3090 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 11, NOVEMBER 2011

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1 090 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO., NOVEMBER 20 Reduced-Compexity Coherent Versus Non-Coherent QAM-Aided Space-Time Shift Keying Shinya Sugiura, Member, IEEE, Chao Xu, Student Member, IEEE, Soon Xin Ng, Senior Member, IEEE, and Lajos Hanzo, Feow, IEEE Abstract A nove reduced-compexity near-optima detection agorithm is proposed for enhancing the recent Coherentydetected Space-Time Shift Keying (CSTSK) scheme empoying arbitrary consteations, such as L-point Phase-Shift Keying (PSK) and Quadrature Ampitude Moduation (QAM). The proposed detector reies on a modified Matched Fiter (MF) concept. More specificay, we expoit both the consteation diagram of the moduation scheme empoyed as we as the Inter-Eement- Interference (IEI)-free STSK architecture. Furthermore, we generaize the Puse Ampitude Moduation (PAM)- or PSK-aided Differentiay-encoded STSK (DSTSK) concept and conceive its more bandwidth-efficient QAM-aided counterpart. Then, the proposed reduced-compexity CSTSK detector is appied to the QAM-aided DSTSK scheme, which enabes us to carry out ow-compexity non-coherent detection, whie dispensing with channe estimation. It is reveaed that the proposed detector is capabe of approaching the optima Maximum Likeihood (ML) detector s performance, whie avoiding the exhaustive ML search. Interestingy, our simuation resuts aso demonstrate that the reduced-compexity detector advocated may achieve the same performance as that of the optima ML detector for the specific STSK scheme s parameters. Another novety of this paper is that the star-qam STSK scheme tends to outperform its square-qam counterpart, especiay for high number of dispersion matrices. Furthermore, we provided both the theoretica anaysis and the simuations, in order to support this unexpected fact. Index Terms Differentia encoding, diversity and mutipexing tradeoff, space-time shift keying, spatia moduation, matched fiter, mutipe antenna array, non-coherent detection. I. INTRODUCTION THE recent Mutipe-Input Mutipe-Output (MIMO) concept of Spatia Moduation (SM) 4 or Space- Shift Keying (SSK) 5 is capabe of increasing the achievabe transmission rate with the aid of mutipe-antenna aided systems, which is ensured without resorting to spatia mutipexing 8, 9. To be more specific, since the SM Paper approved by M. Juntti, the Editor for MIMO and Mutipe-Access of the IEEE Communications Society. Manuscript received March 9, 20; revised June 5, 20. The work of S. Sugiura was partiay supported by KAKENHI (2605). The financia support of the RC-UK under the auspices of the India-UK Advanced Technoogy Centre, as we as of the China-UK 4 th generation wireess systems project and that of the European Union under the auspices of the Optimix project, is gratefuy acknowedged. S. Sugiura is with Toyota Centra R&D Laboratories, Inc., Aichi, , Japan (e-mai: sugiura@mosk.tytabs.co.jp). C. Xu, S. X. Ng, and L. Hanzo are with the Schoo of Eectronics and Computer Science, University of Southampton, Southampton, SO BJ, UK (e-mai: {cxg08, sxn, h}@ecs.soton.ac.uk). Digita Object Identifier 0.09/TCOMM /$25.00 c 20 IEEE transmitter activates one out of M antenna eements for conveying additiona information bits during each symbo interva, no Inter-Eement-Interference (IEI) is imposed on the receiver, hence potentiay enabing ow-compexity singestream detection. Due to the different system architecture of cassic spatia mutipexing and of the SM/SSK schemes, new detection agorithms specific to the SM/SSK schemes have been deveoped, which may be cassified into two fundamenta categories, namey the ow-compexity Matched- Fiter (MF) based detector and the singe-antenna-based optima Maximum-Likeihood (ML) detector 2. In practice, the majority of the previous SM/SSK receivers have adopted the singe-stream ML detector 2, where the optima BER performance is achieved at the cost of an increased decoding compexity. On the other hand, the MF-based detector exhibits a significanty reduced compexity, since the antenna index m and the moduated consteation point are separatey estimated. However, as mentioned in 2, 0, this sub-optima detector ony works under the ideaized assumption of encountering noiseess channes at the antenna-index estimation stage. Recenty, the nove concept of Space-Time Shift Keying (STSK) has been proposed in, where the encoding principe is characterized by the fact that one out of Q space-time dispersion matrices is seected, whie the abovementioned SM and SSK schemes simpy activates one out of M antenna eements. Since the STSK scheme is capabe of expoiting both the space- and time-dimensions, it aows us to strike a fexibe baance between the maximum attainabe diversity order and the throughput. Previous studies of the STSK scheme aso considered the optima singestream-based ML detector, simiary to the SM/SSK schemes 2. One exception is constituted by the soution in 4, where a reduced-compexity detection agorithm was deveoped in the context of Differentiay-encoded STSK (DSTSK) systems, which was assisted by Mutipe-Symbo Differentia Sphere Decoding (MSDSD). 2 The introduction of differen- In order to expound a itte further, both the SM and SSK schemes are subsumed by the STSK arrangement, as demonstrated in,. 2 In 5 a Sphere Detector (SD) was conceived for the SM scheme, in order to cosey approximate the optima ML search. Furthermore, in 4 the matched fiter based non-coherent detector was combined with a SD in the context of a differentiay-encoded STSK scheme, for the sake of further reducing the conventiona SD s compexity without imposing any substantia performance degradation. Athough the same idea may be readiy appicabe to the detector proposed in this paper, the reated investigations wi be eft for our future studies.

