Joint turbo equalisation and carrier synchronisation for SC-FDE schemes

EUROPEAN TRANSACTIONS ON TELECOMMUNICATIONSEur. Trans. Telecomms. 2010; 21:131–141Published online 3 February 2010 in Wiley InterScience(www.interscience.wiley.com) DOI: 10.1002/ett.1377

Transmission Systems

Joint turbo equalisation and carrier synchronisation for SC-FDE schemes

Rui Dinis1,2*, Teresa Araujo1,3, Pedro Pedrosa1,4 and Fernando Nunes1,4

1Instituto de Telecomunicacoes, Avenida Rovisco Pais, 1049-001, Lisbon, Portugal2FCT-UNL, Monte da Caparica, Portugal

3ISEP, Porto, Portugal4IST, Lisbon, Portugal

SUMMARY

In this paper we consider the use of single carrier (SC) modulations combined with frequency-domainequalisation (FDE) in future broadband wireless systems. We propose iterative receiver structures with jointequalisation and carrier synchronisation. The proposed receivers can be regarded as modified frequency-domain turbo equalisers where we perform a decision-directed frequency offset estimation within eachiteration of the turbo equaliser.Our performance results show that the proposed receiver structure has good bit error rate (BER), evenwith moderate frequency offsets and in severely time-dispersive channels. Moreover, our receiver has arelatively low implementation complexity, due to its fast Fourier transform (FFT)-based frequency-domainimplementation. Copyright © 2010 John Wiley & Sons, Ltd.

1. INTRODUCTION

Due to an increased demand for wireless services, futuresystems are required to support high quality of serviceat high data rates. For such high data rates, the time-dispersion effects associated to the multipath propagationcan be severe. In this case, conventional time-domainequalisation schemes are not practical. Block transmissiontechniques, with appropriate cyclic extensions and employ-ing frequency-domain equalisation (FDE) techniques, havebeen shown to be suitable for high data rate transmissionover severely time-dispersive channels without requiringcomplex receivers. The most popular modulations basedon this concept are the orthogonal frequency division mul-tiplexing (OFDM) modulations [1]. Block transmissionsingle carrier (SC) modulations combined with FDE (alsodenoted SC-FDE), are an alternative approach based on thisprinciple [2].

Although OFDM has very poor uncoded performance[3], the achievable performances with appropriate chan-nel coding are similar for OFDM and SC/FDE [4, 5].

* Correspondence to: Rui Dinis, Instituto de Telecomunicacoes, Avenida Rovisco Pais, 1049-001, Lisbon, Portugal. E-mail: [email protected]

The overall implementation complexities for SC-FDE andOFDM schemes are similar, although the OFDM receiversare slightly simpler and the transmitters more complex.Moreover, the OFDM signals have larger envelope fluctua-tions which lead to amplification difficulties. Therefore, theOFDM schemes are clearly preferable for the downlink (i.e.the transmission from the base station (BS) to the mobileterminal (MT)) and the SC-FDE schemes are preferable forthe uplink (i.e. the transmission from the MT to the BS).For this reason, a mixed SC/OFDM air interface was pro-posed [4, 6], with an OFDM scheme in the downlink anda SC-FDE scheme in the uplink. In this paper we consideronly the uplink transmission of this type of system, i.e. anSC-FDE approach.

A promising iterative FDE (IFDE) technique for SC-FDE, denoted iterative block-decision feedback equaliser(IB-DFE), was proposed in Reference [7]. This techniquewas later extended to diversity scenarios [5] and layeredspace-time schemes [8]. These IFDE receivers can beregarded as iterative DFE receivers with the feedforwardand the feedback operations implemented in the frequency

Received 17 September 2007Revised 29 January 2009

Copyright © 2010 John Wiley & Sons, Ltd. Accepted 5 June 2009

132 R. DINIS ET AL.

domain and offer much better performances than the non-iterative methods [5, 7, 8]. Within these IFDE receiversthe equalisation and channel decoding procedures are per-formed separately (i.e. the feedback loop uses the equaliseroutputs instead of the channel decoder outputs). However,it is known that higher performance gains can be achievedif these procedures are performed jointly. An effectiveway of achieving this is by employing the so-called turboequalisation schemes where the equalisation and decodingprocedures are repeated, in an iterative way, with some softinformation being passed between them [9]. Although ini-tially proposed for time-domain receivers, turbo equalisersalso allow frequency-domain implementations [10, 11]‡.

