THE demand for wireless services is constantly increasing.

Transcription

1 2432 IEEE TRANACTION ON COUNICATION, VOL. 61, NO. 6, JUNE 2013 Full-Diversity Preoding Design of Bit-Interleaved Coded ultiple Beamforming with Orthogonal Frequeny Division ultiplexing Boyu Li, emer, IEEE, and Ender Ayanoglu, Fellow, IEEE Astrat ulti-input ulti-output (IO) tehniques have een inorporated with Orthogonal Frequeny Division ultiplexing (OFD) for roadand wireless ommuniation systems. Bit-Interleaved Coded ultiple Beamforming (BICB) an ahieve oth spatial diversity and spatial multiplexing for flat fading IO hannels. For frequeny seletive fading IO hannels, BICB with OFD (BICB-OFD) an e employed to provide oth spatial diversity and multipath diversity, making it an important tehnique. In our previous work, the suarrier grouping tehnique was applied to omat the negative effet of suarrier orrelation. It was also proved that full diversity of BICB-OFD with uarrier Grouping (BICB-OFD-G) an e ahieved within the ondition L 1, where,, andl are the ode rate, the numer of parallel streams at eah suarrier, and the numer of hannel taps, respetively. The full diversity ondition implies that if inreases, may have to derease to maintain full diversity. As a result, inreasing the numer of parallel streams may not improve the total transmission rate. In this paper, the preoding tehnique is employed to overome the full diversity restrition issue of L 1 for BICB-OFD-G. First, the diversity analysis of preoded BICB-OFD-G is arried out. Then, the full-diversity preoding design is developed with the minimum ahievale deoding omplexity. Index Terms IO systems, frequeny division multiplexing, singular value deomposition, diversity methods, suarrier multiplexing, onvolutional odes. I. INTRODUCTION THE demand for wireless servies is onstantly inreasing. At the same time, researh on ulti-input ulti-output (IO) wireless systems has een ongoing for more than a deade, with inreasing suess. IO systems are a way to attak the inreasing apaity demand for wireless servies sine they an offer superior spetral effiieny and improved performane within a given andwidth. A ommonly employed IO approah is known as eamforming via ingular Value Deomposition (VD) of the IO hannel matrix. This approah enales spatial multiplexing 1, enaling inreased data rates. It an also enhane system performane. anusript reeived eptemer 13, 2012; revised January 20 and arh 15, The editor oordinating the review of this paper and approving it for puliation was. Juntti. The authors are with the Center for Pervasive Communiations and Computing, Department of Eletrial Engineering and Computer iene, Henry amueli hool of Engineering, University of California, Irvine, CA UA ( {oyul, ayanoglu}@ui.edu). Digital Ojet Identifier /TCO In this paper, the term spatial multiplexing is used to desrie the numer of spatial suhannels, as in [1]. Note that the term is different from spatial multiplexing gain defined in [2] /13$ IEEE This tehnique requires the Channel tate Information (CI) to e availale at oth the transmitter and reeiver [3]. For flat fading IO hannels, it has een shown that employing eamforming with only one spatial hannel, or transmitting one symol at a time, ahieves the full diversity order provided y the hannel [4], [5]. In addition, employing a IO eamformer with more than one spatial hannel without any hannel oding results in the loss of full diversity [4], [5]. On the other hand, employing Bit Interleaved Coded odulation (BIC) [6] together with VD eamforming restores full diversity [7] [9]. uh a system was analyzed and alled Bit-Interleaved Coded ultiple Beamforming (BICB) in [7] [9]. In the study of BICB, so far, the hannel oding tehnique employed has een onvolutional odes [10]. The output of the onvolutional ode is then interleaved through the multiple suhannels with different diversity orders. BICB an ahieve the full diversity order as long as the ode rate and the numer of spatial hannels satisfy 1 [11], [12]. It has een further shown that when a onstellation preoding tehnique is used in unoded and oded VD eamforming, this latter ondition an e overridden and full diversity and full spatial multiplexing offered y the hannel an e simultaneously ahieved [13] [16]. peifially, without hannel oding, full diversity requires that all spatial hannels are preoded. In the ase of preoding with BICB, partial preoding an ahieve oth full diversity and full spatial multiplexing. Partial preoding is desirale eause it redues the high omplexity of deoding a fully preoded BICB system. These preoders now result in a system that ahieves full diversity even without satisfying the ondition 1. As an alternative to preoding as in [13] [16], Perfet pae- Time Blok Codes (PTBCs) [17] were employed as the preoding tehnique [18] [20]. PTBCs have desirale properties of full rate, full diversity, uniform average transmitted energy per antenna, good shaping of the onstellation, and nonvanishing onstant minimum determinant for inreasing spetral effiieny. The resulting system ahieves almost the same performane as preoded BICB while reduing the deoding omplexity sustantially, for IO dimensions 2 and 4 [18] [20]. If the fading in the hannel is not flat, or when the hannel has frequeny seletive fading, the result is Inter-ymol Interferene (II) for the transmitted symols. Orthogonal Frequeny Division ultiplexing (OFD) is ommonly used to omat II aused y multipath propagation [21]. OFD

2 LI and AYANOGLU: FULL-DIVERITY PRECODING DEIGN OF BIT-INTERLEAVED CODED ULTIPLE BEAFORING WITH ORTHOGONAL transmits data in a parallel fashion on losely spaed suarriers. The suarriers satisfy an orthogonality property in order to redue andwidth. OFD is roust against II. It ahieves this y using equalization in the frequeny domain with the advantage of avoiding the omputational urden and the long onvergene time requirements assoiated with time domain equalization. Therefore, OFD an adapt to severe hannel onditions. In addition, OFD has high spetral effiieny, effiient implementation using Fast Fourier Transform (FFT) and Inverse FFT (IFFT), and low sensitivity to time synhronization errors. With OFD, multipath diversity an e ahieved y adding hannel oding [22], [23]. As a result, OFD is well-suited for roadand data transmission, and it has een seleted as the air interfae for the Institute of Eletrial and Eletronis Engineers (IEEE) Wireless Fidelity (WiFi) standard, the IEEE Worldwide Interoperaility for irowave Aess (WiAX) standard, as well as the Third Generation Partnership Projet (3GPP) Long Term Evolution (LTE) standard [24]. The omination of IO and OFD has een inorporated for all roadand wireless ommuniation standards, i.e., WiFi [25], WiAX [26], and LTE [27]. For frequeny seletive IO hannels, omining eamforming with OFD an omat II and ahieve spatial diversity [28]. oreover, oth spatial diversity and multipath diversity an e ahieved y adding hannel oding, e.g., BICB with OFD (BICB- OFD), [8], [29], [30]. Although more sophistiated odes, suh as turo odes and Low-Density Parity-Chek (LDPC) odes [10], are employed y some of the more modern standards [24], than onvolutional odes employed y BICB- OFD, onvolutional odes are still important eause they an e analyzed and there are a great deal of legay produts using them. Therefore, BICB-OFD an e an important tehnique for roadand wireless ommuniation. The diversity analysis of BICB-OFD was arried out in [31]. In [31], the suarrier grouping tehnique was employed to overome the performane degradation aused y suarrier orrelation and offer multi-user ompatiility. It was proved that full diversity of BICB-OFD with uarrier Grouping (BICB-OFD-G) an e ahieved as long as the ondition L 1 is satisfied, where is the numer of streams transmitted at eah suarrier and L is the numer of hannel taps. The full diversity ondition implies