2 SUGIURA et a.: REDUCED-COMPLEXITY COHERENT VERSUS NON-COHERENT QAM-AIDED SPACE-TIME SHIFT KEYING 09 tia encoding was an important mie-stone, since the corresponding ow-compexity non-coherent receivers are capabe of outperforming their high-compexity coherenty detected counterparts in the presence of reaistic piot-aided channe estimations, as detaied in 6,. However, the appicabiity of this detector is imited to specific ow-order consteations, such as On-Off Keying (OOK), Binary Phase-Shift Keying (BPSK), Quadrature PSK (QPSK) and 8-PSK. Regretfuy, it is not suitabe for bandwidth-efficient Quadrature Ampitude Moduation (QAM) 8. Against this background, the nove contributions of this paper are as foows: Two efficient near-optima detectors are proposed for the Coherenty-detected STSK (CSTSK) scheme. More specificay, both detectors rey on a modified MF concept, whie taking into account the particuar consteation diagram of the moduation scheme empoyed. To be more specific, whie the first detector invokes exhaustive signaspace search at the MF s output, the second detector further reduces the first detector s compexity at the cost of a modest performance penaty. The proposed detectors are appicabe to CSTSK receiver empoying arbitrary moduation schemes, incuding high-order QAM. Furthermore, since the CSTSK scheme subsumes the famiy of SM/SSK schemes as its specia case, the proposed ow-compexity detector is directy appicabe to the cass of SM/SSK schemes. Aso, the computationa compexity imposed by the proposed detector is quantified and compared to that of the conventiona detectors in and 2. Since our previous DSTSK schemes were imited to Puse Ampitude Moduation (PAM) and PSK 4, we extend them to their higher-throughput QAM-aided counterpart. Then, the new reduced-compexity detectors proposed for CSTSK in this paper are further deveoped to its non-coherenty detected QAM-aided DSTSK counterpart. The advantage of the non-coherent DSTSK detector over its coherent counterpart is quantified in scenarios contaminated by channe-estimation errors. An additiona resut of this paper is that the star-qam assisted STSK scheme tends to outperform its square- QAM aided counterpart. This hods true both for the optima ML and for the proposed near-optima detectors. We carried out both the reated anaysis as we as computer simuations, in order to support this somewhat unexpected fact. The remainder of this paper is organized as foows. Section II outines the system mode of the CSTSK scheme. In Section III we commence by reviewing the conventiona detectors and propose the nove near-optima reduced-compexity detection schemes, whie Section IV appies the proposed detectors to the non-coherent STSK scenario and then Section V provides our simuation resuts. Finay, Section VI concudes this paper. II. SYSTEM MODEL OF COHERENT STSK In this section, we briefy review the encoding principe and the received signa mode of the CSTSK scheme. At the CSTSK transmitter, information bits are encoded with the aid of two different operations, namey the dispersionmatrix activation and the cassic L-PSK/QAM moduation. More specificay, the Q space-time dispersion matrices A q C M T (q =,,Q) satisfying the power-constraint reationship of tr ( ) A H q A q = T are assigned to the transmitter in advance of transmissions, where M and T denote the number of transmit antennas and the number of symbos per space-time bock duration, respectivey. Firsty, B = og 2 (Q L) information bits per space-time bock are input to the transmitter, and then the input bits are Seria-to-Parae (S/P) converted to B =og 2 Q and B 2 =og 2 L bits. Next, one out of Q dispersion matrices is seected as A q according to the B =og 2 Q input bits, whie the B 2 =og 2 L input bits are moduated to a PSK/QAM symbo s. Finay, a spacetime codeword S q, = s A q C M T is transmitted from the M transmit antenna eements over T symbo durations. The corresponding bock-based signas Y C N T received at the CSTSK receiver may be expressed as Y = HS + N, () where H C N M represent the channe components, each obeying the compex-vaued Gaussian distribution having a zero mean and a unity variance, i.e. CN(0,), whie each noise eement of N C N T is the compex-vaued Gaussian variabe obeying CN(0,N 0 ). Furthermore, N denotes the number of receive antennas and N 0 represents the noise variance. By impementing the vectoria stacking operation vec( ) at both sides of Eq. () as shown in, we arrive at the equivaent system mode as Ȳ = HK q, + N, (2) where we have Ȳ = vec(y) C NT () H = h,, h Q = (I T H)χ C NT Q (4) χ = vec(a ),,vec(a Q ) C MT Q (5) N = vec(n) C NT (6) and K q, = 0,, 0,s, 0,, 0 T C Q. () qth eement Here, represents the Kronecker product operation. In the rest of this paper, we empoy the parametric notation of STSK(M,N,T,Q) for the sake of space economy. III. LOW-COMLEXITY MF-BASED CSTSKS This section firsty introduces the two conventiona detectors in the context of the CSTSK arrangement, namey the conventiona MF-based detector and the singe-antennnabased ML detector 2. Then we continue by outining the two near-optima receiver architectures advocated, which expoits the properties of the L-PSK/QAM consteations empoyed. Furthermore, we compare the computationa compexity imposed by these four detectors. Note that the aim of the CSTSK detectors is to identify the transmitted index set (q, ) ina reiabe and ow-compexity manner.

