the term cov(x, y) always reduces to zero becausex and y are two uncorrelated variables (their relative timedistance is just one chip). Then, in the best case, we haveE[x] = E[y] and var(x) =var(y) because the two outputs xand y have symmetric moments. They are evaluated from anautocorrelation function (that is symmetric by definition) withtime lags equal to ±Tc/2, respectively (see Fig. 4a). The sameassumptions are also valid in the worst case for the detector’soutputs y and z (i.e. E[y] = E[z] and var(y) =var(z)), where

the time lags are equal to ±Tc/4, respectively (see Fig. 4b).Hence, assuming the correct timing offset equal to zero,

τ = 0 (best case), the performance of the proposed estimatormodifies as follows:

E [g] =116

· 2var [x] − cov [xy]E[z] − E[x]

=116

· 2var [x]E[z] − E[x]

(29)

V ar [g] =12var [z] (30)

Assuming the correct timing offset equal to Tc/4 (worstcase), equations (27)-(28) become (31) and (32) (see (32) onnext page),

E [g] =cov [yz]

2 (E[x] − E[z])+ B (31)


V ar [g] =var [x] + 45var [z] − 12cov [xy] − 24cov [xz] + 72cov [yz]

128

=var [x] + 45var [z] − 12cov [xz] + 72cov [yz]

128(32)

Fig. 6. Worst Case τ = 1/4: Probability of detection (theory: asymptoticcurve; simulation: 106 Monte-Carlo runs) of the conventional and the newmethods for W = 15 and PF A = 10−3 versus SNR (per chip) of practicalinterest.

Fig. 7. Middle Case τ = 1/8: Probability of detection (theory: asymptoticcurve; simulation: 106 Monte-Carlo runs) of the conventional and the newmethods for W = 15 and PF A = 10−3 versus SNR (per chip) of practicalinterest.

where the parabolic misfit B in (31) is evaluated from (10),normalized here to the actual maximum of the ambiguityfunction E[Zk(τ)|H1]. In practice, B reduces to zero in thebest case, while it represents a negligible error (< 3%) in theworst case.

As stated before for (5), the three variables x, y, and zare Gaussian variables with non-zero means for small errors(i.e. variance << mean), and also the testing variable g,expressed by (9), can be assumed asymptotically Gaussian

(as detailed in the previous sections). Hence, in any of thetwo aforementioned (best and worst) cases the asymptoticprobability of detection can be determined by means of theGaussian error function under the in-sync H1 hypothesis, thusobtaining:

PD =12+

12erf

[(−ν + E [g|H1]) · (2var [g|H1])

−1/2]

(33)

where erf1(·) is the inverse of the well known Gaussian errorfunction defined as:

erf (x) =2√π

x∫0

e−t2dt (34)

and where ν is the value of threshold obtained under the out-of-sync hypothesis H0 as explained in the next sub-section..

B. Out-of-Sync: Hypothesis H0

In the out-of-sync condition, the three detector’s outputs{x, y, z} are strictly correlated each other and the probabilityof the statistic z, conditioned to the hypothesis H0, is obtainedas:

Pz|H0 (z|H0) =

+∞∫−∞

+∞∫−∞

P(x,y,z)|H0 (x, y, z) |H0dxdy (35)

where P(x,y,z)|H0(x, y, z)|H0 is the joint probability densityfunction (PDF) of x, y, and z conditioned to H0.

At first glance, in a sequential testing procedure, one couldthink to look for the maximum of each parabola fitted by everytriple sequence of ambiguity samples. This is definitely wrongand not operating. In fact, the fitting may produce timingestimates outside the three samples, upside-down parabolas,and even straight lines (timing variance may go to infinity!).Conversely, we must consider as testing variable the apexof the parabolic function evaluated only for those detector’soutputs in which z is greater than x and y simultaneously. Inother words, the testing variable is not represented by eachoutput sample of the estimated correlation function but onlyby its local maxima. This is in accordance with the operatingmode, where the test is realized not for all the detector’soutputs but only for the local maxima. A similar modificationof the algorithm (that ignores the blocks in which z is notgreater than both x and y) is implicitly taken into accountunder the in-sync hypothesis (detailed in Section IV-A). Thismodification is substantiated under an operating point of view,as well as under a theoretical approach. In the operating case,the algorithm tests only the local maxima (i.e. it is alwaystrue that z > x and z > y simultaneously). On the otherhand, we can formally approximate P (z > x, z > y|H1) ≈ 1and P (x, y, z|H1, z > x, z > y) ≈ P (x, y, z|H1) for the only


