Diversity Gain Region for MIMO Fading Broadcast Channels

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1 ITW4, San Antonio, Texas, October 4 9, 4 Diversity Gain Region for MIMO Fading Broadcast Channes Lihua Weng, Achieas Anastasoouos, and S. Sandee Pradhan Eectrica Engineering and Comuter Science Det. University of Michigan, Ann Arbor, MI weng, anastas, radhanv}@umich.edu Abstract In this work, we introduce the notion of the diversity gain region for a muti-user channe. This region secifies the set of diversity-gain vectors that are simutaneousy achievabe by a users in the muti-user channe. This is done by associating different robabiities of error for different users, contrary to the traditiona aroach where a singe robabiity of system error is considered. We derive an inner bound (achievabe region) and an outer bound for the diversity gain region of a MIMO fading broadcast channe. I. INTRODUCTION It is we-known that the error exonent for a singe-user channe rovides the rate of exonentia decay of the average robabiity of error as a function of the bock ength of the codebooks [1]. The concet of the error exonent was extended to a Gaussian mutie access channe (MAC) in [], where an uer bound on the robabiity of system error (i.e., the robabiity that any user is in error) was derived for random codes. Recenty, Zheng et a. considered error exonents in high signato-noise ratio () aroximation, caed diversity gains, for muti-inut-muti-outut (MIMO) fading singe-user channes [3]. In many aications of muti-user networks, different users might have different reiabiity requirements. For instance, in an uink (or downink) of a ceuar system, a user running an FTP aication might have more stringent reiabiity requirements than a user running a mutimedia aication which is designed for gracefu degradation. Based on the traditiona aroaches [] which consider a singe robabiity of system error, a network can ony be designed to satisfy the most stringent reiabiity requirement. This might resut in a mismatch of resources aocation, and thus, it is inherenty subotima. To address this issue, we introduced the notion of error exonent region (EER) for a genera muti-user channe in [4]. For a given oerating oint, i.e., a rate-air (R 1, R ), the error exonent region consists of a achievabe error exonents when the channe is oerated at that oint. An EER deends on the channe oerating oint, and for a given channe, there are numerous EERs deending on which oerating oint we consider. In this aer, we focus on the EER at high for a MIMO fading broadcast channe. The rest of the aer is structured as foows. In Section II, we introduce the notion of mutiexing gain region (MGR) and diversity gain region (DGR). The MGR and DGR are the channe caacity region (CCR) and the error exonent region at high resectivey. In Section III, we derive the DGR inner bound using two strategies - channe-sitting and suerosition. In Section IV, we roose a unified encoding scheme, generaized-suerosition, which incudes both channe-sitting and suerosition as secia cases. In Section V, we derive the DGR outer bound. Two imortant resuts are aso shown in Section V. First, either one of the two users in the broadcast channe can achieve the singe-user diversity gain if the data rates are ow. Second, the DGR inner bound and DGR outer bound are tight for equa diversity gains if the broadcast channe is symmetric. We concude our work in Section VI. II. MULTIPLEXING GAIN REGION AND DIVERSITY GAIN REGION Consider a MIMO fading broadcast channe with m transmit antennas and n 1 and n receive antennas for user 1 and user. The channe mode is m H 1X + Z 1 (1) Y = m H X + Z. () The channe fading matrices between the transmitter and the receiver 1 and the receiver are reresented by an n 1 m matrix H 1 and an n m matrix H. We assume that H 1 and H remain constant over a bock ength, and change to a new indeendent reaization in the next bock ength. H 1 and H have i.i.d. entries and each entry has a comex Gaussian distribution CN (, 1). We assume