DSTC Layering Protocols in Wireless Cooperative Networks
aa r X i v : . [ c s . N I] M a r DSTC Layering Protocols in Wireless RelayNetworks
Pannir Selvam Elamvazhuthi, Parag Shankar Kulkarni, and Bikash Kumar Dey,
Member, IEEE
Abstract — With multiple antennas at transmitter and receiver,rate of transmission and reliability of information are improved.When there is no possibility of increasing the number of antennas,for example in mobile handsets, sensor networks, etc., the benefitsof multiple antenna systems are obtained by cooperation amongstindividual radio nodes .In literature, cooperation amongst two users having singleantenna each, attempting to send independent data to the samedestination has been studied by many authors and variousstrategies have been formulated. Studies have been carried out touse relays with single antenna each, to convey information froma single source to a destination. Distributed space-time codinghas been proposed which does not require orthogonal channelsto be allocated to various transmitting units, leading to betterutilization of the spectrum. Some latest literature analyze caseswhen the relays have multiple antennas also.Our system model consists of a source-destination pair withtwo layers of relays in which ‘weaker’ links between source andsecond layer and between the first layer and destination are alsoconsidered. We propose five different protocols out of which oneis a straight forward extension of an existing system, which isused for comparison.We have derived the signal-to-noise ratio at the destinationfor all the protocols and by maximizing this, found the optimumpower to be allocated to various relay and source transmissions.We also show that under reasonable channel strength of the‘weaker’ links, the proposed protocols perform ( ≈ dB) betterthan the existing basic protocol. As expected, the degree ofimprovement increases with the strength of the weaker links.We have also shown that if receive channel knowledge isavailable with 50% of the relays, reliability and data rate can beincreased by adopting a technique proposed in this paper. I. I
NTRODUCTION
The enormous potential of multiple-input multiple-output(MIMO) communications has drawn considerable attentionfrom the scientific community since more than a decade. Twoof the benefits of MIMO are the spatial multiplexing anddiversity gains over that of single antenna systems. Installingmore antennas in a small equipment may be infeasible dueto space constraints. Therefore virtual MIMO systems con-stituting multiple wireless systems started to take shape. Asthe name suggests virtual MIMO is to simulate multiple-antenna system without having one. This can be achieved bycooperation amongst multiple radio nodes and is known as
This work has been carried out at Indian Institute of Technology Bombay(IITB), Mumbai,India.P.S. Elamvazhuthi, a research scholar at IITB, India, is with CognizantTechnology Solutions India Pvt. Ltd., Chennai, India (e-mail: [email protected]).P.S. Kulkarni, after completing M.Tech. at IITB in 2008, working forJuniper Networks Inc., Bangalore, India (e-mail: [email protected]).B.K. Dey is with the Department of Electrical Engineering, IITB (e-mail:[email protected]). cooperative communication , as the radio nodes cooperate witheach other to obtain virtual MIMO.In literature, cooperation amongst two users having singleantenna each, attempting to send independent data to the samedestination has been studied by many authors and variousstrategies have been formulated. Studies have been carried outto use relays with single antenna each, to convey informationfrom a single source to one destination. Distributed space-timecoding (DSTC) has been proposed which does not requireorthogonal channels to be allocated to various transmittingunits with single antenna each, leading to better utilizationof the spectrum.The choice of the right space-time coding (STC) in adistributed fashion depends on the requirement. Orthogo-nal space-time block coding (OSTBC) [1] can be selectedto maximize the diversity gain and minimize the receivercomplexity. Codes [2], [3], [4] that maximize both diversitygain and transmission rate, but with a rather high receivercomplexity are also available to choose from. Bell Laboratorieslayered space-time (BLAST) codes [5] or codes with trace-orthogonal design (TOD) [6] can also be selected. BLASTcodes maximize the rate, sacrificing part of the diversity gain,but with intermediate receiver complexity and the TOD has aflexible way to trade complexity, bit rate, and bit error rate.Sendonaris et al. considered a system ([7]) with one desti-nation and two sources cooperating with each other to achievebetter performance. DSTC proposed by Laneman and Wornell,used a space-time code ([8]) at relays and achieved higherspectral efficiency than repetition-based schemes. Jing andHassibi used a system ([9]) with a layer of relays betweensource and destination and obtained the benefits of DSTC.Borade et al. used multiple layers of radio nodes ([10]) torelay information from source to destination. Here the weakerlinks between non-consecutive layers of nodes were neglected.Amplify-and-forward (e.g. [10]), decode-and-forward (e.g.[8]), coded cooperation (e.g. [11]), and simple process-and-forward (e.g. [9]) are some of the strategies used.In this paper, we consider a multihop network, as shownin Fig. 2, of single-antenna radio nodes with two layers ofrelays between source and destination. We adopt the strategyof simple processing and forwarding at the relays proposedby Jing and Hassibi in [9]. However we also make use of theweaker links between the non-consecutive layers shown bydashed lines in Fig. 2.
A. Motivation
In the previous works, the channels from source or onelayer of relays to the next layer of relays or destination were considered to have same power loss, whereas the channel froma radio node to any other radio node (not in the next layer) wasconsidered to have zero gain. We assume that these channels(we shall call them ‘weak’) have a smaller but non zero gain.We consider schemes which make use of these ‘weak’signals as well. After comparing these schemes using sim-ulations, we come up with simple guidelines to select anappropriate scheme depending on the channel strength (powerloss) and transmitted power. We show that the proposedschemes perform better than the simple extension of the basicprotocol proposed by Jing and Hassibi in [9].
B. Contribution
We have • Proposed five different protocols for our system modeland obtained maximum likelihood (ML) decoders. • Calculated the signal-to-noise ratios (SNRs) at the desti-nation and obtained optimum power allocation for trans-mitters by maximizing SNR. • Analyzed and compared the performances of the pro-posed protocols using simulations, and shown that underreasonable strength of the ‘weak’ channels the proposedprotocols perform better than the basic protocol ([9]). • Showed, using simulations, that we can use random realorthogonal matrices instead of random complex unitarymatrices employed in [9] at the relays.This paper is organized as follows. The system model and theprevious work are detailed in Section II. Thereafter in SectionIII the protocols derived from the basic one proposed in [9]have been analyzed, ML decoding rules have been worked out,and receive SNRs have been derived. In Section IV optimumpower allocations are obtained and BER plots of the protocolsare compared using simulations. Finally in Section V we drawconclusion. II. S
YSTEM M ODEL
A general wireless relay network is depicted in Fig. 1. Onesource (S), one destination (D), and 2 N relays constitute thisnetwork. Let us assume that paths from source to N of theserelays have low power loss and that to rest of the N relays havehigher loss. Assume that transmission is carried out in threedifferent phases. Then these can be grouped into two layers,having N relays each, as shown in Fig. 2. Here firm linesindicate stronger paths with identical low power loss while thedashed lines indicate weaker paths having equal high powerloss.Let us introduce the notations used in this paper now. L andL denote the first and the second layers as shown in Fig. 2.R ij is the j th relay in the i th layer. The channel coefficientsare designated h s, j , h j, l and h l,d for S to R j , R j to R l ,and R l to D respectively. Superscript k , if used in channelcoefficients, denotes the phase.For a complex matrix A , | A | , A H , A T and A ∗ denote deter-minant, Hermitian, transpose and conjugate of A respectively. I T denotes the T × T identity matrix. For a vector a , k a k denotes the norm of a . ⌊ δ ⌋ denotes the biggest integer smallerthan or equal to δ . Fig. 1. A general wireless relay network.Fig. 2. System model.
