L2 Orthogonal Space Time Code for Continuous Phase Modulation

L2 Orthogonal Space T ime Code for Continuous Phase Modulation Matthias Hesse, J ´ er ˆ ome Lebrun and Luc Den eire Abstract T o combine the high po wer ef ficiency of Continuous Phase Modulation (CPM) with either high spectral efficienc y or enhanced performance i n low S ignal to Noise conditions, some authors hav e proposed to introdu ce CPM in a MIMO frame, by using Space T ime Codes (STC). In this paper , we address the code design problem of Space T ime Block Codes combined with CPM and introduce a new design criterion based on L 2 orthogonality . This L 2 orthogonality condition, with t he help of simplifying assumption, leads, i n the 2x2 case, to a ne w family of codes. These codes generalize the W ang and Xia code, which was based on pointwise orthogonality . Simulations indicate that the new codes achiev e full di ve rsity and a slightly better coding gain. Moreov er , one of the codes can be interpreted as two antennas fed by two con v entional CPMs using the same data but with dif ferent alphabet sets. Inspection of these alphabet sets lead also to a simple e xplanation of the (small) spectrum broaden ing of Space T ime Coded CPM. I . I N T R O D U C T I O N Since the pio neer work of Alamo uti [1] and T arok h [2], Space Time Coding has been a fast growing field of research wher e numero us coding schemes have been intr oduced. Se veral years later Zhan g and Fitz [3], [4] were the first to apply the idea of STC to continuou s p hase modu lation (CPM) by constru cting trellis codes. In [5] Zaji ´ c and St ¨ uber d erived con ditions for partial respo nse STC-CPM to g et full diversity and optimal coding gain. A STC for nonco herent detectio n based on diag onal blocks was in troduced b y Silvester et al. [6]. The first orthogona l STC for CPM fo r full and partial response was de veloped by W ang and Xia [7], [8]. The scope of this p aper is also the design of an o rthogo nal STC for CPM. But unlike W ang-Xia ap rroach [8] which starts from a QAM orthog onal Space-T ime Code (e.g. Alamo uti’ s sch eme [1]) and m odify it to achieve continuou s phases for th e transmitted signals, we show here that a more gen eral L 2 condition is sufficient to ensure fast max imum likelihood d ecoding with full div ersity . In the considered system m odel (Fig .1), the data sequence d j is define d over the signal constellation set Ω d = {− M + 1 , − M + 3 , . . . , M − 3 , M − 1 } (1) for an alp habet with log 2 M bits. T o obtain th e struc ture f or a Space T ime Block Co de ( STBC) this seque nce is mapp ed to data matrices D ( i ) with elements d ( i ) mr , where m d enotes the tran smitting an tenna, r the time slot into a block and ( i ) a p arameter for p artial respon se CPM. Th e data matrices are then used to m odulate the sending matrix S ( t ) = " s 11 ( t ) s 12 ( t ) s 21 ( t ) s 22 ( t ) # . (2) Each eleme nt is defined for (2 l + r − 1) T ≤ t ≤ (2 l + r ) T as s mr ( t ) = r E s T e j 2 π φ mr ( t ) (3) The work of Matthi as Hesse is supporte d by the EU by a Marie-Curie Fell owshi p (EST -SIGNAL program : http://est-si gnal.i3s.unice .fr ) under contract No MEST -CT -2005-021175. The authors are with Lab . I3S, CNRS, Uni ver sity of Nic e, Sophia Antipolis, France; E-Mail: { hesse,lebrun,d eneire } @i3s.unice. fr L2 OR THOGON AL SP ACE TIME CODE FOR CONTINUOUS PHASE MODUL A TION 2 Fig. 1. Structure of a MIMO T x/Rx system where E s is the symbol e nergy and T the symbo l time. Th e phase φ mr ( t ) is defined