Performance Analysis of Multiple Antenna Multi-User Detection

1 Performance Analysis of Multiple Antenna Multi-User Detec tion Jav ad Kazemitabar Hamid Jafarkhani Abstract W e derive the di versity order of some multiple anten na multi-user can cellation and d etection schemes. T he common pr operty of these detectio n methods is th e u sage of Alamouti and quasi-orthog onal space-time block codes. For detecting J users each having N t ransmit antennas, these schemes require only J antennas at the receiver . Our analysis shows that when ha ving M receive an tennas, the array -processing sch emes provide the div ersity order of N ( M − J + 1) . In ad dition, our results prove that regard less of th e numbe r of users o r receive antennas, when u sing maximum -likelihood decod ing we get the full transmit and re cei ve diversities, i.e. N M , similar to the n o-interfer ence scenario . Index T erms multi-user detection , space-tim e co des, Alamouti code, q uasi-ortho gonal space-time block co de, d i versity . I . I N T R O D U C T I O N Recently , there has b een a lot of a ttention to multi-user de tection s chemes with simple rec ei ver structures. Among the simplest ones are those tha t employ spac e-time co des [1]–[3]. An orthogonal spa ce-time block cod e (OSTBC) has linear Max imum-Likelihood (ML) de coding c omplexity in terms of the number of its symbols [4], [5]. Th is is due to the fact that s uch a code w ith K sy mbols can be mod eled as K scalar channels, eac h bearing information of on ly on e symbo l. When two users employing similar OSTBCs, trans mit data to the same receiver , it is as if we have K s calar channe ls each bearing information of two supe r -imposed symbols. This wo rk was supported in part by an NSF Career A ward CCR-0238042 and a Multi-Univ ersit y Research Initiative (MURI), grant # W911NF-04-1-0224. T he authors are with the Department of EE CS at the Univ ersity of California, Irvine; e-mail: [skazemit,ham idj]@uci.edu . DRAFT 2 Heuristically , to solve two unknowns (symbols), we nee d two independen t linear combinations o f the m. In our case this translates to having two antenna s at the rece i ver . Besides OSTBCs, there are other s pace-time codes tha t allow ap plying the same proc edure. W e ha ve shown in a r ecent paper how one can a pply multi-user detection (MUD) o n any nu mber of us ers with any nu mber of trans mit anten nas [1]. In that work we have used a quasi-orthogona l spac e-time block code (QOSTBC) an d its gen eralization [6]. The ben efit of the MUD s chemes that employ OS TBC o r QOSTBC is that they require very few nu mber o f rece i ve antenna s. For examp le, those using Alamouti code [4] o r g eneralized QOSTBC [1] require a s few as the numbe r of users. Moreover , they provide very simple decod ing. Although the re has bee n a lot of work in this a rea, there is a l ack of performance analysis. T o the be st of ou r knowledge, a ma thematical c alculation of the diversity order of the se MUD s chemes is missing in the literature. Therefore, we were moti vated to find t he exact di versity order of these schemes. In a rece nt work [7] howe ver , the a uthors provide a mathematical model for ca lculating the equiv alent signa l- to-noise-ratio (SNR) of dif ferent MUD methods . Their work gi ves us a tool for ana lyzing the performance of these schemes. In t his paper we wi ll deri ve the diversit y order o f all the multiple anten na mult i-user detection scheme s described in [1] base d on the w ork in [7]. These multi-user schemes include those using Alamouti code for 2, QOSTBC for 4 an d g eneralized QOSTBC for highe r nu mber o f transmit antenn as. In this pa per ,the div ersity orde r is shown to be eq ual to N ( M − J + 1) , where J is t he number of users and N and M are the number of transmit and recei ve antenn as respecti vely . The rest of the pape r is structured as follows. Section II reviews the conce pt of div ersity and discu sses a few methods of deriving it for a s ystem. In Section III we re view the multi-user d etection using Alamouti scheme. W e then de ri ve t he diversity order of that sc heme for two users. In Se ction IV , we review the multi-user detection using QOSTBCs and deri ve the div ersity orde r for it. Section V c oncludes the paper . I I . D I V E R S I T Y O R D E R I N A C O M M U N I C A T I O N S C H E M E Di versity is usually d efined as the expone nt o f t he Signa l-to-Noise-Ratio ( SNR) in the err or rate expression, high-SNR scenario, d = − lim SNR →∞ log P e log SNR (1) where P e represents the prob ability of d ecoding err or . One can de ri ve other variants of the di versity defin ition from the above formula. W e mention one that will be used freque ntly in this work. In [8] the a uthors show DRAFT 3 that in every open-loop MIMO s ystem, the error ev ent is d ominated by Outage . Outage is the scena rio when the instantane ous SNR, due to ba d c hannel