that fading matrices are known by the receivers but unknown by the transmitter. The channe inut X is an m matrix and is normaized such that the average transmit ower at each antenna is one. The noise Z 1 and Z are n 1 and n matrices with i.i.d. entries CN (, 1). In [3], an encoding scheme C() (a famiy of codes) in a MIMO fading singe-user channe is said to achieve mutiexing gain r and diversity gain d if R(SN R) im og = r, im og P e () og = d, (3) where R() and P e () are the rate and the average robabiity of error of the code C(SN R) resectivey. Define R() = r og and P e (). = d if equaities hod in the imit, and,,, are defined simiary. Foowing the same notations in [3], we define an encoding scheme

2 C() to achieve mutiexing gain air (r 1, r ) and diversity gain air (d 1, d ) in a MIMO fading broadcast channe if R 1 () im og = r 1, R () im og = r, im im og P e1 () og og P e () og = d 1, = d, where R 1 (), R (), P e1 (), P e () are the rates and the robabiities of codeword error for user 1 and user. Mutiexing gain region (MGR) is thus defined as the set of a achievabe mutiexing gain air (r 1, r ) for a encoding schemes. The MGR is the CCR at high. As mentioned earier, an EER deends on the oerating oint (R 1, R ). Simiary, given a mutiexing gain air (r 1, r ), we define the diversity gain region (DGR) as the set of a achievabe diversity gain air (d 1, d ). The diversity gain region is the EER at high. Before continuing, we derive a MGR inner bound and a MGR outer bound for a MIMO fading broadcast channe. We summarize the resut in the foowing theorem. Theorem 1 For a MIMO fading broadcast channe with m transmit antennas and n 1, n receive antennas, an MGR inner bound is MGR in = (r 1, r ) : and an MGR outer bound is MGR out = (r 1, r ) : α 1, (4) (5) r 1 min(m, n 1 ) + r 1}, (6) min(m, n ) r 1 (min(m, n 1 ) min(m, n )) + + α minmin(m, n 1 ), min(m, n )}, r (min(m, n ) min(m, n 1 )) + where (x) + = max(x, ). + (1 α) minmin(m, n 1 ), min(m, n )} }, (7) Before deriving the DGR inner and outer bounds, we summarize a the definitions of the diversity gains used in this aer. More detai exanations and exact formuas of these diversity gains are given in ater sections. (1) d m,n, (r): random coding diversity gain with m transmit antennas, n receive antennas, and bock ength. () d ex m,n, (r): exurgated diversity gain with m transmit antennas, n receive antennas, and bock ength. (3) d out m,n(r): outage diversity gain with m transmit antennas and n receive antennas. (4) m,n,, (r): naive singe-user diversity gain with m transmit antennas, n receive antennas, bock ength, and normaized side-interference ower (1 ). (5) d n m,n,, (r): 1,, non-uniform ower random coding diversity gain with m transmit antennas, n receive antennas, bock ength, normaized ower (1 1) for ength, and normaized ower (1 ) for ength (1 ). III. ACHIEVABLE DIVERSITY GAIN REGION Before continuing, et s review the diversity-mutiexing tradeoff for a MIMO fading singe-user channe derived in [3]. It was shown in [3] that both the random coding diversity gain d m,n, (r) and the exurgated diversity gain d ex m,n, (r) are achievabe in a MIMO fading singe-user channe with m transmit antennas, n receive antennas, and bock ength. In addition, the diversity gain was shown to be uer bounded by the outage diversity gain d out m,n(r), where d out m,n(r) is the iecewise inear function connecting the oints (k, d out m,n(k)) = (k, (m k)(n k)), k Z +. Finay, it was aso shown that d m,n, (r) and d out m,n(r) coincide for m + n 1. For the MIMO fading broadcast channe considered in this aer, we roose two encoding strategies - channe-sitting and suerosition. In channe-sitting, we aocate symbos to user 1 and (1 ) symbos to user inside each bock ength, where = k and 1 k 1, k Z (see Fig. 1). Thus, the achievabe diversity gains are d cs 1 = maxd m,n1,( r 1 d cs ), dex,( r 1 r )} (8) = maxd m,n,(1 )( ), dex m,n 1 ( r,(1 ) )}, (9) 1 where the suerscrit cs denotes channe-sitting. ) User 1 User User 1 User User 1 User User 1 User N Figure 1: