Leveraging Coherent Distributed Space-Time Codes for Noncoherent Communication in Relay Networks via Training
aa r X i v : . [ c s . I T ] S e p Leveraging Coherent Distributed Space-TimeCodes for Noncoherent Communication inRelay Networks via Training
G. Susinder Rajan and B. Sundar Rajan
Abstract
For point to point multiple input multiple output systems, Dayal-Brehler-Varanasi have provedthat training codes achieve the same diversity order as that of the underlying coherent space timeblock code (STBC) if a simple minimum mean squared error estimate of the channel formed usingthe training part is employed for coherent detection of the underlying STBC. In this letter, a similarstrategy involving a combination of training, channel estimation and detection in conjunction withexisting coherent distributed STBCs is proposed for noncoherent communication in AF relay net-works. Simulation results show that the proposed simple strategy outperforms distributed differentialspace-time coding for AF relay networks. Finally, the proposed strategy is extended to asynchronousrelay networks using orthogonal frequency division multiplexing.
Index Terms
Cooperative diversity, distributed STBC, noncoherent communication, training.
I. I
NTRODUCTION
Recently the idea of space time coding has been applied in wireless relay networks inthe name of distributed space time coding to extract similar benefit as in point to pointmultiple input multiple output (MIMO) systems. Mainly there are two types of distributedspace time coding techniques discussed in the literature: (i) decode and forward (DF) baseddistributed space time coding [1], wherein a subset (chosen based on some criteria) of therelay nodes decode the symbols from the source and transmit a row/column of a distributed G. Susinder Rajan and B. Sundar Rajan are with the Department of Electrical Communication Engineering, IndianInstitute of Science, Bangalore-560012, India. Email: { susinder,bsrajan } @ece.iisc.ernet.in. Whether the relay transmits a column of a STBC or a row of a STBC depends on the system model.
November 15, 2018 DRAFT space time block code (STBC) and (ii) amplify and forward (AF) based distributed spacetime coding [2], where all the relay nodes perform linear processing on the received symbolsaccording to a distributed space time block code (DSTBC) and transmit the resulting symbolsto the destination. AF based distributed space time coding is of special interest becausethe operations at the relay nodes are greatly simplified and moreover there is no need forevery relay node to inform the destination once every quasi-static duration whether it will beparticipating in the distributed space time coding process as is the case in DF based distributedspace time coding [1]. However, in [2], the destination was assumed to have perfect knowledgeof all the channel fading gains from the source to the relays and those from the relays tothe destination. To overcome the need for channel knowledge, distributed differential spacetime coding was studied in [3], [4], [5], [6], which is essentially an extension of differentialunitary space time coding for point to point MIMO systems to the relay network case. Butdistributed differential space time block code (DDSTBC) design is difficult compared tocoherent DSTBC design because of the extra stringent conditions (we refer readers to [4],[6] for exact conditions) that need to be met by the codes. Moreover, all the codes in [3],[4], [5] for more than two relays have exponential encoding complexity. On the other hand,coherent DSTBCs with reduced maximum likelihood (ML) decoding complexity are availablein [8], [15], [18].Interestingly in [9], it was proved that for point to point MIMO systems, training codes achieve the same diversity order as that of the underlying coherent STBC. This was shown tobe possible if a simple minimum mean squared error (MMSE) estimate of the channel formedusing the training part of the code is employed for coherent detection of the underlying STBC.The contributions of this letter are summarized as follows. • Motivated by the results of [9], a similar training and channel estimation scheme isproposed to be used in conjunction with coherent distributed space time coding in AFrelay networks as described in [2]. An interesting feature of the proposed training schemeis that the relay nodes do not perform any channel estimation using the training symbolstransmitted by the source but instead simply amplify and forward the received trainingsymbols. The proposed strategy is shown to outperform the best known DDSTBCs [3],[4], [5], [6] using simulations. Also, it is shown that appropriate power allocation among Each codeword of a training code consists of a part known to the receiver (pilot) and a part that contains codeword(s)of a STBC designed for the coherent channel (in which receiver has perfect knowledge of the channel)
November 15, 2018 DRAFT the training and data symbols can further improve the error performance marginally. • Finally, this training based strategy is extended to asynchronous relay networks withno knowledge of the timing errors using the recently proposed Orthogonal FrequencyDivision Multiplexing (OFDM) based distributed space time coding [7].The rest of this letter is organized as follows. The proposed training scheme along withchannel estimation is described in Section II. Extension to the asynchronous relay networkcase is addressed in Section III. Simulation results comprise Section IV and a conclusionsare presented in Section V.