in the conventional CPM manner [9 ] with an additional cor rection factor c mr ( t ) and is therewith giv en by φ mr ( t ) = θ m (2 l + r ) + h 2 l + r X i =2 l +1+ r − γ d ( i ) mr q ( t − ( i − 1 ) T ) + c mr ( t ) (4) where h = 2 m 0 /p with m 0 and p relativ e primes is called the modulation index. T he phase smoothing function q ( t ) has to be a continuo us fu nction with q ( t ) = 0 for t ≤ 0 and q ( t ) = 1 / 2 for t ≥ γ T . The mem ory length γ determines the length o f q ( t ) and affects the sp ectral comp actness. For large γ we obtain a compact spectrum but also a higher n umber of possible phase states which increases the decoding effort. For full response CPM, we have γ = 1 an d for partial response sy stems γ > 1 . The cho ice of the correctio n factor c mr ( t ) in Eq. (4) is along with the m apping of d j to D ( i ) , the key element in th e design of ou r codin g scheme. It will be d etailed in Section II. W e then define θ m (2 l + r ) in a most general way θ m (2 l + 3) = θ m (2 l + 2) + ξ (2 l + 2) = θ m (2 l + 1) + ξ (2 l + 1) + ξ (2 l + 2) . (5) The function ξ (2 l + r ) will b e fully d efined fr om the contribution c mr ( t ) to the pha se memory θ m (2 l + r ) . For conv entional CPM system, c mr ( t ) = 0 an d we h av e ξ (2 l + 1) = h 2 d 2 l +1 − γ . The channel coefficients α mn are a ssumed to be Rayleigh distributed and indep endent. Each coefficient α mn characterizes the fading between the m th transmit (Tx) anten na an d the n th receive (Rx) antenn a where n = 1 , 2 , . . . , L r . Further more, th e received signals y n ( t ) = α mn s mr ( t ) + n ( t ) (6) are c orrupted by a complex additive white Gaussian noise n ( t ) with variance 1 / 2 per dimension. At the receiver , the detectio n is done on each of the L r received signals separately . Th erefore, in general, each code block S ( t ) has to be detected by block . E .g. for a 2x2 block , estimatin g the symbols ˆ d j implies computatio nal co mplexity pr oportio nal to M 2 . Now , th is comp lexity can be re duced to 2 M by introd ucing an ortho gonality prop erty as well as simplifyin g assumptions on the code. Criteria for su ch STBC are given in Section II. In Section III, the criteria are used to constru ct OSTBC for CPM. I n Section IV we test th e designed code a nd compare it with the STC from W ang an d Xia [8]. Fin ally , some conclusion s are drawn in Section V. I I . D E S I G N C R I T E R I A The purpose of the design is to achiev e full di versity and a fast m aximum lik elihood d ecoding wh ile m aintaining the continuity of the signal phases. This section shows h ow th e ne ed to perform fast ML decod ing leads to the L 2 orthog onality condition as well as to simplifying assumptio ns, which can b e combined with the continuity co nditions. For con venience we only co nsider one Rx antenna and drop the index n in α mn . L2 OR THOGON AL SP ACE TIME CODE FOR CONTINUOUS PHASE MODUL A TION 3 A. F ast Maximum Likelihood Decoding Commonly , due to the trellis stru cture of CPM, the V iterbi algo rithm is used to perfor m the ML demo dulation . On block l each state in the trellis has M 2 incoming b ranches and M 2 outgoin g b ranches with a distance D l = (2 l +1) T Z 2 lT    y ( t ) − 2 X m =1 α m s m 1 ( t )    2 d t + (2 l +2) T Z (2 l +1) T    y ( t ) − 2 X m =1 α m s m 2 ( t )    2 d t. (7) The numb er of bran ches re sults from the blockwise decoding and the correlatio n b etween th e sent symbols s 1 r ( t ) and s 2 r ( t ) . A way to reduce the nu mber of branches is to