A Codebook-Based Limited Feedback System for Large-Scale MIMO
AA Codebook-Based Limited Feedback Systemfor Large-Scale MIMO
Junyoung Nam
Wireless Communications DivisionElectronics and Telecommunications Research Institute (ETRI), Daejeon, Korea
Abstract —In this paper, we consider limited feedback systemsfor FDD large-scale (massive) MIMO. A new codebook-basedframework for multiuser (MU) MIMO downlink systems isintroduced and then compared with an ideal non-codebook basedsystem. We are particularly interested in the less-known finite-rate feedback regime where the number M of transmit antennasand the number of users are of the same order of magnitude and M is large but finite, which is a typical scenario of large-scaleMIMO. We provide new findings in this regime and identify somebenefits of the new framework in terms of scheduling gain anddownlink dedicated pilot overhead. I. I
NTRODUCTION
Limited feedback MU-MIMO downlink systems may becategorized according to the use of a common codebook(CB). One is non-CB-based precoding with a direct (analog)quantization of channel vectors like zero-forcing beamforming(ZFBF) and the other is CB-based precoding, the latter ofwhich refers to an opportunistic beamforming scheme likerandom beamforming (RBF) [1] where each user terminal(user) selects the channel direction information (CDI) with themaximum signal-to-interference-plus-noise ratio (SINR) andreports the CDI and SINR to the basestation (BS). By theCB-based scheme also employed in some practical systems(e.g., [2]), we stipulate that the BS uses precoding vectorsonly in a common CB also known to all users.In the past few years, we have paid great attention to large-scale MIMO [3]. A typical scenario therein is that the numberof antennas, M , at the BS is larger than the number of users, K . Since the study on MU-MIMO systems has traditionallyfocused on the M (cid:28) K case, new insights may be neededtaking into account the much lesser known case where M and K have the same order of magnitude (denoted by M ∼ K )with M not very large . For frequency division duplex (FDD)large-scale MIMO systems, limited feedback schemes basedon noncoherent trellis-coded quantization [4] or compressivesensing [5] were introduced, where the former is to reduceencoding complexity of channel quantization. More practicalCB-based approaches have been considered not suitable forlarge-scale MIMO due to severe performance degradation. Thegoal of this work is not just to reduce CSI feedback overheadbut rather to improve the performance of CB-based systemsat the cost of a moderate feedback increase compared to theconventional CB-based approach.A main issue for our purpose is as to whether to achieve afull scheduling gain. While ZFBF is a popular sub-optimal linear precoding scheme, its CDI feedback overhead (i.e.,quantization bits) should grow linearly with M log SNR toachieve the full degrees of freedom [6], i.e., unless thenumber of quantization bits is sufficiently large, the non-CB-based approach with ZFBF is subject to severe performancedegradation. Also, it is quite sensitive to noisy channel stateinformation at the transmitter (CSIT) due to channel estimationerrors. However, ZFBF can provide relatively accurate SINRs(relatively tight lower bound on the exact value) of anyselected users [7], which allows flexible scheduling at theBS. The capability of flexible scheduling is very importantunless M (cid:28) K . If M ∼ K , the probability that the schedulerselects min { M, K } users is quite low irrespectively of ZFBFor even the capacity-achieving dirty paper precoding [8] unlessthe SNR is sufficiently high. Thus, the BS should be able toestimate a variety of SINRs per user with the number S ofselected users less than M . Assuming perfect CSI at receivers,the CB-based approach allows users to calculate the exactSINRs for any given subset of beams in CB. However, itrequires each user to report an individual SINR per subset,thus making flexible scheduling infeasible as the number ofall possible subsets becomes extremely large unless M is verysmall. This limited scheduling explains to some extent why aCB-based system suffers significant degradation for M large.The second but more important issue for M large is thecost of downlink dedicated (user-specific) pilot for coherentdetection. For sufficiently large M such that there is nointerference between users at all, downlink dedicated pilot isunnecessary even for the simple maximal-ratio transmission(MRT) scheme [3] with perfect CSIT, since the (cid:96) norm ofuser channel vectors converges to a real positive deterministicvalue due to the channel hardening effect [9] in the i.i.d.Rayleigh fading channel. However, if inter-user interferenceis somehow present, the use of orthogonal dedicated pilotswould be essential for the system performance regardless ofFDD or time division duplex (TDD) and of how min { M, K } is large. For instance, ZFBF suffers from its high sensitivityto imperfect (noisy or outdated) CSIT [10], which causessignificant interference as S increases for M large. There-fore, assuming imperfect CSIT, we need orthogonal downlinkdedicated pilots even in ZFBF as well as MRT and codebook-based schemes. As min { M, K } increases, the number S ofselected users will also grow and then the downlink dedicatedpilots consume more resources, which ironically reduces the a r X i v : . [ c s . I T ] N ov emaining resources for data streams, thus rather degrading thesystem performance. As a consequence, the cost of downlinkdedicated pilot becomes a significant bottleneck to the per-formance of large-scale MIMO systems with imperfect CSIT The resulting performance limit of large-scale MIMO wascharacterized in [11].In order to address the above issues for the large-scalearray regime, we propose a new framework for CB-based MU-MIMO systems. The proposed framework takes advantage ofa new type of channel quality information (CQI) feedback.The framework is shown to enable flexible scheduling withmoderate CQI feedback overhead and to significantly reducethe cost for dedicated pilot.