3 092 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO., NOVEMBER 20 Channe estimation Consteations X N Modified matched fiter Z Dispersionmatrix estimation Symbo estimation Fig.. The structure of the proposed Detectors I and II. A. The Conventiona MF-Based Detector In the conventiona detector of, the Hermitian transpose of the equivaent channe matrix H is mutipied by the equivaent received signa Ȳ in order to formuate the decision metric of G =g,,g Q T C Q, as foows: G = H H Ȳ. (8) Then, the index q of the activated dispersion matrix and the transmitted symbo index are estimated separatey, as foows: ˆq = argmaxg q, (9) q ˆ = D(gˆq )=argmin gˆq hˆq 2 s, (0) where D denotes the demoduation function. Note that since the transmitted-symbo estimation process of Eq. (0) incudes the resut of the dispersion-matrix estimation formuated in Eq. (9), the potentia mis-detection of ˆq may induce error propagation. Importanty, whie the search space of this detector determined by Eqs. (9) and (0), is as ow as the order of (Q + L), this detection scheme tends to exhibit an error foor in fading environments 2, 0. This is mainy due to the fact that ow-compexity MF-operation of Eq. (8) ignores the effect of the channe s fading enveope, as we as because the decision metric of Eq. (9) ony considers the absoute vaue of the matched-fitered symbo, rather than individuay considering each consteation point. In Section III-C beow, our detector wi be further deveoped for the sake of combating these imitations, hence utimatey achieving a higher BER performance than that of the detector of this section. B. The Singe-Stream-Based ML Detector 2, The optima ML performance may be attained by impementing exhaustive search over the egitimate CSTSK codewords K q, ( q Q, L) at the cost of an increased compexity.tobemorespecific, by maximizing the probabiity of P (Ȳ H, K q, ), the ML search may be expressed as (ˆq,ˆ) = argmin HK q, 2 (q,) () = argmin hq 2, (q,) (2) where h q is the qth coumn of H. As shown in Eq. (2), the search space size of the ML detector is the order of (Q L), which is higher than (Q + L) in the detector outined in Section III-A. C. The Proposed MF-Based Detector I In this section, we present the new reduced-compexity CSTSK detector of Fig.. According to Eq. (4), the estimated channes H are firsty transformed to the equivaent channes H. Then, we normaized each coumn of H in order to generate the modified equivaent channes H as h H = h,, hq h Q Then, we have the MF output of. () Z =z,,z Q T = H H Ȳ. (4) Next, we consider the vector-by-vector exhaustive (q, ) search for the MF-output signas of Z as foows: (ˆq,ˆ) = argmin Z h q K q, 2 q. (5) = argmin z q h q s 2 + z q 2 q. (6) q =q ( = argmin z q h q s 2 + Z 2 z q 2) () q. ( = argmax z q 2 ) zq h q s 2 (8) q. = argmax 2 hq {R(z q )R(s )+I(z q )I(s )} q. h q 2 s 2, (9) whie R( ) and I( ) represent the rea and imaginary part, respectivey. In order to reduce the compexity of the detection reying on Eq. (9), we introduce the separate detection of q and. More specificay, assuming that the moduation scheme s has symmetric properties around both the I- andq-axis, the detection of the activated dispersion-matrix index q can be expressed from Eq. (9) as Eqs. (20) and (2) at the top The computations of the modified equivaent channes H are carried out as a part of the detection process in our detector, which has to be updated at intervas corresponding to the channe s coherence time τ. This naturay imposes an additiona decoding compexity and hence wi be taken into account in the compexity evauation of Section III-E.

4 SUGIURA et a.: REDUCED-COMPLEXITY COHERENT VERSUS NON-COHERENT QAM-AIDED SPACE-TIME SHIFT KEYING 09 ˆq = argmax 2 h q {±R(z q )R(s q, )±I(z q) I(s )} h q 2 s 2 (20) = argmax 2 h q {R(z q )R(s q, ) + I(z q) I(s )} h q 2 s 2 (2) of the next page, where s ( =,, L ) represents the consteation points, which corresponds to the points s having the reationship of R(s ), I(s ) 0, as shown in Fig Then, by using the dispersion-matrix index ˆq estimated in Eq. (2), the symbo index is detected according to ˆ = D(zˆq )=argmin zˆq hˆq s. (22) Note that the egitimate search space of Eqs. (2) and (22) is QL ( QL/4) and L, respectivey, which is typicay smaer than the search space QL of Eq. (9). Here, we define the detector proposed in this section as Detector I, which is characterized by Eqs. (4), (2) and (22). D. The Proposed MF-Based Detector II Furthermore, in this section we propose another detector, which is capabe of further reducing the compexity of Detector I of Section III-C. To be specific, the q-detection part of Eq. (2) in Detector I is further approximated, whie aowing a modest performance penaty. Firsty, Eq. (2) is rewritten as: Eq. (2), which is shown at the top of the foowing page. Then, by ony considering the ast term of Eq. (2), we arrive at its approximation of ˆq =argmax R(z q ) R(s ) q, s + I(z q ) I(s ) s. (24) Here, et us define the egitimate pairs of (R(s )/s, I(s )/s ) as V number of unit-norm vectors x v =x v,i, x v,q (v =,,V). More expicity, as exempified in Fig. 2 for both 6 square-qam and 6 star-qam, we have V = vectors. 5 Finay, we have the foowing decision metric of ˆq = argmax q,v) q, v (25) = argmax q)x v,i + I(z q )x v,q. q, v (26) Finay, the approximated version of Detector I is constituted by Eqs. (4), (26) and (22), which is referred to as Detector 4 For exampe, s ( =,,L) is given by s {} for BPSK, whie we have s {/ 2+j/ 2} for QPSK, s {, / 2+j/ 2,j} for 8-PSK and s {/ 0 + j/ 0, / 0 + j/ 0, / 0 + j/ 0, / 0 + j/ 0} for 6 square-qam. Simiary, s ( =,,L) can be defined for any of the moduation schemes, which have symmetric properties around the I- and Q-axis. 5 To eaborate a itte further, this process of generating x v (v =,,V) from the consteation points is appicabe to moduation schemes, which have symmetric properties around both the s I and s Q axes. This indicates that the proposed detection may be used for most of the conventiona PSK/QAM schemes or other consteation schemes, which do not have such symmetric properties, a the consteation points have to be considered for the cacuation of x v (v =,,V). Naturay, this is ony possibe at the cost of an increased vaue of V, hence an increased decoding compexity, as formuated in Eq. (5) of Section III-E. II in this paper. For more detais, we isted in Tabe I a set of exampes for characterizing the mapping of cassic PSK/QAM symbos to the corresponding sets of x v (v =,,V). 6 To be more specific, given a transmitted index set of (q, ) and the corresponding vaue of v, the correct eement g q,v of the decision metric G is represented by ( ) hh g q,v q = x v,i h q R(s )+R h q N ( ) hh q + x v,q h q I(s )+I h q N, (2) whie the incorrect eement g q,v ( q = q, v V ) may be expressed as g q,v = x v,i R + x v,q I Provided that we have ( hh q ) h q h qs + R ( ) hh q h q h qs + I ) h q N ( ) hh q h q N ( hh q. (28) g q,v >g q,v (q = q, v V ), (29) the correct index q is found during each bock interva. Here, et us consider the utimate scenario of N 0 0. Then Eq. (29) becomes x v,i h q {x v,i R(s ) + x v,q I(s )} }{{} ( ) hh q R h q h q s + x v,q φ I > ( ) hh q h q h q s. (0) We note that considering the reationships of x v,i 2 + x v,q 2 = as we as of R(s ), I(s ) 0, φ = x v,i R(s ) + x v,q I(s ) of Eq. (0) may be upper-bounded by φ = x v,i R(s ) + x v,q I(s ) = R(s ) 2 + I(s ) 2 R(s ) xv,i R(s ) 2 + I(s ) 2 I(s ) + x v,q R(s ) 2 + I(s ) 2 R(s ) 2 + I(s ) 2 = s. () 6 Furthermore, for (L 4)-point PSK consteations, we may be abe to formuate a set of parameters (x v,v) as x v = cos π(v ) π(v ) sin 2(V ) 2(V ) (v =,,V) as we as V = L/4+.