Channe-sitting For suerosition encoding, the channe inut is X = X 1 + X, where X 1 and X have i.i.d. entries CN (, 1) and CN (, (1 ) ) ( < < 1) resectivey. We use two decoding strategies - joint maximum-ikeihood (ML) decoding and naive singe-user decoding. In joint ML decoding, user 1 decodes his own message î based on the air (i, j) maximizing P (Y 1 X 1 (i), X (j)) and user decodes his own message ĵ based on the air (i, j) maximizing P (Y X 1 (i), X (j)), where X 1 (i) and X (j) are the i th and j th codewords for user 1 and user resectivey. We can derive the achievabe diversity gains using joint ML decoding as d s 1 = mind out (r 1 ), d out (r 1 + r )} = d out (r 1 + r ) (1) d s = min d out ( r ), dout m,n (r 1 + r )} (11)

3 when the bock ength m + max(n 1, n ) 1, where the suerscrit s denotes suerosition. For the bock ength < m + max(n 1, n ) 1, the outage diversity gain in (1), (11) is reaced by the random coding diversity gain. In (1), d out (r 1 ) accounts for the error event when user 1 decodes X 1 (i) as a wrong codeword, but decodes X (j) correcty, which is referred as the tye 1 error in [], and d out (r 1 +r ) accounts for the error event when user 1 decodes both messages X 1 (i), X (j) as wrong codewords, which is referred as the tye 3 error in []. In (11), d out ( r ) accounts for the tye error when user decodes X (j) as a wrong codeword, but decodes X 1 (i) correcty, and d out (r 1 + r ) accounts for the tye 3 error when user decodes both messages X 1 (i), X (j) as wrong codewords. In naive singe-user decoding, user 1 simy regards user s interference as noise. In this case, user 1 can achieve diversity gain m,n (r 1,, 1), where m,n (r 1,, 1) is caed the naive singe-user diversity gain and its derivation and exact formua are given in the foowing subsection. Since user 1 can choose either joint ML decoding or naive singe-user decoding, the foowing diversity gains are achievabe d s 1 = maxd out (r 1 + r ),,,(r 1 )} (1) d s = min d out ( r ), dout (r 1 + r )}, (13) where we assume m + max(n 1, n ) 1. Simiary, we can exchange the roe of user 1 and user in suerosition encoding, i.e. we may assume X and X 1 have i.i.d. entries CN (, 1) and CN (, (1 ) ) ( < < 1) resectivey. Thus, the foowing diversity gains are aso achievabe d s 1 = min d out ( r 1 ), dout (r 1 + r )} (14) d s = maxd out (r 1 + r ),,,(r )}. (15) In Fig. (a), the soid curve is the boundary of the achievabe DGR by suerosition using joint ML decoding, and the dashed curve is the boundary of the achievabe DGR by suerosition using joint ML decoding and naive singe-user decoding, which merges with the soid curve at (d 1, d ) = (4.5, 9) and (d 1, d ) = (9, 4.5). The dotted curve in Fig. (a) is the boundary of the achievabe DGR by channe-sitting. A. Naive Singe-user Decoding We now derive an achievabe diversity gain using naive singe-user decoding, thus obtaining an exicit exression for m,n,, (r). If we use suerosition encoding for the MIMO fading broadcast channe, i.e., the channe inut X = X 1 + X, we can write the channe outut for user 1 as m H 1(X 1 + X ) + Z 1, (16) where X 1 and X have i.i.d entries CN (, 1) and CN (, (1 ) ) ( < < 1) resectivey. If we decode user 1 s message using naive singe-user decoding, i.e., d (a) d 1 d Figure : Diversity gain region for m = n 1 = n = 4, = 6, r 1 = r =.5 (a) channe-sitting(dotted) and suerosition(soid and dashed), (b) union of channe-sitting and suerosition (soid), and generaized-suerosition (dash-dotted). user 1 simy treats user as noise, we can derive an achievabe diversity gain for user 1. This is equivaent to considering the foowing MIMO fading side-interference singe-user channe Y = H(X + S) + Z, (17) m where H is an n m matrix with i.i.d. entries CN (, 1), and Z and S are n noise and side-interference matrices with i.i.d. entries CN (, 1) and CN (, (1 ) ) resectivey. The channe inut X is an m matrix and is normaized such that the average transmit ower at each antenna is one. Define (x) + = max(x, ) and R n + as the set of rea n-vectors with nonnegative eements. We summarize the resut of the achievabe diversity gain of the side-interference channe in the foowing theorem. Theorem For a MIMO fading side-interference channe oerated at a mutiexing gain r with m transmit antennas, n receive antennas, the otima robabiity of detection error is uer-bounded by where m,n,,(r) = min and B = P e () dns m,n,, (r), (18) α B c d 1 (b) (i 1 + m n )α i + [ (1 α i ( α i ) + ) + r ]}, (19) α R + α 1 α α ; } (1 α i ( α i ) + ) + r () B c = R + B. (1)