purpose of evaluating the mean and variance of the testingvariable g.2

The complexity of the performance analysis is drasticallyreduced since we operate in full analogy with the case of thein-sync hypothesis, because it happens also in this case thatE[x] = E[y], var[x] = var[y], cov[xy] = 0, and B = 0. Asa consequence, all the performance expressions simplify likedetailed in the previous sub-section for the in-sync case.

In order to derive proper performance expressions for theout-of-sync case, let us consider for our convenience the joint(asymptotically) Gaussian statistics of the three random vari-ables {x, y, v}, instead of the former three variables {x, y, z}.In our statistically equivalent model, x and y are again twouncorrelated random variables (their relative time distance isjust one chip), while the third variable v acts as one furtherrandom variable correlated with both x and y (with the relativetime distance of half chip, like z in the former model) thatgenerates the random variable z, here newly defined as z = vif {v > xandv > y}, i.e. if v is greater than x and ysimultaneously.

As a consequence, under the hypothesis of small estimationerrors of the two displacements x and y, (35) modifies as(36) (see next page), where the joint Gaussian probabilitydensity function of {x, y, v} is (37) (also on next page), beingE[v] = E[x] = E[y], while the covariance matrix K of{x, y, v} equals to:

K =

⎡⎣ var (x) 0 cov (xv)

0 var (x) cov (xv)cov (xv) cov (xv) var (x)

⎤⎦ (38)

where we have indeed var[v] = var[x] = var[y] andcov[xv] = cov[yv], because x, y, and v have symmetricmoments evaluated from an autocorrelation function with timelags equal to Tc/4. It has to be noted that, (36) is straightfor-wardly derived as the PDF of the variable z, conditioned toH0, obtained as the ratio between the joint PDF of (x, y, z),integrated over x and y, and the PDF of the conditioningevent (i.e. z > x and z > y simultaneously). Moreover, thehypothesis of small estimation errors of the two displacementx and y we have assumed in our statistical model has beenconfirmed and validated from the simulation trials detailed inSection V.

We are now able to evaluate the mean and variance of theconsidered testing variable g by the expressions (27)-(28), thatinvolve the mean and the variance of both x and z givenH0. These latter statistics can be directly evaluated from theprobability density function expressed by (36). Hence, thethreshold ν can be asymptotically tuned from a straightforwardevaluation of the Gaussian integral for a fixed probability offalse alarm using:

v = E {g|H0} + (2var {g|H0})1/2 erf−1 (1 − 2Pfa) (39)

if we suppose again the approximate Gaussianity of the testingvariable g (or, at least, the cumulative integral of its probabilitydensity function acts like Gaussian around the false alarmoperating point). This latter assumption will be validated bycomputer simulations as detailed in the next section.

2Such weaker assumption means that the mean and the variance of theactual in-synch PDFs are comparable although the PDF tails may be different.

