Remarks on the Criteria of Constructing MIMO-MAC DMT Optimal Codes
Hsiao-feng, Jyrki Lahtonen, Roope Vehkalahti, Camilla Hollanti
aa r X i v : . [ c s . I T ] A ug Remarks on the Criteria of ConstructingMIMO-MAC DMT Optimal Codes
Hsiao-feng (Francis) Lu, Jyrki Lahtonen, Roope Vehkalahti, Camilla Hollanti
Abstract — In this paper we investigate the criteria proposedby Coronel et al. for constructing MIMO MAC-DMT optimalcodes over several classes of fading channels. We first give acounterexample showing their DMT result is not correct when thechannel is frequency-selective. For the case of symmetric MIMO-MAC flat fading channels, their DMT result reduces to exactly thesame as that derived by Tse et al. , and we therefore focus on theircriteria for constructing MAC-DMT optimal codes, especiallywhen the number of receive antennas is sufficiently large. In suchcase, we show their criterion is equivalent to requiring the codesof any subset of users to satisfy a joint non-vanishing determinantcriterion when the system operates in the antenna pooling regime.Finally an upper bound on the product of minimum eigenvaluesof the difference matrices is provided, and is used to show anyMIMO-MAC codes satisfying their criterion can possibly existonly when the target multiplexing gain is small.
I. I
NTRODUCTION
Consider a MIMO multiple-access (MAC) channel with U users, each equipped with n t transmit antennas. The receiverat the base-station is assumed to have n r receive antennas.Code matrices of each user are transmitted synchronouslyand independently in M channel uses. Let H u,m denote thechannel matrix of the u th user at the m th channel use; thengiven transmitted signal x u,m , the received signal vector is y m = r SNR n t U X u =1 H u,m x u,m + z m , m = 0 , , · · · , M − , (1)where z m is the noise vector with i.i.d. C N (0 , entries atthe m th channel use. The transmitted signal x u,m is requiredto satisfy the power constraint of E (cid:13)(cid:13) x u,m (cid:13)(cid:13) = 1 .Following [1], the MIMO-MAC channel is assumed to befrequency selective and has the following assumptions:1) the entries of the channel matrix H u,m are i.i.d. C N (0 , complex Gaussian random variables,2) the channels correspond to different users are statisti-cally independent, and3) the channels { H u,m : m = 0 , · · · , M − } seen by the u th user can be correlated in time.Due to possible time-correlation, Coronel et al. [1] defined an ( M × M ) matrix R H whose entries are given by E { H u,m ( i, j ) H u ′ ,m ′ ( i, j ) } = R H ( m, m ′ ) δ u,u ′ to model the time correlations, where by H u,m ( i, j ) we meanthe ( i, j ) th entry of the channel matrix H u,m .Let X u = { X u } be the block length M code of user u ,consisting of ( n t × M ) code matrices satisfying the averagepower constraint E k X u k F ≤ n t M . The codebook X u has size |X u | . = SNR Mr u such that the user u transmits at multiplexinggain r u .Let U = { , , · · · , U } denote the set of all users. For anysubset of users, S ⊆ U , the probability of users in S being inoutage is lower bounded by Pr {O S } ˙ ≥ SNR − d S ( r ( S )) where r ( S ) := P u ∈S r u . For integral values of r ( S ) ,Coronel et al. showed [1] that d S ( r ( S )) := [ m ( S ) − r ( S )] [ ρM ( S ) − r ( S )] where ρ = rank ( R H ) , m ( S ) := min {|S| n t , n r } , and M ( S ) := max {|S| n t , n r } , and that d S ( r ( S )) is a piecewiselinear function for non-integral values of r ( S ) For any specific choice of codebook X u , Coronel et al. [1] studied the error performance of such code and provideda criterion based on the eigenvalues of the difference codematrices such that X u has error probability upper bounded bythe outage probability. We reproduce their result in the theorembelow. Theorem 1 ( [1], [2]):
For every u ∈ S ⊆ U , let X u haveblock length M ≥ ρ |S| n t . Let the nonzero eigenvalues of R H ⊙ (cid:0)P u ∈S E † u E u (cid:1) , where E u = X u − X ′ u and X u = X ′ u ∈ X u be given in ascending order - at every SNR levelby λ k , k = 1 , , · · · , ρ |S| n t . Furthermore, set Λ ρ |S| n t m ( S ) ( SNR ) := min E u : u ∈S m ( S ) Y k =1 λ k . (2)If there exists an ǫ > independent of SNR and r ( S ) suchthat Λ ρ |S| n t m ( S ) ( SNR ) ˙ ≥ SNR − ( r ( S ) − ǫ ) , (3)then under ML decoding, the event E S of the users of S in er-ror has probability upper bounded by P ( E S ) ˙ ≤ SNR − d S ( r ( S )) . (cid:4) They then derived the optimal MAC-DMT tradeoff of thisMIMO-MAC frequency selective fading channel and at thesame time, provided a sufficient criterion for codes to achievesuch MAC-DMT.