In order to maintain high power and spectral efficiencies,the cyclic prefix, which is longer than the overall channelimpulse response length, should be a small fraction of theblock duration. Therefore, we usually need large blocks forseverely time-dispersive channels, with hundreds or eventhousands of symbols. Typically, the frequency errors can-not exceed a small fraction of the inverse of the blockduration. This means that we have higher sensitivity tofrequency errors for larger blocks, making accurate car-rier synchronisation mandatory. One source of frequencyerrors is the frequency mismatch between the oscillatorsat the transmitter and receiver. Another possible source offrequency errors is the Doppler frequency shift caused byrelative motion between the transmitter and the receiver.

The carrier synchronisation is usually performed in two(or more) stages. Typically a coarse synchronisation, withmoderate accuracy and large acquisition band, is followedby a fine synchronisation, with high accuracy and smallacquisition band (Reference [12], and references therein).The fine carrier synchronisation can be performed afterthe equalisation procedure, namely using decision-directedestimation. This is especially effective for SC modulationssince the frequency offsets produce a progressive constella-tion rotation and the equalised signal resembles the receivedsignal in flat-fading channels [13, 14].

An alternative to a decision-directed carrier frequencyoffset (CFO) estimator, is using known sequences withgood correlation properties. Although yielding more accu-rate estimates, this solution requires extra block overheadthus reducing the bandwidth efficiency [15]. Exploring thesimilarities between the synchronisation requirements ofboth transmission schemes, it is possible to use, in SC-FDEschemes, the frame structure of training symbols originally

‡ The IFDEs of References [5, 7, 8] (or IB-DFEs) can be regarded as specialtypes of frequency-domain turbo equalisers with reduced complexity, sincethe channel decoder is not required in the feedback loop.

designed for synchronisation purposes in OFDM schemes.In Reference [16], Moose proposed a maximum-likelihood(ML) CFO estimator, based on the use of two identical andconsecutive symbols, with a frequency acquisition range±1/(2T ), where T is the ‘useful’ symbol duration. Thisresult was later extended in Reference [17], which uses alsotwo symbols; the first estimates the fractional part of theCFO (|�f | < 1/T ), whereas the second symbol resolvesthe frequency ambiguity inherent in the first symbol, i.e.it estimates the integer part of the CFO (�f multiple of1/T ). Morelli and Mengali proposed in Reference [18] analgorithm exploiting a training symbol with L > 2 identi-cal parts. Its estimation range is ±L/2 times the subcarrierspacing and its accuracy is slightly superior to that of theSchmidl and Cox method. Its main advantage is that it needsjust one training symbol while the Schmidl and Cox methodneeds two symbols. In Reference [19], Morelli and Men-gali improved this estimation technique using algorithmsthat achieve the Cramer–Rao bound at the cost of increasedcomplexity. A comparison between different designs for theframe structure of the pilot symbols is made in Reference[20]. Besides pilot tone-aided algorithms, other techniques,like cyclic-prefix estimation, may be applied to track thefrequency offset (see, for instance, Reference [21]).

In this paper we consider an SC-FDE block transmissionin the presence of residual frequency errors. We propose areceiver structure with joint equalisation and carrier syn-chronisation. We consider iterative receivers that can beregarded as modified frequency-domain turbo equaliserswhere we perform decision-directed frequency offset esti-mation within each iteration of the turbo equaliser.

This paper is organised as follows. The basic iterativeFDE receivers are described in Section 2. Section 3 presentsthe modified receivers with joint equalisation and carriersynchronisation. A set of performance results is presentedin Section 4 and Section 5 is concerned with the conclusions.

2. ITERATIVE FDE RECEIVERS

2.1. Receiver structure

For the sake of simplicity, we will assume in this sectionthat there is perfect carrier synchronisation. Figure 1(A)presents the iterative frequency-domain receiver structureconsidered in this paper. It is assumed that we have L

receiver antennas, i.e. we have L-order space diversity.The received time-domain block associated to the lthantenna, {y(l)

n ; n = 0, 1, . . . , N − 1}, is passed to the fre-quency domain by a DFT operation, leading to the block

Copyright © 2010 John Wiley & Sons, Ltd. Eur. Trans. Telecomms. 2010; 21:131–141DOI: 10.1002/ett

JOINT TURBO EQUALISATION AND CARRIER SYNCHRONISATION 133

Figure 1. (A) IFDE receiver with L-branch space diversity and (B) equivalent receiver structure, since F(l,i)k = H

(l)∗k C

(i)k , l = 1, 2, . . . , L.

{Y (l)k ; k = 0, 1, . . . , N − 1}, with

Y(l)k = SkH

(l)k + N

(l)k (1)

where H(l)k and N

(l)k denote the channel transfer function

and the channel noise, respectively, for the kth subchannelof the lth diversity branch. The block of frequency-domainsymbols {Sk; k = 0, 1, . . . , N − 1} is the DFT of the trans-mitted time-domain block, {sn; n = 0, 1, . . . , N − 1}, withsn denoting the nth data symbol to be transmitted, selectedfrom a given constellation (e.g. a QAM or a PSK constel-lation).

