684 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 2, FEBRUARY 2010...

684 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 2, FEBRUARY 2010

Reduced-Complexity Near-Capacity DownlinkIteratively Decoded Generalized Multi-Layer

Space-Time Coding UsingIrregular Convolutional Codes

Lingkun Kong, Soon Xin Ng, Ronald Y. S. Tee, Robert G. Maunder, and Lajos Hanzo

Abstract—This paper presents a low complexity iterativelydetected space-time transmission architecture based on Gen-eralized Multi-Layer Space-Time (GMLST) codes and IRreg-ular Convolutional Codes (IRCCs). The GMLST combinesthe benefits of the Vertical Bell-labs LAyered Space-Time (V-BLAST) scheme and Space-Time Coding (STC). The GMLSTis serially concatenated with a Unity-Rate Code (URC) andan IRCC which are used to facilitate near-capacity operationwith the aid of an EXtrinsic Information Transfer (EXIT) chartbased design. Reduced-complexity iterative multistage SuccessiveInterference Cancellation (SIC) is employed in the GMLSTdecoder, instead of the significantly more complex MaximumLikelihood (ML) detection. For the sake of approaching themaximum attainable rate, iterative decoding is invoked to achievedecoding convergence by exchanging extrinsic information acrossthe three serial component decoders. Finally, it is shown thatthe SIC-based iteratively detected IRCC-URC-GMLST system iscapable of providing a feasible trade-off between the affordablecomputational complexity and the achievable system throughput.

Index Terms—Generalized Multi-Layer Space-Time Code, it-erative detection, irregular convolutional code, EXIT charts.

I. INTRODUCTION

RECENT information theoretic studies have shownthat the capacity of a Multiple-Input Multiple-

Output (MIMO) system [1]–[4] is significantly higher thanthat of a Single-Input Single-Output (SISO) system. MIMOtechniques are also capable of achieving both multiplexinggain and diversity gain. In [5], Wolniansky et al. proposed thepopular multi-layer MIMO structure, referred to as the VerticalBell-labs LAyered Space-Time (V-BLAST) scheme, which iscapable of increasing the throughput without any increase inthe transmitted power or the system’s bandwidth. Althoughit was primarily designed for attaining transmit multiplexinggain, it is worth noting that upon increasing the number ofantennas, typically the achievable transmit diversity gain alsoincreases at the cost of an increased receiver complexity.

Manuscript received August 18, 2008; revised January 24, 2009; acceptedNovember 23, 2009. The associate editor coordinating the review of this paperand approving it for publication was H. Jafarkhani.

The authors are with the School of Electronics and Computer Science,University of Southampton, SO17 1BJ, United Kingdom (e-mail: {lk06r, sxn,rm, lh}@ecs.soton.ac.uk).

The financial support of the China-UK Scholarship Council, as well asthat of the EPSRC UK, the EU under the auspices of the Optimix project, isgratefully acknowledged.

Digital Object Identifier 10.1109/TWC.2010.02.081117

In contrast to spatial multiplexing techniques, Alamouti [6]discovered a transmit diversity scheme, referred to as a Space-Time Block Code (STBC), where the prime concern wasachieving diversity gain. The attractive benefits of Alamouti’sdesign motivated Tarokh et al. [7] to generalize Alamouti’sscheme to an arbitrary number of transmit antennas. Anothertransmit diversity scheme, referred to as Space-Time TrellisCoding (STTC) was invented by Tarokh et al. in [8], whichis capable of achieving both spatial diversity gain and codinggain or time diversity gain. However, these conventional STBCand STTC schemes achieve at most the same data rate asan uncoded single-antenna system. Hence, a MIMO schemeattaining both multiplexing gain and diversity gain is attrac-tive [9]. Various hybrid BLAST and STTC schemes have beenproposed in [10], [11]. A Generalized Multi-Layer Space-Time (GMLST) code may be constructed as a composite ofthe V-BLAST scheme and Space-Time Coding (STC), whichstrikes a feasible trade-off between the throughput and errorprobability attained. In [11], iterative multistage SuccessiveInterference Cancellation (SIC) was proposed to achieve themaximum receive diversity, attainable by classic MaximumLikelihood (ML) detection at a fraction of its complexity.

Despite the merits mentioned above, the system perfor-mance of a stand-alone uncoded GMLST scheme is far fromthe achievable MIMO channel capacity. Hence, for the sakeof decoding convergence to an infinitesimally low bit errorratio (BER) at near-capacity Signal-to-Noise Ratios (SNRs),the GMLST scheme is serially concatenated with outer codesfor iteratively exchanging mutual information between theconstituent decoders. The decoding convergence of iterativelydecoded schemes can be analysed using EXtrinsic Informa-tion Transfer (EXIT) charts [12]–[14]. Tuchler and Hage-nauer [14], [15] proposed the employment of IRregular Convo-lutional Codes (IRCCs) in serial concatenated schemes, whichare constituted by a family of convolutional codes havingdifferent rates, in order to design a near-capacity system.They were specifically designed with the aid of EXIT chartsto improve the convergence behaviour of iteratively decodedsystems. As a further advance, it was shown in [16]–[19] thata recursive Unity-Rate Code (URC) should be employed as anintermediate code in order to improve the attainable decodingconvergence.

Against this backcloth, the novel contribution of this treatise

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KONG et al.: REDUCED-COMPLEXITY NEAR-CAPACITY DOWNLINK ITERATIVELY DECODED GENERALIZED MULTI-LAYER SPACE-TIME CODING . . . 685

−

−

IRCC

Decoder

EncoderURC

URC

Decoder

BinarySource

HardDecision

(2)(1)(1)

(2)(1)

EncoderIRCC Tx

Encoder

Rx−

−

URC−GMLST Encoder

..

...Decoder

(3)

(3)

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GMLST

GMLST

URC−GMLST Decoder

π1

π1c1 u2

π−11

u1

u1

L1,p(c1) L1,e(c1)

L2,p(u2)

L1,p(u1)

L1,a(c1) L2,e(u2)

π−12

L2,a(c2)

u3c2

L2,e(c2)

L3,e(u3)

π2

π2

L3,p(u3)

L2,p(c2)L2,a(u2) L3,a(u3)

Fig. 1. Schematic of the proposed IRCC-URC-GMLST scheme.

can be summarized as follows:

1) We derive the Discrete-input Continuous-output Mem-oryless Channel (DCMC) [20] capacity formula ofGMLST(STBC) schemes and use EXIT charts to de-sign an iteratively decoded near-capacity concatenatedIRCC-URC-GMLST scheme.

2) In order to achieve a near-capacity performance, wepropose a novel outer IRCC using 36 component codesto accurately match the inner decoder’s EXIT curve.

3) The computational complexity of this concatenatedscheme is substantially reduced at the cost of a modestreduction in the maximum achievable rate comparedto that of ML detection, owing to the employment ofthe low-complexity but suboptimum SIC in the GMLSTdecoder.

The rest of this paper is organised as follows. In Section II,a brief description of the serially concatenated and iterativelydecoded scheme is presented. Section III specifies the encod-ing and decoding processes designed for the GMLST system.In Section IV, we derive the DCMC capacity formula andthe maximum achievable rate of different GMLST schemes.The EXIT chart aided iterative system design is detailed inSection V, while our simulation results and discussions areprovided in Section VI. Finally, we conclude in Section VII.

