Diversity Order Gain with Noisy Feedback in Multiple Access Channels

Di v ersity Order Gain with Noisy Feedback in Multiple Access Channels V aneet Aggarwal Department of Electrical Engineering Princeton Univ ersity Princeton, NJ 08544 Email: vagg arwa@princeton.edu Ashutosh Sabharwal Department of Electrical & Computer Engineering Rice Univ ersity Houston, TX 77005 Email: ashu@rice.edu Abstract —In this paper , we study the effect of feedback channel noise on the diversity-multiplexing tradeoff in multiuser MIMO systems using quantized feedback, where each user has m transmit antennas and the base-station r eceiver has n antennas. W e derive an achievable tradeoff and use it to show that in SNR- symmetric channels, a single bit of imperfect feedback is sufficient to double the maximum diversity order to 2 mn compar ed to when there is no feedback (maximum is mn at multiplexing gain of zero). Further , additional feedback bits do not increase this maximum di versity order beyond 2 mn . Finally , the above diversity order gain of mn over non-feedback systems can also be achieved f or higher multiplexing gains, albeit requiring more than one bit of feedback. I . I N T RO D U C T I O N Channel state information to the transmitters has been extensi vely studied in MIMO systems [1–12] to improv e over the div ersity-multiplexing tradeof f without feedback [13, 14]). While the earlier w ork often assumed noiseless feedback (possibly quantized), recent emphasis has been on studying the performance with noisy feedback [4–7] in single-user MIMO channels. T wo distinct models of feedback ha ve appeared. First is that of two-way training, suitable for symmetric time- division duplex systems and is the focus of study in [5, 6]. The other is that of quantized channel state information [3, 7–10] which is more appropriate for asymmetric frequency-di vision duplex systems. In this paper, we study the impact of errors in the quantized feedback system when used in a multiuser system. W e first model the noise in the feedback which depends on the signal-to-noise ratio of the channel from the transmitter to the recei ver . Thus, we visualize the feedback channel also operating over a noisy communication link. Next, we bound the probability of outage to deri ve the di versity-multiple xing performance of MA C (Multiple Access Channel) for the noisy feedback model for quantized channel state information. The general result leads to the follo wing conclusions about multi- ple access and as special case, single-user MIMO channels. If the forward and feedback channel are SNR-symmetric (true if the nodes have identical po wer constraints operating ov er statistically identical channels), then feedback errors limit the maximum div ersity order to 2 mn , achieved at zero multiplexing point. The div ersity order of 2 mn is double that of what can be achiev ed with no feedback [13, 14] and is identical to that achie ved by tw o-way training method studied in [5, 6]. Thus the two dominant models achieve the same maximum diversity order when the transmitter is mismatched with the recei ver , thus providing a satisfying conclusion. At the same time, it is interesting to note that a single noisy bit of information is same as training the full channel from the point of view of div ersity-multiplexing tradeof f. Howe ver , the picture changes for higher multiplexing gains. While there is no way of controlling diversity order gain with two-way training of [5, 6], more feedback bits lead to different lev el of di versity order gains. W e also show that