3204 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 11, NO. 9, SEPTEMBER 2012

An Initial Ranging Scheme for theIEEE 802.16 OFDMA Uplink

Luca Sanguinetti, Member, IEEE, and Michele Morelli, Senior Member, IEEE

Abstract—The IEEE 802.16 family of standards for no-madic wireless metropolitan area networks adopts orthogonalfrequency-division multiple-access as an air interface. In thesesystems, timing errors between the uplink signals and the basestation time reference give rise to interchannel interference aswell as multiple-access interference with an ensuing degradationof the error-rate performance. To mitigate this problem, usersthat intend to establish a communication link go through asynchronization procedure called Initial Ranging (IR) by whichuplink signals can arrive at the base station synchronously andwith approximately the same power level. In this work, a novelIR scheme compliant with the IEEE 802.16 specifications ispresented. In contrast to existing methods, our solution operateson the basis of a generalized likelihood ratio test (GLRT) andprovides improved timing and power estimates by properlytaking into account the channel correlation across the signalbandwidth. In order to increase the resilience to multiple accessinterference, the GLRT approach is also exploited to derive atwo-stage interference cancellation scheme.

Numerical simulations and theoretical analysis are used todemonstrate the effectiveness of the proposed solutions and tomake comparisons with existing alternatives.

Index Terms—OFDMA, initial ranging, timing and powerestimation, code detection, GLRT.

I. INTRODUCTION

ORTHOGONAL frequency-division multiple-access(OFDMA) is a multiplexing technology in which

clusters of orthogonal subcarriers are exclusively assigned todistinct users for simultaneous data transmission. Due to itsadvantages in terms of dynamic channel allocation, spectralefficiency and robustness to multipath distortions, OFDMAhas been adopted by the WiMAX alliance for efficientbroadband communications in wireless metropolitan areanetworks [1]. To maintain orthogonality among subcarriers ofdifferent users and avoid the occurrence of multiple-accessinterference (MAI), uplink signals arriving at the base station(BS) should be aligned to the local time and frequencyreferences. For this purpose, the IEEE 802.16 family ofstandards specifies a network entry procedure called InitialRanging (IR) by which subscribers can achieve uplinksynchronization and power control. In its basic form, the

Manuscript received July 18, 2011; revised December 2, 2011 and May 2,2012; accepted May 30, 2012. The associate editor coordinating the reviewof this paper and approving it for publication was A. Chockalingam.

This research was supported in part by the Seamless Aeronautical Net-working through integration of Data links, Radios, and Antennas (SANDRA)project co-funded by the European Commission within the “CooperationProgramme” GA No. FP7- 233679.

The authors are with the University of Pisa, Department of Informa-tion Engineering, Via Caruso 56126 Pisa, Italy (e-mail: {luca.sanguinetti,michele.morelli}@iet.unipi.it).

Digital Object Identifier 10.1109/TWC.2012.071612.111356

IR is a contention-based random access procedure operatingas follows. Firstly, ranging terminals (RTs) that intend toestablish a communication link with the BS perform downlinksynchronization through some dedicated control channels.The synch parameters estimated in the downlink are usedby each RT as references in the subsequent uplink phase,during which the RT randomly chooses an available rangingslot and sends a ranging request packet to the BS in order tonotify the request of network entry. The packet consists ofone or two ranging codes, depending on whether the rangingsignal is allowed to span two or four OFDMA symbols [1].If multiple RTs transmit their ranging signal simultaneously,they are allowed to collide on the same ranging channel.As a result of different positions within the radio coveragearea, ranging signals transmitted by different RTs arrive atthe BS with their specific transmission time delay. At thereceiving side, the BS is required to extract timing and powerinformation for each identified code. Then, it will broadcast aresponse message indicating which codes have been detectedand giving the corresponding instructions for timing andpower adjustment. The message will also contain a statusnotification by which the RT is informed as to whether itsranging process has been successfully completed or not. Inthe latter case, the RT repeats the ranging procedure untilsuccess notification.

The above discussion indicates that code identification aswell as multiuser timing and power estimation are the maintasks of the BS during the IR process. These problemshave received great attention in the last few years and somesolutions are available in the open literature. Since a time delayis equivalent to a phase shift in the frequency domain, codedetection and timing recovery is conventionally accomplishedby correlating the received frequency-domain samples withphase-shifted versions of the corresponding ranging sequence[2]. The code is detected if the correlation peak exceeds apre-assigned threshold, with the peak position providing thecorresponding timing information. The methods presented in[3] and [4] are based on a similar approach, but operate in thetime-domain and replace the WiMAX ranging codes by a setof generalized chirp-like sequences in order to get a sharpertiming metric. A simple energy detector is employed in [5] toreveal the presence of a ranging signal, while code detection isaccomplished by exploiting the fact that the ranging sequencesare real-valued. In [6] it is shown that the frequency-domaincorrelation approach outperforms its time-domain counterpartdue to its inherent capability of separating ranging subcarriersfrom data-bearing subcarriers.

All previously discussed schemes have been specifically

1536-1276/12$31.00 c© 2012 IEEE

SANGUINETTI and MORELLI: AN INITIAL RANGING SCHEME FOR THE IEEE 802.16 OFDMA UPLINK 3205

devised under the assumption that the received codes maintaintheir orthogonality at the BS side. In the presence of mul-tipath propagation, however, ranging subcarriers are subjectto different attenuations and phase shifts with an ensuingloss of code orthogonality. This gives rise to MAI plus self-interference (SI) arising from reception of multiple delayedreplicas of the same transmitted code. One possible approachfor MAI mitigation is illustrated in [7] by dividing rangingsignals into several groups with each group being transmittedover exclusively assigned subcarriers. In the signal designdiscussed in [8], the RT randomly chooses a code from anorthogonal set and transmits it over a specified number ofadjacent OFDMA symbols. Since spreading is performed inthe time direction, the code orthogonality is preserved at thereceiver as long as the channel response keeps constant overthe ranging slot. However, spreading across adjacent symbolsincreases the sensitivity to residual carrier frequency offsets(CFOs). Ranging schemes that are robust to frequency errorsare presented in [9] and [10], where users’ CFOs are estimatedby resorting to subspace-based methods.

In spite of their resilience against MAI and SI, the IRschemes proposed in [7] – [10] cannot be applied to commer-cial systems based on the IEEE 802.16 standards. The conceptof successive interference cancellation (SIC) is employed in[11] to design a ranging method that is robust to MAI andcompliant with the WiMAX technology at the same time.The resulting scheme operates in an iterative fashion with thestrongest path of the active RTs being detected and removedfrom the received signal at each new iteration. A possibleshortcoming of this method is that in the earlier iterationsthe detection process is still affected by substantial MAI andSI originated by multipath components yet to be detected. ASIC-based approach for MAI cancellation is also proposedin [12]. The main difference with respect to [11] is thatat each iteration one additional ranging signal is detectedinstead of a single multipath component. This approach hasa couple of advantages. On one hand, it is more robust toSI originated from phase-shifted versions of the same code.On the other hand, it allows significant computational savingby reducing the number of required iterations. However, sincea heuristic procedure is repeatedly applied to get timing andpower estimates, some improvement is expected by resortingto a more specific optimality criterion.

