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Signal Processing

Signal Processing 105 (2014) 30–42

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eduardoantonio

journal homepage: www.elsevier.com/locate/sigpro

Robust differential modulations for asynchronouscooperative systems$

Alfonso Cano, Eduardo Morgado, Javier Ramos, Antonio J. Caamaño n

Department of Signal Theory and Communications, Universidad Rey Juan Carlos, Camino del Molino s/n,28943 Fuenlabrada, Madrid, Spain

a r t i c l e i n f o

Article history:Received 2 December 2013Received in revised form9 May 2014Accepted 18 May 2014Available online 26 May 2014

Keywords:Cooperative communicationsCarrier frequency offsetSynchronizationOFDMDifferential modulationDecode-and-forward

x.doi.org/10.1016/j.sigpro.2014.05.02384/& 2014 Elsevier B.V. All rights reserved.

work was supported by the Spanish Minion (MICINN) through project TEC2010-192hor was also supported by the Spanish Scietion (FECYT) Postdoctoral Grant (2007-0790)esponding author. Tel.: þ34914887248; faxail addresses: alfon.cano@gmail.com (A. Cano.morgado@urjc.es (E. Morgado), javier.ramo.caamano@urjc.es (A.J. Caamaño).

a b s t r a c t

Accurate channel state information and time/frequency synchronization are challengingto acquire in mobile ad hoc cooperative set-ups, where resource-constrained relayingterminals rapidly (dis)join cooperation. This paper derives low-complexity differentialmodulations and cooperative transmission schemes so that detection at the receiver isfeasible without channel knowledge or synchronization information. A simple distributeddoubly-differential (DD) time-division multiplexing (TDM) scheme using diagonal space–time (ST) unitary codes is derived to bypass multiple carrier frequency offsets (CFOs) andcollect space diversity. Precoded differential orthogonal frequency division multiplexing(OFDM) transmissions are derived to additionally bypass timing offsets and collectmultipath diversity using either DD diagonal or single-differential (SD) orthogonal STblock codes (OSTBCs). DD diagonal mappings suffer from considerable coding gain andsignal-to-noise ratio (SNR) loss due to unitary constellation constraints and noiseenhancement at the receiver. SD-OSTBCs achieve higher performance with reduceddecoding complexity, at the cost of reduced CFO mitigation range. Simple and robustnon-coherent selective and adaptive transmissions are also included, showing that fullspace and multipath diversity can still be achieved even with decoding errors at relays.Simulations corroborate theoretical claims.

& 2014 Elsevier B.V. All rights reserved.

1. Introduction

Cooperative schemes using distributed single-antennarelays have been shown to be an effective alternative to ena-ble spatial diversity along with resilience against shadowing

istry of Science and63. The work of thence and Technology.: þ34914887500.),s@urjc.es (J. Ramos),

and coverage enhancement [1,2]. These schemes typicallyoperate under the assumption that cooperating terminals areperfectly synchronized in both time and frequency and thereceiver has exact channel state information from all coop-erating paths. However, this might be difficult to guaranteein uncoordinated mobile ad hoc set-ups, whereby coherencetime is shorter and resource-limited terminals are unable toefficiently mitigate local oscillator drifts [3,4]. Other schemesoperate with non-synchronized terminals but under theassumption of both channel and delay state informationfrom all cooperating paths in the destination node [5]. Without perfect synchronization or accurate channel knowledge,existing coherent cooperative schemes suffer from consider-able performance degradation. This is particularly critical for

A. Cano et al. / Signal Processing 105 (2014) 30–42 31

schemes using distributed orthogonal space–time block codes(OSTBCs) in which all relays transmit simultaneously, requir-ing accurate synchronization at a symbol level [2].

As already recognized in [4,6], orthogonal frequency divi-sion multiplexing (OFDM) transmissions can effectively com-bat time synchronization mismatches by treating them asmultipath and increasing accordingly the cyclic prefix(CP) length. Distributed version of co-located space–time–frequency codes can then be employed to collect space andmultipath diversity [4]. However, if carrier frequency offset(CFO) is also present, OFDM loses its orthogonality. Thepresence of CFO can be caused by (i) relative movement ofterminals, which causes Doppler shifts and (ii) mismatchesbetween local oscillators at relaying and destination terminals.Compared to co-located multi-antenna systems, CFO is parti-cularly challenging in distributed set-ups since different relays(antennas) experience different CFOs. Current works havefocused on algorithms to estimate or mitigate CFO effects viatraining [7–9], but assume perfect time synchronization. Theschemes in [6,10,11] deal with both time and frequencymismatches via equalization assuming single-carrier [10,11],or OFDM transmissions [6], with known channel fading andassuming no relaying errors. The scheme in [12] deals withboth time and frequency mismatches in OFDM-based coop-erative systems, but it performs the estimation of the channeland CFO via training whereas our objective is to bypass bothchannel and CFO estimation at the receiver.

This paper considers a different approach, designingspecific modulations at the relays so that low-complexitydetection at the receiver is feasible without channel knowl-edge or synchronization information. Differential modula-tions are employed, which define a recursion at thetransmitter such that detection at the receiver can beaccomplished using previously-received symbols. Differen-tial modulations have distinctive implementation advan-tages in low-complexity (distributed) dynamic environ-ments [3]. Single-differential (SD) schemes are able tobypass channel knowledge but fail if severe CFO is present.Double-differential (DD) schemes are able to bypass bothchannel and CFO at the cost of reduced signal-to-noise ratio(SNR) at the receiver side. Multi-antenna (OFDM) versions ofSD and DD recursions for point-to-point links (same CFOfor all transmitting antennas) are available in the literature[13–16], showing that differential recursions are able notonly to bypass channel and CFO knowledge but also toachieve space (and multipath) diversity. We must point outthat the coding schemes developed in this paper deal withcooperative systems with different CFOs among relays. Thisproblem of different CFOs is specific in cooperative scenariosand makes the coding schemes previously proposed fornon-cooperative systems, where there is an only CFObetween source and destination nodes, non-directly applic-able to cooperative systems with multiple relays.

In this paper, we design these specific modulations:

�

Assuming relays transmit frames using time-divisionmultiplexing (TDM), a DD-TDM scheme is developed thatdeals with multiple CFOs among relays. Space diversitycan be collected by suitable space–time (ST) mappingand interleaving across relays. Compared to existing

distributed DD designs, this simple DD-TDM schemeallows for distinct CFOs from different cooperating paths[17] and does not require repetition coding [18,19].

�

When both CFO and timing/multipath effects arepresent, a DD-TDM OFDM scheme is derived. Multi-path and space diversity can be collected using codingacross both relays and OFDM blocks, whereas ortho-gonality among OFDM subcarriers is preserved usinga suitable block precoding. The resulting developedscheme can be seen as a distributed extension of theone in [16], designed for point-to-point links.

�

For long multipath spread or increased number of relaysTDM-based schemes suffer from considerable coding gainloss due to diagonal unitary constellation constraintsand reduced SNR at the receiver. For that reason a simpledistributed SD-OSTBC OFDM scheme is also derived.Relays simultaneously transmit signals differentiallyencoded and transmitted within OFDM symbols. Differentfrom ST–frequency systems in which differentially-encoded blocks are transmitted in different OFDM trans-missions [13], the SD-OSTBC OFDM scheme here derivedhas differentially-encoded blocks closer to each other andthus is more robust to CFO effects. This scheme alsofeatures low complexity decoding and collects both multi-path and space diversity.

To deal with decoding errors at relays, simple andpractically-appealing decode-and-forward (DF) protocolsare derived. Analog amplify-and-forward based schemes,as in [20], require analog signal processing and storage atrelays because these relaying terminals amplify thereceived signal and retransmit, but in DF protocols relayingterminals decode frames prior to re-encoding and re-transmitting them to the destination. Two DF schemesare considered: (i) select-and-forward (SF) [2,1], wherebyrelays only transmit frames if correctly decoded; and (ii)link-adaptive-relaying (LAR) [18,2], whereby relays weighthe power of the transmitted frame according to theinstantaneous power of the received signal. SF and LARprotocols were originally developed for coherent frequency-flat cooperative schemes. They are extended here to non-coherent transmissions under multipath channels. Perfor-mance analysis and simulated tests will show that both LARand SF protocols are robust against intermediate decodingerrors and enable the maximum possible diversity availableby the distributed set-up.