structurally d ecorrelate th e sign als sent by the two transmitting ante nnas, i.e. to put to zero the inter-antenna cor relation α 2 α ∗ 1 (2 l +1) T Z 2 lT s 21 ( t ) s ∗ 11 ( t ) d t + α 1 α ∗ 2 (2 l +1) T Z 2 lT s 11 ( t ) s ∗ 21 ( t ) d t + α 2 α ∗ 1 (2 l +2) T Z (2 l +1) T s 22 ( t ) s ∗ 12 ( t ) d t + α 1 α ∗ 2 (2 l +2) T Z (2 l +1) T s 12 ( t ) s ∗ 22 ( t ) d t = 0 . (8) Pointwise orthog onality as defined in [8] is ther efore a su fficient condition but not nece ssary . A less restrictive L 2 orthog o- nality is also sufficient. From Eq . (8), the distance g iv en in Eq. (7) can then be simplified to D l = (2 l +1) T Z 2 lT f 11 ( t ) + f 21 ( t ) − | y ( t ) | 2 d t + (2 l +2) T Z (2 l +1) T f 12 ( t ) + f 22 ( t ) − | y ( t ) | 2 d t (9) with f mr ( t ) = | y ( t ) − α m s mr ( t ) | 2 . When ea ch s mr ( t ) depends only o n d 2 l +1 or d 2 l +2 the branches c an be split a nd calculated separately for d 2 l +1 and d 2 l +2 . The co mplexity of the ML decision is re duced to 2 M . The co mplexity for detecting two symbols is thus red uced from pM γ +1 to pM γ . The STC introd uced by W ang and Xia [8] d idn’t take f ull advantage of the orthog onal design since s mr ( t ) was depen ding on both d 2 l +1 and d 2 l +2 . The gain they obtain ed in [8] was then rely ing on other p roperties of CP M, e.g. some restrictio ns put on q ( t ) a nd p . These restrictions m ay also be applied to o ur desig n code, which w ould lead to ad ditional co mplexity reduction . Howev er, this is not in the scope of this paper and is b e the subject of another up coming paper . B. Orthogonality Conditio n In this section we show how L 2 orthog onality fo r CPM, i.e. k S ( t ) k 2 L 2 = R (2 l +2) T 2 lT S ( t ) S H ( t ) d t = 2 I , can be o btained. As such, the correlation between the two transmitting antennas per cod ing block is cancelled if (2 l +2) T Z 2 lT s 1 r ( t ) s ∗ 2 r ( t ) d t = (2 l +1) T Z 2 lT s 11 ( t ) s ∗ 21 ( t ) d t + (2 l +2) T Z (2 l +1) T s 12 ( t ) s ∗ 22 ( t ) d t = 0 . (10) Replacing s mr ( t ) by the corre sponding CPM symbo ls from Eq. (4), we get (2 l +1) T Z 2 lT exp n j 2 π ˆ θ 1 (2 l + 1) + h 2 l +1 X i =2 l +2 − γ d ( i ) 1 , 1 q ( t − ( i − 1) T ) + c 1 , 1 ( t ) − θ 2 (2 l + 1) − h 2 l +2 X i =2 l +3 − γ d ( i ) 2 , 1 q ( t − ( i − 1) T ) − c 2 , 1 ( t ) ˜ o d t + (2 l +2) T Z (2 l +1) T exp n j 2 π ˆ θ 1 (2 l + 2) + h 2 l +2 X i =2 l +3 − γ d ( i ) 1 , 2 q ( t − ( i − 1) T ) + c 1 , 2 ( t ) − θ 2 (2 l + 2) − h 2 l +1 X i =2 l +2 − γ d ( i +1) 2 , 2 q ( t − iT ) − c 2 , 2 ( t ) ˜ d t o = 0 . (11) L2 OR THOGON AL SP ACE TIME CODE FOR CONTINUOUS PHASE MODUL A TION 4 The phase m emory θ m (2 l + r ) is independ ent of time and has not to be considered for integration . Using Eq. (5) to replace phase mem ory θ m (2 l + 2) of the second time slot, we obtain (2 l +1) T Z 2 lT exp n j 2 π ˆ h 2 l +1 X i =2 l +2 − γ d ( i ) 1 , 1 q ( t − ( i − 1) T ) + c 1 , 1 ( t ) − h 2 l +1 X i =2 l +2 − γ d ( i ) 2 , 1 q ( t − ( i − 1) T ) − c 2 , 1 ( t ) ˜ o d t + exp n j 2 π ˆ ξ 1 (2 l + 1) − ξ 2 (2 l + 1) ˜ o · (2 l +1) T Z 2 lT exp n j 2 π ˆ h 2 l +1 X i =2 l +2 − γ d ( i +1) 1 , 2 q ( t − ( i − 1) T ) + c 1 , 2 ( t + T ) − h 2 l +1 X i =2 l +2 − γ d ( i +1) 2 , 2 q ( t − ( i − 1) T ) − c 2 , 2 ( t + T ) ˜ o d t = 0 . (12) C. Simplifying assump tions T o simplify this e xpression , we factor Eq. (12) in to a time in depend ent an d a time depe ndent part. For merging the two integrals to one time depend ent p art, we h av e to map d ( i ) m 2 to d ( i ) m 1 and c mr ( t ) to