In the present work, a novel IR scheme is derived whichis compliant with the WiMAX specifications and employs ageneralized likelihood ratio test (GLRT) to decide whether agiven code is present or not in the considered ranging slot.In formulating our testing problem, we consider the channelfrequency response and timing error of the hypothesized codeas unknown parameters. For simplicity, the contribution of theother ranging signals is modeled as a Gaussian disturbanceterm whose power is jointly estimated with the code parame-ters by resorting to maximum likelihood (ML) arguments. Thepower level of the received code is eventually extracted fromthe estimated channel frequency response and its accuracy iscompared with the relevant Cramer-Rao bound (CRB). Ourmethod is inherently robust to SI originated by multipathpropagation and achieves better performance than the heuristicscheme in [12] at the price of an increased system complexity.

In order to improve the resilience to MAI, we also illustratea multiuser extension operating in two stages. In the first one,the contribution of the strongest RT is iteratively detected andremoved from the received signal in a way similar to [12]. Inthe second stage, the detected codes are re-checked again byapplying a parallel interference cancellation (PIC) approach.

The remainder of this paper is organized as follows. Nextsection describes the investigated OFDMA system and for-mulates the problem. The proposed IR scheme is derived inSection III using the GLRT approach. The extension to themultiuser case is discussed in Section IV. Simulation resultsare provided in Section V while some conclusions are drawnin Section VI.

The following notation is used throughout the paper. Matri-ces and vectors are denoted by boldface letters, with IN beingthe identity matrix of order N and 0N representing the N -dimensional null vector, respectively. A = diag{a(n) ; n =1, 2, . . . , N} denotes an N ×N diagonal matrix with entriesa(n) along its main diagonal, while B−1 and tr{B} are theinverse and the trace of a square matrix B. We use E{·},(·)∗, (·)T and (·)H for expectation, complex conjugation,transposition and Hermitian transposition, respectively. Thenotation ‖·‖ represents the Euclidean norm of the enclosedvector while [·]k,� denotes the (k, �)th entry of the enclosedmatrix.

II. SYSTEM MODEL AND PROBLEM FORMULATION

A. System model

Our system model is compliant with the IEEE 802.16estandard for OFDMA-based local and metropolitan area net-works (MANs) [1]. We denote by N the number of subcarrierswith frequency spacing Δf . To avoid aliasing problems atthe receiver, a specified number of null subcarriers is placedat both edges of the signal spectrum. A ranging channelconsists of Q subcarriers with indices {iq; q = 1, 2, . . . , Q}that are chosen so as to effectively exploit the frequencydiversity offered by the multipath channel. After downlinksynchronization, each RT selects an available ranging slotand sends a request packet to the BS in order to notify therequest of network entry. The packet consists of one binaryphase-shift keying (BPSK) ranging code, which is randomlychosen from a specified set C. Each code has length Q andmodulates the available ranging subcarriers in a way similar tomulti-carrier code-division multiple-access [13]. Without lossof generality, we assume that there is one ranging channel atthe beginning of the uplink frame and let K be the number ofRTs that are simultaneously active on the same slot. The time-domain samples of the kth RT (k = 1, 2, . . . ,K) are repeatedtwice in a phase continuous manner and transmitted over twoconsecutive OFDMA symbols [5]. They are expressed by

sk(m) =1√N

Q∑q=1

ck(q)e−j2πiqm/N 0 ≤ m ≤ 2(N+Ng)−1

(1)where ck = [ck(1), ck(2), . . . , ck(Q)]T is the ranging codeselected by the kth RT and Ng is the cyclic prefix (CP) length.Since the first symbol acts as a prolonged CP for the secondsymbol, orthogonality among ranging subcarriers is preserved

3206 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 11, NO. 9, SEPTEMBER 2012

at the BS even in the presence of large propagation delays.The latter only appear as linear phase shifts at the output ofthe discrete Fourier transform (DFT) unit.

The waveform transmitted by the kth RT propagates througha multipath channel and arrives at the BS with specifictiming and frequency misalignments with respect to the localreferences. After down-conversion to baseband, the samplesreceived from the kth RT are given by

xk(m) = ej2πmεk/NL−1∑�=0

hk(�)sk(m− �− θk) (2)

where hk = [hk(0), hk(1), . . . , hk(L−1)]T is the channel im-pulse response (CIR) of order L, while θk and εk represent thetiming error (normalized by the sampling period Ts) and theCFO (normalized by the subcarrier spacing Δf ), respectively.Timing errors are related to the different distances betweenthe RTs and the BS. Their maximum value corresponds tothe round trip propagation delay for a RT located at the cellboundary and is given by θmax = 2R/(cTs), with R beingthe cell radius and c denoting the speed of light. In practice,R should be designed such that θmax does not exceed theOFDMA symbol length N . On the other hand, since duringthe IR period the CFOs are mainly due to Doppler shiftsand downlink synchronization errors, they are expected to besignificantly smaller than Δf . In this work, we assume thatfrequency estimation errors in the downlink are within 2% ofΔf as specified in [1]. In such a case, the impact of CFOs onthe IR process can reasonably be neglected.

At the receiver side, the first OFDMA symbol in the consid-ered ranging slot is discarded as it only contains incompleteparts of the transmitted codes, while the second symbol isfed to a DFT unit for further processing. We assume thatusers other than those performing IR have been successfullysynchronized to the BS and do not generate interference overthe ranging channel. Denoting by X(iq) the DFT output overthe iqth subcarrier, we may write

X(iq) =

K∑k=1

ck(q)Hk(iq)e−j2πiqθk/N + n(iq) (3)

where Hk(iq) is the kth channel frequency response over theiqth subcarrier and is related to hk by

Hk(iq) =

L−1∑�=0

hk(�)e−j2π�iq/N (4)

while n(iq) accounts for background noise and is modeled as acircularly-symmetric complex Gaussian random variable withzero mean and variance σ2

n. The power that the BS receivesfrom the kth RT is denoted by

Pk =‖Hk‖2

Q(5)

with Hk = [Hk(i1), Hk(i2), . . . , Hk(iQ)]T collecting the kth

channel frequency response over the ranging channel. The BSexploits the DFT vector X = [X(i1), X(i2), . . . , X(iQ)]

T

to detect the active codes and for extracting the associatedtiming and power information. Then, for each detected codeit checks whether the estimated synchronization parameters

are adequately close to the specified requirements. In the af-firmative case, it transmits a response message with ”success”notification so as to inform the RT that its IR procedure hasbeen successfully completed. Otherwise, it sends a responsemessage with “continue” status, indicating the detected codeand providing instructions for timing and power adjustment.In the latter case, the RT shall continue the ranging processas done on the first entry by choosing a new code from theset C, which is sent on the next ranging slot with updatedsynchronization parameters. If the RT does not receive anyresponse message from the BS, after a specified time it per-forms another ranging attempt using a new randomly selectedcode with increased power level.