For a given iteration i, the frequency-domain samples atthe output of the FDE are given by§

S(i)k =

L∑l=1

F(l,i)k Y

(l)k − B

(i)k S

(i−1)k (2)

§ Our IFDE receiver is slightly different from the IB-DFE receivers ofReferences [5] and [7], since there the correlation factor is incorporated inthe feedback coefficients.

where {F (l,i)k ; k = 0, 1, . . . , N − 1} (l = 1, 2, . . . , L) are

the feedforward coefficients and {B(i)k ; k = 0, 1, . . . ,

N − 1} are the feedback coefficients. {S(i−1)k ; k = 0, 1,

. . . , N − 1} denotes the DFT of the block of time-domain average symbol values associated to the previ-ous iteration, {s(i−1)

n ; n = 0, 1, . . . , N − 1}. The methodfor obtaining these average values is described inSection 2.2.

It can be shown that the optimum feedback coefficientsare [5, 7]‖

B(i)k =

L∑l=1

F(l,i)k H

(l)k − 1 (3)

and the feedforward coefficients are given by

F(l,i)k = F

(l,i)k

γ (i) (4)

‖ Contrarily to References [5] and [7], we are considering normalised

equalisers, i.e. 1N

∑N−1k=0

∑L

l=1 F(l,i)k

H(l)k

= 1.


134 R. DINIS ET AL.

with

F(l,i)k = H

(l)∗k

α + (1 − (ρ(i−1))2)∑L

l=1 |H (l)k |2

(5)

where α = E[|N(l)k |2]/E[|Sk|2]

In Equation (4) we have

γ (i) = 1

N

N−1∑k=0

L∑l=1

F(l,i)k H

(l)k (6)

and the correlation factor ρ(i−1) in Equation (5) is defined as

ρ(i−1) = E[s(i−1)n s∗n]

E[|sn|2](7)

where the block {s(i−1)n ; n = 0, 1, . . . , N − 1} denotes the

data estimates associated to the previous iteration, i.e.the hard-decisions associated to the time-domain blockat the output of the FDE, {s(i)

n ; n = 0, 1, . . . , N − 1} =IDFT {S(i)

k ; k = 0, 1, . . . , N − 1}.It should be noted that

F(l,i)k = H

(l)∗k C

(i)k (8)

with

C(i)k = 1/γ (i)

α + (1 − (ρ(i−1))2)∑L

l=1 |H (l)k |2

(9)

Therefore, the receiver structure of Figure 1(A) is equiv-alent to the receiver structure of Figure 1(B), where onlyC

(i)k changes with the iteration order.

2.2. Computation of the average values and thecorrelation factor

Clearly, ρ(i) can be regarded as the blockwise reliability ofthe estimates {s(i)

n ; n = 0, 1, . . . , N − 1}. This means thatwe can define the ‘blockwise average’ symbols as {s(i)

n =ρ(i)s(i)

n ; n = 0, 1, . . . , N − 1}.In order to improve the performance of the receiver, we

may replace the ‘blockwise averages’ by ‘symbol averages’.In addition, this provides an expedite way of computing theblockwise reliabilities.

Assume that the transmitted symbols are selected from aQPSK constellation under a Gray mapping rule (the gener-alisation to other cases is straightforward). We define sn =±1 ± j = sIn + js

Qn , with sIn = Re{sn} = ±1 and s

Qn =

Im{sn} = ±1, n = 0, 1, . . . , N − 1 (similar definitions canbe made for sn = sIn + js

Qn , sn = sIn + js

Qn and sn = sIn +

jsQn ).The loglikelihood ratios (LLRs) of the ‘in-phase bit’ and

the ‘quadrature bit’, associated to sI(i)n and s

Q(i)n , respec-

tively, are given by¶

LI(i)n = 2

σ2i

sI(i)n (10)

and

LQ(i)n = 2

σ2i

sQ(i)n (11)

respectively, where

σ2i = 1

2E[|sn − s(i)

n |2] ≈ 1

2N

N−1∑n=0

E[|s(i)n − s(i)

n |2] (12)

Under a Gaussian assumption, it can be shown that themean value of sn is

s(i)n = tanh

(LI(i)

n

2

)+ j tanh

(L

Q(i)n

2

)(13)

Clearly, the hard decisions sI(i)n = ±1 and s

Q(i)n = ±1 are

defined according to the signs of LI(i)n and L

Q(i)n , respec-

tively. Therefore, s(i)n = ρI(i)

n sI(i)n + jρ

Q(i)n s

Q(i)n , where

ρI(i)n = E[sIns

I(i)n ]

E[|sIn|2]

= 1 − 2 Prob(sI(i)n �= sIn) = tanh

( |LI(i)n |2

)(14)

and

ρQ(i)n = E[sQn s

Q(i)n ]

E[|sQ(i)n |2]

= 1 − 2 Prob(sQ(i)n �= sQn ) = tanh

(|LQ(i)

n |2

)(15)

Quantities ρI(i)n and ρ

Q(i)n can be regarded as the reliabili-

ties associated to the ‘in-phase’ and ‘quadrature’ bits of thenth symbol (naturally, 0 � ρI(i)

n � 1 and 0 � ρQ(i)n � 1).