that if the numer of streams transmitted at eah suarrier inreases, the ode rate may have to derease in order to keep full diversity. Hene, inreasing the numer of parallel streams may not improve the total transmission rate, whih is a similar issue to the full diversity ondition 1 of BICB for flat fading IO hannels [11], [12]. ine preoding tehniques have een suessfully used to solve the full diversity restrition issue of 1 for BICB in the ase of flat fading IO hannels [13] [16], [18] [20], it may e possile to apply these tehniques to BICB-OFD-G so that its full diversity ondition is not restrited to L 1 for frequeny seletive IO hannels. Nevertheless, the design riteria and diversity analysis annot e generalized in a straightforward manner eause of the inreased system omplexity. In this paper, the main ontriution is that the preoding tehnique is employed to solve the full diversity restrition Fig. 1. R Conv. En. ^ Bit Interleaver Viteri De. od. etri Cal. V(1) xk(1) V() xk() yk(1) truture of BICB-OFD. H U(1) H U() yk() Nt Nt Nr Nr IFFT FFT Add CP Remove CP issue of L 1 for BICB-OFD-G proposed in [31]. First, diversity analysis of preoded BICB-OFD- G is arried out. Based on the analysis, a full diversity ondition related to the omination of the preoding matrix, the onvolutional ode, and the it interleaver is provided. Then, the full diversity preoding design is developed. This design provides a suffiient method to guarantee full diversity while minimizing the inreased deoding omplexity aused y preoding. The remainder of this paper is organized as follows: etion II riefly desries BICB-OFD and BICB-OFD-G. In etion III, the system model of BICB-OFD-G employing preoding is proposed. In etion IV, the diversity analysis of preoded BICB-OFD-G is arried out. Then, etion V develops the full-diversity preoding design with the minimum ahievale deoding omplexity. In etion VI, simulation results are provided. Finally, a onlusion is drawn in etion VII. II. BACKGROUND KNOWLEDGE OF BICB-OFD In this setion, a rief desription of BICB-OFD is provided in order to offer suffiient akground knowledge for the following setions. ore details of BICB-OFD an e found in [31]. A. BICB-OFD For frequeny seletive IO hannels, BICB-OFD was proposed to provide oth spatial diversity and multipath diversity [8], [29], [30]. Fig. 1 presents the struture of BICB-OFD. First, the it odeword is generated from the information its y the onvolutional enoder of ode rate, whih is possily omined with a perforation matrix for a high rate puntured ode [32]. After that, a random it interleaver is applied to generate an interleaved it sequene, whih is then modulated, e.g., Quadrature Amplitude odulation (QA), to a symol sequene. The numer of transmit and reeive antennas are denoted y N t and N r respetively. Assume that suarriers are employed to transmit the symol sequene, and min{n t,n r } parallel streams realized y VD in the frequeny domain for eah suarrier are transmitted at the same time. Hene, an 1 symol vetor x k (m) is arried on the mth suarrier at the kth time instant with m,...,. The length of Cyli Prefix (CP), whih is employed for OFD to omat II aused y Nt Nr

3 2434 IEEE TRANACTION ON COUNICATION, VOL. 61, NO. 6, JUNE 2013 multipath propagation, is assumed to e L p where L p L with L denoting the numer of hannel taps. The L-tap frequeny seletive fading IO hannel is assumed to e Rayleigh quasi-stati and known y oth the transmitter and reeiver, whih is denoted y H(l) C Nr Nt with l = 1,...,L where C stands for the set of omplex numers. Let L ( H(m) = H(l)exp i 2π(m 1)τ ) l (1) T l denote the quasi-stati flat fading IO hannel oserved at the mth suarrier, where T denotes the sampling period, τ l indiates the lth tap delay, and i = 1 [33]. Then, the eamforming matries at the mth suarrier are determined y VD of H(m), i.e.,h(m) =U(m)Λ(m)V H (m), where the N r N r matrix U(m) and the N t N t matrix V(m) are unitary, and the N r N t matrix Λ(m) is a retangular diagonal matrix whose sth diagonal element, λ s (m) R +, is a singular value of H(m) in dereasing order with s,..., where R + denotes the set of positive real numers. When streams are transmitted for eah suarrier at the same time, the first olumns of U(m) and V(m), i.e., U (m) and V (m), are hosen as eamforming matries at the reeiver and transmitter for the mth suarrier respetively. For eah suarrier, the multipliations with eamforming matries V (m) and U H (m) are arried out at eah suarrier efore exeuting IFFT and adding CP at the transmitter, and after exeuting FFT and removing CP at the reeiver, respetively. Therefore, the input-output relation of BICB- OFD for the mth suarrier at the kth time instant is y s,k (m) =λ s (m)x s,k (m)+n s,k (m), (2) for s = 1,...,,wherey s,k (m) and x s,k (m) are the sth element of the 1 reeived symol vetor y k (m) and the transmitted symol vetor x k (m) respetively, and n s,k (m) is the additive white Gaussian noise with zero mean and variane N 0 = N t /γ [34] with γ denoting the reeived ignal-to- Noise Ratio (NR) over all the reeive antennas. Note that the total transmitted power is saled y N t in order to make the reeived NR γ. The loation of the oded it within the transmitted symol is denoted as (k, m, s, j), meaning that the oded it is mapped onto the jth it position on the lael of x s,k (m).letχdenote the signal set of the modulation sheme, and let χ j denote a suset of χ whose laels have {0, 1} at the jth it position. By using the loation information (k, m, s, j) and the input-output relation in (2), the reeiver alulates the aximum Likelihood (L) it metris for = as Δ(y s,k (m), )= min x χ j y s,k (m) λ s (m)x 2. (3) Finally, the L deoder, whih applies the soft-input Viteri deoding [10] to find a odeword with the minimum sum weight, makes deisions ased on the rule given y [6] as ĉ =argmin Δ(y s,k (m), ). (4) In [31], the diversity analysis of BICB-OFD was arried R Conv. En. R Conv. En. Fig. 2. R Conv. En. ^ ^ Bit Interleaver Bit Interleaver Viteri De. Viteri De. od. od. etri Cal. etri Cal. truture of BICB-OFD-G. Bit Interleaver od. L L L L L T1 T2 xk(mg,1) xk(mg,l) V(1) xk(1) V() xk() Nr H U(1) yk(1) H U() yk() 1(mg,1) (mg,1) 1(mg,L) (mg,l) Nt Nt Nr yk(mg,1) yk(mg,l) L IFFT FFT etri Cal. Add CP Remove CP Nt Nr Viteri De. Fig. 3. truture of BICB-OFD-G in the frequeny domain for one it stream transmission of the gth suarrier group. out. Aording to the analysis, the maximum ahievale diversity of BICB-OFD was derived and the full diversity restrition of L 1 was proved. In addition, the performane degradation due to suarrier orrelation was investigated, whih showed that although the maximum ahievale diversity is the same when NR is relatively high, strong suarrier orrelation an result in signifiant performane loss for NRs in the pratial range. B. uarrier Grouping In order to omat the performane degradation of BICB- OFD aused y suarrier orrelation, the suarrier grouping tehnique was employed in [31]. Instead of transmitting one stream of information through all suarriers of OFD, suarrier grouping tehnique transmits multiple streams of information through multiple group of suarriers, whih was also suggested for multi-user interferene elimination [35], Peak-to-Average Ratio (PAR) redution [36], and omplexity redution [37]. For BICB-OFD-G, assuming that G = /L Z where Z denotes the set of integer numers, then G streams of it odewords are arried on G different groups of L unorrelated or weakly orrelated suarriers at the same time. Example: Consider the ase of L =2and =64. Then, the gth and the (g + 32)th suarriers are unorrelated or weakly orrelated for g,...,32. Then, the suarrier grouping tehnique an transmit G = 32 streams of it odewords simultaneously through the 32 groups of two unorrelated or weakly orrelated suarriers without or with only small performane degradation. Fig. 2 presents the struture of BICB-OFD-G. In Fig. 2, T 1 is a permutation matrix at the transmitter that distriutes the modulated symols to their orresponding suarriers, ^