It is assumed that channels are Rayleigh fading and quasistatic with a coherence interval of at least T symbol duration.The scheme proposed by Jing and Hassibi ([9]) consideredonly S, L , and D in Fig. 2. The channel variance from Sto L and L to D was assumed to be constant at unity bythe authors. The scheme consisted of two phases; in phase1, S transmits and in phase 2, L layer relays encode theirreceived signals using a matrix of their own and transmit toD. The authors proved that this effectively obtains a DSTCand achieves the same diversity as that of a multiple-antennasystem with little degradation. Let us call this basic protocolas Jing Hassibi Scheme (JHS). Now let us prepare to derivedifferent protocols from our model shown in Fig. 2 based onJHS.Assume that s = [ s (1) · · · s ( T )] T is the transmitted sig-nal from S during a block of length T , when the channelcoefficients are assumed to remain constant. Assume alsothat the signal s ∈ Ω = { s , · · · , s L } ⊂ C T × is selectedfrom Ω , whose cardinality is L , for transmission and that s is normalized with E [ s H s ] = 1 . Let r ( k ) ij denote the vectorreceived by the relay R ij in phase k , t ( k ) ij denote the vectortransmitted by R ij in phase k multiplied with a factor and r ( k ) d denote the received vector at destination in phase k in a blockduration T . Let σ be the variance corresponding to the channel coeffi-cients, h s, j , h j, l , h l,d and σ be the variance correspondingto the channel coefficients h s, j , h j,d for ≤ j, l ≤ N. i.e. E [ | h s, j | ] = E [ | h j, l | ] = E [ | h l,d | ] = σ and E [ | h s, j | ] = E [ | h j,d | ] = σ . σ > σ as discussed earlierand also assume with no loss of generality that σ = 1 .Assume that u ( k ) ij and u ( k ) d are the noise vectors added atthe relay R ij and the destination, D respectively during the k th phase. Let the components of these vectors be zero–meanwhite Gaussian independent random variables with variance σ n . By keeping σ n = 1 throughout, SNR is varied by varying P , the total average power per symbol duration of the system.Each of the relays, R ij , have their own matrices, A ij , givenby A ij = (cid:2) a klij (cid:3) (1)which they use to finally produce a distributed space-timecode [9]. Here k and l denote the row and column numbersrespectively. These matrices are random real orthogonal with A ij A T ij = I T and each of the components, a klij is zero meanGaussian independent random variable with variance /T . Theperformance of the system, in fact, has been proved to be thesame, using simulations in Chapter IV, with real orthogonalinstead of complex unitary matrices considered in [9]. Thevector notations used are defined below: r ( k ) ij = r ( k ) ij (1) ... r ( k ) ij ( T ) , t ( k ) ij = t ( k ) ij (1) ... t ( k ) ij ( T ) , r ( k ) d = r ( k ) d (1) ... r ( k ) d ( T ) , u ( k ) ij = u ( k ) ij (1) ... u ( k ) ij ( T ) , and u ( k ) d = u ( k ) d (1) ... u ( k ) d ( T ) . In the next chapter we will derive different protocols from JHSsuggested by Jing and Hassibi [9].III. P
ROTOCOLS DERIVED FROM
JHSFive protocols have been derived from the one proposed in[9]. Let us assume that all these protocols operate in threephases of T symbol duration each, with an available totalaverage power of P T . As the first phase is the same for allthe protocols, we will see it here and see the second and thirdphases in corresponding sections.Refer to the System Model discussed in Chapter II shown inFig. 2. In phase 1, S transmits c s ( τ ) at time τ , for ≤ τ ≤ T .i.e. S transmits c s during T symbol duration, where s =[ s (1) · · · s ( T )] T . R ij receives r (1) ij ( τ ) = c s ( τ ) h s,ij + u (1) ij ( τ ) at time τ and in vector form r (1) ij = c s h s,ij + u (1) ij . (2)To find c the power transmitted by S is to be known. If weassume that p is the power transmitted per symbol durationby S, then p T = E ( c s H s c ) = c ⇒ c = p p T . (3)
Fig. 3. Various phases in RMC/RSC/RMCKC.
The average power received by R j in T symbol duration is E [ r (1) H j r (1)1 j ] = E [( c s H h ∗ s, j + u (1) H j )( c s h s, j + u (1)1 j )]= c E [ | h s, j | ] + E [ u (1) H j u (1)1 j ]= c + T. (4)Equation (4) is arrived at with the assumption that signal,noise, and channel are uncorrelated amongst each other withzero mean. Similarly it can be proved that the power receivedby R j is E [ r (1) H j r (1)2 j ] = σ c + T. (5)Let us see a detailed description of each one of the five derivedprotocols, while also discussing their second and third phases,in the following sections. A. Relay Matrix Combining (RMC)
Different phases of transmission and reception of this pro-tocol are shown in Fig. 3 and explained below: • Phase 1: S transmits; L and L layer relays receive. • Phase 2: L layer relays transmit; L layer relays and Dreceive. • Phase 3: L layer relays transmit and D receives.As the name suggests, this system combines the two vectorsreceived by L in phases 1 and 2, using a matrix beforetransmission in third phase. Let p /N and p /N be the powertransmitted per symbol duration by each of the relays in thesecond and third phases respectively.