B. Problem formulation

From (4) it follows that

Hk = Fhk (6)

where F is a Q × L matrix with entries [F]q,� =e−j2π(�−1)iq/N for 1 ≤ q ≤ Q and 1 ≤ � ≤ L. Then, lettingCk = diag{ck(1), ck(2), . . . , ck(Q)} we may rewrite (3) inmatrix form as

X =∑k∈C

CkΓ(θk)Fhk + n (7)

where Γ(θk) = diag{e−j2πi1θk/N , e−j2πi2θk/N , . . . , e−j2πiQθk/N}and n = [n(i1), n(i2), . . . , n(iQ)]

T is the noise contribution.Since the BS has no prior knowledge as to which codesare actually present in the considered ranging slot, thesummation in (7) is extended over the entire code set Cwith the assumption that hk = 0L if the kth code is notactive. To proceed further, we denote by |C| the cardinalityof C and observe that there is a total of J = 2|C| possibledifferent combinations of codes that might be transmittedin the considered ranging slot. Then, denoting by Hj thehypothesis associated to the jth combination, the problem ofcode detection could be formulated as a composite multiplehypothesis test in which the BS must decide which ofthe mutually exclusive hypotheses {H0,H1, . . . ,HJ−1} isactually the true one. Unfortunately, this approach cannot beimplemented with affordable complexity as it requires thecomputation of a suitable decision metric for each of the Jhypotheses, which is unfeasible when |C| exceeds a few units.To overcome this difficulty, we adopt a single-user strategyin which code detection is performed individually for anycm ∈ C. More precisely, for each m = 1, 2, . . . , |C| the BSdecides in favour of one of the following two hypotheses:H0) the code cm is not present in the observation vectorX; H1) cm is present in X. In doing so, the contributionof the active codes ck with indices k �= m is treated asa disturbance term, which inevitably degrades the systemperformance. Although suboptimal, this approach has theadvantage of allowing a simple formulation of the detectionproblem as a composite binary hypothesis test:

H0 : X = wm (8)

H1 : X = CmΓ(θm)Fhm +wm (9)

SANGUINETTI and MORELLI: AN INITIAL RANGING SCHEME FOR THE IEEE 802.16 OFDMA UPLINK 3207

where wm accounts for the contribution of MAI plus thermalnoise. To further simplify the derivation, the entries of wm aremodeled as independent Gaussian random variables with zeromean and unknown power σ2 = σ2

MAI + σ2n, where σ2

MAI isthe average MAI power.

Vector X can be exploited to make a decision between thetwo hypotheses H0 and H1. Inspection of (8) and (9) revealsthat this task is complicated by the presence of the unknownparameters hm, θm and σ2. Recalling that the GLRT providesan effective solution to detect the presence of a partiallyunknown signal embedded in additive Gaussian noise, in thesequel the GLRT is applied to the observation vector X tocheck the presence of cm in the ranging slot and to retrievethe corresponding timing and power information.

III. IR ALGORITHM BASED ON THE GLRT CRITERION

A. Code detection and timing estimation

We denote by pH�the probability density function (pdf) of

X under the hypothesis H� for � = 0, 1. Then, the GLRT canbe formulated as

pH1

(X; σ2

H1, θm, hm

)pH0

(X; σ2

H0

) H1

≷H0

λ (10)

where λ is a suitable threshold, (θm, hm) is the ML estimateof (θm,hm) and σ2

H�is the ML estimate of σ2 conditioned

to H�. From (8) and (9) it follows that

pH1

(X;σ2, θm,hm

)=

1

(πσ2)Qe−

1σ2 ‖X−CmΓ(θm)Fhm‖2

(11)and

pH0

(X;σ2

)=

1

(πσ2)Qe−

1σ2 ‖X‖2

. (12)

The ML estimate of σ2 under the hypothesis H0 is obtainedby maximizing pH0

(X;σ2

)in (12) with respect to σ2. This

yields σ2H0

= ‖X‖2 /Q, from which it follows that

pH0

(X; σ2

H0

)=

(Q

πe ‖X‖2)Q

. (13)

On the other hand, the ML estimate of (σ2, θm,hm) in thehypothesis H1 is the location where pH1

(X;σ2, θm,hm

)achieves its global maximum. Keeping σ2 and θm fixed andmaximizing with respect to hm yields

hm =(FHF

)−1zm(θm) (14)

where zm(θm) is defined as

zm(θm) = FHΓH(θm)Ym (15)

with Ym = CmX. Substituting this result back into (11)produces

pH1

(X;σ2, θm, hm

)=

1

(πσ2)Qe− 1

σ2

[‖X‖2−zHm(θm)(FHF)−1

zm(θm)]

(16)

which is then maximized with respect to θm to find the MLtiming estimate. This yields

θm = arg max0≤ θ≤ θmax

Ψm(θ) (17)

where Ψm(θ) is expressed by

Ψm(θ) = zHm(θ)(FHF

)−1zm(θ). (18)

After replacing θm with θm in (16), an estimate of σ2 isobtained as

σ2H1

=1

Q

[‖X‖2 −Ψm(θm)

]. (19)

Substituting (17) and (19) into (16) yields

pH1

(X; σ2

H1, θm, hm

)=

⎛⎝ Q

πe[‖X‖2 −Ψm(θm)

]⎞⎠

Q

(20)which is eventually divided by pH0

(X; σ2

H0

)in (13) so as to

obtain the GLRT in the form(‖X‖2

‖X‖2 −Ψm(θm)

)QH1

≷H0

λ (21)

or, equivalently,Ψm(θm)

‖X‖2H1

≷H0

λ′ (22)

with the threshold λ′ = 1− λ−1/Q being a design parameterthat must be chosen so as to achieve the desired trade-offbetween the false-alarm and mis-detection probabilities.

B. Power estimation

Using the invariance property of the ML estimator [14], theestimate of the power Pm associated to cm can be retrievedfrom (5) as

Pm =

∥∥∥Hm

∥∥∥2Q

(23)

where Hm = Fhm is the ML estimate of Hm. Using (14)and (18), we may rewrite (23) in the equivalent form

Pm =Ψm(θm)

Q. (24)

Assuming ideal timing estimation (i.e., θm = θm), in Ap-pendix A it is shown that Pm and σ2

H1are biased estimators

withE{Pm} = Pm +

L

Qσ2 (25)

andE{σ2

H1} =

Q− L

Qσ2. (26)

From these results, an unbiased estimate of Pm is found to be

P (f)m = Pm − L

Q− Lσ2H1

(27)

which can also be rewritten as

P (f)m =

QΨm(θm)− L ‖X‖2Q(Q− L)

(28)

3208 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 11, NO. 9, SEPTEMBER 2012

after using the expressions of σ2H1

and Pm given in (19) and(24), respectively. In the sequel, the code detector (22) andthe associated timing and power estimators (17) and (28) arereferred to as the single-user GLRT-based scheme (SU-GLRT).