This work extends our results obtained in [21], where weexplored the use of DDmodulations in cooperative systems toobtain a low-complexity detection at the receiver, by-passingchannel and CFO estimation. In the present work, we considernot only different CFOs among relays, but also timing/multi-paths effects, again without channel knowledge or synchro-nization information. The rest of the paper is organized asfollows. Section 2 presents the general system model, includ-ing CFO, timing offsets and fading effects. Section 3 considersthe case when CFO is only present, derives the DD-TDMscheme and analyzes its performance in the error-free case.Section 4 considers the case when both CFO and timing/multipaths effects are present, and derives the main novelcontributions of the present paper: two distributed differen-tial OFDM modulations that collect both space and multipath

A. Cano et al. / Signal Processing 105 (2014) 30–4232

diversity in the error-free case. The DD-TDM OFDM scheme,derived in Section 4.1, is robust to any CFO range. The SD-OSTBC OFDM scheme, derived in Section 4.2, achieves highercoding gain and features simpler decoding complexity, at theexpense of limited CFO mitigation range. Section 5 presentscooperative protocols to deal with decoding errors at relayingnodes, and further analyzes their performance to prove that,also in the non-error-free case, our two differential OFDMschemes keep collecting full diversity. Simulated results areprovided in Section 6; and Section 7 concludes the paper.

Notation: Upper (lower) bold face letters will be used formatrices (column vectors); calligraphic letters will be used forsets; ð�ÞT , ð�Þn and ð�ÞH are the transpose, conjugate and trans-pose conjugate (hermitian) of a vector or matrix respectively;½��k is the kth entry of a vector; � denotes Kronecker product;IN denotes the N�N identity matrix; 1N (1N�M) is the N � 1(N�M) all-one vector; 0N (0N�M) is the N � 1 (N�M) all-zero vector; diagðX1;…;XNÞ is a matrix with matricesX1;…;XN in its diagonal; Dx is a diagonal matrix with theelements of vector x in its diagonal; J � J is the Frobeniusnorm; j � j is the cardinality of a set; FN is the N�N FastFourier Transform (FFT) matrix with ½FN�kþ1;nþ1≔N�1=2

expð2πnk=NÞ; and CN ðμ; s2Þ denotes the complex Gaussiandistribution with mean μ and variance s2.

2. System model and problem statement

With reference to Fig. 1, consider a set of R relayingterminals fTrgRr ¼ 1 collaborating to transmit a block ofsymbols s to an access point or destination (D). Eachelement of this block belongs to a set As with cardinalityjAsj and thus transports log2jAsj information bits. Vector scan be made available to fTrgRr ¼ 1 through a previousbroadcast transmission by a source S (as in Fig. 1) orthrough successive broadcast phases among relayingterminals as in [2]. For simplicity, it will be assumed blocks is the same at all terminals; i.e., no transmission erroroccurred during the broadcasting phase. Section 5 willdevelop and analyze cooperation protocols that considerfTrgRr ¼ 1 have different decoded blocks f~srgRr ¼ 1.

Block s is encoded at each relay Tr and cooperativelysent to the destination. The nth sample at D, namely y(n),is the noisy output of the fading-and-CFO equivalentchannel, with input being the transmitted symbol tr(n)per relay Tr; i.e.,

yðnÞ ¼ ∑R

r ¼ 1ejωrn ∑

L�1

ℓ ¼ 0hr;ℓtrðn�ℓÞþzðnÞ ð1Þ

Fig. 1. R-relay cooperation network.

where ωr and hr;ℓ are the normalized CFO and theequivalent channel for link Tr�D respectively, whereasz(n) is the channel noise. The CFO ωr is given by ωr ¼2πTsf r , with Ts being the sampling period and fr being thephysical frequency offset in Hertz. The CFO is differentacross terminals and is assumed to remain constant alongthe duration of the transmission and modeled as uniformwithin the interval ð�1=2Ts;1=2Ts�. The channel hr;0;…;

hr;L�1 is the effective impulse response per relay Tr, withL¼ ⌈ðτmax

c þτmaxs Þ=Ts⌉þ1 including both maximum channel

delay spread (τmaxc ) and maximum sampling position error

(τmaxs ). Note that in general hrℓ might be correlated across

index ℓ. However, to simplify performance analysis hr;ℓwill be assumed independent and identically distributed(i.i.d.) zero-mean complex Gaussian; i.e., hr;ℓ � CN ð0; s2r;ℓγ Þ,with γ r≔γ∑L�1

ℓ ¼ 0s2r;ℓ the average SNR of link Tr�D. Simu-

lated tests in Section 6 will consider different Tr�D linkprofiles. The noise term z(n) is modeled as white GaussianzðnÞ � CN ð0;1Þ 8n.

Note that the model in (1) is similar to that of [22],where a basis expansion model is employed to modelDoppler shifts. Also note however that here the frequencyof the complex exponentials (basis) is unknown. If known,only the fading coefficients would need to be bypassed,and so simple SD schemes developed for TV channelscould be applied. The challenge here is to design modula-tion strategies at fTrgRr ¼ 1 such that detection at D can beaccomplished without knowledge of either hr;ℓ or ωr 8rwhile at the same time exploiting the maximum degreesof freedom that the R independent fades enable. In otherwords, the modulation scheme should also be designed soas to achieve the maximum spatial diversity enabled bythe distributed set-up.

Next section will deal with a simpler model in whichmultipath effects are not considered. It will be shown thatin this case diagonal space–time matrices can robustlybypass CFO and channel knowledge. The scheme describedbelow will serve as a basis for the most general I/O modeldescribed in (1).

3. DD-TDM transmissions under multiple CFOs

In the present section and in Section 4 we focused onthe relaying phase under the assumption that symbols sare available error-free at all relaying terminals. The base-band system model is depicted in Fig. 2. We will start fromthe outer (DD encoder) to the inner (parallel-to-serial (P/S)

+

Fig. 2. DD-TDM system model.

A. Cano et al. / Signal Processing 105 (2014) 30–42 33

conversion) stages at the transmitters' side, and proceedthrough the channel to the inner (serial-to-parallel (S/P)conversion) and outer (DD decoder) stages at the receiver.

3.1. DD recursion and interleaving

Assume that the block s at the input of the DDmodulator is of length K�2 information symbols. Eachsymbol in s is mapped to a unitary diagonal matrix Dvk ,with k¼ 3;…;K , size R�R picked from a constellation Vwith jVj ¼ jAsj as in [14]. The constellation V criticallyaffects the system performance, as will be shown at theend of this section. The considered scenario in [14] is non-distributed, with multiple antennas in transmission andreception; therefore, we suggest the straightforward appli-cation of the constellations designed in a distributedscenario, where the multiple antennas are replaced withthe relays and there is an only antenna in the destinationnode. Matrix Dvk is used to yield K DD modulated blocks xk

according to these recursions

gk ¼Dvkgk�1; k¼ 3;…;K

1R; k¼ 2

(ð2aÞ

and

xk ¼Dgkxk�1; k¼ 3;…;K

1R; k¼ 1;2

(; ð2bÞ

similar recursions are also used in [14,23], where, remem-bering our notation, Dgk is a diagonal matrix with theelements of vector gk in its diagonal. Note that xk rangesfrom k¼ 1;…;K , whereas Dvk ranges from k¼ 3;…;K asit transports the K�2 symbols in s. Elements of xk aretransmitted by different relays so as to enable spatialdiversity. In practice, however, transmission is arrangedin frames (or blocks) and so interleaving becomes neces-sary as shown next.

Symbols xk are concatenated to build a block x≔½xT

1 ;…; xTK �T size KR� 1 that is block-interleaved using a

matrix ΘK to form t≔ΘKx. The KR� 1 matrix ΘK is definedso that ½t�ðr�1ÞKþk ¼ ½x�ðk�1ÞRþ r and can be compactlywritten as

ΘK≔½IR � e1;…; IR � eK � ð3Þwith ek being the kth column of IK . The interleaved block ist≔½tT1 ;…; tTR�T . Relay Tr transmits the subblock tr during therth time slot. Thus, after P/S conversion, the n-th sampleof the transmitted signal tr(n) contains the entries of trduring time slot ðr�1ÞKþ1;…; rK and is zero outside thatinterval. The duration of the entire transmission is N¼KR.