4 LI and AYANOGLU: FULL-DIVERITY PRECODING DEIGN OF BIT-INTERLEAVED CODED ULTIPLE BEAFORING WITH ORTHOGONAL R Conv. En. Bit Interleaver od. L P P L-NpP L T L 1(mg,1) (mg,1) 1(mg,L) (mg,l) yk(mg,1) yk(mg,l) L etri Cal. Viteri De. Fig. 4. truture of BICB-OFD-G with preoding in the frequeny domain for one it stream transmission of the gth suarrier group. while T 2 = T 1 1 is a permutation matrix at the reeiver that distriutes the reeived symols to their related streams for deoding. Note that the struture of BICB-OFD-G in Fig. 2 an also e onsidered as Orthogonal Frequeny- Division ultiple Aess (OFDA) [24] version of BICB- OFD. OFDA is a multi-user version of OFD and it has een employed in moile WiAX [26] as well as the downlink of LTE [27]. OFDA assigns susets of suarriers to individual users to ahieve multiple aess, whih is similar to the suarrier grouping tehnique. Consequently, BICB- OFD an offer multi-user ompatiility with suarrier grouping. Fig. 3 presents the struture of BICB-OFD-G in the frequeny domain for one it stream transmission of the gth suarrier group with g {1,...,G}, and the assoiated suarrier index for the lth suarrier of the gth group is denoted in the figure as m g,l =(l 1)G+g with l,...,l. Note that Fig. 3 an also present the struture of BICB- OFD in the frequeny domain when L =. Compared to BICB-OFD without suarrier grouping, BICB-OFD- G ahieves etter performane with the same transmission rate and the same deoding omplexity while also provides multi-user ompatiility. III. YTE ODEL OF BICB-OFD-G WITH PRECODING As disussed in etion II-B, BICB-OFD-G is oviously a muh etter hoie than BICB-OFD without suarrier grouping eause it provides etter performane with the same transmission rate and deoding omplexity while also offers multi-user ompatiility. Therefore, the preoding tehnique disussed in the following parts of this paper is employed on top of BICB-OFD-G. ine the G groups of it streams are transmitted separately in the frequeny domain for BICB-OFD-G and the only differene is the orresponding singular values of suhannels as shown in Fig. 3, the preoding tehnique an e applied to eah suarrier group independently. Therefore, it is suffiient to onsider one suarrier group to illustrate the system model. Fig. 4 presents the struture of BICB-OFD-G with preoding in the frequeny domain for one it stream transmission of the gth suarrier group. Compared to BICB- OFD-G without preoding as shown in Fig. 3, the hannel oding, it interleaver, and modulation remain the same, while two more preoding loks are added at the transmitter. peifially, Θ is defined as a P P preoding matrix where P L denotes the dimension of Θ, whih is applied to preode P of L suhannels employed for the gth suarrier ^ group. The P preoded suhannels are defined as one preoded suhannel set. Let N p denote the numer of preoded suhannel sets employed for the gth suarrier group, where N p P L. As a result, N p P suhannels are preoded while the remaining N n = L N p P suhannels are non-preoded. The seletions of preoded suhannel sets and non-preoded suhannels are predefined y a permutation matrix T. Note that there are L suarriers of eah group and eah suarrier inludes suhannels realized y VD. For the sth suhannel at the lth suarrier for the gth group, its singular value is λ s (m g,l ) where m g,l =(l 1)G+g. ine the G suarrier groups are independent, the group index is omitted for revity in the following, and λ s (m g,l ) is rewritten as λ l,s where the two-dimensional index {l, s} denotes the sth suhannel at the lth suarrier. For the sake of onveniene, the two-dimensional index is further onverted to a single dimensional one following the rule {l, s} q =(l 1) + s with q {1,...,L}, and the orresponding inverse onversion is q {l, s} = {[ (q 1)/ +1], [(q 1) mod +1]}. Define η z =[η1 z...ηz P ] as a vetor whose elements ηz p denote the suhannel indies of the zth preoded suhannel set with z {1,...,N p }, and are ordered inreasingly suh that ηu z <ηz v for u<v.inthesameway,ω =[ω 1...ω Nn ] is defined as an inreasingly ordered vetor whose elements are the indies of the non-preoded suhannels. At the kth time instant, after modulation, the serial-toparallel onverter of the transmitter organizes the L 1 symol vetor x k as x k =[x T η 1,k......x T..x T η Np,k ω,k ]T where x ηz,k = [x η z 1,k...x η z P,k] T and x ω,k = [x ω1,k...x ωnn,k] T with x q,k denoting the modulated symol supposed to e transmitted through the qth suhannel. Then, at the reeiver, the L 1 reeived symol vetor y k = [yη T 1,k....yT η Np,k.yT ω,k ]T,wherey ηz,k = [y η z 1,k...y η z P,k] T and y ω,k = [y ω1,k...y ωnn,k] T with y q,k denoting the reeived symol of the qth suhannel, is written as y k = Λ Θx k + n k, (5) where Λ denotes a L L lok diagonal matrix Λ = diag[ Λ η Λ η Np. Λ ω ] with diagonal singular matries defined as Λη z = diag[λ η z 1...λ η z P ] and Λ ω = diag[λ ω1...λ ωnn ], Θ is a L L lok diagonal matrix Θ = diag[θ......θ..inn ] with I Nn defined as the N n - dimensional identity matrix, and n k is an L 1 vetor n k =. [n T η 1,k n T.n T η Np,k ω,k ]T,wheren ηz,k =[n η z 1,k...n η z P,k] T and n ω,k =[n ω1,k...n ωnn,k] T with n q,k denoting the additive white Gaussian noise with zero mean and variane N 0 = N t /γ at the qth suhannel. As a result, the input-output relation in (5) an e deomposed into N p +1 equations as y ηz,k = Λ η z Θx ηz,k + n ηz,k, y ω,k = Λ (6) ω x ω,k + n ω,k. In a similar way to BICB-OFD introdued in etion II, the loation of the oded it within the transmitted symol is denoted as (k, q, j), whih means that is mapped onto the jth it position on the lael of x q,k.byusingthe

5 2436 IEEE TRANACTION ON COUNICATION, VOL. 61, NO. 6, JUNE 2013 loation information and the input-output relation in (5), the reeiver alulates the L it metris for = {0, 1} as where ξ q,j ξ q,j Δ(y k, )= min y k Λ Θx 2, (7) x ξ q,j is a suset of χ L,definedas = {x =[x 1...x L ] T : x u=q χ j and x u q χ}. Based on the deomposition of (5) to (6), the it metris equivalent to (7) are min y ηz,k Λ η zθx 2, if q η z, x ψ Δ(y k, )= ˆq,j min y q,k λ q x 2, if q ω, x χ j (8) where ˆq {1,...,P} is the assoiated index of the qth suhannel in η z,andψ ˆq,j is a suset of χ P defined as ψ ˆq,j = {x =[x 1...x P ] T : x p=ˆq χ j and x p ˆq χ}. Finally, the L deoder, whih applies the soft-input Viteri deoding to find a odeword ĉ with the minimum sum weight and its orresponding information it sequene ˆ, uses the it metris alulated y (8) and makes deisions as ĉ =argmin Δ(y k, ). (9) IV. DIVERITY ANALYI OF BICB-OFD-G WITH PRECODING The performane of BICB-OFD-G with preoding of eah suarrier group is ounded y the union of the Pairwise Error Proaility (PEP) orresponding to eah error event [8], [29], [30]. In partiular, the overall diversity order is dominated y the pairwise errors whih have the smallest negative exponent value of NR in their PEP representations. As a result, the alulation of eah PEP is needed. In this setion, an upper ound to eah PEP is derived. Based on the it metris in (7), the instantaneous PEP etween the transmitted odeword and the deoded odeword ĉ is alulated as ( Pr ( ĉ H(m), m) =Pr min y k Λ Θx 2 x ξ q,j min x ξ q,j ĉ 2 y k Λ Θx, (10) where and ĉ are the oded its of and ĉ, respetively. Let d H denote the Hamming distane [10] etween and ĉ. ine the it metris orresponding to the same oded its etween the pairwise errors are the same, (10) is rewritten as Pr ( ĉ H(m), m) =Pr min y k Λ Θx 2 x ξ q,j,d H 2 min y k Λ Θx, x ξ,d q,j H ĉ (11) where,d H stands for the summation of the d H values orresponding to the different oded its etween the it odewords. Define x k and ˆx k as x k =argmin x ξ q,j y k Λ Θx 2, ˆx k =argmin x ξ q,j y k Λ Θx (12) 2, where is the omplement of in inary. It is easily found that x k is different from ˆx k sine the sets that x q elong to are disjoint, as an e seen from the definition of ξ q,j.inthe same manner, it is lear that x k is different from ˆx k. With x k and ˆx k, (11) is rewritten as Pr ( ĉ H(m), m) =Pr y k Λ Θ x k 2,d H y k Λ Θˆx k 2.,d H (13) Based on the fat that y k Λ Θx k 2 y k Λ Θ x k 2,and the relation in (5), equation (13) is upper ounded y Pr ( ĉ H(m), m) Pr y k Λ Θx k 2,d H y k Λ Θˆx k 2,d H =Pr ɛ,,d H Λ Θ(x k ˆx k ) 2 (14) where ɛ =,d H Tr[ (x k ˆx k ) H Θ H Λ H n k n H Λ Θ(x k k ˆx k )]. ineɛ is a zero-mean Gaussian random variale with variane 2N 0,d H Λ Θ(x k ˆx k ) 2, (14) is given y the Q funtion as Pr ( ĉ H(m), m) Q Λ Θ(x k ˆx k ) 2. 2N 0 (15) By using the upper ound on the Q funtion Q(x) 0.5exp( x 2 /2), the average PEP an e upper ounded as Pr ( ĉ) =E[Pr( ĉ H(m), m)] [ ( 1 k E 2 exp,d H Λ Θ(x )] k ˆx k ) 2. 4N 0 (16) Aording to (6), the negative numerator of the exponent in (16) is rewritten as κ =,d H Λ Θ(x k ˆx k ) 2