Theorem 2 ( [1]):
The optimal tradeoff of the frequencyselective-fading MIMO MAC channel defined in (1) is givenby d ∗ ( r ) = d S ∗ ( r ( S ∗ )) , where S ∗ := arg min S⊆U d S ( r ( S )) is the dominant outage set for error performance. That is, theoptimal MAC-DMT is given by d ∗ ( r ) = ( m ( S ∗ ) − r ( S ∗ )) ( ρM ( S ∗ ) − r ( S ∗ )) , (4) where r = ( r , · · · , r U ) is the vector of the multiplexing gainsof all users. Moreover, if the overall family of codes X = X × · · · × X U satisfies (3) for the dominant outage set S ∗ andfor every S 6 = S ∗ , there exists ǫ > such that Λ ρ |S| n t m ( S ) ( SNR ) ˙ ≥ SNR − ( γ S − ǫ ) (5)where ≤ γ S ≤ Γ S := d − S ( d S ∗ ( r ( S ∗ ))) (6)then X achieves the optimal DMT d ∗ ( r ) . (cid:4) We remark that in a preceding publication [2] Coronel etal. had provided a different criterion that is stronger thancondition (5). The difference between this criterion and (5)is that the former replaces γ S by r ( S ) . Obviously since S ∗ isthe dominant outage set, we have r ( S ) ≤ Γ S for all S ⊆ U ,and (5) is more relaxed than that in [2]. Moreover, we notethat (5) can be further relaxed by replacing γ S with Γ S . Asa result, in all subsequent discussions we will use Γ S insteadof γ S . A. DMT of Frequency Selective Channels: A Counterexample
Unfortunately, the MAC-DMT (4) claimed by Coronel et al. [1] for the frequency selective fading channel is not correct.This can be seen from the counterexample below.
Example 1:
Consider a single-user, point-to-point, multi-block fading MIMO channel [3], [4]. In terms of the channelmodel (1) we assume the channel has the following input-output relation y m = r SNR n t H m x m + z m , m = 0 , , · · · , M − where M = 2 n t and the channel matrices are given by H m = (cid:26) H , m = 0 , , · · · , n t − H n t , m = n t , · · · , n t − . (7)The entries of H and H n t are i.i.d. C N (0 , random vari-ables. Thus (7) models a quasi-static MIMO Rayleigh blockfading channel in which coding is allowed to be spread overtwo independent fading blocks, each fixed for n t channel uses.Given multiplexing gain r , it is well known [3], [4] that theoptimal MAC-DMT of this channel equals d ∗ ( r ) = 2 ( n t − r ) ( n r − r ) (8)since the two fading blocks are statistically independent.On the other hand, from Theorem 2 we note the followings.1) The time correlation matrix R H is R H = (cid:20) (cid:21) where is the all-one matrix of size ( n t × n t ) . Therefore ρ = rank ( R H ) = 2 .2) The dominant outage set S ∗ = { } since this is asingle-user, point-to-point channel. Hence m ( S ∗ ) =min { n t , n r } and M ( S ∗ ) = max { n t , n r } .Substituting the above into (4) of Theorem 2, Coronel et al. claimed however the DMT of this channel is d ∗ Coronel ( r ) = (min { n t , n r } − r ) (2 max { n t , n r } − r ) . (9) D i v e r s i t y ga i n d Multi−Block DMTCoronel DMT
Fig. 1. The two MAC-DMTs (8) (in solid line) and (9) (in dotted line) forsingle-user multi-block fading channel.