II. SYSTEM OVERVIEW

The schematic of the proposed serially concatenated systemis illustrated in Fig. 1. At the transmitter side, IRCCs [14], [15]are employed for encoding specifically optimized fractions ofthe input stream, where each fraction’s code rate was designedfor achieving a near-capacity performance with the aid ofEXIT charts [13]. Again, a recursive URC was amalgamatedwith the above-mentioned GMLST in Fig. 1 as the inner codefor assisting the non-recursive GMLST scheme in achievingdecoding convergence to an infinitesimally low BER at near-capacity SNRs. The GMLST encoder partitions the long bitstream emanating from the intermediate URC encoder intoseveral substreams and each substream is separately space-time encoded, as will be detailed in Section III. Two differenthigh-length bit interleavers are introduced between the threecomponent encoders so that the input bits of the URC andGMLST encoders can be rendered independent of each other,

which is one of the required conditions for EXIT chartsanalysis [13].

At the receiver side, according to Fig. 1, an iterative decod-ing procedure is operated, which employs three A PosterioriProbability (APP)-based decoders. The received signals arefirst decoded by the APP-based GMLST decoder in orderto produce the a priori Log-Likelihood Ratio (LLR) valuesL2,a(c2) of the coded bits c2. The URC decoder processes theinformation forwarded by the GMLST decoder in conjunctionwith the a priori LLR values L2,a(u2) of the information bitsu2 in order to generate the a posteriori LLR values L2,p(u2)and L2,p(c2) of the information bits u2 and the coded bits c2,respectively. In the scenario when iterations are needed withinthe amalgamated “URC-GMLST” decoder so as to achievea near-capacity performance, the a priori LLRs L2,a(c2) aresubtracted from the a posteriori LLR values L2,p(c2) and thenthey are fed back to the GMLST decoder as the a prioriinformation L3,a(u3) through the interleaver π2. Similarly,the a priori LLR values of the URC decoder are subtractedfrom the a posteriori LLR values produced by the MaximumAposteriori Probability (MAP) algorithm [21], for the sakeof generating the extrinsic LLR values L2,e(u2). Next, thesoft bits L1,a(c1) are passed to the IRCC decoder in orderto compute the a posteriori LLR values L1,p(c1) of the IRCCencoded bits c1. During the last iteration, only the LLR valuesL1,p(u1) of the original information bits u1 are required, whichare passed to the hard-decision block in order to estimate thesource bits. As seen in Fig. 1, the extrinsic information L1,e(c1)is generated by subtracting the a priori information from the aposteriori information, which is fed back to the URC decoderas the a priori information L2,a(u2) through the interleaver π1.For the sake of clarity, in the iterative detection procedure, wedefine the initial decoding process passing information fromthe inner decoder to the outer decoder as the first iterationbetween the two decoders.

III. GENERALIZED MULTI-LAYER SPACE-TIME CODE

The transmitter and receiver schematics of the GMLSTscheme are shown in Fig. 2 and Fig. 3, respectively. TheGMLST transmitter shown in Fig. 2 divides the antennas intoseveral groups and each group utilizes an STC encoder. Atthe receiver side of Fig. 3, we avoid the potentially excessivecomplexity of jointly detecting all groups. This complexity-



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S/P

C1

Cq

N1t

N qt

Nt

N2t

C2

Ψq

Ψ2

Ψ1STC1

STC2

STCq

B1

B2

Bq

B

Fig. 2. GMLST transmitter schematic using the vector-based temporalinterleaver Ψ j .

reduction may be achieved in the spirit of the V-BLASTdetection algorithm [5], [22] using the reduced-complexitySIC based detection scheme of [11], where the space-timecode of each individual group was processed successively,as will be detailed later in Section III-B. Therefore, it isclearly seen that the GMLST scheme can be viewed as abeneficial amalgam of a V-BLAST and a STC scheme. Withthe advent of the STC employed, we will show that thisGMLST architecture is capable of achieving a higher spatialdiversity compared to the conventional V-BLAST scheme. Asan added benefit, the overall system’s throughput becomessignificantly higher than that of the STC scheme, owing toits BLAST-like layered architecture.

A. Encoding

Consider now a point-to-point wireless communication linkequipped with Nt transmit and Nr receive antennas. Whencomplex-valued M -ary PSK/QAM is employed, the receivedsignal vector of the MIMO system can be written as:

y = Hc+n , (1)

where y= [y1, . . . ,yNr ]T is an Nr-element vector of the received

signals, H is an (Nr ×Nt )-element channel matrix, the entriesof which are independent and identically complex Gaussiandistributed with a zero mean and a variance of 0.5 perdimension, c = [c1, . . . ,cNt ]

T is an Nt -element vector of thetransmitted signals and n = [n1, . . . ,nNr ]

T is an Nr-elementnoise vector. Each element of n is an Additive White GaussianNoise (AWGN) process having a zero mean and a variance ofN0/2 per dimension.

The GMLST encoding process is illustrated in Fig. 2. Weassume that a block of B information bits is input to a serial-to-parallel (S/P) converter, which partitions this bit streaminto q groups, which we refer to as layers having lengthsof B1,B2, . . . ,Bq, where we have B1 +B2+, . . . ,+Bq = B bitsin total. Then, each group of Bj bits, for 1 ≤ j ≤ q, isseparately encoded in Fig. 2 by a component encoder STCj

associated with N jt number of transmit antennas, where we

have N1t +N2

t +, . . . ,+Nqt = Nt . The resultant (N j

t ×K)-elementcodeword matrix C j of STCj will be transmitted by the N j

ttransmit antennas of Fig. 2 during K symbol intervals. Werefer to the kth column c j,k of C j as the symbol vectorgenerated by group j at time instant k. Following space-time encoding, the symbol vectors of each group are passedthrough an independent vector-based temporal interleaver Ψ j.The vector-based temporal interleavers Ψ j represented by the

dashed block of Fig. 2 are used for the codewords generatedby the different groups, for the sake of eliminating the effectsof bursty error propagation among different groups duringthe decoding iterations [11]. Note that when the STBC isemployed as the component space-time code, the vector-basedinterleavers are not needed at all, because they will destroy thestructure of the STBC codeword matrix and would be of nouse in eliminating the effects of bursty error propagation, sincethe different STBC codeword matrices are independent in aspecific group. We can rewrite Eq. (1) at time instant k as:

yk = H1,kc1,k +H2,kc2,k + ⋅ ⋅ ⋅+Hq,kcq,k +nk, (2)

where H j,k denotes the (Nr ×N jt )-element subchannel matrix

of group j at time instant k.

B. Iterative Multistage SIC Detection

Again, for the sake of maintaining an affordable computa-tional complexity, a reduced-complexity SIC based detectionscheme was proposed in [11] instead of ML detection. Forthe sake of low complexity, the signals are detected layer-by-layer in a multistage manner, instead of high-complexityML-style joint-detection. In the same spirit of the V-BLASTscheme [23], the decoding order of the SIC-based schemehas a significant effect on the performance of the GMLSTsystem. Similarly to the classic decoding scheme using theoptimum decoding method developed in [5], in our per-group-based optimally ordered decoding scheme, the higher the post-detection SNR of a specific layer, the earlier the layer is chosento be detected. Without loss of generality, the decoding orderin Fig. 3 is assumed to be {1′,2′, . . . ,q′} and we present theSIC-based decoding algorithm as follows.