Notations: A ∼ B means the quantities A and B have thesame order of magnitude.II. P RELIMINARIES
A. System Model
We consider a single cell MIMO downlink channel wherethe BS has M transmit antennas and K users have a singleantenna each. Let HHH denote the M × K system channel matrixgiven by stacking the K users channel vectors hhh by columns.The signal vector received by the users is given by yyy = HHH H xxx + zzz (1)where xxx is the input signal with S the rank of the inputcovariance Σ = E [ xxxxxx H ] (i,e., the total number of independentdata streams) and zzz ∼ CN (000 , III ) is the Gaussian noise(plus other-cell interference in multi-cell environment) at thereceivers. The system has the total power constraint such thattr ( Σ ) ≤ P , where P implies the total transmit SNR. Denotingby p k , k , and d k the power, linear precoding vector, andinformation symbol of user k , respectively, the signal receivedby the selected user k can be represented as y k = √ p k hhh H k k d k + (cid:88) j ∈S ,j (cid:54) = k √ p j hhh H k j d j + z k , k ∈ S (2)where S ⊆ [1 : K ] is the set of users selected by the BSscheduler such that |S| = S ≤ min( M, K ) and (cid:80) k ∈S p k ≤ P . The SINR of user k is given by γ k = | p k hhh H k k | σ k + (cid:80) j ∈S ,j (cid:54) = k | p j hhh H k j | . (3)A popular limited feedback system model is based on theuser feedback in terms of CDI and CQI. While CDI of user k conveys CSI on the channel direction hhh k / (cid:107) hhh k (cid:107) , CQI is oftenrepresented by the received SINR or the norm of hhh k .For ZFBF with perfect CSIT, the CDI of user k becomes thechannel vector hhh k itself and the BS needs no CQI feedback but σ k (i.e., noise plus other-cell interference power) of users forrate adaptation of each user. In this case, the BS can calculatethe exact SINR in (3) of user k . We assume no quantizationfor ZFBF in this paper to provide a more clean view, eventhough the comparison with a limited feedback scheme isunfair. Without perfect CSIT, we often resort to a CB-basedscheme in practice, as will be reviewed in the sequel. B. Typical Framework for CB-Based MU-MIMO Systems1) CDI Feedback:
For convenience, a CB C of size M T based on unitary beamforming is considered for the CDIquantization. One papular way is to use the well-knowndiscrete Fourier transform (DFT) based approach alreadyemployed in real-world systems like LTE as follows: The CBis constructed by truncating the first M rows of the M T -point DFT matrix and by normalizing to satisfy the unit-normproperty of precoding vector k ∈ C for all k . The resultingprecoding vector ccc (cid:96) is given by ccc (cid:96) = 1 √ M (cid:104) e − j π(cid:96) MT e − j π(cid:96) MT · · · e − j π(cid:96) M − MT (cid:105) T (4)where (cid:96) = (cid:96) T + (cid:96) with (cid:96) = 0 , , · · · , T − and (cid:96) =0 , , · · · , M − , thus yielding the expression of (cid:96) (cid:44) ( (cid:96) , (cid:96) ) .We also denote by CCC t = { ccc (cid:96) : (cid:96) = t, ∀ (cid:96) } (5)a subset of C which forms a unitary matrix. For notationalconvenience we will use the notation ccc ( t ) m , where t = (cid:96) and m = (cid:96) , instead of ccc (cid:96) , if necessary.We follow a similar way to [2], [12]. Assuming all M beamsfor each CCC t are scheduled with the equal power allocation, i.e., p k = PM , user k selects the best beam index (cid:96) k = ( t k , m k ) that maximizes the corresponding SINR such that ( t k , m k ) = argmax t,m γ k ( t, m ) . (6)where γ k ( t, m ) = (cid:12)(cid:12) hhh H k ccc ( t ) m (cid:12)(cid:12) MP σ k + (cid:80) j ∈ [1: M ] ,j (cid:54) = m (cid:12)(cid:12) hhh H k ccc ( t ) j (cid:12)(cid:12) . Each user is assumed to report its CDI as a (cid:100) log M T (cid:101) -bitindex (cid:96) k through error-free and delay-free uplink to the BS.
2) CQI Feedback:
The CQI feedback is intended for the BSto estimate the SINRs experienced by users for schedulingand rate adaptation. An accurate estimate of the SINRs isgenerally complicated in the limited feedback case as theexact interference cannot be evaluated at users without apriori information on the scheduling decision. Notice that theCDI in (6) is just the result on a best-effort basis at theindividual user side. Therefore, scheduling with such CDI andCQI is restrictive especially when K is small, thus yieldingperformance degradation. In general each user is requiredto report either the SINR γ k ( (cid:96) k ) corresponding to (cid:96) k or itsachievable rate log (1 + γ k ( (cid:96) k )) .