C. Remarks

The following remarks are of interest.i) The accuracy of the power estimate P

(f)m is assessed

in terms of mean square error (MSE), which is defined asMSE(Pm) =E{[P (f)

m − Pm]2}. Assuming θm = θm, inAppendix A it is shown that

MSE(Pm) =2Pm

Qσ2 +

L

Q (Q− L)σ4. (29)

The CRB for the estimation of Pm is computed in AppendixB and is given by

CRB(Pm) =2Pm

Qσ2. (30)

Since MSE(Pm) approaches the CRB for small values of σ2,we may conclude that P (f)

m is asymptotically efficient.ii) As explained in [15], the fractional part of the timing

error can be incorporated into the CIR. This allows one tomodel θm as an integer-valued parameter, thereby reducingthe complexity of the grid-search in (17).

iii) So far we have tacitly assumed that SU-GLRT isprovided with ideal knowledge of the channel length L. Inpractice, L is unknown and must be estimated in some manner.A possible solution is to replace it with a design parameter andlet SU-GLRT operate in a mismatched mode. Some guidelinesfor the selection of such a parameter will be provided later onthe basis of numerical simulations. In particular, it turns outthat different values of L, say Lθ and LP , should be usedfor the timing and power estimation tasks, respectively. Thisresults into a modified scheme wherein Lθ is employed toevaluate the timing metric Ψm(θ) for θ = 0, 1, . . . , θmax. Afterobtaining the timing estimate θm, we recompute Ψm(θm) from(18) by replacing L with LP . Finally, Ψm(θm) is used in (19)and (24) to get the quantities σ2

H1and Pm, which are next

employed to evaluate the power estimate as indicated in (28).iv) Setting Lθ = 1 in (18) yields Ψm(θ) = |zm(θ)|2, which

is just the metric employed by the conventional single-userscheme (CSU) discussed in [6] for code detection and timingestimation. This means that SU-GLRT reduces to CSU underthe assumption of a flat fading channel.

v) The computational load of SU-GLRT is mainly in-volved in the evaluation of the timing metric Ψm(θ) forθ = 0, 1, . . . , θmax. An efficient way to compute Ψm(θ) isas follows. First, the right-hand-side of (18) is reformulatedas

Ψm(θ) =∥∥∥Uzm(θ)

∥∥∥2 (31)

where U is the Lθ×Lθ upper triangular matrix obtained fromthe Cholesky factorization of

(FHF

)−1. Observing that U

can be pre-computed and stored in the receiver, evaluating theright-hand-side of (31) for each θ starting from zm(θ) requiresa total of 4L2

θ floating point operations (flops), including realmultiplications and real additions. Second, from (15) it isseen that zm(θ) = [zm(θ), zm(θ + 1), . . . , zm(θ + Lθ − 1)]T

is a segment of an N−dimensional sequence {zm(n);n =0, 1, . . . , N − 1} representing the inverse discrete Fouriertransform (IDFT) of

Zm(n) =

{Ym(iq)

0if n = iq

0 ≤ n ≤ N − 1 and n �= iq .

This means that all vectors zm(θ) for θ = 0, 1, . . . , θmax areobtained from a single N−point IDFT operation requiring5N log2 N flops. It follows that approximately 5N log2 N +4θmaxL

2θ flops are needed by SU-GLRT for any tested code,

while 5N log2 N flops are required by CSU. The overallcomplexity of these schemes is summarized in the first tworows of Table I.

IV. MULTIUSER EXTENSION OF THE RANGING SCHEME

In detecting the presence of a given code, the SU-GLRTsimply treats the contribution of other active codes as ad-ditional noise without attempting any MAI mitigation. Thismethod results in some performance degradation as thenumber of RTs increases, thereby reducing the capabilityof detecting low-power codes overwhelmed by high-powerusers. In order to improve the resilience to MAI, we suggestthe adoption of an interference cancellation (IC) procedureoperating in two stages as follows. In the first stage, theactive codes are detected by means of the GLRT criterionand successively cancelled from the received signal accordingto the SIC approach. The detected codes and their associatedtiming and power information are next exploited in the secondstage to further mitigate the interference by resorting to a PICstrategy.

For illustration purposes, we concentrate on the jth iterationof the SIC-stage (with j = 1, 2, . . .) and call X(j)

SIC the residualobservation vector, which is obtained from X after cancelingthe contribution of codes detected up to the (j−1)th iteration.

These codes are collected into a set C(j−1), while C(j−1)=

C − C(j−1) denotes the set of IR codes that have not beenchecked yet. Clearly, at the initialization step we set C(0) = ∅

and X(1)SIC = X. For each cm ∈ C(j−1)

, the SIC obtains theL−dimensional vector z(j)SIC,m(θ) from X

(j)SIC in the same way

as the SU-GLRT computes zm(θ) from X. Then, followingthe same steps leading to (17) and (28), the timing and powerestimates associated to cm are obtained as

θ(j)SIC,m = arg max

0≤ θ≤ θmax

Ψ(j)SIC,m(θ) (32)

and

P(j)SIC,m =

QΨ(j)SIC,m(θ

(j)SIC,m)− LP

∥∥∥X(j)SIC

∥∥∥2Q(Q− LP )

(33)

where Ψ(j)SIC,m(θ) is expressed as in (18) after replacing zm(θ)

with z(j)SIC,m(θ). At this stage, the index m(j) of the code with

the highest received power is identified according to

m(j) = arg maxm∈I(j−1)

P(j)SIC,m (34)

where I(j−1)collects the indices of all codes cm ∈ C(j−1)

. AGLRT decision is eventually taken between the two hypothe-ses H1 and H0 corresponding to cm(j) being active or not.

SANGUINETTI and MORELLI: AN INITIAL RANGING SCHEME FOR THE IEEE 802.16 OFDMA UPLINK 3209

Mathematically, we have

ΨSIC,m(j)(θ(j)

SIC,m(j))∥∥∥X(j)SIC

∥∥∥2H1

≷H0

λ′. (35)

If the hypothesis H0 is chosen, no further code is detectedand the iterative process is terminated. Otherwise, the iterationindex is incremented to j + 1 and the observation vector isupdated as

X(j+1)SIC = X

(j)SIC −Cm(j)Γ(θ

(j)

SIC,m(j))H(j)

SIC,m(j) (36)

where H(j)

SIC,m(j) = F(FHF)−1z(j)

SIC,m(j)(θ(j)

SIC,m(j)) is the

channel estimate associated to cm(j) . Then, after letting C(j) =C(j−1) ∪ {cm(j)}, a new iteration is performed.

Once the SIC stage is completed, we get the list of detectedcodes CSIC = {cm(j) ; j = 1, 2, . . . , J}, with J being thenumber of performed iterations. From (36) it is seen that,while detecting cm(j) , only the interference generated by codescm(i) with i < j is canceled from the observation vectorX. Intuitively speaking, better results are expected if thecontribution of codes cm(i) with i > j is also subtracted fromX. For this purpose, we may adopt the approach outlined in[12] according to which the codes belonging to CSIC are re-checked again and their timing and power estimates refined bymeans of a PIC strategy. More precisely, let {θ(i)

PIC,m(i) ; i =

1, 2, . . . , j − 1} and {H(i)

PIC,m(i) ; i = 1, 2, . . . , j − 1} bethe refined timing and channel estimates available at the jthPIC iteration. Then, the contribution of the detected codes iscanceled from X by evaluating the expurgated vector

X(j)PIC =X−

j−1∑i=1

Cm(i)Γ(θ(i)

PIC,m(i))H(i)

PIC,m(i)

−J∑

i=j+1

Cm(i)Γ(θ(i)

SIC,m(i))H(i)

SIC,m(i) (37)

which is used in place of X(j)SIC to get the quantities

z(j)PIC,m(θ) and the timing metric Ψ(j)

PIC,m(θ). The refined tim-ing and power estimates associated to cm(j) are obtained from(32) and (33) after replacing Ψ

(j)SIC,m(θ) with Ψ

(j)PIC,m(θ).