1) Protocol Analysis:
In phase 2 the relays R j , ≤ j ≤ N, transmit c t (2)1 j ( τ ) at time τ for ≤ τ ≤ T where t (2)1 j ( τ ) = P Tp =1 a τp j r (1)1 j ( p ) and in vector form t (2)1 j = A j r (1)1 j (6)where A j is shown in (1) with i = 1 . The relays in L receive r (2)2 j and D receives r (2) d . These can be proved to be r (2)2 j = S h s, , j + c N X i =1 h i, j A i u (1)1 i + u (2)2 j (7)and r (2) d = S h s, ,d + u x (8)where S = c c [ A s . . . A N s ] , h s, , j = h s, h , j ... h s, N h N, j , h s, ,d = h s, h ,d ... h s, N h N,d , and u x = c N X i =1 h i,d A i u (1)1 i + u (2) d . (9)Like in JHS [9], it has been proved in (8) that the distributedspace-time code in this case is S and the equivalent channelmatrix is h s, ,d with the equivalent noise vector u x . To find c we require to get an expression for the power transmitted byeach relay, p T /N , which is E h c t (2) H ij t (2) ij c i . The availablepower p T is equally divided amongst N relays as the varianceof the channel coefficients are the same for all of them. Thepower transmitted by each relay can be proved to be c ( p +1) T , which leads to c = r p N ( p + 1) . (10)In phase 3, the two received vectors r (1)2 j and r (2)2 j aretransmitted, by R j , after a matrix combining operation onthe stacked vector r j = " r (1)2 j r (2)2 j , (11)namely t (3)2 j = A ′ j r j . The matrix A ′ j (size T × T ) is therelay matrix of R j and is also orthogonal like its counterpartin L relays, and given by A ′ j = 1 √ a j · · · a , T j ... . . . ... a T j · · · a T, T j . A ′ j ’s can also be written in the submatrix form as A ′ j = 1 √ (cid:2) A j (1) | A j (2) (cid:3) (12)where A j (1) and A j (2) are the submatrices of A ′ j withfirst T columns and the last T columns respectively. Alsothese submatrices are chosen to be orthogonal.Hence the vector transmitted by R j is c t (3)2 j and thereforethe vector received by D is r (3) d where the components aregiven by r (3) d ( τ ) = N X i =1 c t (2)2 i ( τ ) h i,d + u (3) d ( τ ) . It can be proved after some calculations that r (3) d = c c √ S (1) h s, ,d + u z + c c c √ S , (2) h s, , . . . S N, (2) h s, , N ] h ,d (13)where S (1) = [ A (1) s . . . A N (1) s ] , h s, ,d = h s, h ,d ... h s, N h N,d , (14) S n, (2) = [ A n (2) A s . . . A n (2) A N s ] , (15) h s, , n = h s, h , n ... h s, N h N, n , h ,d = h ,d ... h N,d , (16)and u z = c √ N X j =1 h A j (1) u (1)2 j + A j (2) u (2)2 j i h j,d + c c √ N X i =1 N X j =1 h i, j h j,d A j (2) A i u (1)1 i + u (3) d . (17)Here A j ( l ) , l = 1 , are given in (12) and ≤ n ≤ N in (16).To find c let us find the power transmitted by each relay inL . This is given by p T /N = E (cid:16) c t (3) H j t (3)2 j c (cid:17) . The totalavailable power p T is equally divided amongst N relays asthe variance of the channel coefficients are same for all ofthem. The power transmitted by each relay can be proved tobe c T (cid:2) p σ + p (cid:3) , which leads to c = r p N (2 + p σ + p ) . (18)All the transmission vectors and the multiplication factorsare summarized in Table I. It can be seen from (13) that thespace-time code here has been mingled up with the channel.Nevertheless an ML decoder has been derived for this protocol. TABLE IT
RANSMITTED VECTORS AND MULTIPLICATION FACTORS - RMC
Vector Factor Transmitted by s c = √ p T S in phase 1 t (2)1 j = A j r (1)1 j c = q p N ( p +1) L relays in phase 2 t (3)2 j = A ′ j r j c = q p N (2+ p σ + p ) L relays in phase 3
2) ML Decoder:
D has two received vectors namely, r (2) d = x , say and r (3) d = z , say as shown in (8) and (13) respectively.These two vectors are stacked as y = (cid:20) xz (cid:21) . (19)The likelihood function that s is transmitted is Pr( y | s ) . Tofind an expression for this (given in (26)) we have to knowthe nature of the joint density function. Let us first consider x and z separately. It can be seen from (8) that x is jointlyGaussian and from (13) that z is jointly Gaussian. Also themean of x | s is E [ x | s ] = c c [ A s . . . A N s ] h s, ,d = m x , say (20)and the covariance matrix of x | s can be worked out to be E (cid:2) ( x − m x )( x − m x ) H | s (cid:3) = c N X j =1 | h j,d | I T = P x , say. (21) Similarly we can obtain m z and P z as m z = E [ z | s ] = c c √ S (1) h s, ,d + c c c √ S h h ,d (22)and P z = E h ( z − u z ) ( z − u z ) H i = c N X j =1 | h j,d | I T + c c N X i =1 N X j =1 N X k =1 h i, j h j,d h ∗ i, k h ∗ k,d A j (2) A T k (2) . (23)From the above discussions we can see that y is also jointlyGaussian with mean vector and covariance matrix given by[12] m y = (cid:20) m x m z (cid:21) and P y = (cid:20) P x P xz P zx P z (cid:21) (24)respectively. Here P xz and P zx are the cross covariancematrices given by P xz = E (cid:2) ( x − m x )( z − m z ) H | s (cid:3) = P H zx and we can derive P xz = c c √ N X i =1 N X j =1 h i,d h ∗ i, j h ∗ j,d A T j (2) . (25)Now as y is complex Gaussian we can write [13] Pr( y | s ) = exp (cid:2) − ( y − m y ) H P − y ( y − m y ) (cid:3) π T | P y | (26)where m y and P y are given in (24). Hence we can write thedecoded vector as [12] b s = arg max s Pr( y | s ) = arg min s k y ′ k (27)where y ′ = P − y ( y − m y ) .
3) Receive SNR:
Let us derive an expression for receiveSNR. We have two received signal vectors at the destinationnamely, r (2) d shown in (8) and r (3) d shown in (13). The receivedsignal power and noise power in second phase can be writtenas P (2) s = E [ m H x m x ] and P (2) n = E [ u H x u x ] respectively.Hence from (9) and (20) P (2) s = E c c N X j =1 N X i =1 h ∗ j,d h ∗ s, j s H A H ij A i s h s, i h i,d and P (2) n = E c N X j =1 N X i =1 h ∗ j,d u (1) H j A H ij A i u (1)1 i h i,d + E h u (2) H d u (2) d i . Fig. 4. Various phases in EJHS.
Now as the channel coefficients are all independent and zeromean, unless i = j , the expected values will be zero. So theabove equations simplify to, P (2) s = c c N X j =1 E h | h j,d | i E h | h s, j | i E (cid:2) s H A H ij A j s (cid:3) = c c N σ and P (2) n = c N X j =1 E h | h j,d | i E (cid:2) u H j A H ij A j u j (cid:3) + E h u (2) H d u (2) d i = c T N σ + T. (28)Similarly, P (3) s and P (3) n can be derived from (13) as P (3) s = 12 (cid:2) c c N σ + c c c N (cid:3) and P (3) n = c T N + c c T N T. (29)The receive SNR is thensnr RMC = P (2) s + P (3) s P (2) n + P (3) n = 2 c c N σ + c c N σ + c c c N c N T σ + 2 c N T + c c N T + 4 T .