At the first decoding stage ( j = 1), let y1k = yk, for all 1 ≤

k ≤ K.At the jth decoding stage, given that the previous ( j− 1)

groups 1′,2′, . . . ,( j−1)′ have already been decoded and theireffects have been cancelled out from the received signals,the resultant received signal y j

k for k = 1, . . . ,K, which stillcontains interference imposed by the not-yet-decoded groups( j + 1)′,( j + 2)′, . . . ,q′, can now be written as

y jk = H j′,kc j′,k +H( j+1)′,kc( j+1)′,k + ⋅ ⋅ ⋅+Hq′,kcq′,k +n j

k. (3)

Then at time instance k, we can find a set of orthonormalcolumn vectors (not necessarily unique) in the null space of[H( j+1)′,k, . . . ,Hq′,k]

T , and amalgamate their transposes into an

[(Nr −Nt +N1′t + ⋅ ⋅ ⋅+N j′

t )×Nr]-element nulling matrix W jk

1

based on zero-forcing (ZF). Then, let both sides of Eq. (3)be left-multiplied by W j

k, which suppresses all the interferingsignals imposed by the groups ( j+1)′ to q′ and generates theinterference-free equivalent ‘sliced’ signal of group j′

y jk = W j

kyjk = W j

kH j′,kc j′,k + 0 +W jkn

jk = H j′,kc j′,k + n j

k. (4)

We can see that H j′,k is the equivalent channel matrix having

(Nr−Nt +N1′t + ⋅ ⋅ ⋅+N j′

t )×N j′t elements at time instance k for

group j′, which has the same statistical properties as H j′,k dueto the orthonormalizing effect of W j

k, i.e., the entries of H j′,kare independent and identically complex Gaussian distributed

1The computation of the nulling matrix W jk is described in the Appendix.



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adder

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GMLST Decoder

. . .

kth iteration2nd iteration

ST Dec 1′

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ST Dec 1′

ST Dec 2′

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ST Dec 1′

ST Dec q′

1st iteration

B1′H1′

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La(B2′)

La(Bq′)

C2′

ST Dec 2′

Nr

B2′

(a) main block diagram

Ψj′Ψ−1j′

Bj′

Cj′

La(Bj′)STTC−1

j′yj′ Demapper

(b) sub-block diagram for the “ST Dec j′” component for STTC,where STTC−1

j′ represents soft-input soft-output APP-based STTCdecoder and La(Bj′ ) correspond to the a priori LLR values of theinformation bits Bj′

STBC−1j′

Cj′

Bj′

STBCj′yj′

La(Bj′)

(c) sub-block diagram for the “ST Dec j′” component for STBC,where STBC−1

j′ represents linear STBC decoder and STBC j′ is thecorresponding STBC recoder. The a priori LLR values of the infor-mation bits Bj′ are not utilized by the corresponding STBC decoder

Fig. 3. GMLST receiver schematic employing iterative multistage SIC detection, where “IN” and “IC” denote the group interference nulling module andthe group interference cancellation module respectively.

with a zero mean and a variance of 0.5 per dimension asmentioned in Section III-A. Assuming that perfect interfer-ence cancellation is achieved, the resultant noise vector n j

kalso contains independent and identically complex Gaussiandistributed entries with mean zero and variance N0/2 perdimension [10].

Based on Eq. (4), the j′-layer codeword which is denotedby C j′ can be decoded using the corresponding space-time de-coder of Fig. 3. Prior to moving on to the next decoding stage,interference cancellation is carried out according to Fig. 3 bysubtracting the contribution of the just-decoded group j′ fromy j

k, which results in the ‘partially decontaminated’ receivedsignal y j+1

k

y j+1k = y j

k −H j′,kc j′,k, 1 ≤ k ≤ K, (5)

where c j′,k is the kth column of C j′ . Then the above procedureof Fig. 3 is repeated for j = j+1, until all groups are decoded( j = q).

In order to reduce the detrimental effects of error propa-gation, soft interference cancellation can be invoked, whichmeans that instead of using the hard decisions c j′,k in Eq. (5),the expectation value

cij′,k =

M

∑m=1

xm ⋅ p(cij′,k = xm∣y j

k), 1 ≤ i ≤ N j′t (6)

is used, where cij′,k is the i-th element of vector c j′,k and

p(cij′,k = xm∣y j

k) is the probability that the M -ary constellationpoint xm was transmitted, given the equivalent received signalvector y j

k. Since soft-values close to the legitimate hard-decision-based constellation points are subtracted in Eq. (5) incase of reliable decisions, whereas small values are subtractedin case of unreliable decisions, the effects of potential errorpropagation are significantly reduced.

However, it is clearly seen that this SIC-based decodingscheme fails to achieve the maximum attainable receive diver-sity, because layer j′ decoded at the jth decoding stage hasan antenna diversity order of (Nr −Nt +N1′

t + ⋅ ⋅ ⋅+N j′t )×N j′

t .Clearly, the earlier a layer is detected, the lower its diversityorder is. In order to maximize the attainable receive diversitygain, the SIC-based iterative decoding scheme of [11] wasinvoked, which is depicted in Fig. 3. We can see that thefirst iteration is the same as described above, except thattemporal interleaving and deinterleaving should be carried outaccordingly, which substantially reduces the effects of bursterror propagation among groups. In the subsequent iterations,since all groups have already been decoded, the interferencenulling (IN) stage of Eq. (4) and Fig. 3 is no longer needed,which results in a theoretical receive diversity order of Nr

for each group. Each iteration consists of q layers and eachSTC-protected layer is processed successively, in the sameorder as during the first iteration. After a number of iterations,the maximum attainable receive diversity order of Nr may beachieved for all layers, when the residual error propagation



..

....Demapper

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ML

GMLST Decoder

P/S

Ψ1

Ψ−12

Ψ2

Ψq

Ψ−1q

Ψ−11

NrSTTC−1

2

STTC−1q

B1

B2

Bq

STTC−11 La(B1)

La(B2)

La(Bq)

Fig. 4. Iterative ML detection of the GMLST(STTC) scheme, where Ψ j isthe corresponding temporal interleaver, STTC−1

j represents j-layer soft-inputsoft-output APP-based space-time decoder and La(Bj) correspond to the apriori LLR values of the information bits Bj .

among different groups becomes negligible. Compared toML detection, this iterative decoding scheme is capable ofapproaching the same receive diversity order, despite imposingonly a fraction of the computational complexity of ML-stylejoint detection.

C. Iterative ML Detection for GMLST(STTC)

For comparison, we also present a significantly more com-plex iterative ML detection scheme in Fig. 4 designed forGMLST schemes using STTC [8] as the component STCs,which we refer to as the GMLST(STTC) arrangement. Theiterative ML detection procedure is shown in Fig. 4. Firstly,the Nt transmitted M -ary symbols are jointly detected asa combined [Nt × log2M ]-bit symbol by an ML demapper.Then, the probability vector of each [Nt × log2M ]-bit symbolis converted to q number of probability vectors correspondingto the q number of [N j

t × log2M ]-bit symbols of the differentlayers. The resultant probability vectors are then fed to thecomponent STTC decoders. An iterative detection gain canbe attained, since the EXIT curve of the ML demapper isa sloping line, although this is not explicitly shown hereowing to space limitations. However, it is not guaranteed thatthe DCMC capacity, as detailed later in Section IV, can beachieved with the aid of iterative ML detection, because themaximum achievable rate of the iterative ML detection schemeis dependent on the EXIT curve shapes of the various space-time trellis codes.