realization, is unable to support the desired rate. The result from [8] states that lim SNR →∞ log P e log SNR = lim SNR →∞ log P out log SNR (2) Therefore, when finding the di versity order , it is sufficient to kn ow the outage behavior of the sy stem [9] d = lim ǫ → 0 + log P r { Insta ntaneous SNR < ǫ } log ǫ (3) I I I . M U LT I - U S E R D E T E C T I O N U S I N G A L A M O U T I Consider two users transmitting da ta simultaneo usly to a sing le receiver . Assume also, that they are using the Alamouti scheme . W e den ote the first use r’ s message by c = ( c 1 , c 2 ) T , a nd the second u ser’ s messag e by s =( s 1 , s 2 ) T . Whe n using Alamouti the original co de transmitted will be in the form o f   c 1 c 2 − c ∗ 2 c ∗ 1   and   s 1 s 2 − s ∗ 2 s ∗ 1   . As described in [1] h owe ver , one can derive an equiv alent no tation as follo wing r = H · c + G · s + n (4) where r has en tries r i = [ r 1 i − r ∗ 2 i ] T with r 1 i and r 2 i being the s ignals recei ved at the i th receiv e anten na over two consecutive sy mbol periods. n has a Gaussian distrib ution with E [ nn ∗ ] = 2 S N R I . Also, H and G are the equiv alent cha nnel matrices from the first and se cond user to the receiv er , respec ti vely . As suming 2 receiv e antennas , H and G will hav e an Alamouti structure a s foll ows H =   H 1 H 2   and G =   G 1 G 2   H i =   h 1 i h 2 i − h ∗ 2 i h ∗ 1 i   and G i =   g 1 i g 2 i − g ∗ 2 i g ∗ 1 i   for i =1,2 (5) In order to deco de the mess age of e ach use r , one ca n use several techniques as mentioned in [3], [7]. T he mos t tri v ial and comp utationally complex method is decoding b oth u sers toge ther . This method, also kno wn as ML, DRAFT 4 finds c and s as follows. argmax p ( r | c , s ) = argmax 1 π 2 σ 4 exp  − 1 2 σ 2 k r − Hc − Gs k 2  (6) The second me thod is Array-Proc essing (AP) and is so metimes na med a s Ze ro-Forcing (ZF) or soft interference cance llation. It req uires very little computation and has linear decoding c omplexity . Th e following shows the the first step of this decoding metho d to separate c and s ,   I 2 − G 1 G − 1 2 − H 2 H − 1 1 I 2     r 1 r 2   =   H ′ 0 0 G ′     c s   +   n ′ 1 n ′ 2   (7) Note that the in verse of the Ala mouti matrix is a mult iple of its Hermitian and therefore easy to compute. In what follo ws, first, we prov e a lemma that we use in the calculation of t he div ersity order . Lemma 1: The follo wing equ ality is valid for all H and G ma trices of the f orm (5): k H k 2 k G k 2 − k H † G k 2 =  a 5 b 1 − a 6 b 2 − a 7 b 3 − a 8 b 4 + a 1 b 5 + a 2 b 6 + a 3 b 7 + a 4 b 8 − 2( a 1 b 4 + a 3 b 2 + a 4 b 1 − a 2 b 3 )( b 1 b 8 + b 2 b 7 − b 3 b 6 + b 4 b 5 ) b 2 1 + b 2 2 + b 2 3 + b 2 4  2 +  a 6 b 1 + a 5 b 2 − a 8 b 3 + a 7 b 4 + a 1 b 6 − a 2 b 5 + a 3 b 8 − a 4 b 7 − 2( a 1 b 4 + a 3 b 2 + a 4 b 1 − a 2 b 3 )( − b 1 b 7 + b 2 b 8 + b 3 b 5 + b 4 b 6 ) b 2 1 + b 2 2 + b 2 3 + b 2 4  2 +  a 7 b 1 + a 8 b 2 + a 5 b 3 − a 6 b 4 + a 1 b 7 − a 2 b 8 − a 3 b 5 + a 4 b 6 + 2( a 1 b 4 + a 3 b 2 + a 4 b 1 − a 2 b 3 )( − b 1 b 6 + b 2 b 5 − b 3 b 8 − b 4 b 7 ) b 2 1 + b 2 2 + b 2 3 + b 2 4  2 +  a 8 b 1 − a 7 b 2 + a 6 b 3 + a 5 b 4 + a 1 b 8 + a 2 b 7 − a 3 b 6 − a 4 b 5 + 2( a 1 b 4 + a 3 b 2 + a 4 b 1 − a 2 b 3 )( b 1 b 5 + b 2 b 6 + b 3 b 7 − b 4 b 8 ) b 2 1 + b 2 2 + b 2 3 + b 2 4  2 (8) where, h 11 = a 1 − j a 2 , h 21 = − a 3 + j a 4 , h 12 = − a 5 − j a 6 , h 22 = − a 7 − j a 8 g 11 = b 1 + j b 2 , g 21 = b 3 + j b 4 , g 12 = b 5 + j b 6 , g 22 = b 7 − j b 8 (9) Proof : Can be checked e asily after plugging in t he auxilary vair ables. DRAFT 5 A. Di versity or d er of ML method Consider the system de scribed in Eq. (4) with M receiv e antenna s. When using ML, the rec ei ver finds the codeword that satisfies the minimum distance criteri on for the follo wing system   r 11 r 12 · · · r 1 M r 21 r 22 · · · r 2 M   =   c 1 c 2 s 1 s 2 − c ∗ 2 c ∗ 1 − s ∗ 2 s ∗ 1          h 11 h 12 · · · h 1 M h 21 h 22 · · · h 2 M g 11 g 12 · · · g 1 M g 21 g 22 · · · g 2 M        +   n 11 n 12 · · · n 1 M n 21 n 22 · · · n 2 M   (10) The diversit y of the ab ove sys tem is equal to the minimum rank of all the difference code matrices times the number of rece i ve antennas [10]. For the a bove system this value will be 2 M . For more than two use rs, the div ersity order will remain the same since the minimum rank does no t c hange 1 . A similar argument ap plies to any full-rank code designe d for N transmit antennas , including codes design ed in [1], as our rea soning is independ ent of N . Th erefore, in general, the di versity of the ML decoding me thod is equal to M N . B. Diversity order of the array-pr ocessing method with 2 r e ceive antennas When there are two Alamo uti-equipped trans mitters, the effecti ve SNR for user number one when us ing array-process ing (ze ro-forcing) has b een deri ved in [7] to be SNR AP = k H k 2 σ 2 (1 − k Λ k 2 ) (11) where Λ is defined as Λ = H † G k H kk G k (12) W e now apply the formula in Eq. (3) to deri ve the diversity order . d AP = lim ǫ → 0 + log P r { SNR AP <ǫ } log ǫ = lim ǫ → 0 + log P r n k H k 2 . k G k 2 −k H † G k 2 σ 2 k G k 2 <ǫ o log ǫ (13) W e can us e (8) to simplify t he numerator as shown in Eq. (17) on top of the next page, where b = [ b 1 b 2 · · · b 8 ] . In tha t equation, c onditioned on b , eac h of the terms ins ide the 4 main pa rentheses is a zero-mean rea l Gaussian random variable d ue to independe nce of a i s. Once divided by the square root of the denominator 1 The rank of J concatenated Alamoutis-a 2 J × 2 matrix-is always 2 DRAFT 6 their variance wil l b ecome equal to one. Moreover , it ca n be easily che cked that thes e Gaussian random variables are independe nt. Therefore, the sum of their squares is Chi-squ are distributed with 4 degrees of freedom an d has the follo wing d ensity function f ( x ) = xe − x x > 0 (14) For s mall enough ǫ , Z σ 2 ǫ 0 f ( x ) dx = σ 4 ǫ 2 + O ( σ 4 ǫ 2 ) (15) where f ( x ) = O ( g ( x )) me ans there is a positi ve c onstant c such that f ( x ) ≤ cg ( x ) for t he desired range of x . Since the quantity in Eq. (15) is independent o f b , its expected v a lue with r espec t t o b will remain the same. Therefore, we ha ve d = lim ǫ → 0 + log ( σ 4 ) + log ( ǫ 2 ) log ( ǫ ) = 2 (16) P r                             a 5 b 1 − a 6 b 2 − a 7 b 3 − a 8 b 4 + a 1 b 5 + a 2 b 6 + a 3 b 7 + a 4 b 8 − 2( a 1 b 4 + a 3 b 2 + a 4 b 1 − a 2 b 3 )( b 1 b 8 + b 2 b 7 − b 3 b 6 + b 4 b 5 ) b 2 1 + b 2 3 + b 2 4 + b 2 2  2 +  a 6 b 1 + a 5 b 2 − a 8 b 3 + a 7 b 4 + a 1 b 6 − a 2 b 5 + a 3 b 8 − a 4 b 7 − 2( a 1 b 4 + a 3 b 2 + a 4 b 1 − a 2 b 3 )( − b 1 b 7 + b 2 b 8 + b 3 b 5 + b 4 b 6 ) b 2 1 + b 2 2 + b 2 3 + b 2 4  2 +  a 7 b 1 + a 8 b 2 + a 5 b 3 − a 6 b 4 + a 1 b 7 − a 2 b 8 − a 3 b 5 + a 4 b 6 + 2( a 1 b 4 + a 3 b 2 + a 4 b 1 − a 2 b 3 )( − b 1 b 6 + b 2 b 5 − b 3 b 8 − b 4 b 7 ) b 2 1 + b 2 2 + b 2 3 + b 2 4  2 +  a 8 b 1 − a 7 b 2 + a 6 b 3 + a 5 b 4 + a 1 b 8 + a 2 b 7 − a 3 b 6 − a 4 b 5 + 2( a 1 b 4 + a 3 b 2 + a 4 b 1 − a 2 b 3 )( b 1 b 5 + b 2 b 6 + b 3 b 7 − b 4 b 8 ) b 1 + b 2 2 + b 2 3 + b 2 4  2 σ 2 ( b 2 1 + b 2 2 + b 2 3 + b 2 4 + b 2 5 + b 2 6 + b 2 7 + b 2 8 ) < ǫ                            = E b                           P r                                                     a 5 b 1 − a 6 b 2 − a 7 b 3 − a 8 b 4 + a 1 b 5 + a 2 b 6 + a 3 b 7 + a 4 b 8 − 2( a 1 b 4 + a 3 b 2 + a 4 b 1 − a 2 b 3 )( b 1 b 8 + b 2 b 7 − b 3 b 6 + b 4 b 5 ) b 2 1 + b 2 3 + b 2 4 + b 2 2  2 +  a 6 b 1 + a 5 b 2 − a 8 b 3 + a 7 b 4 + a 1 b 6 − a 2 b 5 + a 3 b 8 − a 4 b 7 − 2( a 1 b 4 + a 3 b 2 + a 4 b 1 − a 2 b 3 )( − b 1 b 7 + b 2 b 8 + b 3 b 5 + b 4 b 6 ) b 2 1 + b 2 2 + b 2 3 + b 2 4  2 +  a 7 b 1 + a 8 b 2 + a 5 b 3 − a 6 b 4 + a 1 b 7 − a 2 b 8 − a 3 b 5 + a 4 b 6 + 2( a 1 b 4 + a 3 b 2 + a 4 b 1 − a 2 b 3 )( − b 1 b 6 + b 2 b 5 − b 3 b 8 − b 4 b 7 ) b 2 1 + b 2 2 + b 2 3 + b 2 4  2 +  a 8 b 1 − a 7 b 2 + a 6 b 3 + a 5 b 4 + a 1 b 8 + a 2 b 7 − a 3 b 6 − a 4 b 5 + 2( a 1 b 4 + a 3 b 2 + a 4 b 1 − a 2 b 3 )( b 1 b 5 + b 2 b 6 + b 3 b 7 − b 4 b 8 ) b 1 + b 2 2 + b 2 3 + b 2 4  2 b 2 1 + b 2 2 + b 2 3 + b 2 4 + b 2 5 + b 2 6 + b 2 7 + b 2 8 < σ 2 ǫ                               b                                                                            (17) DRAFT 7 C. The cas e with mor e than 2 r ec eive an tennas Let us now assu me the pre viou s system with the exc eption that there are 3 recei ve an tennas rather than tw o. For this s ystem we ha ve r 1 = H 1 · c + G 1 · s + n 1 r 2 = H 2 · c + G 2 · s + n 2 r 3 = H 3 · c + G 3 · s + n 3 (18) After applying the array proces sing algorithm a nd ca ncelling the ef fect of user c orresponding to message s we get r ′ 1 =  G † 2 H 2 k G 2 k 2 − G † 1 H 1 k G 1 k 2  c + n ′ 1 r ′ 2 =  G † 3 H 3 k G 3 k 2 − G † 1 H 1 k G 1 k 2  c + n ′ 2 (19) Conditioned on G i s, the noise terms n ′ 1 and n ′ 2 are correlated Gaussian random variables. Similar statement applies to the n ew channe l matrices  G † 2 H 2 k G 2 k 2 − G † 1 H 1 k G 1 k 2  and  G † 2 H 2 k G 2 k 2 − G † 1 H 1 k G 1 k 2  . In [11] it is shown that in a Rayleigh fading system with recei ve correlation, lik e the one w e have he re, the diversity order will be N M as long as the correlation matrix o f the channel is full-rank. Since, [11] assumes white n oise, the equi valent correlation matr ix in our case will be correlation matr ix of the channel mult iplied by the in verse of that of the noise. Clearly , the in verse of the correlation matrix of the noise acc ounts for the no ise-whitening o peration. Therefore, if we s how tha t bo th of these two co rrelation matrices are full-rank, w e can co nclude that the sys tem in Eq. (19) pro vides a di versity orde r of 4 ( N = 2 , M = 2 ). The correlation matrix of noise is equal t o     σ 2 k G 2 k 2 + σ 2 k G 1 k 2  I 2 σ 2 k G 1 k 2 I 2 σ 2 k G 1 k 2 I 2  σ 2 k G 3 k 2 + σ 2 k G 1 k 2  I 2    (20) where I 2 is the 