Substituting the values of c , c , and c we can obtainequation (30) shown at the top of next page. Now allocationof p , p , and p can be done by maximizing the receive SNRshown in (30). But as it is quite tedious a fine computer searchis resorted to as discussed in Section IV-B. B. Extended Jing Hassibi Scheme (EJHS)
This is named so, as the JHS suggested by [9] has beenextended here to have one extra layer. The derivation and anal-ysis of this simple protocol is warranted as the performanceof EJHS forms a base line for comparison with other derivedprotocols.Different phases of transmission and reception in this pro-tocol are shown in Fig. 4 and explained below: • Phase 1: S transmits; L layer relays receive. • Phase 2: L layer relays transmit and L layer relaysreceive. • Phase 3: L layer relays transmit and D receives.
1) Protocol Analysis:
Phase 2 is exactly similar to that ofRMC, except that D neglects any signal received. Let p /N be the average power transmitted per symbol duration by eachof the L relays in this phase. Hence r (2)2 j is the same as thatof RMC and is given by equation (7). snr RMC = p (cid:2) p σ + (1 + p ) p σ + p ( p + 4 σ + 2 p σ ) (cid:3) p σ + (2 + p )(4 + p + 2 p σ + 2 p (4 + p + 2 σ + p (2 + σ )) . (30)In phase 3, let p /N be the average power transmitted persymbol duration by each of the L relays. The vector that istransmitted by R j is c t (3)2 j = c A j r (2)2 j . (31)The vector received by destination is r (3) d where the compo-nents are given by r (3) d ( τ ) = N X i =1 c t (2)2 i ( τ ) h i,d + u (3) d ( τ ) . It can be proved after some calculations that r (3) d = m z + u z = z , say (32)where m z = c c c [ S h s, , . . . S N h s, , N ] h ,d , (33)and S n =[ A n A s . . . A n A N s ] , ≤ n ≤ N. (34)Here h s, , n and h ,d are defined earlier in equation (16). Also u z = c c N X i =1 N X j =1 h i, j h j,d A j A i u (1)1 i + c N X j =1 h j,d A j u (2)2 j + u (3) d . (35) c is the same as that of RMC shown in (10). To find c thepower transmitted by each relay in L is to be found out. Thisis given by p T /N = E (cid:16) c t (3) H j t (3)2 j c (cid:17) . This can be provedto be c T [1 + p ] which implies c = r p N (1 + p ) . (36)The transmission vectors and the corresponding multiplicationfactors are summarized in Table II. TABLE IIT
RANSMITTED VECTORS AND MULTIPLICATION FACTORS - EJHS
Vector Factor Transmitted by s c = √ p T S in phase 1 t (2)1 j = A j r (1)1 j c = q p N ( p +1) L relays in phase 2 t (3)2 j = A j r (2)2 j c = q p N (1+ p ) L relays in phase 3
2) ML Decoder:
Unlike in RMC case, EJHS has only onereceive vector at D. We can prove that this vector, z , shownin (32), is complex Gaussian with mean m z , shown in (33),and covariance matrix P z , where P z can be derived to be P z = c N X j =1 | h j,d | I T + c c N X i =1 N X j =1 N X k =1 A i A T k h i,d h ∗ k,d h j, i h ∗ j, k . (37)Now the decoded vector can be proved to be b s = arg max s Pr( z | s ) = arg min s k z ′ k (38)where z ′ = P − z ( z − m z ) .
3) Receive SNR:
On similar lines as was done in RMC inSection III-A.3, we can prove that the receive SNR in this caseis snr
EJHS = c c c N c c N T + c N T + T = p p p (1 + p )(1 + p ) + p (1 + p + p ) . (39)It can also be derived that snr EJHS attains the maximum valueof P P + P ) (40)when p = p = p = 1 / . This has also been verified inSection IV-B using simulations. Hence in the BER simulationsin Section IV-C for EJHS, the total power is divided equallyamongst the three phases accordingly.If σ is very low ( < . ), then EJHS is expected to performbetter than all protocols as it neglects these weaker signals.This is verified in Section IV-C using simulations. C. Modified Jing Hassibi Scheme (MJHS)
As the name suggests the JHS has been modified in thisprotocol. Different phases of transmission and reception inMJHS case are shown in Fig. 5 and explained below: • Phase 1: S transmits; L and L layer relays receive. • Phase 2: L layer and L layer relays transmit; and Dreceives. • Phase 3: L layer and L layer relays transmit; and Dreceives.
1) Protocol Analysis:
In this protocol, we have phase 3exactly similar to phase 2 so as to keep the total time durationto be 3 T , similar to the other protocols. Let p T / be thepower transmitted by L and p T / by L relays in the secondphase. As the vectors to be transmitted by L and L relays inthe second and third phases are identical and that the channelis assumed to have the same statistics, we have equally divided Fig. 5. Various phases in MJHS. the power between the second and third phases. Let c t ( k )1 j and c t ( k )2 j be the vectors transmitted by R j and R j relaysrespectively, in k th phase, with k = 2 , . Average powertransmitted by R j in T channel uses during the k th phaseis E [ c t ( k ) H j c t ( k )1 j ] = c E [ t ( k ) H j t ( k )1 j ]= c E [( r (1) H j A H j )( A j r (1)1 j )]= c ( c + T ) . (41)Equation (41) is arrived from (4) and A H j A j = I T . Hencetotal power transmitted by R j alone in phase k , with k = 2 , is p T N = c ( c + T ) ⇒ c = r p N (1 + p ) . (42)Here c is substituted from (3). Similarly it can be proved thatthe power transmitted by R j in T channel uses is E [ c t ( k ) H j c t ( k )2 j ] = c ( σ c + T ) . (43)Hence total power transmitted by R j alone in phase k , with k = 2 , can be worked out to be p N = c (1 + σ p ) ⇒ c = r p N (1 + σ p ) . (44)It can be proved that the received vector at D in phase k is r ( k ) d = c S h s,d + w dk , k = 2 , , (45)where S = [ c A s . . . c A N s c A s . . . c A N s ] , h s,d = h s, h ,d ... h s, N h N,d h s, h ,d ... h s, N h N,d , and w dk = N X j =1 c A j u (1)1 j h j,d + N X j =1 c A j u (1)2 j h j,d + u ( k ) d . (46)The transmission vectors and the corresponding multiplicationfactors are summarized in Table III. TABLE IIIT
RANSMITTED VECTORS AND MULTIPLICATION FACTORS - MJHS
Vector Factor Transmitted by s c = √ p T S in phase 1 t (2)1 j = A j r (1)1 j c = q p N (1+ p ) L relays in phase 2 t (2)2 j = A j r (1)2 j c = q p N (1+ σ p ) L relays in phase 2 t (3)1 j = t (2)1 j c = c L relays in phase 3 t (3)2 j = t (2)2 j c = c L relays in phase 3
2) ML Decoder:
The two received vectors at D for MJHSare as shown in (45), which we shall call x for k = 2 and z for k = 3 . Let y be the concatenated vector of x and z namely y = [ x T | z T ] T . It can be proved as in RMC that y isjointly Gaussian and that the mean vector, m y and covariancematrix, P y of y are given in (24). The mean, covariance, andcross-covariance of the received vectors can be proved to be m x = c S h s, ,d , m z = m x , P x = c N X j =1 | h j,d | + c N X j =1 | h j,d | I T , P z = c N X j =1 | h j,d | + c N X j =1 | h j,d | I T , and P xz = c c N X j =1 | h j,d | + c c N X j =1 | h j,d | I T . The decoded vector is given by b s = arg max s Pr( y | s ) = arg min s k y ′ k (47)where y ′ = P − y ( y − m y ) .