V. NUMERICAL RESULTS

Several trials were performed to validate the theoreticalexpressions and the statistical model assumptions (derived inthe previous sections) on the asymptotic performance of boththe conventional procedure, based on the power detection,and the novel acquisition method, based on the parabolicinterpolation. In the reported results, we have considered thefalse alarm probability PFA = 10−3, the roll-off γ = 0.22,the code length 256 chip, and the observation window madeof only W = 15 blocks. In fact, fast acquisition methodsare based on processing few observed data. As discussed inthe following, a consequence of such a choice is that theasymptotic (W → +∞) Gaussianity of the detector’s outputs,used in the mathematical derivations of Sections III and IV, isless valid but is still effective for evaluating the system per-formance bounds [23]. In our trials, the theoretical probabilityof false alarm and the theoretical probability of detection havebeen determined by means of the asymptotic expressions (33)and (39), for both the considered detectors. Several Monte–Carlo simulation trials have been implemented to estimate theactual probability of false alarm (that determines the actualthreshold) and the probability of detection in order to validatethe (asymptotic) results obtained by theory. In particular, wehave numerically verified that the effect of the estimationvariance of the detector’s outputs (6) largely overcomes thenumerical error due to the parabolic misfit (with a sub-chipparabolic interpolation).

In the following, we focus on the three cases of our interest:τ = 0 (best case), τ = 1/4 (worst case) and the intermediatevalue of τ = 1/8 (middle case), as explained in Fig. 4.

The results of the probability of detection are shown inFig. 5 versus different values of the SNR (per chip) ofpractical interest for the “best” operating case with a falsealarm probability PFA = 10−3. Simulated results (from 106

independent runs) are also reported in the presented plots.The results of the theoretical analysis are very similar to theones obtained from simulation, which ensures the correctnessof the approximate expressions. In particular, As expected,the conventional detector outperforms the new method inthe “best” case because the power detector’s output, Zk, issynchronized with the correct timing offset (see again Fig.4a). In practice, both analyzed detectors reach a near unitaryprobability of detection in the “best” case, which is nothowever the typical operating case.

In order to stress the different behavior of these two detec-tors, we took the “worst” operating scenario into account (seeFig. 4b). In such a case, the interpolation-based acquisitionmethod significantly outperforms the conventional approach.The resulting probability of detection for the worst operatingcase is depicted in Fig. 6 (with PFA = 10−3) versus theSNR (per chip). As it can be easily seen from the graphs,the correlation with offset (τ = Tc/4) has a non-negligibleimpact on the acquisition performance of the conventionaldetector. Also in this case, the results of the simulation arevery similar to the ones obtained from the theoretical analysis,which ensures their correctness. As it can be easily seen fromthe figures, the curves referring to the conventional methodare always lower of the ones referring to the new method.

However, the above mentioned results seem contradictory,


Pz|H0 (z|H0, z > x, z > y) =z∫

−∞

z∫−∞

px,y,z (x, y, z)

∞∫−∞

∞∫−∞

∞∫x+y+|x−y|

2 px,y,v (x, y, v) dx dy dv

dx dy (36)

px,y,v (x, y, v) =

(2π)−32 (det [K])−

12 e−

12 [x−E[x] y−E[y] v−E[v]]K−1[x−E[x] y−E[y] v−E[v]]T (37)

since the new detector is slightly poorer in the best case, whileit evidences better performance in the worst case. Actually,the best and worst cases represent the (upper and lower)bounds of the system’s performance. In order to assess theperformance in intermediate cases, typical of asynchronousacquisition systems, let us now focus our attention on themiddle timing offset τ = 1/8 (see Fig. 4c). Fig. 7 reportshere the detection probability for the middle case versusthe SNR (per chip) for the same probability of false alarmPFA = 10−3. According to our results, the new acquisitionmethod outperforms the conventional approach also in theintermediate case. The results of computer simulation areagain similar to the ones obtained from the theoretical analysis.

No noteworthy increase of computational complexity andprocessing throughput is required to implement the new acqui-sition system. In serial search, all the methods need to evaluatethe detector’s output for all the possible time offsets. Then, theconventional approach tests one detector’s sample per timeoffset, while the proposed technique tests one simple functionof three detector’s samples per time offset. In practice, nearthe totality of the amount of the required computation is dueto the former stage, since the latter logic testing is negligiblefrom the implementation and computation viewpoint.