6 LI and AYANOGLU: FULL-DIVERITY PRECODING DEIGN OF BIT-INTERLEAVED CODED ULTIPLE BEAFORING WITH ORTHOGONAL = = N p z,d H,η z N n u N p Λ η zθ (x ηz,k ˆx ηz,k) 2 + λ ωu (x ωu,k ˆx ωu,k) 2,d H,ωu P θ T p (x η z,k ˆx ηz,k) 2 + λ 2 ηp z z p,d H,η z N n λ 2 ω u u (x ωu,k ˆx ωu,k) 2, (17),d H,ωu where,d H,η and z,d H,ωu stand for the summation related to the d H,η z and d H,ωu different oded its arried on the suhannels in η z and suhannel ω u, respetively, and θ T p denotes the pth row of Θ. By reordering the indies of singular values, (17) an e rewritten as the following form κ = L q ρ q λ 2 q, (18) where θ Ṱ q (x η z,k ˆx ηz,k) 2, if q η z, k ρ q =,d H,η z (x ωu,k ˆx ωu,k) 2, if q = ω u,,d H,ωu (19) with ˆq denoting the assoiated index of the qth suhannel in its preoded suhannel set. For BICB-OFD-G, the suarriers of eah group are unorrelated or weakly orrelated [31]. In that ase, the singular value matries Λ(m) an e onsidered independent for eah suarrier group [31]. Therefore, y onverting the one-dimensional suhannel indies ak to their orresponding two-dimensional indies, (16) is further rewritten as Pr ( ĉ) [ ( s E exp ρ )] l,sλ 2 l,s. (20) 4N 0 l For eah suarrier, the terms inside the expetation in (20) an e upper ounded y employing the theorem proved in [38], whih is given elow. Theorem. Consider the largest min{n t,n r } eigenvalues μ s of the unorrelated entral N r N t Wishart matrix [39] that are sorted in dereasing order, and a weight vetor ρ =[ρ 1...ρ ] T with non-negative real elements. In the high NR regime, an upper ound for the expression E[exp( γ s ρ sμ s )], whih is used in the diversity analysis of a numer of IO systems, is [ ( )] E exp γ ρ s μ s ζ (ρ min γ) (Nr δ+1)(nt δ+1), s (21) where γ is NR, ζ is a onstant, ρ min =min ρi 0 {ρ i } i, and δ is the index to the first non-zero element in the weight vetor. Proof: ee [38]. By applying the aforementioned theorem to (20), an upper ound of PEP is Pr ( ĉ) l,ρ l 0 ( ) Dl ρl,min ζ l γ, (22) 4N t with D l =(N r δ l +1)(N t δ l +1),whereρ l,min denotes the minimum non-zero element in ρ l whose element ρ l,s denotes the weight of λ 2 l,s, δ l denotes the index of the first non-zero element in ρ l,andζ l is a onstant. Therefore, the diversity an e easily found from (22), whih is D = D l. (23) l,ρ l 0 Based on the results of (22) and (23), full diversity an e ahieved if and only if ρ l,1 0, l for all error events. ine the error events are related to the onvolutional ode and the it interleaver, the full diversity ondition is related to the omination of the preoding matrix, the onvolutional ode, and the it interleaver. V. FULL-DIVERITY PRECODING DEIGN OF BICB-OFD-G WITH PRECODING The preoding design satisfying the full diversity ondition ρ l,1 0, l of all error events may not e unique. In this setion, a suffiient method of preoding design is developed for BICB-OFD-G whih guarantees full diversity while minimizing the inreased deoding omplexity aused y preoding. A. Choie of Preoding atrix An upper ound of PEP for BICB-OFD-G without preoding an e written as in [31] in a similar form as (20) only with different weights of ρ l,s = d 2 min α l,s, (24) where d min is the minimum Eulidean distane [21] in the onstellation, and α l,s denotes the numer of distint its transmitting through the sth suhannel of the lth suarrier for an error path whih implies that L l s α l,s = d H. The diversity an e derived in a similar fashion to (22) and (23), and the full diversity ondition is α l,1 0, l for all error events. As proved in [31], the full diversity ondition an e ahieved only if the ondition of L 1 is satisfied. Otherwise, full diversity annot e provided. The reason is that, in the ase of L > 1, there always exists at least one error path with no errored it of the error event transmitted through the first suhannel of a suarrier. It is ovious that when α l,1 = 0, then ρ l,1 = 0, if the {l, 1}th suhannel is non-preoded. However, if the {l, 1}th suhannel is preoded, ρ l,1 ould e non-zero even if α l,1 =0, whih depends on θ Ṱ q and eah error event as shown in (19). Therefore, BICB-OFD-G with preoding ould ahieve full diversity even if L > 1 y proper preoding design. When designing the preoding matrix, it is inonvenient to onsider all error events whih ould e large in numer. However, sine an error event only affets x ηz,k ˆx ηz,k in (19), a suffiient ondition of the preoding

7 2438 IEEE TRANACTION ON COUNICATION, VOL. 61, NO. 6, JUNE 2013 design is given y θ Ṱ q (x ˆx) 2 0, for (q mod ), (25) of all different x and ˆx. It is not hard to find θ Ṱ q whih satisfies (25). In fat, as long as every element in θ Ṱ q is non-zero, the ondition (25) is satisfied. Note that the ondition (25) is designed for ertain rows of Θ orresponding to the first suhannel of all suarriers. Although other suhannels do not affet the diversity as shown in (22) and (23), the ondition (25) an e further simplified to θ T p (x ˆx) 2 0, p, (26) of all different x and ˆx. Assume that the average transmitted power at eah transmit antenna is the same, then the preoding matrix is hosen as θ u,v 0, u, v and θ T p 2, p. (27) In fat, the preoding matries in [13] all satisfy the ondition (27), whih are onsidered in the next three susetions. B. inimum Effetive Dimension of Preoding atrix When the preoding matries in [13] are applied, the weights of (19) an e simplified to ρ q = d 2 minβ q, (28) where { β q = θˆq,min 2 α ηz,min, if q η z, (29) α q, if q = ω u, with θˆq,min denoting the element in θ ˆq having the smallest asolute value and α ηz,min denoting the minimum non-zero α element orresponding to the P preoded suhannels for the zth set. Note that (28) is a lower ound for (19). As a result, an upper ound with a form similar to (22) and (23) an e derived with the simplified weights (28). Compared to the weights of (24) for BICB-OFD-G without preoding, the weights of the N n non-preoded suhannels are the same. For the N p P preoded suhannels, eah weight now depends on the α elements of the P preoded suhannels of the orresponding set instead of only one suhannel. Therefore, if an errored it is transmitted through a preoded suhannel, then all weights in (28) for the P preoded suhannels of the orresponding set is non-zero. However, if no errored it is transmitted through a preoded suhannel set, then all weights of the P preoded suhannels are zero, whih are the same as BICB-OFD-G without preoding. If that happens, preoding is meaningless sine the PEPs with the worst diversity dominate the overall performane. Therefore, the preoding design requirement is that at least an errored it of eah error event is transmitted through eah preoded suhannel set. The aforementioned preoding design requirement is related to the onvolutional ode, the it interleaver, and the dimension of the preoding matrix. In fat, if P = L, whih means all suhannels are preoded y only