Diversity-Multiplexing Tradeoff in Selective-Fading Multiple-Access MIMO Channels

Di v ersity-Multiplexing T radeof f in Selecti v e- F ading Multiple-Access MIMO Channels Pedro Coronel, Markus G ¨ artner and Helmut B ¨ olcskei Communication T echnology Laboratory ETH Zurich, 8092 Zurich, Switzerland E-mail: { pco, gaertner , boelcskei } @nari.ee.ethz.ch Abstract — W e establish the optimal diversity-multiplexing (DM) tradeoff of coherent selectiv e-fading multiple-access multiple-input multiple-output (MIMO) channels and pro vide corresponding code design criteria. As a byproduct, on the concep- tual level, we find an interesting relation between the DM tradeoff framework and the notion of dominant error event regions which was first introduced in the A WGN case by Gallager , IEEE T rans. IT , 1985. This relation allows to accurately characterize the error mechanisms in MIMO fading multiple-access channels. In partic- ular , we find that, for a given rate tuple, the maximum achievable diversity order is determined by the error event that dominates the total error probability exponentially in SNR. Finally , we show that the distributed space-time code construction proposed recently by Badr and Belfiore, Int. Zurich Seminar on Commun. , 2008, satisfies the code design criteria derived in this paper . I . I N T RO D U C T I O N The div ersity-multiplexing (DM) tradeoff framework intro- duced by Zheng and Tse allows to efficiently characterize the information-theoretic performance limits of communication over multiple-input multiple-output (MIMO) fading channels both in the point-to-point [1] and in the multiple-access (MA) case [2]. For coherent 1 point-to-point flat-fading channels, DM tradeoff optimal code constructions have been reported in [3]–[6]. The optimal DM tradeoff in point-to-point selective-f ading MIMO channels was recently characterized in [7]. In the MA case, the optimal DM tradeoff is known only for flat-fading channels [2]. Corresponding DM tradeoff optimal code constructions were recently reported in [8], [9]. Contributions: The aim of this paper is to characterize the DM tradeoff in selecti ve-fading MIMO multiple-access channels (MA Cs) and to deriv e corresponding code design criteria. As a byproduct, on the conceptual le vel, we find an interesting relation between the DM tradeoff frame work and the notion of error e vent regions which was first introduced in the A WGN case by Gallager in [10] and recently applied to MIMO fading MA Cs in [11]. This relation leads to an accurate characterization of the error mechanisms in MIMO fading MACs. Furthermore, we extend the techniques introduced in [7] for computing the DM tradeoff in This work was supported in part by the Swiss National Science Foundation (SNF) under grant No. 200020-109619 and by the STREP project No. IST -026905 MASCO T within the Sixth Frame work Programme of the European Commission. 1 Throughout the paper , we shall consider the coherent case, where the recei ver has perfect channel state information (CSI) and the transmitter does not have CSI, but is aware of the channel law . point-to-point selecti ve-fading channels to the MA case. Finally , we prov e that the distributed space-time block codes proposed in [9] satisfy the code design criteria deriv ed in this paper . Notation: M T and M R denote, respectiv ely , the number of transmit antennas for each user and the number of receiv e antennas. The set of all users is U = { 1 , 2 , . . . , U } , S is a subset of U with ¯ S and |S | denoting its complement in U and its cardinality , respectively . The