3) Scheduling:
A user selection algorithm can be given asfollows: Let S ( t ) m = { k : (cid:96) k = ( t, m ) } and S ( t ) = ∅ . For each ( t, m ) , select the best user such that k ∗ = argmax k ∈S ( t ) m log (1 + γ k ( (cid:96) k )) S ( t ) = S ( t ) ∪ { k ∗ } . (7)hen, select the best t such that t ∗ = argmax t (cid:88) k ∈S ( t ) log (1 + γ k ( (cid:96) k )) S = S ( t ∗ ) . (8)It is easy to see that the above scheduling is restrictivedue to limited CQI feedback. Moreover, while CQI is fedback assuming all M beams are scheduled, the number S ofselected users is often less than M , thus implying the powerallocation of PM does not make sense but users have no a priori information on this. Even if we restrict to schedule beamsless than M , the BS hardly knows which beams are better toselect and how many beans are good enough. As a result, thescheduler for DFT-SINR makes use of (6) as a lower boundon the exact SINR γ k in (3).We refer to the above scheme as DFT-SINR in this paperas it uses the DFT-based beamforming with SINR feedback.III. N EW F RAMEWORK BASED ON
INR F
EEDBACK
In the typical framework, each user is required to feed backthe maximum SINR in (6), assuming that the BS schedules andtransmits all the M beams for each CCC t and that all users havethe equal transmit power P/M . Although these assumptionsare inevitable due to feedback overhead, they do not makemuch sense in realistic systems unless K (cid:29) M , which iseven more infeasible for M large. Therefore, we would likethe scheduler to select users in a fully flexible manner over allpossible s -combinations of [1 : M T ] for s = 1 , · · · , M , whosesize is (cid:80) Ms =1 C MTs , where C MTs = (cid:0) MTs (cid:1) . Denote by M s theset of each s -combinations of [1 : M T ] and by M s,n its n thelement (it is also a set of size s ), where n = 1 , , · · · , C MTs .Assuming the equal power allocation again such that p k = Ps ,the SINRs of user k with respect to precoding vectors in M s,n can be expressed as γ k ( s, n, q ) = (cid:12)(cid:12) hhh H k ccc q (cid:12)(cid:12) sP σ k + (cid:80) j ∈M s,n \ q (cid:12)(cid:12) hhh H k ccc j (cid:12)(cid:12) , q ∈ M s,n . (9)Although the above SINRs of all users allow the BS tooptimally select users for a given CB C , the overhead of (cid:80) Ms =1 C MTs
CQIs per user is prohibitively large even for M small. In order to overcome this difficulty, we propose a newframework based on the following CQI feedback. A. INR Feedback
For each beam ccc (cid:96) in C , we define the correspondinginterference to noise ratio (INR) as INR k,(cid:96) = (cid:12)(cid:12) hhh H k ccc (cid:96) (cid:12)(cid:12) σ k , (cid:96) ∈ [1 : M T ] . (10)If the above INRs are available at the BS, the scheduler canselect one of them as the numerator of SINR and then calculatethe corresponding all possible SINRs in (9) for user k , as easily shown by (cid:12)(cid:12) hhh H k ccc q (cid:12)(cid:12) sP σ k + (cid:80) j ∈M s,n \ q (cid:12)(cid:12) hhh H k ccc j (cid:12)(cid:12) = (cid:12)(cid:12) hhh H k ccc q (cid:12)(cid:12) σ k sP + (cid:80) j ∈M s,n \ q (cid:12)(cid:12) hhh H k ccc j (cid:12)(cid:12) σ k = INR k,qsP + (cid:80) j ∈M s,n \ q INR k,j . (11)Therefore, the BS needs only M T
INRs per user to accuratelyestimate all possible (cid:80) Ms =1 C MTs
SINRs per user for a given C . This new type of CQI feedback indeed enables fully flexiblescheduling for a given C . It is easy to see that the INR feedbackin (10) also suggest no need for any CDI feedback on beamindex (cid:96) k like (6). The above INR feedback is referred to asthe full INR feedback as it allows fully flexible scheduling.However, M T
INRs per user may incur too much overheadespecially for M large, even if small T (e.g., 2) is generallysuitable for CB-based MU-MIMO schemes.In order to reduce the feedback overhead, we proposeanother INR feedback scheme that allows the BS to selectusers only with the same subset CCC t ⊆ C in (5). We call it the partial INR feedback . Selecting t k such that t k = argmax t ∈ [1: T ] max m ∈ [1: M ] (cid:12)(cid:12) hhh H k ccc ( t ) m (cid:12)(cid:12) user k feeds back INR ( t k ) k,m = (cid:12)(cid:12) hhh H k ccc ( t k ) m (cid:12)(cid:12) σ k , m ∈ [1 : M ] (12)along with the selected subset index t k . Thus the partial INRscheme entails the feedback overhead of M INRs and one (cid:100) log T (cid:101) -bit index t k per user. Denoting by M ( t k ) s the set ofall s -combinations of [1 : M ] (i.e., the power set of [1 : M ] )and by M ( t k ) s,n its n th element with |M ( t k ) s,n | = s , where n =1 , , · · · , C Ms , the BS with this partial INR feedback fromeach user can calculate the following SINRs γ k ( t k , s, n, q ) = (cid:12)(cid:12) hhh H k ccc ( t k ) q (cid:12)(cid:12) sP σ k + (cid:80) j ∈M ( tk ) s,n \ q (cid:12)(cid:12) hhh H k ccc ( t k ) j (cid:12)(cid:12) , q ∈ M ( t k ) s,n (13)for all k, s, n, and q . B. Benefits of INR Feedback1) Flexible Scheduling:
The fully flexible scheduling en-abled by the full INR feedback (10) is to select users overall possible combinations with SINRs γ k ( s, n, q ) in (9) forall k, s, n, and q . Clearly, this is the optimal scheduling fora given CB C within the typical framework in Section II. Incontrast, the scheduling with the partial INR feedback (12) andSINRs in (13) is naturally under the constraint of schedulingusers only with the same CCC t such that their precoding vectorsare orthogonal to each other. simple algorithm for the partial INR feedback can begiven as follows. We need to eventually find the set S ofselected users and the set Q of their precoding vectors in C .Initializations: S ( t )0 = { k : t k = t } and S ( t ) s,n = ∅ , Q ( t ) s,n = ∅ , µ ( t ) s,n = 0 for all ( t, s, n ) .Step 1) For each ( t, s, n ) , where t = 1 , · · · , T , s =1 , · · · , M , and n = 1 , · · · , C Ms , calculate the sum rate µ ( t ) s,n as the following loop with i = 1 : q = M ( t ) s,n ( i ) (14) k ∗ = argmax k ∈S ( t )0 log (1 + γ k ( t, s, n, q )) . (15)If S ( t ) s,n ∩ { k ∗ } = ∅ and i < s , then S ( t ) s,n = S ( t ) s,n ∪ { k ∗ } (16) Q ( t ) s,n = Q ( t ) s,n ∪ { q } (17) µ ( t ) s,n = µ ( t ) s,n + log (1 + γ k ∗ ( t, s, n, q )) (18) i = i + 1 . (19)Otherwise, finish this loop and continue Step 1) until we get S ( t ) s,n , Q ( t ) s,n , and µ ( t ) s,n for all ( t, s, n ) .Step 2) Select S and Q as follows: ( t ∗ , s ∗ , n ∗ ) = argmax t,s,n µ ( t ) s,n (20) S = S ( t ∗ ) s ∗ ,n ∗ (21) Q = Q ( t ∗ ) s ∗ ,n ∗ . (22)The condition in Step 1) is to prevent allocating multiplebeams to a single user and for simplicity here we ignoredfinding a best alternative user for such a beam. Similarly,the fully flexible (optimal) scheduling algorithm can be givenwith γ k ( s, n, q ) in (9) and taking into account finding a bestalternative user for the multiple beams case.
