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Decode-and-Forward Cooperation AidedTriple-Layer Turbo Trellis-Coded Hierarchical

ModulationHua Sun, Soon Xin Ng, Senior Member, IEEE, Chen Dong, and Lajos Hanzo, Fellow, IEEE

Abstract—Hierarchical Modulation (HM) is widely employedacross the telecommunication industry. The potential applicationof the coded HM scheme in cooperative communications hasdrawn a lot of interests. In this paper, a twin-relay aided triple-layer cooperative communication system is proposed. The systemamalgamates rate-1/2 TTCM, triple-layer HM-64QAM and twin-layer SPM-16QAM schemes in the context of cooperative com-munications. We have optimized the entire system based on theHM ratio pair (R1, R2), the Superposition Modulation (SPM)weighting pair (α, β) and the positions of the two relays. Thesimulation results show that our optimized system is capableof reliably transmitting a triple-layer HM-64QAM signal withthe aid of two time slots at an average Signal-to-Noise-Ratio of6.94 dB per time slot.

Index Terms—Hierarchical Modulation, Superposition Modu-lation, Turbo Trellis-Coded Modulation, Cooperative Communi-cation, Soft Decoding and Power Efficiency.

I. INTRODUCTION

Hierarchical Modulation (HM) constitutes an integral partof the DVB-T/-H standard [1], but it has also been widelyemployed by the IT industry for upgrading diverse telecom-munication services [2], [3]. Compared to a system usingconventional modulation, the system employing HM has ahigher flexibility, while maintaining backward compatibility.Explicitly, both the original and the upgraded new servicesmay be combined by the HM scheme and broadcast to the re-ceivers without requiring any additional bandwidth. Althoughthe services is upgraded to a higher data rate, the originaldevices are still supported by the upgraded broadcast systemwithout requiring software or hardware upgrade [4].

HM has been developed for combining independent infor-mation streams at bit-level layer by layer, which are thenmapped onto HM constellations. The information containedin different layers may also be demapped/detected separately.The general performance of the HM scheme have been detailedin [5]–[8]. It can be observed from the simulation resultsof [5], [6] that different layers in HM constellations receivedifferent protection levels. Consequently, the required receivedSignal-to-Noise Ratio (SNRr) for receiving the layer withhigher protection level requires lower SNRr than that of theless protected layers. The author of [9] employed HM inhis system to provide unequal error protection (UEP) forthe information contained in different layers. It has drawn

The financial support of the European Research Council’s Advanced FellowGrant as well as that of the EPSRC under the India-UK Advanced TechnologyCentre and of the EU’s Concerto project in gratefully acknowledged.

a lot of interests [10]–[15]. More specifically, the authorsof [13], [14] invoked a HM scheme for providing UEP forvideo and image encoding, where the information bits aremapped to specific protection layers according to their error-sensitivity-based priority. Moreover, the HM scheme has alsobeen combined with sophisticated channel coding schemesin [13], [14], for protecting the most important information.The simulation results of [13], [14] have shown that receivingthe information having the highest priority requires a lowerSNRr compared to conventional modulation schemes at agiven target BER performance.

A typical relay aided coded HM scheme was introducedin [16], where Hausl and Hagenauer combined Turbo Coding(TC) [17] with a HM scheme conceived for cooperative com-munications, where the original signal sequence was broadcastby the Source Node (SN) by ensuring that the layer with higherprotection may be received by the Destination Node (DN)directly, while the less protected layer will be received andretransmitted by the Relay Node (RN). However, the authorsof [16] only considered the specific scenario, when the positionof the RN is right in the middle of the SN-DN path and invokeda specific bit-to-symbol mapping scheme. The performance ofthe coded HM schemes was then further discussed in [18]–[21] in the context of cooperative communications, where thecommon choice is to employ multiple encoders at the SN andcombine all the coded bit sequences layer-by-layer to create aHM signal sequence. Again, the less well-protected layers areassisted by the RN of the cooperative network.

The bit-to-symbol mapping optimization of the HM schemewas considered in [20], [22]. More specifically, by appropri-ately designing the constellation mapping, the HM schemeis capable of enhancing the protection of the higher-priorityinformation at the expense of providing a weaker protectionfor the other layers. In [21], the specific position of the RN wasexplicitly considered in the BER analysis. For a specific codedHM scheme aided system, the receive power at the RN shouldbe sufficiently high for guaranteeing that the RN becomes ca-pable of receiving the information in the lower protected layerwith an acceptable integrity. Therefore, the position of theRN may influence the power allocation of the entire system.Several parameters have to be taken into consideration, whenoptimizing a coded HM aided cooperative communicationsystem. On one hand, distorting the HM constellation for thesake of improving the BER of its high-priority layers at thedetriment of its low-priority layers degrades its average BER,compared to conventional modulation schemes. On the other



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hand, sophisticated channel coding schemes, such as Trellis-Coded Modulation (TCM), Turbo Trellis-Coded Modulation(TTCM) and Bit-Interleaved Coded Modulation (BICM) [17],[23]–[25] are required for protecting each HM layer at theexpense of an increased complexity. Hence, giving cognizanceboth to the complexity and to the power efficiency of theoverall system, while maintaining its flexibility becomes achallenging task.

In [26], a cooperative communication system assisted bya TTCM based HM-16QAM scheme was proposed, wherewe proposed the optimum constellation mapping for the HM-16QAM scheme in the context of single RN aided cooperativecommunications. However, the position of the RN was fixedto be right in the middle of the SN-DN path. Hence, thescheme of [26] was suboptimal, because its power allocationwas suboptimal. Against this background, in this treatisewe intrinsically amalgamate HM, Superposition Modulation(SPM) [27] and TTCM for creating an attractive cooperativecommunication system. Our goal is to increase the time-efficiency and reduce the total power consumption of the entiresystem, while maintaining a low complexity. This cooperativecommunication system model may be readily used for assist-ing multilayer video transmission for example [13], [28] or formultilayer image transmission [14]. We exploit the idealizedsimplifying assumption that the system benefits from perfectChannel State Information (CSI), including both the fadingand path-loss. Hence, according to the receiver’s SNR, we arecapable of determining the transmission power required at thetransmitter, which we defined as the transmit SNR (SNRt)1.We proposed a Turbo Trellis-Coded Hierarchical Modulation(TTCHM) aided twin-relay based cooperative communicationscheme, where three rate-1/2 TTCM encoders are employedat the SN for constructing a 64QAM-based triple-layer HMscheme. The TTCM scheme detailed in [17] has a betterperformance for transmission over Rayleigh fading channelsthan other joint coding and modulation schemes, such asTCM and BICM. An excellent performance can be attainedby TTCM without expanding the bandwidth. A rate-k/(k+1)TTCM scheme can be used for protecting a k-bit HM layerby expanding the number of constellation points from 2k to2k+1. We considered k = 1 in this contribution.