superscripts T and H stand for transposition and conjugate transposition, respectiv ely . A ⊗ B and A  B denote, respectiv ely , the Kronecker and Hadamard products of the matrices A and B . If A has the columns a k ( k = 1 , 2 , . . . , m ), v ec( A ) = [ a T 1 a T 2 . . . a T m ] T . A 1 / 2 de- notes the positiv e semidefinite square root of A . For index sets I 1 ⊆ { 1 , 2 , . . . , n } and I 2 ⊆ { 1 , 2 , . . . , m } , A ( I 1 , I 2 ) stands for the (sub)matrix consisting of the rows of A index ed by I 1 and the columns of A index ed by I 2 . The nonzero eigen v alues of the n × n Hermitian matrix A , sorted in ascending order , are denoted by λ k ( A ) , k = 1 , 2 , . . . , rank( A ) . The Kronecker delta function is defined as δ n,m = 1 for n = m and zero otherwise. If X and Y are random variables (R Vs), X ∼ Y denotes equiv alence in distribution. E X is the expectation op- erator with respect to (w .r .t.) the R V X . The random vector x ∼ C N ( 0 , C ) is multiv ariate circularly symmetric complex Gaussian with E  xx H  = C . The functions f ( x ) and g ( x ) are said to be exponentially equal, denoted by f ( x ) . = g ( x ) , if lim x →∞ log f ( x ) log x = lim x →∞ log g ( x ) log x . Exponential inequality , denoted by ˙ ≥ and ˙ ≤ , is defined analogously . I I . C H A N N E L A N D S I G N A L M O D E L W e consider a selective-f ading MA C where U users, with M T transmit antennas each, communicate with a single receiv er with M R antennas. The corresponding input-output relation is given by y n = r SNR M T U X u =1 H u,n x u,n + z n , n = 0 , 1 , . . . , N − 1 , (1) where the index n corresponds to a time, frequency or time- frequency slot and SNR denotes the per-user signal-to-noise ratio at each receive antenna. The vectors y n , x u,n and z n denote, respectively , the M R × 1 receiv e signal vector , the M T × 1 transmit signal v ector corresponding to the u th user , and the M R × 1 circularly symmetric complex Gaussian noise vector satisfying E  z n z H n 0  = δ n,n 0 I M R , all for the n th slot. W e assume that the receiver has perfect knowledge of all channels and the transmitters do not have CSI but are aware of the channel law . W e restrict our analysis to spatially uncorrelated Rayleigh fading channels so that, for a given n , H u,n has i.i.d. C N (0 , 1) entries. The channels corresponding to dif ferent users are assumed to be statistically independent. W e do, howe ver , allow for correlation across n for a giv en u , and assume, for simplicity , that all scalar subchannels have the same correlation function so that, in summary , E { H u,n ( i, j ) ( H u 0 ,n 0 ( i 0 , j 0 )) ∗ } = R H ( n, n 0 ) δ u,u 0 δ i,i 0 δ j,j 0 , for i, i 0 = 1 , 2 , . . . , M R , j, j 0 = 1 , 2 , . . . , M T , and n, n 0 = 0 , 1 , . . . , N − 1 . The cov ariance matrix R H is obtained from the channel’ s time-frequency correlation function [12]. In the sequel, we let ρ , rank( R H ) . For any set S = { u 1 , u 2 , . . . , u |S | } , we stack the corresponding users’ channel matrices for a given slot index n according to H S ,n = [ H u 1 ,n H u 2 ,n . . . H u |S | ,n ] . (2) W ith this notation, it follows that E  v ec( H S ,n ) (vec( H S ,n 0 )) H  = R H ( n, n 0 ) I |S | M T M R . (3) I I I . P R E L I M I NA R I E S Assuming that all users employ i.i.d. Gaussian codebooks 2 , the set of achiev able rate tuples ( R 1 , R 2 , . . . , R U ) for a giv en channel realization { H u,n } is given by R = ( ( R 1 , R 2 , . . . , R U ) : ∀S ⊆ U , R ( S ) ≤ 1 N N − 1 X n =0 log det  I + SNR M T H S ,n H H S ,n  ) (4) where R ( S ) = P