as the number of feedback bits grow , the div ersity order of ( mn + div ersity order achiev able without feedback ) can be achiev ed. For example, in single user MIMO channel, a div ersity order of mn + ( m − r )( n − r ) for integer multiplexing gains 0 ≤ r < min( m, n ) can be achie ved with finite number of error-prone feedback bits, a number which we quantify . W e highlight the fact that all our results are deriv ed for a multiuser system with L users, each with m transmit antennas and a receiver with n receiv e antennas. This in contrast to most of the earlier work which has considered noisy channel state feedback in the context of single user systems [1, 2, 4, 5, 7]. The rest of the paper is outlined as follows. In Section II we give background on the channel model, introduce feedback model and div ersity-multiplexing tradeoff. In Section III, we find the div ersity-multiplexing tradeoff for Multiple Access Channel. In Section IV , we discuss these results. Section V concludes the paper . I I . S Y S T E M M O D E L A. Channel Model Consider a multiple access channel with L users where the transmitters ha ve an array of m transmit antenna and the receiv er has an array of n receive antenna. The channel is constant during a fading block of T channel uses, but changes independently from one block to the next. During a fading block l , the channel is represented by n × m random matrices H s,l ( 1 ≤ s ≤ L ), and the recei ved signal can be written in the matrix form as Y l = P 1 ≤ i ≤ L H i,l X i,l + W l . Here, W l of size n × T represents additive white Gaussian noise at the receiv er with all entries i.i.d. C N (0 , 1) . W e consider a richly scattered Rayleigh fading environment, i.e. elements of H s,l are assumed to be i.i.d C N (0 , 1) . The transmitters are subject to an av erage power constraint such that the long-term power is upper bounded, i.e, E h X 2 s,l i ≤ SNR i for 1 ≤ s ≤ L . B. F eedback Model W e will assume that the receiv er has perfect kno wledge of the channel coefficients H s,l ( 1 ≤ s ≤ L ). W e denote H l = ( H 1 ,l , H 2 ,l , ...H L,l ) . The receiver then uses the knowledge of channel coefficients to compute a feedback signal I( H l ) which is sent to the transmitters. Furthermore, we will assume that this feedback signal takes on only finite number of values from the set { 1 , 2 , . . . , K } , where K > 1 . Note that when K = 1 , there is no feedback, and hence the case reduces to that in [13]. Finally , the mapping I( H l ) : H l 7→ { 1 , 2 , . . . , K } is a deterministic function which can potentially depend on the SNR and the rate of transmission. Due to the error in the feedback, the users do not receiv e the same signal as is sent by the receiv er . The feedback channel is modeled as follows. Let I( H l ) = i be transmitted from the receiver . User s receiv es an index I s which takes on only finite number of values from the set { 1 , 2 , . . . , K } and is given by I s =  i with probability 1 −  i 0 6 = i with probability  K − 1 for 1 ≤ s ≤ L, where  depends on SNR . C. Diversity-Multiplexing T radeoff Definitions A codeword X s,l is assumed to span a single fading block. Since we do not consider coding over multiple fading blocks, the block index l will be omitted whenev er this does not cause any confusion. Conditioned on indices I s = i s , the transmitter s chooses a codeword X s from the codebook C s,i s = { X s,i s (1) , X s,i s (2) , ..., X s,i s ( E s ) } of rate R s for 1 ≤ s ≤ L . All the X s,i ( k ) ’ s are matrices of size m × T . In this paper , we will only consider