For sake of comparison, we have considered in our plotsthe ideal (but computationally unpractical) analog detector(corresponding to the case Qc → +∞, i.e. an infinite numberof samples per chip of the ambiguity function). In practice,it can be obtained by a large value of Qc (Qc = 20 in ourtrials of Figs. 5–7), that requires to dramatically increase thecomputational cost of the same factor Qc. As a remark, theconventional detector outperforms the analog method in Fig.5, because the power detector employs a priori informationin the best case. In fact, it tests the ambiguity function at thecorrect timing offset, unlike an ideal interpolator that combinesthe possibly wrong tails of the ambiguity function.

In order to investigate for which timing offsets the newmethod overcomes the conventional power detector, the de-pendence of performance on the timing offset τ has beenanalyzed in its whole possible range from best (τ = 0) toworst (τ = 1/4) cases. In particular, the ratio Rd betweenthe detection probabilities of the two methods (for the sameprobability of false alarm) is depicted in Fig. 8 versus theactual timing offset τ . As it can be easily seen, the detectionprobability of the new method is greater then the one achieved

Fig. 8. Ratio (Rd) between the detection probabilities of parabolic andconventional methods (theory: asymptotic curve; simulation: 106 Monte-Carloruns versus the actual code offset for W = 15, PF A = 10−3, and SNR (perchip) = −15 dB.

by the conventional approach (i.e. the ratio Rd is greater thanone) in about the 80% of cases.

In fact, the devised method outperforms the conventionaldetector while the actual code offset (randomly distributed inan asynchronous system) is not close to an integer multiple ofthe sampling instants. In other words, detection performanceslightly decreases when it is good (“best” case), but increaseswhen it starts worsening for a wide range of typical operatingcases (from “middle” to “worst” cases). In practice, theparabolic acquisition in asynchronous systems tends to turninto a synchronous-like behavior. This happens because ittests for the actual maximum of the cross-correlation function,estimated by the apex of the fitted parabola.

VI. CONCLUSION

This paper has presented a new acquisition method to obtaina finer synchronization in the initial cell search, based onthe parabolic interpolation of cross-correlation samples. Theparabolic estimator provides a fast estimate by interpolatingonly three samples of the ambiguity function in the neighbor-hood of the coarse estimate.

The performance of the new method has been analyzed incomparison with the conventional power detector. The the-oretical results, substantiated by computer simulations, haveevidenced that the devised method is well suited for asyn-chronous spread-spectrum communications. The dependence


of the system performance on the actual timing offset (whichis randomly distributed in asynchronous systems) has beeninvestigated. In particular, best, middle, and worst cases for theactual code offset have been analyzed in the considered op-erating sets. According to the presented results, the parabolicinterpolation technique outperforms the conventional detectorfor a wide range of timing offsets.

REFERENCES

[1] R. L. Pickholtz, L. B. Milstein, and D. L. Schilling, “Spread spectrumfor mobile communications," IEEE Trans. Veh. Technol., vol. 40, pp.313–322, Feb. 1991.

[2] Y.-P. E. Wang and T. Ottosson, “Cell search in W-CDMA," IEEE J.Select. Areas Commun., vol. 18, no. 8, pp. 1470–1482, Aug. 2000.

[3] R. Esmailzadeh and M. Nakagawa, “Pre-Rake diversity combinationfor direct sequence spread spectrum mobile communications systems,"IEICE Trans. Commun., pp. 1008–1015, Aug. 1993.

[4] Y. H. Lee and S. Tantaratana, “Sequential acquisition of PN sequencesfor DS/SS communications: design and performance," IEEE J. Select.Areas Commun., vol. 10, pp.750–759, May 1992.

[5] S. Tantaratana, A. W. Lam, and P. J. Vincent, “Noncoherent sequentialacquisition of PN sequences for DS/SS communication with/withoutchannel fading," IEEE Trans. Commun., vol. 43, pp. 1738–1746, 1995.

[6] K. M. Chugg and M. Zhu, “A new approach to rapid PN code acquisitionusing iterative message passing techniques," IEEE J. Select. AreasCommun., vol. 23, no. 5, pp. 884–897, May 2005.