u ∈S R u . If a giv en rate tuple ( R 1 , R 2 , . . . , R U ) / ∈ R , we say that the channel is in outage w .r .t. this rate tuple. Denoting the corresponding outage ev ent as O , we have P ( O ) = P   [ S ⊆ U O S   (5) where the S -outage ev ent O S is defined as O S = ( { H S ,n } N − 1 n =0 : 1 N N − 1 X n =0 log det  I + SNR M T H S ,n H H S ,n  < R ( S ) ) . (6) Our goal is to characterize (5) as a function of the rate tuple ( R 1 , R 2 , . . . , R U ) in the high-SNR regime and to find criteria on the users’ codebooks guaranteeing that the corresponding error probability behaves exponentially in SNR like P ( O ) . T o this end, we shall emplo y the DM tradeoff framew ork [1], which, in its MA version [2], will be briefly summarized next. 2 A standard argument along the lines of that used to obtain [1, Eq. 9] shows that this assumption does not entail a loss of optimality in the high SNR regime, relev ant to the DM tradeoff. In the DM tradeoff framework, the data rate of user u scales with SNR as R u ( SNR ) = r u log SNR , where r u denotes the mul- tiplexing rate. Consequently , a sequence of codebooks C r u ( SNR ) , one for each SNR, is required. W e say that this sequence of codebooks constitutes a family of codes C r u operating at multi- plexing rate r u . The family C r u is assumed to have block length N . At any giv en SNR, C r u ( SNR ) contains codewords X u = [ x u, 0 x u, 1 . . . x u,N − 1 ] satisfying the per-user power constraint T r  X u X H u  ≤ M T N , ∀ X u ∈ C r u , u = 1 , 2 , . . . , U. (7) Since we are dealing with a MA C, the o verall family of codes is given by C r = C r 1 × C r 2 × · · · × C r U , where r = ( r 1 , r 2 , . . . , r U ) denotes the multiplexing rate tuple. At a giv en SNR, the corresponding codebook C r ( SNR ) contains SNR N r ( U ) codew ords with r ( U ) = P U u =1 r u . The DM tradeof f realized by C r is characterized by the function d ( C r ) = − lim SNR →∞ log P e ( C r ) log SNR where P e ( C r ) is the total error probability (that is, the probability for the recei ver to make a detection error for at least one user) ob- tained through maximum-likelihood (ML) detection. The optimal DM tradeoff curve d ? ( r ) = sup C r d ( C r ) , where the supremum is taken over all possible families of codes satisfying the power constraint (7), quantifies the maximum achiev able div ersity gain as a function of the multiplexing rate tuple r [1]. Since the outage probability P ( O ) is a lo wer bound (exponentially in SNR) on the error probability of any coding scheme [2, Lemma 7], we hav e d ? ( r ) ≤ − lim SNR →∞ log P ( O ) log SNR (8) where the outage ev ent, defined in (5) and (6), is w .r .t. the rates R u ( SNR ) = r u log SNR , ∀ u . As an extension of the result for the flat-fading case [2], we shall show in this paper that (8) holds with equality also for selectiv e-fading MA Cs. Ho wev er, just like in the case of point-to-point channels, a direct characterization of the right-hand side (RHS) of (8) for the selective-f ading case seems analytically intractable since one has to deal with the sum of correlated (recall that the H u,n are correlated across n ) terms in (6). In the ne xt section, we sho w how the technique introduced in [7] for characterizing the DM tradeof f of point-to-point selecti ve- fading MIMO channels can be extended to the MA case. I V . C O M P U T I N G T H E O P T I M A L D M T R A D E O FF C U RV E A. Lower bound on P ( O S ) First, we derive a lower bound on the individual terms P ( O S ) that will be key in establishing the optimal DM tradeof f. W e start