single rate transmission where the rate of the codebooks does not depend on the feedback index and is known to the receiver . Therefore, regardless of which feedback index the transmitters receiv e, the recei ver attempts to decode the recei ved code word from the same codebook. Outage occurs when the transmission power is less than the power needed for successful (outage-free) transmission. The av erage power constraint at each transmitter can be giv en along the lines of [8] as follows. First define average power per codeword P i s , 1 T E s E s X k =1 || X s,i ( k ) || 2 F , 1 ≤ s ≤ S which leads to av erage power constraints E H [ P I s ( H ) s ] , K X i =1 P i s Π(I s ( H ) = i ) ≤ SNR s , 1 ≤ s ≤ L (1) where Π( α ) denotes probability of ev ent α . Since our focus is asymptotic performance behavior in the form of div ersity-multiplexing tradeoff, we will assume that SNR s . = SNR for all 1 ≤ s ≤ L . 1 Note that all the index mappings, codebooks, rates, powers are dependent on SNR s. The dependence of rates on the SNR s is explicitly given by R s = r s log SNR s . W e refer to r , ( r s ) 1 ≤ s ≤ L as the multiplexing gains. In point-to-point channels, outage is defined as the e vent that the mutual information of the channel, I ( X ; Y | H ) is less than the desired rate R , where I ( X ; Y | H ) = log det  I + P m H QH †  is the mutual information of a point- to-point link with m transmit and n recei ve antennas, transmit power P and input distribution Gaussian with cov ariance matrix Q [14]. Since I ( X ; Y | H ) depends on transmit po wer , we write this dependence explicitly as I ( X ; Y | H, P ) . In a multiple access channel, corresponding outage ev ent is defined as the ev ent that the channel cannot support target data rate for all the users [13]. Hence, for a multiple access channel with L users, each equipped with m transmit antennas, and a receiv er with n receiv e antennas, the outage e vent is O ( R , P ) , ∪ S O S ( R , P ) where P = ( P 1 , P 2 , .., P L ) and R = ( R 1 , R 2 , ..., R L ) . The union is taken over all subsets S ⊆ { 1 , 2 , ..., L } , and O S ( R , P ) , { H ∈ C n × Lm : I ( X S ; Y | X S c , H , P ) < P S R i } where X S contains the input signals from the users in S with powers P . As before, I ( X S ; Y | X S c , H , P ) represents I ( X S ; Y | X S c , H ) when the transmit powers are P . Let Π( O ) denote the probability of outage. The system is said to hav e div ersity order of d if Π( O ) . = SNR − d . The diversity multi- plexing for the multiple users can be described as: giv en the multiplexing gains r for all the users, the div ersity order that can be achieved describes the diversity-multiple xing tradeof f region. The probability of outage with rate R = ( R 1 , R 2 , ..., R L ) and transmit power P = ( P 1 , P 2 , ..., P L ) is denoted by Π( R , P ) , Π( O ( R , P )) = Π( ∪ S O S ( R , P )) . Also let U ( R , P ) be defined as the indicator function of ∪ S O S ( R , P ) . Then, Π( R , P ) is the probability of event { U ( R , P ) = 1 } ov er the randomness of channel matrices. Let P s . = SNR p s for all 1 ≤ s ≤ L . Further let R = ( R 1 , R 2 , ..., R L ) . = ( r 1 log SNR , r 2 log SNR , ..., r L log SNR ) . Let D ( r , p ) be de- fined as Π( R , P ) . = SNR − D ( r , p ) where r = ( r 1 , r 2 , ..., r L ) and p = ( p 1 , p 2 , ..., p L ) . Lemma 1. Let p s = p for all 1 ≤ s ≤ L . Also, let P i ∈ S r i ≤ min( | S | m, n ) for all non-empty subsets S of { 1 , 2 , ..., L } . Then, D ( r , p ) = min S G | S | m,n X i ∈ S r i , p ! (2) wher e G m,n ( r , p ) , inf α