4 Proof : See Aendix B. C 1 C 1,1 C,1 IV. IMPROVED ACHIEVABLE DIVERSITY GAIN REGION BY GENERALIZED-SUPERPOSITION ENCODING In this section, we introduce a new encoding scheme, generaized-suerosition, which incudes channe-sitting and suerosition as secia cases. Before introducing generaizedsuerosition, we first derive the non-uniform ower random coding diversity gain d n m,n,, (r). 1,, Consider a random codebook CB with M codewords (see Fig. 3 (a)). Denote C i the i th codeword with bock ength in the codebook CB. Denote C i (k) the k th eement in the codeword C i. Each random variabe C i (k) is i.i.d. with CN (, (1 1) ) for 1 k, and CN (, (1 ) ) for + 1 k, where = q and q, q Z. If we ot E C i (k) } versus k, we have a function with vaue (1 1) for 1 k and with vaue (1 ) for + 1 k (see Fig. 3(a)). Extending the derivation of the random coding diversity gain d m,n, (r) in [3], we can derive the non-uniform ower random coding diversity gain d n m,n,, 1,, (r) for a non-uniform ower random codebook CB as d n m,n,, 1,,(r) = min α B c [ ( 1 α i ) + + (1 ) and B = (i 1 + m n )α i + ( α i ) + r ]}, α R + α 1 α α ; ( 1 α i ) + + (1 ) () } ( α i ) + r (3) B c = R + B. (4) The derivation for d n m,n,, 1,, (r) is a straight forward extension of the derivation for d m,n, (r) and is omitted. We are now ready to introduce generaized-suerosition. In generaized-suerosition, we construct two indeendent random codebooks CB 1 and CB. Denote C 1,i and C,j the i th and j th codewords with bock ength in codebooks CB 1 and CB resectivey. Denote C 1,i (k) the k th eement in the codeword C 1,i and C,j (k) the k th eement in the codeword C,j. Each random variabe C 1,i (k) is i.i.d. with CN (, 1) for 1 k, and CN (, (1 1) ) for + 1 k. Simiary, each random variabe C,j (k) is i.i.d. with CN (, (1 ) ) for 1 k and CN (, 1) for +1 k (see Fig. 3(b)). If we ot E C 1,i (k) } versus k, we have a function with vaue 1 for 1 k and with vaue (1 1) for + 1 k. Simiary, E C,j (k) } is a function with vaue (1 ) for 1 k and with E C i (k) } ( 1 1) ( 1 ) C CB C M (a) ) k E C 1,i (k) } E C,j (k) } ( 1 1) C 1, CB 1 C 1, M ) ( 1 Figure 3: (a) Non-uniform ower random codebook, (b) Generaizedsuerosition. vaue 1 for +1 k. It is cear that both suerosition and channe-sitting are secia cases of generaized-suerosition. In the receivers, we use a mixture of joint ML and naive singe-user decoding. We summarize the resut in the foowing theorem. Theorem 3 For a MIMO fading broadcast channe with m transmit antennas, n 1, n receive antennas for user 1, user, and bock ength m + max(n 1, n ) 1, an achievabe DGR gs is given by DGR gs (r 1, r ) = k (b) ) C, CB C, M (d 1, d ) : = k, k, k Z, ) 1 1, 1, d 1 max min d n m,n (r 1,,1, 1, 1), d out (r 1 + r ) },,, ( r } 1 ) (5) d max min d n,,,1, (r ), d out (r 1 + r ) }, r }},(1 ), 1 ( 1 ). (6) Note that,, ( r 1 ) shoud be interreted as for = and,(1 ), 1 ( r 1 ) shoud be interreted as for = 1 in the above theorem, since the diversity gain is zero for any scheme with encoding bock ength. In Fig. (b), the soid curve is the boundary of the union of the channe-sitting and suerosition achievabe DGRs, and the dash-dotted curve is the boundary of the achievabe DGR by generaized-suerosition, which merges with the soid curve at (d 1, d ) = (3.6, 1.5) and (d 1, d ) = (1.5, 3.6). V. OUTER BOUND FOR DIVERSITY GAIN REGION k