by noting that for any set S ⊆ U , Jensen’ s inequality provides the following upper bound: 1 N N − 1 X n =0 log det  I + SNR M T H S ,n H H S ,n  ≤ log det  I + SNR M T N H S H H S  , J S (9) where the “Jensen channel” [7] is defined as H S = ( [ H S , 0 H S , 1 . . . H S ,N − 1 ] , if M R ≤ |S | M T , [ H H S , 0 H H S , 1 . . . H H S ,N − 1 ] , if M R > |S | M T . (10) Consequently , H S has dimension m( S ) × N M( S ) with m( S ) , min( |S | M T , M R ) and M( S ) , max( |S | M T , M R ) . In the follo wing, we say that the ev ent J S occurs if the Jensen channel H S is in outage w .r .t. the rate r ( S ) log SNR , where r ( S ) = P u ∈S r u , i.e., J S , { J S < r ( S ) log SNR } . From (9) we can conclude that, obviously , P ( J S ) ≤ P ( O S ) . W e shall next characterize the Jensen outage probabil- ity analytically . Recalling (3), we start by writing H S = H w ( R T / 2 ⊗ I M( S ) ) , where R = R H , if M R ≤ |S | M T , and R = R T H , if M R > |S | M T , and H w is the i.i.d. C N (0 , 1) matrix with the same dimensions as H S giv en by H w = ( [ H w, 0 H w, 1 . . . H w,N − 1 ] , if M R ≤ |S | M T , [ H H w, 0 H H w, 1 . . . H H w,N − 1 ] , if M R > |S | M T . Here, H w,n denotes i.i.d. C N (0 , 1) matrices of dimension M R × |S | M T . Since H w U ∼ H w , for an y unitary U , and R H and R T H hav e the same eigen values, we get H S H H S ∼ H w ( Λ ⊗ I M( S ) ) H H w , where Λ = diag { λ 1 ( R H ) , λ 2 ( R H ) , . . . , λ ρ ( R H ) , 0 , . . . , 0 } . With H w = H w ([1 : m( S )] , [1 : ρ M( S )]) , it was shown in [7] that P ( J S ) is nothing but the outage probability of an effecti ve MIMO channel with ρ M( S ) transmit and m( S ) receiv e antennas and satisfies P ( J S ) . = P  log det  I + SNR H w H H w  < r ( S ) log SNR  . = SNR − d S ( r ( S )) (11) where we infer from the results in [1] that d S ( r ) is the piecewise linear function connecting the points ( r , d S ( r )) for r = 0 , 1 , . . . , m( S ) , with d S ( r ) = (m( S ) − r )( ρ M( S ) − r ) . (12) Since, as already noted, P ( O S ) ≥ P ( J S ) , it follows from (11) that P ( O S ) ˙ ≥ SNR − d S ( r ( S )) . (13) W e shall see below that (13) is a key ingredient for establishing the optimal DM tradeoff. B. Err or event analysis Follo wing [2], [10], we decompose the total error probability into 2 U − 1 disjoint error events according to P e ( C r ) = X S ⊆ U P ( E S ) (14) where the S -error event E S corresponds to all the users in S being decoded incorrectly and the remaining users being decoded correctly . More precisely , we have E S , n ( ˆ X u 6 = X u , ∀ u ∈ S ) ∧ ( ˆ X u = X u , ∀ u ∈ ¯ S ) o (15) where X u and ˆ X u are, respectively , the transmitted and ML-decoded codewords corresponding to user u . The following result establishes a DM tradeoff optimal code design criterion for a specific error ev ent E S . Theor em 1: For ev ery u ∈ S , let C r u hav e block length N ≥ ρ |S | M T , and set λ n = λ n  ( R T H  ( P u ∈S E H u E u )  for n = 1 , 2 , . . . , ρ |S | M T , where E u = X u − X 0 u and X u , X 0 u ∈ C r u ( SNR ) . Furthermore, define Λ ρ |S | M T m( S ) ( SNR ) , min E u = X u − X 0 u , ∀ u ∈S X u , X 0 u ∈C r u ( SNR ) m( S ) Y k =1 λ k . (16) If there exists an  > 0 such that Λ ρ |S | M T m( S ) ( SNR ) ˙ ≥ SNR − ( r ( S ) −  ) , (17) then, under ML decoding, P ( E S ) ˙ ≤ SNR − d S ( r ( S )) . Pr oof: W e start by deri ving an upper bound on the average (w .r .t. the random channel) pairwise error probability (PEP) of an S -error event. Let the codewords of C r ( SNR ) be given by X = [ X T 1 X T 2 . . . X T U ] T . Based on (15), we note that E u = X u − X 0 u is nonzero for u ∈ S and E u = 0 for u ∈ ¯ S . Assuming, without loss of generality , that S = { 1 , 2 , . . . , |S |} , the probability of the ML decoder mistakenly deciding in fa vor of the codeword X 0 when X was actually transmitted can be upper bounded in terms of X − X 0 = [ E T 1 E T 2 . . . E T |S | 0 . . . 