Then, the jth code is checked again according to the GLRTrule yielding

ΨPIC,m(j)(θ(j)

PIC,m(j) )∥∥∥X(j)PIC

∥∥∥2H1

≷H0

λ′. (38)

The refined channel estimate is eventually obtained asH

(j)

PIC,m(j) = F(FHF)−1z(j)

PIC,m(j) (θ(j)

PIC,m(j)) in the hypoth-

esis H1, while we let H(j)

PIC,m(j) = 0Q in the hypothesis H0.In the sequel, the illustrated two-stage iterative procedure

is referred to as the multiuser GLRT-based scheme (MU-GLRT). It is worth observing that during the jth iterationthe SIC evaluates |C| − j + 1 timing metrics Ψ

(j)SIC,m(θ)

for θ = 0, 1, . . . , θmax. Once the SIC procedure has beencompleted, additional J metrics are computed in the PICstage. This means that MU-GLRT approximately needs thecomputation of J |C| different timing metrics, leading to the

200

150

100

50

0

Exp

ecte

d tim

ing

met

ric

160140120100806040�~

SU-GLRT, SNR = 10 dB K = 1, L

�

Fig. 1. Expected SU-GLRT timing metric with Lθ = 40, K = 1 and SNR= 10 dB.

200

150

100

50

0

Exp

ecte

d tim

ing

met

ric

160140120100806040�~

SU-GLRT, SNR = 10 dB K = 1, L

�

Fig. 2. Expected SU-GLRT timing metric with Lθ = 16, K = 1 and SNR= 10 dB.

overall complexity listed in the third row of Table I where wehave assumed that J = K .

V. SIMULATION RESULTS

The performance of the proposed ranging algorithms hasbeen assessed by computer simulations in terms of codedetection capability as well as timing and power estimationaccuracy. Comparisons are made with the CSU scheme dis-cussed in [6] and with the alternative solution presented byLin and Su (LS) in [12].

A. System parameters

The simulation parameters are chosen in compliance withthe IEEE 802.16 family of standards for wireless MANs [1].

3210 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 11, NO. 9, SEPTEMBER 2012

The DFT size is N = 1024, with a CP composed of 64samples. The sampling period is Ts = 89.28 ns, correspondingto a subcarrier spacing of 10.94 kHz. In addition to the nullDC subcarrier, there are 92 and 91 unmodulated subcarriers forthe left and right guard bands, respectively. The remaining 840subcarriers are grouped into tiles, each containing 4 adjacentsubcarriers. Six non-consecutive tiles constitute a subchannel,which is the minimum frequency resource assigned to anuplink user. The ranging slot is composed by 6 subchannels,meaning that a total of Q = 144 subcarriers are reservedfor IR. The indices of the ranging subcarriers are obtained byapplying a specific permutation rule distributing them over theavailable spectrum in a pseudo-random fashion. The numberof codes reserved for IR is fixed to |C| = 32, while theCFO of each RT is Gaussian distributed with a standarddeviation of 1% of the subcarrier spacing. The modulationpulse is a root-raised-cosine function with roll-off α = 0.22and duration Tg = 10Ts. Unless otherwise specified, at eachsimulation run the mobile speed v varies in the interval [0,10]m/s with uniform distribution. A mixed channel model is usedthroughout simulations. Specifically, if v ≤ 5 m/s we select thepower delay profile specified by the ITU IMT-2000 Ped-A orPed-B channel models with equal probability, while the Veh-A or Veh-B delay profile is chosen when v > 5 m/s. In anycase, the path gains are generated by passing statistically in-dependent and circularly symmetric white Gaussian processesthrough a third-order low-pass Butterworth filter. The 3-dBbandwidth of the filter is taken as a measure of the Dopplerbandwidth BD = vf0/c, where f0 = 5.1 GHz denotesthe carrier frequency. The CIRs have a maximum order ofL = 40. A cell radius of 3 km is assumed, which amountsto setting θmax = 224. Accordingly, the RT timing offset θkis uniformly distributed in the interval [0, 224]. Although anew channel initialization is generated at the beginning of theranging slot, the ranging signal, which is used to measure thesystem performance, is normalized so that its signal-to-noiseratio (SNR) is fixed irrespective of the channel realization.

The performance of the investigated IR algorithms is firstassessed in the presence of a single RT, while the case ofmultiple RTs is considered later.

B. Performance evaluation in the presence of a single RT

1) Timing recovery: As mentioned previously, the truechannel length L is unknown in practice and must be replacedwith a suitable design parameter Lθ. For this purpose, in Figs.1 and 2 we show the expected value of Ψm(θ) as obtainedwith Lθ = 40 and 16, respectively, with θ varying in theinterval [40, 160]. Measurements are obtained by numericallyaveraging 1000 Monte-Carlo trials for each value of θ underthe assumption that a single RT with a fixed timing errorθ1 = 100 and SNR = 10 dB is present in the rangingsubchannel. Inspection of Fig. 1 indicates that for Lθ = 40the timing metric is plagued by a large plateau region whichmay deteriorate the system performance. As shown in Fig. 2,this problem is alleviated by reducing Lθ to 16 since in sucha case Ψm(θ) exhibits a sharper peak. From the above results,we argue that in evaluating the timing metric it might beconvenient to choose Lθ < L. Such an intuition is validated by

225

200

175

150

125

100

75

50

25

0

MSE

of t

imin

g es

timat

es

403632282420161284 L�

SU-GLRT, K = 1

SNR = 5 dB SNR = 10 dB SNR = 15 dB

Fig. 3. MSE(θ1) versus Lθ for the SU-GLRT with K = 1 and differentSNR values per ranging attempt.

80

70

60

50

40

30

20

10

0

MSE

of t

imin

g es

timat

es

2520151050 SNR per ranging attempt

CSU LS SU-GLRT

K = 1

Fig. 4. MSE(θ1) versus SNR per ranging attempt for the investigatedschemes with K = 1.

the numerical analysis shown in Fig. 3, where the MSE of thetiming estimation error defined as MSE(θ1) =E{(θ1−θ1)

2} isillustrated as a function of Lθ for three different SNR values. Itturns out that the best results are obtained for 16 ≤ Lθ ≤ 24,while a rapid degradation of the estimation accuracy is ob-served for Lθ > 24. As expected, some advantage is achievedwith respect to Lθ = 1, which corresponds to the CSU schemeillustrated in [6]. Since the number of flops required by SU-GLRT increases with Lθ, this parameter must be designedso as to achieve a reasonable tradeoff between computationalcomplexity and estimation accuracy. From the results of Fig.3 it turns out that a good compromise can be achieved bysetting Lθ = 16. For this reason, such a value is used in allsubsequent simulations.

SANGUINETTI and MORELLI: AN INITIAL RANGING SCHEME FOR THE IEEE 802.16 OFDMA UPLINK 3211

10-4

10-3

10-2

10-1

100 M

SE o

f pow

er e

stim

ates

403632282420161284 LP

SNR = 10 dB

SNR = 15 dB

SNR = 5 dB

Simulated Theoretical CRB

SU-GLRT, K = 1

Fig. 5. MSE(P1) versus LP for the SU-GLRT with K = 1 and differentSNR values per ranging attempt.

10-5

10-4

10-3

10-2

10-1

100

MSE

of p

ower

est

imat

es

2520151050 SNR per ranging attempt

CSU LS SU-GLRT CRB

K = 1

Fig. 6. MSE(P1) versus SNR per ranging attempt for the investigatedschemes with K = 1.