one L L preoding matrix Θ, the requirement an e easily satisfied. However, a larger dimension for Θ results in higher omplexity for alulating the metris assoiated with the preoded its in (8). As a result, the minimum effetive dimension of Θ should e found. Assume that N = LJ information its are transmitted, then J oded its are transmitted y eah of the L parallel suhannels. Hene, PJ oded its are transmitted y a preoded suhannel set. Note that N information its an provide 2 N different it odewords. Hene, if PJ is smaller than N, there always exists at least a pair of it odewords whose PJ oded its transmitted y a preoded suhannel set are the same. The reason is that the total possile numer of it sequenes for a preoded suhannel set, whih is 2 PJ, is smaller than the total possile it odewords 2 N. As a result, the preoded suhannel set is non-effetive. Therefore, PJ annot e smaller than N, whih implies that P L. inep is an integer, the minimum effetive dimension of Θ is P = L. Note that P = L is only proved in this susetion to e a neessary ondition eause the requirement, i.e., at least an errored it of eah error event is transmitted through eah preoded suhannel set, is also related to the onvolutional ode and the it interleaver. C. inimum Effetive Numer of Preoding uhannel ets Assume that at least an errored it of eah error event is transmitted through eah preoded suhannel y a properly designed omination of the onvolutional ode and the it interleaver. Then, every preoded suhannel set is effetive. However, it still does not guarantee full diversity. Note that the full diversity ondition derived in etion IV requires that ρ l,1 0, l of all error events. It is also illustrated in etion V-B that the non-preoded suhannels result in the same weights for oth preoded and non-preoded BICB- OFD-G. As a result, if a first suhannel of a suarrier is not preoded, there always exists at least one error path with no errored its transmitted through that suhannel when L > 1, as proved in [31]. In that ase, full diversity annot e ahieved even if all preoded suhannel sets are effetive. Therefore, the first suhannels of all suarriers should e preoded. ine there are L suarriers whih offer L first suhannels in total, and eah Θ an preode P L suhannels, then the minimum effetive numer of preoding suhannel sets is N p = L/P. The aforementioned full diversity requirement is that the first suhannels of all suarriers should e preoded. However, if L/P P >L, the full diversity requirement annot e satisfied eause not all first suhannels of all suarriers an e preoded y Θ with effetive dimension P L. In fat, the ase of L/P P > L an only happen when = 1. In other words, when 2, L/P P L is always valid, whih is proved in the following. Proof: Note that L/P 1. If L/P, L/P P L is always valid eause P L. On the other hand, if L/P 2, thenp<l. Beause 2, then L/P P <2L L. (30) This onludes the proof. As a result, when 2, the minimum effetive numer of Θ is N p = L/P with P = L. imilar seletion of

8 LI and AYANOGLU: FULL-DIVERITY PRECODING DEIGN OF BIT-INTERLEAVED CODED ULTIPLE BEAFORING WITH ORTHOGONAL N p and P an e applied for if N p P L. Otherwise, if N p P>Lfor, the dimension P of Θ needs to e inreased so that the produt of P and its orresponding N p satisfies N p P L. Example: Consider the parameters N t = N r =2, L =4,,and =2/3, then the minimum effetive dimension of Θ is P = L =3. Hene, the minimum effetive numer of preoding suhannel sets is N p = L/P =2.As a result, N p P =6suhannels are required to e preoded y two sets. However, there are only four suhannels. Therefore, full diversity annot e ahieved with the seletion of P =3 and N p =2. Hene, P =4should e applied instead. In that ase, N p,andn p P =4 L. Therefore, P =4and N p is the minimum effetive seletion. Note that similar to the disussion on P in etion V-B, the minimum effetive seletion of P and its orresponding N p is only a neessary ondition, eause the onvolutional ode and the interleaver also need to e onsidered to satisfy the requirement, i.e., at least one errored it of eah error event is transmitted through eah preoded suhannel set. D. eletion of Preoded uhannels Aording to (22), (23), (28), and (29), the diversity of BICB-OFD-G with preoding also depends on the α- spetra of BICB-OFD-G without preoding. In fat, the α-spetra are related with the it interleaver and the trellis struture of the onvolutional ode, and are independent of the preoding matrix. Note that the α-spetra an e derived y a similar approah to BICB in the ase of flat fading IO hannels presented in [11], or y omputer searh. Based on the α-spetra for a ertain omination of the onvolutional ode and the it interleaver, the seletion of preoded suhannels should e properly designed in order to satisfy the ondition of ρ l,1 0, l for all error events. Example: Consider the 4-state /2 onvolutional ode with generator polynomials (5, 7) in otal representation, in a suarrier group of BICB-OFD-G without preoding with parameters N t = N r = = L =2and =64.Two types of spatial interleavers are onsidered to demonstrate the way to selet preoded suhannels for eah set. The 1st, 2nd, 3rd, and 4th suhannels are symolially represented as a,, and and d, respetively. The spatial interleaver used in T 1 is a simple it-y-it rotating swith on four suhannels. For T 2, the spatial interleaver is simply rotated 6-its-y-6- its on four suhannels. In the following transfer funtions, eah term represents an α-spetrum, and the exponents of a,,, andd of a term indiate its orresponding values of the α-spetrum. T 1 = Z 5 (a 2 2 d + 2 d 2 )+ Z 6 (a 2 2 d 2 +2a 2 2 d d 2 )+ Z 7 (a a d + a 2 2 d 3 +2a 2 2 d 2 + a 2 2 d d 2 )+ Z 8 (a a d +4a d 2 + a 2 2 d 4 + 4a 2 2 d 3 + a 2 4 d d 2 )+. (31) T 2 = Z 5 (a 5 + a a 3 d 2 + a a 2 d d d 3 + d 5 )+ Z 6 (a a 4 d 2 +3a a 3 d 3 + a a a 2 2 d 2 + a 2 2 d 2 + a 2 d d d d d 4 )+ Z 7 (2a a 4 d 3 +2a a a 3 2 d 2 + a a 3 3 d +2a 3 2 d 2 +2a 3 d 4 +2a a 2 3 d 2 +2a a 2 2 d 3 +3a 2 3 d 2 + 2a 2 2 d 3 + a 3 d d d d d d d 4 )+. (32) Without preoding, to satisfy the full diversity ondition α l,1 0, l of all error events, eah term in the transfer funtion should inlude oth a and. ForT 1,theα-spetra A =[01;22]and A =[22;02]without full diversity dominate the performane. ine eah term of the transfer funtion inludes at least a or for eah term, a two-dimensional Θ preoding a and an satisfy the full diversity ondition ρ l,1 0, l of all error events, for whih the seletion of P = L =2and N p = L/P is the minimum effetive hoie derived in etion V-B and etion V-C. On the other hand, for T 2, the α-spetra A = [05;00] and A =[00;05]without full diversity dominate the performane. ine the transfer funtion also inludes α-spetra A = [5 0; 0 0] and A =[00;50], a four-dimensional Θ preoding all suhannels is required to provide full diversity, for whih the seletion of P = L =4and N p = L/P is not the minimum effetive hoie derived in etion V-B and etion V-C. The aforementioned example shows that the minimum effetive seletion of P and N p may not e effetive when the it interleaver is not properly designed. As a result, the preoded suhannels and the it interleaver should e jointly designed to provide full diversity. In etion V-B and etion V-C, the minimum effetive seletion of P and N p is provided as a neessary full diversity ondition. In the following, the minimum effetive seletion of P and N p is proved to e suffiient to provide full diversity with the joint design of preoded suhannels and the it interleaver. Proof: Consider the rate of the onvolutional ode = k /n where k and n are positive integers with k <n, whih implies that eah k ranhes in the trellis of onvolutional oded generates n oded its. If the spatial demultiplexer is not a random swith for the whole paket, the period of the spatial de-multiplexer is an integer multiple of the Least Common ultiple (LC) of n and L. Note that a period of the interleaver is restrited to an integer multiple of trellis ranhes. Define Q = LC(n,L) as the numer of oded its for a minimum period, whih is onsidered elow. ine eah suhannel needs to e evenly employed for a period, Q/(L) oded its are assigned on eah suhannel. Therefore, QP/(L) oded its are transmitted through one preoded suhannel set, whih offer the same effet on the diversity. Note that the trellis struture of onvolutional ode an e designed suh that the oded its generated from the first ranh splitting from the zero state are all errored its of an error event. Consequently, to guarantee ρ l,1 0, l of all error events, it is suffiient to onsider only the first