min( m,n ) 1 ∈ A min( m,n ) X i =1 (2 i − 1 + max( m, n ) − min( m, n )) α i 1 W e adopt the notation of [14] to denote . = to represent exponential equality . W e similarly use . < , . > , . ≤ , . ≥ to denote exponential inequalities. with A , { α min( m,n ) 1 | α 1 ≥ . . . α min( m,n ) ≥ 0 , min( m,n ) X i =0 ( p − α i ) + < r } . Proof : Note that Π( R , P ) = Π H  S S O S ( R , P )  . Hence, Π H ( O S ( R , P ) 6 Π( R , P ) 6 P S Π H ( O S ( R , P )) . As, Π H ( O S ( R , P )) is probability of outage for single user with | S | m transmit antennas, n recei ve antennas, rate . = P i ∈ S r i log( SNR ) , power . = SNR p , by [8], Π H ( O S ( R , P )) . = SNR − G | S | m,n ( P i ∈ S r i ,p ) . Hence, Π( R , P ) . = SNR − D ( r , p ) .  Remark 1. G m,n ( r , p ) is a piece wise linear curv e connecting the points ( r , G m,n ( r , p )) = ( k p, p ( m − k )( n − k )) , k = 0 , 1 , . . . , min( m, n ) for fixed m , n and p > 0 . This follows directly from Lemma 2 of [8]. D. F eedback-based P ower Contr ol In this section, we describe the power control policy for the optimum receiv er for which successful decoding occurs if the transmission power is greater than or equal to the power needed for outage-free transmission. Recall that the sent feedback signal I and the receiv ed feedback signal I s takes values ov er a finite set as described in Section II-B. For each receiv ed index I s = i s at User s , the transmitted po wer is denoted by P i s s . W e assume that P 1 s ≤ P 2 s ≤ · · · ≤ P K s . W e denote the power tuple as P i = ( P i 1 , P i 2 , ..., P i L ) . Follo wing [8, 11], I = i is calculated for as i =  1 if U ( R , P K ) = 1 min k ∈{ 1 ,...,K } { U ( R , P k ) = 0 } otherwise . According to the scheme, we transmit at minimum power le vel needed for outage-free transmission in case outage can be av oided, and send at minimum power level in case it cannot be avoided. Using the scheme, we can compute the probability of occurrence of ev ent ( I = i ) as Π(I = i ) = ( 1 + Π( R , P K ) − Π( R , P 1 ) , i = 1 Π( R , P i − 1 ) − Π( R , P i ) , 2 ≤ i ≤ K . (3) The po wer le vels are chosen to minimize the outage probabil- ity Π( O ) subject to the power constraint (1). I I I . D I V E R S I T Y - M U LT I P L E X I N G T R A D E O FF In this section, we will gi ve an achiev able div ersity mul- tiplexing tradeof f with errors in feedback with certain cases when this is the best achiev able. W e assume that the feedback errors decays with SNR as  . = SNR − y . When K = 1 , there is no feedback and hence no im- perfection. The diversity for any multiplexing vector r = ( r 1 , r 2 , .., r L ) is giv en by D ( r , 1 ) for P i ∈ S r i < min( | S | m, n ) for all non-empty subsets S of { 1 , 2 , ..., L } where 1 is a vector of length L containing all ones [13]. So, we only consider the case K > 1 in this section. Let C m,n,K ( r ) be giv en by a recursive equation C m,n,j ( r ) =  0 when j = 0 D ( r , 1 (1 + C m,n,j − 1 ( r ))) when j ≥ 1 . Theorem 1. Suppose that K > 1 and  . = SNR − y for some y > 0 . Further suppose that P i ∈ S r i < min( | S | m, n ) for all non-empty subsets S of { 1 , 2 , ..., L } . Then, the lower bound for diversity-multiplexing tradeoff is given by d K opt = min( C m,n,K ( r ) , y + C m,n, 1 ( r )) where C m,n,K ( r ) is given by a r ecursive equation C m,n,j ( r ) =  0 when j = 0 D ( r , 1 (1 + min( y , C m,n,j − 1 ( r )))) when j ≥ 1 Proof : The probability of outage for this scheme can be bounded as Π( O ) ≤ Π( R , P K ) + L X s =1 K X i =2 Π( I s < i | I = i )Π( I = i ) = (1 − L )Π( R , P K ) + L