3.2. DD receiver and performance analysis

In the absence of multipath effects, the I/O relationshipin (1) reduces to

yðnÞ ¼ ∑R

r ¼ 1e� jωrnhrtrðnÞþzðnÞ; n¼ 1;…;RK: ð4Þ

Let the received block at D after S/P conversion be denotedas ~y . This block can be partitioned into R subblocks~y≔½ ~yT

1 ;…; ~yTR�T , each corresponding to a different relay,

with

~yr ¼ hrejωr ðr�1ÞKDωr trþ ~zr ð5Þwhere Dωr≔diagðejωr ;…; ejωrK Þ and ~zr is the noise term.With reference to Fig. 2, the KR� 1 block ~y is passedthrough the de-interleaving matrix ΘT

K to form y≔ΘTK ~y≔

½y1;…; yK �T . Each R� 1 subblock yk is given by

yk ¼DhDωDωkxkþzk ð6Þwith ½Dh�r;r≔hr , ½Dω�r;r≔ejωr ððr�1ÞKÞ and ½Dωk �r;r≔ejωrk acc-ounting for channel and CFO. The noise vector zk remainswhite Gaussian with the same power, as permutations donot affect its distribution. Observe that Dh and Dω areindependent of k, whereas matrix Dωk changes with k.Consider three consecutive received blocks yk, yk�1, yk�2,expressed as [cf. (2) and (6)]

yk ¼DhDωDωk� 1Dω1D2gk� 1

Dvkxk�2þzk ð7aÞ

yk�1 ¼DhDωDωk� 1Dgk� 1xk�2þzk�1 ð7bÞ

yk�2 ¼DhDωDωk� 1Dω� 1xk�2þzk�2: ð7cÞCompared to [14], here the relative phase is expressed interms of the diagonal matrix Dω instead of a scalar. Asmentioned in [14] in the context of DD designs for co-located systems, the maximum-likelihood (ML) decoderfor Dvk given yk, yk�1 and yk�2 in (7) may depend on thefrequency offsets. To avoid this problem, a heuristic detec-tor to decode Dvk by-passing CFO knowledge, and whoseperformance is surprisingly close to the ML detector, isderived next. Note that, as it is shown in Appendix A,

Dykyn

k�1 ¼DvkDyk� 1yn

k�2þz0k; ð8Þwhere, remembering our notation, Dyk is a diagonal matrixwith the elements of vector yk in its diagonal, and

z0k ¼Dzkyn

k�1þDn

zk� 1ykþDn

zk� 1zk�Dvk ½Dzk� 1y

n

k�2

þDn

zk� 2yk�1þDn

zk� 2zk�1�: ð9Þ

Discarding high-order noise terms, z0k can be approximatedas Gaussian with covariance matrix Σk≔E½z0kz0Hk � ¼DykD

n

ykþ

2Dyk� 1Dn

yk� 1þDyk� 2

Dn

yk� 2. This Gaussian approximation

allows one to write a detector for Dvk from (7) as

Dvk ¼ argminDv

fJΣ�1=2k ðDyky

n

k�1�DvDyk� 1yn

k�2ÞJ2g ð10Þ

8k¼ 3;…;K . De-mapping Dvk , a decoded block s isobtained at the destination. Note that this receiver onlyrequires knowledge of previously-received blocks Dyk ,Dyk� 1

and Dyk� 2, by-passing channel or CFO knowledge.

The average Pairwise Error Probability (PEP) will beused as the performance metric. The PEP is defined as theprobability that the detector (10) confuses s for another saveraged over all channel realizations, and is denoted asPrðs-sÞ. We are interested in its diversity order, defined asthe rate of decay in a logarithm scale as a function of theSNR (in dB); see, e.g., [2,14]. The following propositionstates the diversity order achieved by (10).

Proposition 1 (Diversity of error-free DD-TDM). The diver-sity order achieved by the detector in (10) assuming error-free relays is R; i.e., Prðs-sÞr ðGcγ Þ�R.

Fig. 4. DD-TDM OFDM transmitted frame.

A. Cano et al. / Signal Processing 105 (2014) 30–4234

The coding gain coefficient Gc ¼ Gcðs21;…; s2R;VÞ absorbsdifferent relative SNRs among Tr�D links, constellationdistances and potential noise correlation effects, and willbe analyzed via simulations in Section 6. The diagonalstructure of the coding matrices effectively separate indifferent CFOs from the relays in distinct time slotsand thus the resulting system in (10) is also diagonal.

The diversity order depends on the rank of Dvk �Dvk . If Vis designed to guarantee that Dvk �Dvk is full rank, or

equivalently ½Dvk �r;r�½Dvk �r;ra0 8r and 8Dvk ; Dvk in V,then the detector in (10) achieves the maximum possiblediversity R. The details of the proof are similar to those in[14], where diagonal matrices are also employed, and thusare omitted here. Note however that when relays sufferfrom decoding errors the transmitted block is differentfrom Dvk , and thus full rank may be lost. Section 5 will dealwith this scenario in detail. Before that, we extend theanalysis of this section to OFDM transmissions.

4. Differential OFDM transmissions under multiple CFOsand timing/multipath effects

In this section, two differential schemes are derived tocope with both CFO and timing/multipath effects usingOFDM transmissions. The first one is based on differentialencoding across OFDM blocks and can be seen as a dis-tributed generalization of the scheme in [16] using diagonalST codes as in the previous section. The second schemedifferentially encodes symbols within OFDM blocks usingOSTBCs and allows for simultaneous relay transmissions.

4.1. DD-TDM OFDM scheme: DD-TDM using OFDM

Assume the block of symbols s at Tr is now of sizeMðK�2Þ, that is, it has K�2 subblocks of size M each. Themth symbol at the kth subblock is mapped to an R�Runitary diagonal matrix Dvk and differentially encoded asin (1)

gm;k ¼Dvm;kgm;k�1; k¼ 3;…;K

1R; k¼ 2

(ð11aÞ

xm;k ¼Dgm;k

xm;k�1; k¼ 3;…;K

1R; k¼ 1;2

(: ð11bÞ

The block xk≔½xT1;k;…; xT

M;k�T is first permuted as ~xk≔

ΘMxk≔½ ~xT1;k;…; ~xT

R;k�T , with ~xr;k size M � 1; see Fig. 3.

Fig. 3. DD-TDM OFDM m

Different from the previous section, ~x r;k is now precodedprior to transmission as

x r;k≔ð1L � FMÞ ~x r;k ð12Þ

where FM is an M�M FFT matrix. Note that the precodingin (12) amounts to repeating FM ~x r;k L times. This will beinstrumental to diagonalize the resulting CFO-plus-multipath channel. The ML� 1 block x r;k in (12) is trans-mitted using OFDM. Mathematically, this is expressed usingan IFFT matrix followed by an L-length CP-inserting matrixTML. The resulting OFDM block tr;k is thus given by

tr;k≔TMLFHMLx r;k ð13Þ

where TML ¼ ½ ~TML; IML�T and ~TML ¼ ½0L;ðM�1ÞL; IL�T . Fig. 3shows the overall scheme. Subblocks tr;k are concatenatedacross index k to form the block tr of length ðMþ1ÞLK to betransmitted by relay r through the channel in (1). Fig. 4shows the resulting frame structure.

Upon CP removal and S/P conversion the receivedsignal is ~y≔½ ~yT

1;1;…; ~yT1;K ;…; ~yT

R;K �T , with each ML� 1 sub-block ~yT

r;k is given by

~yr;k ¼ ejωr Ir;kDωrHrFHMLxr;kþ ~zr;k ð14Þ

where Ir;k ¼ ðMLþLÞððr�1ÞKþk�1ÞþL; ½Dωr �m;m≔ejωrm; Hr

is an ML�ML circulant matrix with first column ½hTr ;

0TðM�1ÞL�T where hr≔½hr;0;…;hr;L�1�T ; and ~zr;k is the AWGN

noise. Note that the number of subcarriers in this case issimply M� L. In the absence of CFO, HrFHML in (14) can bediagonalized by passing ~yr;k through an FFT block atthe receiver side. However, the presence of Dωr destroysthe orthogonality among subcarriers. In Appendix B it isshown that by exploiting the precoding in (12) theexpression in (14) can be rewritten as

~yr;k ¼ffiffiffiL

pejωr Ir;kDωr ð ~xr;k � ILÞhrþ ~zr;k: ð15Þ

This expression shows that the equivalent channel seen atthe receiver is now diagonal; i.e., orthogonality is pre-served. Note however that entries in ~xr;k are coded acrossindex r. Since the differential recursions in (11) are carried

across indexes m; k, the block ~yk≔½ ~yT1;k;…; ~yT

R;K �T is de-

interleaved to obtain yk≔ΘTML ~yk ¼ ½yT1;k;…; ~yT

M;K �T where

odulation scheme.