Depending on the specific symbol-to-bit demapping ar-rangement of the HM scheme, different HM layers havedifferent protections. Explicitly, the information in the higher-protection layers may require a lower SNRr at the DN thanthat of the information in the lower-protection layers. Hence,when the SN transmits a multi-layer HM signal in our cooper-ative communication network, the SNRt at the SN (SNRSNt )may be reduced to the minimum value that can ‘just’ guaranteethe successful detection of the base layer (highest priority) ofthe HM signal at the DN. By contrast, the information in thelower priority layers may be received and retransmitted bythe RN. Since we proposed a triple-layer HM scheme, twoRNs are activated for retransmitting the information of the

1The definition of transmit SNR was proposed in [29], which is convenientfor simplifying the discussions, although this is not a physically measurablequantity, because it relates the power at the transmitter to the noise at thereceivers.

two lower layers. More specifically, a linear SPM scheme isemployed by the two RNs for simultaneously transmitting thetwo signal frames to the DN during the second time slot (TS).Hence, two TSs are required for the transmission of all thethree layers from the SN to the DN. Since the transmissionsbetween the SN and RN (or RN and DN) only deal with asingle 4QAM layer of the triple-layer HM-64QAM signals,the decoding complexity imposed on the two RNs (and DN)is reduced. Moreover, the SNRt at the RN (SNRRNt ) canalso be minimized, because both RNs will only retransmitusing 4QAM. If only one RN is available for assisting thetransmissions, the RN would have to detect both enhancementlayers from the HM-64QAM signals. The position of thisRN would be near to the SN and the transmission betweenthe RN and the DN will be based on 16QAM modulation.This would require a high SNRt due to the transmissionof a higher-order modulation scheme over a longer distance.Due to the flexibility of HM, the lowest-protection layer thatcontains the least important information can be discarded inthe adverse situation, when none of the RNs is capable ofdetecting it. Nonetheless, the DN can still receive the pairof more important layers of the HM-64QAM. We foundthat apart from reducing the power dissipation of the entiresystem, the processing complexity of the twin-relay aidedcooperative communication network may also be mitigated,when an appropriate design is invoked.

The main contributions of this paper are as follows:• A triple-layer HM-64QAM scheme is designed for aid-

ing a Decode-and-Forward (DF) based cooperative com-munications, which involves intrinsically amalgamatingTTCM, HM and SPM schemes;

• Based on our Monte-Carlo simulations, a power-allocation is conceived and it is demonstrated that thepower consumption of the entire system may be readilyoptimized by relying on the related variables, namely bythe HM-based symbol-energy ratio pair (R1, R2), by theSPM weighting pair (α, β), and by the relay’s geographicposition.

The rest of the paper is organized as follows. Section IIintroduces both the system model and our cooperative com-munication strategy. The specific HM-64QAM mapping ruledesigned for cooperative communication is detailed in Sec-tion III. The protocol of the symbol-to-bit demapper of the HMsymbols is discussed in Section IV. The triple-layer TTCHM-64QAM system design is detailed in Section V and thesimulation results are displayed in Section VI , our conclusionsand future research ideas are discussed in Section VII.

II. SYSTEM MODEL

Our TTCHM aided DF RN based cooperative communica-tion system is illustrated in Fig. 1. During the first transmissionTS, the SN will broadcast a sequence of TTCHM symbolsx1 to RN1, RN2 and DN. In the following TS, RN1 willtransmit a signal frame x2 to the DN, another signal framex3 will also be sent to the DN by the RN2 simultaneously.Again, the entire system would require two TSs to convey thetriple-layer TTCHM-64QAM symbol based signal frame x1to DN.



3

SN DN

RN2

RN1

dSD

x1

dSR2dR2D

dR1DdSR1

x1

x1

x2

x3

Fig. 1: The model of a two-relay cooperative system.

We considered an uncorrelated Rayleigh flat-fading channel,both the transmitters and receivers were assumed to acquireperfect CSI. During the first TS, each symbol received by theDN may be expressed as:

ySD =√GSDhSDx1 + nSD , (1)

where each of the symbols received by the RN1 and RN2 are:

ySR1=√GSR1

hSR1x1 + nSR1

, (2)

ySR2 =√GSR2hSR2x1 + nSR2 , (3)

where the subscript SD denotes the SN-DN link and thesubscript SRk represents the SN-RNk link. By contrast, thesymbols received at the DN during the second TS, which aresent by the two RNs, may be expressed as:

yRD = ρ1α√GR1DhR1Dx2 + ρ2β

√GR2DhR2Dx3 + nRD ,

(4)where ρ1 and ρ2 are the pre-coding parameters and (α, β) isthe SPM ratio pair. The subscript RkD represents the RNk-DNlink. Additionally, the notations hSD, hSRk and hRkD denotethe complex-valued coefficients of the uncorrelated Rayleighfading for the different links, nSD, nSRk and nRkD denote theAdditive White Gaussian Noise (AWGN) having a varianceof N0/2 per dimension. Moreover, the variables GSD, GSRkand GRkD represent the Reduced-Distance-Related-Pathloss-Reduction (RDRPLR) for each link, which we also refer to asthe path-gain [25], [29]–[31]. We consider an inverse-powerlaw based non-free-space path-loss model and naturally andwe define the path-loss exponent to be 3 which is usuallyused to simulate the path-loss in urban areas [32]. The path-gain GSD of the SD link is assumed to be unity. Thereforethe path-gains of the two SR links are:

GSRk =

(dSDdSRk

)3

, (5)

and similarly, the path-gains of the two RD links are:

GRkD =

(dSDdRkD

)3

, (6)

We assume that every node in the cooperative networkhas perfect CSI. Hence, given a specific path-loss and a RNposition, we may both compensate the effect of the path-lossas well as that of the Rayleigh fading with the aid of transmitpre-coding. Specifically, the ρ1 and ρ2 pre-coding parametersshould satisfy:

ρ1 =h∗R1D

| hR1D |2√GR1D

, (7)

ρ2 =h∗R2D

| hR2D |2√GR2D

. (8)

Hence, during the second TS, the signal received by the DNmay be written as:

yRD = αx2 + βx3 + nRD . (9)

In a realistic situation, there is always a path-loss between theSN and DN, but in order to simplify the system model, wenormalized this path-loss to 0 dB. Hence the transmit powerat the SN (SNRSNt ) would be identical to the power receivedat the DN (SNRDNr ). If the transmissions between the SNand DN are on a frame-by-frame basis over an uncorrelatedRayleigh fast fading channel, the average received SNR at DN(SNRDNr ) would be given by:

SNRDNr = E(|h|2 SNRt) = E(|h|2)SNRSNt , (10)

where the SNRSNt is the transmit SNR defined as the ratio ofthe transmit power to the noise power at the DN:

SNRSNt =E(|x|2)

N0=

1

N0, (11)

where E(|x|2) = 1. Furthermore, the uncorrelated Rayleighfading coefficient h is generated by the complex-valued Gaus-sian distribution having a zero mean and a variance of one.When the number of uncorrelated Rayleigh fading coefficientswe generated is large, we have [33]:

E(|h|2) =1

N

N∑k=1

|hk|2 ≈ 1 . (12)

Hence, for a large frame size of N symbols, we may as-sume that SNRDNr is equal to the SNRSNt , or equivalentlySNRDNr = SNRSNt .