5 For a MIMO fading broadcast channe, the robabiity of decoding error for user i can aways be ower bounded by the robabiity of decoding error for user i oerating over a ointto-oint channe defined by the margina distribution P (Y i X), for i = 1,. Further, we use the fact that the erformance of a broadcast channe deends ony on the margina distributions P (Y 1 X) and P (Y X), not on the joint distribution P (Y 1, Y X). To be secific, consider another broadcast channe with margina distributions the same as those in the origina broadcast channe, i.e., P (Y 1 X) = P (Y 1 X) and P (Y X) = P (Y X), but with P (Y 1, Y X) P (Y 1, Y X) in genera. The DGR of this new broadcast channe is the same as the DGR of the origina broadcast channe, since the robabiity of error of each user deends ony on the corresonding margina distributions [5]. If we now aow the two receivers in the new broadcast channe to cooerate, we have a singe-user channe, whose robabiity of error (using an otima receiver), P e, shoud be ess than or equa to the robabiity of system error P e in the origina broadcast channe. Using the union bound, it is aso easy to show that P e maxp e1, P e }, where P ei denotes the robabiity of error for user i in the origina broadcast channe. Coecting a these ideas, we have the foowing outer bound for the DGR d 1 d out (r 1 ) (7) d d out (r ) (8) mind 1, d } maxd out (r 1 + r ), d out (r 1 + r )}. (9) In Fig. 4, the soid curve is the boundary of the DGR inner bound and the dash-dotted curve is the boundary of the DGR outer bound. Two imortant resuts are observed in Fig. 4: (i) the DGR inner bound and the DGR outer bound are tight at the right, ower corner and at the eft, uer corner; (ii) the DGR inner bound and the DGR outer bound are tight at d 1 = d. Resut (i) imies that the aearance of the second (first) user does not affect the first (second) user since the first (second) user achieves the singe-user diversity gain. We summarize these two resuts in the foowing theorem. Theorem 4 Consider a MIMO fading broadcast channe with bock ength > m + min(n 1, n ) 1. Either user 1, with a mutiexing gain r 1 < 1, can achieve his maximum singe-user diversity gain if user s mutiexing gain r < (1 r 1 )(1 m+n 1 1 ) min(m, n ), or user, with a mutiexing gain r < 1, can achieve his maximum singe-user diversity gain if user 1 s mutiexing gain r 1 < (1 r )(1 m+n 1 ) min(m, n 1 ). If the MIMO fading broadcast channe is symmetric, i.e., n 1 = n, the DGR inner bound and the DGR outer bound are tight at d 1 = d. Proof: The resut that the DGR inner bound and the DGR outer bound are tight at d 1 = d for a symmetric MIMO fading broadcast channe is a direct consequence of the DGR uer bound (9). Consider user 1 with a mutiexing gain r 1 < 1 and define 1 = (1 r 1 )(1 m+n 1 1 ). For suerosition encoding with channe inut X = X 1 + X, where X 1 and X have i.i.d. entries CN (, 1) and CN (, (1 1) ) resectivey, the achievabe diversity gain by naive singe-user decoding for user 1 is,, 1 (r 1 ) = d out (r 1 ) (note that 1 is the argest vaue this equation,, 1 (r 1 ) = d out (r 1 ) hods, i.e.,,, (r 1) < d out (r 1 ) for 1 < ). The achievabe diversity gain for user is min 1 d out ( r 1 ), d out (r 1 + r )}, so we have a ositive diversity gain for user if r < 1 min(m, n ) = (1 r 1 )(1 m+n 1 1 ) min(m, n ). The roof for user, with a mutiexing gain r < 1, achieving his maximum singe-user diversity gain if r 1 < (1 r )(1 m+n 1 ) min(m, n 1 ) is simiar. d Figure 4: Diversity Gain Region for m = n 1 = n = 4, = 6, r 1 = r =.5: inner (soid) and outer (dash-dotted) bounds. VI. CONCLUSION In this work, we introduce the notion of mutiexing gain region and diversity gain region, which are the channe caacity region and the error exonent region at high. We derive the DGR inner and outer bounds for a MIMO fading broadcast channe. We rove that when the data rates are ow, either user 1 or user can achieved the singe-user diversity gain. For a symmetric MIMO fading broadcast channe, the DGR inner bound and the DGR outer bound are tight at equa diversity gains (d 1 = d ). The concet of the MGR and the DGR is very genera and can be aied to other muti-user channes, such as a MIMO fading mutie access channe. VII. APPENDIX A. Proof of Theorem 1 It is sufficient to rove that (7) is an outer bound for the MGR, since (6) is achieved by suerosition encoding (see (1), (13)). Without oss of generaity, we may assume that a the nonzero eigenvaues of H 1H 1 and H H are one, where H 1 and H are the conjugate transose of H 1 and H. The robabiity of the event A = λ [ ɛ ɛ ] minm,n} } goes to one as, where ɛ is any ositive constant, λ is a vector of the nonzero eigenvaues of H 1H 1 (or H H ), and n = n 1 (or d 1