The above disagrees completely with the well-known multi-block DMT result (8) of [3], [4]. In Fig. 1 we plot the twoMAC-DMTs, (8) and (9), for the case of n t = n r = 2 . It isseen that the latter (9) is too optimistic on the diversity gain atall values of r . Hence we conclude that (4) claimed by Coronel et al. is not correct for the case of frequency selective fadingchannels. (cid:4) The above example shows the MAC-DMT result (4) claimedby Coronel et al. [1], [2] is not correct for the case of single-user MIMO frequency-selective channel. As their result failsin the point-to-point scenario, it will not hold in the MAC caseeither.
B. DMT of Flat-Fading MIMO-MAC Channels
For the flat fading case let us consider a MIMO-MACchannel with U users, communicating independently and syn-chronously over a quasi-static Rayleigh flat fading channel.Because of the quasi-static assumption we shall have H u,m = H u , m = 0 , , · · · , M − . Entries of H u are i.i.d. C N (0 , random variables. The time-correlation matrix R H in this case is given by R H = M × M ,where M × M is the all-one matrix of size ( M × M ) , hence ρ = rank ( R H ) = 1 .Here we are interested in the symmetric MIMO-MAC flatfading channel [5] in which all users transmit at the same levelof multiplexing gain, i.e. r = · · · = r U = r . Applying theseassumptions to Theorem 2 the resulting MAC-DMT is givenby d ∗ ( r ) = min ≤ s ≤ U ( sn t − sr ) ( n r − sr ) which is exactly the same as that shown by Tse et al. [5].In the remaining of this paper we will focus on the investiga-tion of criterion (5) in Theorem 2 for constructing MAC-DMToptimal codes. Specifically, when n r ≥ U n t ,1) if the system operates in antenna pooling regime, inSection II we will show the relaxation of γ S in (5)is not possible, and we must have γ S = |S| r in (5).Also, we will relate criterion (5) to the non-vanishingdeterminant (NVD) criterion [6]–[8] that is well-known for constructing approximately universal point-to-pointspace-time codes.2) in Section III, based on an analysis of the minimumdeterminant we will provide a stronger result that canbe applied to all range of multiplexing gain r .II. C ONNECTION T O NVD C
RITERION
In this section, we will focus on the case of MIMO-MACflat fading channels and will assume1) n r ≥ U n t and2) the multiplexing gain r u = r for all users.Following Theorem 1, let X u be the codebook of the u th userwith block length M ≥ U n t and consist of ( n t × M ) codematrices. As R H = M × M , for any subset S of users thematrix in Theorem 1 simplifies to R H ⊙ (cid:0)P u ∈S E † u E u (cid:1) = (cid:0)P u ∈S E † u E u (cid:1) , where E u = X u − X ′ u and X u , X ′ u ∈ X u .For the ease of presentation, below we define a notation forconcatenating matrices that will be used frequently in the laterdiscussions. Definition 1:
Let X , · · · , X s be matrices with the samenumber of columns; then we define the vertical concatenationof these matrices as M ( X , · · · , X s ) := X ... X s . With the above notation, set ∆ S := M ( E u , · · · , E u S ) (10)where S = { u , · · · , u S } ; then the nonzero eigenvalues of (cid:0)P u ∈S E † u E u (cid:1) are the same as those of ∆ S ∆ †S . Clearly,rank (∆ S ) ≤ |S| n t . Since n r ≥ U n t by assumption, we have m ( S ) = |S| n t . As Λ |S| n t m ( S ) ( SNR ) is comprised of the productof the least m ( S ) nonzero eigenvalues of (cid:0)P u ∈S E † u E u (cid:1) , (2)forces rank (∆ S ) = m ( S ) = |S| n t . Hence we can rewrite (2)as Λ |S| n t m ( S ) ( SNR ) = min E u : u ∈S m ( S ) Y k =1 λ k = min E u : u ∈S det (cid:16) ∆ S ∆ †S (cid:17) , where λ k are the nonzero eigenvalues of (cid:0)P u ∈S E † u E u (cid:1) ,or equivalently, the eigenvalues of ∆ S ∆ †S , arranged in theascending order. Moreover, condition (3) can be reformulatedas Λ |S| n t m ( S ) ( SNR ) = min E u : u ∈S det (cid:16) ∆ S ∆ †S (cid:17) ˙ ≥ SNR − ( |S| r − ǫ ) , (11)and similarly condition (5) can be rewritten as Λ |S| n t m ( S ) ( SNR ) = min E u : u ∈S det (cid:16) ∆ S ∆ †S (cid:17) ˙ ≥ SNR − (Γ S − ǫ ) . (12)Our first goal in this section is to relate the above conditions,(11) and (12), to the well-known NVD condition [6]–[8], [8]for constructing approximately universal space-time codes. Tothis end, we recall the transmit-receive channel model of [6]is Y = κ U X u =1 H u C u + Z (13) where Y is the ( n r × M ) received signal matrix, H u is the ( n r × n t ) channel matrix of user u , and C u ∈ C u is thecorresponding code matrix with power constraint E k C u k ≤ SNR rnt . κ is a scaling parameter with κ . = SNR − rnt such that E k κC u k . = E (cid:13)(cid:13)(cid:13)q SNR n t X u (cid:13)(cid:13)(cid:13) . = SNR and the twomodels, (1) and (13), agree on the same level of input-SNR.In [6], [8], the NVD condition states for any single code C u , if min D u det (cid:0) D u D † u (cid:1) ≥ SNR for any D u = C u − C ′ u and C u = C ′ u ∈ C u , then the error probability of C u isupper bounded by the corresponding outage probability. Weremark that this NVD condition can be relaxed such that onlyexponential inequality min D u det (cid:0) D u D † u (cid:1) ˙ ≥ SNR is neededwithout affecting the proof in [6]. The NVD condition can beextended to the MAC case as well (see (15) below).Contrast to the channel model (1) we see X u = SNR − r nt C u (14)With the above in mind, substituting (14) into (10) gives ∆ S = SNR − r nt ∆ C where ∆ C = M ( D u , · · · , D u S ) , D u = C u − C ′ u , and C u and C ′ u are the code matrices associated with X u and X ′ u ,respectively. Hence det (cid:16) ∆ S ∆ †S (cid:17) = h SNR − r nt i |S| n t det (cid:16) ∆ C ∆ † C (cid:17) . After clearing the common terms, condition (11) is equivalentto D ( S ) := min D u : u ∈S det (cid:16) ∆ C ∆ † C (cid:17) ≥ SNR ǫ (15)for some ǫ > . Remark 2.1:
We remark that condition (15) is exactly theNVD condition shown in [6], [8]. Hence, Theorem 1 whenrestricted to the case of flat fading channels, is equivalent toan earlier result of [6], see Theorems 2 and 3 of [6]. (cid:4)
Next, we turn our attention to the condition (12). Againapplying the key relation (14) to (12) and after clearing thecommon terms, condition (12) can be reformulated in termsof the difference matrix ∆ C . Thus in order to achieve theMAC-DMT optimality Theorem 2 requires for every S ⊆ U D ( S ) = min D u : u ∈S det (cid:16) ∆ C ∆ † C (cid:17) ˙ ≥ SNR − (Γ S −|S| r − ǫ ) . (16)For the ease of handling the parameter Γ S , below we willrestrict ourselves to the case of n r = U n t . Recall thesymmetric MIMO-MAC DMT of this case [5] is given by d ∗ ( r ) = d S ∗ ( r ( S ∗ )) = d ( r ) , r ∈ h , Un t U +1 i d U ( U r ) , r ∈ h Un t U +1 , n t i (17)where d ( r ) = ( n t − r )( n r − r ) represents the DMT for S ∗ = { } when r ∈ h , Un t U +1 i . This interval of r is coined the single-user performance regime by Tse et al. [5]. d U ( r ) = ( U n t − U r )( n r − U r ) is the DMT for S ∗ = U and dominates theMAC-DMT when r ∈ h Un t U +1 , n t i . Such regime is called the antenna-pooling regime . Here we focus on the latter regime,i.e. the case when S ∗ = U , and distinguish two kinds of outagesets.