[7] A. Polydoros and C. L. Weber, “A unified approach to serial searchspread-spectrum code acquisition," IEEE Trans. Commun., vol. 32, no.5, pp. 542–560, May 1984.

[8] M. K. Simon, J. K. Omura, R. A. Scholtz, and B. K. Levitt, SpreadSpectrum Communications Handbook. New York: McGraw-Hill, 1994.

[9] J.-C. Lin, “Noncoherent sequential PN code acquisition using slidingcorrelation for chip-asynchronous direct-sequence spread-spectrum com-munications," IEEE Trans. Commun., vol. 50, no. 4, pp. 664–676, Apr.2002.

[10] W. R. Braun, “PN acquisition and tracking performance in DS-CDMAsystems with symbol-length spreading sequences," IEEE Trans. Com-mun., vol. 45, pp. 1595–1601, Dec. 1997.

[11] E. Moulines and K. Choukri, “Time-domain procedures for testing thata stationary time series is Gaussian," IEEE Trans. Signal Processing,vol. 44, pp. 2010–2025, Aug. 1996.

[12] D. M. Di Carlo and C. L. Weber, “Multiple dwell serial search:performance and application to direct-sequence code acquisition," IEEETrans. Commun., vol. 31, pp. 650–659, May 1983.

[13] G. Giunta, “Generalized Q-functions for application to non-coherent se-rial detection of spread-spectrum communication signals," IEEE Trans.Signal Processing, vol. 48, no. 5, pp. 1506–1513, May 2000.

[14] G. Giunta, A. Neri, and M. Carli, “Constrained optimization of non-coherent serial acquisition of spread-spectrum code by exploiting thegeneralized Q-functions," IEEE Trans. Veh. Technol., pp. 1378–1385,Sept. 2003.

[15] J. G. Proakis, Digital Communications, 3rd ed. McGraw-Hill, 1995.[16] K. Amleh, H. Li, and T. Li, “Blind and training-assisted subspace code-

timing estimation for CDMA with bandlimited chip waveforms," IEEETrans. Veh. Technol., vol. 53, no. 6, pp. 1735–1745, Nov. 2004.

[17] G. Giunta, “Fine estimators of two-dimensional parameters and appli-cation to spatial shift estimation," IEEE Trans. Signal Processing, vol.47, no. 12, pp. 3201–3207, Dec. 1999.

[18] A. Mantravadi and V. V. Veeravalli, “On chip-matched filtering anddiscrete sufficient statistics for asynchronous band-limited CDMA sys-tems," IEEE Trans. Commun., vol. 49, no. 8, pp. 1457–1467, Aug. 2001.

[19] A. J. van der Veen, M. C. Vanderveen, and A. Paulraj, “Joint angle anddelay estimation using shift-invariance techniques," IEEE Trans. SignalProcessing, vol. 46, pp. 405–418, Feb. 1998.

[20] F. Benedetto, M. Carli, G. Giunta, and A. Neri, “Performance benefits offractional sampling in the initial code synchronization for the wirelessaccess of 3G mobile communications," in Proc. IEEE 61st Veh. Technol.Conf., 2005, vol. 1, pp. 377–381.

[21] H. L. Van Trees, Detection, Estimation, and Modulation Theory. NewYork: Wiley, 1971.

[22] G. Jacovitti and G. Scarano, “Discrete time techniques for time delayestimation," IEEE Trans. Signal Processing, vol. 41, pp. 525–533, Feb.1993.

[23] G. Giunta and L. Vandendorpe, “A “Rayleigh-ness" test for DS/SS codeacquisition," IEEE Trans. Commun., vol. 51, no. 9, pp. 1492–1501, Sept.2003.

[24] S. Yoon, I. Song, and S. Y. Kim, “Code acquisition for DS/SS commu-nications in non-Gaussian impulsive channels," IEEE Trans. Commun.,vol. 52, no. 2, pp. 187–190, Feb. 2004.