¶ Once again, it is assumed that γ (i) = 1, i.e. we have a normalised FDE.



For the first iteration, ρI(i)n = ρ

Q(i)n = 0 and s(i)

n = 0; aftersome iterations and/or when the signal-to-noise ratio(SNR) is high, typically ρI(i)

n ≈ 1 and ρQ(i)n ≈ 1, leading

to sn ≈ s(i)n . The feedforward coefficients are still obtained

from Equation (4) to (5), but with the blockwise reliabilitygiven by

ρ(i) = 1

N

N−1∑n=0

E[s∗ns(i)n ]

E[|sn|2]= 1

2N

N−1∑n=0

(ρI(i)n + ρQ(i)

n ) (16)

The receiver with ‘blockwise reliabilities’, denoted in thefollowing as iterative FDE with hard decisions (IFDE-HD),and the receiver with ‘symbol reliabilities’, denoted in thefollowing as IFDE with soft decisions (IFDE-SD), employthe same feedforward coefficients; however, in the former,the feedback loop uses the ‘hard-decisions’ on each datablock, weighted by a common reliability factor, while in thelatter, the reliability factor changes from symbol to symbol(in fact, the reliability factor is different in the real and imag-inary component of each symbol). The IB-DFE schemeconsidered in References [5, 7] corresponds to the IFDE-HD, since ‘blockwise reliabilities’ were considered there.

2.3. Use of channel decoder outputs in thefeedback loop

We can define a frequency-domain turbo equaliser thatemploys the channel decoder outputs instead of the uncoded‘soft decisions’ in the feedback loop. The receiver structure,that will be denoted Turbo FDE, is similar to the IFDE-SD, but with a soft-in, soft-out (SISO) channel decoderemployed in the feedback loop. The SISO block, that canbe implemented as defined in Reference [22], providesthe LLRs of both the ‘information bits’ and the ‘codedbits’. The input of the SISO block are LLRs of the ‘codedbits’ at the FDE output, given by Equations (10) and (11).Once again, the feedforward coefficients are obtained fromEquation (4) to (5), with the blockwise reliability given byEquation (16).

3. JOINT EQUALISATION AND CARRIERSYNCHRONISATION

3.1. Receiver structure

Let us assume now that there is a residual frequency offset�f between the transmitter and the receiver local oscilla-tors. In this case, the received time-domain block associated

to the lth diversity branch is {y′(l)n ; n = 0, 1, . . . , N − 1} and

the corresponding frequency-domain blocks are {Y ′(l)k ; k =

0, 1, . . . , N − 1}, with

Y ′k

(l) = S′kH

(l)k + N

(l)k (17)

where the block of frequency-domain symbols{S′

k; k = 0, 1, . . . , N − 1} is the DFT of the effec-tively transmitted block of time-domain data symbols,{s′n; n = 0, 1, . . . , N − 1}, with

s′n = sn exp

(j2π

�fnT

N

)(18)

For the sake of simplicity, it is assumed that the resid-ual frequency offset results exclusively from the transmitter(i.e. the MT, since we are considering the uplink); we alsoassume that the phase rotation is 0 for n = 0.

Clearly, the residual frequency offset leads to a progres-sive phase rotation of the time-domain symbols at the FDEoutput. Since this phase rotation might lead to significantperformance degradation, we will modify our receiver soas to estimate and compensate the phase rotation associatedto the frequency offset. Figure 2 presents a receiver withjoint equalisation and frequency offset estimation andcompensation. This receiver is based on the basic FDEdescribed in Figure 1(A) (the extension to the receiverformat of Figure 1(B) is straightforward), and the receiverparameters are obtained in a similar way. However,for each iteration we perform the frequency offset esti-mation and compensation before the detection procedure.Once again, we can have either hard decisions or softdecisions (IFDE-HD or IFDE-SD, respectively); we canalso use the channel decoder outputs in the feedback loop(Turbo FDE). For each iteration, the frequency offset isestimated as described in the following subsection.

3.2. Frequency offset estimation

Let us assume a slowly varying scenario. In this case, theoverall channel at the output of the FDE can be regarded asa stationary flat fading channel (at least for the duration ofa given block). Therefore, we can describe the frequencyoffset estimation procedure assuming an ideal Gaussianchannel.