2) Dedicated Pilot Overhead Reduction:
Apart from flex-ible scheduling, a probably more important benefit of theproposed INR feedback is that it can reduce the dedicatedpilot overhead. Note that the INR gives us a side informationas to mutual interference between users. Depending on theINR feedback, the BS scheduler can figure out how muchthe intended user k with beam k is interfered by the otheruser using beam j ∈ C with j (cid:54) = k . Thus the scheduler canallocate a common resource to multiple dedicated pilots ofusers with their beams sufficiently low interfering each other.In sum, the INR feedback provides not only flexible schedulingbut it can also significantly reduce the overhead for dedicatedpilot.
3) Arbitrary Power Allocation:
Although we have assumedthe equal power allocation to calculate SINRs in (9) and (13)for the proposed framework, in fact the INR feedback allowsunequal power allocation of the total power P over any subsetof users. Although the resulting power optimization problemis non-convex, an appropriate sub-optimal power allocationalgorithm (e.g., water-filling) will bring a noticeable per-formance improvement especially when users have different SNRs. Furthermore, we may jointly optimize user selectionand power allocation. However, developing power allocationalgorithms is for further study. C. Feedback Overhead of INR
The feedback overhead of the full INR feedback is
M T
INRs (positive real values) per user, while that of the partialfeedback is M INRs and (cid:100) log T (cid:101) -bit subset index t k peruser. In contrast, DFT-SINR requires only a single SINR and (cid:100) log M T (cid:101) -bit beam index (cid:96) k per user. However, the lattershows non-negligible performance degradation due to lack ofscheduling flexibility (i.e., multiuser diversity gain reduction)as M increases and hence the ratio K/M (the number of usersper beam) decreases for K fixed. Notice that the quantizedfeedback of SINR generally requires only a small number ofbits per SINR in practice (e.g., 2 or 4 bits in [13]), which isalso the case with INR. For instance, the partial INR feedbackmay consume only (2 M + 1) bits per user for INR and t k ,lending itself to large-scale MIMO unless M is too large.IV. T HROUGHPUT OF
INR-B
ASED S CHEMES
In this section, we show how large a flexible schedulinggain owing to the INR feedback is in the i.i.d. Rayleigh fadingchannel. We focus on the M ∼ K case and for simplicity let T = 1 so that the full and partial feedback schemes becomeequivalent. It is well known that the achievable throughput ofRBF is [1] R RBF = E (cid:20) M (cid:88) m =1 log (cid:16) ≤ k ≤ K γ k ( m ) (cid:17)(cid:21) + o (1) ≈ M E (cid:20) log (cid:16) ≤ k ≤ K γ k ( m ) (cid:17)(cid:21) (23)where o (1) takes into account the fact that user k may bethe strongest user of two or more beams with a vanishingprobability as K goes to infinity. The equality follows fromthe fact that γ k ( m ) is i.i.d. over both k and m since the beamvectors of RBF forms a unitary matrix. For sufficiently large K , we have R RBF = M log log K + M log PM + o (1) . (24)Assuming that ccc , · · · , ccc M are random orthonormal vectorslike RBF and that all users have the same SNR with σ k =1 , ∀ k , the SINR in (10) can be rewritten as γ k ( s, n, q ) = v sP + y , q ∈ M s,n (25)where v = | hhh H k ccc q | and y = (cid:80) j ∈M s,n \ q | hhh H k ccc j | . Since { ccc q : q ∈ M s,n } are orthonormal, v is i.i.d. over k and q with χ (2) distribution and y is χ (2 s − distributed, thus resultingin that γ k ( s, n, q ) is i.i.d. over k . The distribution of such γ k ( s, n, q ) is well known to satisfy the sufficient condition in[14] (also [1, Corollary A.1] for extreme value theory.Given s and q ∈ [1 : M ] , it is easy to see that the number ofoccurrences of q in M s is C M − s − due to the flexible schedulingover the s -combinations of [1 : M ] . Denote by M s ( q ) such subset of M s for q . As a consequence, γ k ( s, n (cid:48) , q ) is i.i.d.over n (cid:48) ∈ M s ( q ) as well by the same argument as above.The flexible scheduling is to select the maximum sum rate µ s,n over s and n in (20), implying that the maximum µ s,n does not necessarily correspond to the sum of the maxima of γ k ( s, n (cid:48) , q ) ’s over k and n (cid:48) for q ∈ M s,n . Nevertheless, weassume that the scheduler selects the maximum γ k ( s, n (cid:48) , q ) for simplicity. Otherwise, it is very difficult to analyze thethroughput of the INR-based scheme by using extreme valuetheory since the standard approach requires the calculation ofthe distribution of (cid:80) q ∈M s,n log (cid:0) k γ k ( s, n, q ) (cid:1) , whichis i.i.d. over n for s fixed but log (cid:0) k γ k ( s, n, q ) (cid:1) ’s arenot independent over q . Suppose that the maximum µ s ∗ ,n ∗ corresponds to the sum of the i q th largest γ k ( s ∗ , n (cid:48) , q ) ’s over k and n (cid:48) for each q ∈ M s ∗ ,n ∗ . Letting i = max q i q we notice that the asymptotic behaviors of the largest γ k ( s ∗ , n (cid:48) , q ) and the i th largest one would be quite similar forall q since K × C M − s − is sufficiently large relative to i withhigh probability, i.e., the former quantity grows much fasterthan the latter as M increases. Using this approximation, forlarge K × C M − s − , we can write the throughput of INR as R INR = E (cid:104) max s,n µ s,n (cid:105) + o (1) ≈ E (cid:20) max s,n (cid:88) q ∈M s,n log (cid:16) k γ k ( s, n, q ) (cid:17)(cid:21) ≈ E (cid:20) max s s log (cid:16) k,n (cid:48) γ k ( s, n (cid:48) , q ) (cid:17)(cid:21) = max s E (cid:20) s log (cid:16) k,n (cid:48) γ k ( s, n (cid:48) , q ) (cid:17)(cid:21) (26)where s = 1 , · · · , M , n (cid:48) = 1 , · · · , C M − s − , and k = 1 , · · · , K .The first approximation is analogous to (23) and the secondapproximation follows from the equivalence of the asymptoticbehaviors of the largest γ k ( s, n (cid:48) , q ) and the i th largest onefor given s and q . The last equality is due to the fact that γ k ( s, n (cid:48) , q ) is i.i.d. over n (cid:48) as well as k . Consequently, theflexible scheduling in the i.i.d. Rayleigh fading channel is toselect the maximum of K × C M − s − i.i.d. random variables γ k ( s, n (cid:48) , q ) over both k and n (cid:48) for each s and q . Then, itchooses the optimal s that maximizes (26). Eventually, weobtain a simple result on the sum-rate scaling of INR asfollows. Theorem 1.
For M finite, the achievable throughput of theINR scheme behaves in the i.i.d. Rayleigh fading channel as R INR = max ≤ s ≤ M s log log (cid:0) K × C M − s − (cid:1) + s log Ps + o (1) . (27) Proof:
The proof of (27) follows the same line of [1]. Fora given s , using the fact that γ k ( s, n (cid:48) , q ) is i.i.d. over both k Recall that we consider the M ∼ K case with M not small. Accordingly, s ∗ near either M or will almost surely not happen. SNR (dB) S u m R a t e ( bp s / H z ) ZFBF−SUSINRRBF
Fig. 1. Achievable sum rate curves of MU-MIMO downlink schemes versusSNR in the independent fading case for M = 16 and K = 20 . and n (cid:48) , we recall that the i th largest γ k ( s, n (cid:48) , q ) behaves likethe largest one for K × C M − s − sufficiently large, where i (cid:28) K × C M − s − . Then, applying extreme value theory [15], it turnsout that the maximum of γ k ( s, n (cid:48) , q ) over ( k, n (cid:48) ) behaves like Ps log( K × C M − s − ) , thereby yielding (27). A notable differencefrom the proof of RBF lies in that K need not be necessarilylarge to invoke extreme value theory, unless M is small.The above result shows that, letting S = s ∗ , the flexiblescheduling algorithm due to the INR feedback provides themultiuser diversity gain of S log log (cid:0) K × C M − s − (cid:1) and SNRgain of MS (i.e., SNR is PS instead of PM ) at the cost ofmultiplexing gain reduction by a factor of SM , compared to(24) of RBF that requires sufficiently large K . Note that thisresult holds true as long as K is too small, as mentionedearlier. Therefore, Theorem 1 shows that flexible schedulingis of use for our goal that improves the conventional CB-basedapproach like RBF in the M ≥ K regime with M large. Also,it is easy to see that the flexible scheduling gain vanishes as K goes to infinity with M finite, i.e., lim K →∞ R INR = R RBF with S = M. Fig. 1 shows the achievable throughputs of ZFBF-SUS [10],RBF, and INR schemes in the i.i.d. Rayleigh fading channelfor M = 16 and K = 20 . We can see that the rate gapbetween RBF and INR is quite large, advocating the flexiblescheduling gain is remarkable. Even if INR outperforms RBF,it suffers severe performance degradation compared to ZFBFin the independent fading case.Extending Theorem 1, where we assumed T = 1 , into themultiple T case, we have the following corollary. Corollary 1.