IV. CAPACITY AND MAXIMUM ACHIEVABLE RATE

In the context of discrete-amplitude QAM [20] andPSK [20] modulation, a Discrete-input Continuous-outputMemoryless Channel (DCMC) is encountered. In order todesign a near-capacity coding scheme, in this section we de-rive the DCMC capacity formula for GMLST(STBC), whichrefers to the GMLST schemes using STBC [6], [7] as thecomponent STCs. Additionally, the bandwidth efficiency ofvarious SIC and ML based GMLST(STTC) schemes is derivedfor transmission over the DCMC based on the properties ofEXIT charts [24].

A. DCMC Capacity of GMLST(STBC)

A specific GMLST scheme using an identical STBC to thecomponent STBCs is designed for encoding over L transmis-sion symbols, assuming that the channel’s envelope may be

considered quasi-static over this period. Based on Eq. (1), thesignal received during L symbol periods can be written as:

Y = HC+N , (7)

where Y = [y1, . . . ,yL]∈ℂNr×L is the sampled received signalmatrix, H ∈ ℂ

Nr×Nt is the quasi-static channel matrix, whichis constant over L symbol periods, C = [c1, . . . ,cL] ∈ ℂNt×L

is the GMLST(STBC) transmitted signal matrix and N =[n1, . . . ,nL] ∈ ℂ

Nr×L represents the additive white Gaussiannoise matrix.

In this contribution, we only consider STBCs having square-shaped codeword matrices [6], [25]–[27] as the componentSTBCs of the GMLST schemes, where we have L = Nt/q.For example, when Alamouti’s G2 space-time scheme [6] isused as the component STBC, the GMLST(STBC) codewordmatrix is as follows:

C =

(c1,1 c1,2 . . . c j,1 c j,2 . . . cq,1 cq,2

−c1,2 c1,1 . . . −c j,2 c j,1 . . . −cq,2 cq,1

)T

,

(8)

where the columns represent L = 2 different time slots,while the rows represent qL different transmit antennas, and(

c j,1 c j,2

−c j,2 c j,1

)T

is the j-layer STBC codeword matrix. Hence,

when complex-valued M -ary PSK/QAM is employed in aGMLST(STBC) scheme, we have a total of M =M qL numberof possible GMLST(STBC) codeword matrix combinationsfor L consecutive symbol periods. Based on Eq. (7), theconditional probability of receiving a signal matrix Y, giventhat an M-ary GMLST(STBC) codeword matrix Cm, m ∈{1, . . . ,M}, was transmitted over uncorrelated flat Rayleighfading channels is determined by the Probability DensityFunction (PDF) of the noise, yielding:

p(Y∣Cm) =1

(πN0)LNr

exp

(−∥Y−HCm∥2

N0

), (9)

=L

∏k=1

Nr

∏r=1

1πN0

exp

(−∣yr,k −∑Ntt=1 hr,tcm

t,k∣2N0

).

The channel capacity per symbol period evaluated for theGMLST(STBC) scheme when using complex-valued M -aryPSK/QAM for transmission over the DCMC can be shown tobe:

CGMLST(STBC)DCMC =

1L

maxp(C1)...p(CM)

M

∑m=1

∫Y

p(Y∣Cm)p(Cm)

⋅ log2

(p(Y∣Cm)

∑Mn=1 p(Y∣Cn)p(Cn)

)dY, (10)

where the right hand side of Eq. (10) is maximized, when wehave p(Cm) = 1/M for m ∈ {1, . . . ,M}. Hence, Eq. (10) canbe simplified to:

CGMLST(STBC)DCMC =

log2(M)

L−

1ML

M

∑m=1

E

[log2

M

∑n=1

exp(Ψm,n)

∣∣∣∣ Cm

], (11)

where E[A∣Cm] is the expectation of A conditioned on Cm andthe expectation in Eq. (11) is taken over H and N, while Ψm,n



mimo-glst-4x4-4qam.gle

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CCMC4x4DCMC4x4-4QAMDCMC4x4-4QAM-GMLST(STBC-G2)GMLST(STBC-G2)-4QAM, 2 Soft SICGMLST(STBC-G2)-4QAM, 2 Hard SICGMLST(STBC-G2)-4QAM, 1 Hard SICGMLST(STTC-16)-4QAM, ML w/ 3 IterGMLST(STTC-16)-4QAM, ML w/o IterGMLST(STTC-16)-4QAM, 3 Soft SICGMLST(STTC-16)-4QAM, 3 Hard SICGMLST(STTC-16)-4QAM, 1 Hard SIC

Nt=4, Nr=4, =2 bit/s/HzEb/N0=-6.20dB:CCMC4x4Eb/N0=-6.10dB:DCMC4x4-4QAMEb/N0=-5.40dB:DCMC4x4-4QAM-GMLST(STBC-G2)Eb/N0=-5.00dB:GMLST(STBC-G2)-4QAM, 2 Soft SICEb/N0=-4.70dB:GMLST(STBC-G2)-4QAM, 2 Hard SICEb/N0=-3.40dB:GMLST(STBC-G2)-4QAM, 1 Hard SICEb/N0=-6.05dB:GMLST(STTC-16)-4QAM, ML w/ 3 IterEb/N0=-5.70dB:GMLST(STTC-16)-4QAM, ML w/o IterEb/N0=-5.90dB:GMLST(STTC-16)-4QAM, 3 Soft SICEb/N0=-5.70dB:GMLST(STTC-16)-4QAM, 3 Hard SICEb/N0=-3.80dB:GMLST(STTC-16)-4QAM, 1 Hard SIC

Fig. 5. The capacity and maximum achievable rate of various SIC and ML based GMLST schemes, when communicating over uncorrelated flat Rayleighfading channels employing Nt = 4 transmit antennas and Nr = 4 receive antennas.

is given by:

Ψm,n =−∥H(Cm−Cn)+N∥2 + ∥N∥2

N0. (12)

Based on the DCMC capacity formula of Eq. (11) we cancompute the capacity of any GMLST(STBC) scheme by sub-stituting the corresponding GMLST(STBC) codeword matrixC ∈ ℂ

Nt×L into Eq. (12).The resultant bandwidth efficiency is computed by normal-

ising the channel capacity given by Eq. (11), with respect tothe product of the bandwidth W and the signalling period T :

η =C

WT[bit/s/Hz] , (13)

where WT = 1 for PSK/QAM schemes, when assuming zeroNyquist excess bandwidth. The bandwidth efficiency, η, istypically plotted against the SNR per bit given by: Eb/N0 =SNR/η. For simplicity, we will refer to η as the capacity.

B. Maximum Achievable Rate Based on EXIT Charts

The concept of EXIT charts was first proposed in [12]. Itwas then stated in [15], [24] that the maximum achievablebandwidth efficiency/rate of the system is equal to the areaunder the EXIT curve of the inner code, provided that thechannel’s input is independently and uniformly distributed,assuming furthermore that the inner code rate is 1 and thatthe MAP decoding algorithm is used. This area propertyof the EXIT charts may be exploited by considering Fig.2 of [24]. Explicitly, the area under the EXIT curve ofthe inner code quantifies the capacity of the communicationchannel (the upper a priori Channel 1 in Fig. 2 of [24]),when the communication channel’s input is independently anduniformly distributed, while the a priori channel (the lowera priori Channel 2 in Fig. 2 of [24]) is modeled by a BinaryErasure Channel (BEC). This area property was shown to bevalid for arbitrary inner codes and communication channels,

provided that the a priori Channel 2 is a BEC. Furthermore,there is experimental evidence that the area property of EXITcharts is also valid when the a priori Channel 2 is modeledby an AWGN channel, as it was originally done in [13].