In the presence of a frequency offset �f , the receivedblock is {y′

n = s′n + νn; n = 0, 1, . . . , N − 1}, where s′n isgiven by Equation (18) and νn is the Gaussian noise compo-nent, with E[νn] = 0 and E[|νn|2] = 2σ2 (σ2 denotes thevariance of both real and imaginary parts of νn).


136 R. DINIS ET AL.

Figure 2. Proposed receiver for joint equalisation and carrier synchronisation.

If the transmitted symbols are known we can estimate thefrequency offset in the following way:

�f = N

2πMTarg {ξ} (19)

with

ξ =N−M−1∑

n=0

y′n+My′∗

n s∗n+Msn = ξ + ε (20)

for an appropriate M, where ξ = E[ξ] and ε is an error term.Clearly, s∗n+M sn is used to wipe out the phase modulation iny′n+My′∗

n . For the sake of simplicity, we are assuming |sn| =s, i.e. we are employing constant amplitude constellations(e.g. a PSK constellation), thus yielding

y′n+My′∗

n s∗n+Msn = s4 exp

(j2π

�fMT

N

)+ εn (21)

where the channel noise contribution is allotted to εn; sinceE[εn] = 0,

ξ = E[|sn|2]2(N − M) exp

(j2π

�fMT

N

)(22)

It can be shown that, for high SNR, the frequency offsetestimate given by Equation (19) is unbiased, with variance

σ2�f = E[|�f − �f |2] =

(N

2πMT

)2 σ2εQ

|ξ|2 (23)

with σ2εQ denoting the variance of the quadrature component

of εn (for �f = 0 this corresponds to the variance of the

imaginary part of the noise). It is shown in Appendix that,for high SNR, this variance is approximately given by

σ2εQ ≈

{s8(N − M)/SNR, M > N/2

s8M/SNR, M < N/2(24)

with

SNR = E[|sn|2]

E[|νn|2](25)

This means that

σ2�f ≈ 1

SNR(2πT )2

N2

M2(N − M)(26)

for M > N/2 and

σ2�f ≈ 1

SNR(2πT )2

N2

M(N − M)2 (27)

for M < N/2 (thus, σ2�f takes the same values for M = M0

and M = N − M0).It can easily be shown that there are two optimum values

for M, M = 2N/3, and M = N/3, both corresponding to

σ2�f ≈ 1

SNR(2πT )2

27

4N(28)

Figure 3 shows the impact of M/N on σ�f T whenN = 512. Clearly, we have almost the same performancefor a wide range of values of M/N, provided that M is nottoo close to N nor too small. For moderate-to-high SNR val-ues, the theoretical values of σ�f T , given by Equation (28),are close to the simulated ones; for low SNR the theoretical



Figure 3. Impact of M/N on σ�f T .

values are slightly better than the simulated ones (the differ-ences are higher for M < N/2, especially for smaller valuesof M). Moreover, the optimum value of M is independentof the actual SNR.

Since the nth data symbol is rotated by nθ, with θ =2π�fT/N, the corresponding bit error rate (BER) for aQPSK constellation and an ideal Gaussian channel is**

Pb,n = 1

2Q

(√2Eb

N0(cos(nθ) − sin(nθ))

)

+ 1

2Q

(√2Eb

N0(cos(nθ) + sin(nθ))

)(29)

(2Eb/N0 = SNR) and the overall BER is the average ofPb,n over n. Figure 4 shows the impact of the frequencyoffset on the BER. Clearly, we can have an almost idealperformance if �fT � 0.025; with �fT = 0.05 we haveabout 2 dB of degradation at BER = 10−4. This means thatthe accuracy of the proposed frequency offset estimationmethod is more than enough for typical SNR and moderate-to-long blocks.

Naturally, the frequency-offset estimation methoddescribed above requires the knowledge of the transmit-ted symbols sn. Since they are not known at the receiver,we can replace the symbol estimates by their average val-ues from the previous iteration, i.e. for the ith iteration (20)

** Naturally, this means that there is no compensation of the frequencyoffset.

Figure 4. BER for QPSK with several normalised frequency off-sets �fT .

takes the form

ξ =N−M−1∑

n=0

y′n+My′∗

n s(i−1)∗n+M s(i−1)

n (30)