5 094 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO., NOVEMBER 20 =argmax q, { ˆq =argmax 2 h q R(z q ) R(s ) q, s ( h q s { R(z q ) R(s ) s + I(z q ) I(s ) s } s h q 2 s 2, } 2 + I(z q ) I(s ) s }) 2 { + R(z q ) R(s ) s + I(z q ) I(s ) s (2) x x x 2 x 2 x x 6 square QAM 6 star QAM Fig. 2. Sets of s {s,,s L } and X =x T,, xt V for 6 square-qam and star-qam consteations. TABLE I MAPPING EXAMPLES FROM MODULATION SCHEMES TO POINTS x v (v =,,V) Moduation x v (v =,,V) Moduation x v (v =,,V) Scheme Scheme BPSK x =, 0 64-square-QAM x =, QPSK x =, 0 x 2 = 5, x 2 = 0, x =, PSK x =, 0 x 4 =, star-QAM 9 x 2 =, 2 2 x 5 = 5, 4 4 x = 0, x 6 =, PSK x =, 0 x =, star-QAM 20 x 2 = cos ( ) ( π 8, sin π ) 8 5 x8 =, 4 4 x =, x 9 =, x 4 = cos ( ) ( π 8, sin π ) 8 x0 =, x 5 = 0, x =, square-QAM x =, 2 2 x 2 =, square-QAM x =, 0 0 x =, x 2 =, 2 2 x =, 0 0 In Eq. () we have equaity if and ony if x v,i = R(s ) x v,q I(s ). (2) Therefore, the projection operation from the consteation points s ( L) to the corresponding points x v ( v V ) on the unit circe, as shown in Fig. 2, is aimed for having x v, which satisfies Eq. (2), hence maximizes the eft hand side of Eq. (0). As a resut, Eq. (0) is aways satisfied and the conventiona detector s error foor can be eiminated. E. Compexity Anaysis In this section, we compare the computationa compexity imposed by the four detectors presented in Section III, i.e. the origina MF-based detector, the optima ML detector 2, the proposed Detector I of Section III-C and the proposed Detector II of Section III-D. Here, we quantified the compexity as the number of rea-vaued mutipications, where a singe compex-vaued mutipication is deemed to be equivaent to four rea-vaued mutipications. The corresponding compexity per bit may be expressed, respectivey, as Eqs. () (6), which are shown at the top of the foowing page. Here, the coherence bock interva

6 SUGIURA et a.: REDUCED-COMPLEXITY COHERENT VERSUS NON-COHERENT QAM-AIDED SPACE-TIME SHIFT KEYING 095 C Meseh = (4MNTQ+2NTQ)/τ +4NTQ+2Q +4L, og 2 (Q L) () C ML = (4MNTQ+4NTQL)/τ +2NTQL, og 2 (Q L) (4) C I = (4MNTQ+4NTQ+ QL )/τ +4NTQ+QL +4L, og 2 (Q L) (5) C II = (4MNTQ+4NTQ)/τ +4NTQ+2VQ+4L og 2 (Q L) (6) τ is defined as the number of space-time bocks, where the channe matrix H remains constant. This indicates that for exampe τ = represents an instantaneousy fading scenario and that upon increasing the coherence interva τ, the associated compexity of Eqs. () (6) may decrease, because for exampe the cacuations of H, s h q, h q 2 and H can be reused. Not that the compexity evauations of the proposed Detectors I and II, as formuated in Eqs. (5) and (6), are detaied in the Appendix. TABLE II COMPARISON OF SQUARE-QAM AND STAR-QAM CONSTELLATIONS Minimum of Minimum of s 2 s s 2 6 square-qam star-qam square-qam star-qam square-qam star-qam F. Effects of Consteations on High-Q STSK Arrangements In this section, we provide a quaitative anaysis of the effects of the specific consteation on the achievabe coding gain of the QAM-aided STSK scheme. Let us define the Pairwise Error Probabiity (PEP) as P (S,q S,q ), which quantifies the probabiity that a codeword S,q is erroneousy decoded as S,q. According to 2, at high SNRs the Chernoff bound of the PEP may be expressed as ( ) ( RN R ) N P (S,q S,q ), () 4N 0 μ }{{} r= r }{{} diversity gain coding gain where we have Δ = S,q S,q (8) R = rank(δδ H ). (9) Furthermore, μ r represents the rth non-zero eigenvaue of ΔΔ H. Assuming that codewords are designed to satisfy a rank of R = min(m,t), the performance difference is essentiay determined by the minimum vaue of the mutipicative coding-gain term in Eq. () for the egitimate combinations of (S,q, S,q ). For simpicity, et us consider a scenario of ( R ) N M = T. Then, the coding gain r= μ r of Eq. () corresponds to det(δδ H ) N, which may be cassified by the reationship between (, q) and (,q ) as Eqs. (40a) (40c), which are shown at the top of the next page. Typicay, the minimum vaue of det(δδ H ) may be given by Eq. (40a) or Eq. (40b), rather than by Eq. (40c). We note, furthermore, that s s in Eq. (40a) corresponds to the distance between the consteation points and, whie s indicates the absoute vaue of the th consteation point. For high Q vaues it becomes more chaenging to achieve ahighvauefordet (A q A q )(A q A q ) H for a the egitimate (q, q ) combinations in Eq. (40b), than to maintain a high minimum vaue for det ( ) A q A H q for a q in Eq. (40a). Hence empoying specific moduation schemes, which have a high minimum vaue of s, may resut in a high coding gain. For exampe, et us compare in Tabe II two important consteations, namey 6 square-qam 8 and 6 star-qam 9. Observe in Tabe II that star-qam attains a higher vaue of s than square-qam for L = 6, 2 and 64. Hence somewhat unexpectedy, the star-qam assisted STSK scheme has the potentia of outperforming its square-qam counterpart for high-q scenarios. This wi