2 × 2 ide ntity matrix. This matrix is c learly full -rank for almost (surely) all G i realizations. It remains no w to find the correlation matrix of the e quiv a lent chan nel. Since both l ines in Eq. (19) represent an Alamouti scheme, we can con vert them b ack into the re gular Alamouti representation as f ollows y 1 =   c 1 c 2 − c ∗ 2 c ∗ 1   ·   A 1 + j A 2 A 3 + j A 4   + noise y 2 =   c 1 c 2 − c ∗ 2 c ∗ 1   ·   B 1 + j B 2 B 3 + j B 4   + noise (21) DRAFT 8 where the coefficients are normalized so that the noise terms have unit power . Us ing the SNR result from [7] and Eq. (8) we can wri te A 1 = a 5 b 1 − a 6 b 2 − a 7 b 3 − a 8 b 4 + a 1 b 5 + a 2 b 6 + a 3 b 7 + a 4 b 8 − 2( a 1 b 4 + a 3 b 2 + a 4 b 1 − a 2 b 3 )( b 1 b 8 + b 2 b 7 − b 3 b 6 + b 4 b 5 ) b 2 1 + b 2 2 + b 2 3 + b 2 4 σ √ b 2 1 + ··· + b 2 8 A 2 = a 6 b 1 + a 5 b 2 − a 8 b 3 + a 7 b 4 + a 1 b 6 − a 2 b 5 + a 3 b 8 − a 4 b 7 − 2( a 1 b 4 + a 3 b 2 + a 4 b 1 − a 2 b 3 )( − b 1 b 7 + b 2 b 8 + b 3 b 5 + b 4 b 6 ) b 2 1 + b 2 2 + b 2 3 + b 2 4 σ √ b 2 1 + ··· + b 2 8 A 3 = a 7 b 1 + a 8 b 2 + a 5 b 3 − a 6 b 4 + a 1 b 7 − a 2 b 8 − a 3 b 5 + a 4 b 6 + 2( a 1 b 4 + a 3 b 2 + a 4 b 1 − a 2 b 3 )( − b 1 b 6 + b 2 b 5 − b 3 b 8 − b 4 b 7 ) b 2 1 + b 2 2 + b 2 3 + b 2 4 σ √ b 2 1 + ··· + b 2 8 A 4 = a 8 b 1 − a 7 b 2 + a 6 b 3 + a 5 b 4 + a 1 b 8 + a 2 b 7 − a 3 b 6 − a 4 b 5 + 2( a 1 b 4 + a 3 b 2 + a 4 b 1 − a 2 b 3 )( b 1 b 5 + b 2 b 6 + b 3 b 7 − b 4 b 8 ) b 2 1 + b 2 2 + b 2 3 + b 2 4 σ √ b 2 1 + ··· + b 2 8 B 1 = a 9 b 1 − a 10 b 2 − a 11 b 3 − a 12 b 4 + a 1 b 9 + a 2 b 10 + a 3 b 11 + a 4 b 12 − 2( a 1 b 4 + a 3 b 2 + a 4 b 1 − a 2 b 3 )( b 1 b 12 + b 2 b 11 − b 3 b 10 + b 4 b 9 ) b 2 1 + b 2 2 + b 2 3 + b 2 4 σ √ b 2 1 + ··· b 2 4 + b 2 9 + ··· + b 2 12 B 2 = a 10 b 1 + a 9 b 2 − a 12 b 3 + a 11 b 4 + a 1 b 10 − a 2 b 9 + a 3 b 12 − a 4 b 11 − 2( a 1 b 4 + a 3 b 2 + a 4 b 1 − a 2 b 3 )( − b 1 b 11 + b 2 b 12 + b 3 b 9 + b 4 b 10 ) b 2 1 + b 2 2 + b 2 3 + b 2 4 σ √ b 2 1 + ··· b 2 4 + b 2 9 + ··· + b 2 12 B 3 = a 11 b 1 + a 12 b 2 + a 9 b 3 − a 10 b 4 + a 1 b 11 − a 2 b 12 − a 3 b 9 + a 4 b 10 + 2( a 1 b 4 + a 3 b 2 + a 4 b 1 − a 2 b 3 )( − b 1 b 10 + b 2 b 9 − b 3 b 12 − b 4 b 11 ) b 2 1 + b 2 2 + b 2 3 + b 2 4 σ √ b 2 1 + ··· b 2 4 + b 2 9 + ··· + b 2 12 B 4 = a 12 b 1 − a 11 b 2 + a 10 b 3 + a 9 b 4 + a 1 b 12 + a 2 b 11 − a 3 b 10 − a 4 b 9 + 2( a 1 b 4 + a 3 b 2 + a 4 b 1 − a 2 b 3 )( b 1 b 9 + b 2 b 10 + b 3 b 11 − b 4 b 12 ) b 2 1 + b 2 2 + b 2 3 + b 2 4 σ √ b 2 1 + ··· b 2 4 + b 2 9 + ··· + b 2 12 (22) The above v alues are real and imagina ry parts of the cha nnel coefficients. Instead of fin ding the comp lex correlation matrix we can find t he follo wing real correlation matrix C = E { [ A | B ] T [ A | B ] } (23) where A = [ A 1 A 2 A 3 A 4 ] and B = [ B 1 B 2 B 3 B 4 ] . It ca n be easily shown that i f C is full-rank so will be the complex channe l c orrelation ma trix. W e already know that { A i } a nd { B i } are eac h unc orrelated among themselves. Calculating E { A i B j } we will ha ve C = 1 σ 2   I X X I   (24) where X = 1 √ ( b 2 1 + ··· + b 2 8 )( b 2 1 + ··· + b 2 4 + b 2 9 + ··· + b 2 12 )        b 5 b 6 b 7 b 8 b 6 − b 5 b 8 − b 7 b 7 − b 8 − b 5 b 6 b 8 b 7 − b 6 − b 5        ·        b 9 b 10 b 11 b 12 b 10 − b 9 − b 12 b 11 b 11 b 12 − b 9 − b 10 b 12 − b 11 b 10 − b 9        (25) DRAFT 9 From [12] we ha ve det ( C ) = 1 σ 16 det ( I − X T · X ) = 1 σ 16 (1 − ( b 2 5 + ··· + b 2 8 )( b 2 9 + ··· + b 2 12 ) ( b 2 1 + ··· + b 2 8 )( b 2 1 + ··· + b 2 4 + b 2 9 + ··· + b 2 12 ) ) I 6 = 0 (26) Therefore, the system des cribed in E q. (21) will provide full diversity , i.e. 2 × 2 =4. This mea ns that the desc ribed array process ing s cheme provides a di versity order equa l t o N × ( M − J + 1) for the c ase of N = 2 , J = 2 , and M = 3 . W e now further inspect the diversit y o rder of the scheme by considering the gene ral ca se of M recei ve antennas , while keeping the s ame n umber of use rs a nd transmit antennas. After canceling the e f fect of t he us er correspond ing to messa ge s we get r ′ 1 =  G † 2 H 2 k G 2 k 2 − G † 1 H 1 k G 1 k 2  c + n ′ 1 r ′ 2 =  G † 3 H 3 k G 3 k 2 − G † 1 H 1 k G 1 k 2  c + n ′ 2 . . . r ′ M − 1 =  G † M H M k G M k 2 − G † 1 H 1 k G 1 k 2  c + n ′ M − 1 (27) W e will again form the correlation matrix for n oise a nd the e quiv a lent Alamouti cha nnel coefficients and examine whether they are full-rank. The noise correlation matri x will be           σ 2 k G 2 k 2 + σ 2 k G 1 k 2  σ 2 k G 1 k 2 · · · σ 2 k G 1 k 2 σ 2 k G 1 k 2  σ 2 k G 3 k 2 + σ 2 k G 1 k 2  · · · σ 2 k G 1 k 2 . . . . . . . . . . . . σ 2 k G 1 k 2 σ 2 k G 1 k 2 · · ·  σ 2 k G 2 k 2 + σ 2 k G M k 2           ⊗ I 2 (28) The matrix o n the left hand s ide of the ten sor product is full-rank since it has M − 1 no nzero eigen values as follo wing 2 σ 2 k G 2 k 2 + σ 2 k G 1 k 2 , σ 2 k G 3 k 2 + σ 2 k G 1 k 2 , · · · , σ 2 k G M k 2 + σ 2 k G 1 k 2 (29) W e s