Fig. 4 illustrates MSE(θ1) for the investigated IR schemesas a function of the SNR. We see that SU-GLRT has the bestaccuracy and achieves a remarkable gain with respect to CSUand LS.

2) Power level estimation: We now turn our attention tothe power estimation task and assess the impact of LP onthe accuracy of P (f)

1 . Fig. 5 illustrates MSE(P1) =E{(P (f)1 −

P1)2} as a function of LP under the same operating conditions

of Fig. 3. The theoretical analysis expressed in (29) and therelevant CRB in (30) are also shown for comparison and theyare practically the same for all the investigated SNR values.As expected, the agreement between numerical results andtheoretical analysis is achieved only when LP is adequatelylarge. Specifically, at SNR = 5 dB we see that LP shouldbe chosen greater than or equal to 16, while larger values are

����

����

����

���

P fa a

nd P

md

�������������������������������

Threshold, �’

SNR = 10 dB

CSU SU-GLRT

Pfa Pfa

Pmd

Pmd

Fig. 7. Pfa and Pmd versus λ′ as obtained with SU-GLRT and CSU atSNR=10 dB.

needed as the SNR increases. The results of Fig. 5 indicatethat LP = 24 is a good choice irrespective of the operatingSNR.

Fig. 6 illustrates MSE(P1) as a function of the SNR forthe considered ranging schemes. As it is seen, the SU-GLRTattains the CRB at all SNR values smaller than 20 dB, whilethe accuracy of CSU and LS is virtually independent of theSNR and exhibits a significant loss compared to SU-GLRT.

3) Code detection: The code detection capability is as-sessed in terms of false alarm and mis-detection probabilities,say Pfa and Pmd. The former is defined as the probability ofdeclaring the presence of a code which is turned off, whilethe latter represents the probability of missing a code whichis actually present. Fig. 7 illustrates Pfa and Pmd versus thethreshold λ′ as obtained with SU-GLRT and CSU at SNR= 10 dB. These results indicate that SU-GLRT guaranteesa large range of threshold values for which both Pfa andPmd are adequately small, say less than 10−3. The situationis remarkably worse with the CSU as in such a case the rangeof threshold values leading to Pfa and Pmd smaller than 10−3

is significantly reduced with respect to SU-GLRT. As for LS,we recall that the threshold employed in [12] is the cross pointbetween two distributions and depends on the tested code cmand the iteration index j according to

λ(j)LS,m =

σ2m,j

2QI−10

(eQ

2/σ2m,j

)(39)

where I−10 (x) in the inverse of the modified Bessel function

of zero order, while σ2m,j is the noise-plus-interference power

measured at the jth iteration on the basis of z(j)SIC,m(θ). Settingthe threshold as indicated in (39), our simulations indicate thatin the presence of a single RT with SNR = 10 dB the LSscheme is able to provide a Pfa smaller than 10−4 while thePmd is equal to 4 · 10−2, which is larger than that guaranteedby SU-GLRT.

4) Impact of user mobility: All previous measurementshave been conducted assuming that the RT speed is uniformly

3212 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 11, NO. 9, SEPTEMBER 2012

10-4

10-3

10-2

10-1

100

101

102 M

SE o

f pow

er a

nd ti

min

g es

timat

es

70605040302010 Mobile speed, v (m/s)

MSE of power estimates

MSE of timing estimates

SNR = 5 dB SNR = 10 dB SNR = 15 dB

SU-GLRT, K = 1

Fig. 8. MSE of the power and timing estimates versus the mobile speed forSU-GLRT with different SNR values per ranging attempt.

80

70

60

50

40

30

20

10

0

MSE

of t

imin

g es

timat

es

2520151050 Average SNR

CSU LS SU-GLRT

K = 1

Fig. 9. MSE(θ1) versus average SNR for the investigated schemes withK = 1.

distributed within the interval [0,10] m/s. In order to assessthe impact of an increased user mobility on the systemperformance, in Fig. 8 the MSE of the timing and powerestimates is shown as a function of the mobile speed v for threedifferent SNR values. Note that each point of the illustratedcurves is obtained by keeping v fixed to the value reported onthe x-axis. It turns out that the accuracy of the power estimatesdegrades with v, especially at high SNR values. On the otherhand, the timing estimator is only marginally affected by usermobility.

5) Impact of SNR fluctuations: So far, the SNR of theranging signal has been kept fixed irrespective of the channelrealization. Our choice is motivated by the fact that anyranging attempt only spans two OFDMA blocks and during

10-5

10-4

10-3

10-2

10-1

100

MSE

of p

ower

est

imat

es

2520151050 Average SNR

CSU LS SU-GLRT CRB

K = 1

Fig. 10. MSE(P1) versus average SNR for the investigated schemes withK = 1.

this short time period the CIR cannot exhibit large powerfluctuations. Although signals of different ranging terminals(as discussed later) are normally received by the BS at muchdifferent power levels, during the ranging slot the powerreceived from each terminal is expected to remain unchangedand, accordingly, it is reasonable to measure the accuracy ofthe proposed ranging algorithms at a fixed SNR rather than atan average SNR. For the sake of completeness, however, inFigs. 9 and 10 we report the accuracy of the timing and powerestimate obtained in the same operating conditions of Figs. 4and 6, except that now the CIR power is not normalized andfluctuates at each new simulation run according to the channelstatistics. As we can see, the novel results are qualitativelysimilar to those in Figs. 4 and 6, with SU-GLRT still exhibitingmuch better performance than LS and CSU. Specifically, theaccuracy of the power estimates obtained with the proposedscheme still approaches the relevant CRB, which is nowcomputed by numerically averaging the bound in (30) withrespect to the power fluctuations.

C. Performance evaluation in the presence of multiple RTs

We now assess the performance of SU-GLRT and MU-GLRT in the presence of multiple RTs. In our simulations,the maximum number of users that simultaneously access thesame ranging slot is fixed to Kmax = 4. Without loss ofgenerality, the system performance is measured on the basis ofthe signal received from the first RT. This signal is normalizedso as to keep its SNR fixed to 10 dB, while the SNR valuesof the other RTs are independently generated at each newsimulation run according to a uniform distribution over theinterval [0,20] dB. Such large power fluctuations may arise asa consequence of shadowing effects and/or different distancesof the RTs from the BS. The speed of the user of interest isstill randomly chosen in the interval [0,10] m/s. However, inorder to model a realistic scenario characterized by a largerange of user mobility, the speeds of the remaining RTs are

SANGUINETTI and MORELLI: AN INITIAL RANGING SCHEME FOR THE IEEE 802.16 OFDMA UPLINK 3213

����

����

����

���

P fa a

nd P

md

�������������������������������

Threshold, �’

MU-GLRT SNR = 10 dB, K = 4

Pfa

Pmd

Fig. 11. Pfa and Pmd versus λ′ as obtained with MU-GLRT for K = 4and SNR = 10 dB per ranging attempt.