K − 1 K − 1 X i =1 Π( R , P i ) (4) W e next calculate the probability that I s = i as Π(I s = i ) =  K − 1 +  1 − K K − 1  Π(I = i ) . (5) The power le vels are selected to minimize outage probabil- ity subject to power constraints (1). Consider the power lev els as: e P i s = ( SNR s K when i = 1 SNR s K (  K − 1 +(1 − K K − 1 )Π( R , e P i − 1 )) when i > 1 ∀ 1 ≤ s ≤ L. These power lev els satisfy the SNR constraints, and hence the optimal outage probability is ≤ the outage probability with these po wer le vels. Let e Π( O ) be the outage probability using power lev els e P i s . Then, Π( O ) . ≤ e Π( O ) . From these power lev els, we find that e P i s . = SNR 1+min( y ,C m,n,i − 1 ( r )) (6) Hence, e Π( O ) . ≤ (1 − L )Π( R , e P K ) + L K − 1 K − 1 X i =1 Π( R , e P i ) . = SNR − C m,n,K ( r ) + SNR − y − C m,n, 1 ( r ) . = SNR − min( C m,n,K ( r ) ,y + C m,n, 1 ( r )) Hence, we find that the outage probability is . ≤ SNR − min( C m,n,K ( r ) ,y + C m,n, 1 ( r )) . Noting that C m,n, 1 ( r ) = C m,n, 1 ( r ) prov es the theorem.  Theorem 1 gi ves a lo wer bound to the di versity-multiplexing tradeoff performance. W e will now consider some special cases when this bound is tight. Lemma 2. Suppose that K > 1 ,  = 0 and P i ∈ S r i < min( | S | m, n ) for all non-empty subsets S of { 1 , 2 , ..., L } . Then, the diversity-multiplexing tradeoff with K indices of global feedback is given by C m,n,K ( r ) for given multiplexing gain r = ( r 1 , r 2 , ..., r L ) . Proof : The lo wer bound of the div ersity multiplexing tradeof f follows from Theorem 1 by taking limit as y → ∞ . W e will no w prov e the upper bound for the di versity multiplexing tradeoff by finding the lower bound for outage probability . Note that Π( O ) = Π( R , P K ) when there is no error in the feedback. T o calculate the lo wer bound for outage probability , we first weaken the abo ve optimization problem as min Π( O ) subject to the following power constraint Π( I = i ) P i s ≤ SNR s ∀ 1 ≤ s ≤ L (7) The solution of (7) is denoted by P i s . As the constraint set is bigger compared to the original problem, it follo ws that Π( O ) . ≥ Π( O ) where Π( O ) is the outage probability taking powers P i s . Note from (7) that P 1 s ≤ K SNR s which gi ves P 1 s . ≤ SNR . Using (7) and (3) recursiv ely , we find that P i s . ≤ SNR 1+ C m,n,i − 1 ( r ) (8) Hence, the outage probability Π( O ) . ≥ Π( R , P K ) . ≥ SNR D ( r , 1 (1+ C m,n,K − 1 ( r ))) . = SNR − C m,n,K ( r )  (9) Remark 2. Lemma 2 has earlier been prov ed in [8] for L = 1 and [11] for L = 2 . Lemma 3. Suppose that K > 1 , L = 1 and  . = SNR − y for some y > 0 . Further suppose that r 1 < min( m, n ) . Then, the diversity-multiple xing tradeoff for the optimal r eceiver with K indices of feedback is given by d K opt = min( C m,n,K ( r ) , y + C m,n, 1 ( r )) . Proof : The lo wer bound follo ws by Theorem 1. W e will no w prov e the upper bound for the diversity multiplexing tradeoff by finding the lower bound for outage probability . For this, we first weaken the abov e optimization problem as min Π( O ) subject to the follo wing power constraint  K − 1 P i 1 +  1 − K K − 1  Π( I = i ) P i 1 ≤ SNR 1 (10) The solution of (10) is denoted by P i 1 . As the constraint set is bigger compared to the original problem, it follo ws that Π( O ) . ≥ Π( O ) where Π( O ) is the outage probability taking powers P i 1 . Note that P 1 1 ≤ K SNR . From (10), it follows that SNR P j 1 ≥  K − 1 +  1 − K K − 1  Π( I = j ) . Hence, P j 1 . ≤ SNR 1+ y . Assuming that P j 1 . ≤ SNR 1+ y , we find using (10) that