be verified ater in Section V. IV. REDUCED-COMPLEXITY NON-COHERENT STSK RECEIVER In this section, we firsty formuate the QAM-aided DSTSK scheme by further deveoping the DSTSK architecture of, 4. Then, we appy the proposed reduced-compexity detectors of Sections III-C and III-D to our QAM-aided DSTSK receiver. A. System Mode of QAM-Aided DSTSK Scheme We arrange for Q dispersion matrices B q C T T (q =,,Q) to be pre-assigned at the transmitter, simiary to the CSTSK scheme, where B q is a unitary-matrix, which satisfies the reationship of B q B H q = B H q B q = I. It is aso assumed for the sake of simpicity that the number of transmit antennas To eaborate a itte further, in the cassic QAM modems the minimum distance min(s s ) affects the achievabe performance, whie the minimum absoute vaue min(s ) does not. On the other hand, the recent STSK scheme does not obey this we-known rue, as mentioned above. For this reason, the star-qam aided STSK scheme tends to outperforms its square- QAM counterpart, which is in contrast to cassic QAM modems.

7 096 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO., NOVEMBER 20 s s 2 det ( A q A H ) q, for =,q = q (40a) det(δδ H )= s 2 det (A q A q )(A q A q ) H, for =,q = q (40b) det (s A q s A q )(s A q s A q ) H, for =,q = q (40c) M is the same as the number of symbo durations T per transmission bock. In each bock duration, B information bits are input and S/P converted to B and B 2 bits, simiary to the CSTSK transmitter. According to the B bits, one out of Q dispersion matrices is activated as B (n) q, whie the B 2 bits are mapped to a PSK/QAM symbo s (n). Here, the superscript n denotes the space-time bock index. Finay, the transmitted space-time matrix S (n) of the nth bock may be formuated with the aid of differentia encoding as foows: S (n) = s (n) s (n ) S (n ) B (n) q (n ), (4) where we have S (0) = I and s (0) =. 8 Furthermore, the number of successivey encoded bocks per frame is defined as ζ in this paper. Assuming that the channe matrix H remains constant over two successive bock intervas, the corresponding received signas at the (n )st and the nth bocks may be expressed as Y (n ) = HS (n ) + N (n ), (42) Y (n) = HS (n) + N (n) (4) = s (n) s (n ) From Eqs. (4), (42) and (44), we arrive at Y (n) = s (n) s (n ) HS (n ) B (n) q + N (n). (44) Y (n ) B (n) q + N (n), (45) where we have the equivaent noise components of N (n) = N (n) s(n) s (n ) N (n ) B (n) q. (46) Hence, the ML detection at the nth bock interva may be represented by (ˆq,ˆ) =argmin (q,) Y(n) s(n) ν Y(n ) B (n) q, (4) where we assume that the estimate ν = s (n ) is obtained ˆ from the ML detection resut of the (n )st bock, acknowedging that this may potentiay induce error propagation to the detection of the nth bock. We aso note that the DSTSK scheme s ML detector formuated in Eq. (4) does not 8 Athough we assumed for the QAM-aided DSTSK scheme to have the reationship of M = T, a ower transmit-ae scenario of M < T may be readiy used by choosing M<Trows from the signas S (n) in Eq. (4). This impementation is simiar to that of the G -OSTBC 22. incude any channe eements, hence dispensing with channe estimation. B. The Proposed DSTSK Detector In order to enabe us to impement the reduced-compexity MF detectors proposed for the CSTSK scheme in the context of the above-mentioned QAM-aided DSTSK scheme, we firsty derive a inearized DSTSK mode, which corresponds to that of the CSTSK scheme of Eq. (2). More specificay, by appying the vectoria stacking operation vec( ) at both sides of Eq. (45), we get Ȳ (n) = H (n) K (n) q, + N (n), (48) where we have ( Ȳ (n) = vec Y (n)), (49) H (n) = I T Ȳ(n ) s (n ) χ, (50) χ = vec(b ),,vec(b Q ), (5) ( N ) (n) = vec N (n) (52) and K (n) q, = 0,, 0,s (n), 0,, 0 T. (5) qth eement Since Eq. (48) obeys the same system mode as Eq. (2), the detectors proposed in Sections III-C and III-D can be directy appied to our PSK/QAM-aided DSTSK scheme. This impies that higher-order PSK/QAM consteations can be readiy empoyed in a simiar manner to the CSTSK scenario, whie those of the previous DSTSK scheme 4 are imited to consteations up to 8-PSK. Again, as shown in Eq. (50), the matrix H (n) does not contain any channe eements. Additionay, unike the CSTSK scenarios of Section III-C, the proposed DSTSK detector has to update H (n) at each bock interva, regardess of the channe s coherence time τ. By contrast, whie the performance of the CSTSK scheme is routiney degraded by channe estimation errors, the proposed DSTSK detector aows us to dispense with channe estimation, hence potentiay outperforming its CSTSK counterpart. We aso note that our DSTSK detector has the expicit benefit of dispensing with the channe power estimation, whie this woud be necessary for the QAM-aided Differentia Orthogona Space-Time Bock Codes (OSTBC) detector of 2. This contributes towards the mitigation of error propagation to the forthcoming signaing bocks.