0 ] T as P ( X → X 0 ) ≤ E { H S ,n } ( exp − SNR 4M T N − 1 X n =0 T r  H S ,n e n e H n H H S ,n  ! ) (18) where T r  H S ,n e n e H n H H S ,n  =   P u ∈S H u,n e u,n   2 with H S ,n defined in (2) and e n = [ e T u 1 ,n e T u 2 ,n · · · e T u |S | ,n ] T , where e u,n = x u,n − x 0 u,n . Defining H S = [ H S , 0 H S , 1 · · · H S ,N − 1 ] , we get from (18) P ( X → X 0 ) ≤ E H S  exp  − SNR 4M T T r  H S diag  e n e H n  N − 1 n =0 H H S   = E H w  exp  − SNR 4M T T r  H w ΥΥ H H H w   (19) where we used H S = H w ( R T / 2 H ⊗ I |S | M T ) with H w an M R × N |S | M T matrix with i.i.d. C N (0 , 1) entries and Υ = ( R T / 2 H ⊗ I |S | M T ) diag { e n } N − 1 n =0 . (20) Noting that Υ H Υ = R T H  ( P u ∈S E H u E u ) and using the fact that the nonzero eigen v alues of ΥΥ H in (19) equal the nonzero eigen values of Υ H Υ , it follows, by assumption, that ΥΥ H has precisely ρ |S | M T nonzero eigen values. The remainder of the proof proceeds along the lines of the proof of Theorem 1 in [13] 3 . In particular, we split and subsequently bound the S -error 3 For the point-to-point case, the criterion in [13, Theorem 1] requires the m = min(M T , M R ) smallest eigenv alues of the effectiv e codeword difference matrix to satisfy Q m k =1 λ k ˙ ≥ SNR − ( r −  ) , whereas Theorem 1 in [7] requires λ m min ˙ ≥ SNR − ( r −  ) . It can readily be seen that the latter condition implies the former and, hence, the criterion in [13] provides a relaxed optimality condition. probability as P ( E S ) = P ( E S , J S ) + P  E S , ¯ J S  = P ( J S ) P ( E S |J S ) | {z } ≤ 1 + P  ¯ J S  | {z } ≤ 1 P  E S | ¯ J S  ≤ P ( J S ) + P  E S | ¯ J S  . (21) As detailed in [13], the code design criterion (17) yields the following upper bound on the second term in (21): P  E S | ¯ J S  ≤ SNR N r ( S ) exp − SNR / m( S ) 4M T ! . (22) In contrast to the Jensen outage probability which satisfies P ( J S ) . = SNR − d S ( r ( S )) , (22) decays exponentially in SNR. Hence, upon inserting (22) into (21), we get P ( E S ) ˙ ≤ P ( J S ) , and can therefore conclude that P ( E S ) ˙ ≤ SNR − d S ( r ( S )) . In summary , for ev ery E S , we have a sufficient condition on {C r u : u ∈ S } for P ( E S ) to be exponentially upper bounded by P ( J S ) . Based on this result, we shall next establish the optimal DM tradeoff for the MA C and provide corresponding design criteria on the family C r . C. Optimal code design W e start by noting that (5) implies P ( O ) ≥ P ( O S ) for any S ⊆ U , which combined with (13) giv es rise to 2 U − 1 lower bounds on P ( O ) . F or a given multiplexing rate tuple r , the tightest lower bound (exponentially in SNR) corresponds to the set S that yields the smallest SNR exponent d S ( r ( S )) . More precisely , the tightest lower bound is characterized by P ( O ) ˙ ≥ SNR − d S ? ( r ( S ? )) (23) where the dominant outage ev ent corresponds to the set S ? = arg min S ⊆ U d S ( r ( S )) . (24) Next, we sho w that, for an y multiplexing rate tuple, the total error probability P e ( C r ) can be made exponentially equal to the lo wer bound in (23) by appropriate design of the users’ codebooks. As a direct consequence thereof, using P e ( C r ) ˙ ≥ P ( O ) [2, Lemma 7] and (23), we then obtain