Based on the area property of EXIT charts, the maximumachievable rate curves of various SIC and ML based GMLSTschemes are shown in Fig. 5 together with the DCMCcapacity curves of the multiplexing-based MIMO scheme,which employed a ML detector. The capacity curve of theunrestricted Continuous-input Continuous-output MemorylessChannel (CCMC) [4], [20] is also depicted in Fig. 5 forcomparison. In this contribution, we consider the scenariowhen Nt = 4 transmit and Nr = 4 receive antennas are used,where N1

t = N2t = 2. For simplicity, the component STCs

utilized for all groups are assumed to be identical. Specifically,we use STBC-G2 [6] and 16-state based STTC-16 [8, Fig. 5]as the component STCs of the GMLST schemes, respectively.As shown in Fig. 5, the DCMC capacity of the multiplexing-based MIMO scheme employing 4QAM (DCMC4×4-4QAM)is higher than that of all the GMLST schemes and may beregarded as the tight capacity upper bound of the (4× 4)-element 4QAM MIMO systems. On the other hand, theDCMC capacity curve of the GMLST scheme using STBC-G2 is plotted against Eb/N0 according to Eq. (11), which isindicated by the label DCMC4×4-4QAM-GMLST(STBC-G2)in Fig. 5. Note that the DCMC4×4-4QAM-GMLST(STBC-G2)capacity curve is below that of the ML-based GMLST(STTC-16) scheme using three iterations. This is due to the loss oftemporal diversity within the STBC-G2 orthogonal code ofeach group [4]. It is also seen in Fig. 5 that when invokingtwo SIC operations in the GMLST(STBC-G2) scheme, themaximum achievable rate curve approaches the DCMC4×4-4QAM-GMLST(STBC-G2) rate curve, since the maximumattainable receive diversity has been approached. However,no substantial further improvements are attained after twoSICs, since the remaining inter-group interference propagating



across the groups is independently distributed across K/2consecutive G2 codeword block periods during K symbolintervals and hence cannot be eliminated by SIC operations.By contrast, for the GMLST(STTC-16) scheme of Fig. 5,the maximum achievable rate improves steadily for three SICoperations, although the maximum attainable receive diversityhas already been achieved during the second iteration. Thisis due to the strong error-correcting capability of the STTCassociated with temporal vector-based interleavers. Observein Fig. 5 that beyond three SICs, the additional improvementsremain marginal. For further improvements, soft interferencecancellation can be invoked for both the GMLST(STBC) andGMLST(STTC) schemes, which is explicitly demonstrated inFig. 5 as well.

Since no iterations can be carried out between the SIC-baseddemodulator and the STTC decoder when using SIC-baseddetection, some mutual information or throughput loss occurs,which cannot be recovered. For comparison, the maximumachievable rate of the ML based iterative detection scheme ofGMLST(STTC-16) is also quantified in Fig. 5. The numberof iterations between the ML demapper and the GMLSTcomponent STTC decoders was fixed to three, which is thesame as the number of SIC iterations. Hence, both schemesinvoke the same number of STTC decoder operations, butthe ML demapper exhibits a higher complexity than the SICoperation. As shown in Fig. 5, the ML-based scheme providesa higher maximum achievable rate than that of the SIC-basedarrangement.

V. EXIT CHART AIDED SYSTEM DESIGN AND ANALYSIS

The main objective of employing EXIT charts [13] is toanalyse the convergence behaviour of iterative decoders byexamining the evolution of the input/output mutual informa-tion exchange between the inner and outer decoders during theconsecutive iterations. As mentioned in Section IV, the areaunder the EXIT curve of the inner decoder is approximatelyequal to the channel capacity, when the channel’s input isindependently and uniformly distributed. Similarly, the areaunder the EXIT curve of the outer code is approximately equalto (1-R), where R is the outer code rate. Furthermore, ourexperimental results show that an intermediate URC changesonly the shape, but not the area under the EXIT curve ofthe inner code. We assume that the normalised area underthe EXIT curve of the inner decoder is represented by AE

throughout this paper. A narrow, but marginally open EXIT-tunnel in an EXIT chart indicates the possibility of achievinga near-capacity performance. Therefore, we invoke IRCCsfor the sake of appropriately shaping the EXIT curves byminimizing the area in the EXIT-tunnel with the aid of theoptimization algorithm of [14].

The original IRCC constituted by a set of P = 17 subcodeswas constructed in [15] from a systematic, rate-1/2, memory-4mother code defined by the generator polynomial (1,g1/g0),where g0 = 1 + D + D4 is the feedback polynomial andg1 = 1 + D2 + D3 + D4 is the feedforward one. Higher coderates may be obtained by puncturing, while lower rates arecreated by adding more generators and by puncturing underthe constraint of maximizing the achievable free distance.In the proposed system the two additional generators are

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Decoding trajectory for GMLST(STBC-G2)Hard-2-URCEb/N0 = -4.2 dB

Nt=4, Nr=4, =2 bit/s/HzGMLST(STBC-G2)Hard-2-URC: Eb/N0 = -4.2 dB (AE=0.53)GMLST(STBC-G2)Hard-2: Eb/N0 = -4.7 dB (AE=0.50). IRCC : R=0.5 (AE=0.50)

Fig. 6. The EXIT chart curves for the GMLST(STBC-G2)-URC, 17-component IRCC having weighting coefficients [α1, ⋅ ⋅ ⋅ ,α17] = [0, 0,0.0131465, 0.01553, 0, 0, 0.48839, 0.215005, 0, 0, 0, 0, 0.198844, 0, 0,0, 0.0691192] and 17 original IRCC subcodes, when communicating overuncorrelated flat Rayleigh fading channels using Nt = 4 and Nr = 4. Thenotation GMLST(STBC-G2)Hard-2 indicates 2 hard SIC iterations in GMLSTdecoder and no iteration is needed between the GMLST(STBC-G2) and URCdecoders, since the EXIT curve of the GMLST(STBC-G2) detector is ahorizontal line.


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Decoding trajectory for GMLST(STBC-G2)Soft-2-URCEb/N0 = -4.5 dB

Nt=4, Nr=4, =2 bit/s/HzGMLST(STBC-G2)Soft-2-URC: Eb/N0 = -4.5 dB (AE=0.53)GMLST(STBC-G2)Soft-2: Eb/N0 = -5.0 dB (AE=0.50). IRCC : R=0.5 (AE=0.50)

Fig. 7. The EXIT chart curves for the GMLST(STBC-G2)-URC, 17-component IRCC having weighting coefficients [α1, ⋅ ⋅ ⋅ ,α17] = [0, 0,0.0131465, 0.01553, 0, 0, 0.48839, 0.215005, 0, 0, 0, 0, 0.198844, 0, 0,0, 0.0691192] and 17 original IRCC subcodes, when communicating overuncorrelated flat Rayleigh fading channels using Nt = 4 and Nr = 4. Thenotation GMLST(STBC-G2)Soft-2 indicates 2 soft SIC iterations in GMLSTdecoder and no iteration is needed between the GMLST(STBC-G2) and URCdecoders, since the EXIT curve of the GMLST(STBC-G2) detector is ahorizontal line.



g2 = 1+D+D2+D4 and g3 = 1+D+D3+D4. The resultantP = 17 subcodes have coding rates spanning 0.1,0.15,0.2, . . . ,to 0.9.