[25] R. De Gaudenzi, C. Elia, and R. Viola, “Bandlimited quasi-synchronousCDMA: a novel satellite access technique for mobile and personalcommunication systems," IEEE J. Select. Areas Commun., vol. 10, no.2, pp. 328–343, Feb. 1992.

[26] R. De Gaudenzi, F. Giannetti, and M. Luise, “Signal recognition andsignature code acquisition in CDMA mobile packet communications,"IEEE Trans. Veh. Technol., vol. 47, no. 1, pp. 196–208, Feb. 1998.

[27] C. Caini, G. E. Corazza, and A. Vanelli-Coralli, “DS-CDMA codeacquisition in the presence of correlated fading—part II: applicationto cellular networks," IEEE Trans. Commun., vol. 52, no. 8, pp. 1397–1407, Aug. 2004.

[28] I. N. Psaromiligkos, S. N. Batalama, and M. J. Medley, “Rapid combinedsynchronization/demodulation structures for DS-CDMA systems—partI: algorithmic developments," IEEE Trans. Commun., vol. 51, pp. 983–994, June 2003.

[29] I. N. Psaromiligkos and S. N. Batalama, “Rapid combined synchro-nization/demodulation structures for DS-CDMA systems—part II: finitedata record performance analysis," IEEE Trans. Commun., vol. 51, pp.1162–1172, July 2003.

Francesco Benedetto (SM’04-AM’08) received theDr. Eng. degree in Electronic Engineering from theUniversity of Roma Tre, Rome, Italy, in May 2002,and the PhD degree in Telecommunication Engi-neering from the University of Roma Tre, Rome,Italy, in April 2007. The research interests of Dr.Benedetto are in the field of digital signal and imageprocessing in telecommunications, code acquisitionand synchronization for the 3G mobile commu-nication systems, and multimedia communication.In particular, he has published numerous research

articles on multimedia communications and video coding, ground penetratingradar (GPR) signal processing, spread-spectrum code synchronization for 3Gcommunication systems and satellite systems (GPS and GALILEO), correla-tion estimation, and spectral analysis. He is a member of The Institution ofEngineering and Technology (IET, formerly IEE), a member of the Institutionof Electrical and Electronic Engineers (IEEE), and a member of CNIT (ItalianInter-Universities Consortium for Telecommunications). He is a reviewerfor IEEE COMMUNICATION LETTERS and the IEEE TRANSACTIONS ONVEHICULAR TECHNOLOGY and he also served as a reviewer for many IEEEinternational conferences.

Gaetano Giunta (M’88) received the ElectronicEngineering degree from the University of Pisa,Italy, and the Ph.D. degree in Information andCommunication Engineering from the Universityof Rome La Sapienza, Italy, in 1985 and 1990,respectively. In 1986, he obtained a research grantfrom the Italian Research Council (CNR) of Pisa,Italy. He was also (since 1989) a Research Fellowof the Signal Processing Laboratory (LTS), SwissFederal Institute of Technology (EPFL), Lausanne,Switzerland. In 1992, he became an Assistant Pro-

fessor with the INFO-COM Department, University of Rome La Sapienza.Since 1998, he has taught Digital Signal Processing in the Third Univer-sity of Rome. From 2001 to 2005, he was with the Third University ofRome as an Associate Professor of Telecommunications. Since 2005, hehas been a Full Professor of Telecommunications with the same University.His research interests include signal processing for mobile communications,video communications and security, spread-spectrum systems, satellite andwireless networks. Prof. Giunta has been a representative member of CNIT(Italian Inter-Universities Consortium for Telecommunications) and the IEEESocieties of Communications, Signal Processing, and Vehicular Technology.He has also served as a reviewer for several IEEE transactions, IET (formerlyIEE) Proceedings, and EURASIP journals, and a TPC member for severalinternational conferences and symposia in the same fields.

A self-synchronizing method for asynchronous code acquisition in band-limited spread spectrum...

Documents

Transcript of A self-synchronizing method for asynchronous code acquisition in band-limited spread spectrum...