For M finite, the achievable throughput of thefull INR feedback scheme behaves in the i.i.d. Rayleigh fadinghannel as R full INR = max ≤ s ≤ M s log log (cid:16) KT × C M − s − (cid:17) + s log Ps + o (1) . (28) Assuming the same number of users for each subset C t in (5),the achievable throughput of the partial INR feedback schemebehaves like R partial INR = max ≤ s ≤ M s log log (cid:16) KT × C M − s − (cid:17) + s log Ps + o (1) . (29)Comparing (28) with (29), we can see that the full INRfeedback gives only a vanishing gain over the partial INR as K increases for T fixed.V. F EEDBACK O VERHEAD R EDUCTION
One may argue that even the M -bit INR feedback per useris still burdensome for large-scale MIMO, where both M and K are large. We present here a simple and heuristic approachto reduce the cost of INR feedback, referred to as one-bit INRfeedback .For each user k , first select ( t k , m k ) such that ( t k , m k ) = argmax ≤ t ≤ T, ≤ m ≤ M (cid:12)(cid:12) hhh H k ccc ( t ) m (cid:12)(cid:12) σ k . (30)We consider the resulting | hhh H k ccc ( t k ) m k | /σ k as the SNR (denotedby SNR k ) of user k with respect to the selected beam ( t k , m k ) and hence the partial INR feedback is assumed here. Then, weimpose a threshold γ on INR ( t k ) k,j , j (cid:54) = m k in (12). If INR ( t k ) k,j SNR k ≥ γ then one bit is assigned to beam j by ‘1’ and otherwise by ‘0’.By setting γ sufficiently small (e.g., γ = 0 . ( − dB)), thisfeedback information allows the BS to just know whether theinterference between any selected users is negligible or not,instead of the accurate INR. Since at high SNR the receiverperformance is more sensitive to the channel estimation error, γ slightly depends on SNR k . Therefore, γ may be a functionof SNR k rather than a fixed quantity.Clearly, the one-bit INR feedback is to impose a restrictionon the scheduler by reducing the number of hypotheses. Atthe sacrifice of an inevitable performance loss, this techniqueprovides feedback overhead reduction by a factor of / asit consumes only one SNR (e.g., bits) and ( M − -bitINRs per user. The performance loss will be evaluated laterby simulation results.[OPEN ISSUES]1) We definitely need a low complexity scheduling algo-rithm for the one-bit INR feedback scheme, which may becomputationally much more efficient than others for large-scale MIMO.2) Note that ‘1’ would be sparse when the AS of usersare large. On the other hand, ‘0’ would become more sparseas the number M of antennas and/or the center frequency f c of the system increases. To exploit these aspects and hencefurther reduce the feedback overhead, we need to derive orat least employ a compressive sensing technique for binarysparse signals. VI. N UMERICAL R ESULTS
We evaluate achievable throughputs of several schemes ofinterest in correlated fading MIMO broadcast channels, forwhich we use the following one-ring channel model.
A. One-Ring Channel model
In the typical cellular downlink case, the BS is elevatedand free of local scatterers, and the users are placed atground level and are surrounded by local scatterers. Thischannel scenario corresponds to the one-ring model [16], forwhich a user located at azimuth angle θ and distance s issurrounded by a ring of scatterers of radius r such that AS ∆ ≈ arctan( r / s ) . Assuming the uniform linear array (ULA)with a uniform distribution of the received power from planarwaves impinging on the BS array, we have that the correlationcoefficient between BS antennas ≤ p, q ≤ M is given by[17] [ RRR ] p,q = 12∆ (cid:90) ∆ − ∆ e j πD ( p − q ) sin( α + θ ) dα (31)where D is the normalized distance between antenna elementsby the wavelength. B. System Throughput
We will present here the achievable throughput of MU-MIMO downlink systems discussed in this paper. Considerthe BS has M = 4 , , antennas with ULA of antennaspacing D = 1 / , and users are uniformly distributed overthe range [ − o , o ] with ∆ k uniformly distributed in therange [5 o , o ] , where ∆ k is angular spread (AS) of user i .The transmit correlation matrices are generated by the one-ring channel model (31).
1) Flexible vs. Limited Scheduling:
For M = 4 (i.e., small M ), Fig. 2 compares the sum rates of ZF-DPC with greedyuser selection [18] (ZF-DPC-GUS), which approximates thesum capacity, ZFBF-SUS, DFT-based precoding with full INRfeedback (DFT-INR-Full), DFT-based precoding with partialINR feedback (DFT-INR-Partial), DFT-based precoding withSINR feedback (DFT-SINR), random beamforming (RBF) [1].While ZF-DPC-GUS and ZFBF-SUS assume perfect CSIT,the others employ the DFT-based CB (4). Throughout thissection, the number T of subsets in the CB was set to ,which is appropriate as a large CB size usually underminesmultiuser diversity gain. The beneficial impact of transmitcorrelation on the capacity of MU-MIMO downlink channelswas addressed in [11]. We show that spatially correlated fadingis more beneficial to the CB-based schemes of interest thanZFBF with full CSI. Fig. 3 and Fig. 4 present the sum-ratecomparison for M = 8 , (slightly large M ), respectively.From Figs. 2–4, we observe that the proposed frameworkbased on INR feedback improves the system throughput due to Number of users (K) S u m R a t e ( bp s / H z ) ZF−DPC−GUSZFBF−SUSDFT−INR (Full)DFT−INR (Partial)DFT−SINRRBF
Fig. 2. Achievable sum rate curves of various MU-MIMO downlink schemesversus K for M = 4 , SNR = 10 dB, and ∆ k ∈ [5 o , o ] . While the solidlines indicate spatially correlated Rayleigh fading channels, the dash-dot linesdenote the i.i.d. Rayleigh fading channel. Number of users (K) S u m R a t e ( bp s / H z ) ZF−DPC−GUSZFBF−SUSDFT−INR (Full)DFT−INR (Partial)DFT−SINRRBF
Fig. 3. Achievable sum rate curves of various MU-MIMO schemes versus K for M = 8 , SNR = 10 dB, and ∆ k ∈ [5 o , o ] . While the solid lines indicatespatially correlated Rayleigh fading channels, the dash-dot lines denote thei.i.d. Rayleigh fading channel. flexible scheduling, compared to the conventional frameworkbased on CDI and SINR feedback (i.e, DFT-SINR), especiallywhen K is small. Therefore, the flexible scheduling gain ofthe new framework is readily expected to become even muchlarger as M increases at the cost of a moderate increasein CSI feedback overhead. It is also shown that DFT-SINRapproaches DFT-INR for K sufficiently large. This is becausethe optimal scheduling selects all M orthogonal beams in asubset t with high probability as K → ∞ , which correspondsto the SINR feedback case. Number of users (K) S u m R a t e ( bp s / H z ) ZFBF−SUSDFT−INR (Full)DFT−INR (Partial)DFT−SINRRBF
Fig. 4. Achievable sum rate curves of various MU-MIMO schemes versus K for M = 16 , SNR = 10 dB, and ∆ k ∈ [5 o , o ] .