that d S ? ( r ( S ? )) constitutes the optimal DM tradeoff of the selectiv e-fading MIMO MAC. Theor em 2: The optimal DM tradeoff of the selective-f ading MIMO MAC in (1) is giv en by d ? ( r ) = d S ? ( r ( S ? )) , that is d ? ( r ) = (m( S ? ) − r ( S ? ))( ρ M( S ? ) − r ( S ? )) . (25) Moreov er , if the family of codes C r satisfies (17) for every S ⊆ U , then d ( C r ) = d ? ( r ) . (26) Pr oof: Inserting the upper bound (21) into (14), we get P e ( C r ) ≤ X S ⊆U  P ( J S ) + P  E S | ¯ J S  ˙ ≤ X S ⊆U P ( J S ) (27) . = SNR − d S ? ( r ( S ? )) (28) where (27) is a consequence of the assumption that C r satisfies (17) for e very S ⊆ U and (28) follo ws from (11) together with the definition (24). With P e ( C r ) ˙ ≥ P ( O ) [2, Lemma 7], combining (23) and (28) yields P e ( C r ) . = P ( O ) . = SNR − d S ? ( r ( S ? )) . (29) Since, by definition, d ( C r ) ≤ d ? ( r ) , using (8), we can conclude from (29) that d ( C r ) = d ? ( r ) = d S ? ( r ( S ? )) . As a consequence of Theorem 2, the maximum achiev able div ersity order is determined by the error ev ent that dominates the total error probability exponentially in SNR. T o see this, let E S 0 denote the S -error event that dominates the ov erall error probability so that, based on (14), P e ( C r ) . = P ( E S 0 ) . By (29), we necessarily hav e d S ? ( r ( S ? )) = d ( E S 0 ) , where d ( E S 0 ) = − lim SNR →∞ log P ( E S 0 ) log SNR . Since C r satisfies (17), Theorem 1 yields d ( E S 0 ) ≥ d S 0 ( r ( S 0 )) and, hence, we get d S ? ( r ( S ? )) ≥ d S 0 ( r ( S 0 )) . Howe ver , by the definition of S ? , we also have d S ? ( r ( S ? )) ≤ d S 0 ( r ( S 0 )) which implies d S ? ( r ( S ? )) = d S 0 ( r ( S 0 )) . Thus, P e ( C r ) . = P ( E S ? ) , which is to say that the optimal DM tradeoff is gi ven by the SNR exponent corresponding to the dominant error ev ent. Example: W e assume M T = 3 , M R = 4 , and ρ = 2 . For U = 2 , the 2 2 − 1 = 3 possible error events are denoted by E 1 (user 1 only is in error), E 2 (user 2 only is in error) and E 3 (both users are in error). The SNR exponents of the corresponding error probabilities are obtained from (12) as d u ( r u ) = (3 − r u )(8 − r u ) , u = 1 , 2 , d 3 ( r 1 + r 2 ) =  4 − ( r 1 + r 2 )  12 − ( r 1 + r 2 )  . (30) Based on (30), we can now explicitly determine the dominant error ev ent for e very multiplexing rate tuple r = ( r 1 , r 2 ) . In Figure 1, we plot the rate regions dominated by the dif ferent error e vents. Note that the SNR exponent of the error probability is zero whene ver r 1 > 3 , r 2 > 3 or r 1 + r 2 > 4 . In the rate re gion domi- nated by E 1 , we hav e d 1 ( r 1 ) < d 2 ( r 2 ) and d 1 ( r 1 ) < d 3 ( r 1 + r 2 ) , implying that the SNR exponent of the total error probability equals d 1 ( r 1 ) , i.e., the SNR exponent that would be obtained in a point-to-point selective-f ading MIMO channel with M T = 3 , M R = 4 , and ρ = 2 . The same reasoning applies to the rate region dominated by E 2 and, hence, we can conclude that, in the sense of the DM tradeoff, the performance in regions E 1 and E 2 is not affected by the presence of the second user . In contrast, in the area dominated by E 3 , we hav e d 3 ( r 1 + r 2 ) < d u ( r u ) , u = 1 , 2 , which is to say that the multiuser interference does hav e an impact on the DM tradeoff and reduces the di versity gain that would be obtained if only one user were present. V . A N O P T I M A L C O D E F O R T H E FL AT - FA D I N G C A S E Satisfying the code design criterion (17) for every S ⊆ U is non-tri vial and systematic procedures for designing DM tradeoff optimal codes are an important open problem. In this section, we show that the algebraic code construction proposed recently in [9] for flat-fading MA Cs with single-antenna users satisfies (17) for ev ery S ⊆ U and any multiplexing rate tuple 4 . 