hould now examine the channe l correlation matrix. In the general case of M recei ve antennas , we will 2 The eigen vectors of this matrix are standard unit vectors e 1 , e 2 , · · · , e M − 1 . DRAFT 10 have C = 1 σ 2          I X 12 · · · X 1( M − 1) X 21 I · · · X 2( M − 1) . . . . . . . . . . . . X ( M − 1)1 X ( M − 1)2 · · · X ( M − 1)( M − 1)          (30) where X ij = B i B T j with B i = 1 √ b 2 1 + ··· + b 2 4 + b 2 4 i +1 + ··· + b 2 4( i +1)        b 4 i +1 b 4 i +2 b 4 i +3 b 4( i +2) b 4 i +2 − b 4 i +1 b 4( i +2) − b 4 i +3 b 4 i +2 − b 4( i +2) − b 4 i +1 b 4 i +2 b 4( i +1) b 4 i +3 − b 4 i +2 − b 4 i +1        (31) It ca n be chec ked e asily that B i · B T i = b 2 4 i +1 + ··· + b 2 4( i +1) b 2 1 + ··· + b 2 4 + b 2 4 i +1 + ··· + b 2 4( i +1) I = β i I . It proves that C is full-rank if w e can find a 4( M − 1) × 4( M − 1) matrix U such that the rank of U T CU (32) is equal to 4 M . W e will try to construct U based o n the follo wing structure U =  u 1 | · · · | u M − 1  (33) where u i s are 4( M − 1) × 4 ma trices. The foll owi ng two lemmas wil l lead us to construct the m atrix U . Lemma 2: Gi ven a i = 1 λ ∗ + β i − 1 where λ ∗ is a root of P M − 1 i =1 β i λ + β i − 1 = 1 we have C ·  a 1 B 1 | a 2 B 2 | · · · | a M − 1 B M − 1  T = λ ∗  a 1 B 1 | a 2 B 2 | · · · | a M − 1 B M − 1  T (34) Proof : See Appendix. Lemma 3: The follo wing e quation has M − 1 distinct non-zero roots M − 1 X i =1 β i λ + β i − 1 = 1 (35) Proof : See Appendix. DRAFT 11 W e name these distinct non-zero roots λ ∗ 1 , · · · , λ ∗ M − 1 . Let us now defin e u i vectors by u i =  a 1 i B T 1 | a 2 i B T 2 | · · · | a ( M − 1) i B T M − 1  T (36) where a mi = 1 λ ∗ i + b m − 1 for i, m = 1 , · · · , M − 1 . From Lemma 3 and properties of B j s it is clear t hat C · u i = λ ∗ i u i u T i u i = P j β j a 2 j i I = S i I (37) Also, since λ ∗ i s are distinct we hav e u T i Cu j = u T i λ ∗ j u j = λ ∗ j u T i u j and u T i Cu j = u T i C T u j = ( C u i ) T u j = λ ∗ i u T i u j = ⇒ ( λ ∗ i − λ ∗ j ) u T i u j = 0 = ⇒ u T i u j = 0 given i 6 = j. (38) W e are now ready to show why C is full-rank as follo ws     u T 1 u T 2 . . . u T M − 1     · C · ( u 1 | u 2 | · · · | u M − 1 ) =     u T 1 u T 2 . . . u T M − 1     · ( Cu 1 | Cu 2 | · · · | Cu M − 1 ) = diag ( S 1 λ ∗ 1 , S 1 λ ∗ 1 , S 1 λ ∗ 1 , S 1 λ ∗ 1 , S 2 λ ∗ 2 , S 2 λ ∗ 2 , S 2 λ ∗ 2 , S 2 λ ∗ 2 , · · · , S M − 1 λ ∗ M − 1 , S M − 1 λ ∗ M − 1 , S M − 1 λ ∗ M − 1 , S M − 1 λ ∗ M − 1 ) (39) which is clea rly f ull-rank and it proves the same property f or the ma trix C . The refore, the channe l correlation matrix is full- rank an d the p rovided diversity for the sch eme described i n Eq. (27) is 2( M − 1) . D. The cas e with mor e than 2 users Let us now ass ume the multi-user s ystem with 3 users a nd 3 antennas at t he receiver as follo ws r 1 = H 1 · c + G 1 · s + K 1 · x + n 1 r 2 = H 2 · c + G 2 · s + K 2 · x + n 2 r 3 = H 3 · c + G 3 · s + K 3 · x + n 3 (40) DRAFT 12 Once we apply the cancellation technique on the us er corresponding t o message x we get r ′ 1 = K − 1 1 r 1 − K − 1 3 r 3 = ( K − 1 1 H 1 − K − 1 3 H 3 ) c + ( K − 1 1 G 1 − K − 1 3 G 3 ) s + z 1 r ′ 2 = K − 1 2 r 2 − K − 1 3 r 3 = ( K − 1 2 H 2 − K − 1 3 H 3 ) c + ( K − 1 2 G 2 − K − 1 3 G 3 ) s + z 2 (41) W e note that K − 1 i = K † i k K i k 2 . Conditioned on K i s, the above system represents a Rayleigh fading channel with 2 users and 2 recei ve antenn as. The refore, s imilar to the s ystem in (19) a ll the diversity claims of a 2 user systems (conditionally) apply . 3 In other words, the di versity order will be equal to 2. T aking the expectation over all K i s will not ch ange this con stant value a nd the diversit y will remain 2. Similarly , when having M receiv e a ntennas for multi-user d etection of 3 us ers we g et the diversity orde r of 2( M − 3 + 1) . Using indu ction on the number of users the n, we can prov e t he follo w ing theorem. Theorem 1: Suppose we hav e J Alamouti-equippe d users tr ansmitting to the same receiver in the same frequency ba nd that are time s ynchronized . Let us a lso a ssume tha t at the receiver we have M antennas and we us e array proces sing as explained in [1]. Th e diversity p rovided to e ach use r will be e qual to 2( M − J + 1) . I V . M U LT I - U S E R D E T E C T I O N F O R M O R E T H A N T W O T R A N S M I T A N T E N N A S In this section we first briefly explain t he sche me in [1] a nd then find its di versity o rder . Su ppose, we ha ve two users each with 4 trans mit antennas using a QOSTBC. They are synchrono usly transmitting data to a receiv er with two recei ve antenn as as following         r 11 r 21 r 31 r 41         =         c 1 c 2 c 3 c 4 − c ∗ 2 c ∗ 1 − c ∗ 4 c ∗ 3 c 3 c 4 c 1 c 2 − c ∗ 4 c ∗ 3 − c ∗ 2 c ∗ 1         ·         h 11 h 21 h 31 h 41         +         s 1 s 2 s 3 s 4 − s ∗ 2 s ∗ 1 − s ∗ 4 s ∗ 3 s 3 s 4 s 1 s 2 − s ∗ 4 s ∗ 3 − s ∗ 2 s ∗ 1         ·         g 11 g 21 g 