To be more specific, the information flow of the entire sys-tem is illustrated in the block diagram shown in Fig. 2. In oursystem, the SN employs three rate-1/2 4QAM-TTCM encodersand combines the three independent codeword sequences intoa HM signal stream. Thus, the signal frame x1 is formedby HM-64QAM symbols. When the transmit power at thesource is relatively low, the DN may opt for decoding onlythe information from Encoder 1 during the first TS, wherethe information contained in the other two layers would bedecoded and retransmitted by the RNs. In order to reduce thecomplexity of the entire system, two RNs are activated forassisting the transmissions and each of the two RNs is usedfor retransmitting only one information layer of the triple-layer HM-64QAM symbols. With aid of the pre-coding andSPM schemes, the two RNs become capable of transmittingsimultaneously. Hence, the system now needs two TSs tocomplete its transmissions between the SN and DN. Morespecifically, each RN will only deal with a single layer of thetriple-layer HM-64QAM scheme, so the signal frames theytransmit are all 4QAM symbol frames. In this way, both theprocessing complexity of the entire system and the transmitpower of the RN is reduced.



4

Encoder 2

Encoder 3

Encoder 1

64QAM

HM

Mapper

HM16QAM

De−Mapper

LLR

converter 1Decoder 2

Encoder 24QAM

Mapper

LLR

converter 2Decoder 3

Encoder 34QAM

Mapper

Decoder 1

Decoder 2

Decoder 3

T1

SNT2

HM64QAM

De−Mapper

T2

+

+

De−Mapper

HM4QAM

SPM

16QAM

De−Mapper

DN

d3 c3

d2 c2

c1d1

p(c1, c2) d2

p(c2)

c2

RN1

ρ1, α

RN2

p(c1, c2, c3) d3

c3

ρ2, βp(c3)

p(c1) d1

d2

d3p(c3)

p(c2)

Fig. 2: The system diagram of a twin-relay HM and SPM aided cooperative system.

00−11 00−10

00−00 00−01

00−00−10

00−00−01

00−11−10

00−11−01

01−00 01−01

01−11 01−10

01−00−00 01−00−01

01−00−11 01−00−10

01−01−00 01−01−01

01−01−11 01−01−10

0100

1011

01−11−00 01−11−01

01−11−11 01−11−1000−11−11

00−11−00

00−00−11

00−00−00

00−01−11 00−01−10

00−01−00 00−01−01

00−10−00 00−10−01

00−10−11 00−10−1000−10−11 01−10−11

01−10−00 01−10−01

01−10−10

11−00 11−01

11−1011−11

11−00−00 11−00−01

11−00−11 11−00−10

11−01−00 11−01−01

11−01−11 11−01−10

11−11−11 11−11−10

11−10−0011−11−00 11−11−01 11−10−01

11−10−11 11−10−10

10−00 10−01

10−10

10−11

10−00−00 10−00−01

10−00−11 10−00−10

10−01−00 10−01−01

10−01−11 10−01−10

10−11−00 10−11−01

10−11−11 10−11−10

10−10−00 10−10−01

10−10−11 10−10−10

d1 d0

δ2

δ1

HM 16QAM: b5b4 − b3b2

HM 64QAM: b5b4 − b3b2 − b1b0

Base Layer: b5b4

d3 d2

Fig. 3: The constellation map of the triple-layer HM-64QAMscheme, where R1 = d1/d0, R2 = d3/d2.

III. HM AND SPM MODULATION SCHEMES

A. Triple Layer HM Scheme

Our triple-layer model of the HM-64QAM constellationseen in Fig. 3 was originally introduced in [26]. Since TTCMis used, where the symbol-based decoder’s performance isdetermined by the Symbol Error Rate (SER), hence set-partition based mapping is invoked by the HM constellationinstead of Gray mapping.

We define the six bits in a HM-64QAM symbol as(b5b4b3b2b1b0), where the base layer or first layer (L1) isoccupied by (b5b4), (b3b2) belong to the second layer (L2) and(b1b0) are contained in the third layer (L3). The generation ruleof the triple-layer HM-64QAM symbols may be expressed as:

SHM−64QAM = β[S4QAM ±

√2δ1e

±π4j ±√2δ2e

±π4j]. (13)

The parameter β is used for normalizing the average symbolenergy to unity, which given by β = 1/

√1 + 2δ2

1 + 2δ22 . Fur-

thermore, the ratios R1 = d1/d0 and R2 = d3/d2 are definedfor controlling the shape of the HM-64QAM constellations, asshown in Fig. 3, where all the three parameters β, δ1 and δ2will be directly controlled by the HM ratio pair (R1, R2) andtheir relationship may be expressed as follows:

δ1 =1√

2 (1 +R1), (14)

δ2 =R1 −R2√

2 (1 +R1) (1 +R2). (15)

The constraint of the HM ratio pair (R1, R2) in the simulationsare:

0 < R2 < R1 if R1 < 11

2(R1 − 1) < R2 < R1 if R1 > 1 .

(16)

The derivations of Eq. (14), Eq. (15) and Eq. (16) are detailedin appendices A, B, C and D respectively. The entire HM-64QAM constellation point arrangement is directly controlledby the HM ratios R1 and R2. Upon increasing the value ofthese two HM ratios, the constellation points in each quadrantwould move closer to each other. Hence it is necessary tohave a higher receive SNR at the RN (SNRRNr ) in order toadequately detect the information contained both in L2 as wellas in L3, but a lower SNRDNr is necessitated for detecting thetwo bits in L1. Our design-goal is to find the optimum HMratios and RN position based on a given SNRSNt .

B. Twin Layer SPM Scheme

The twin layer SPM scheme is detailed in [34], where weobserve from this Eq. (9) that the reception of the signal atthe DN is identical to that of detecting a twin-layer linearSPM signal received over AWGN channels. Note that theperformance of a specific modulation scheme in the AWGNchannel is directly determined by the Euclidean distanceamong the constellation points. The relationship between αand β is given by [34]:

α2 + β2 = 1 . (17)



5

If we only focus our attention on the relationship between theEuclidean distance dmin and α, we have:

dmin =

1√2

min(

2√

1− α2, 2(α−√

1− α2)) (

α ≥√

1/2)

1√2

min(

2α, 2(√

1− α2 − α)) (

α <√

1/2).

(18)

Theoretically the largest Euclidean distance is achieved,when α is

√1/5 or

√4/5. Hence, we anticipate that the best

performance of an uncoded twin-layer linear SPM schemewould appear when α equals to

√1/5 or

√4/5. However,

when TTCM is used, it depends on dmin of the entire TTCMset partitioning scheme [17].

IV. DEMAPPER AND RN POSITION

In [34], we have discussed the receiving of the two signalframes using SPM schemes, so in this section, we only focusour attention on the receiving of the triple-layer TTCHM-64QAM symbols. The symbol-to-bit Demapper block ofFig. 2 will produce a (N ×M)-element Probability DensityFunction (PDF) matrix of receiving y given x(i) transmit-ted. x(i) is the hypothetically transmitted M -ray symbol fori ∈ 0, 1, · · · ,M − 1 and the element in the matrix isp(y|x(i)), which is the soft-input to the TTCM decoder, N isthe number of symbols in a transmission block. The generalequation of calculating the PDF of receiving y, given that x(i)

is transmitted may be expressed as:

p(y|x(i)) =1

πN0exp

(−|y −

√Ghx(i)|2N0

)i ∈ 0, 1, . . . ,M − 1 , (19)

where h is the fading coefficient and G is the path-gain.