3) Receive SNR:
From equation (45) we can derive thereceive SNR of this protocol at D to besnr
MJHS = N c ( c + c ) σ c N T σ + c N T + T .
This can be simplified tosnr
MJHS = p σ (cid:2) (1 + p ) p + p (1 + p σ ) (cid:3) p + 2 p σ + p σ + p (2 + p + 2 σ + p σ ) . (48)Maximizing the receive SNR shown in (48) became quitetedious and hence a fine computer search has been resortedto, as discussed in Section IV-B. Optimum power allocationequations have been obtained by curve fitting as a function ofthe total average power in that Section. D. Relay SNR Combining (RSC)
Various phases of RSC are similar to that of RMC shownin Fig. 3 and explained in the first paragraph of Section III-A.As the name suggests, in this protocol the relays in the L layer combine the two received vectors using the respectiveSNRs.
1) Protocol Analysis:
Here every operation till secondphase is the same like RMC, but at Layer L the relaysR j (1 ≤ j ≤ N ) combine the two vectors r (1)2 j and r (2)2 j in a different fashion for transmission. The vector that istransmitted is c t (3)2 j = c A j h γ r (1)2 j + γ r (2)2 j i . Here γ and γ are the SNRs of the received signals r (1)2 j and r (2)2 j respectively at R j . These can be derived to be γ = p σ and γ = p p p + p . (49) c is the same as that shown in (3) and c is similar to that ofRMC protocol as shown in equation (10). Let us work out c with the restriction that the power transmitted by each of theL relays is p T /N in T duration. The power transmitted is p TN = E [ c t (3) H j t (3)2 j ]= c (cid:2) γ p T σ + ( γ + γ ) T + γ p T (cid:3) ⇒ c = r p N [ γ (1 + p σ ) + γ (1 + p )] . (50)In phase 2 the received vectors are the same as that shownin equations (7) and (8) for L layers and D respectively. Inphase 3, it can be shown that the destination receives r (3) d = c c γ S h s, ,d + u z + c c c γ [ S , h s, , . . . S N, h s, , N ] h ,d (51)where S = [ A s . . . A N s ] , S n, =[ A n A s . . . A n A N s ] and u z = c γ N X j =1 A j u (1)2 j h j,d + c γ N X j =1 A j u (2)2 j h j,d + c c γ N X i =1 N X j =1 A i A j u (1)1 j h j, i h i,d + u (3) d (52)with h s, , n given in (16). The transmission vectors and the TABLE IVT
RANSMITTED VECTORS AND THE MULTIPLICATION FACTORS - RSC
Vector Factor Transmitted by s c = √ p T S in phase 1 t (2)1 j c = q p N ( p +1) L relays in phase 2 t (3)2 j c = q p N [ γ (1+ p σ )+ γ (1+ p ) ] L relays in phase 3 corresponding factors are summarized in Table IV.
2) ML Decoder:
The two received vectors at D for RSCare as shown in (8) and (51) which we shall call x and z respectively. Let y be the concatenation of these vectors,namely, y = [ x T | z T ] T . It can be proved as in RMC that y isjointly Gaussian and that the mean vector, m y and covariancematrix, P y of y are given in (24). Here m x and P x arethe same as that shown in (20) and (21) respectively. Alsothe mean vector and covariance matrix of z along with crosscovariance matrix can be proved to be m z = c c γ S h s, ,d + c c c γ [ S , h s, , . . . S N, h s, , N ] h ,d (53) P z = c ( γ + γ ) N X j =1 | h j,d | I T + c c γ N X i =1 N X j =1 N X k =1 A i A T k h j, i h i,d h ∗ j, k h ∗ k,d , (54)and P xz = c c γ N X j =1 N X k =1 h k,d h ∗ k, j h ∗ j,d A H j = P H zx . (55)The decoded vector is given by b s = arg max s Pr( y | s ) = arg min s k y ′ k (56)where y ′ = P − y ( y − m y ) .
3) Receive SNR:
The receive SNR can be derived for thisprotocol to besnr
RSC = c c N σ + c c γ N σ + c c c γ N c N T σ + c ( γ + γ ) N T + c c γ T N + 2 T which is simplified and shown in equation (57) at the top ofnext page. Maximizing the receive SNR shown in (57) is quitetedious and hence a fine computer search was resorted to, asdiscussed in Section IV-B. E. RMC with Known Channel (RMCKC)
Various phases of RMCKC are similar to that of RMCshown in Fig. 3 and explained in the first paragraph ofSection III-A. In this protocol the relays R ij are presumed toknow the receive channels; R j knows h s, j , and R j knows h s, j and h i, j , i ∈ { , . . . , N } . Note that in RMCKC therelays do not know the transmit channels h ij,d .1) Protocol Analysis: In phase 2, the L relays transmit c t (2)1 j where t (2)1 j = A j r (1)1 j h ∗ s, j . Here c is similar to thatof RMC shown in (10). Now L layer relays would transmit c t (3)2 j where t (3)2 j = A ′ j r j and r j is a concatenated vectorgiven by r j = " r (1)2 j h ∗ s, j r (2)2 j k h , j k . snr RSC = p p σ (cid:2) p (1 + p ) + ( σ + p σ )(1 + p + p ) (cid:3) + p p p + p p σ (1 + p )(1 + p + p ) p ) (cid:2) p (1 + p ) + ( σ + p σ )(1 + p + p ) (cid:3) + p σ (cid:2) p (1 + p ) + ( σ + p σ )(1 + p + p ) (cid:3) + p p . (57)Also the received vector at R j in phase 2 is r (2)2 j = N X i =1 c t (2)1 i h i, j + u (2)2 j = c c N X i =1 | h s, i | h i, j A i s + c N X i =1 h ∗ s, i h i, j A i u (1)1 i + u (2)2 j . (58)Here A ′ j is the same as that of RMC shown in (12). Themultiplying factor h ∗ s, j is the conjugate of the channel thetransmitted signal would have gone through when r (1)2 j isreceived. Similarly, the