8 SUGIURA et a.: REDUCED-COMPLEXITY COHERENT VERSUS NON-COHERENT QAM-AIDED SPACE-TIME SHIFT KEYING 09 BER 6-QAM STSK(4,4,4,6) 0 0 ML 0 - Proposed I Proposed II Bounds star- QAM SNR db 6 square- QAM 64-QAM STSK(4,4,4,64) 64 star-qam SNR db ML Proposed I Proposed II 64 square- QAM Fig.. Achievabe BER performance of the 6-QAM CSTSK(4,4,4,6) and 64-QAM CSTSK(4,4,4,64) schemes, where the optima ML detector as we as the proposed Detectors I and II were considered, whie comparing starand square-qam moduations. BER ML Proposed I Proposed II Meseh et a. Bounds BPSK-mod. SM(4,4) SNR db 8-PSK mod. STSK(4,4,2,8) Fig. 4. Achievabe BER performance of the 8-PSK moduated CSTSK(4,4,2,8) scheme and the BPSK-moduated SM scheme empoying M = N = 4 transmit and receive antennas, where the optima ML detector, the proposed Detectors I and II and Meseh s MF-based detector were considered. Here, a the arrangements have the throughput of bits/symbo and are equipped with M = N =4transmit and receive antennas. V. PERFORMANCE RESULTS In this section we provide our performance resuts for characterizing both the achievabe BER and the computationa compexity of the above-mentioned four detectors. 9 Throughout the simuations, we considered frequency-fat bock Rayeigh fading channes. In order to generate a dispersion-matrix set for each STSK arrangement, Monte Caro simuations were carried out, where the dispersion-matrix set was optimized according to the rankand determinant criterion, by generating random matrices. Additionay, for the CSTSK and DSTSK schemes, having the reationship of M = T, we imposed a unitarymatrix constraint on the randomy generated matrices. 9 Since the performance advantage of the cass of STSK schemes over other MIMO arrangements has been shown in the previous studies of,, 24, in this paper we focus our attaintion on the comparisons between the proposed and conventiona STSK detectors. For readers who are interested in the performance differences between the STSK scheme and the conventiona MIMO arrangements, pease refer to,, 24 Figs. and 4 compare the BER of the STSK famiy, namey of the SM and the CSTSK schemes, empoying M =4transmit and N =4receive antenna eements. In Fig. we considered the 6-QAM assisted CSTSK(4, 4, 4, 6) and 64-QAM assisted CSTSK(4, 4, 4, 64) schemes, empoying square- and star-qam consteations, whie in Fig. 4 the BER curves of the BPSK-moduated SM and the 8-PSK moduated CSTSK(4, 4, 2, 8) schemes having R =bits/symbo were portrayed. Here, we aso potted the corresponding tight BER upper-bound curves cacuated based on the Moment- Generating Function (MGF) 25, in order to confirm the ML detector s BER resuts. Observe in Fig. that both the L =6and 64 scenarios showed the simiar resuts. More specificay, for star-qam the three detectors, i.e. the ML detector and the proposed Detectors I and II, exhibited the aomost identica performance, whie for square-qam Detector II showed a fraction of one db worse performance than those of the ML detector and Detector I. This margina performance penaty of Detector II is owing to the fact that Detector II is designed to approximate the q-estimation part of Detector I for the sake of substantiay reducing the compexity, whose effects wi aso be verified ater in our simuations. Aso, in Fig. 4 whie the origina MF detector exhibited an error foor as predicted from 2, 0, the proposed Detector II achieved a near-optima performance for CSTSK, simiary to the resuts in Fig.. In order to provide further insights, in Fig. 5 we compared the effective SNRs recorded for BER = 0 4 in the context of the 6-square-QAM and 6-star-QAM aided CSTSK(4, 4, T, Q) schemes empoying both the ML and the proposed MF Detectors I and II. More specificay, in Fig. 5(a) we varied the number of dispersion matrices from Q = to 64, whie maintaining the number of symbo durations at T =4. Furthermore, in Fig. 5(b) we varied the number of symbo durations per bock as T =, 2,, 4 and 5. Observein Fig. 5(a) that since an increase of Q increases the normaized throughput, this naturay increased the effective SNR required for both QAM modems. The findings from Figs. 5(a) and 5(b) are isted beow: As predicted from Section III-F, the 6-star-QAM aided CSTSK scheme outperformed its 6-square-QAM aided counterpart for Q 2 and its performance advantage increased upon increasing the vaue of Q regardess of the specific choice of the detectors empoyed. The ML detector and Detector I exhibited an identica performance for a the scenarios considered. For M T 6-star-QAM scenarios the proposed reduced-compexity MF Detector II achieved a performance comparabe to that of the optima ML detector, whist there was an observabe performance gap between Detector II and the other two detectors for 6-square- QAM and for 6-star-QAM scenarios associated with M>T. 0 Observe in Fig. 5(b) that the ower the vaue of T,the 0 An information-theoretic proof of the proposed detector s capabiity of attaining the exact scenario-specific ML performance remains an open issue, which wi be investigated in our future studies.