Notation:
Vectors and matrices are represented by lowercase and uppercase boldfacecharacters respectively. An identity matrix of size N × N will be denoted by I N . A complexGaussian vector with zero mean and covariance matrix Ω will be denoted by CN (0 , Ω ) .II. P ROPOSED T RAINING B ASED S TRATEGY
In this section, we briefly review the distributed space time coding protocol for AF relaynetworks in [2], make some crucial observations and then proceed to describe the proposedtraining based strategy.Consider a wireless relay network consisting of a source node, a destination node and R relay nodes U , U , . . . , U R which aid the source in communicating information to thedestination. All the nodes are assumed to be equipped with a half duplex constrained, singleantenna transceiver. The wireless channels between the terminals are assumed to be quasi-static and flat fading. The channel fading gains from the source to the i -th relay, f i and thosefrom the i -th relay to the destination g i are all assumed to be independent and identicallydistributed (i.i.d) complex Gaussian random variables with zero mean and unit variance.Symbol synchronization and carrier frequency synchronization are assumed among all thenodes. A. Observations from Coherent Distributed Space Time Coding
In order to explain coherent distributed space time coding, we shall assume in this sub-section alone that the destination has perfect knowledge of all the channel fading gains f i , g i , i = 1 , . . . , R . Every transmission cycle from the source to the destination is comprisedof two phases. In the first phase, the source transmits a vector z = h z z . . . z T i T composed of T complex symbols z i , i = 1 , . . . , T to all the R relays using a fraction π ofthe total power P d for data transmission. The vector z satisfies E[ z H z ] = T and P d denotes November 15, 2018 DRAFT the total average power spent by the source and the relays for communicating data to thedestination. The received vector at the i -th relay is then given by r i = √ π P d f i z + v i where, v i ∼ CN (0 , I T ) represents the additive noise at the i -th relay.In the second phase, the i -th relay transmits t i = q π P d π P d +1 B i r i or t i = q π P d π P d +1 B i r ∗ i tothe destination, where B i ∈ C T × T is called the ‘relay matrix’. Without loss of generalitywe may assume that the first M relays linearly process r i and the remaining R − M relayslinearly process r i ∗ . Under the assumption that the quasi-static duration of the channel is muchgreater than R channel uses, the received vector at the destination can be expressed as y = P Ri =1 g i t i + w = q π π P d π P d +1 Xh + n where, X = h B z . . . B M z B M + z ∗ . . . B R z ∗ i , h = h f g f g . . . f M g M f ∗ M +1 g M +1 . . . f ∗ R g R i T , (1) n = q π P d π P d +1 (cid:16)P Mi =1 g i B i v i + P Ri = M +1 g i B i v ∗ i (cid:17) + w and w ∼ CN (0 , I T ) represents theadditive noise at the destination. The power allocation factors π and π are chosen to satisfy π P d + π P d R = 2 P d . The covariance matrix of n is given by Γ = E[ nn H ] = I T + π P d π P d +1 ( P Ri =1 | g i | B i B Hi ) . Let the DSTBC C denote the set of all possible codeword matrices X . Then the ML decoder is given by ˆX = arg min X ∈ C k Γ − ( y − s π π P d π P d + 1 Xh ) k F . (2)Note from (2) that the ML decoder in general requires the knowledge of all the channelfading gains f i , g i , i = 1 , . . . , R . Consider the following decoder: ˆX = arg min X ∈ C k y − s π π P d π P d + 1 Xh k F . (3) Remark 1:
The decoder in (3) is suboptimal in general and coincides with the ML decoderfor the case when Γ is a scaled identity matrix. The relay matrices for all the codes in [2],[15], [17], [18] and some of the codes in [8] are unitary. For the case when B i B Hi is adiagonal matrix for all i = 1 , , . . . , R ( Γ is a diagonal matrix for this case), the performanceof the suboptimal decoder in (3) differs from that of the ML decoder (2) only by codinggain and the diversity gain is retained. This can be proved on similar lines as in the proof ofTheorem 7 in [8]. The class of DSTBCs from precoded co-ordinate interleaved orthogonaldesigns in [8] is an example for the case of diagonal Γ matrix. Γ requires knowledge of the g i ’s and h requires knowledge of f i g i , i = 1 , . . . , M and f ∗ i g i , i = M + 1 , . . . , R whichtogether imply knowledge of f i , g i , i = 1 , . . . , R . November 15, 2018 DRAFT
The decoder in (3) requires only the knowledge of h and not necessarily the knowledgeof all the individual channel fading gains f i , g i , i = 1 , , . . . , R . The training strategy to bedescribed in the sequel essentially exploits this crucial observation. B. Training cycle
Note from the previous subsection that one data transmission cycle comprises of T + T channel uses. In the proposed training strategy, we introduce a training cycle comprising of R + 1 channel uses for channel estimation before the start of data transmission cycle. Weassume that the quasi-static duration of the channel is greater than ( R + 1) + F ( T + T ) channel uses where, F denotes the total number of data transmission cycles that can beaccommodated within the channel quasi-static duration. Thus, for F = 1 , T = T = R ,the minimum channel quasi-static duration required for the proposed strategy is R + 1 channel uses. Let P t be the total average power spent by the source and the relays duringthe training cycle. Thus, the total average power P used by the source and the relays is P = P t ( R +1)+ P d F ( T + T ) R +1+ F ( T + T ) .In the first phase of the training cycle, the source transmits the complex number to all therelays using a fraction π of the total power P t dedicated for training. The received symbolat the i -th relay denoted by ˆ r i is given by ˆ r i = √ π P t f i + ˆ v i where ˆ v i ∼ CN (0 , is theadditive noise at the i -th relay.The second phase of the training cycle comprises of R channel uses, out of which onechannel use is assigned to every relay node. Without loss of generality, we may assume thatthe i -th time slot is assigned to the i -th relay. Furthermore, we assume that the value of M to be used during the data transmission cycle is already decided. During its assigned timeslot, the i -th relay transmits ˆ t i = q π P t Rπ P t +1 ˆ r i = q π π P t Rπ P t +1 f i + q π P t Rπ P t +1 ˆ v i , if i ≤ M q π P t Rπ P t +1 ˆ r ∗ i = q π π P t Rπ P t +1 f ∗ i + q π P t Rπ P t +1 ˆ v ∗ i , if i > M .At the end of the training cycle, the received vector ˆ y at the destination is given as follows: ˆy = s π π P t Rπ P t + 1 I R h + ˆn (4)where, ˆn = q π P t Rπ P t +1 h g ˆ v . . . g M ˆ v M g M +1 ˆ v ∗ M +1 . . . g R ˆ v ∗ R i T + ˆw , h is same asthat given in (1) and ˆw ∼ CN (0 , I R ) is the additive noise at the destination. The entiretransmission from source to destination is illustrated pictorially in Fig. 1 and Fig. 2.Note that the entries of h as well as ˆn are not complex Gaussian distributed since theyinvolve terms that are product of complex Gaussian random variables. To be precise, the November 15, 2018 DRAFT entries of h are i.i.d random variables with mean and variance . Similarly, the entriesof n t are i.i.d random variables with mean and