10 20 30 40 50 60 70 80 90 1005101520253035
Number of users (K) S u m R a t e ( bp s / H z ) ZFBF−SUSDFT−INR (Full)DFT−INR (Partial)DFT−SINR
Fig. 5. Impact of transmit correlation on sum rates of various MU-MIMOschemes for varying K where M = 8 and SNR = 10 dB. The solid, dash-dot, and dotted lines indicate ∆ k ∈ [5 o , o ] , ∆ k ∈ [10 o , o ] , and ∆ k ∈ [20 o , o ] , respectively. Finally, we observe that the DFT-based schemes signif-icantly outperform RBF in correlated fading channels, thelatter of which is known to be quite inferior to ZFBF withlimited feedback (e.g., [19]). The reason why RBF showssevere degradation in the correlated fading channel is thatit uses isotropic orthonormal beams even though the userchannels have long-term channel directions, i.e., some ofbeams transmitted by RBF are outside the channel directionsof users.
2) Impact of Transmit Correlation:
Fig. 5 shows the effectof three different transmit correlation scenarios captured by thedistribution of ∆ k on achievable rates of ZFBF-SUS, DFT- Number of users (K) S u m R a t e ( bp s / H z ) ZFBF−SUSDFT−INR (Full)DFT−INR (Partial)DFT−SINR
Fig. 6. Impact of channel estimation error on sum rates of various MU-MIMOschemes for varying K where M = 16 , SNR = 10 dB, and ∆ k ∈ [5 o , o ] .The solid lines and dash-dot lines indicate σ err = 0 . and . , respectively. INR, and DFT-SINR for M = 8 . As transmit correlationincreases (i.e., ∆ k decreases), the rate gap between ZFBF-SUSand DFT-INR reduces, thus implying DFT-INR lends itself tohighly correlated fading channels.
3) Sensitivity to Noisy CSIT:
Figure 6 shows the sensitivityof ZFBF-SUS, DFT-INR, and DFT-SINR to noisy CSIT ˜ hhh k ,which is modeled as ˜ hhh k = (cid:112) − σ err hhh k + σ err nnn , where nnn ∼ CN (000 , III ) and ≤ σ err < . As readily expected,ZFBF-SUS with noisy CSIT (i.e., channel estimation errordue to imperfect CSIR) suffers relatively large performancedegradation, while DFT-INR is less sensitive. Notice thatthis channel estimation error incurs additional performancedegradation, aside from channel quantization error.
4) Impact of Dedicated Pilot Overhead:
So far, we haveassumed the scheduler can use all the resources during thecommunication phase. However, we should consider a costfor the downlink dedicated pilot phase in realistic systems,regardless of FDD or TDD. For convenience, we use somenumerology of LTE standards to study the overhead for dedi-cated pilots in a more realistic fashion. Given a resource block(RB) consisting of 14 OFDM symbols with 12 subcarrierseach, the LTE-advanced system allocates roughly 3 symbolsfor control channel and others (CSI-RS, etc) and 1 symbol per2 data streams (DM-RS ports) for MU-MIMO downlink. Forthe conventional framework with SINR feedback (includingZFBF-SUS with noisy CSIT), the ratio of the remainingOFDM symbols to convey user data streams to the totalsymbols can be expressed as κ no-INR ( i ) = 11 − (cid:100) S ( i )2 (cid:101) (32)where S ( i ) ≤ min( M, K ) is the number of selected users atthe i th scheduling time interval. In contrast, we let the ratio κ INR of DFT-INR fixed to / . Say, the BS schedules up
10 20 30 40 50 60 70 80 90 10046810121416182022
Number of users (K) S u m R a t e ( bp s / H z ) ZFBF−SUSDFT−INR (Full)DFT−INR (Partial)DFT−SINR
Fig. 7. Impact of dedicated pilot overhead on sum rates of various MU-MIMO schemes for varying K where M = 16 , SNR = 10 dB, ∆ k ∈ [5 o , o ] , and σ err = 0 . . to S ( i ) users only when one group of S ( i ) users can sharea common resource for their dedicated pilots with negligibleinterference to each other (e.g., − dB lower than eachintended signal power) and the other group does so, resultingin that two user groups consume only a single symbol.The curves in Fig. 7 show the adjusted system throughputsof the schemes of interest for M = 16 with noisy CSIT ( σ err =0 . ), where the throughputs of ZFBF-SUS and DFT-SINRare adjusted by the scaling factor κ no-INR ( i ) every schedulinginterval, while DFT-INR is scaled by κ INR = 10 / . Thisresult points out that the dedicated pilot overhead significantlyaffects the system performance. The performance of DFT-INRscales with K , while the others do not. Noticing that κ no-INR ( i ) keeps growing as both M and K increase, it is expected thatDFT-INR will show larger performance benefits when M ismoderately large (e.g., up to ).Finally, Fig. 8 shows the average system throughput of theone-bit INR feedback scheme with some large-scale channelparameters like the mean of ∆ k and path loss exponent(heterogeneous SNRs among users) obeying 3GPP case 1(urban macro scenario) [20] with σ err = 0 . and γ = 0 . ( − dB). Compared to the full INR feedback, the one-bitINR exhibits remarkably marginal degradation in the morerealistic situation. VII. C ONCLUSION