9 098 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO., NOVEMBER 20 SNR at BER = 0-4 db ML Proposed I Proposed II 6 square-qam 6 star-qam og 2 Q bits SNR at BER = 0-4 db square-qam 6 star-qam ML proposed I proposed II Number of symbo durations per bock T (a) T =4 (b) Q =6 Fig. 5. Comparison of effective SNRs recorded for BER = 0 4 between the 6-square-QAM and 6-star-QAM aided CSTSK(4, 4,T,Q) schemes empoying the ML detector as we as the proposed Detector I and Detector II. (a) og 2 Q = 6, T =4and (b) T = 5, Q =6. 6 star-qam STSK(4,4,4,6) 64 star-qam STSK(4,4,4,6) 0 0 DSTSK 0 4 ML Proposed I Proposed II Meseh et a CSTSK Compexity 0 ML Proposed I Proposed II Meseh et a. BER (perfect CSI) Coherence interva τ Coherence interva τ SNR db Fig. 6. Compexity comparison between the four detectors, which are presented in Section III, for the 6-star-QAM as we as 64-star-QAM assisted CSTSK(4, 4, 4, 6) arrangements. higher the performance gap became between Detector II and the other two detectors. Hence, it may be concuded that star-qam consteations may be more suitabe for the high-q CSTSK scheme than the cassic square-qam consteations. Let us emphasize that this concusion is somewhat surprising, since it is different from the conventiona QAM modems. Moreover, Fig. 6 shows the computationa compexity imposed by the above-mentioned four detectors for the 6-star as we as 64-star-QAM assisted CSTSK(4, 4, 4, 6) schemes. Observe in Fig. 6 that upon increasing the coherence interva τ, the compexity of each detector was reduced towards a certain minimum vaue. As expected, the conventiona MF detector as we as the Detectors I and II attained a significanty ower compexity than that of the ML detector. To be more specific, the conventiona MF detector and Detector II exhibited amost the same compexity, which is approximatey Fig.. BER comparison of the 6-square-QAM aided CSTSK(4, 4, 4, 6) and the 6-star-QAM aided DSTSK(4, 4, 4, 6) schemes, where the proposed MF-based Detector II was empoyed for both the schemes. We aso considered the effects of CSI error associated with an equivaent channe estimation s noise variance of ω = 0, 0., /N 0 and /N 0. The number of successive bocks ζ for the DSTSK scheme was set to ζ =0. twice ower than that of Detector I for the 64-QAM scenario. In Fig. we compared the achievabe BER performance of the 6-square-QAM aided CSTSK(4, 4, 4, 6) and the 6-star- QAM aided DSTSK(4, 4, 4, 6) schemes, both achieving the normaized throughput of 2 bits/symbo, where the proposed MF-based Detector II was empoyed for both the schemes. Here, we aso considered the effects of CSI errors, where the estimated channes were contaminated by the additive Gaussian noise of CN(0,ω) having a power of ω =0, 0. and /N 0 db 26 as we as ω =/N 0 in comparison to the average signa power. Observe in Fig. that athough the CSTSK scheme outperformed its DSTSK counterpart for the perfect-csi and ω =/N 0 scenarios, upon introducing CSIestimation errors, it exhibited an error foor. This is because

10 SUGIURA et a.: REDUCED-COMPLEXITY COHERENT VERSUS NON-COHERENT QAM-AIDED SPACE-TIME SHIFT KEYING 099 the DSTSK scheme remained unaffected by the potentia CSI-estimation errors owing to the expicit benefit of noncoherent detection. Other impicit benefits of the DSTSK scheme are that it does not require any piot overhead and that is competey dispensing with the compexity associated with channe estimation unike its CSTSK counterpart. Next, we compared in Fig. 8 the achievabe BER performance of the 6-star-QAM and 6-square-QAM DSTSK(4, 4, 4, 6) schemes, where the number of successive bocks ζ per frame was varied as ζ =5, 0, 25, 50 and 00. It can be seen from Fig. 8 that an increase of ζ eads to a performance degradation for both schemes, since the mis-detection of s (n ) at the (n )st bock affects the foowing bock, as shown in Eq. (4). However, the degradation was found to be margina, especiay for high SNRs. Surprisingy, the 6 star- QAM scenario tended to attain a better performance than that of 6 square-qam, simiary to the CSTSK systems. This is owing to the fact that the system mode of the DSTSK scheme has the same structure as that of the CSTSK scheme, as shown in Eqs. (2) and (48). Hence the discussions of Section III-F are aso vaid for the DSTSK scheme. Based on the resuts of our simuations, it was found that the proposed Detectors I and II have their own performance versus compexity tradeoffs, depending on the particuar STSK scenarios considered. More specificay, the foowing guideines may be provided: For the M T STSK schemes empoying star-qam or PSK moduations, the adoption of Detector II is preferabe to that of Detector I, since the compexity of Detector II is ower than that of Detector I, whie both the proposed schemes are capabe of attaining the optima ML performance. For other scenarios, we have an option of choosing either of the proposed Detectors I or II, according to the receiver s design poicy regarding its decoding compexity and the achievabe performance. This is because the BER performance of Detector I is marginay better than that of Detector II, whie Detector II s decoding compexity is ower than that of Detector I. Again, athough predominanty the exhaustive ML search has been empoyed for the SM, SSK and STSK schemes, the proposed reduced-compexity detectors have the potentia of repacing it without any substantia performance oss. VI. CONCLUSIONS In this paper, we proposed a reduced-compexity nearoptima detector for the STSK scheme empoying an arbitrary PSK/QAM consteation, which expoits the STSK-specific IEI-free system mode, rather than that of spatia mutipexing. More specificay, the proposed MF detector takes into account the specific consteation diagram considered. As a resut, our detector is capabe of achieving a ower compexity than that of ML detection, whie avoiding any substantia BER performance oss. Interestingy, our simuation resuts demonstrated that the proposed reduced-compexity detector may achieve a performance identica to that of the optima ML detector for the specific STSK s parameters. Therefore, the empoyment of this detector further augments the benefits BER star-qam 6 square-qam SNR db Fig. 8. Achievabe BER performance of the 6-star-QAM and 6-square- QAMDSTSK(4, 4, 4, 6) schemes empoying the proposed Detector II, where the number of successive bocks ζ per frame was varied from ζ =5to 00. of the CSTSK scheme. Furthermore, we generaized the previous PAM- or PSK-aided DSTSK concept to conceive its more bandwidth-efficient QAM-aided counterpart. Then, the proposed reduced-compexity CSTSK detector was appied to the QAM-aided DSTSK scheme, which enabed us to carry out ow-compexity