A. Cano et al. / Signal Processing 105 (2014) 30–42 35

the RL� 1 subblock yTm;k is given by

ym;k ¼ffiffiffiL

p~Dω

~Dωm;k ðIL � Dxm;k Þhþzm;k ð16Þ

where now ½ ~Dω�m;m≔ejωmodfm;Rg⌈m=R⌉, ½ ~Dωm;k �m;m≔ejωmodfm;Rg ~Im;k

with ~Im;k≔ðMLþLÞððmodfm;Rg�1ÞKþk�1ÞþmL and h≔½h1;0;…;hR;0;…;hR;L�1�T . Subblocks ym;k can be further

partitioned as ym;k ¼ ½ym;k;1;…;ym;k;L�T , and so can be ~Dω,~Dωm;k and zm;k. Following similar steps as in the previoussection, it is not difficult to show that Dym;k;ℓ

yn

m;k�1;ℓ ¼Dvm;k;ℓDym;k� 1;ℓ

yn

m;k�2;ℓþz0m;k;ℓ. Approximating the noisez0m;k;ℓ as Gaussian with covariance matrix Σm;k;ℓ≔Dym;k;ℓ

yn

m;k;ℓþ2Dym;k� 1;ℓyn

m;k�1;ℓþDym;k� 2;ℓyn

m;k�2;ℓ, the followingdetector can be derived [cf. (10)]:

Dvm;k¼ argmin

Dv

∑L�1

ℓ ¼ 0JΣ1=2

m;k;ℓðDym;k;ℓyn

m;k�1;ℓ�DvDym;k� 1;ℓyn

m;k�2;ℓÞJ2� �

ð17Þ

8 k¼ 3;…;K; m¼ 1;…;M. As in (10), the search is overthe same alphabet V, and so the detection rule in (17)incurs in the same complexity as the one in (10). De-

mapping Dvk , a decoded block s is obtained at thedestination. The detector in (17) collects both multipathand space diversity as highlighted in the followingproposition.

Proposition 2 (Diversity of error-free DD-TDM OFDM). Thediversity order achieved by the detector in (17) assumingerror-free relays is RL; i.e., Prðs-sÞrðGcγ Þ�RL.

Multipath diversity is enabled thanks to the repetitionstructure in (12). To show this, notice that the detector in(17) can be alternatively written as

Dvm;k ¼ argminDv

fJΣ1=2m;kðDym;k

yn

m;k�1�ðIL � DvÞDym;k� 1yn

m;k�2ÞJ2g

ð18Þ

with Σ1=2m;k formed diagonally concatenating matrices Σ1=2

m;k;ℓ.

The diversity now is given by the rank of ðIL � ðDv� ~DvÞÞ.Since the Kronecker product of a full-rank matrix is stillfull rank, the rank of ðIL � ðDv� ~DvÞÞ is LR. It is worthstressing that the diversity result holds true when the Lentries of hr are i.i.d. Gaussian, as stated in Section 2. If noi.i.d. multipath is present and the spread L is only given bysynchronization mismatches, then the entries of hr arecorrelated and multipath diversity is lost. Still, the repeti-tion structure in (12) is required to bypass CFO mis-matches. Simulations in Section 6 with further elaborateon this.

As far as throughput is concerned, a total of 2KRchannel uses (KR channel uses to share information amongcooperating terminals, plus another KR channel uses totransmit to the destination) are required to transmit K�2information symbols. This is 1/2 of that of a point-to-pointMIMO system with T¼R transmit antennas, which woulduse KT channel uses to transmit same K�2 symbols. Thisreduction is understood as the price to pay for havingcheap distributed cooperating terminals instead of a sin-gle, more expensive multi-antenna transmitter.

Note that the two DD-TDM strategies described so farseparate transmissions in time slots to avoid inter-terminalinterference. This is effected using the ST mappings in (2) and(11), constrained to be unitary diagonal. However, if thenumber of antennas or transmission rate increases, the mini-mum constellation distance decreases considerably fast.Moreover, diagonal mappings incur in decoding complexitythat increases exponentially with the number of the transmitrelays R. Next, it is shown that one can design an efficientdifferential scheme by having terminals simultaneously trans-mit signals within OFDM symbols with low complexitydecoding using SD-OSTBC modulations.

4.2. SD-OSTBC OFDM scheme: simultaneous transmissionsusing OSTBCs

In the following it will be assumed that the CFO is lessthan half of the subcarrier spacing. Note that for largervalues one can consider null-subcarrier based algorithmsto correct the integer part of the offset prior to transmis-sion [24]. The premise here is that, if information is codedwithin an OFDM block instead of across blocks as istypically done in space–time–frequency codes, the CFOseen is considerably reduced, and so SD-OSTBCs can beemployed.

As before, we start from the outer differential encoder.The block of symbols s is now split into K�1 subblockssizeM each, with its kth entry of themth subblock mappedto a set of P unitary fvm;k;1;…; vm;k;Pg complex scalars eachtransporting 1=P log2jAsj bits, with which the followingmatrix can be constructed [15]:

Vm;k ¼1ffiffiffiP

p ∑P

p ¼ 1Φp Refvm;k;pgþ jΨp Imfvm;k;pg ð19Þ

where Φp and Ψp are unitary matrices given in [15].Having mapped s to blocks ffVm;kgMm ¼ 1g

K

k ¼ 2, the latter aredifferentially encoded as

Xm;k ¼VTm;kXm;k�1; k¼ 2;…;K

IR; k¼ 1

(: ð20Þ

The resulting KR�R coded block Xm≔½XTm;1;…;XT

m;K �T isfurther processed as follows:

~Xm ¼ ð1L � FK ÞXm: ð21ÞEach relay Tr will transmit the rth column of ~Xm, namely~xm;r , using OFDM. The mth transmitted block by the rthrelay is given by

tm;r ¼ TKLFHKL ~xm;r : ð22ÞObserve that, different from the DD-TDM scheme, tm;r doesnot depend on index k used for differential recursions.This is because differential encoding is carried within (asopposed to across) OFDM blocks.

Subblocks tm;r are stacked to obtain tr≔½tT1;r ;…; tTM;r �T . Allrelays fTrgRr ¼ 1 transmit ftrgRr ¼ 1 simultaneously throughthe channel in (1). After S/P conversion and CP removal,the mth KL� 1 OFDM block at the receiver, labeled as ~ym isgiven by [cf. (1)]

~ym ¼ ∑R

r ¼ 1ejωr ððm�1ÞKLþmLÞDωrHrFHKL ~xm;rþ ~zm ð23Þ

A. Cano et al. / Signal Processing 105 (2014) 30–4236

where ½Dωr �k;k ¼ ejωrk. Each row ~xm;r was repeated L timesin (21). Note that in this case the number of subcarriers isR� K � L. Mimicking the steps in Appendix B, each termwithin the sum (23) can be rewritten as

~ym ¼ffiffiffiL

p∑R

r ¼ 1ejωr ððm�1ÞKLþmLÞDωr ð ~xm;r � ILÞhrþ ~zm: ð24Þ

Vector ~ym can be partitioned in K subgroups size R each;i.e., ~ym≔½ ~yT

m;1;…; ~yTm;K �T . In the absence of CFO, ~ym;k ¼ffiffiffi

Lp

ðXm;k � ILÞhþ ~zm;k. Clearly, the ST matrix Xm;k isrepeated L times. Let us further split ~ym;k into L subgroupsby defining ym;k≔ΘT

L ~ym;k≔½yTm;k;1;…; yTm;k;L�T . In the absenceof CFO, each subblock ym;k;ℓ is given by

ym;k;ℓ ¼ffiffiffiL

pXm;k

~hℓþzm;k;ℓ ð25Þwith ~hℓ≔½h1;ℓ;…;hR;ℓ�T . From (25), the following ML SDdetector can be derived:

Vm;k ¼ argminV

∑L�1

ℓ ¼ 0Jym;k;ℓ�VTym;k�1;ℓ J

2� �

ð26Þ

which searches over all jAsj possible matrices V. Afterstandard manipulations, and exploiting the structure of V in(19), a simple ML detector for each vm;k;p can be derived [15]

vm;k;p ¼ argmaxv

(∑L�1

ℓ ¼ 0ReftrðΦpyn

m;k;ℓyTm;k�1;ℓÞRefvgg

þReftrðjΨpyn

m;k;ℓyTm;k�1;ℓÞImfvgg

)ð27Þ

where the search is now carried over jvj ¼ jAsj1=P symbols invm;k;p. De-mapping vm;k;p a decoded block s is obtained at thedestination. Interestingly, the diversity achieved is still thesame, as summarized next.