a different c m ′ r ′ ( t ) . Con sequently , f or the data symbols d ( i ) mr there exist th ree po ssible ways of mapping: • cr osswise mapp ing with d ( i ) 1 , 1 = d ( i ) 2 , 2 and d ( i ) 1 , 2 = d ( i ) 2 , 1 ; • r epetitive m apping with d ( i ) 1 , 1 = d ( i ) 1 , 2 and d ( i ) 2 , 1 = d ( i ) 2 , 2 ; • parallel mapp ing with d ( i ) 1 , 1 = d ( i ) 2 , 1 and d ( i ) 1 , 2 = d ( i ) 2 , 2 . The same approach can be applied to c mr ( t ) : • cr osswise mapp ing with c 11 ( t ) = − c 22 ( t − T ) and c 12 ( t ) = − c 21 ( t − T ) ; • r epetitive m apping with c 11 ( t ) = c 12 ( t − T ) and c 21 ( t ) = c 22 ( t − T ) ; • parallel mapp ing with c 11 ( t ) = c 21 ( t ) a nd c 12 ( t ) = c 22 ( t ) . For each combination of map pings, Eq. (12) is now the product of tw o facto rs, one c ontaining the integral and the other a time in depend ent pa rt. T o fu lfill Eq. (12) it is suf ficient if one factor is zero, namely 1 + e j 2 π [ ξ 1 (2 l +1) − ξ 2 (2 l +1)] = 0 , i.e. if k + 1 2 = ξ 1 (2 l + 1) − ξ 2 (2 l + 1) (13) with k ∈ N . W e thus g et a very sim ple cond ition which o nly dep ends on ξ m (2 l + 1) . D. Continuity of Phase In this section we determine the fu nctions ξ m (2 l + 1) to ensure the ph ase continu ity . Precisely , the phase of the CPM sym bols has to be equ al at all intersections of symbols. For an ar bitrary block l , it means that φ m 1 ((2 l + 1 ) T ) = φ m 2 ((2 l + 1 ) T ) . Using Eq. (4), it results in ξ m (2 l + 1) = h 2 l +1 X i =2 l +2 − γ d ( i ) m, 1 q ((2 l + 2 − i ) T ) + c m, 1 ((2 l + 1 ) T ) − h 2 l +2 X i =2 l +3 − γ d ( i ) m 2 q ((2 l + 2 − i ) T ) − c m 2 ((2 l + 1 ) T ) . (14) For the second intersection at (2 l + 2) T , since φ m 2 ((2 l + 2 ) T ) = φ m 1 ((2 l + 2 ) T ) , we get ξ m (2 l + 2) = h 2 l +2 X i =2 l +3 − γ d ( i ) m 2 q ((2 l + 3 − i ) T ) + c m 2 ((2 l + 2 ) T ) − h 2( l +1)+1 X i =2( l +1)+2 − γ d ( i ) m, 1 q ((2 l + 3 − i ) T ) − c m, 1 ((2 l + 2 ) T )] . (15) Now , by choosing one of the mapp ings d etailed in Section III, these equ ations can be greatly simplified. Hence, we have all the tools to construct our cod e. I I I . O RT H O G O N A L S PAC E T I M E C O D E S In this section we will have a closer look at two codes constructed under the afore-mentio ned con ditions. L2 OR THOGON AL SP ACE TIME CODE FOR CONTINUOUS PHASE MODUL A TION 5 A. Existing Code As a first example, we will give an alternative con struction of the co de given by W ang and Xia in [8]. In deed, the pointwise orthog onality co ndition used by W ang and Xia is a sp ecial case of the L 2 orthog onality cond ition, hence, the ir ST -code can be o btained within our framework. For the first anten na W ang a nd Xia use a con ventional CPM with d ( i ) 1 r = d i for i = 2 l + r + 1 − γ , 2 l + r + 2 − γ , . . . , 2 l + r and c 1 r ( t ) = 0 . The symbols of the secon d antenn a are mapped cr osswis e to th e first d ( i ) 21 = − d i +1 for i = 2 l + 2 − γ , 2 l + 3 − γ , . . . , 2 l + 1 and d ( i − 1) 22 = − d i − 1 for i = 2 l + 3 − γ , 2 l + 4 − γ , . . . , 2 l + 2 . Using th is cr oss mapp ing makes it d iffi cult to compute ξ m (2 l + 1 ) since th e CPM typical order of the data symbols is mixed. W ang and Xia circu mvent this by introducin g another cor rection factor f or the seco nd an tenna c 2 r ( t ) = γ − 1 X i =0 ( h ( d 2 l +1 − i + d 2 l +2 − i ) + 1) q 0 ( t − (2 l + r − 1 − i ) T ) (16) By first computing ξ m (2 l + 1) with Eq. (17) a nd then Eq. (13), we get the L 2 orthog onality of the W ang-Xia-STC. B. P arallel Code T o get