31 g 41         +         n 11 n 21 n 31 n 41                 r 12 r 22 r 32 r 42         =         c 1 c 2 c 3 c 4 − c ∗ 2 c ∗ 1 − c ∗ 4 c ∗ 3 c 3 c 4 c 1 c 2 − c ∗ 4 c ∗ 3 − c ∗ 2 c ∗ 1         ·         h 12 h 22 h 32 h 42         +         s 1 s 2 s 3 s 4 − s ∗ 2 s ∗ 1 − s ∗ 4 s ∗ 3 s 3 s 4 s 1 s 2 − s ∗ 4 s ∗ 3 − s ∗ 2 s ∗ 1         ·         g 12 g 22 g 32 g 42         +         n 12 n 22 n 32 n 42         (42) 3 The only difference is that the noise an d the channel co ef ficients are correlated. Howe ver , this will not af fect the div ersity results since the correlation matrices are exactly li ke those i n E qs. (20 ) and (24) and therefore full-rank. DRAFT 13 W e then define r 1 =   r 11 + r 31 − r ∗ 21 − r ∗ 41   and r ′ 1 =   r 11 − r 31 − r ∗ 21 + r ∗ 41   (43) Assuming similar definitions for r 2 and r ′ 2 we will ha ve r 1 = H 1 c + + G 1 s + + n 1 , r ′ 1 = H ′ 1 c − + G ′ 1 s − + n ′ 1 r 2 = H 2 c + + G 2 s + + n 2 , r ′ 2 = H ′ 1 c − + G ′ 1 s − + n ′ 2 (44) where H 1 =   h 11 + h 31 h 21 + h 41 − h ∗ 11 − h ∗ 31 h ∗ 21 + h ∗ 41   , H ′ 1 =   h 11 − h 31 h 21 − h 41 − h ∗ 11 + h ∗ 31 h ∗ 21 − h ∗ 41   c + =   c 1 + c 3 c 2 + c 4   , c − =   c 1 − c 3 c 2 − c 4   (45) and the rest of the matri ces are defined similarl y . Eq. (44 ) r eminds u s of Eq. (4) an d (5). Using the same array-process ing algorithm one can cancel the effect of s and get the f ollo wing r + = G − 1 1 r 1 − G − 1 2 r 2 = ( G − 1 1 H 1 − G − 1 2 H 2 ) c + + ( G − 1 1 n 1 − G − 1 2 n 2 ) = As + + z r − = G ′ − 1 1 r ′ 1 − G ′ − 1 2 r ′ 2 = ( G ′ − 1 1 H 1 − G ′ 2 − 1 H ′ 2 ) c − + ( G ′ − 1 1 n 1 − G ′ 2 − 1 n ′ 2 ) = A ′ c − + z ′ (46) where A and A ′ can be sho wn to be i n the form of A =   α 1 α 2 − α ∗ 2 α ∗ 1   A ′ =   α ′ 1 α ′ 2 − α ′ 2 ∗ α ′ 1 ∗   (47) Conditioned on G i and G ′ i values, the noise t erms will b e i.i.d. complex Gaussia n random variables. A similar argument a pplies to α i and α ′ i . Now , if we perform t he re verse o f the con version i n (42)-(44) we get R =          c 1 c 2 c 3 c 4 − c ∗ 2 c ∗ 1 − c ∗ 4 c ∗ 3 c 3 c 4 c 1 c 2 − c ∗ 4 c ∗ 3 − c ∗ 2 c ∗ 1                   α 1 + α ′ 1 2 α 2 + α ′ 2 2 α 1 − α ′ 1 2 α 2 − α ′ 2 2          + i.i.d noise (48) Conditioned on G i and G ′ i values, the above system is e qui valent to a single-user QOSTBC with independent DRAFT 14 noise an d Rayleigh fading channel coefficients. T his sy stem clearly p rovides a di versity o rder of four 4 , even after taking the expec tation. Therefore, in the case of J = 2 users, N = 4 transmit, an d M = 2 a ntennas the div ersity order i s 4= N ( M − J + 1) . A. The cas e w ith mor e t han 2 r eceive a ntennas Let us c onsider the abov e system with the exception that there a re three r eceive a ntennas ins tead of tw o. For this system we ha ve r 1 = H 1 c + + G 1 s + + n 1 , r ′ 1 = H ′ 1 c − + G ′ 1 s − + n ′ 1 r 2 = H 2 c + + G 2 s + + n 2 , r ′ 2 = H ′ 2 c − + G ′ 2 s − + n ′ 2 r 3 = H 3 c + + G 3 s + + n 3 , r ′ 3 = H ′ 3 c − + G ′ 3 s − + n ′ 3 (49) After cance ling out s we get r + 1 = G − 1 2 r 2 − G − 1 1 r 1 = ( G − 1 2 H 2 − G − 1 1 H 1 ) c + + ( G − 1 2 n 2 − G − 1 1 n 1 ) = A 1 s + + z 1 r − 1 = G ′ − 1 1 r ′ 1 − G ′ − 1 2 r ′ 2 = ( G ′ − 1 1 H 1 − G ′ 2 − 1 H ′ 2 ) c − + ( G ′ − 1 1 n 1 − G ′ 2 − 1 n ′ 2 ) = A ′ 1 c − + z ′ 1 r + 2 = G − 1 3 r 3 − G − 1 1 r 1 = ( G − 1 3 H 3 − G − 1 1 H 1 ) c + + ( G − 1 3 n 3 − G − 1 1 n 1 ) = A 2 s + + z 2 r − 2 = G ′ − 1 3 r ′ 3 − G ′ − 1 1 r ′ 1 = ( G ′ − 1 3 H 3 − G ′ 1 − 1 H ′ 1 ) c − + ( G ′ − 1 3 n 3 − G ′ 1 − 1 n ′ 1 ) = A ′ 2 c − + z ′ 2 (50) where A 1 =   α 11 α 21 − α ∗ 21 α ∗ 11   , A ′ 1 =   α ′ 11 α ′ 21 − α ′ 21 ∗ α ′ 11 ∗   A 2 =   α 12 α 22 − α ∗ 22 α ∗ 12   , A ′ 2 =   α ′ 12 α ′ 22 − α ′ 22 ∗ α ′ 12 ∗   (51) Although Gaussian, n either the no ise terms, nor the chann el fades are uncorrelated. The correlation matrix for the ( z 1 z 2 ) T will be equal to     σ 2 k G 2 k 2 + σ 2 k G 1 k 2  I 2 σ 2 k G 1 k 2 I 2 σ 2 k G 1 k 2 I 2  σ 2 k G 3 k 2 + σ 2 k G 1 k 2  I 2    (52) 4 This is assuming rotated constellation for c 3 and c 4 DRAFT 15 and for ( z ′ 1 z ′ 2 ) T it will be     σ 2 k G ′ 2 k 2 + σ 2 k G ′ 1 k 2  I 2 σ 2 k G ′ 1 k 2 I 2 σ 2 k G ′ 1 k 2 I 2  σ 2 k G ′ 3 k 2 + σ 2 k G ′ 1 k 2  I 2    (53) The correlation matrix of ( A 1 A 2 ) and ( A ′ 1 A ′ 2 ) will be of t he form 1 σ 2   I X X I   (54) where X is similar to E q. (25) 5 .Clearly , all the se c orrelation ma trices are full-rank. Now , similar to the 2-rec ei ve antenna case, we can perform t he rev erse con version a nd write the abov e equation in the following form R =          c 1 c 2 c 3 c 4 − c ∗ 2 c ∗ 1 − c ∗ 4 c ∗ 3 c 3 c 4 c 1 c 2 − c ∗ 4 c ∗ 3 − c ∗ 2 c ∗ 1                   α 11 + α ′ 11 2 α 12 + α ′ 12 2 α 21 + α ′ 21 2 α 21 + α ′ 22 2 α 11 − α ′ 11 2 α 12 − α ′ 12 2 α 21 − α ′ 21 2 α 22 − α ′ 22 2          + noise (55) The correlation matrix of the new channel and noise terms can b e derived via row operations and block- concate