transmitted signal would have gonethrough a vector of channel coefficients h , j when r (2)2 j isreceived, and hence k h , j k is the multiplying factor.The received vector r (2) d at D can be proved to be r (2) d = c c [ A s . . . A N s ] h ′ s, ,d + u x (59)where h ′ s, ,d = | h s, | h ,d ... | h s, N | h N,d and u x = c N X j =1 h ∗ s, j h j,d A j u (1)1 j + u (2) d . (60)The received vector r (3) d at D can be proved to be r (3) d = c c √ N X j =1 | h s, j | h j,d A j (1) s + c c c √ N X i =1 N X j =1 k h , j k h j,d h i, j | h s, i | A j (2) A i s + u z (61)where u z = c √ N X j =1 h ∗ s, j h j,d A j (1) u (1)2 j + c c √ N X j =1 N X i =1 k h , j k h j,d h ∗ s, i h i, j A j (2) A i u (1)1 i + c √ N X j =1 k h , j k h j,d A j (2) u (2)2 j + u (3) d . (62) With the total average power transmitted per symbol durationfixed at p /N in phase 3, c can be derived to be c = vuuut (1 + p ) p N [8 p p + N (1 + p + p )+(1 + p ) σ + σ p (1 + p )] (63)Expressions for c , c , c , and the transmission vectors aresummarized in Table V. TABLE VT
RANSMITTED VECTORS AND MULTIPLICATION FACTORS - RMCKC
Vector Factor Transmitted by s c = √ p T S in phase 1 t (2)1 j c = q p N ( p +1) L relays in phase 2 t (3)2 j c , shown in (63) L relays in phase 3
2) ML Decoder:
The two received vectors at D for RM-CKC are as shown in (59) and (61), which we shall call x and z respectively. Let y be the concatenated vector of x and z namely y = [ x T | z T ] T . It can be proved as in RMC that y isjointly Gaussian and that the mean vector, m y and covariancematrix, P y of y are given in (24). The mean vector, covariance,and cross covariance matrices can be proved to be m x = c c N X i =1 | h s, i | h i,d A i s , (64) P x = " c N X i =1 | h s, i | | h i,d | I T , (65) m z = c c √ N X j =1 | h s, j | h j,d A j (1) s + c c c √ N X i =1 N X j =1 k h , j k h j,d h i, j | h s, i | A j (2) A i s , (66) P z = c N X j =1 (cid:16) | h s, j | + k h , j k (cid:17) | h j,d | I T + c c N X i =1 N X j =1 N X k =1 k h , j k k h , k k h j,d h ∗ k,d | h s, i | h i, j h ∗ i, k A j (2) A T k (2) , and (67) P xz = c c √ N X j =1 N X i =1 k h , j k h ∗ j,d | h s, i | h i,d h ∗ i, j A T j (2) . (68) The decoded vector is given by b s = arg max s Pr( y | s ) = arg min s k y ′ k (69)where y ′ = P − y ( y − m y ) .
3) Receive SNR:
From equations (59) and (61), we canderive the receive SNR of this protocol at D to besnr
RMCKC = 16
N c c σ + 2 N c c (3 σ + σ ) + 8 N c c c N T c σ + 4 T + N c c T + N c T + N c σ T . snr
RMCKC is simplified and shown in equation (70) at thetop of next page. Maximizing the receive SNR shown in (70)became quite tedious and hence a fine computer search hasbeen resorted to, as discussed in Section IV-B.IV. S
IMULATIONS
We have seen five different protocols, namely RMC, EJHS,MJHS, RSC, and RMCKC. In all these protocols matrices atrelays have been used, for generating a distributed space-timecode. Using simulations the performance of the system hasbeen compared, when these matrices are real orthogonal andcomplex unitary. Optimum power allocation to all transmis-sions using simulations have been found out. Finally BERs forvarious protocols have been plotted while using the optimumpower allocations obtained.In the simulations, a block size of length T = 5 symbolduration and number of relays in each layer, N = 5 for arun of 10,000 data blocks have been used. As defined earlier s = [ s (1) · · · s ( T )] T , and s ( k ) = s r ( k ) + js i ( k ) , ≤ k ≤ T .Let us also assume that the real part s r ( k ) and the imaginarypart s i ( k ) of s ( k ) are equally likely selected from the M -PAMsignal set K (cid:26) − M − , · · · , − , , · · · , M − (cid:27) , where K is the normalizing factor so that E (cid:2) s H s (cid:3) = 1 . Hencethe cardinality, L , of Ω is M T . The value of K is found from E (cid:2) s H s (cid:3) = E T X k =1 | s ( k ) | = T E (cid:2) | s ( k ) | (cid:3) = T E (cid:2) s r ( k ) + s i ( k ) (cid:3) =2 T E (cid:2) s r ( k ) (cid:3) = T K M M/ X j =1 (2 j − = T K M ( M − M ( M + 1)6=1 ⇒ K = s T ( M − .M = 2 has been used in all the simulations. A. Relay Matrices
The relay matrices A ij have been selected to be realorthogonal as the performance in terms of BER is the sameas that when complex unitary matrices are used [9]. To provethis, simulations were carried out with the simple JHS system.Fig. 6 shows a plot of transmitted power vs. BER achieved
12 14 16 18 20 22 2410 −3 −2 −1 Real and Complex relay matricesPower P in dB B i t E rr o r R a t e Real orthogonalComplex unitary
Fig. 6. Comparison of performance of real orthogonal matrices with complexunitary matrices.Fig. 7. Power distribution surface where optimum power allocation pointresides. where there are two curves one representing that of usingreal orthogonal and the other complex unitary matrices at therelays. It is clear that the BER for all SNRs using real is thesame as that while using complex matrices. Hence in all thesimulations, real orthogonal matrices have been used to makeDSTC.