a simpler co rrection facto r as in [ 8], we d esigned a new c ode b ased on the p arallel structure which permits to maintain th e conventional CPM m apping for b oth an tennas. Hence we choose the follo wing mapping : d ( i ) m 1 = d ( i − 1) m 2 = d i for i = 2 l + r + 1 − γ , 2 l + r + 2 − γ , . . . , 2 l + r . Then, Eq. (14) and (1 5) can be simplified into ξ m (2 l + 1) = h 2 d 2 l +2 − γ + c m 1 ((2 l + 1 ) T ) − c m 2 ((2 l + 1 ) T ) (17) ξ m (2 l + 2) = h 2 d 2 l +3 − γ + c m 2 ((2 l + 2 ) T ) − c m 1 ((2 l + 2 ) T ) . (18) W ith this simplified function s, the or thogon ality conditio n only depen ds on the start and end values of c mr ( t ) , i.e. k + 1 2 = c 11 ((2 l + 1 ) T ) − c 12 ((2 l + 1 ) T ) − c 21 ((2 l + 1 ) T ) + c 22 ((2 l + 1 ) T ) . (19) T o merge the two integrals in Eq. (12) not only the mapping of d ( i ) mr is necessary but also an equ ality b etween different c mr ( t ) . From the three possible m appings, we choose the r epeat mapp ing because of the possibility to set c mr ( t ) to zero fo r one antenna. Hen ce we are able to send a conventional CPM signal o n one antenna an d a mo dified o ne on the second. Using Eq. (19) and the equa lities for the mapping, we can formulate the following cond ition k + 1 2 = c 12 (2 l T ) − c 12 ((2 l + 1 ) T ) − c 22 (2 l T ) + c 22 ((2 l + 1 ) T ) . (20) W ith c 11 ( t ) = c 12 ( t ) = 0 , we can take for c 21 ( t ) = c 22 ( t ) any continu ous function which is zero at t = 0 and 1 / 2 at t = T . Another po ssibility is to choo se th e co rrection factor of the second anten na with a structure similar to CPM modulation, i.e. c 2 r ( t ) = 2 l +1 X i =2 l +1 − γ q ( t − ( i − 1 ) T ) (21) for (2 l + r − 1) T ≤ t ≤ (2 l + r ) T . W ith this approac h, the corre ction factors can be in cluded in a classical CPM modulation with co nstant offset of 1 /h . Th is offset m ay also be expressed as a modified alph abet for th e secon d antenna Ω d 2 = {− M + 1 + 1 h , − M + 3 + 1 h , . . . , M − 3 + 1 h , M − 1 + 1 h } (22) Consequently , th is L 2 -ortho gonal design may be seen a s two co n ventional CPM designs with different alphabet sets Ω d and Ω d 2 for each antenna. Howe ver , in th is method, the constant offset to the ph ase may cau se a shift in frequency . But as shown by ou r simulation s in th e next section, this shift is quite moder ate. I V . S I M U L A T I O N S In this section we verify the proposed algo rithm by simulations. Therefore a STC-2REC-CPM-sender w ith two transmitting antennas has b een implemented in MA TLAB. For the signal of the first antenna we use conventional Gray-cod ed CPM with L2 OR THOGON AL SP ACE TIME CODE FOR CONTINUOUS PHASE MODUL A TION 6 P S f r a g r e p l a c e m e n t s E b / N 0 BER 2 T x , 2 R x 1Tx, 1Rx 2Tx, 1Rx 2Tx, 1Rx, W ang 2Tx, 2Rx 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 5 10 15 20 25 30 35 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 10 − 4 10 − 3 10 − 2 10 − 1 10 0 Fig. 2. Simulated BE R for dif ferent numbers of Tx and Rx antennas of the proposed STC and of the W ang-Xia-STC P S f r a g r e p l a c e m e n t s Normalized F requenzy f · T b Po wer 2 n d T x a n t . W a n g 1 st Tx ant. 2 nd Tx ant. 1 st Tx an t.W ang 2 nd Tx an t.W ang 1 st antenna 2 nd antenna 1 st antenna 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 -6 -4 -2 0 2 4 6 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 10 − 6 10 − 5 10 − 4 10 − 3 10 − 2 10 − 1 Fig. 3. Simulated psd for each Tx antenna of the proposed S TC (continuous line) and the W ang-Xia-ST C (dashed line) a modulation index h = 1 / 2 , the length of th e ph ase re sponse fun ction γ = 2 an d an alphabe t size of M = 8 . The sign al o f the second an tenna is mo dulated by a CPM with the same parameter s but a different alphabet Ω d 2 , correspon ding to Eq. (22). The channel used is a f requency flat Rayleigh fading model with additi ve wh ite Gaussian n oise. The fading co efficients α mn are constant for the duration of a code block (block fading) an d known at receiver (co herent d etection). The received signal y n ( t ) is de modulated by two filterbanks with pM 2 filters, which are u sed to calculate the corre lation b etween the received and candidate signals. D ue to the o rthogo nality of the an tennas each filterbank is independen tly ap plied to th e co rrespond ing time slot k of the block code. T he correlation is used as metric for the V iterbi algorithm ( V A) which has pM states and M paths leaving ea ch state. In our simulation , the V A is truncated to a path memory of 10 co de block s, which means that we get a de coding delay o f 2 · 10 T . From the simulation results gi ven in Figu re 2, we can reason ably assume that th e proposed code ac hiev es full di versity . Indeed , the curves f or th e 2x1 and 2 x2 systems re spectiv ely show a slop e of 2 and 4. Mor eover , th e curve of the 2x1 systems follows the sam e slope as the ST cod e propo sed by W ang and Xia [8], which was proved to hav e f ull diversity . Note also that the n ew co de provid es a slightly better performance. A main reason of using CPM for STC is the spectral efficiency . Figure 3 show the simulated power spectral density (p sd) L2 OR THOGON AL SP ACE TIME CODE FOR CONTINUOUS PHASE MODUL A TION 7 for b oth Tx antenn as of the proposed ST code (c ontinuo us line) and the ST code propo sed by W ang and Xia [8]. The fir st antenna of our app roach u ses a conventional CPM sign al and hence shows an eq ual psd. The spectrum of the seco nd antenna is shif ted due to adding an offset c mr ( t ) with a non zero mean . Minimizing th e difference between the two spectra by shif ting one, result in a p hase d ifference of 0 . 3 7 5 measured in nor malized freque ncy f · T d , where T d = T / log 2 ( M ) is the bit symbol length. The first antenna o f the W ang- Xia-algorith m has almost the sam e psd while th e spectrum of the second antenn a is shifted by app roximately 0 . 56 f · T d . This means that the OSTC by W an g and Xia requires a slightly larger bandwidth than our OSTC. V . C O N C L U S I O N In app lications where the p ower efficiency is crucial, com bination of Continu ous Phase Modu lation and Space Time Coding has the p otential to provide h igh sp ectral e fficienc y , thanks to spatial div ersity . T o address this p ower efficiency , ST code d esign for CPM has to ensure both lo w c omplexity d ecoding an d f ull di versity . T o fulfill these requirem ents, we hav e p roposed a new L 2 orthog onality co ndition. W e hav e shown that this c ondition is sufficient to ach iev e low co mplexity M L decod ing a nd leads, with the help of simplifying assumptio n to a simple code. Moreover , simu lations in dicate th at the co de most proba bly achieves full diversity . Further work w ill be concentrated on th e d esign of o ther cod es based on L 2 orthog onality as in the meanwhile, we ha ve been able to ob tain the d esign of full div ersity , full rate L 2 orthog onal codes for 3 antennas [1 0]. R E F E R E N C E S [1] S. M. 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