nation of the correlation matrices in (20,25). Therefore, they will also b e full-rank and the di versity order of the equi valent scheme shown in (55) will be 4 × 2 =8. For the genera l case of M rece i ve antenn as, one can perform similar operations and ge t to the noise a nd channe l correlation matrices like those in (28 ) and (30). After re verse con version, the equiv alent sing le-user system will look lik e R =          c 1 c 2 c 3 c 4 − c ∗ 2 c ∗ 1 − c ∗ 4 c ∗ 3 c 3 c 4 c 1 c 2 − c ∗ 4 c ∗ 3 − c ∗ 2 c ∗ 1                   α 11 + α ′ 11 2 α 12 + α ′ 12 2 · · · α 1( M − 1) + α ′ 1( M − 1) 2 α 21 + α ′ 21 2 α 22 + α ′ 22 2 · · · α 2( M − 1) + α ′ 2( M − 1) 2 α 11 − α ′ 11 2 α 12 − α ′ 12 2 · · · α 1( M − 1) − α ′ 1( M − 1) 2 α 21 − α ′ 21 2 α 22 − α ′ 22 2 · · · α 2( M − 1) − α ′ 2( M − 1) 2          + n oise (56) which provides the diversity order o f 4( M − 1) due to the full-rank correlation argument. Therefore, for the case of J = 2 us ers, N = 4 transmit antennas a nd general M receiv e ante nnas, the diversity order will be 5 The ne w b i terms are different, but can be calculated using Eqs. (21-25). DRAFT 16 N ( M − J + 1) . B. The cas e w ith mor e t han 2 users Let us now as sume the multi- user system with 3 use rs a nd 3 antennas. Let us repres ent their c hannel coefficients by { h i } 4 i =1 , { g i } 4 i =1 , a nd { k i } 4 i =1 respectively . Naturally the code e ach of them will transmit will be in the form a QOS TBC. No w , if we perform the c on version to the Alamouti form, we derive the followi ng equations r 1 = H 1 c + + G 1 s + + n 1 , r ′ 1 = H ′ 1 c − + G ′ 1 s − + K ′ 1 x − + n ′ 1 r 2 = H 2 c + + G 2 s + + n 2 , r ′ 2 = H ′ 2 c − + G ′ 2 s − + K ′ 2 x − + n ′ 2 r 3 = H 3 c + + G 3 s + + n 3 , r ′ 3 = H ′ 3 c − + G ′ 3 s − + K ′ 3 x − + n ′ 3 (57) Once we apply the cancellation technique on the us er corresponding t o message x we get r + 1 = K − 1 2 r 2 − K − 1 1 r 1 = A 1 s + + B 1 x + + z 1 r − 1 = K ′ − 1 1 r ′ 1 − K ′ − 1 2 r ′ 2 = A ′ 1 c − + B ′ 1 x − + z ′ 1 r + 2 = K − 1 3 r 3 − K − 1 1 r 1 = A 2 s + + B 2 x + + z 2 r − 2 = K ′ − 1 3 r ′ 3 − K ′ − 1 1 r ′ 1 = A ′ 2 c − + B ′ 2 x − + z ′ 2 (58) Conditioned on K i s an d K ′ i s, the a bove system represe nts a Rayleigh fading cha nnel with 2 us ers an d 2 receive antennas . Therefore, similar to the system in (19) all the di versity claims of a 2-user systems (conditionally) apply . 6 In o ther words, the di versity o rder will be equal to 4. T ak ing the expectation over a ll K i s an d K ′ i s will not change this constant v a lue and the d i versity will remain 4 . Simi larly , when ha ving M receive an tennas f or multi-user detection of 3 use rs we g et the diversity o rder of 4 ( M − 3 + 1) . Using i nduction on the nu mber of users then, we can prov e the following theorem. Theorem 2: Suppose we have J QOSTBC-equippe d use rs transmitting to the s ame receiver in the same frequency ba nd that are time s ynchronized . Let us a lso a ssume tha t at the receiver we have M antennas and we us e array proces sing as explained in [1]. Th e diversity p rovided to e ach use r will be e qual to 4( M − J + 1) . In [1] we s howed ho w , using   A B B A   one can generalize the a rray proce ssing method to any numbe r of transmit ante nnas. The trick whe n N = 2 k is to b reak the system into t wo systems with N = 2 k − 1 and then 6 The only difference is that the noise an d the channel co ef ficients are correlated. Howe ver , this will not af fect the div ersity results since the correlation matrices are exactly li ke those i n E qs. (25 ) and (30) and therefore full-rank. DRAFT 17 perform the i nterference c ancellation tech nique on ea ch of them separa tely . The n, one can combine them to g et the original system. Similar to the method we showed for con verting the N = 4 to N = 2 systems, one c an perform the same diversity a nalysis on any N = 2 k -transmit an tenna system with   A B B A   structure. In addition, the res ult can be extended to n on-power -of-2s with co lumn removal me thod explained in [1] to p rove the follo wing corollary . Corollary: Assume we have J users each with N trans mit antennas using the   A B B A   structure explained above. They are all sending data synch ronously to a receiv er with M ≥ J rec ei ve antenna s. The d i versity of the array processing method explained in [1] will be equal to N ( M − J + 1) . V . D E C O D I N G O F A N I N T E R F E R E N C E C A N C E L E D S Y S T E M The a lgorithm we desc ribed in [1] provides a method to remove un wanted eff ect of other users and leaves us with a single use r system. Howev er , as we noticed in the cases wh ere the number of re ceiv e anten nas is more than that of use rs, b oth the c hannel and n oise coef ficients are correlated. W e describe the optimal decoding of the system -after interference cance llation- in this section. W e prove that this method, which requ ires “noise - whitening” operation, will