We have proposed a new framework for CB-based MU-MIMO systems, which requires no explicit feedback of CDIand rather relies on a new type of CQI feedback referred to asINR. The benefits of the new framework can be summarizedas flexible scheduling, dedicated pilot overhead reduction, andarbitrary power allocation.Assuming a CB of size M , limited scheduling of RBFselects the largest one of K SINRs per beam, while flexible
Number of users (K) S u m R a t e ( bp s / H z ) ZFBF−SUSDFT−INR (Partial)DFT−INR (One−bit)DFT−SINR
Fig. 8. System throughput of the one-bit INR feedback scheme for M = 16 with 3GPP case 1 parameters. scheduling due to INR is asymptotically equivalent to selectthe largest one of K × C M − S − SINRs per beam, where S is thenumber of selected users. In the i.i.d. Rayleigh fading channel,the resulting multiuser diversity gain is S log log( K × C M − S − ) for large K × C M − S − but not requiring large K . In contrast,the well-known M log log K of RBF is valid only when K is sufficiently large with K (cid:29) M . Therefore, if K ∼ M with M not small, the performance improvement thanks toINR feedback was shown to be considerable. It is arguablymost interesting and surprising that taking the cost of downlinkdedicated pilot into account, DFT-INR under the new limitedfeedback framework may significantly outperform the idealZFBF-SUS without CDI quantization even for the large-scalearray regime. R EFERENCES[1] M. Sharif and B. Hassibi, “On the capacity of MIMO broadcast channelwith partial side information,”
IEEE Trans. on Inform. Theory , vol. 51,no. 2, pp. 506–522, 2005.[2] S. Sesia, I. Toufik, and M. Baker,
LTE – The UMTS Long TermEvolution: From Theory to Practice . Wiley Online Library, 2009.[3] T. L. Marzetta, “Noncooperative cellular wireless with unlimited num-bers of base station antennas,”
IEEE Trans. on Wireless Commun. , vol. 9,no. 11, pp. 3590–3600, Nov. 2010.[4] D. J. L. J. Choi, Z. Chance and U. Madhow, “Noncoherent trellis codedquantization: A practical limited feedback technique for massive MIMOsystems,”
IEEE Trans. on Commun. , vol. 61, no. 12, pp. 5016–5029,Nov. 2013.[5] P.-H. Kuo, H. T. Kung, and P.-A. Ting, “Compressive sensing basedchannel feedback protocols for spatially-correlated massive antennaarrays,” in Proc. IEEE Wirel. Comm. and Net. Conf. (WCNC) , pp. 492–497, Apr. 2012.[6] N. Jindal, “MIMO broadcast channels with finite-rate feedback,”
IEEETrans. on Inform. Theory , vol. 52, no. 11, pp. 5045–5060, Nov. 2006.[7] T. Yoo, N. Jindal, and A. Goldsmith, “Multi-antenna downlink chan-nels with limited feedback and user selection,”
IEEE J. Select. AreasCommun. , vol. 25, no. 7, pp. 1478–1491, Sep. 2007.[8] G. Caire and S. Shamai, “On the achievable throughput of a multiantennaGaussian broadcast channel,”
IEEE Trans. on Inform. Theory , vol. 49,no. 7, pp. 1691–1706, 2003. [9] B. Hochwald, T. Marzetta, and V. Tarokh, “Multi-antenna channel-hardening and its implications for rate feedback and scheduling,”
IEEETrans. on Inform. Theory , vol. 50, no. 9, pp. 1893–1909, Sep. 2004.[10] T. Yoo and A. Goldsmith, “On the optimality of multi-antenna broadcastscheduling using zero-forcing beamforming,”
IEEE J. Select. AreasCommun. , vol. 24, no. 3, pp. 528–541, 2006.[11] J. Nam, “Fundamental limits in correlated fading MIMO broadcastchannels: Benefits of transmit correlation diversity,” 2014. [Online].Available: http://arxiv.org/abs/1401.7114[12] D. Yang, L.-L. Yang, and L. Hanzo, “DFT-based beamforming weight-vector codebook design for spatially correlated channels in the unitaryprecoding aided multiuser downlink,”
Proc. IEEE Int. Conf. on Commun.(ICC) , 2010.[13] X. Zhang and X. Zhou,
LTE-Advanced Air Interface Technology . CRCPress, 2012.[14] N. T. Uzgoren, “The asymptotic development of the distribution of theextreme values of a sample,” in Studies in Mathematics and MechanicsPresented to Richard von Mises , pp. 346–353, 1954.[15] M. R. Leadbetter and H. Rootzen, “Extremal theory for stochasticprocesses,”
The Annals of Probability , vol. 16, pp. 431–478, 1988.[16] D. Shiu, G. Foschini, M. Gans, and J. Kahn, “Fading correlation and itseffect on the capacity of multielement antenna systems,”
IEEE Trans.on Commun. , vol. 48, no. 3, pp. 502–513, 2000.[17] A. Adhikary, J. Nam, J.-Y. Ahn, and G. Caire, “Joint spatial divisionand multiplexing: The large-scale array regime,”
IEEE Trans. on Inform.Theory , vol. 59, no. 10, pp. 6441–6463, 2013.[18] G. Dimic and N. Sidiropoulos, “On downlink beamforming with greedyuser selection: performance analysis and a simple new algorithm,”
IEEETrans. on Sig. Proc. , vol. 53, no. 10, pp. 3857–3868, 2005.[19] N. Ravindran and N. Jindal, “Multi-user diversity vs. accurate channelfeedback for MIMO broadcast channels,”
Proc. IEEE Int. Conf. onCommun. (ICC) , pp. 3684–3688, 2008.[20] 3GPP, “Spatial channel model for mimo simulations,”