P j 1 . ≤ SNR 1+ G m,n ( r,p j − 1 ) , for P j − 1 1 . ≤ SNR p j − 1 1 . Using this recursiv ely , we find that P j 1 . ≤ SNR 1+min( y ,C m,n,j − 1 ( r )) and hence Π( R , P j 1 ) . ≥ C m,n,j ( r ) . Also, since L = 1 , Π( O ) = Π( R , P K ) + K X i =2 Π( I 1 < i | I = i )Π( I = i ) = (1 −  )Π( R , P K ) +  K − 1 K − 1 X i =1 Π( R , P i ) (11) Hence, the outage probability Π( O ) . ≥ (1 −  ) SNR − C m,n,K ( r ) +  K − 1 K − 1 X i =1 SNR − C m,n,i ( r ) . = SNR − min( C m,n,K ( r ) ,y + C m,n, 1 ( r ))  (12) Remark 3. It was recently observed in [7] that for K > 1 and L = 1 , we do not gain in div ersity order with feedback if the feedback errors do not decay with SNR . This also follows as a special case of Lemma 3 with y → 0 in which case, d K opt = C m,n, 1 ( r ) is the same as the diversity order without feedback. I V . D O U B L I N G O F D I V E R SI T Y O R D E R Theorem 1 gives achiev able div ersity-multiplexing trade- off for MAC with imperfect feedback. Note that the theo- rem considered  . = SNR − y for any y > 0 . W e saw in Lemma 2 the performance of MAC with perfect feedback. When the feedback error does not decay with SNR , we get d K opt = C m,n, 1 ( r ) = D ( r , 1 ) . This is same as the div ersity- multiplexing tradeoff without feedback [13] and hence if the feedback error does not decay with SNR , the feedback do not help in getting any increase in diversity order . When the forward and the backward channel are SNR - symmetric, the feedback error from the transmitter to the receiv er scales as  . = SNR − mn . 2 Thus, y = mn . Next, we analyze the performance loss with imperfection in feedback. Lemma 4 (Doubling of Diversity Order) . Let y = mn . When r → 0 , the diversity or der is given by d K opt =  mn when K = 1 2 mn when K > 1 . Furthermor e, as r → 0 , C m,n,K and C m,n,K behave as C m,n,K ( 0 ) = mn  ( mn ) K − 1 mn − 1  for K ≥ 1 . C m,n,K ( 0 ) =  mn when K = 1 mn (1 + mn ) when K > 1 . When the feedback is perfect, diversity order increases exponentially with the number of feedback indices [11] while Lemma 4 sho ws that if the feedback is imperfect, the di versity order do not increase with feedback (for K > 1 ). Lemma 4 also sho ws that diversity order of 2 mn is achiev able as multiplexing gains go to zero for any number of indices of feedback ( K > 1 ), and hence also for single-bit of feedback. This further means that achiev able div ersity order doubles with just a single-bit of feedback compared to the case of no feedback for zero multiplexing gains. In [5, 6], it was shown for MIMO channels that div ersity of 2 mn can be achieved by training. In this paper , by Lemma 4, we hav e sho wn that div ersity order of 2 mn can be achiev ed with just a single bit of feedback. Hence, the training can be replaced with just a single bit of feedback if the objectiv e is just to achiev e diversity of 2 mn for zero multiplexing gains. 2 Note that the recei ver which is sending back the feedback is assumed to operate without an y channel state information, especially when operating in an FDD system. T ill no w , we focussed on zero multiplexing gains. Now , we consider general multiplexing gains satisfying P i ∈ S r i < min( | S | m, n ) . Lemma 5. Let r = ( r 1 , ..., r L ) with P i ∈ S r i < min( | S | m, n ) . Then, the following holds: D ( r , 1 (1 + mn )) ≥ mn + D ( r , 1 ) (13) Proof : W e will be done if we prov e that G | S | m,n ( P i ∈ S r i , 1 + mn ) ≥ mn + G | S | m,n ( P i ∈ S r i , 1) . Hence, it is enough need to prove that G m,n ( t, 1 + mn ) ≥ mn + G m,n ( t, 1) for t < min( m, n ) . Note that G m,n ( d t e , 1 + mn ) = mn (1 + mn ) − d t e ( m + n − 1) ≥ mn + ( m − d t e )( n − d t e ) = mn + G m,n ( d t e , 1) . Similarly , G m,n ( b t c , 1 + mn ) ≥ mn + G m,n ( b t c , 1) . Since, both G m,n ( x, 1 + mn ) and mn + G m,n ( x, 1) are linear in x for b t c ≤ x ≤ d t e , the result follows.  