variance ( π P t Rπ P t +1 + 1) . For the point topoint MIMO case, where the channel and additive receiver noise are modeled as complexGaussian, Dayal-Brehler-Varanasi in [9] have proposed a simple linear channel estimator. Inthis letter, we propose to employ a similar estimator for the equivalent channel h as follows: ˆ h = s π π P t Rπ P t + 1 (cid:18) π P t R + π π P t Rπ P t + 1 + 1 (cid:19) − ˆy (5)Now using the estimate ˆ h , coherent DSTBC decoding can be done in every data transmis-sion cycle, as ˆX = arg min X ∈ C k y − q π π P d π P d +1 X ˆ h k F . Thus, coherent DSTBCs [8], [15],[16], [17], [18] can be employed in noncoherent relay networks via the proposed trainingscheme. We would like to mention that there may be better channel estimation techniquesthan the one described by (5), but this is beyond the scope of this letter. However, thesimulation results in section V show that a simple channel estimator as in (5) is good enoughto outperform the best known DDSTBCs.III. T RAINING S TRATEGY F OR A SYNCHRONOUS R ELAY N ETWORKS
The training strategy described in the previous section assumes that the transmissionsfrom all the relays are symbol synchronous with reference to the destination. In this section,we relax this assumption and extend the proposed training strategy to asynchronous relaynetworks with no knowledge of the timing errors of the relay transmissions. However weshall assume that the maximum of the relative timing errors from the source to the destinationis known. An asynchronous wireless relay network is depicted in Fig.5. Let τ i denote theoverall relative timing error of the signals arrived at the destination node from the i -th relaynode. Without loss of generality, we assume that τ = 0 , τ i +1 ≥ τ i , i = 1 , . . . , R − .Recently there have been several works [7], [8], [10], [11], [12], [13], [14] on distributedspace time coding for asynchronous relay networks, some of which employ OFDM. Theproposed scheme relies on the OFDM based distributed space time coding in [7], [8], whichis essentially distributed space time coding over OFDM symbols and the cyclic prefix (CP)of OFDM is used to mitigate the effects of symbol asynchronism. The number of sub-carriers N and the length of the cyclic prefix (CP) l cp are chosen such that l cp ≥ max i =1 , ,...,R { τ i } .The channel quasi-static duration assumed for this strategy is (( R + 1) + F (2 R )) ( N + l cp ) channel uses. November 15, 2018 DRAFT
As for the synchronous case, there will be a training cycle before the start of datatransmission from the source. In the first phase of the training cycle, the source takes the N point inverse discrete Fourier transform (IDFT) of the N length vector p = h . . . i T and adds a CP of length l cp to form a OFDM symbol ¯p . This OFDM symbol is transmitted tothe relays using a fraction π of the total power P t . The i -th relay receives ˆr i = √ π P t f i ¯p + ¯ˆv i where ¯ˆv i ∼ CN (0 , I N + l cp ) is the additive noise at the i -th relay. The second phase ofthe training cycle comprises of R OFDM time slots and the i -th relay is allotted the i -thOFDM time slot for transmission. During its scheduled time slot, the i -th relay transmits ˆt i = q π RP t π P t +1 ˆr i , if i ≤ M q π RP t π P t +1 ζ (( ˆr i ) ∗ ) , if i > M where ζ ( . ) denotes the time reversal operation, i.e., ζ ( r ( n )) , r ( N + l cp − n ) . The destination receives R OFDM symbols which are processedas follows:1) Remove the CP for the first M OFDM symbols.2) For the remaining OFDM symbols, remove CP to get a N -length vector. Then shiftthe last l cp samples of the N -length vector as