A. L1 Detection at DN

The DN of Fig. 2 will demap the HM-64QAM signal framesreceived from the SN as 4QAM symbols for detecting theinformation contained in the base layer of the triple-layerHM-64QAM constellation. According to the HM-64QAMgeneration rule of Eq. (13), Eq. (19) may be rewritten as:

p(ySD|x(i)) =1

πN0exp

(−|ySD −

√GSDhSDx

(i)|2N0

)x(i) ∈ βejπ/4, βej3π/4, βej−3π/4, βej−π/4 , (20)

where we have i ∈ 0, 1, 2, 3.

B. L2 Detection at RN1

The information output of Encoder 2 in Fig. 2 is mappedonto L2, RN1 will demap the received signal frame x1 as theHM-16QAM symbols shown in Fig. 3 and will obtain the jointsymbol probability of L1 and L2 in the HM-64QAM symbolstreams for producing a (N × 16)-element PDF matrix. Then,the Log Likelihood Ratio (LLR) converter 1 of Fig. 2, willextract the PDF of L2 from the (N × 16)-element PDF matrix.Therefore, RN1 can decode L2 even when L1 is received with

errors. When demapping the HM-64QAM symbol as HM-16QAM, Eq. (19) may be reformulated as:

p(ySR1 |x(i)) =1

πN0exp

(−|ySR1 −

√GSR1hSR1x

(i)|2

N0

)x(i) ∈ β

[S4QAM ±

√2δ1e

±π4j] , (21)

where i ∈ 0, 1, · · · , 15. The HM-16QAM constellationpoints of x(i) are shown as hollow circles in Fig. 3. We definedL

(0)2 as the pair of bits (00) in L2, L(1)

2 as (01), L(2)2 (10) and

finally L(3)2 for (11), where the corresponding generation rule

is given by:

p(ySR1|L(l)

2 ) =p(ySR1|x(l)) + p(ySR1

|x(l+4))+

p(ySR1 |x(l+8)) + p(ySR1 |x(l+12)) , (22)

where l ∈ 0, 1, 2, 3. The input PDF matrix of Decoder 2 isformulated in Eq. (22).

C. L3 Detection at RN2

RN2 requires the highest receive power, because it has toconvey the information of L3. In order to receive L3, RN2 hasto fully demap the whole HM-64QAM symbol stream. HenceEq. (19) may be represented as:

p(ySR2 |x(i)) =1

πN0exp

(−|ySR2 −

√GSR2hSR2x

(i)|2

N0

)x(i) ∈ β

[S4QAM ±

√2δ1e

±π4j ±√2δ2e

±π4j] , (23)

where i ∈ 0, 1, · · · , 63. Let L(0)3 denote the pair of bits (00)

in L3, L(1)3 represent (01), L(3)

2 (10) and finally L(3)3 for (11).

Then the LLR converter 2 of Fig. 2 may produce the PDF ofL3 according to:

p(ySR2|L(l)

3 ) =

15∑k=0

p(ySR1|x(i=4k+l)) , (24)

where l ∈ 0, 1, 2, 3. The (N × 4)-element PDF matrixgenerated by the LLR converter 2 is then fed to Decoder 3of RN2, as seen in Fig. 2, for detecting L3. Furthermore, weconsider logarithmic probabilities, so that the approximate logMAP algorithm [17] may be directly employed by the decoderblock.

D. RN position

In our simulations, the same rate-1/2 encoder is employedfor all three SN encoders. Hence we only focus our attentionon the specific SNR values for achieving a BER of 10−6.Multiple values of the two HM ratios R1 and R2 had beentested. At a given HM constellation ratio pair (R1, R2), theminimum receive SNR required SNRL1

r for decoding L1 atthe DN, SNRL2

r for receive L2 at the RN1 and SNRL3r for

receiving L3 at RN2 may be computed. The SNR differencesamong the three layers are:

GL1,L2

SNR = SNRL2r − SNRL1

r [dB] , (25)

GL1,L3

SNR = SNRL3r − SNRL1

r [dB] , (26)



6

where, GL1,LjSNR is the SNR difference between SNRL1

r andSNR

Ljr , for j ∈ 2, 3. If we set SNRSNt to be identical to the

SNR required for receiving L1 from the HM-64QAM symbol,namely to SNRSNt = SNRL1

r , this would guarantee that theBER of decoding L1 would reach an arbitrarily low value. Inthis situation, if we want the BER performance of receivingL2 to become sufficiently low, the channel gain GSR1

of theSN-RN1 link should satisfy:

10 log10GSR1+ SNRL1

r = SNRL2r . (27)

If we use the distance-ratio dSR1/dSD to represent the position

of the RN, we arrive at:

GL1,L2

SNR = 10 log10

(dSDdSR1

)3

, (28)

where GL1,L2

SNR is given by Eq. (25) and hence we have:

dSR1

dSD= 10−

GL1,L2SNR30 . (29)

Similarly, the position of RN2 is related to:

dSR2

dSD= 10−

GL1,L3SNR30 . (30)

V. TRIPLE-LAYER TTCHM-64QAM COOPERATIVESYSTEM DESIGN

In practice we do not have any control over the position ofmobile relays, but the relay-selection algorithm would appointa relay close to the optimum location. In this section, we willoptimize this practical system regarding the position of the RN,as well as the HM ratio pair (R1, R2) and the SPM weightingpair (α, β). Additionally, in this investigation, the simulationsare carried out by IT++ and the number of iterations of ourrate-1/2 TTCM decoder is ζ = 4, the block size is η = 12, 000symbols. Using a large number of iterations allows the TTCMdecoder to more closely approach capacity and a large blocklength assists in avoiding error propagation, but also imposesan increased complexity. In the simulations, we observed thatno substantial BER performance improvement is achieved formore than four iterations (ζ > 4) or for a block size of η >12, 000 symbols.

A. Optimum SPM ratio

In our simulations, the pair of signal sequences receivedfrom RN1 and RN2 will be multiplied by a specific SPM ratioα or β and be received by the DN simultaneously. Here, weonly focus our attention on the performance of receiving asingle signal sequence having a SPM ratio α in the simulationsand the related results are shown in Fig. 4(a) and Fig. 4(b).

From the combined signal sequences, we set the signalassociated with a larger SPM ratio α to be the dominantsignal and the signal with a smaller SPM ratio β to be theauxiliary signal. Based on the simulation results of Fig. 4(a)and Fig. 4(b), we may observe that as expected, for thedominant signal, a larger SPM ratio would result in a betterBER performance. However, for the auxiliary signal, thebest BER performance appears at the SPM ratio of 0.45

10-6

10-5

10-4

10-3

10-2

10-1

1

BE

R

6 9 12 15 18

SNR

SPM =0.1

SPM =0.2

SPM =0.3

SPM =0.45

SPM =0.5

SPM =0.6

(a) α <√

1/2

10-6

10-5

10-4

10-3

10-2

10-1

1

BE

R0 3 6 9 12 15

SNR

SPM =0.71

SPM =0.8

SPM =0.85

SPM =0.89

SPM =0.95

(b) α >√

1/2

Fig. 4: The BER versus SNR performance of receiving a singlesignal sequence with SPM ratio α in the twin layer SPMschemes. Both of the two signal sequences are encoded byrate-1/2 TTCM encoder. The number of iterations of the rate-1/2 TTCM decoder is ζ = 4, the block size is η = 12, 000symbols and the channel is uncorrelated Rayleigh fadingchannel.