1) For S with |S| = 1 , it is easy to verify that Γ := Γ S (cid:12)(cid:12) |S| =1 = d − ( d U ( U r )) > r for U > since r in the antenna pooling regime and d ( r ) > d U ( U r ) . Hence (16) requires the individualcodes C u of every user u to satisfy D ( { u } ) = min D u det (cid:0) D u D † u (cid:1) ˙ ≥ SNR − (Γ − r ) , i.e. allowing vanishing determinant. In the above, wehave dropped the constant ǫ for simplicity.2) For the case of S = U , we have Γ U := Γ S (cid:12)(cid:12) |S| = U = d − U ( d U ( U r )) =
U r and (16) requires the overall code C = C × · · · × C U tosatisfy D ( U ) = min D u : u ∈U det (cid:16) ∆ C ∆ † C (cid:17) ˙ ≥ SNR , (18)where ∆ C = M ( D , · · · , D U ) .The above analysis shows that if n r = U n t and if r is in theantenna pooling regime, the criterion in Theorem 2 allows eachuser to use codes with vanishing determinant, but it expectsthe overall code C to satisfy the NVD criterion. A furtherinvestigation of this will show that the latter constraint actuallyleads to the total-NVD criteria for all subsets of users, and inparticular that the individual codes must be NVD as well. Theorem 3:
For H u,m = H u , n r ∈ [ U n t , ( U + 1) n t ) , r u = r ∈ h n r U +1 , n t i , i.e. the system operates in the antenna poolingregime, the condition (5) of Theorem 2 for S 6 = S ∗ is the sameas D ( S ) = min D u : u ∈S det (cid:16) ∆ C ∆ † C (cid:17) ˙ ≥ SNR . Thus (5) holds only for ≤ γ S ≤ r |S| , and the resultingcriterion is therefore called the joint NVD criterion . Proof:
First, let S = U and let ∆ C = M ( D , · · · , D U ) be the matrix comprising of the difference matrices fromall users. Fisher’s inequality [9] on positive definite matricesshows det (cid:16) ∆ C ∆ † C (cid:17) ≤ U Y u =1 det (cid:0) D u D † u (cid:1) . Combining the above with (18) yields the conditionSNR ˙ ≤ D ( U ) ≤ U Y u =1 min D u det (cid:0) D u D † u (cid:1) = U Y u =1 D ( { u } ) (19)where the second inequality follows from the fact that theusers do not cooperate. If every individual code has vanish-ing determinant, then the condition (18) cannot be satisfied.Thus there exists some user v ∈ U such that D ( { v } ) =min D v det (cid:0) D v D † v (cid:1) . = SNR p v for some p v > . Notethat user v transmits at multiplexing gain r . Simply alongthe lines of the proof of approximately universal codes in [6]it can be shown that the single user DMT achieved by C v is d ( r − p v ) . This means user v transmits at multiplexing gain r and achieves diversity gain d ( r − p v ) > d ( r ) , a contradictionto the point-to-point DMT. Therefore p v = 0 , and the aboveimplies that every user u achieves D ( { u } ) . = SNR , i.e. the individual codes must be NVD, forced by the condition (18).The rest of the proof proceeds with induction on the size of S . Assume we have shown for any S ⊆ U with |S| = s and D ( S ) ˙ ≥ SNR . Let S = S ′ ∪ { v } and we will show the sameNVD criterion holds for S ′ . To establish this claim, againapplying Fisher’s inequality gives D ( S ) ≤ D ( S ′ ) · D ( { v } ) .As D ( { v } ) . = SNR , we conclude that D ( S ′ ) ˙ ≥ SNR . Theclaim on the range of γ S follows obviously.The above theorem shows that when the symmetric MIMO-MAC system operates in the antenna pooling regime, condition(5) in Theorem 2 on the product of the minimum eigenvaluesof the difference matrices Λ ρ |S| n t m ( S ) ( SNR ) is equivalent toasking the minimum determinant D ( S ) for all subsets of usersto be nonvanishing. Furthermore, it shows the relaxation on γ S is not possible. In the next section, we will further investigatethe condition (16) and provide a general bound on D ( U ) thatcan be applied to all values of r .III. G ENERAL B OUNDS O N D ( U ) F ROM P IGEON H OLE P RINCIPLE