Proposition 3 (Diversity of CFO/error-free SD-OSTBC OFDM).The diversity order achieved by the detector in (17) assumingerror-free relays and in the absence of CFO is RL; i.e.,Prðs-sÞr ðGcγÞ�RL.

In the absence of CFO ym;k;ℓ ¼VTm;kXm;k�1

~hℓþzm;k;ℓ.Stacking across index ℓ, ym;k ¼ ðIL � VT

m;kÞðIL � Xm;k�1Þ~hþzm;k with ~h≔½hT

1;…;hTL �T . Using the decoder in (26),

which is equivalent to the one in (27), it is not difficult tosee that once again the diversity order depends on therank of ðIL � ðVm;k� Vm;kÞÞ. Since OSTBCs are full rankmappings, Proposition 3 follows.

Simulations in Section 6 will show that SD-OSTBC OFDMefficiently bypasses a wide range of CFO values. Tests willalso show that while maximum diversity is achieved,SD-OSTBC OFDM will feature higher coding gains comparedto the DD-TDM OFDM in Section 4.1. This is because: (i)OSTBC matrices are not constrained to be diagonal, and sohave larger minimum distance among constellation pointsand (ii) SD detection as in (26) is incurred in only 3 dB SNRloss, as opposed to DD detection as in (10) or (17), whichincur in at least 6 dB SNR loss; see (9), [14].

Note that SD-OSTBC OFDM transmissions differentiallycode information within OFDM blocks and thus the phasedifference due to CFO seen between consecutive sub-blocks is considerably reduced with respect to differen-tially code across OFDM blocks. Specifically, with OFDMblocks of size RKL, the CFO seen from subblock ym;k to

subblock ym;k�1 is 1=K less than what would be seen ifdifferential recursions were carried across OFDM blocks.Hence, for sufficiently large K, CFO effects are negligible.This is because the CFO is assumed to be less than halfof the subcarrier spacing, and integer deviations can beefficiently corrected in OFDM transmissions via null sub-carrier methods [24].

About the possibility of exploiting time diversity, theCFO-and-multipath model in (1) can also include timevarying channels as a special case. Note however that ingeneral this scheme would not be able to collect Dopplerdiversity. For that matter a rather more sophisticatedapproach would be needed, involving differential modula-tion across space, frequency, and time as in [22]. However,simulations will show that for large L or R, the extradiversity that could be gained using additional codingmay be negligible for typical SNR ranges, while at thesame time coding gain loss would exponentially increasedue to higher constellation mappings. This suggest thatcoding across space and frequency while bypassing fre-quency drifts strikes the right trade-off between diversityand coding gain advantages.

As far as computational complexity is concerned, as itcan be seen for both DD-TDM (10) and DD-TDM OFDM (17)decoders, a ML demodulation scheme is derived, therefore,the computational complexity of both methods exponen-tially grows with the size of Dvk (therefore the numbermultiplications/additions exponentially grows with thenumber of cooperating relays R). It is worth noting thatcompared to existing point-to-point MIMO non-coherentmodulation schemes, this design entails in the samedecoding complexity. Furthermore, if needed, the compu-tational complexity of both algorithms may be broughtdown to polynomial complexity (therefore the numberof multiplications/additions grows with R3), without anymajor change in the setting, by using lattice reductionmethods, e.g. such as the one proposed in [25]. In the caseof DD-TDM OFDM, the multipath diversity increases theexponential complexity of the decoder, and the number ofmultiplications/additions exponentially grows with RL.

For the special case of SD-OSTBC OFDM transmissions,the search in (27) is reduced to be linear the number ofcooperating relays, as is also the case with point-to-pointMIMO systems as in, e.g., differential Alamouiti schemes(thus the number of operations grows with R). Thecomputational complexity of the decoding scheme of[17] is also linear with the number of relays. However, asit can be seen in Fig. 5, the performance of such method isless than optimal compared to the one proposed here,because it assumes the same CFO across terminals. In otherwords, our proposed design necessitates equal complexityto solve a more general and complex problem (differentCFOs). The same can be said for the detector in [13] that ispolynomial in complexity with the number of relays(therefore the number of multiplications/additions growswith R3), but the performance of such detector is lackingand practically insensitive to changes in SNR.

Finally, when compared with coherent designs, thedifferential modulation schemes proposed in this workdo not require channel estimation, nor CFO correctionblocks at the receiver. Thus, they considerably simplify

Fig. 5. DD-TDM vs. no cooperation and [17] with no errors for R¼2,3.Also included asymptotes with diversity of order 2 (narrow gray line) and3 (thick gray line).

A. Cano et al. / Signal Processing 105 (2014) 30–42 37

the receiver complexity compared to coherent designs.Although this is also true for point-to-point systems, itbecomes more critical in cooperative setups, which arebased on the premise of cheap low-complexity cooperat-ing relays.

5. Relaying errors

The synchronization problem in the broadcasting phase isa well known problem because this phase consists in a set ofpoint-to-point links. Therefore, we focused on the relayingphase, and differential modulations robust to CFO andmultipath were derived in the previous section under theassumption that symbols s are available error-free at allterminals. In this section it is shown that these modulationscan also be robust against errors at intermediate steps; i.e.,when s is erroneously decoded at the relays because of thepresence of CFO in the source-relay links or any other reason.For that matter, two DF cooperating protocols will be appliedto the previously developed differential modulations:(i) selective forwarding (SF) [1,2] and (ii) link-adaptiveregenerative (LAR) transmissions [2,18]. OFDM transmissions(either DD-TDM or SD-OSTBC) for the Tr�D link will beconsidered. Non-OFDM transmissions in Section 2 will fall asspecial case of the analysis here and thus are omitted. TheS�Tr link will be assumed multipath with channel coeffi-cients hðsÞr;0;…;hðsÞ

r;L�1. Since the S�Tr channel is non-coop-erative, the source S can employ point-to-point DD-OFDMtransmissions as in [16].

5.1. Selective and adaptive transmissions

In SF, cooperating relays decode the source's block sand, if correctly decoded, they transmit the re-encodedsignal tr in (13) or (22). If relay Tr incurs in an error, tr isnot transmitted. Decoding errors can be verified using, e.g.,cyclic redundancy check (CRC) codes, which are relativelybandwidth-efficient. Mathematically, the SF scheme canbe described using a scalar αSFr that multiplies tr and is

given by

αSFr ¼ 1 if sr ¼ s0 if sras

(ð28Þ

where sr is the decoded block at relay Tr. Note that αrSF

could be defined for each entry of s, instead of the entireblock. However, in practice transmission and error detec-tion via CRC codes is carried in frames, and so the defini-tion in (28) is more practically-appealing. Also, note thatdiscarding entire frames instead of individual symbolsalways increases the end-to-end error rate, and so thedefinition in (28) can be seen as a “worst-case” coopera-tion scenario in terms of error performance.

In LAR, the decoded frame is always transmitted by allrelays regardless of whether sr equals s or not, but withpower weighted according to an estimate of the instanta-neous channel reliability using coefficient

αLARr ¼1 if ϕrZγ rϕr

γ rif ϕroγ r

8><>: ð29Þ

where γ r≔γ∑L�1ℓ ¼ 0s

2r;ℓ is the average Tr�D SNR (cf. (1)) and

ϕr≔ðyðsÞr ÞHyðsÞr =P�1 with yðsÞ

r being the received signal at Trfrom the source, which is assumed of length P. With this

definition, and since yðsÞr is defined as in [16], it holds that

at high SNR ðyðsÞr ÞHyðsÞr P∑L�1ℓ ¼ 0jhðsÞr;ℓj2þP (the variance of

the noise at Tr is assumed without loss of generality 1). Or

equivalently, ϕr ∑L�1ℓ ¼ 0jhðsÞr;ℓj2, which is the instantaneous

SNR of link S�Tr . In words, relays transmit frames at fullpower when the S�Tr is more reliable than the averageTr�D link; otherwise, αr

LARscales power down to mitigate

errors. Note that since γ r varies at a large scale, having itavailable at relays is certainly affordable.