10

100

MSE

of t

imin

g es

timat

es

4321 Number of ranging terminals, K

SU-GLRT LS MU-GLRT

SNR = 10 dB

Fig. 12. MSE of the timing estimates versus K with SNR = 10 dB perranging attempt.

uniformly distributed in the interval [0,30] m/s.Fig. 11 illustrates Pfa and Pmd vs. the threshold λ′ as

obtained with MU-GLRT. Compared to Fig. 7, we see thatthe situation is much more critical than in the single-usercase, since in the present scenario there exists only a verylimited range of threshold values centered around λ′ = 0.32for which both Pfa and Pmd are in the order of 10−3. Usingthe threshold in (39) in the presence of four RTs, the LSscheme provides Pfa = 10−3 and Pmd = 5 · 10−2 at SNR= 10 dB. On the other hand, from Fig. 11 it is seen thatthe MU-GLRT achieves a Pmd in the order of 10−3 when thePfa is set to 10−3, thereby exhibiting improved code detectioncapability with respect to LS.

The MSE of the timing estimates for the considered ranging

10-4

10-3

10-2

10-1

100

101

MSE

of p

ower

est

imat

es

4321 Number of ranging terminals, K

SU-GLRT LS MU-GLRT CRB

SNR = 10 dB

Fig. 13. MSE of the power estimates versus K with SNR = 10 dB perranging attempt.

TABLE ICOMPUTATIONAL LOAD OF THE INITIAL RANGING SCHEMES

Algorithm Required flops

CSU 5|C|N log2 N

SU-GLRT |C|(5N log2 N + 4θmaxL2θ)

MU-GLRT K|C|(5N log2 N + 4θmaxL2θ)

LS 5K|C|N log2 N

schemes is shown in Fig. 12 as a function of the number K ofactive users. Although SU-GLRT and MU-GLRT exhibit thesame performance with K = 1, the latter is only marginallyaffected as K grows from 1 to 4, while larger degradationsoccur with SU-GLRT. Both schemes outperform LS, which isplagued by relatively large timing errors.

Fig. 13 illustrates the accuracy of the power estimates inthe same operating conditions of Fig. 12. Again, we see thatMU-GLRT is the best scheme, even though it departs fromthe CRB as K increases. Compared to LS, both MU-GLRTand SU-GLRT provide a remarkable gain.

D. Computational complexity

It is interesting to compare the investigated ranging schemesin terms of computational requirement. From the results inTable I, it follows that for the considered values of N , Lθ,θmax and |C| the complexity of SU-GLRT is approximately 5.5times greater than that involved by CSU. The same factor 5.5arises when comparing the number of flops needed by LS andMU-GLRT. This means that the improved performance of theproposed GLRT-based schemes over the existing alternativesis achieved at the price of an increased computational load.However, the results shown in Fig. 3 indicate that parameterLθ can be reduced from 16 to 12 with only a marginal lossin the estimation accuracy, while the shape of the timingmetric in Fig. 2 reveals that the peak of Ψm(θ) can bereliably detected even when Ψm(θ) is not computed for anyθ ∈ {0, 1, . . . , θmax}. All these facts may help the designer to

3214 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 11, NO. 9, SEPTEMBER 2012

achieve the desired trade-off between computational require-ment and system performance. For example, letting Lθ = 12and evaluating Ψm(θ) only for θ ∈ {0, 2, 4, . . . , θmax} reducesthe complexity of SU-GLRT and MU-GLRT by a factor 2.5without incurring any significant degradation of the estimationaccuracy.

VI. CONCLUSIONS

We have presented a novel approach for initial ranging inOFDMA systems in which the GLRT criterion is applied todecide whether a given code is present or not in the rangingsubchannel. The proposed scheme is fully compliant with theIEEE 802.16 specifications and inherently takes into accountthe multipath distortions introduced by the propagation chan-nel. The timing error and power level of the detected codesare recovered by means of ML arguments after modelingthe MAI as white Gaussian noise. Computer simulations andnumerical analysis indicate that the resulting scheme (SU-GLRT) performs better than conventional IR methods derivedunder the simplifying assumption of a flat-fading channel.

After combining the GLRT approach with the IC principle,we have derived a multiuser ranging scheme (MU-GLRT) withimproved resilience to MAI. Compared to existing multiusermethods, the MU-GLRT exhibits better performance at theprice of a certain increase of the system complexity. However,a judicious design of the algorithm parameters allows one toreduce the computational requirement without incurring anysignificant performance degradation. The choice between SU-GLRT and MU-GLRT depends on the different weigths thatmay be given to various performance indicators, includingestimation accuracy, code detection capability and processingload. While the MU-GLRT is a promising candidate forhighly interfered scenarios, the SU-GLRT may be the favouritescheme in applications where the probability of having twoor more users that simultaneously access the same rangingchannel is relatively low.

APPENDIX A

In this Appendix, we evaluate the expectation of the esti-mates σ2

H1and Pm expressed in (19) and (24), respectively.

In addition, the accuracy of the power estimate P(f)m shown

in (28) is assessed. For simplicity, we assume ideal timinginformation (i.e., θm = θm) and omit the code index m.

Our analysis starts from the observation that computingE{P} and E{σ2

H1} requires the expectation of the quantities

Ψ(θ) and ‖X‖2. For this purpose, we recall that Y = CXand use (9) to obtain

Y = Γ(θ)(H + ξ) (40)

where H = Fh and ξ = ΓH(θ)Cw is statistically equivalentto w. Substituting the above result into (15) and (18) yields

Ψ(θ) = ‖H‖2 + ξHH+HHξ + ξHAξ (41)

with A = F(FHF)−1FH and ‖H‖2 = QP . At this stage,we observe that ξ is a random vector with zero mean andcovariance matrix Cξ = σ2IQ. Hence, using the property

E{ξHAξ} = tr{ACξ} = σ2tr{A} and bearing in mind thattr{A} = tr{(FHF)−1FHF} = tr{IL} = L, produces

E {Ψ(θ)} = QP + Lσ2. (42)

The quantity E{‖X‖2} is computed from the identity

‖X‖2 = ‖H‖2 + ξHH+HHξ + ‖ξ‖2 (43)

after observing that E{‖ξ‖2} = tr{Cξ} = σ2Q. This yields

E{‖X‖2

}= Q(P + σ2). (44)

Finally, taking the expectation of both sides of (19) and (24)and using the results (42) and (44), we get the expressions ofE{Pm} and E{σ2

H1} as given in (25) and (26), respectively.

We now turn our attention to the power estimate P(f)m and

highlight the major steps leading to the evaluation of thecorresponding MSE. Substituting (41) and (43) into (28) andletting B = QA− LIQ, yields

P (f) = P +ξHH+HHξ

Q+

ξHBξ

Q (Q− L)(45)

from which it follows that

MSE(P ) =E{(ξHH+HHξ)2

}Q2

+E{(ξHBξ)2

}Q2(Q− L)2

. (46)

Using standard computations, it is easily found that

E{(ξHH+HHξ)2

}= 2σ2PQ (47)

while E{(ξHBξ)2} = tr{BCξBCξ} [14]. Recalling thatCξ = σ2IQ and observing that B2 = Q(Q− 2L)A+ L2IQ,yields

E{(ξHBξ)2} = σ4tr{B2} = σ4QL (Q − L) . (48)

The expression (29) of MSE(P ) is eventually found aftersubstituting (47) and (48) into (46).