The EXIT functions of these 17 original subcodes are shownin Figs. 9-11, indicated by the dotted lines. Observe in Figs. 9-11 that the original memory-4 IRCC exhibits a near-horizontalportion in the EXIT chart, which is typical of strong CCshaving a memory of 4 associated with 16 trellis states. Asdiscussed in [14], [15], the EXIT function of an IRCC canbe obtained by superimposing those of its subcodes. Morespecifically, the EXIT function of the target IRCC is theweighted superposition of the EXIT functions of its subcodes.Hence, a careful selection of the weighting coefficients2 couldproduce an outer code EXIT curve that closely matches theEXIT curve shape of the inner code. When the area betweenthe two EXIT curves is minimized, decoding convergence toan infinitesimally low BER would be achieved at the lowestpossible SNR. On the other hand, in order to match the shapeof the inner codes’ EXIT curves more accurately, the shapeof the outer codes’ EXIT functions can be adjusted in a way,which allows us to match a more diverse-shaped set of innercodes’ EXIT functions. Hence we introduce a more diverserange of EXIT functions, particularly near the diagonal of theEXIT chart. This can be achieved by invoking weaker codeshaving a lower memory.

Accordingly, memory-1 CCs are incorporated into the orig-inal IRCC scheme, which have a simple two-state trellisdiagram. The generator polynomial of this rate-1/2 memory-1mother code is defined by (1,g1/g0), where g0 =D and g1 = 1.For a lower code rate, an extra output generator polynomial,namely g2 is used, where g2 = g1. For a higher code rate,the puncturing pattern of the original memory-4 IRCC isemployed [15]. This way we generate 10 additional EXITfunctions as shown by the dashed lines in Figs. 9-11, spanningthe range of [0.45, 0.9] with a stepsize of 0.05. Furthermore, arepetition code is a simple memoryless code, which consists ofonly two codewords, namely the all-zero and the all-one word.Since it has no memory, the EXIT functions of such repetitioncodes are diagonally-shaped. Hence, we also incorporate 9different-rate repetition codes in the novel IRCC scheme, asindicated by the solid lines in Figs. 9-11 and spanning thecode-rate range of [0.1, 0.5] with the rate-stepsize of 0.05.

Similarly to [15], each of these P = 36 subcodes encodes aspecific fraction of the information bit stream according to aspecific weighting coefficient αi, where i = 1,2, . . . ,36. Morespecifically, assume that there are N number of encoded bits,where each subcode i encodes a fraction of αiriN informationbits and generates αiN encoded bits using a coding rate ofri. As for the P number of subcodes, given the target overallaverage code rate of R ∈ [0,1], the weighting coefficients αi

must satisfy:

1 =P

∑i=1

αi, R =P

∑i=1

αiri, and αi ∈ [0,1], ∀i. (14)

In this paper, we consider an average coding rate of R = 0.5for the outer IRCC code. Hence the effective throughput is

2The optimum selection was found by computer search using the iterativeprocedure of the optimization algorithm [14].


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Decoding trajectory for GMLST(STBC-G2)ML-URC2Eb/N0 = -5.0 dB

Nt=4, Nr=4, =2 bit/s/HzGMLST(STBC-G2)ML-URC2: Eb/N0 = -5.0 dB (AE=0.52)GMLST(STBC-G2)ML: Eb/N0 = -5.4 dB (AE=0.50). IRCC : R=0.5 (AE=0.50)

Fig. 8. The EXIT chart curves for the GMLST(STBC-G2)-URC, 17-component IRCC having weighting coefficients [α1, ⋅ ⋅ ⋅ ,α17] = [0, 0, 0, 0,0, 0.327442, 0.186505, 0.113412, 0, 0.0885527, 0, 0.0781214, 0.0962527,0.0114205, 0.0346015, 0.0136955, 0.0500168] and 17 original IRCC sub-codes, when communicating over uncorrelated flat Rayleigh fading channelsusing Nt = 4 and Nr = 4. The notation GMLST(STBC-G2)ML indicates theML decoder for the GMLST(STBC-G2) scheme and the subscript of URCdenotes the number of iterations between the GMLST(STBC-G2)ML and URCdecoders.

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Decoding trajectory for GMLST(STTC-16)Hard-3-URC6Eb/N0 = -5.0 dB

Nt=4, Nr=4, =2 bit/s/HzGMLST(STTC-16)Hard-3-URC6: Eb/N0 = -5.00 dB (AE=0.56)GMLST(STTC-16)Hard-3: Eb/N0 = -5.70 dB (AE=0.50). IRCC : R=0.5 (AE=0.50)

Fig. 9. The EXIT chart curves for the GMLST(STTC-16)Hard-URC and36-component IRCC having weighting coefficients [α1, ⋅ ⋅ ⋅ ,α36] = [0, 0, 0, 0,0, 0, 0, 0, 0, 0, 0.0238076, 0.0654278, 0.0108539, 0, 0.0736835, 0.0251597,0.124231, 0.0829128, 0, 0.140577, 0.0983615, 0.134327, 0.109642, 0, 0, 0,0, 0, 0, 0, 0, 0, 0, 0, 0.00875876, 0.102257] and 36 IRCC subcodes, whencommunicating over uncorrelated flat Rayleigh fading channels using Nt =4 and Nr = 4. The notation GMLST(STTC-16)Hard-3 indicates 3 hard SICiterations in GMLST decoder, and the subscript of URC denotes the numberof iterations between the GMLST(STTC-16)Hard and URC decoders.



2× R log2 4 = 2 bit/s/Hz, when 4QAM is employed by theabove-mentioned two-layer GMLST schemes. The maximumachievable rates of different GMLST schemes computed ac-cording to the properties of EXIT charts [15], [24] at athroughput of η = 2 bit/s/Hz are depicted in Fig. 5. Theexchange of extrinsic information in the schematic of Fig. 1is visualised by plotting the EXIT characteristics of theinner amalgamated “URC-GMLST” decoder and the specificoptimized outer IRCC decoder in Figs. 6-11.

Note that for GMLST(STBC-G2) scheme of Figs. 6 and 7, anear-capacity performance may be achieved without extrinsicinformation exchange using decoding iterations between theURC decoder and the GMLST(STBC-G2) decoder, since theEXIT curve of the SIC-based GMLST(STBC-G2) decoder is ahorizontal line. As seen in Figs. 6, 7 and 8, when we employthe 17 original outer subcodes of [15] in our IRCC design,the outer IRCC is capable of accurately matching the EXITcurve of the inner amalgamated “URC-GMLST” decoder withthe aid of the matching algorithm of [14]. On the otherhand, observe in Figs. 9, 10 and 11, for the GMLST(STTC-16) scheme that the EXIT curve of the GMLST(STTC-16)decoder is a slanted line, hence extrinsic information exchangeusing decoding iterations between the URC decoder and theGMLST(STTC-16) decoder is needed in order to achieve anear-capacity performance. When there is no iteration betweenthe URC decoder and the GMLST(STTC-16) decoder, theEXIT curve shape of the URC decoder depends on the IEvalue of the GMLST(STTC-16) decoder observed in Figs. 9,10 and 11 at IA = 0. Hence, the URC-GMLST(STTC-16)scheme requires a higher Eb/N0 value in order to maintain anormalised area of AE = 0.5, as shown in Figs. 9, 10 and 11.In other words, a certain mutual information i.e. throughputloss will occur, if there is no iteration between the URCdecoder and the GMLST(STTC-16) decoder. Furthermore, dueto the “S”-shape of the inner amalgamated “URC-GMLST”decoder’s EXIT curve, the original IRCC using 17 subcodescannot match it accurately [28, Figs. 7 and 8]. As a benefit,the proposed 36-component IRCC has a more diverse range ofEXIT function shapes, hence it is capable of providing moreaccurately matching EXIT curves to fit the inner amalgamated“URC-GMLST” decoders, as depicted in Figs. 9, 10 and 11.