The LAR protocol can be further improved by settingαLARr ¼ 1 if no error is detected as in the SF protocol (sincethe wireless protocol may be employing CRC codes atupper layers anyway), otherwise the block is transmittedwith αr

LARas in (29). This strategy outperforms LAR,

because it guarantees that those blocks correctly decodedwill always be transmitted with full power.

Both SF and LAR protocols only affect the transmittedpower, the former does it in an on–off fashion whereas thelatter does it adaptively. In any case, the destination “sees”an equivalent channel fading hα

r≔ffiffiffiffiffiαr

phr per relay Tr, with

αr as in (28) or (29). Thus, decoding as in (10) and (17) or(27) can still be performed without knowledge of fαrgRr ¼ 1;i.e., the decoding is still non-coherent. Most interestingly,both SF and LAR strategies prevent these detectors fromincurring in diversity loss even when relays have errors, asshown next.

5.2. Performance analysis

Define the set E of terminals with erroneously decodedblock ~sr and its complementary set E containing theterminals that successfully decoded s. The instantaneouserror probability of the S�Tr link can be bounded asPrðTrAEÞrexpð�δ2r∑

L�1ℓ ¼ 0jhðsÞr;ℓj2Þ, where the non-zero con-

stant δr2depends on the constellation employed and the

Fig. 6. DD-TDM OFDM (solid line) vs. SD-OSTBC OFDM (dashed line) withno errors for R¼2,3, L¼2,3 and ρ¼ 0:5;2=3. Also included asymptoteswith diversity of order 4 (narrow gray line), 6 (thick gray line) and 9(thickest gray line).

A. Cano et al. / Signal Processing 105 (2014) 30–4238

number of repetitions L [16]. Since δr2is independent of the

fading realizations, it will be henceforth ignored. Assum-ing hðsÞ

r;ℓ is independent across S�Tr links, the probability ofhaving a set E of relaying terminals incorrectly decoding sis, with the exception of a multiplicative constant,

Pe;hðEÞrexp � ∑R

r ¼ 1∑L�1

ℓ ¼ 0jhðsÞr;ℓj2

� �: ð30Þ

Whenever Tr is in E, the ST mapping at Tr entailsin errors. We will henceforth treat both DD-TDM andDD-OSTBC OFDM systems in a unified manner. Let~s≔½~sT1 ;…; ~sTR�T be the set of all blocks estimated at allrelays. If TrAE , then ~sr ¼ s; however if TrAE, then ~sras.Appendix C shows that using either DD-TDM or DD-OSTBCOFDM transmissions, the conditional probability of decod-ing s provided that s was transmitted by S and ~s wastransmitted by fTrgRr ¼ 1 is given by, with the exception ofmultiplicative constants,

Prðs; ~s-s h�� ÞrQ

∑rAEαr∑L�1ℓ ¼ 0jhr;ℓj2�∑rAEαr∑L�1

ℓ ¼ 0jhr;ℓj2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi∑R

r ¼ 1αr∑L�1ℓ ¼ 0jhr;ℓj2

q0B@

1CA:

ð31ÞConsidering all possible error events E, each with prob-

ability Pe;hðEÞ, the overall conditional PEP can be written as

Prðs-sjhÞr∑EPe;hðEÞPrðs; ~s-sjh; EÞ ð32Þ

with Pe;hðEÞ and Prðs; ~s-sjh; EÞ given in (30) and (31),respectively. It is worth noting that when E ¼ | and αr ¼ 1,the expression in (32) reduces to

Prðs-sjhÞrQ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi∑R

r ¼ 1∑L�1

ℓ ¼ 0jhr;ℓj2

s !ð33Þ

whose expectation over the channel fading clearly achievesdiversity RL, indirectly proving Propositions 2 and 3. Mostinterestingly, the expectation of (32) achieves this samediversity order even when Ea∅, as shown in the followingproposition.

Proposition 4 (Diversity order of SF and LAR OFDM trans-missions). The average of the PEP in (32) using the weight αras defined in (28) or (29) has diversity of order RL; i.e.,Prðs-sÞr ðGcγ Þ�RL.

Proof. See Appendix D.

Thus, the diversity order is independent of the inter-mediate errors ~sr per relay Tr and preserves the multipathdiversity. Note that if one single link has reduced diversitythe overall diversity order is affected. For example, ifthe S�Tr link has diversity 1 (instead of L), the overalldiversity is reduced to R [cf. Appendix D]. Simulations inSection 6 will also include this case. Since both SF/LARschemes achieve the same diversity order, we will com-pare their relative coding gain performance through simu-lations in the ensuing section.

6. Simulations

This section compares the performance of the distributeddifferential schemes in Sections 3, 4.1 and 4.2 with andwithout relaying errors under different system assumptions.

6.1. Error-free DD-TDM transmissions

Fig. 5 compares the symbol-error-rate (SER) vs. SNRcurves of the DD-TDM scheme for R¼2,3 and N¼500. Theconstellation size is jAsj ¼ 2ρR where ρ is the transmissionrate (not including the source transmission) in bits perchannel use (bpcu). The rates considered here are ρ¼ 0:5;1for R¼2 and ρ¼ 2=3;1 for R¼3. The CFO ωr is uniformbetween ½�π; π�. Scalar non-cooperative DD (i.e. a simpleDD transmission, with only a source and a destinationnode, and without any space–time block coding) and theDD-OSTBC scheme in [17] for R¼2, ρ¼ 0:5 are alsoincluded for comparison. Fig. 5 also includes two asymp-totes with diversity order 2 and 3, respectively. You canuse these asymptotes to corroborate that DD-TDMachieves diversity of order R for high SNR (from 15 dBand 20 dB respectively with R¼2 and R¼3), whereasscalar non-cooperative DD transmissions are stuck withdiversity 1. The DD-OSTBC scheme in [17] considerablydegrades as it is designed assuming all CFOs are the same.It can also be observed that coding gain losses of DD-TDMincrease with R and ρ. This is because the constellation sizejAsj ¼ 2ρR increases and the minimum constellation dis-tance reduces; see [14].

6.2. Error-free OFDM transmissions

Figs. 6 and 7 compare the SER vs. SNR curves of theDD-TDM OFDM and SD-OSTBC OFDM schemes for R¼2,3and L¼2,3. For comparison, the same transmission rates asin Fig. 5 are employed, with Fig. 6 fixing ρ¼ 0:5;2=3 andFig. 7 fixing ρ¼ 1. SD-OSTBC OFDM uses slightly higherrates so that scalar PSK constellations can be employed forvm;k;p in (19). Note that because of the L-fold repetition inthe precoding (12) and (21), the constellation size is nowjAsj ¼ 2ρRL. The CFO is set to ½�π; π�=Nc where Nc is thenumber of subcarriers. For DD-TDM OFDM Nc ¼ LM¼ 500;for SD-OSTBC OFDM Nc ¼ RKL¼ 900. The (multipath)channel is i.i.d. Rayleigh with s2r;ℓ ¼ 1 8r;ℓ. SD space–frequency codes using OSTBCs or diagonal matrices as in

Fig. 8. DD-TDM OFDM (solid line) vs. SD-OSTBC OFDM (dashed line) withdifferent CFO ranges β for SNR¼ 10;15 and R¼2.

Fig. 9. DD-TDM with and without OFDM with different multipath profilesfor SNR¼20 and R¼2.

Fig. 7. DD-TDM OFDM (solid line) vs. SD-OSTBC OFDM (dashed line) withno errors for R¼2,3, L¼2,3 and ρ¼ 1. Also included asymptotes withdiversity of order 4 (narrow gray line), 6 (thick gray line) and 9 (thickestgray line).

A. Cano et al. / Signal Processing 105 (2014) 30–42 39

[13] coding across OFDM blocks are included for compar-ison. As shown, both DD-TDM OFDM and SD-OSTBC OFDMachieve higher diversity orders, whereas non-CFO awareSD schemes severely degrade. Figs. 6 and 7 also includethree asymptotes with diversity order 4, 6 and 9. You canuse these asymptotes to corroborate that SD-OSTBC OFDMtends to achieve diversity of order RL for medium SNR (inthe range from 10 dB to 15 dB), whereas DD-TDM OFDMneeds higher SNR to approach this diversity order (espe-cially with ρ¼ 1) because of its higher coding gain losses.Coding gain losses become considerably large as R and Lgrow, especially for the DD-TDM OFDM scheme, as seenin Fig. 7, since the constellation size grows exponentiallywith ρRL. In contrast, SD-OSTBC OFDM offers reducedcoding gain losses, which in turn allow for diversity gainsto “kick-in” earlier. This is because: (i) SD decoding isemployed, reducing the SNR loss by 3 dB and (ii) OSTBCmappings are not constrained to be unitary diagonal.