(√

1/5 ≈ 0.45) or 0.5. We have found from our simulationsthe SNRr value required for achieving BER=10−6 for differentvalues of the SPM ratio α, as shown in Fig. 5. Note that 0.7071(√

1/2 ≈0.7071) is a bound, where we have α = β =√

1/2,which would collapse the 16-point constellation to a 9-pointconstellation making unambiguous decoding to be impossible.Fig. 5 shows that when α = 0.5, the SNRr required forreceiving the auxiliary signal from the SPM-16QAM signalis about 10.31 dB which is the lowest value of receiving theauxiliary signal. Meanwhile, when α = 0.5, the correspondingSPM ratio β will be

√1− α2 ≈ 0.87, and the SNRr required

for receiving the dominant signal from the SPM-16QAMsignal using β = 0.87 is about 5.74 dB. Hence, based onthe results of Fig. 5, we find that the optimum SPM ratiopair is (α = 0.87, β = 0.5) (we assume α > β in thispaper). In this combination, the SNRr required for receivingthe two signal sequences is the lowest. Consequently, the SNRtrequired at the two RNs would also be the lowest at the givenRN positions.

B. SNRt of the two RNs

In order to evaluate the power-efficiency of our cooperativecommunications scheme, we have to calculate the average



7

0

4

8

12

16

20

24

28

SN

Rr[d

B]

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

SPM ratio

.

.

.

. ..

.

.

.

.

.

.

..

.

.

Fig. 5: The SNRr versus SPM ratio performance of receivinga single signal sequence with SPM ratio α in the twin layerSPM schemes. Both of the two signal sequences are encodedby rate-1/2 TTCM encoder. The number of iterations of therate-1/2 TTCM decoder is ζ = 4, the block size is η = 12, 000symbols and the channel is uncorrelated Rayleigh fadingchannel. The SNRr here is the required receive SNRr forachieving a performance of BER=10−6.

SNRt (SNRt) defined as the SNRt per TS. Based on thestatistics seen in Fig. 5, we opted for the SPM ratio pair of(α = 0.87, β = 0.5) and hence we have SNRDNr = 10.31 dB.Next, we have to find the relationship among SNRRN1

t ,SNRRN2

t and SNRDNr . Let us denote the SNR by γ, which isexpressed as 10 log10(γ) in dB. Furthermore, since L2 of thetriple-layer HM-64QAM symbols has a higher priority thanthat of L3, the signal frame x2 transmitted from RN1 willbe multiplied by a higher SPM weighting factor of α = 0.87,while another SPM weighting factor of β = 0.5 is used forthe signal frame x3 at RN2. Hence we have:

E[γRN1t

]= E

[ | αρ1x2 |2N0

], (31)

whereE[| x2 |2

]= 1 . (32)

Therefore, it can be observed that:

E[γRN1t

]=E

[α2|ρ1|2N0

]=α2E

[ | h∗R1D|2

| hR1D |4 GR1DN0

]=

α2

GR1DN0E

[1

|hR1D|2

], (33)

where h obeys the Rayleigh distribution of [32]:

f(h) =2h

Ωexp

(−h

2

Ω

). (34)

Note that the mean square value of h is given by Ω = 1,Let Z = |h|2, then the distribution of the variable Z may beexpressed as [32]:

fZ(z) =1

Ωexp

(− z

Ω

). (35)

Let us denote the Cumulative Distribution Function (CDF) asFZ(Γ) = Pr(Z < Γ), which can be expressed as:

FZ(Γ) = Pr(Z < Γ) = 1− exp

(−Γ

Ω

), (36)

where Pr denotes probability. Upon introducing Γ′ = 1/Γ, wemay further express Eq. (36) as:

Pr(Z < Γ) = Pr

(1

Z>

1

Γ

)= 1− Pr

(1

Z< Γ′

). (37)

Hence, it can be observed that:

Pr

(1

Z< Γ′

)= exp

(− 1

ΩΓ′

). (38)

If we let Θ = 1Z , the PDF of the variable Θ may be expressed

as [35]:

fΘ(θ) =1

Ωθ2exp

(− 1

Ωθ

). (39)

Therefore, the expectation E[

1|h|2

]may be derived as:

E

[1

|h|2]

=

∫ Θup

Θlow

Θ

ΩΘ2exp(− 1

ΩΘ)dΘ , (40)

where Θlow and Θup are the lower and upper limits of theintegration. Let us define ξ = 1

Θ , then Eq. (40) may beconverted to:

−1

Ω

∫ 1/Θlow

1/Θup

1

ξexp

(−ξ

Ω

)dξ =

1

Ω

[Ei

(−1

ΩΘup

)− Ei

(−1

ΩΘlow

)],

(41)where Ei is the Euler function:

Ei(u) =

∫ ∞−u

e−t

tdt . (42)

Theoretically, we have |h| ∈ [0,+∞), which gives usΘlow = 0 and Θup = +∞. When Θlow = 0 , we mayhave Ei

(−1

ΩΘlow

)= 0. However, if Θup = +∞, the term

Ei

(−1

ΩΘup

)becomes infinite and we are unable to derive the

value of E[1/|h|2

]. To resolve this dilemma, we defined an

outage threshold, which is given by [|h|2]min = 0.03. Accord-ing to Eq. (35), the probability of Pr(|h|2 > 0.03) ≈ 0.97,indicates that our system will halt its transmissions, whenthe fading obeys |h|2 < 0.03. Hence, we may have 1

Θup=

[|h|2]min = 0.03. In this situation, we have E[1/|h|2

]= 2.96.

Hence, Eq. (33) may be expressed as:

E[γRN1t

]=

2.96α2

GR1DN0. (43)

Note that this assumption will lead to a 3% throughputreduction for the entire system. Additionally, we have:

E[γDNr

]= E

[ | αx2 + βx3 |2N0

]=

1

N0. (44)

Therefore, we find that:

E[γRN1t

]=

2.96α2

GR1DE[γDNr

], (45)



8

which may be expressed in dB as:

E[SNRRN1

t

]= E

[SNRDN

r

]+ 10 log10

(2.96α2

GR1D

). (46)

Similarly, the relationship between SNRRN2t and SNRDNr

may be formulated as:

E[SNRRN2

t

]= E

[SNRDN

r

]+ 10 log10

(2.96β2

GR2D

). (47)

Finally, SNRt may be expressed as:

SNRt = 10 log10 (γt) = 10 log10

(γSNt + γRN1

t + γRN2t

2

). (48)

VI. SIMULATION RESULTS

Based on Section V, we characterized our cooperativecommunications system for multiple values of the HM ratiopairs (R1, R2) for the sake of generating the power dissipationsurface of the cooperative system, which is shown in Fig. 6.Explicitly, 64 pairs of (R1, R2) have been simulated, whereR1 is chosen from 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0 andfor each R1, multiple R2 values were selected based on theconstraints illustrated in Section III. Given a specific HMratio pair (R1, R2), the receive SNR required for adequatelyreceiving L1, L2 and L3, namely SNRL1

r , SNRL2r and SNRL3

r

respectively, may be derived for a target BER of 10−6. Morespecifically, the optimum position of the two RNs and theoptimum average transmit SNR (SNRt) of the entire systemmay be calculated according to SNRL1

r , SNRL2r and SNRL3

r

based on the discussions in Section IV and Section V. Hence,we can compute the optimum SNRt for each of the HM ratiopairs (R1, R2), as shown in Fig. 6. The bold line marked bydots in Fig. 6 illustrates the lowest power consumption pointfor a specific HM ratio pair (R1, R2) and the correspondingdata is recorded in Table I.