VI. SIMULATION RESULTS AND DISCUSSIONS

As we can see from Figs. 9 and 10, the Monte-Carlosimulation based decoding trajectory of the IRCC-URC-GMLST(STTC-16) schemes using SIC detection within theGMLST(STTC-16) decoder has a slight mismatch with theirEXIT curves. This is due to the trellis structure of STTCs,which results in correlated non-zero error propagation be-tween the different layers in the process of SIC operation.By contrast, the decoding trajectory of the IRCC-URC-GMLST(STBC-G2) scheme of Fig. 6 and 7 using hard andsoft SIC accurately matches both the inner and the outer EXITcurves owing to the uncorrelated non-zero error propagation.For comparison, Fig. 8 presents the decoding trajectory ofthe IRCC-URC-GMLST(STBC-G2) scheme using an ML de-coder. On the other hand, the decoding trajectory of the IRCC-URC-GMLST(STTC-16) scheme employing ML detection ispresented in Fig. 11. Again, we use three iterations between


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Decoding trajectory for GMLST(STTC-16)Soft-3-URC6Eb/N0 = -5.1 dB

Nt=4, Nr=4, =2 bit/s/HzGMLST(STTC-16)Soft-3-URC6: Eb/N0 = -5.10 dB (AE=0.57)GMLST(STTC-16)Soft-3: Eb/N0 = -5.90 dB (AE=0.50). IRCC : R=0.5 (AE=0.50)

Fig. 10. The EXIT chart curves for the GMLST(STTC-16)Soft-URC and 36-component IRCC having weighting coefficients [α1, ⋅ ⋅ ⋅ ,α36] = [0, 0, 0, 0, 0,0, 0, 0, 0, 0, 0.0238076, 0.0654278, 0.0108539, 0, 0.0736835, 0.0251597,0.124231, 0.0829128, 0, 0.140577, 0.0983615, 0.134327, 0.109642, 0, 0,0, 0, 0, 0, 0, 0, 0, 0, 0, 0.00875876, 0.102257] and 36 IRCC subcodes,when communicating over uncorrelated flat Rayleigh fading channels usingNt = 4 and Nr = 4. The notation GMLST(STTC-16)Soft-3 indicates 3 soft SICiterations in GMLST decoder, and the subscript of URC denotes the numberof iterations between the GMLST(STTC-16)Soft and URC decoders.


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Decoding trajectory for GMLST(STTC-16)ML-3-URC6Eb/N0 = -5.3 dB

Nt=4, Nr=4, =2 bit/s/HzGMLST(STTC-16)ML-3-URC6: Eb/N0 = -5.30 dB (AE=0.53)GMLST(STTC-16)ML-3: Eb/N0 = -6.05 dB (AE=0.50). IRCC : R=0.5 (AE=0.50)

Fig. 11. The EXIT chart curves for the GMLST(STTC-16)ML-URC and 36-component IRCC having weighting coefficients [α1, ⋅ ⋅ ⋅ ,α36] = [0, 0, 0, 0, 0,0, 0, 0, 0, 0, 0.0238076, 0.0654278, 0.0108539, 0, 0.0736835, 0.0251597,0.124231, 0.0829128, 0, 0.140577, 0.0983615, 0.134327, 0.109642, 0, 0,0, 0, 0, 0, 0, 0, 0, 0, 0, 0.00875876, 0.102257] and 36 IRCC subcodes,when communicating over uncorrelated flat Rayleigh fading channels usingNt = 4 and Nr = 4. The notation GMLST(STTC-16)ML-3 indicates 3 iterationsbetween ML demapper and GMLST component decoders, and the subscriptof URC denotes the number of iterations between the GMLST(STTC-16)MLand URC decoders.



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2)D

CM

CC

apac

ity

GM

LST(S

TTC

)M

axA

chie

vabl

eR

ate

0.9dB

1.0dB1.2dB1.4dB

Fig. 12. The BER performance comparison of the IRCC-URC-GMLST schemes and of the stand-alone GMLST schemes with various SIC and ML basedGMLST decoders.

the ML demapper and GMLST component decoders so thatthe number of STTC decoder operations invoked remains thesame as that of the scheme employing three SIC iterations.Since there is no error propagation in the ML-based scheme,the decoding trajectory of the ML-based scheme closelymatches its EXIT curve. However, the error propagation wasavoided in the ML-based scheme at the price of a highercomplexity.

Fig. 12 presents the BER performance of both the near-capacity IRCC-URC-GMLST and of the stand-alone GMLSTschemes employing various space-time codes combined withSIC and ML detection. It is observed that for three SICoperations, the performance of the GMLST(STTC-16) schemesignificantly improves due to the attainment of the maxi-mum achievable receive diversity, whereas the soft-SIC basedscheme approaches the performance of the ML-detectedGMLST(STTC-16) scheme more closely since the residualerror propagation is further mitigated. On the other hand asseen in Fig. 12, the GMLST(STBC-G2) scheme using twoSIC operations exhibits a 1.6 dB gain over a single hardSIC, but no more improvements are attained beyond two SICoperations, since the residual interference propagating amonggroups is independently distributed across K/2 consecutiveG2 codeword block periods during K symbol intervals andcannot be eliminated by SIC operations. Observe in Fig. 12that the throughput of the stand-alone GMLST schemes isfar from the corresponding DCMC capacity. For the concate-nated systems, it is clearly shown in Fig. 12 that the IRCC-URC-GMLST(STBC-G2) scheme is capable of performingwithin 0.9 dB of the corresponding DCMC capacity of theGMLST(STBC-G2) scheme, when soft SIC is employed. Onthe other hand, observe in Figs. 9 and 10 that the IRCC-URC-GMLST(STTC-16) scheme exhibits a mismatch between thedecoding trajectory and the EXIT curve, but performs closer tothe DCMC4×4-4QAM capacity as shown in Fig. 12 at the priceof a significantly higher complexity than that of the IRCC-

URC-GMLST(STBC-G2) scheme. The BER performance ofthe most complex ML-detected IRCC-URC-GMLST(STTC-16) scheme is also depicted in Fig. 12, where we can seea 0.4 dB gain over the soft SIC based scheme. However,the complexity of the ML-based scheme is unaffordable,especially when the number of GMLST layers is high.

VII. CONCLUSIONS

In this contribution, we have proposed a low-complexitySIC-based iteratively decoded IRCC-URC-GMLST schemefor achieving a near-capacity performance with the aid ofEXIT chart analysis. According to the simulation results, wefound that the iterative IRCC-URC-GMLST scheme using SICdetection strikes an attractive trade-off between the complexityimposed and the effective throughput achieved. In conclusion,we summarize our design procedure as follows:

1) Derive the DCMC capacity formula of GMLST(STBC)schemes and quantify the maximum achievable rates ofthe GMLST schemes using 16-state STTCs with the aidof EXIT charts.