6.3. Robustness to CFO drifts

As stated in Section 4.2, SD-OSTBC OFDM schemesoperate under the assumption that the CFO is less thanhalf of the subcarrier spacing. This assumption is tested inFig. 8, which shows the SER as a function of the CFO,measured in number of subcarriers β; i.e., ωrA ½�π; π�β=Nc

with Nc¼512, for SNR¼10,15 dB. The DD-TDM OFDMscheme is included for comparison. As shown, with smallCFO deviations (i.e. when CFO is less than half of thesubcarrier spacing; and, in this case, when the CFO varieswithin 20 subcarriers' range (βo20)), SD-OSTBC OFDMoutperforms DD-TDM OFDM because of its reduced codinggain losses, recently justified in the previous section.DD-TDM OFDM, on the other hand, is robust to any CFOrange. The same curves were obtained for Nc¼64, showingthat a relatively small number of subcarriers suffice toachieve robust performance against CFO.

6.4. Robustness to multipath effects

The major drawback of the precoding employed in (12)is that coding gain losses become considerably large as Lincreases, as already seen in Figs. 6 and 7. We furtherelaborate on this effect by comparing the performance ofDD-TDM with and without precoding and OFDM transmis-sions for different values of L and multipath profiles. Theper-path variance is now set to s2r;ℓ ¼ dℓ�1, with 0rdr1.Fig. 9, shows the SER as a function of d for different valuesof L with the SNR fixed to 20 dB. When d¼0, then½s2r;0;…; s2r;L�1� ¼ ½1;0;…;0�; i.e., there is no multipatheffect and the SER of the DD-TDM scheme coincides withthat of Fig. 5 at 20 dB. DD-TDM OFDM is outperformed inthis case since no multipath diversity is present to mitigatethe coding gain loss. The other extreme case is when d¼1,in such case the SER of the DD-TDM OFDM schemecoincides with that of Fig. 7 at 20 dB. DD-TDM withoutOFDM suffers from severe degradation as it was notdesigned to deal with multipath. For intermediate valuesof d it can be observed that DD-TDM rapidly degrades and

A. Cano et al. / Signal Processing 105 (2014) 30–4240

is outperformed by DD-TDM OFDM, even for small L or d.DD-TDM OFDM experiences faster performance improve-ment for small L, because diversity gains “kick-in” earlier;for large L coding gain losses predominate.

Fig. 11. SF vs. LAR vs. DF SD-OSTBC OFDM for R¼2,3 and L¼2. Alsoincluded asymptotes with diversity of order 4 (narrow gray line) and 6(thick gray line).

6.5. SF and LAR protocols

SF and LAR protocols to combat relaying errors aretested here. Fig. 10 shows the SER vs. SNR curves ofDD-TDM OFDM for R¼2,3, L¼2 and ρ¼ 1;2=3. The S�Tr

link is also assumed multipath with L¼2 with scalar OFDMtransmissions as in [16] and SNR 3 dB larger than that ofthe Tr�D link. Recall that SF and LAR adjust the trans-mitted power using different coefficients αr

SFand αr

LARas in

(28) and (29). Thus, for a fair comparison, the overalltransmission power is normalized to be the same. This isdone by numerically computing the power consumption ofeach scheme, and adjusting the overall SNR accordingly.Compared to Fig. 5, relaying errors inevitably degradesystem performance. The good news here is that diversityRL is still achievable, as predicted in Section 5.2. LAR-basedtransmissions attain larger coding gains because theyconsider “soft” information at the receiver. SF, however,discards entire blocks, here of size ðK�2ÞLM¼ 2940, evenwhen only a few symbols were erroneously decoded at therelays. Frequency-flat (L¼1) scalar DD transmissions forlink S�Tr are also included for comparison. In this casediversity is reduced from RL to R, as the S�Tr link onlycollects diversity 1, and consequently becomes moreunreliable (there are more relaying errors). Fig. 11 repeatsthis test using now SD-OSTBC OFDM transmissions. Notethat in some cases (R¼3 or R¼2 from 17 dB) LAR isoutperformed by SF. This is because transmissions witherrors affect the OSTBC matrix orthogonality. This effect isnot present when using DD-TDM OFDM transmissions (seeFig. 11) because the diagonal matrix structure is alwayspreserved; the transmitted matrix is always diagonal. Inany case, LAR transmissions are still able to combat errorsmore efficiently than the non-adaptive DF [2] (settingαr ¼ 1 8r), which is also included for comparison. Figs. 10and 11 also include two asymptotes with diversity order 4

Fig. 10. SF vs. LAR DD-TDM OFDM for R¼2,3; and L¼2. Also includedasymptotes with diversity of order 4 (narrow gray line) and 6 (thickgray line).

and 6, respectively. You can use these asymptotes tocorroborate that SF and LAR SD-OSTBD OFDM tend toachieve diversity of order RL for high SNR (from 16 dB and20 dB respectively with R¼2 and R¼3), whereas SF andLAR DD-TDM OFDM need higher SNR to approach thisdiversity order because of its higher coding gain losses.

7. Conclusions

Differential modulation schemes that do not requirechannel state or synchronization information are developed.With the DD-TDM scheme, TDM transmissions using diagonalDD ST mappings are shown to effectively bypass channel andCFO knowledge at the destination. Precoded differentialOFDM transmissions can additionally bypass timing offsetsusing either: (i) DD-TDM OFDM scheme (diagonal DD codingacross OFDM blocks); and (ii) SD-OSTBC OFDM scheme(SD-OSTBC within OFDM blocks).

DD-TDM has the lowest complexity, but it is vulnerable totiming errors among relays or multipath effects. The differ-ential OFDM schemes deal with timing/multipath effects andcollect space and multipath diversity: DD-TDM OFDM trans-missions are robust to any CFO range, whereas SD-OSTBCOFDM achieve higher coding gains and have simpler decodingcomplexity, at the expense of limited CFO mitigation range.

Performance analysis assuming adaptive/selective decode-and-forward relaying is also provided, showing that diversitycan still be achieved even when errors are present. Simula-tions corroborated theoretical claims and showed that the SD-OSTBC OFDM scheme using SF, due to its simple decoding,relative robustness, and considerably higher coding gain canbe preferable in applications with small CFO deviations.

Appendix A. Derivation of (7) and (8)

Eqs. (7a)–(7c) are derived from (6) by substituting andexpanding the recursive equations in (2a) and (2b). Spe-cifically, from (2b) we can expand xk, for kZ3, as

xk ¼Dgkxk�1 ¼DgkDgk� 1xk�2

A. Cano et al. / Signal Processing 105 (2014) 30–42 41

and, from (2a), we can expand gk as

DgkDgk� 1xk�2 ¼DvkDgk� 1

Dgk� 1xk�2 ¼DvkD

2gk� 1

xk�2;

where the latter equality is possible because Dgk� 1is a

diagonal matrix. Substituting xk ¼DvkD2gk� 1

xk�2 and rewrit-ing Dωk ¼Dωk� 1Dω1 yields (7a). To derive (7b), we only needto expand xk�1 ¼Dgk� 1

xk�2. To derive (7c), we only need torewrite Dωk� 2 ¼Dωk� 1Dω� 1 .

Eq. (8) is derived by expanding the products Dykyn

k�1and Dyk� 2

yn

k�2. Substituting yk and yk�1 from (7a) and(7b), respectively, and ignoring the noise term for simpli-city, yields

Dykyn

k�1 ¼DhDωDωk� 1Dω1DvkD2gk� 1

Dxk� 2 ðDhDωDωk� 1Dgk� 1xk�2Þn

¼DvkDjhj2Dω1gk�1: ðA:1Þ

The last equality follows because, except for Dh, allmatrices involved are diagonal and unitary, and so: (1)we can freely apply commutative property and (2) con-jugate multiplication is the identity vector/matrix:

Dωk� 1Dn

ωk� 1¼ I;

Dxk� 2xn

k�2 ¼ 1;

D2gk� 1

Dn

gk� 1¼Dgk� 1

:

Likewise, substituting yk�1 and yk�2 from (7b) and (7c),respectively, we obtain:

Dyk� 1yn

k�2 ¼DhDωDωk� 1Dgk� 1Dxk� 2 ðDhDωDωk� 1Dω� 1xk�2Þn

¼Djhj2Dω1gk�1: ðA:2Þ

As seen, (A.1) and (A.2) are the same except for the Dvkterm, hence (in the absence of noise):

Dykyn

k�1 ¼DvkDyk� 1yn

k�2: ðA:3ÞIntroducing noise, and after a straightforward calculations,the term z0k in (9) can be likewise derived.