6 n ). The integra of mutiexing gains over the range A c can be shown to be negigibe, so we may assume ɛ λ i ɛ, where λ i is any entry of λ and i = 1,,, min(m, n). Since ɛ is any ositive constant, we can make ɛ arbitrariy sma and assume that λ is a vector with every entry equa to one. Based on the assumtion that a nonzero eigenvaues are in the vaue of one, we can consider the foowing equivaent broadcast channe (after singuar vaue decomosition of the fading matrices) m V 1I 1 W 1X + Z 1 (3) Y = m V I W X + Z. (31) I 1 = [I 1,ij ] is an n 1 m matrix with I 1,ii = 1 for 1 i min(m, n 1 ) and I 1,ij = otherwise, where I 1,ij is the eement on the i th row and the j th coumn of I 1. I = [I,ij ] is an n m matrix with I,ii = 1 for 1 i min(m, n ) and I,ij = otherwise, where I,ij is the eement on the i th row and the j th coumn of I. V 1 and V are n 1 n 1 and n n unitary random matrices resectivey. W 1 and W are m m unitary random matrices. Z 1 and Z are min(m, n 1 ) and min(m, n ) matrices with i.i.d. entries CN (, 1). It is we-known that the caacity region of a broadcast channe deends ony on the margina distributions, so we can consider the caacity region of the foowing broadcast channe m V 1I 1 W 1X + Z 1 (3) Y = m V I W 1X + Z. (33) If we now assume that the channe matrices V 1, V, W 1 are known both at the transmitter and the receivers in the broadcast channe defined in (3), (33), we can consider the foowing equivaent broadcast channe Ỹ 1 = m I X 1 + Z 1 (34) Ỹ = m I X + Z, (35) where X = W 1X, Ỹ1 = V 1Y 1, Ỹ = V Y, Z 1 = V 1Z 1, and Z = V Z. The MGR of the broadcast channe defined in (34), (35) is an outer bound of the MGR of the origina fading broadcast channe defined in (1), (). However, it is easy to see that the MGR of the broadcast channe defined in (34), (35) is exacty (7). This cometes the roof. B. Proof of Theorem At high, we can ignore the integra of the robabiity of error over the range H / R +, so We rove that P (error, H B c ) is uer bounded by dns m,n,, (r). The roof that P (H B) is aso uer bounded by dns m,n,, (r) is trivia. Assume X(), X(1) are two ossibe transmitted codewords and X = X(1) X(). Suose X() is transmitted, then the robabiity that a receiver wi make a detection error in favor of X(1), conditioned on a certain reaization of H, is P (X() X(1) H) = P ( 1 m m ( ex 4m m ) 1 HH + I H X F ) 1 HH + I H X w F, (37) where w is the additive noise with variance 1/, I is an identity matrix, and F is the Frobenius norm. Average over the ensembe of random codes, we have the average airwise error robabiity (PEP) given the channe reaization H P EP (H) I +. = m ( m HH + I) 1 HH ( m HH + I) 1 (1 αi ( αi)+ ) +, (38) where λ i = α i (α 1 α ) and λ i s are the nonzero eigenvaues of HH. Ay the union bound and integrate over B c, we have P (error, α B c ) dns m,n,, (r). (39) REFERENCES [1] R. G. Gaager, A sime derivation of the coding theorem and some aications, IEEE Trans. Information Theory, vo. 11, no. 1,. 3 18, Jan [], A ersective on mutiaccess channes, IEEE Trans. Information Theory, vo. 31, no., , Mar [3] L. Zheng and D. N. C. Tse, Diversity and mutiexing: A fundamenta tradeoff in mutie-antenna channes, IEEE Trans. Information Theory, vo. 49, no. 5, , May 3. [4] L. Weng, A. Anastasoouos, and S. S. Pradhan, Error exonent region for Gaussian mutie access channes, in Proc. Internationa Symosium on Information Theory, Chicago, IL, June 4. [5] H. Sato, An outer bound on the caacity region of broadcast channes, IEEE Trans. Information Theory, vo. 4, no. 3, , May P e () P (H B) + P (error, H B c ). (36)