Fig. 6: The 3D plot of the simulation-based transmit powerdissipation surface of the entire system versus the HM-64QAMratio pair (R1, R2). The two RNs employ SPM schemes asso-ciated with the SPM weighting pair of (α = 0.87, β = 0.5).Explicitly, the number of iterations of the rate-1/2 TTCMdecoder is ζ = 4, the block size is η = 12, 000 symbolsand an uncorrelated Rayleigh fading channel is considered.

Based on the results of Fig. 6, the best performanceof our cooperative communication system obeying this ar-rangement is achieved, when the HM ratio pair is given by(R1 = 1.5, R2 = 0.8) and the SPM weighting factor pair is(α = 0.87, β = 0.5), where the optimum SNRt per TS is6.94 dB. In this situation, the positions of RN1 and RN2 aredSR1/dSD = 0.53 and dSR2/dSD = 0.31, where SNRSNt is6.81 dB, SNRRN1

t is 4.01 dB and SNRRN2t is 4.08 dB, as

shown in Table I. The throughput per TS for this scheme is3/2×0.97 = 1.455 bps owing to the 3% throughput reductionimposed by the threshold of [|h|2]min = 0.03.

A. EXIT chart analysis

The Extrinsic Information Transfer (EXIT) chart is ca-pable of visualizing the input/output characteristics of theconstituent MAP decoders in terms of the achievable averagemutual information transfer [36], [37]. It may be used forpredicting whether a soft decision based decoder is capableof decoding the information with an arbitrarily low BERbased on the available mutual information provided by thesymbol-to-bit demapper. Since we do not invoke iterationsbetween the demapper and the TTCM decoder in our symbol-based scheme, we need a sufficiently high receive SNRr forguaranteeing that the mutual information gleaned from thedemapper is sufficiently high for the decoder to attain a lowBER. More specifically, the EXIT curves of the HM-64QAMand SPM-16QAM demappers are illustrated in Fig. 7.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

I E(i) ,

I A(o

)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

IA(i)

, IE(o)

HM-64QAM L1, SNRr=6.81 dB



SPM-16QAM L2, SNRr=5.74 dB

SPM-16QAM L3, SNRr=10.31 dB

Outer, =2

Outer, =4

Outer, =6

Outer, =8

Inner

Fig. 7: The symbol based EXIT chart of our rate-1/2 TTCMaided triple-layer HM-64QAM scheme and twin-layer SPM-16QAM scheme. The number of iterations in the rate-1/2TTCM decoder is from the set of ζ ∈ 2, 4, 6, 8. TheHM-64QAM ratio pair is (R1 = 1.5, R2 = 0.8) and theSPM weighting coefficient pair is (α = 0.87, β = 0.5). Thereceive SNR required for achieving a BER of 10−6 based onsimulations is denoted as SNRr. An uncorrelated Rayleighfading channel is considered.

In Fig. 7, the notation ‘Inner’ iteration represents the infor-mation exchange between the demapper and decoder, while‘Outer’ iteration refers to the information exchange betweenthe two components of the TTCM decoder. Since there are



9

R1 R2 Pos. RN1

[dSR1/dSD]Pos. RN2

[dSR2/dSD]SNRSN

t[dB]

SNRRN1t

[dB]SNRRN2

t[dB]

SNRt

[dB]0.5 0.3 0.82 0.33 11.18 -8.81 3.88 8.941.0 0.6 0.65 0.30 7.98 -0.04 4.27 6.961.5 0.8 0.53 0.31 6.81 4.01 4.08 6.942.0 1.2 0.47 0.26 5.91 5.61 5.12 7.322.5 1.6 0.42 0.22 5.42 6.71 5.76 7.763.0 2.0 0.38 0.19 5.06 7.52 6.20 8.143.5 2.4 0.35 0.17 4.91 8.10 6.50 8.464.0 2.8 0.33 0.16 4.79 8.58 6.73 8.74

TABLE I: The simulation based average transmission power SNRt per TS of the HM-64QAM and SPM based cooperativesystem. The SPM weighting pair is (α = 0.87, β = 0.5), the number of iterations of the rate-1/2 TTCM decoder is ζ = 4, andthe block size η = 12, 000 symbols. The channel is an uncorrelated Rayleigh fading channel.

no iterations between the demapper and the TTCM decoder,we may observe in Fig. 7 that the inner curve is a straightline. Specifically, the inner curve shows the mutual informationreceived by the decoder from the demapper, which is increasedupon increasing the receive SNR. In order to guarantee thatthe decoder becomes capable of decoding the information withan arbitrarily low BER, an open EXIT chart tunnel has to bemaintained between the ‘Inner’ and ‘Outer’ curves all the wayto the (x, y) = (2, y) point, where we have x = I

(i)A = I

(o)E

and y = I(i)E = I

(o)A . Note that the subscript i and o denote

‘Inner’ and ‘Outer’ respectively, whilst IA and IE denote the apriori and extrinsic information. When the HM-64QAM ratiopair is (R1 = 1.5, R2 = 0.8) and the number of TTCMiterations is ζ = 4, the receive SNR required for achievinga BER of 10−6 for each HM layer is: SNRL1

r = 6.81 dB,SNRL2

r = 15.15 dB and SNRL3r = 21.87 dB. It also can be

observed in Fig. 7 that there are open tunnels between the three‘Inner’ curves upon receiving triple-layer HM symbols and theζ =4-iteration ‘Outer’ curve. Note that the ‘Inner’ curve ofreceiving L3 of the HM scheme is closer to the ‘Outer’ curvecompared to the other two ‘Inner’ curves of receiving L1 andL2. When L1 of the HM scheme is detected at the DN, eachHM-64QAM symbol received will be detected as a 4QAMsymbol. Similarly, L2 of the HM scheme will be detectedin the form of 16QAM symbols at RN1. This HM-specificsimplifying demapping assumption shifts the ‘Inner’ curvescorresponding to L1 and L2 upwards to higher values thanthe ‘Inner’ curve recorded for receiving L3. Hence, we inferthat we can reduce the SNR required for receiving L1 and L2

to let the two ‘Inner’ curves to be closer to the ζ =4-iterationbased ‘Outer’ curve. However, according to the simulations,even though there might be an open EXIT tunnel, the integrityof L1 and L2 will be degraded, if we reduce SNRr.