Appendix B. Derivation of (15)

Matrix Hr is circulant and can be diagonalized (pre)multiplying by (I)FFT matrices; i.e., HrFHML ¼ FHMLDhr

with

Dhr≔

ffiffiffiffiffiffiffiML

pdiagðFð1:LÞML hrÞ and Fð1:LÞML being a matrix formed by

the first L columns of FML. Using this fact in (14), andsubstituting (12), it holds that

HrFHMLx r;k ¼ffiffiffiffiffiffiffiML

pFHML diagð1L � ðFM ~x r;kÞÞFð1:LÞML hr : ðB:1Þ

In [16, Appendix A] it is shown that for any M � 1 vector uit further holds thatffiffiffiffiffiffiffiML

pFHMLðIL � diagðuÞÞFð1:LÞML ¼

ffiffiffiL

pððFHMuÞ � ILÞ: ðB:2Þ

Substituting u¼ FM ~x r;k, Eq. (15) follows.

Appendix C. Derivation of (31)

Consider first the DD-TDM OFDM scheme. For simpli-city, we set M¼1 and thus subindex m will be henceforthremoved. Define vector rk;ℓ≔Σ1=2

k;ℓ Dyk;ℓyn

k�1;ℓ. The detectorin (17) can be written as

Dvk ¼ argminDv

∑L�1

ℓ ¼ 0Jrk;ℓ�Dvrk�1;ℓ J2

� �: ðC:1Þ

Let ~Dvk be the transmitted matrix by the relays, which isdifferent from the error-free Dvk . With appropriate con-stants, the block error probability Prðs; ~s-sjhÞ can bebounded by the symbol error probability. Since there's aone-to-one mapping from each symbol (entry) in s, ~s and sto Dvk , ~Dvk and Dvk , the block error probability is thusgiven by

Prðs; ~s-sjhÞrPr ∑L�1

ℓ ¼ 0Jrk;ℓ�Dvkrk�1;ℓ J2o ∑

L�1

ℓ ¼ 0Jrk;ℓ�Dvkrk�1;ℓ J2

� �

ðC:2Þwith ~Dvk being the equivalent matrix transmitted by allrelays. We approximate rk;ℓ as rk;ℓ

ffiffiffiL

pDvk Σ�1=2

k;ℓ~Dωk;ℓ

hðαÞℓ þz″k;ℓ, with ½hðαÞ

ℓ �r≔ffiffiffiffiffiαr

p jhr;ℓj and z″k;ℓ Gaussian. Ignor-ing Σk;ℓ, the probability of error in (C.2) can be written asPrðX40Þ, where X is Gaussian with respective mean andvariance

μ≔ ∑L�1

ℓ ¼ 0J ð ~Dvk �Dvk ÞhðαÞ

ℓ J2� J ð ~Dvk �Dvk ÞhðαÞℓ J2

s≔ ∑L�1

ℓ ¼ 0J ð ~Dvk �Dvk ÞhðαÞ

ℓ J2:

The probability of error in (C.2) can then be bounded as

Prðs; ~s-s h�� ÞrQ

∑L�1ℓ ¼ 0 Jð ~Dvk �Dvk ÞhðαÞ

ℓ J2�∑L�1ℓ ¼ 0 J ð ~Dvk �Dvk ÞhðαÞ

ℓ J2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2∑L�1

ℓ ¼ 0 Jð ~Dvk �Dvk ÞhðαÞℓ J2þ2∑L�1

ℓ ¼ 0 J ð ~Dvk �Dvk ÞhðαÞℓ J2

q0B@

1CA:

ðC:3ÞThe terms inside the exponential can be bounded as, up tomultiplicative constants,

J ð ~Dvk �Dvk ÞÞhðαÞℓ J2Z ∑

rACαr ∑

L�1

ℓ ¼ 0jhr;ℓj2

J ð ~Dvk �Dvk ÞÞhðαÞℓ J2r ∑

rAEαr ∑

L�1

ℓ ¼ 0jhr;ℓj2:

Substituting in (C.3) and using the inequality

Qa�bffiffiffiffiffiffiffiffiffiffiaþb

p� �

rQa0 �b0ffiffiffiffiffiffiffiffiffiffiffiffiffia0 þb0

p !

whenever a0ra and b0Zb, Eq. (31) yields.A similar argument can be carried for OSTBCs by simply

substituting rk;ℓ≔yk;ℓ in (C.1) and approximating it (inthe absence of CFO) after Eq. (C.2) as rk;ℓ

ffiffiffiL

pVTkXk�1

hðαÞℓ þz″k;ℓ.

Appendix D. Proof of Proposition 4

In the SF protocol, αr ¼ f0;1g and thus the expectationof (32) over the channel can be split as the product of twoexpectations

Eh½Prðs-sjhÞ�r∑EEh exp � ∑

rAE∑L�1

ℓ ¼ 0jhðsÞr;ℓj2

� ��

�Eh exp � ∑rAE

∑L�1

ℓ ¼ 0jhr;ℓj2

!" #: ðD:1Þ

Using the Rayleigh channel model, the expectations in(D.1) can be easily computed yielding, up to multiplicativeconstants,

Eh½Prðs-sjhÞ�r∑Eγ �jEjLγ �ðR�jEjÞLrγ �RL ðD:2Þ

A. Cano et al. / Signal Processing 105 (2014) 30–4242

The second term shows that diversity at the destination isreduced to ðR�jEjÞL. However the probability of thathappening behaves with diversity jEjL, as seen in the firstterm. The product of both terms shows full diversity oforder RL is achieved.

Next we deal with αr defined as in (29). In this caseboth Pe;hðEÞ and Prðs; ~s-sjh; EÞ are coupled through αrand the expectation of (32) cannot be partitioned into aproduct of two terms. Alternatively, one can use thedefinition of αr in (29) to bound the terms inside (31) asfollows:

∑rAE

αr ∑L�1

ℓ ¼ 0jhr;ℓj2Z

minℓ;rAE fjhr;ℓj2; γgγ

!∑rAE

∑L�1

ℓ ¼ 0jhr;ℓj2

ðD:3Þ

∑rAE

αr ∑L�1

ℓ ¼ 0jhr;ℓj2r ∑

rAEαr

� �∑rAE

∑L�1

ℓ ¼ 0jhr;ℓj2

r ∑rAE

∑L�1

ℓ ¼ 0jhðsÞr;ℓj2

� �∑rAE

∑L�1

ℓ ¼ 0

jhr;ℓj2γ

: ðD:4Þ

For compactness, define the terms inside (D.3) and (D.4) as

λE≔ ∑rAE

∑L�1

ℓ ¼ 0jhr;ℓj2; λE≔ ∑

rAE∑L�1

ℓ ¼ 0jhðsÞr;ℓj2

κE≔ ∑rAE

∑L�1

ℓ ¼ 0

jhr;ℓj2γ

; κE≔minℓ;rAE fjhr;ℓj2; γg

γ

With these definitions, the product in (32) can bebounded as

Pr s-sjh �rexp �λEð ÞQ κE

λE �κEλEffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2κE λE þ2κEλE

p!

ðD:5Þ

where the inequality

Qa�bffiffiffiffiffiffiffiffiffiffiaþb

p� �

rQa0 �b0ffiffiffiffiffiffiffiffiffiffiffiffiffia0 þb0

p !

whenever a0ra and b0Zb have been used. The randomvariable λE (λE) is Gamma-distributed with jE jL (jEjL)degrees of freedom. The random variables κE and κE areindependent of the SNR γ , since they are divided by it.In [2, Lemma 2] it is shown that a expression of theform (D.5) with the conditions above-mentioned can bebounded by a Gamma-distributed random variable withjE jLþjEjL¼ RL degrees of freedom. Averaging over thechannel distribution the diversity result of Proposition 4follows.

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