Note that C m,n,j +1 ( r ) > C m,n,j ( r ) for P i ∈ S r i < min( | S | m, n ) . Also, C m,n, 1 ( r ) ≤ mn . Hence, there exist a k ≥ 1 such that: C m,n,j ( r )  ≤ mn for j ≤ k > mn for j > k . Lemma 6. Let y = mn . Further assume that k ≥ 1 be the maximum j such that C m,n,j ( r ) ≤ mn . Then, the achie vable diversity-multiple xing tradeof f in Theor em 1 reduces to: d j opt =    C m,n,j ( r ) for j ≤ k min( C m,n,j ( r ) , mn + C m,n, 1 ( r )) for j = k + 1 mn + C m,n, 1 ( r ) for j > k + 1 . Proof : This follows directly from Theorem 1 & Lemma 5.  Thus, we find from Lemma 6 that the div ersity increases in the same fashion with imperfect feedback as it does with perfect feedback till number of feedback indices ≤ k . For number of indices > k + 1 , the di versity is limited by mn + the di versity without feedback. Hence, the gain in di versity order with imperfect feedback over no feedback is limited by mn for any multiplexing gain. The maximum div ersity order that can be achiev ed with feedback is more than double as compared to the diversity order without feedback for non-zero multiplexing gains since the div ersity order without feedback is less than mn . T o get this maximum gain in di versity order , k + 2 feedback indices are sufficient. Hence, although single bit was enough for multiplexing gains going to 0 , for general multiplexing gains we need more feedback to attain maximum div ersity . This can also be seen in Figure 1 for MIMO channel that di versity order of 2 mn can be achie ved with single bit of feedback which is the maximum possible. Higher amount of feedback indices help at higher multiplexing to get a gain in div ersity order of mn above no-feedback di versity order . W e also see in Figure 2 for MA C with two transmitters ( L = 2 ) that the di versity order is mn more than the di versity order without feedback after a certain number of feedback levels. From these figures, we see that as the multiplexing increases, more indices are needed to get the maximum gain in di versity order of mn . 0 0.5 1 1.5 2 2.5 3 3.5 0 5 10 15 20 25 Multiplexing r Diversity Order K=1 (No feedback) K=2 (1 bit of feedback) K=3 (log 2 (3) bits of feedback) K=4 (2 bits of feedback) min(m,n)=3 2mn=24 mn=12 Fig. 1. Diversity-Multiplexing T radeoff for MIMO Channel ( m = 3 , n = 4 , y = mn ). 0 0.5 1 1.5 2 2.5 0 2 4 6 8 10 12 14 16 Multiplexing r 2 Diversity Order K=1 (No feedback) K=2 (1 bit of feedback) K=3 (log 2 (3) bits of feedback) K=4 (2 bits of feedback) Fig. 2. Diversity-Multiplexing T radeoff for MA C ( m = 3 , n = 4 , y 1 = y 2 = mn , r 1 = 1 . 5 ). V . C O N C L U S I O N Channel state information at the transmitters is imperfect due to noise. Inspired by this fact, we constructed a feedback error model and characterized the di versity multiplexing trade- off performance for MA C systems. V I . A C K N O W L E D G E M E N T This work is supported in part by NSF under Grants ANI- 0338807, CRI-0551692, MRI-0619767 and TF-0635331. R E F E R E N C E S [1] V . K. N. Lau and Y . R. Kwok, Channel-Adaptive T echnologies and Cross- Layer Designs for W ir eless Systems with Multiple Antennas , John W iley & Sons, Inc., 2006. [2] W . Shin, S. Chung, Y . H. 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