In this section we will restrict ourselves to the case of n r ≥ U n t receiving antennas. Assume that we are to designa MIMO-MAC system over a flat fading channel for U users,each transmitting synchronously and independently with n t antennas at multiplexing gain r . We can describe each user’ssignals as ( n t × M ) complex matrices with M = U n t required by Theorem 2. It is natural to assume that each useris maximally using all the degrees of freedom available tohim/her. Therefore, the lattice of the individual code shouldbe of full rank n = 2 n t M and the corresponding code C u isgiven by C u = ( C u = n X k =1 a u,k B u,k : a u,k ∈ Z , − N ≤ a u,k ≤ N ) where the set { B u,k : k = 1 , . . . , n } is the basis of lattice L u of user u . In other words, the parameter a u,k is the N -PAM coordinate of the lattice L u of user u . Equivalently, aQAM-oriented reader may then view C u as a linear dispersionof n = n t M independently chosen N -QAM symbols. As |C u | = | N | n t M , we shall set N . = SNR r nt (20)such that user u transmits at multiplexing gain r . The code C u has average power E k C u k . = N = SNR rnt , hence ascaling constant κ = SNR − rnt is need such that κ C u meetsthe power constraint E k κC u k . = SNR. Moreover, it shouldbe noted that the code C u is the same as that discussed in thechannel model (13). For any such code C u and for any numberof users, in this section we will aim to provide an upper boundon D ( U ) = min D u : u ∈U det (cid:16) ∆ C ∆ † C (cid:17) of condition (16).To describe the idea we lead off with the simpler case n t = 1 , where user uses only single antenna, and is thustransmitting a vector c u ∈ C u ⊂ L u ∈ C U since M = U . Set d u = c u − c ′ u for some c u = c ′ u ∈ C u . Let us fix the differencesignals d u for all but one user, say, fix the vectors d , · · · , d U .We want to keep the coefficients { a u,k , u = 2 , · · · , U ; k = , · · · , n } of these as small as possible. If d , · · · , d U arelinearly dependent, then D ( U ) = 0 . We shall assume thatthese vectors form a linearly independent set. Therefore theyspan a complex vector space W of dimension U − .We shall be applying the pigeon hole principle in thequotient space V = C U /W . To make the calculations morespecific we may identify V with an orthogonal complement(with respect to the Euclidean inner product of C U identifiedwith R U ) of W in C U . The mapping f from C U → C givenby f : c det M ( c , d , · · · , d U ) is linear in c with con-stant coefficients of a bounded size (as we selected d , · · · , d U with minimal coefficient a u,k ). Furthermore, f ( c − c ′ ) = 0 whenever ( c − c ′ ) ∈ W , so we can view f as a linear functionfrom V to C . Let π : C U → V be the natural projectionthat we may also think of as an orthogonal projection, i.e. amapping that can only shrink a vector in length.The assumption about the rank of the lattice L says thatthere are O ( N U ) code vectors in C . The coordinates of all ofthem in C U are of the size O ( N ) . As π is a shrinking map, thecoordinates of their images in V are also of the size O ( N ) , sothey fall into a square shaped region R ⊂ V with side length O ( N ) , as a real vector space V has dimension 