B. Optimum Power Allocation
Allocation of power to various transmissions, namely, p , p , and p are to be done in such a way that it minimizesthe transmission errors. Ideally one should minimize probabil-ity of error (PE) or pairwise error probability (PEP) and obtainthe optimum power allocation. Computation of PEP was foundto be complicated. In [9] the authors proved that the optimumpower allocation obtained by minimizing PEP also maximizesreceive SNR for their system model and protocol. Hence inthis work, receive SNR has been selected as the parameter tobe maximized and expect that this gives near optimum powerallocation. Maximizing this receive SNR analytically becametoo complex and hence a fine computer search has been carriedout as explained here.We have 3 variables namely p , p , and p which are thepowers allocated to the three transmissions used in the pro-tocols discussed. These three variables have two constraints,namely, p + p + p ≤ P and p , p , p ≥ . Let us considerthe best case constraint of p + p + p = P . This can be + snr RMCKC = 16( N + 8 p ) p p σ + 2 p p σ (1 + p ) (1 + 3 σ ) + 8 p p (1 + p )[ p + 2 σ ( N + σ + p σ )](1 + p ) [ N (1 + p )(4 + p ) + p (32 p + N (4 + p ))] + 4(1 + p ) σ + 2 N (1 + p ) p σ +2( N + 8 p ) p σ + (1 + p ) p σ + 2(1 + p ) [2 p (1 + p ) + p ] σ + 2 p (1 + p ) p σ . (70) p in fraction of P RMC − Optimum allocation is p =0.26,p =0.38,p =0.36 and SNR=11.9823 dB, for σ =0.15 and P=24 dB p in fraction of P R e c e i v e S NR a t D i n W a tt s Fig. 8. Plot of receive SNR for RMC. geometrically expressed as shown in Fig. 7, where AB is in p − p , BC in p − p and AC in p − p planes. We can select p and keep varying p with p automatically getting fixed.All the points on this plane need to be considered to find theoptimum power allocation. As it is impossible to consider allthe points on this plane we can select them with a granularity.Consider the straight line shown on the plane in Fig. 7, DE,which is parallel to BC. The equation of this straight line is p + p + p = P ; p = p ′ where ≤ p ′ ≤ P. By varying p ′ ,we will get more straight lines parallel to BC. With a certaingranularity we will vary p ′ . i.e. p ′ = nδP where ≤ δ ≤ and ≤ n ≤ (cid:4) δ (cid:5) , n being an integer. Once p ′ is selected,let us select p with a granularity as for the case of p as p ′ = mǫP with ≤ m ≤ (cid:4) − δǫ (cid:5) , m being an integer and ≤ ǫ ≤ . Then p is fixed as p ′ = P − p ′ − p ′ . Hence we canget the point G ( p ′ , p ′ , p ′ ) as shown in Fig. 7. The completeregion of the plane ABC is scanned fully and receive SNRs arecalculated for each point. That point which has the maximumSNR is selected as the optimum point.In the calculations, δ = 1 / and ǫ = 1 / havebeen used. Hence the region has been scanned with a gran-ularity of 0.001 in all the three axes. The optimum points ( p opt , p opt , p opt ) differed for various powers and σ . In allthe protocols p , p , and p represent the powers allocatedto the three phases except in MJHS, where p represents thepower transmitted by L relays in both second and third phasesand p represents that of L relays in both phases . 3Dplots of receive SNR for all the protocols for various totalaverage power P , when σ = 0 . , . , and 0.5 have beengenerated and obtained the optimum points when the receiveSNR is maximum. These plots for RMC, EJHS, MJHS, RSC,and RMCKC are shown in Figures 8 to 12 respectively for σ = 0 . and P = 24 dB. The plots show receive SNR forvarious possible combinations of p , p , and p . It can be seenthat the maximum SNR is achieved at p opt = 0 . , p opt =0 . , and p opt = 0 . for RMC. Also for EJHS the fact that in fraction of PEJHS − Optimum allocation is p =0.34,p =0.33,p =0.33 and SNR=14.4049 dB, for σ =0.15 and P=24 dBp in fraction of P R e c e i v e S NR a t D i n W a tt s Fig. 9. Plot of receive SNR for EJHS. in fraction of PMJHS − Optimum allocation is p =0.64,p =0,p =0.36 and SNR=11.9042 dB, for σ =0.15 and P=24 dBp in fraction of P R e c e i v e S NR a t D i n W a tt s Fig. 10. Plot of receive SNR for MJHS. in fraction of PRSC − Optimum allocation is p =0.3,p =0.34,p =0.36 and SNR=13.5574 dB, for σ =0.15 and P=24 dBp in fraction of P R e c e i v e S NR a t D i n W a tt s Fig. 11. Plot of receive SNR for RSC. p opt = 1 / , p opt = 1 / , and p opt = 1 / seen in subsectionIII-B.3 is verified from the 3D plot shown in Fig. 9. Figures13 to 15 show plots of p , p , and p of four of the protocolsRMC, MJHS, RSC, and RMCKC to achieve maximum receiveSNR at D for σ = 0 . , 0.1 and 0.5 respectively. (Plot forEJHS is left out as the power allocation remains the same asshown in subsection III-B.3, for any P .) The following can beobserved from Fig. 13: • RMC, RSC & RMCKC in fraction of PRMCKC − Optimum allocation is p =0.22,p =0.78,p =0 and SNR=19.6168 dB, for σ =0.15 and P=24 dBp in fraction of P R e c e i v e S NR a t D i n W a tt s Fig. 12. Plot of receive SNR for RMCKC.
12 14 16 18 20 22 240.20.250.30.350.40.45
RMC
Power
P in dB F r a c t i on o f P For σ =0.01 p p p
10 15 2000.20.40.60.81
MJHS
Power
P in dB F r a c t i on o f P
12 14 16 18 20 22 240.20.30.4
RSC
Power
P in dB F r a c t i on o f P
12 14 16 18 20 22 240.20.30.40.5
RMCKC
Power
P in dB F r a c t i on o f P Fig. 13. Plot of optimum power allocations for RMC, MJHS, RSC, andRMCKC for σ = 0 . . – p , i.e. power transmitted by L relays, needs to beincreased with the increase in P , whereas that ofsource, p , and L relays, p are to be reduced. – As P increases the rate at which p and p are to bereduced or p to be increased is less in the case ofRSC compared to that of RMCKC and RMC. – Power transmitted by source and L relays are almostthe same in the case of RSC. • MJHS – It does not transmit using L relays to achieve highreceive SNR. i.e. p remains zero. – The source needs to increase its power whereas L relays are to decrease their powers for increase inthe total power. – The plots can be curve fitted by minimizing meansquared error with quadratic setting as e p =0 .
71 + 0 . P − . × − P , (71)and e p =0 . − . P + 4 . × − P . (72)The following can be observed from Fig. 14: • RMC – The powers transmitted by source and L layersare to be reduced while that of L layers is to beincreased as P increases. – At P ≈ dB the powers transmitted by S and L layers are the same, above which L layer transmitsmore power than S.
12 14 16 18 20 22 240.250.30.350.4
RMC
Power
P in dB F r a c t i on o f P For σ =0.15
12 14 16 18 20 22 2400.20.40.6
MJHS
Power
P in dB F r a c t i on o f P
12 14 16 18 20 22 240.280.30.320.340.36
RSC
Power
P in dB F r a c t i on o f P
12 14 16 18 20 22 2400.20.40.60.8
RMCKC
Power
P in dB F r a c t i on o f P p p p Fig. 14. Plot of Optimum power allocations for MJHS, RMC, RSC, andRMCKC for σ = 0 . . • MJHS – To get maximum receive SNR, this protocol keepsthe L relays mute throughout. i.e. this protocoldoes not require those relays that are nearer to thesource. The power transmitted by the source, p isto be increased while that of L relays, p is to bedecreased as the total power, P increases. • RSC – Source and L relays are to decrease their powerswhile L relays are to increase their powers toobtain maximum receive SNR, as the total power, P increases. – At P ≈ dB the powers transmitted by L and L layers are the same, above which L layer transmitsmore power than S. • RMCKC – This protocol, unlike MJHS, does not require L relays throughout. i.e. it keeps p to be zero through-out. The power of the source, p , is to be decreasedwhile that of L layers, p , is to be increased as thetotal power, P , increases.The following can be observed from Fig. 15: • RMC/RMCKC – Like in the case of RMCKC for σ = 0 . , theseprotocols, for σ = 0 . , do not require L relaysthroughout. i.e. p remains to be zero throughout.The source power, p , is to be decreased while thatof L layers, p , is to be increased as the total power, P , increases. • MJHS/RSC – Unlike RMC and RMCKC, these protocols do notrequire L relays throughout. i.e. p remains to bezero throughout. Source and L relays are allocatedhalf the total power each, to get maximum receiveSNR.We can infer the following from all the above observationsmade on figures 13 to 15: • As the channels from source to L layer and L layerto destination improve, (i.e. σ > . ) RMC reducesthe importance to the relays in L while giving moreweightage to source and L layer relays.