4. PERFORMANCE RESULTS

In this section we present a set of performance results con-cerning the joint equalisation and carrier synchronisationtechnique proposed in this paper. We consider SC-FDEmodulations with blocks of N = 512 ‘useful’ modulationsymbols (corresponding to a duration of 4 �s), plus anappropriate cyclic prefix. The modulation symbols belongto a QPSK constellation and are selected from the trans-mitted data according to a Gray mapping rule. The channelencoder is the well-known rate-1/2 64-state convolutionalcode†† with generators 1 + D2 + D3 + D5 + D6 and 1 +D + D2 + D3 + D6. Additionally, the coded bits are inter-leaved before being mapped into the constellation points.We consider linear power amplification and perfect chan-nel estimation. The propagation channel is characterised bythe power delay profile type C for HIPERLAN/2 (HIghPERformance Local Area Network) [23], with uncorre-lated Rayleigh fading on the different paths (similar resultscould be obtained for other severely time-dispersive chan-

†† It should be pointed out that more powerful coding schemes, such asthe well-known turbo codes, could be employed. We selected a convolu-tional code due to its good performance with small blocks, not requiringinterblock interleaving.


138 R. DINIS ET AL.

Figure 5. Uncoded BER performance for IFDE-HD and IFDE-SD, together with the MFB.

nel models with rich multipath propagation). The channelsassociated to different diversity branches are uncorrelated.

Let us first consider the performance of the differentiterative FDE receivers under perfect carrier synchroni-sation. Figure 5 shows uncoded BER performances forIFDE-HD and IFDE-SD, without diversity (L = 1) andwith two-branch space diversity (L = 2). We also includethe matched filter bound (MFB) performance, defined as [5]

Pb,MFB = E

Q

√√√√2Eb

N0

1

N

N−1∑k=0

L∑l=1

|H (l)k |2

(31)

Figure 6. Coded BER performance for IFDE-HD and IFDE-SD.

From this figure, we can observe that there is a signif-icant gain associated to the iterative procedure, especiallywithout diversity (L = 1). Moreover, the asymptotic per-formances can be very close to the MFB after just a fewiterations. Notice that the performances with hard decisions(IFDE-HD) and soft decisions (IFDE-SD) are very simi-lar, especially with diversity (L = 2). Figure 6 shows thecoded BER performances for IFDE-SD and Turbo FDE.Clearly, the performance of the Turbo FDE is better, withgains above 1 dB.

Let us consider now the impact of frequency offsets. Forthe sake of simplicity, it is assumed that the linear phase

Figure 7. Evolution of E[�f − �f ] for IFDE-SD, when L = 1and �fT = 0.1 or 0.15.

Figure 8. Uncoded BER performance for IFDE-SD, when L = 1and �fT = 0.05.



rotation associated to the frequency offset starts from zero,as in Equation (18).

Figure 7 presents the evolution of E[�f − �f ] forL = 1 and an IFDE-SD, when �fT = 0.05 or 0.1 (similarperformances were obtained for IFDE-HD; for L = 2we could obtain similar conclusions, since the curves areessentially shifted to the left). Clearly, the frequency offsetestimates are biased, although the bias decreases withthe number of iterations and with Eb/N0. This bias is aconsequence of the decision errors before estimating �f .Figures 8 and 9 present the corresponding uncoded BER

Figure 9. Uncoded BER performance for IFDE-SD, when L = 1and �fT = 0.1.

Figure 10. Coded BER performance for IFDE-SD, when L = 1and �fT = 0.05 or 0.1.

performances. For the sake of comparison, we also includeresults without frequency offset estimation and withperfect frequency offset estimation. Clearly, the IFDE-SDhas good asymptotic performances when �fT = 0.05,approaching the performances with perfect frequencyoffset estimation after just a few iterations; for �fT = 0.1we have a degradation of about 1 dB relative to the casewith perfect carrier synchronisation. Regarding the codedBER, depicted in Figure 10, the difference relative to thecase with perfect carrier synchronisation is less than 0.5 dBand about 2 dB for �fT = 0.05 and 0.1, respectively.

Figure 11. Evolution of E[�f − �f ] for IFDE-SD, when L = 1and �fT = 0.1 or 0.15.

Figure 12. Coded BER performance for the Turbo FDE, whenL = 1 and �fT = 0, 0.1 or 0.15.


140 R. DINIS ET AL.

Let us consider now the Turbo FDE, with coded feedbackusing a SISO decoder, implemented through the Max-Log-MAP approach [22]. Figure 11 presents the evolution ofE[�f − �f ] for a Turbo FDE with L = 1, when �fT =0.1 or 0.15 (i.e. higher frequency offsets). Compared withthe results of Figure 11, the bias of the frequency offsetestimates are much lower at low SNR. The correspondingBER is depicted in Figure 12. Clearly, the Turbo FDE cancope with larger frequency offsets.

5. CONCLUSIONS

In this paper we considered the use of SC-FDE with jointequalisation and frequency offset estimation. We have aniterative receiver that estimates the residual frequency off-set using estimates of the transmitted symbols for eachiteration.