9 2440 IEEE TRANACTION ON COUNICATION, VOL. 61, NO. 6, JUNE 2013 ranhes that split from the zero state in one period eause of the repetition property of onvolutional ode. In other words, if at least one oded it for eah preoded suhannel set is assigned for eah ranh, full diversity is ahieved. Note that there are Q ranhes in a minimum period. Beause P L, thenqp/(l) Q. As a result, all ranhes in a minimum period an e assigned at least one oded it transmitted through eah preoded suhannel set, whih guarantees full diversity. This onludes the proof. E. Complexity With preoding, BICB-OFD-G without the full diversity restrition of L 1 an ahieve full diversity with the trade-off of an inreased deoding omplexity. Assume that square QA with onstellation size N m is employed. peifially, the omplexity of L metri alulation for (7) depends on only one of the real and imaginary parts orresponding to the oded it [40], [41]. If quantization is applied, the omplexity is proportional to 1, denoted y O(1). With preoding matries introdued in [13], the worst-ase omplexity of L metri alulation for the preoded its in (8) is O(Nm P 1 ) y using a real-valued phere Deoding (D) ased on the real lattie representation in [42], [43], plus quantization of the last two layers. ine the omplexity for L metri alulation of the preoded part dominates the overall omplexity, the L deoding omplexity of preoded BICB-OFD-G is onsidered as O(Nm P 1 ) for the worst ase. In [18] [20], PTBCs, whih have the properties of full rate, full diversity, uniform average transmitted energy per antenna, good shaping of the onstellation, and nonvanishing onstant minimum determinant for inreasing spetral effiieny whih offers high oding gain, have een onsidered as an alternative sheme to replae the onstellation preoding tehnique for oth unoded and oded VD eamforming with onstellation preoding in the ase of flat fading IO hannels. By doing so, the deoding omplexity in dimensions 2 and 4 an e redued while the performane is almost the same. The reason of the omplexity redution is that, due to the speial property of the generation matries in dimensions 2 and 4, the real and imaginary parts of the reeived signal an e separated, and only the part orresponding to the oded it is required to alulate one it metri for Viteri deoder. For BICB-OFD-G, PTBCs an also e applied. As a result, the worst-ase deoding omplexity of L metri alulation for the preoded its in (8) is O(Nm 0.5 ) for P = 2 and O(Nm 1.5 ) for P =4. Therefore, the L deoding omplexity of preoded BICB-OFD-G is onsidered as O(Nm 0.5) for P =2and O(Nm 1.5 ) for P =4in the worst ase. Tale I summarizes the worst-ase L deoding omplexity of BICB-OFD-G with different dimensions of preoding matries when square N m -QA is employed. Note that PTBCs are only availale in dimensions 2, 3, 4, and6, and dimensions 3 and 6 have no omplexity advantage and do not employ QA. Tale I shows that the omplexity of P =4 an e lower than P =3y employing PTBC. As a result, if the minimum effetive dimension of the preoding matries TABLE I WORT-CAE L DECODING COPLEXITY OF PRECODED BICB-OFD-G. [13] [18] [20] P =2 O(N m ) O(Nm 0.5 ) P =3 O(Nm) 2 N/A P =4 O(N 3 m) O(N 1.5 m ) P 5 O(N P 1 m ) N/A is P =3,thenP =4and its orresponding N p should e applied if they are a valid seletion. In the sequel, PTBCs are inorporated into a design algorithm, and in etion VI, simulation results are provided with systems employing PTBCs. F. Full-Diversity Preoding Design ummary Based on the disussion of the previous susetions, a suffiient method of the full-diversity preoding design for BICB-OFD-G with L > 1 is summarized as the following steps. 1) Calulate P = L. etflag = 0. 2) If P = 3 and flag = 0,setP = 4.IfP = 3 and flag = 1, setp =5.Otherwise,goto3). 3) Calulate N p = L/P. 4) Calulate N p P.IfN p P > L and P =3,goto2). If N p P > L,andP =4,setP =3and flag = 1, then go to 3). If N p P>Land P 3and P 4,set P = P +1 andgoto2).otherwise,goto5). 5) If P =2or P =4, PTBCs are applied as in [18] [20]. Otherwise, onstellation preoding is applied with preoding matries introdued in [13]. 6) elet N p P preoded suhannels whih inlude all the L first suhannels of all suarriers. 7) Design a it interleaver pattern of Q = LC(n,L) oded its for a period y assigning one preoded suhannel from eah set to eah ranh. G. Disussion In this paper, the dimension P of Θ for eah preoded suhannel set is assumed to e the same, whih is atually not neessary. Applying preoded suhannel sets with different numer of suhannels may ahieve lower deoding omplexity eause the omplexity inreases exponentially as the dimension inreases, as shown in Tale I. Example: Consider the parameters N t = N r =2, L =7, = 1,and = 1/3, thenp = 7 and N p = 1 is the minimum effetive seletion if different dimensions of the preoded suhannel sets are not onsidered. On the other hand, if different dimensions of the preoded suhannel sets are onsidered, two sets with P 1 = 3 and P 2 = 4 an also provide full diversity, whih an ahieve lower deoding omplexity. In this paper, the numer of employed suhannels y VD for eah suarrier is assumed to e the same, whih is. However, they ould e different in pratie. In that ase, the

10 LI and AYANOGLU: FULL-DIVERITY PRECODING DEIGN OF BIT-INTERLEAVED CODED ULTIPLE BEAFORING WITH ORTHOGONAL full diversity ondition for one suarrier group of BICB- L OFD-G without preoding is l l 1 where l denotes the numer of employed suhannels y VD for the lth suarrier of the group. With preoding, the minimum L effetive seletion of P l l and N p = L/P an e derived applying the same method as summarized in etion V-B and etion V-C respetively for BICB-OFD- L G with l l > 1. In fat, if L l l < N p P min{n t,n r }L, instead of retrying different seletions of P and N p with higher deoding omplexity, the urrent seletion an eome valid y inreasing the numer of employed suhannels at eah suarrier so that L l l = N p P.Insuha way, the minimum deoding omplexity an e ahieved with inreased numer of employed parallel suhannels. As shown in Tale I, the worst-ase L deoding omplexity of preoded BICB-OFD-G is O(Nm 0.5 ), O(Nm 1.5 ), and O(Nm) 2 for dimensions 2, 4, and3, respetively. Note that the worst-ase omplexity of O(Nm 4 ) for P =5has a signifiant inrease. As a result, if P 5, instead of applying the preoding diretly, employing a onvolutional ode with smaller rate so that the minimum effetive dimension P 4 may e a more reasonale option. In [14] [16], onstellation preoding is applied to BICB without the full diversity restrition 1 for flat fading IO hannels. It was presented that partial preoding ould ahieve oth full diversity and full multiplexing with the properly designed omination of the onvolutional ode, the it interleaver, and the onstellation preoder. However, the general full-diversity preoding design was not provided. ine BICB of flat fading IO hannels an e onsidered as a suarrier of BICB-OFD-G in the frequeny domain with L, the full-diversity preoding design proposed in this paper an e applied to BICB with >1for flat fading IO hannels. VI. IULATION REULT To verify the diversity analysis and the full-diversity preoding design, 2 2 =64BICB-OFD-G with L =2 and L =4using 