the first l cp samples.Discrete Fourier transform (DFT) is then applied on the resulting R vectors to obtain ˆx j = h ˆ y ,j ˆ y ,j . . . ˆ y N − ,j i T , j = 1 , , . . . , R . Let ˆw j = h ˆ w ,j ˆ w ,j . . . ˆ w N − ,j i T represent the additive noise at the destination node in the j -th OFDM time slot and let ˆv j = h ˆ v ,j ˆ v ,j . . . ˆ v N − ,j i T denote the DFT of ¯ˆv j after CP removal. Note that a delay τ in the time domain translates to a phase change of e − i πkτN in the k -th sub carrier. Now usingthe identities (DFT( x )) ∗ = IDFT( x ∗ ) , (IDFT( x )) ∗ = DFT( x ∗ ) , DFT( ζ (DFT( x ))) = x , p ∗ = p we have in the j -th OFDM time slot ˆx j = f j g j q π π RP t π P t +1 p ◦ d τ j + q π RP t π P t +1 g j ˆv j ◦ d τ j + ˆw j if j ≤ Mf ∗ j g j q π π RP t π P t +1 p ◦ d τ j + q π RP t π P t +1 g j ˆv ∗ j ◦ d τ j + ˆw j if j > M where, d τ j = h e − i πτjN . . . e − i πτj ( N − N i T and ◦ denotes Hadamard product. Thus, ineach sub-carrier k, ≤ k ≤ N − , we get ˆy k = h ˆ y k, ˆ y k, . . . ˆ y k,R i T = s π π RP t π P t + 1 I R h k + ˆn k (6)where, h k = h f g u τ k f g . . . u τ M k f M g M u τ M +1 k f ∗ M +1 g M +1 . . . u τ R k f ∗ R g R i T , (7) November 15, 2018 DRAFT u τ i k = e − i πkτiN and ˆn k = q π P t Rπ P t +1 h u τ k g ˆ v k, . . . u τ M k g M ˆ v k,M u τ M +1 k g M +1 ˆ v ∗ k,M +1 . . . u τ R k g R ˆ v ∗ k,R i + h ˆ w k, ˆ w k, . . . ˆ w k,R i T . Analogous to the synchronous case, we propose to estimate the equivalent channel matrix h k from (6) as ˆh k = q π π RP t Rπ P t +1 (cid:16) π P t R + π π P t Rπ P t +1 + 1 (cid:17) − ˆy k . After the training cycle, thedata transmission cycle starts for which refer the readers to [7] and section IV of [8] fora detailed explanation. In essence, a DSTBC is seen by the destination in every sub-carrierand the equivalent channel seen by the destination in the k -th sub-carrier is precisely thematrix h k , whose estimated value is available at the end of the training cycle. As for thesynchronous case (see (3)), we propose to ignore the covariance matrix of the equivalentnoise while performing data detection .IV. S IMULATION R ESULTS
In this section, simulations are used to compare the error performance of the proposedstrategy against the best known DDSTBC for relays[4] and relays[6]. Note that for relays, the DDSTBCs in [6] were shown to outperform the codes reported in [3], [4], [5] inboth complexity as well as performance. For all the simulations, we set π = 1 , π = R (assuggested in [2]), T = T = 4 and F = 50 . The channel fading gains f i , g i , i = 1 , . . . , R areeach generated independently following a complex Gaussian distribution with mean andunit variance . The decoder used for the proposed scheme is the one described by (3) andfor the DDSTBC case, the decoder proposed in [6] has been used. We chose P t = (1 + α ) P d ,where α denotes the power boost factor to allow for power boosting to the pilot symbols.In order to quantify the loss in error performance due to channel estimation errors in theproposed strategy, the performance of the corresponding coherent DSTBC (assuming perfectchannel knowledge) is taken as the reference.For a relay network, the Alamouti code is applied both as a DDSTBC[4] and as theunderlying coherent STBC in the proposed training scheme. The signal constellation ischosen to be 4-QAM and 16-QAM for rates of and bpcu respectively. Fig. 3 shows theerror performance of the proposed strategy in comparison with Alamouti DDSTBC and the This is a suitable assumption for the case when the relays are approximately equidistant from both the source as wellas the destination.