Additionally, the ‘Inner’ curves associated with receivingthe pre-coding based twin-layer SPM-16QAM scheme are alsoshown in Fig. 7. In the simulations, the L2 and L3 of the triple-layer HM-64QAM will be mapped to base layer and auxiliarylayer of the SPM symbols, respectively. It can be observed inFig. 7 that the SNRr required for achieving a BER of 10−6

for both layers of our twin-layer SPM scheme can provideopen tunnels between the two ‘Inner’ curves and the ζ =4-iteration ‘Outer’ curve. Moreover, both ‘Inner’ curves are veryclose to the ζ =4-iteration ‘Outer’ curve, since the DN fullydetects the SPM-16QAM symbols for the sake of receiving

10-6

10-5

10-4

10-3

10-2

10-1

1

BE

R

0 5 10 15 20 25 30 35

SNRr [dB]

HM-64QAM L1

HM-64QAM L2

HM-64QAM L3

Perfect CSI

CSI var=0.01

CSI var=0.1

Fig. 8: BER versus SNR performance of the triple-layer codedHM-64QAM scheme with imperfect CSI, where the HM-64QAM ratio pair is (R1 = 1.5, R2 = 0.8). Explicitly, thenumber of iterations of the rate-1/2 TTCM decoder is ζ = 4,the block size is η = 12, 000 symbols and an uncorrelatedRayleigh fading channel is considered.

the soft-information of L2 and L3. We have investigated theoptimum number of iterations for the TTCM decoder. It canbe observed from Fig. 7 that increasing ζ beyond 4 only givesus a marginal gain, while significantly increasing the decodingcomplexity. Hence we have opted for ζ = 4 for our design.

B. Imperfect CSI for receiving triple-layer HM and pre-codingbased twin-layer SPM

When considering the impact of imperfect CSI at all nodesin cooperative communications, the performance of our co-herent scheme is expected to be degraded. To investigatethe robustness of our triple-layer coded HM-64QAM andpre-coding based twin-layer coded SPM-16QAM schemes,we model the CSI estimation errors by a Gaussian processsuperimposed on each channel coefficient at the two RNs andDN, where the CSI estimation error variance is denoted by σ.

The BER performance of receiving triple-layer coded HM-64QAM recorded for diverse CSI estimation error variancesσ is shown in Fig. 8. We can observe that a CSI estimationerror of σ = 0.01 only slightly impedes the performance ofreceiving L1, but will cause a 2 dB SNR degradation (at a BERof 10−6) for receiving L2. However, the system excessivelydegrade L3 for σ = 0.01. By contrast, a CSI estimation



10

error of σ = 0.1 would impose a 5 dB SNR degradation onthe performance of L1, whilst the cooperative communicationsystem will have completely lost L2 and L3 in this situation. Itis shown in Fig. 8 that a CSI estimation error variance belowσ = 0.001 is required for receiving L3. Hence, we find that therobustness of each layer in the HM-64QAM symbols againstimperfect CSI is different, where L1 is the most robust layer,while L3 has the highest sensitivity to CSI errors.

10-6

10-5

10-4

10-3

10-2

10-1

1

BE

R

0 3 6 9 12 15

SNR [dB]

SPM-16QAM L2

SPM-16QAM L3

Perfect CSI

CSI var=10-5

CSI var=0.001

CSI var=0.01

Fig. 9: BER versus SNR performance of the twin-layer codedSPM-16QAM scheme with imperfect CSI, where the SPM-16QAM weighting pair is (α = 0.87, β = 0.5). Explicitly, thenumber of iterations of the rate-1/2 TTCM decoder is ζ = 4,the block size is η = 12, 000 symbols and an uncorrelatedRayleigh fading channel is considered.

The performance of our pre-coding based coded SPMscheme associated with CSI estimation errors are shown inFig. 9. If there is CSI estimation error for the R1N (R2N )and the DN link, the resultant ρ1 in Eq. (7) and ρ2 in Eq. (8)become:

ρ1 =h∗R1D

| h∗R1D|2√GR1D

, (49)

ρ2 =h∗R2D

| h∗R2D|2√GR2D

. (50)

Hence, the pre-coding based twin-layer coded SPM-16QAMsymbols received by the DN during the second TS is expressedas:

yRD =αh∗R1D

hR1D

| h∗R1D|2

x2 +βh∗R2D

hR2D

| h∗R2D|2

x3 + nRD . (51)

More specifically, we have:

hR1D = hR1D + ∆1 , (52)

hR2D = hR2D + ∆2 , (53)

where ∆1 is the CSI estimation error imposed on hR1D,and ∆2 is the CSI error contaminating hR2D. In order tosimplify our discussions, we assume that both ∆1 and ∆2

obey the Gaussian distribution with zero mean and a varianceof σ1 = σ2 = σ. It can be seen from Fig. 9 that our pre-codingbased twin-layer coded SPM-16QAM scheme is sensitive to

the CSI estimation errors. Explicitly, an error floor aroundBER of 10−4 exists for the detection of L2 and L3 of the twin-layer SPM scheme when σ = 0.001. Upon increasing the CSIestimation error variance σ, the performance of the pre-codingbased coded SPM scheme will be dramatically reduced. Theerror floor will be eliminated at BER of 10−6 by employingsophisticated channel estimation schemes, which could reduceσ below a level of 10−5, as shown in Fig. 9.

C. Comparisons with other systems

Fig. 10 compares our rate-1/2 TTCM aided triple-layer HM-64QAM scheme to that of the same system but aided by arate-1/2 LDPC code [38]. It can be observed that the bit-based LDPC coding scheme performs slightly worse than thesymbol based TTCM coding scheme, even though the numberof LDPC decoder iterations is much higher than that of theTTCM decoder. This is because a symbol-based scheme tendsto have a lower convergence SNR than an equivalent bit-basedscheme, as detailed in [39].

10-6

10-5

10-4

10-3

10-2

10-1

1

BE

R

0 5 10 15 20 25 30 35

SNR [dB]

HM-64QAM L1

HM-64QAM L2

HM-64QAM L3

Rate 1/2 TTCM

Rate 1/2 LDPC

Fig. 10: The BER versus SNR performance of our triple-layer64QAM TTCHM scheme and triple-layer 64QAM LDPCscheme. The triple-layer HM-64QAM ratio pair is (R1 =1.5, R2 = 0.8). The block size of the decoder of the twocoding schemes is the same, which is η = 12, 000 symbols,the code-rate of the two coding schemes’ encoder is 1/2.For TTCM decoder, the iteration number is ζ = 4 and themaximum iteration number of LDPC decoder is ζl = 20.The system communicates over uncorrelated Rayleigh fadingchannel.

The BER performance curves of the non-cooperative rate-2/3 8PSK TTCM and of our scheme in [26] are shown inFig. 11. The throughputs of the two schemes are 2 bps and1.5 bps (the scheme in [26] requires 2 TSs for cooperativetransmission). Note that the throughput of the optimizedscheme proposed in this paper is 1.455 bps. By contrast, theachievable throughput of the system proposed in this paper is3×0.97 = 2.91 bps, when the channel SNR is sufficiently highfor the DN to detect all the three layers of the HM symbolsin a single TS without the assistance of the RN. The SNRSNtrequired for achieving a BER of 10−6 for the non-cooperativerate-2/3 8PSK is 13.2 dB, while that of the optimized schemein [26] is 14.8 dB. By contrast, the SNRSNt of the optimizedscheme proposed in this paper is 6.81 dB and SNRt is 6.94 dBper TS. Additionally, the simulation results of [13] show that



11

10-6

10-5

10-4

10-3

10-2

10-1

1

BE

R

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

SNRr

rate-2/3 8PSK TTCM

Opt. scheme in [27]

Fig. 11: The BER versus SNR performance of non-cooperativerate-2/3 8PSK TTCM and the optimized scheme in [26]. Thenumber of iterations of the TTCM decoder in each of the threeschemes is ζ = 4, the block size is η = 12, 000 symbols. Anuncorrelated Rayleigh fading channel is considered.

in order to transmit a twin-layer HM-16QAM signal withthe aid of two rate-1/2 H264/AVC encoders over an AWGNchannel for achieving a BER of 10−6, their SNRSNt should behigher than 14 dB, which is about 7.19 dB higher than that ofour optimized scheme transmitting a triple-layer HM-64QAMsignal over uncorrelated Rayleigh channels. It can be observedthat the scheme optimized in this paper reduced the SNRSNtand SNRt of the cooperative system.