4-QA, as well as 4 4 =64BICB- OFD-G with L = 2 using 4-QA, are onsidered for simulations. The numer of employed suhannels for eah suarrier and the dimension of preoding matrix for eah preoded suhannel set are assumed to e the same, respetively. The generator polynomials in otal for the onvolutional odes with /4 and /2 are (5, 7, 7, 7) and (5, 7) respetively, and the odes with =2/3 and =4/5 are puntured from the /2 ode [32]. The length of CP is L p 6. Eah OFD symol has 4μs duration, of whih 0.8μs is CP. Equal power hannel taps are onsidered. The it interleaver employs simple rotation for BICB-OFD-G with non-effetive preoding seletions and without preoding. For BICB-OFD-G with effetive preoding seletions, the proposed full-diversity preoding design in this paper is employed. In the figures, NP indiates non-preoded. Note that unequal power hannel taps are not onsidered eause they do not affet the maximum ahievale diversity as disussed in [31]. Also note that simulations of 2 2 BICB-OFD with L =2and L =4as well as 4 4 BICB-OFD with L =2 are shown in this setion eause the diversity values ould Referene, D=8 2x2, L=2, /2, 2x2, L=2, /4, =2 Referene, D6 2x2, L=4, /4, Referene, D=32 4x4, L=2, /2, NR in db Fig. 5. vs. NR for non-preoded BICB-OFD-G ahieving full diversity. e investigated expliitly through figures. In our simulations, we will show that the maximum diversity of N r N t L [3] is ahieved. A. Non-Preoded BICB-OFD-G Fig. 5 shows the Bit Error Rate () performane of non-preoded BICB-OFD-G ahieving full diversity for different system parameters. ystem onfigurations of 2 2 L =2 /2, 2 2 L =2 /4 =2, 2 2 L =4 /4,and4 4 L =2 /2 are onsidered. Aording to [31], they all ahieve their orresponding full diversity orders eause they all satisfy the full diversity ondition of L 1 for non-preoded BICB-OFD-G. The full diversity orders of 2 2 L =2, 2 2 L =4,and4 4 L =2systems are 8, 16, and32 respetively. In the figure, the theoretial proaility of error for aximum Ratio Comining (RC) diversity systems with N r = D {8, 16, 32} reeive antennas using Binary Phase- hift Keying (BPK) over Rayleigh flat fading hannels are drawn as referenes to the ases of diversity orders D [21]. Note that Fig. 5 provides full diversity referenes for this setion. ine this paper fouses on full diversity, referenes for the non-full diversity orders are not offered in figures. Note that the non-full diversity orders in this setion are derived y the results of (22) and (23) as disussed in etion IV. B. 2 2 L =2Preoded BICB-OFD-G Fig. 6 shows the performane of 2 2 L =2 = 64 BICB-OFD-G with and without preoding for different.for /2, full diversity of 8 an e ahieved even if preoding is not applied as shown in Fig 5, eause L 1 [31]. On the other hand, in the ase of =2/3, the diversity order is 4 instead of full diversity sine L > 1. However, with the full-diversity preoding design proposed in this paper of P =2and N p,full diversity of 8 is suessfully reovered. imilarly, Fig. 7 shows the performane of 2 2 L =2 =64 =2BICB-OFD-G with and without preoding for different.for /4, full diversity of 8 an e ahieved even without preoding as shown in Fig

11 2442 IEEE TRANACTION ON COUNICATION, VOL. 61, NO. 6, JUNE 2013 /2, NP =2/3, P=2, N p =2/3, NP P=4, N p P=3, N p P=2, N p =2 NP NR in db NR in db Fig. 6. vs. NR for 2 2 L =2 =64 BICB-OFD-G with and without preoding. Fig. 8. vs. NR for 2 2 L =2 =64 =2 =2/3 BICB- OFD-G with different preoding seletions and without preoding. /4, NP /2, P=2, N p /2, NP =2/3, P=4, N p =2/3, NP =4/5, P=4, N p =4/5, NP P=4, N p P=3, N p P=2, N =2 p NP NR in db Fig. 7. vs. NR for 2 2 L =2 =64 =2BICB-OFD-G with and without preoding NR in db Fig. 9. vs. NR for 2 2 L =2 =64 =2 =4/5 BICB- OFD-G with different preoding seletions and without preoding. 5, sine L 1 [31]. On the other hand, in the ases of /2, =2/3, and =4/5, the diversity orders are 5, 2, and1 respetively, and the full diversity degradations result from L > 1. Nevertheless, full diversity of 8 an e restored y employing the full-diversity preoding design proposed in this paper. The orresponding seletions are P =2N p, P =4N p,andp =4N p for /2, =2/3, and =4/5, respetively. Fig. 8 shows the performane of 2 2 L = 2 =64 =2 =2/3 BICB-OFD-G with preoding for different seletions of P and N p and without preoding. ine L > 1, full diversity annot e ahieved without preoding, and the diversity is 2. On the other hand, oth P =4N p shown in Fig. 7 and P =3N p are effetive preoding seletions to provide the full diversity of 8, while P = 2 N p = 2 with diversity of 5 annot offer full diversity eause L =3. As disussed in etion V-E, P =4has lower worst-ase L deoding omplexity of O(Nm 1.5 ) than P =3of O(Nm). 2 However,P =3ahieves slightly etter performane, whih is less than 0.5dB, than P =4in this ase. Note that in the ase of P =2N p =2,in order to ahieve relatively high diversity, the first suhannel of the first suarrier is preoded with the seond suhannel of the seond suarrier, while the seond suhannel of the first suarrier is preoded with the first suhannel of the seond suarrier. imilarly, Fig. 9 shows the performane of 2 2 L =2 =64 =2 =4/5 BICB-OFD-G with preoding for different seletions of P and N p and without preoding. Due to the fat that L > 1, full diversity annot e provided without preoding and the diversity is 1. Onthe other hand, P =4N p shown in Fig. 7 is an effetive preoding seletion to restore the full diversity of 8, while P =3N p and P =2N p =2with diversity orders of 4 and 5 respetively annot reover full diversity sine L =4. Note that in order to ahieve relatively high diversity, in the ase of P =3N p, the first suhannel of the seond suarrier is non-preoded, while for P =2 N p =2, the first suhannel of the first suarrier is preoded with the seond suhannel of the seond suarrier, and the seond suhannel of the first suarrier is preoded with the first suhannel of the seond suarrier. C. 2 2 L =4Preoded BICB-OFD-G Fig. 10 shows the performane of 2 2 L =4 = 64 BICB-OFD-G with and without preoding for different.for /4, full diversity of 16 an e provided even without preoding as shown in Fig 5, eause

12 LI and AYANOGLU: FULL-DIVERITY PRECODING DEIGN OF BIT-INTERLEAVED CODED ULTIPLE BEAFORING WITH ORTHOGONAL /4, NP /2, P=2, N p =2 R /2, NP =2/3, P=4, N p R =2/3, NP =4/5, P=4, N p =4/5, NP P=4, N p P=3, N p P=2, N p =2 NP NR in db Fig. 10. vs. NR for 2 2 L =4 =64 BICB-OFD-G with and without preoding NR in db Fig. 12. vs. NR for 2 2 L =4 =64 =4/5BICB- OFD-G with different preoding seletions and without preoding. P=4, N p P=3, N p P=2, N p =2 NP /2, NP =2/3, P=2, N p =2/3, NP NR in db Fig. 11. vs. NR for 2 2 L =4 =64 =2/3 BICB- OFD-G with different preoding seletions and without preoding. L 1 [31]. On the other hand, in the ases of /2, = 2/3, and = 4/5, the diversity orders are 12, 8, and 4 respetively, and full diversity annot e ahieved sine L > 1. However, full diversity of 16 an e reovered y applying the full-diversity preoding design proposed in this paper. The orresponding seletions are P =2N p =2, P =4N p,andp =4N p for /2, =2/3, and =4/5, respetively. Fig. 11 shows the performane of 2 2 L = 4 =64 =2/3 BICB-OFD-G with preoding for different seletions of P and N p and without preoding. Beause L > 1, full diversity annot e ahieved without preoding, and the diversity is 8. On the other hand, P =4 N p shown in Fig. 10 is an effetive preoding seletion to provide the full diversity of 16, while P = 3 N p = 1 and P = 2 N p = 2 with diversity orders of 12 and 8 respetively annot offer full diversity eause L/P P >L and L =3respetively. Note that in order to ahieve relatively high diversity, in the ase of P =3N p,the suhannel of the seond suarrier is non-preoded, while for P =2N p =2, the suhannel of the first suarrier is preoded with the suhannel of the third suarrier, and the suhannel of the seond suarrier is preoded with