still keep the se parate decoding property o f the 2-transmit antenna case. As shown, a fter canceling the first user we ha ve r ′ 1 =  G † 2 H 2 k G 2 k 2 − G † 1 H 1 k G 1 k 2  c + n ′ 1 = H ′ 1 c + n ′ 1 r ′ 2 =  G † 3 H 3 k G 3 k 2 − G † 1 H 1 k G 1 k 2  c + n ′ 2 = H ′ 2 c + n ′ 2 (59) where the correlation matrix of the n oise is C n =     σ 2 k G 2 k 2 + σ 2 k G 1 k 2  I 2 σ 2 k G 1 k 2 I 2 σ 2 k G 1 k 2 I 2  σ 2 k G 3 k 2 + σ 2 k G 1 k 2  I 2    (60) Let us define r = ( r 1 r 2 ) T H =  H ′ 1 H ′ 2  T (61) Then, the maximum-likeli hood decoding metric will be arg min c ( r − Hc ) † C n − 1 ( r − Hc ) . (62) DRAFT 18 It can be shown [14] that C − 1 n =   x I y I y I t I   (63) where x, y , t are real numbers. Therefore, t he ML criterion will be to mi nimize ( r † 1 − c † H 1 † r † 2 − c † H 2 † )   x I y I y I t I     r 1 − H 1 c r 2 − H 2 c   = x k r 1 − H 1 c k 2 + t k r 2 − H 2 c k 2 + 2 y Re { ( r † 1 − c † H † 1 )( r 2 − H 2 c ) } = x k r 1 − H 1 c k 2 + t k r 2 − H 2 c k 2 + 2 y Re { r † 1 r 2 − r † 1 H 2 c − c † H † 1 r 2 + c † H † 1 H 2 c } (64) The o nly part in the above eq uation that could generate cros s-terms and the refore caus e non-sepa rate dec oding is c † H † 1 H 2 c . Before we expand this term, we note that H † 1 H 2 is in the form of an Alamouti matrix and can be written as   h 1 h 2 − h ∗ 2 h ∗ 1   (65) Having tha t in mind the las t term in Eq. (64) can be written as Re { h 1 | c 1 | 2 + h ∗ 1 | c 2 | 2 + h 2 c ∗ 1 c 2 − h ∗ 2 c ∗ 2 c 1 } = R e { h 1 } ( | c 1 | 2 + | c 2 | 2 ) + 2 Re { j · I m { h 2 c ∗ 1 c 2 }} = R e { h 1 } ( | c 1 | 2 + | c 2 | 2 ) (66) which clearly does not ha ve any cross-terms and therefore c 1 and c 2 can be decoded separately . V I . C O N C L U S I O N W e deri ved the d i versity order of some mult iple antenna multi-user cancellation and detection scheme s. The common property of these detection methods is the usage of Alamouti and quasi-orthogona l spa ce-time block codes. For dete cting J users each h aving N transmit antenna s, these schemes require only J a ntennas at the receiv er . Our analysis s howed that when having M rec ei ve a ntennas, the array-process ing s cheme provides the div ersity orde r of N ( M − J + 1) . In addition, we proved that regardless of the nu mber of u sers or recei ve antennas , when u sing maximum-likelihood deco ding we get the full trans mit and rece i ve div ersities, i.e. N M , similar to the no-interference scen ario. DRAFT 19 A P P E N D I X Proof of Lemma 2 : Plugg ing in C by C =          I B 1 B T 2 · · · B 1 B T M − 1 B 2 B T 1 I · · · B 2 B T M − 1 . . . . . . . . . . . . B M − 1 B T 1 B M − 1 B T 2 · · · I          (67) we ge t C ·          a 1 B 1 a 2 B 2 . . . a M − 1 B M − 1          =          ( a 1 + a 2 β 2 + · · · + a M − 1 β M − 1 ) B 1 ( a 1 β 1 + a 2 + · · · + a M − 1 β M − 1 ) B 2 . . . ( a 1 β 1 + a 2 β 2 + · · · + a M − 1 ) B M − 1          (68) and s olving for a i and λ we g et a i = P M − 1 i =1 a i β i λ + β i − 1 for i = 1 , 2 , · · · , M − 1 (69) W e ca n always normalize a i coefficients s uch that P M − 1 i =1 a i β i = 1 . The refore, a i = 1 λ + β i − 1 and P M − 1 i =1 a i β i = P M − 1 i =1 β i λ + β i − 1 = 1 (70) which proves the lemma.  Proof of Lemma 3 : It is clear w hy no ne of the roots can be zero. Becaus e, if it is so, we will have P M i =1 β i β i − 1 = 1 which is impos sible since β i − 1 < 0 an d β i > 0 by definition. Also, from the definition we know β i s are distinct. Therefore, without loss of ge nerality we ca n a ssume β 1 < · · · < β M − 1 . It will the n be easy to show that f ( λ ) = P M − 1 i =1 β i λ + β i − 1 is monoton ic over the following M − 1 intervals (1 − β M − 1 , 1 − β M − 2 ) , · · · , (1 − β 2 , 1 − β 1 ) , (1 − β 1 , ∞ ) (71) For the first M − 2 intervals, f ( λ ) takes all the values from − ∞ to + ∞ . For the last interval, it takes ∞ wh en λ is at the proximity of 1 − β 1 and 0 whe n λ goe s to ∞ . The refore, it takes the value of 1 in all of these M − 1 intervals exactly o nce, wh ich proves the lemma.  DRAFT 20 A C K N O W L E D G E M E N T The a uthors would like to thank Seyed Jalil Ka zemitabar from Sharif University of T e chnology for his useful comments. R E F E R E N C E S [1] J. Kazemitabar and H. Jafarkhani, “Multiuser Interference Cancellation and Detection for Users with More than T wo Transmit Antennas, ” IEEE T rans. on Comm. , pp. 574-583, April 2008. [2] N. Al-Dhahir and A. R. Calderbank, “F urther results on interference cancellation and space-time block codes, ” Pr oc. 35th Asilomar conf. on Signals, Systems and Computers , pp. 257-262, October 2001. [3] A. F . Naguib, N. Seshadri, and A. R. Calderbank, “ Applications of space-time block codes and interference suppression for high capacity and high data rate wir eless systems, ” Pr oc. 32nd Asilomar Conf. Signals, Systems and Computers , pp. 1803-1810, 1998. [4] S .M. 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