November 15, 2018 DRAFT corresponding coherent DSTBC for α = 0 and transmission rates of bits per channel use(bpcu) and bpcu respectively. It can be observed that the proposed scheme has marginallybetter performance compared to the DDSTBC strategy for transmission rates of and bpcu.Note that the performance advantage of the proposed strategy over the DDSTBC strategy ismore for the 2 bpcu case.For a relay network, the coherent DSTBC employed in the proposed strategy for simula-tions is z z − z ∗ − z ∗ z z − z ∗ − z ∗ z z z ∗ z ∗ z z z ∗ z ∗ where { Re( z ) , Re( z ) } , { Re( z ) , Re( z ) } , { Im( z ) , Im( z ) } and { Im( z ) , Im( z ) } take values from quadrature amplitude modulation (QAM) rotated by . ◦ (QAM constellation size chosen based on transmission rate). The relay matricescorresponding to this coherent DSTBC are unitary and M = 2 . The DDSTBC taken forcomparison is the one reported recently in [6]. It can be observed from Fig. 4 that for arate of bpcu and codeword error rate (CER) of − , the proposed strategy outperformsthe DDSTBC of [6] by approximately dB for α = 0 . For a transmission rate of bpcu,the performance gap between the proposed strategy and the DDSTBC of [6] increases to dB. Finally, observe that a power boost to the pilot symbols gives marginally betterperformance (gain of . dB).From all the above simulations, we infer that the performance advantage of the proposedstrategy over DDSTBCs increases as the transmission rate increases. Also, note that theproposed strategy is better than the DDSTBCs of [6], [4] at all signal to noise ratio (SNR).In spite of the simple channel estimation method employed (Eq. (5)), note from Fig.3 andFig.4 that the performance loss due to channel estimation errors is only about dB fortransmission rates of and bpcu respectively. We can attribute three reasons for the proposedstrategy to outperform DDSTBCs as follows: (1) lesser equivalent noise power seen by thedestination during data transmission cycle as compared to distributed differential space timecoding [3], [4], [5], [6], (2) no restriction of coherent DSTBC codewords to unitary/scaledunitary matrices as is the case with DDSTBCs [3], [4], [5], [6] and (3) the relay matrices B i , i = 1 , , . . . , R need not satisfy certain algebraic relations involving the codewords (see[4], [6] for exact relations), thus giving more room to optimize the minimum determinant of When calculating transmission rate, the rate loss due to initial few channel uses for training is ignored ( R + 1 forproposed strategy and R for DDSTBC [3], [4], [5], [6]). November 15, 2018 DRAFT0 difference matrices (coding gain).Simulation results are not reported for the asynchronous case because the use of OFDMessentially makes the signal model in every sub-carrier similar to the synchronous case. Exceptfor a rate loss due to CP, the performance will thus be same as that for the synchronous case.V. C
ONCLUSION
Similar to the results of [9] for point to point MIMO systems, a simple training and channelestimation scheme combined with the protocol in [2] was shown to outperform distributeddifferential space time coding at all SNR. The proposed strategy leverages existing coherentDSTBCs [8], [15], [16], [17], [18] for noncoherent communication in AF relay networks.Finally, the proposed strategy is extended for application in asynchronous relay networks withno knowledge of the timing errors using OFDM. Some of the interesting directions for furtherwork are: (1) design of optimal training sequences, (2) better channel estimation techniquesand (3) optimal power allocation between the training cycle and the data transmission cycle.R
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November 15, 2018 DRAFT2
Terminal Slot Slot . . . Slot M + 1 Slot M + 2 . . . Slot R + 1 Source √ π P t Relay q π π P t Rπ P t +1 f + q π P t Rπ P t +1 n ... . ..Relay M q π π P t Rπ P t +1 f M + q π P t Rπ P t +1 n M Relay M + 1 q π π P t Rπ P t +1 f ∗ M +1 + q π P t Rπ P t +1 n ∗ M +1 ... .. .Relay R q π π P t Rπ P t +1 f ∗ R + q π P t Rπ P t +1 n ∗ R Fig. 1. Training cycleTerminal Data transmission Data transmissioncycle . . . cycle F Phase I Phase IISlots R + 2 Slots R + 2 Slots R (2 F −