Our results shown that the proposed receiver struc-tures have excellent performances, allowing good BER,even with moderate frequency offsets and severely time-dispersive channels.

APPENDIX

Without loss of generality, let us assume that there is nofrequency error. For the sake of simplicity we will alsoassume that sn is constant and real (i.e. sn = s). In this case,yn = s + νn and

ymy∗ns

∗msn = s4 + s3(ν∗

n + νm) + s2ν∗nνm

≈ s4 + s3(ν∗n + νm) (32)

where the approximation is valid for moderate and highSNR values. If we write νn (and νm) as νn = νI

n + jνQn ,

with νIn = Re{νn} and ν

Qn = Im{νn}, then

ymy∗ns

∗msn ≈ s4 + s3(νI

n − jνQn + νI

m + jνQm ) (33)

This means that the variance of the real and imaginaryparts of the noise component in Equation (33) are approxi-mately given by 2s6σ2.

IfM > N/2 the different noise components in the sum areuncorrelated and, since we have (N − M) terms, the vari-ance of the real and imaginary parts of ξ is approximatelygiven by

2(N − M)σ2s6 = s8 N − M

SNR(34)

with SNR = E[|sn|2]/E[|νn|2] = s2/E[|νn|2].When M < N/2, ξ can be written as

ξ =M−1∑n=0

∑l

yn+M+lMy∗n+lMs2 (35)

where l ranges from 0 to �N/M� if 0 � n � N Mod M andfrom 0 to �N/M� − 1 otherwise (�x� denotes ‘larger inte-ger not higher than x’ and x Mod y denotes the ‘reminderof the division between x and y’). Although the M termsassociated to the first sum are uncorrelated, this is not truefor the terms associated to the second sum. In fact, the noisecomponent associated to

t∑l=0

yn+M+lMy∗n+lM (36)

is approximately given by

t∑l=0

s(νIn+lM − jν

Qn+lM + νI

n+lM+M + jνQn+lM+M)

= s

(νIn + νI

n+tM+M + 2t∑

l=1

νIn+lM − jνQ

n + jνQn+tM+M

)(37)

This means that the noise quadrature component of ξ hasvariance

2Mσ2s6 = s8 M

SNR(38)

ACKNOWLEDGEMENT

This work was partially supported by Fundacao para aCiencia e Tecnologia (FCT; pluriannual funding, U-BOATproject PTDC/EEA-TEL/67066/2006 and PhD grantsSFRH/BD/29682/2006 and SFRH/BD/40265/2007). Part ofthis research was published in IEEE GLOBECOM’04.

REFERENCES

1. Cimini L, Jr. Analysis and simulation of a digital mobile channel usingorthogonal frequency division multiplexing. IEEE Transactions onCommunications 1985; 33: 400–411.



2. Sari H, Karam G, Jeanclaude I. An analysis of orthogonal frequency-division multiplexing for mobile radio applications. In Proceedings ofIEEE Vehicle Technology Conference , VTC’94, Stockholm, Sweden,June 1994; 1635–1639.

3. Gusmao A, Dinis R, Esteves N. On frequency-domain equalization anddiversity combining for broadband wireless communications. IEEETransactions on Communications 2003; 51: 1029–1033.

4. Gusmao A, Dinis R, Conceicao J, Esteves N. Comparison of twomodulation choices for broadband wireless communications. In Pro-ceedings of IEEE Vehicle Technology Conference, VTC’2000 (Spring),Tokyo, Japan, May 2000; 1300–1305.

5. Dinis R, Gusmao A, Esteves N. On broadband block transmissionover strongly frequency-selective fading channels. In Wireless 2003,Calgary, Canada, July 2003.

6. Falconer D, Ariyavisitakul S, Benyamin-Seeyar A, Eidson B. Fre-quency domain equalization for single-carrier broadband wirelesssystems. IEEE Communications Magazine 2002; 4(4): 58–66.

7. Benvenuto N, Tomasin S. Block Iterative DFE for Single CarrierModulation. IEE Electronic Letters 2002; 39(19): 1144–1145.

8. Dinis R, Kalbasi R, Falconer D, Banihashemi A. Iterative layeredspace-time receivers for single-carrier transmission over severe time-dispersive channels. IEEE Communications Letters 2004; 8(9): 579–581.

9. Tuchler M, Koetter R, Singer A. Turbo equalization: principles andnew results. IEEE Transactions on Communications 2002; 50: 754–767.

10. Tuchler M, Hagenauer J. Turbo equalization using frequency domainequalizers. Allerton Conference, October 2000.

11. Tuchler M, Hagenauer J. Linear time and frequency domain turboequalization. IEEE VTC’01 (Fall), Atlantic City, NJ, USA, October2001; 2773–2777.