2) Generate the EXIT curve of the inner decoder, whichis constituted here by the GMLST decoder and URCdecoder.

3) Design an IRCC outer code to match the EXIT curveof the inner code and hence to approach the capacitydetermined for the high-complexity, but optimum MLdetector.

4) Reduce the complexity imposed by the ML detector byusing the lower-complexity SIC detector, while avoidingany substantial performance degradation by redesigningthe IRCC scheme with more component codes, whichhence matches the EXIT curve of the SIC inner decodermore accurately.

5) This design procedure is generically applicable, re-gardless of the specific choice of the inner and outer



decoder components, as exemplified by other detectors,such as Sphere Detector (SD), Markov chain MonteCarlo (MCMC) detector as well as by other outer codes,such as an irregular low-density parity-check (LDPC)code, etc.

APPENDIX

THE NULLING MATRIX

In Section III-B, for the sake of nulling the interferinggroups in Eq. (3), the following condition must be satisfied:

W jk ⋅ [H( j+1)′,k, . . . ,Hq′,k] = 0, (15)

which can be reformulated in the transpose notation as:

[H( j+1)′,k, . . . ,Hq′,k]T ⋅ [W j

k]T = 0. (16)

Since the null space of a matrix A is the set of all vectorsv, which solves the equation Av = 0, where v is also referredto as the kernel of A, in set-construction notation we have:

Null(A) = {v ∈ V : Av = 0}.

Hence, we can mathematically deduce the nulling matrixW j

k in Eq. (4) from the null space of [H( j+1)′,k, . . . ,Hq′,k]T .

For the sake of maximizing the receive diversity order, wechoose a nulling matrix W j

k with the largest rank, which isequal to the dimension of the null space. There are severalcomputational approaches for determining the null space of Aand the choice of the specific method used depends on boththe structure and size of A. A popular approach which can beused, even when A is large, is to compute the singular valuedecomposition (SVD) of A and use the resultant right singularvectors corresponding to singular values of zero as a basis forthe null space. Below we generate the nulling matrix W j

k indetail as follows.

As [H( j+1)′,k, . . . ,Hq′,k]T is an [(Nt −N1′

t ⋅ ⋅ ⋅ −N j′t )×Nr]-

element matrix, its SVD is given by:

[H( j+1)′,k, . . . ,Hq′,k]T = U

[Σr×r 0

0 0

]VH , (17)

where U∈ℂ(Nt−N1′t ⋅⋅⋅−N j′

t )×(Nt−N1′t ⋅⋅⋅−N j′

t ), V∈ℂNr×Nr and Σr×r

is a diagonal matrix having singular values on the diagonaland r = rank([H( j+1)′,k, . . . ,Hq′,k]

T ). Since the subchannels areuncorrelated and random, the matrix [H( j+1)′,k, . . . ,Hq′,k]

T is

of full rank and r = Nt −N1′t ⋅ ⋅ ⋅ −N j′

t < Nr. As a result, the

matrix

[Σr×r 0

0 0

]can be reformulated as

[Σr×r 0r×(Nr−r)

].

Upon substituting Eq. (17) into Eq. (16), we have

U[Σr×r 0r×(Nr−r)

]VH [W j

k]T = 0 (18)

if and only if

[Σr×r 0r×(Nr−r)

]VH [W j

k]T = 0. (19)

When choosing VH [W jk]

T =

[0r×(Nr−r)

I(Nr−r)×(Nr−r)

], the nulling ma-

trix W jk is obtained as

W jk =

{V[

0r×(Nr−r)

I(Nr−r)×(Nr−r)

]}T

=

{[VNr×rVNr×(Nr−r)

][ 0r×(Nr−r)

I(Nr−r)×(Nr−r)

]}T

(20)

= VTNr×(Nr−r).

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[28] L. Kong, S. X. Ng, and L. Hanzo, “Near-capacity three-stage downlinkiteratively decoded generalized layered space-time coding with lowcomplexity,” in Proc. GLOBECOM’08, New Orleans, LA, USA, pp. 1–6, 30 Nov. 2008.

Lingkun Kong received the B.Eng. degree in in-formation engineering and the M.Eng. degree incommunications and signal processing from South-east University, Nanjing, China, in 2004 and 2006,respectively. He is currently working towards thePh.D. degree with the Communications ResearchGroup, School of Electronics and Computer Sci-ence, University of Southampton, UK. His researchinterests include channel coding, space-time coding,iterative detection techniques as well as co-locatedand distributed MIMO systems.

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 lecturer in wireless com-munications at the University of Southampton. Hehas been part of a team working on the OPTIMIX

European project since March 2008. His research interests include adaptivecoded modulation, channel coding, space-time coding, joint source andchannel coding, OFDM, MIMO, cooperative communications and distributedcoding. He has published numerous papers and coauthored a book in thisfield.

Ronald Yee Siong Tee received the B.Eng. degree(with first-class honors) in Electrical and Electronicsengineering from the University of Manchester In-stitute of Science and Technology (1999), the M.Sc.degree from the National University of Singapore(2004) and the PhD. degree from the Universityof Southampton (2008). From 2000 to 2003, hewas with Nortel Networks Switzerland and a localSingapore IT company, where he worked in dataand optical networks. His research interests includeiterative decoding, sphere packing modulation, and

coded modulation schemes. Dr. Tee was the recipient of several academicawards, including the Overseas Research Scheme, the ASEAN scholarship,and the Malaysian Government studentships. He is currently with Ernst YoungLondon, working in the area of computer forensic investigation and electronicdiscovery.

Robert G. Maunder has studied with the Schoolof Electronics and Computer Science, Universityof Southampton, UK, since October 2000. He wasawarded a first class honours BEng in ElectronicEngineering in July 2003, as well as a PhD in Com-munications and a lectureship in December 2007.Rob’s research interests include joint source/channelcoding, iterative decoding and irregular coding. Hehas published a number of IEEE papers in theseareas.

Lajos Hanzo (M’91–SM’92–F’04) FREng, FIEEE,FIET, DSc received his degree in electronics in1976 and his doctorate in 1983. During his 31-yearcareer in telecommunications he has held variousresearch and academic posts in Hungary, Germanyand the UK. Since 1986 he has been with the Schoolof Electronics and Computer Science, Universityof Southampton, UK, where he holds the chair intelecommunications. He has co-authored 17 bookson mobile radio communications totalling in excessof 10 000 pages, published in excess of 800 research

papers, acted as TPC Chair of IEEE conferences, presented keynote lecturesand been awarded a number of distinctions. Currently he is directing anacademic research team, working on a range of research projects in the field ofwireless multimedia communications sponsored by industry, the Engineeringand Physical Sciences Research Council (EPSRC) UK, the European ISTProgramme and the Mobile Virtual Centre of Excellence (VCE), UK. He isan enthusiastic supporter of industrial and academic liaison and he offers arange of industrial courses. He is also an IEEE Distinguished Lecturer as wellas a Governor of both the IEEE ComSoc and the VTS. He is the acting Editor-in-Chief of the IEEE Press. For further information on research in progressand associated publications please refer to http://www-mobile.ecs.soton.ac.uk.


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