2. Thereforewe can partition the set R into at most |C | − O ( N U ) smaller squares with side length O ( N/N U ) = O (1 /N U − ) .The pigeon hole principle then tells us that there exists apair of distinct vectors c = c ′ ∈ C such that π ( c ) and π ( c ′ ) fall into the same small square. By linearity, theprojection of their difference vector π ( d ) = π ( c ) − π ( c ′ ) then has coordinates of size O (1 /N U − ) . Therefore also f ( d ) = det M ( d , d , · · · , d U ) = det(∆ C ) with ∆ C = M ( d , d , · · · , d U ) has value of the size O (1 /N U − ) . Wehave proven the following result. Theorem 4: (Pigeon hole bound, single antenna case) Forany MIMO-MAC lattice code of U users, each transmitting atmultiplexing gain r with n t = 1 , then there exists a constant K > such that D ( U ) ≤ (cid:12)(cid:12)(cid:12)(cid:12) KN U − (cid:12)(cid:12)(cid:12)(cid:12) . = SNR − ( U − r . (cid:4) We turn our attention to the case of multiple transmitantennas. The application of the pigeon hole principle is verysimilar in spirit. We simply need to keep track of the dimesionsof various vector spaces.Let us, again, begin by fixing non-zero difference sig-nal matrices D u = C u − C ′ u with C u = C ′ u ∈ C u for users u = 2 , , . . . , U . We want to find a large sub-space W ⊆ M n t × M ( C ) with M = U n t such that det M ( D, D , D , . . . , D U ) = 0 whenever D ∈ W . Weassume M ( D , · · · , D U ) is of full rank and the ( U − n t rowsof the blocks D , . . . , D U are linearly independent, otherwise D ( U ) = 0 . Let W ′ be their complex linear span, and let W bethe vector space consisting of complex ( n t × M ) matrices withrows in W ′ . Obviously dim W = n t · dim W ′ = ( U − n t ,so the quotient space V = M n t × Un t /W has real dimension n t . When we restrict the selection of D = C − C ′ into C their projections in V are confined to a hypercube R of sidelength O ( N ) . The size of the constellation C is O ( N Un t ) as the lattice L was assumed to be of a full rank U n t .Again we partition the n t -dimensional hypercube R into O ( N Un t ) smaller cubes of side length O ( N/N U ) . As in thesingle antenna case we can then produce a non-zero differencevector d such that all the coordinates of its projection in V are of the size O (1 /N U − ) . This time the determinant f : C det M ( C , D , · · · , D U ) is a polynomial function (withconstant size coordinates) of degree n t of these coordinates of C , so by pigeon hole principle there must exist C = C ′ ∈ C such that f ( D = C − C ′ ) has value (cid:2) O (1 /N U − ) (cid:3) n t . Wehave proven the following. Theorem 5: (Pigeon hole bound, multi-antenna case). Forany MIMO-MAC lattice code of U users, each transmitting atmultiplexing gain r with n t = transmit antennas, there existsa constant K > such that D ( U ) ≤ (cid:12)(cid:12)(cid:12)(cid:12) KN ( U − n t (cid:12)(cid:12)(cid:12)(cid:12) . = SNR − ( U − r . (cid:4) Applying the above theorem to Theorem 3 we immediatelysee that there does not exist any MIMO-MAC codes satisfy-ing condition (5) when the MIMO-MAC system operates inthe antenna pooling regime. For the single-user performanceregime, the dominant outage set S ∗ = { } and condition(5) requires D ( { } ) ˙ ≥ SNR for the individual code and D ( U ) ˙ ≥ SNR − (Γ U − Ur − ǫ ) for the overall code. However,Theorem 5 shows that this is possible only if Γ U ≥ (2 U − r ,i.e. we need d U ((2 U − r ) ≥ d ( r ) . Corollary 6:
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