the suhannel of the fourth suarrier. imilarly, Fig. 12 shows the performane of NR in db Fig. 13. vs. NR for 4 4 L =2 =64 BICB-OFD-G with and without preoding. L =4 =64 =4/5 BICB-OFD-G with preoding for different seletions of P and N p and without preoding. ine L > 1, full diversity annot e offered without preoding and the diversity is 4. On the other hand, P =4N p showninfig.10isaneffetivepreoding seletion to reover the full diversity of 16 while P =3N p = 1 and P =2N p =2with diversity orders 4 and 8 respetively annot restore full diversity eause L =4. Note that in order to ahieve relatively high diversity, in the ase of P =3N p, the suhannel of the seond suarrier is non-preoded, while for P =2N p =2, the suhannel of the first suarrier is preoded with the suhannel of the third suarrier, and the suhannel of the seond suarrier is preoded with the suhannel of the fourth suarrier. D. 4 4 L =2Preoded BICB-OFD-G Fig. 13 shows the performane of 4 4 L =2 =64 BICB-OFD-G with and without preoding for different.for /2, full diversity order of 32 an e ahieved even if preoding is not applied as shown in Fig 5, sine L 1 [31]. On the other hand, in the ase of = 2/3, the diversity order is 16 instead of full diversity eause L > 1. Nevertheless, full diversity of 32 is reovered with

13 2444 IEEE TRANACTION ON COUNICATION, VOL. 61, NO. 6, JUNE 2013 the full-diversity preoding design proposed in this paper of P =2and N p. E. Disussion For a versus Nurve, the negative diversity value is the slope while the oding gain [3] and array gain [44] reflet the position. As a result, although larger diversity provides etter performane, est oding gain and array gain with full diversity an provide optimal performane. In this paper, the fous is drawn on ahieving full diversity in terms of performane for BICB-OFD-G without speifi onentration on oding gain and array gain. In addition, equal power for eah transmit antenna is assumed in this paper. Unequal power distriution with orresponding preoding design to ahieve optimal performane is onsidered as future works. In the figures of this setion, some urves of BICB- OFD-G ahieve full diversity ut with different gains. The performane differenes result from different reasons. In Fig. 5, oth /2 and /4 =2for 2 2 L =2 non-preoded BICB-OFD-G ahieve full diversity of 8 with the same it data rate. The performane disadvantage of /4 =2isaused mainly y the usage of suhannels without only the largest eigenvalues for eah suarrier. Note that the referene urves are for flat fading IO hannels and the it data rates are muh less than BICB-OFD- G. In Fig. 6, Fig. 7, Fig. 10, and Fig. 13, the performane differenes of full diversity urves result from the different employed onvolutional odes, as a onvolutional ode with higher rate provides worse performane [10] ut greater it data rate in general. oreover, in Fig. 8, oth P =4N p and P = 3 N p = 1 are effetive preoding seletions to ahieve the full diversity of 8 for 2 2 L =2 =64 =2 =2/3BICB-OFD-G. The performane differene is aused y the different oding gains of the employed preoding tehniques. As presented in etion II, BICB-OFD-G requires the knowledge of CI at the Transmitter (CIT), whih is usually partial and imperfet in pratie due to the andwidth limitation and the hannel estimation errors, respetively. Reently, limited CIT feedak tehniques have een introdued to ahieve a performane lose to the perfet CIT ase for oth unoded and oded VD-ased eamforming systems [45] [48]. For these tehniques, a odeook of preoding matries is known oth at the transmitter and reeiver. The reeiver selets the preoding matrix that satisfies a desired riterion, and only the index of the preoding matrix is sent ak to the transmitter. In pratie, similar tehniques an e applied to BICB-OFD-G. As disussed in etion IV, the PEPs with the worst diversity order dominate the overall performane. For BICB- OFD-G without the ondition L 1, the PEPs without full diversity an e improved y applying the fulldiversity preoding design proposed in this paper so that full diversity is ahieved. If the diversity without full-diversity preoding is relatively small ompared to full diversity, sustantial improvement an e ahieved y the full-diversity preoding design, e.g., the ases of =2/3and =4/5in Fig. 7. On the other hand, if the diversity without full-diversity preoding is lose to full diversity, the advantage of the fulldiversity preoding design may start at the NR providing very low. In that ase, its value depends on the s of different appliations. Take the ase of /2 in Fig. 10 as an example, if the requirement is, then preoding may not e neessary. However, if the requirement is 10 9, preoding may e worthwhile. With the full-diversity preoding design proposed in this paper, more hoies of BICB-OFD-G with different trade-offs among performane, transmission rate, and deoding omplexity are provided. Take Fig. 7 as an example, without preoding, the ase of /4 ahieves full diversity. However, inreasing the transmission rate y employing onvolutional odes with higher rates /2, =2/3, and =4/5 results in the loss in performane. By applying the proposed full-diversity preoding design, full diversity an e reovered whih trades off with higher deoding omplexity. oreover, higher rates of =2/3 and =4/5 ause more inreased deoding omplexity than the ase of /2. The most proper hoie varies whih depends on the different requirements on performane, throughput, and deoding omplexity. VII. CONCLUION In this paper, a full-diversity preoding design is developed for BICB-OFD-G without the full diversity restrition of L 1. The design provides a suffiient method to guarantee full diversity while minimizing the inreased deoding omplexity aused y preoding. With this method, more hoies are offered with different trade-offs among performane, transmission rate, and deoding omplexity. As a result, BICB-OFD-G eomes a more flexile roadand wireless ommuniation tehnique. 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Commun., vol. 57, no. 5, pp , ay Boyu Li ( 10-13) reeived the B.. degree from Beijing Institute of Tehnology, Beijing, China, in 2007, and the.. and Ph.D. degrees from University of California, Irvine, Irvine, CA, in 2009 and 2012, respetively, all in eletrial engineering. From 2009 to 2012, he was a Graduate tudent Researher with the Center for Pervasive Communiations and Computing, University of California, Irvine. He is urrently an Assistant peialist with the Department of Eletrial Engineering and Computer iene, University of California, Irvine. His researh experiene inludes full spatial multiplexing and full diversity ultiple-input ultiple-output wireless ommuniation systems with low omplexity implementation, as well as single- and multi-mirophone derevereration. Ender Ayanoglu ( F 98) reeived the B.. degree from the iddle East Tehnial University, Ankara, Turkey, in 1980, and the.. and Ph.D. degrees from tanford University, tanford, CA, in 1982 and 1986, respetively, all in eletrial engineering. He was with the Communiations ystems Researh Laoratory, AT&T Bell Laoratories, Holmdel, NJ (Bell Las, Luent Tehnologies after 1996) until 1999 and was with Ciso ystems until ine 2002, he has een a Professor in the Department of Eletrial Engineering and Computer iene, University of California, Irvine, where he served as the Diretor of the Center for Pervasive Communiations and Computing and held the Conexant-Broadom Endowed Chair during Dr. Ayanoglu is the reipient of the IEEE Communiations oiety tephen O. Rie Prize Paper Award in 1995 and the IEEE Communiations oiety Best Tutorial Paper Award in ine 1993, he has een an Editor of the IEEE TRANACTION ON COUNICATION and served as its Editor-in-Chief from 2004 to From 1990 to 2002, he served on the Exeutive Committee of the IEEE Communiations oiety Communiation Theory Committee, and from 1999 to 2001, was its Chair.