12. Meyr H, Moeneclaey M, Fetchel S. Digital Communication Receivers(2nd edn), Wiley Series on Telecommunications and Signal Process-ing. Wiley: New York, 1998.

AUTHORS’ BIOGRAPHIES

Rui Dinis was an IEEE Student in 1996 and an IEEE Member in 2000. He received the PhD degree from Instituto Superior Técnico,Technical University of Lisbon, Portugal, in 2001. From 2001 to 2008 he was Professor at IST. Since 2008 he is a professor at Faculdadede Ciências e Tecnologia da Universidade Nova de Lisboa. He was a member of the research centre Centro de Análise e Processamentode Sinais from 1992 to 2005; from 2006 to 2008 he was a member of the research centre Instituto de Sistemas e Robótica. Since 2008 heis a researcher at Instituto de Telecomunicações in Lisbon. He has been involved in several research projects in the broadband wirelesscommunications area. His main research interests include modulation, equalisation, and channel coding.

Teresa Araújo received the Licenciatura degree in Applied Maths/Computer Science from University of Porto, Porto, Portugal and theMSc in Electrical and Computer Engineering from Instituto Superior Técnico, Technical University of Lisbon, Portugal, in 1991 and2004, respectively. She is a teacher at the Math Department from Instituto Superior de Engenharia do Porto, Porto since 1993. From2006 to 2008 she did research at the Signal and Image Processing Group from Institute of Systems and Robotics and in 2009 she joinedthe research centre Instituto de Telecomunicações in Lisbon. She is pursuing a PhD degree in Electrical and Computer Engineering atInstituto Superior Técnico. Her research interests are in the area of digital communications, with emphasis on multicarrier transmissionand nonlinear distortion of digital signals.

Pedro Pedrosa received the MSc degree in electrical and computer engineering from Instituto Superior Técnico, Technical Universityof Lisbon, Portugal, in 2007. Currently he is pursuing the PhD degree in Electrical and Computer Engineering, also in IST. From2007 to 2008 he was a student member at Instituto de Sistemas e Robótica and since 2008 he is a student member at Instituto deTelecomunicações in Lisbon. His general field of interest is on digital wireless communications and his current research is focused onfrequency-domain equalisation schemes.

Fernando D. Nunes received his EE, MSc, and PhD degrees in Electrical Engineering all from the Instituto Superior Técnico, TechnicalUniversity of Lisbon, Portugal. He is currently an assistant professor at the Department of Electrical and Computer Engineering, InstitutoSuperior Técnico, and researcher with the Communication Theory and Pattern Recognition Group at the Instituto de Telecomunicaçõesin Lisbon. His current research interests include communication theory and signal processing in mobile communications and radio-navigation systems.

13. Olmos J, Agusti R, Casadevall F. Decision feedback equalizationand carrier recovery in 140Mbit QAM digital radio systems. In Pro-ceedings of IEEE ICC’88, Philadelphia, PA, USA, June 1988; 1336–1342.

14. Maryan S, Falconer D. Joint turbo frequency domain equalization andcarrier synchronization. IEEE Transactions on Wireless Communica-tions, January 2008.

15. Czylwik A. Low overhead pilot-aided synchronization for single car-rier modulation with frequency domain equalization. In Proceedingsof IEEE GLOBECOM’98, Sydney, Australia, November 1998; 2068–2073.

16. Moose P. A technique for orthogonal frequency division multiplexingfrequency correction. IEEE Transactions on Communications 1994;42: 2908–2914.

17. Schmidl T, Cox D. Robust frequency and timing synchronizationfor OFDM. IEEE Transactions on Communications 1997; 45: 1613–1621.

18. Morelli M, Mengali U. An improved frequency offset estimatorfor OFDM applications. IEEE Communications Letters 1999; 3:75–77.

19. Morelli M, Mengali U. Carrier-frequency estimation for transmissionsover selective channels. IEEE Transactions on Communications 2000;48: 1580–1589.

20. Reinhardt S, Weigel R. Pilot aided timing synchronization for SC-FDE and OFDM: a comparison. In Proceedings of IEEE ISCIT’04,Sapporo, Japan, October 2004; 628-633.

21. Chen H, Pottie G. A comparison of frequency offset tracking algo-rithms for OFDM. In Proceedings of IEEE Globecom’03, SanFrancisco, CA, December 2003; 1069–1073.

22. Vucetic B, Yuan J. Turbo Codes: Principles and Applications. KluwerAcademic Publisher: Boston, 2002.

23. ETSI. Channel models for HIPERLAN/2 in different indoor scenarios.ETSI EP BRAN 3ERI085B, March 1998; 1–8.


Documents

Joint turbo equalisation and carrier synchronisation for SC-FDE schemes