APPENDIX B

In this Appendix, the CRB for the joint estimation of hm,θm, σ2 is computed and subsequently used to derive theCRB to the accuracy of Pm. For notational simplicity, thecode index m is omitted and the real and imaginary partsof h are called hR and hI , respectively. Then, denoting byζ = [hR,hI , θ, σ

2]T the set of unknown parameters, thecomponents of the Fisher information matrix FIM are givenby

[FIM ]i,j = −E

{∂2 ln pH1 (X; ζ)

∂ζi∂ζj

}1 ≤ i, j ≤ 2L+ 2

(49)where ζ� denotes the �th element of ζ and pH1 (X; ζ) is thepdf of X under the hypothesis H1. Substituting (11) into (49)and using standard computations, yields

FIM =2

σ2

⎡⎣ G u 0L

uT ‖MFh‖2 0L

0TL 0T

L Q/(2σ2)

⎤⎦ (50)

SANGUINETTI and MORELLI: AN INITIAL RANGING SCHEME FOR THE IEEE 802.16 OFDMA UPLINK 3215

where M = diag{2πiq/N ; q = 1, 2, . . . , Q} and we havedefined

G =

[ �e{FHF} −m{FHF}m{FHF} �e{FHF}

](51)

and

u =

[ m{FHMFh}−�e{FHMFh}

]. (52)

Letting γ = hHFHM[IQ−F(FHF)−1FH ]MFh, the inverseof FIM is found to be

F−1IM =

σ2

2

⎡⎣

G−1 + (G−1uuTG−1)/γ −G−1u/γ 0L

−uTG−1/γ 1/γ 0L

0TL 0T

L 2σ2/Q

⎤⎦

(53)

from which it follows that CRB(θ) = σ2/(2γ) andCRB(σ2) = σ4/Q. At this stage, we recall that P =‖Fh‖2 /Q and, accordingly, the CRB to the accuracy of Pis given by [14]

CRB(P ) = vTF−1IMv (54)

where

v =

⎡⎣ �e{FHFh}

m{FHFh}02

⎤⎦ (55)

is the gradient of P with respect to ζ. Finally, substituting (53)and (55) into (54) provides CRB(P ) as expressed in (30).

REFERENCES

[1] “IEEE 802.16 2009 - IEEE Standard for local and metropolitan areanetwork part 16: Air Interface for Broadband Wireless Access Systems,”http://standards.ieee.org/getieee802/802.16.html, Tech. Rep., May 2009.

[2] J. Krinock, M. Singh, M. Paff, V. Tien, A. Lonkar, L. Fung, and C.-C. Lee, “Comments on OFDMA ranging scheme described in IEEE802.16ab-01/01r1,” IEEE 802.16abc-01/24, Aug. 2001.

[3] X. Fu and H. Minn, “Initial uplink synchronization and power control(ranging process) for OFDMA systems,” in Proc. 2004 IEEE GlobalCommun. Conf., pp. 3999–4003.

[4] D. H. Lee, “OFDMA uplink ranging for IEEE 802.16e using modifiedgeneralized chirp-like polyphase sequences,” in Proc. 2005 Int. Conf. inCentral Asia on Internet, pp. 1–5.

[5] H. Mahmoud, H. Arslan, and M. Ozdemir, “Initial ranging for WiMAX(802.16e) OFDMA,” in Proc. 2006 Military Commun. Conf., pp. 1–7.

[6] Y. Zhou, Z. Zhang, and X. Zhou, “OFDMA initial ranging for IEEE802.16e based on time-domain and frequency-domain approaches,” inProc. 2006 Int. Conf. on Commun. Techn., pp. 1–5.

[7] X. Zhuang, K. Baum, V. Nangia, and M. Cudak, “Ranging enhancementfor 802.16e OFDMA PHY,” IEEE C802.16e-04/143, June 2004.

[8] X. Fu, Y. Li, and H. Minn, “A new ranging method for OFDMAsystems,” IEEE Trans. Wireless Commun., vol. 6, no. 2, pp. 659–669,Feb. 2007.

[9] M. Morelli, L. Sanguinetti, and H. V. Poor, “A robust ranging schemefor OFDMA-based networks,” IEEE Trans. Commun., vol. 57, no. 8,pp. 2441–2452, Aug. 2009.

[10] L. Sanguinetti, M. Morelli, and H. V. Poor, “An ESPRIT-based approachfor initial ranging in OFDMA systems,” IEEE Trans. Commun., vol. 57,no. 11, pp. 3225–3229, Nov. 2009.

[11] M. Ruan, M. C. Reed, and Z. Shi, “Successive multiuser detection andinterference cancelation for contention based OFDMA ranging channel,”IEEE Trans. Wireless Commun., vol. 9, no. 2, pp. 481–487, Feb. 2010.

[12] C.-L. Lin and S.-L. Su, “A robust ranging detection with MAI cancella-tion for OFDMA systems,” in Proc. 2011 Int. Conf. on Adv. Commun.Technol., pp. 937–941.

[13] K. Fazel and S. Kaiser, Multi-Carrier and Spread Spectrum Systems.Wiley, 2003.

[14] S. M. Kay, Fundamentals of Statistical Signal Processing: EstimationTheory. Prentice Hall, 1993.

[15] M. Morelli, “Timing and frequency synchronization for the uplink of anOFDMA system,” IEEE Trans. Commun., vol. 52, no. 2, pp. 296–306,Feb. 2004.

Luca Sanguinetti (S’04, M’06) received the LaureaTelecommunications Engineer degree (cum laude)and the Ph.D. degree in information engineeringfrom the University of Pisa, Italy, in 2002 and2005, respectively. Since 2005 he has been withthe Department of Information Engineering of theUniversity of Pisa. In 2004, he was a visiting Ph.D.student at the German Aerospace Center (DLR),Oberpfaffenhofen, Germany. During the period June2007 – 2008, he was a postdoctoral associate in theDepartment of Electrical Engineering at Princeton.

During the period June 2010 – Sept. 2010, he was selected for a researchassistantship at the Technische Universitat Munchen. He is currently anAssistant Professor at the Department of Information Engineering of theUniversity of Pisa. His expertise and general interests span the areas ofcommunications and signal processing, estimation and detection theory. Heis currently serving as an Associate Editor for the IEEE TRANSACTIONS ON

WIRELESS COMMUNICATIONS.

Michele Morelli (M’01, SM’07) received the Lau-rea (cum laude) in electrical engineering from theUniversity of Pisa, Italy, in 1991. From 1992 to1995 he was with the Department of InformationEngineering of the University of Pisa, where hereceived the Ph.D. degree in telecommunicationengineering. In 1996 he joined the Italian NationalResearch Council (CNR) where he held the positionof Research Fellow for five years. Since 2002 he hasbeen with the Department of Information Engineer-ing of the University of Pisa, where he is currently

Professor of Digital Transmissions and Telecommunications.He published approximately 60 journal papers and co-authored the book

Multicarrier Techniques for Broadband Wireless Access: A Signal ProcessingPerspective (Imperial College Press, London, 2007). He also contributed tothe John Proakis Telecommunications Encyclopedia with a chapter entitled“Synchronization in digital communication systems.” His research interestsare in the field of digital communications, with emphasis on synchronizationmethods, equalization schemes and precoding techniques.

He is a member of the Communication Theory Committee and was aco-recipient of the “Best Student Paper Award” at the IEEE VehicularTechnology Conference VTC ’06, Fall. He served as a Technical ProgramCommittee (TPC) member for several international conferences and wasthe principal investigator in many research projects funded by internationaltelecommunications companies. He is currently an Editor for the IEEEWIRELESS COMMUNICATIONS LETTERS and served as Associate Editor forthe IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS from 2007 to2011.