4 In [9], the DM tradeof f optimality of the proposed code is shown for r 1 = r 2 . 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 Fig. 1. Dominant error ev ent regions for the two-user MIMO MA C with M T = 3 , M R = 4 , and ρ = 2 . W e start by briefly re viewing the code construction described in [9] for a system with M T = 1 , M R = 2 , U = 2 , N = 2 , and ρ = 1 . For each user u , let A u denote a QAM constellation with 2 R u ( SNR ) points carved from Z [ i ] = { k + il : k , l ∈ Z } , where i = √ − 1 . The proposed code spans two slots so that the vector of information symbols corresponding to user u is giv en by s u = [ s u, 0 s u, 1 ] , where s u, 0 , s u, 1 ∈ A u . Using the unitary transformation matrix U underlying the Golden Code [5], the 1 × 2 code word X u is obtained as X T u = U s T u =  x u σ ( x u )  with U = 1 √ 5  α αϕ ¯ α ¯ α ¯ ϕ  (31) where ϕ = 1+ √ 5 2 denotes the Golden number with corresponding conjugate ¯ ϕ = 1 − √ 5 2 , α = 1 + i − iϕ and ¯ α = 1 + i − i ¯ ϕ . Here, σ denotes the generator of the Galois group of the quadratic extension Q ( i, √ 5) o ver Q ( i ) = { k + il : k , l ∈ Q } given by σ : Q ( i, √ 5) → Q ( i, √ 5) a + b √ 5 7→ a − b √ 5 . (32) Moreov er , one of the users, say user 2, multiplies the symbol corresponding to the first slot by a constant γ ∈ Q ( i ) , resulting in the ov erall 2 × 2 codeword X =  x 1 σ ( x 1 ) γ x 2 σ ( x 2 )  . (33) As shown in [9], for any γ 6 = ± 1 and any two X , X 0 according to (33), it holds that det( ∆ ) 6 = 0 , where ∆ = X − X 0 . For the so-defined construction we have the following result. Theor em 3: For any multiplexing rate tuple r , the algebraic code construction in [9] satisfies (17) for any S ⊆ U . Pr oof: W e start by assuming that at any giv en SNR, user u carves out 2 R u ( SNR ) −  log SNR points from Z [ i ] for some  > 0 , i.e., |A u | = SNR r u −  . In order to satisfy the po wer constraint (7), we scale A u by SNR − ( r u −  ) / 2 so that, due to the linearity (over C ) of the transformation in (31), the codeword corresponding to user u is gi ven by SNR − ( r u −  ) / 2 X u . From (33) and the linearity of the mapping σ ov er Q ( i, √ 5) , the codew ord difference matrix is obtained as E =  SNR − ( r 1 −  ) / 2 e 1 SNR − ( r 1 −  ) / 2 σ ( e 1 ) SNR − ( r 2 −  ) / 2 γ e 2 SNR − ( r 2 −  ) / 2 σ ( e 2 )  (34) where e u = x u − x 0 u , u = 1 , 2 . Next, we note that in the flat- fading case R T H  ( E H E ) = E H E . In particular , considering user 1, i.e., S = { 1 } , we hav e |S | = 1 and m( S ) = 1 so that, from (16), we obtain Λ 1 1 ( SNR ) = SNR − ( r 1 −  ) min e 1 ( | e 1 | 2 + | σ ( e 1 ) | 2 ) . Letting X T 1 = Us T 1 and ( X 0 1 ) T = U ( s 0 1 ) T and since U is unitary , we get ( | e 1 | 2 + | σ ( e 1 ) | 2 ) = || s 1 − s 0 1 || 2 ≥ 2 d 2 min , where d min is the (nonzero) minimum distance in A 1 . W e there- fore conclude that Λ 1 1 ( SNR ) . = SNR − ( r 1 −  ) . For user 2, a similar argument shows that Λ 1 1 ( SNR ) . = SNR − ( r 2 −  ) and, hence, the construction satisfies the criteria arising from (17) for S = { 1 } and S = { 2 } . For S = { 1 , 2 } , note that |S | = 2 and m( S ) = 2 so that Λ 2 2 ( SNR ) = min E | det( E ) | 2 . From (34), we get | det( E ) | 2 = SNR − ( r 1 + r 2 − 2  ) | det( ∆ ) | 2 . 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