12 14 16 18 20 22 2400.20.40.6
RMC
Power
P in dB F r a c t i on o f P For σ =0.5 p p p
12 14 16 18 20 22 2400.20.40.6
MJHS
Power
P in dB F r a c t i on o f P
12 14 16 18 20 22 2400.20.40.6
RSC
Power
P in dB F r a c t i on o f P
12 14 16 18 20 22 2400.20.40.6
RMCKC
Power
P in dB F r a c t i on o f P Fig. 15. Plot of Optimum power allocations for RMC, MJHS, RSC, andRMCKC for σ = 0 . .
10 12 14 16 18 20 22 24051015202530
Maximum Receive SNRs when σ = 0.01Power P in dB S NR i n ab s o l u t e un i t EJHSMJHSRMCRSCRMCKC
Fig. 16. Plot of Maximum Receive SNRs for σ = 0 . . • Irrespective of the power loss condition (i.e. for any valueof σ ), MJHS keeps the L layer relays muted and doesnot use them throughout. Also the difference between thepowers divided between the source and L layer relaysnarrows down and finally becomes zero as the channelvariance from source to L layer along with L layer todestination improves and reaches 0.5. • Unlike MJHS, which shuts down L layer relays com-pletely in any power loss condition, RMCKC mutes L layer relays when the signal from the source to the secondlayer or the L layer relays to destination undergoes lowerattenuation.All the plots shown in figures 13 to 15 can be curve fitted asshown in (71) and (72), so that they can be readily used forpower allocations.Fig. 16 shows the maximum receive SNRs of the protocolsdiscussed in this paper for various values of P with σ = 0 . . It can be observed that the performance in terms of receiveSNR of the protocols almost matches the performance in BERshown in Fig. 17 in the next Section.
C. BER Plots
Finally for comparison of various protocols, BER plotsshown in Figures 17 to 19 for σ = 0 . ,
10 12 14 16 18 20 22 2410 −3 −2 −1 BERs when σ = 0.01Power P in dB B i t E rr o r R a t e EJHSMJHSRMCRSCRMCKC
Fig. 17. Comparison of BER of protocols when σ = 0 . .
12 14 16 18 20 22 2410 −2 −1 BERs when σ = 0.15Power P in dB B i t E rr o r R a t e EJHSMJHSRMCRSCRMCKC
Fig. 18. Comparison of BER of protocols when σ = 0 . .
12 14 16 18 20 22 2410 −2 −1 BERs when σ = 0.5Power P in dB B i t E rr o r R a t e EJHSMJHSRMCRSCRMCKC
Fig. 19. Comparison of BER of protocols when σ = 0 . . • The performance of RMCKC for P ≤ dB is the sameas that of EJHS. For P > dB EJHS is the best. • Amongst RMC, MJHS, and RSC, RMC performs better.From Fig. 18 we can observe the following: • All the protocols proposed by us except MJHS performbetter than EJHS for P ≤ dB when σ = 0 . . • Further the performance of RMCKC is the best for P ≤ dB.From Fig. 19 we observe the following: • All the proposed protocols perform better than EJHS for σ = 0 . throughout the usable range of transmittedpower P . • As expected, RMCKC attains the lowest BER using thereceive channel knowledge. • For P ≥ dB, RSC, RMC, and MJHS work better thanRMCKC. D. Discussion and Observations
All the protocols proposed by us except MJHS displaybetter performance than EJHS when σ > . for usablerange of transmitted power P . Only for σ ≤ . , EJHSworks better than all the proposed protocols. The reason forthis is that when σ reduces to a low value, say 0.01, thesignals which reach L in phase 1 and D in phase 2 arehighly attenuated. Hence the proposed protocols, which usethese attenuated signals, do not perform as good as EJHS, assome power is expended with no particular advantage in thesesignals. It is also observed that for P ≥ dB, RSC, RMC,and MJHS outperform RMCKC implying that RMCKC doesnot use whatever channel information it has to its best andthere is a possible scope for improvement. However with justthe receive channel knowledge of h s, j , RMCKC performsbest; it has lowest BER and high data rate.Further when σ → expectedly RMC and RSC have beenfound to perform similar to that of MJHS in which all the N relays are merged into one layer. An interesting result whichis to be emphasized is that when the signal from source to thesecond layer reaches with less attenuation (channel variance, σ > . ), then we can opt for RMCKC which selects onlythose relays that are closer to the source and does not usethose that are closer to the destination. This implies that weneed only two phases of transmission leading to higher datarate compared to those which use three phases.To summarize, when σ ≤ . we can select EJHS andno considerable gain would be obtained in going for theschemes which use ‘weak’ links. However, when σ > . the proposed protocols perform better for usable range oftransmitted power P . Here we can select either RMC or RSCwhen there is no channel knowledge at the relays for lowervalues of P depending upon σ (e.g. for P < dB when σ = 0 . ). But if the relays have just the receive channelknowledge, we can use RMCKC for most values of P (e.g.for P < dB when σ = 0 . , above which RMC/RSC to beused). Also it is beneficial to select RMCKC as it gets betterreliability with increased data rate, as it uses only two phases.V. C ONCLUSION
In this paper, the simple relay processing system usingmatrices suggested by Jing and Hassibi in [9] to achievebenefits of DSTC has been modified and enlarged. Alsorandom orthogonal matrices have been used at relays and wehave shown that BER performance achieved is the same asthat when complex unitary matrices as suggested in [9] areused.Four new protocols have been derived from the one pro-posed in [9]. We have made use of the signals from ‘weak’channels (which are received by relays and destination withhigh power loss) in these protocols and shown that theyperform better than the basic protocol proposed in [9] withreasonable strength of the ‘weak’ channels.An interesting result when the relays have the receive channel knowledge in the protocol RMCKC is shown in Fig.20. Above σ ≥ . , RMCKC uses only two phases. Hencethe data rate is improved by 1/3 compared to all other protocolsand it gets the lowest BER also for most of the usable rangeof the transmitted power. Fig. 20. Best scheme when σ > . with receive channel knowledge atrelays. R EFERENCES[1] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time blockcodes from orthogonal designs,”
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