VII. CONCLUSIONS

In this paper, we proposed a TTCHM aided cooperativecommunication system. The system employed both pre-codingand SPM schemes at the two RNs, as detailed in Section II.The results have demonstrated that the best performanceof the system is achieved at a HM-64QAM ratio pair of(R1 = 1.5, R2 = 0.8) and the optimum SPM ratio pair is(α = 0.87, β = 0.5). The optimized system requires an SNRtof 6.94 dB per TS. It can be concluded that by employing HMin cooperative communications, both the SNRSNt may indeedbe reduced, along with the SNRt of the entire system. Wenote that spatial modulation [40] may also be employed forfurther reducing the transmit power dissipation of the entiresystem. The benefit of employing spatial modulation will beinvestigated in our future work.

APPENDIX

A. Derivation of Eq. (14)

Before normalization, the L1 is represented as conventionalsquare 4QAM symbols, hence we may have d0 + d1 =

√2.

Therefore, it can be expressed that:

δ1 =d0

2=

d0√2√

2=

d0√2 (d0 + d1)

=1√

2 (1 +R1). (54)

B. Derivation Eq. (15)

As shown in Fig. 3, we have:

R1 = d1/d0 and R2 = d3/d2 . (55)

Hence, we can write d1, d2 and d3 as:

d1 = d0R1 and d3 = d2R2 , (56)

d1 = d3 + 2δ2 and d2 = d0 + 2δ2 , (57)

It may be observed from Eq. (56) and Eq. (57) that

δ2 =d1 − d3

2=d0R1 − d2R2

2=d0R1 − (d0 + 2δ2)R2

2,

(58)where δ2 here may be expressed as:

δ2 =d0(R1 −R2)

2(1 +R2). (59)

Upon substituting Eq. (54) into Eq. (59) we have:

δ2 =(R1 −R2)√

2(1 +R1)(1 +R2). (60)

C. Restrictions on R1

It can be observe that:

R1:max =d1max

d0min

=

√2

0⇒∞ , (61)

R1:min =d1min

d0max

=0√2⇒ 0 , (62)

so we have 0 < R1 <∞.

D. Restrictions on R2

Note that R2 is directly restricted by R1 as follows:

R2max =d3max

d2min

=d1

d0⇒ R1 . (63)

If R1 > 1, then max (δ2)→ d0/2 and we have:

R2min =d3min

d2max

=d1 − d0

2d0⇒ 1

2(R1 − 1) . (64)

By contrast, if R1 < 1, then we have max (δ2) → d3/2 andhence:

R2min =d3min

d2max

=0

d0 + d3⇒ 0 . (65)

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Hua Sun received the B.Eng. degree in electronicsand information engineering from the HuazhongUniversity of Science & Technology (HUST),Wuhan, China, in 2009. In 2010, he obtained a MScdegree with distinction in Wireless Communicationsfrom the University of Southampton, Southampton,UK. He is currently working towards the PhD de-gree in the Research Group of Communications,Signal Processing and Control, School of Electronicsand Computer Science, University of Southampton,Southampton, UK. His research interests include

Superposition Modulation, Hierarchical Modulation, Turbo Trellis-CodedModulation as well as cooperative communications.

Dr Soon Xin Ng (S’99-M’03-SM’08) received theB.Eng. degree (First class) in electronics engineeringand the Ph.D. degree in wireless communicationsfrom the University of Southampton, Southampton,U.K., in 1999 and 2002, respectively. From 2003 to2006, he was a postdoctoral research fellow workingon collaborative European research projects knownas SCOUT, NEWCOM and PHOENIX. Since Au-gust 2006, he has been a member of academicstaff in the School of Electronics and ComputerScience, University of Southampton. He is involved

in the OPTIMIX and CONCERTO European projects as well as the IU-ATC and UC4G projects. He is currently an associate professor in wirelesscommunications at the University of Southampton.

His research interests include adaptive coded modulation, coded modula-tion, channel coding, space-time coding, joint source and channel coding,iterative detection, OFDM, MIMO, cooperative communications, distributedcoding, quantum error correction codes and joint wireless-and-optical-fibercommunications. He has published over 180 papers and co-authored two JohnWiley/IEEE Press books in this field. He is a Senior Member of the IEEE,a Chartered Engineer and a Fellow of the Higher Education Academy in theUK.



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Chen Dong received his BS degree in electronicinformation sciences and technology fromUniversity of Science and Technology ofChina(USTC), Hefei, China in 2004, and hisMEng degree in pattern recognition and automaticequipment from the University of Chinese Academyof Sciences, Beijing, China in 2007. In 2014, hewas awarded PhD degree from the University ofSouthampton, UK. Now he is a post-doc in thesame University. He was the recipient of scholarshipunder the UK-China Scholarships for Excellence

programme and he has been awarded Best Paper Award at IEEE VTC2014-Fall. His research interests include applied math, relay system, channelmodelling and cross-layer optimization.

Lajos Hanzo (http://www-mobile.ecs.soton.ac.uk)FREng, FIEEE, FIET, Fellow of EURASIP, DScreceived his degree in electronics in 1976 andhis doctorate in 1983. In 2009 he was awardedthe honorary doctorate “Doctor Honoris Causa” bythe Technical University of Budapest. During his38-year career in telecommunications he has heldvarious research and academic posts in Hungary,Germany and the UK. Since 1986 he has been withthe School of Electronics and Computer Science,University of Southampton, UK, where he holds

the chair in telecommunications. He has successfully supervised about 100PhD students, co-authored 20 John Wiley/IEEE Press books on mobile radiocommunications totalling in excess of 10 000 pages, published 1400+ researchentries at IEEE Xplore, acted both as TPC and General Chair of IEEEconferences, presented keynote lectures and has been awarded a number ofdistinctions. Currently he is directing a 100-strong academic research team,working on a range of research projects in the field of wireless multimediacommunications sponsored by industry, the Engineering and Physical SciencesResearch Council (EPSRC) UK, the European Research Council’s AdvancedFellow Grant and the Royal Society’s Wolfson Research Merit Award. He isan enthusiastic supporter of industrial and academic liaison and he offers arange of industrial courses. He is also a Governor of the IEEE VTS. During2008 - 2012 he was the Editor-in-Chief of the IEEE Press and a ChairedProfessor also at Tsinghua University, Beijing. His research is funded bythe European Research Council’s Senior Research Fellow Grant. For furtherinformation on research in progress and associated publications please referto http://www-mobile.ecs.soton.ac.uk Lajos has 20 000+ citations.

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