non-coherent detection, whie dispensing with any channe estimation as we as eiminating the piot overhead. Moreover, it was found from our theoretica and numerica anaysis that the star-qam assisted STSK scheme tends to outperform its square-qam counterpart, especiay for high-q scenarios. The proposed detector designed for the cass of co-ocated STSK schemes readiy ends itsef to cooperative communications 2, 28 as we as reying on semi-bind joint channe estimation and data detection 2. APPENDIX The compexity of the proposed MF-based Detectors I and II is represented by Eqs. (5) and (6), respectivey. This appendix provides more detaied information for this assessment. A. The Compexity of Detector I Firsty, we assume that the receiver stores the power of each consteation point s 2 ( =,, L). After estimating the channes H, the equivaent channes H =(I T H)χ of Eq. (4) are cacuated. The associated compexity is given by comp(i T H)χ =4MNTQ, (54) where comp represents the number of rea-vaued mutipications, which is required for cacuating. Then, the norm of each coumn of H, namey h q (q =,,Q), is computed at a compexity of comp h q (q =,,Q) =2NTQ. (55) Then the cacuation of H = h / h h Q / h Q in Eq. () imposes a compexity of comp H =2NTQ. (56)

11 00 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO., NOVEMBER 20 Since Eqs. (54), (55) and (56) ony have to be updated at intervas determined by the channe s coherence time τ, the average compexity may be expressed as 4MNTQ+4NTQ. (5) τ Furthermore, the MF operation of Eq. (4) imposes a compexity of comp H H Ȳ =4NTQ, (58) whie the (.q) detection of Eq. (2) has a compexity of Eqs. (59) and (60). Finay, considering that each bock of the STSK scheme carries og 2 (Q L) bits, the tota per-bit compexity of the proposed Detector I is given by Eq. (5), aso taking into account Eqs. (5), (58), (59) and (60). See Eqs. (59) and (60) at the top of the next page. B. The Compexity of Detector II By contrast, in the proposed Detector II the detection of dispersion-matrix q, as formuated in Eq. (2), is repaced by the approximated version of Eq. (26), whose compexity is given by comp arg max R(zq)xv,I + I(zq)xv,Q q, v =2VQ. (6) Hence, based on Eqs. 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Shinya Sugiura (M 06) received the B.S. and M.S. degrees in aeronautics and astronautics from Kyoto University, Kyoto, Japan, in 2002 and 2004, respectivey, and the Ph.D. degree in eectronics and eectrica engineering from the University of Southampton, Southampton, UK, in 200. Since 2004, he has been with Toyota Centra R&D Laboratories, Inc., Japan. His research has covered a range of areas in communications, incuding space-time moduation/demoduation, turbo coding, cooperative communications, mutiuser detection, and automotive antenna design, as we as vehicuar ad hoc networking. Dr. Sugiura has pubished over 40 research papers in various journas and conference proceedings. He was awarded the IEEE AP-S Japan Chapter Young Engineer Award in December 2008.

12 SUGIURA et a.: REDUCED-COMPLEXITY COHERENT VERSUS NON-COHERENT QAM-AIDED SPACE-TIME SHIFT KEYING 0 comp arg max 2 hq {R(z q )R(s q, ) + I(z q) I(s )} h q 2 s 2 =QL + QL τ, (59) comp arg min zˆq hˆq s =4L (60) Chao Xu (S 09) received a B.Eng. degree from Beijing University of Posts and Teecommunications, Beijing, China, and a B.Sc. (Eng) with First Cass Honours from Queen Mary, University of London, London, UK, through a Sino-UK joint degree program in 2008, both in teecommunications engineering with management. In 2009, he obtained a M.Sc. degree with distinction in radio frequency communication systems from the University of Southampton, Southampton, UK, and he was awarded the IEEE Communications Society UK&RI Chapter Best MSc Student in Broadband and Mobie Communication Networks. He is currenty working towards the Ph.D. degree with the Communications Research Group, Schoo of Eectronics and Computer Science, University of Southampton, UK. His research interests incude reduced-compexity MIMO design, noncoherent space-time moduation detection, and EXIT-chart-aided turbo detection, as we as cooperative communications. Soon Xin Ng (S 99-M 0-SM 08) received the B.Eng. degree (First Cass) in eectronics engineering and the Ph.D. degree in wireess communications from the University of Southampton, Southampton, U.K., in 999 and 2002, respectivey. From 200 to 2006, he was a postdoctora research feow working on coaborative European research projects known as SCOUT, NEWCOM, and PHOENIX. Since August 2006, he has been a member of the academic staff in the Schoo of Eectronics and Computer Science, University of Southampton. He is invoved in the OPTIMIX European project, as we as the IU-ATC and UC4G projects. His research interests incude adaptive coded moduation, coded moduation, channe coding, space-time coding, joint source and channe coding, iterative detection, OFDM, MIMO, cooperative communications, and distributed coding. He has pubished over 20 papers and co-authored two John Wiey/IEEE Press books. He is a senior member of the IEEE and a feow of the Higher Education Academy in the UK. Lajos Hanzo (M 9-SM 92-F 04) FREng, FIEEE, FIET, Eurasip Feow, DSc received his master s degree in eectronics in 96 and his doctorate in 98. In 200, he was awarded the Doctor Honaris Causa honorary doctorate by the Technica University of Budapest. During his 5-year career in teecommunications he has hed various research and academic posts in Hungary, Germany, and the UK. Since 986, he has been with the Schoo of Eectronics and Computer Science, University of Southampton, UK, where he hods the chair in teecommunications. He has co-authored 20 John Wiey - IEEE Press books on mobie radio communications totaing in excess of 0,000 pages, pubished 200+ research papers and book chapters (which can be found on IEEE Xpore), acted as TPC Chair of IEEE conferences, presented keynote ectures, and has been awarded a number of distinctions. Currenty, he is directing a 00-strong academic research team, working on a range of research projects in the fied of communications, signa processing, and contro. The team is sponsored by industry, the Engineering and Physica Sciences Research Counci (EPSRC) UK, the European IST Programme, and the Mobie Virtua Centre of Exceence (VCE), UK. He is an enthusiastic supporter of industry and an academic iaison, and he offers a range of industria courses. He is aso an IEEE Distinguished Lecturer as we as a Governor of the IEEE VTS. He is the Editor-in-Chief of the IEEE Press and a Chaired Prof. at Tsinghua University, Beijing. For further information on research in progress and associated pubications pease refer to