Genetic and Greedy User Scheduling for MultiuserMIMO Systems with Successive Zero-Forcing

Robert C. Elliott1, Shreeram Sigdel1, Witold A. Krzymien1, Mazin Al-Shalash2 and Anthony C. K. Soong21University of Alberta / TRLabs, Edmonton, Canada

2Huawei Technologies, Plano, TX, USA

AbstractIn this paper we consider efficient and low complex-ity scheduling algorithms for multiuser multiple-input multiple-output (MIMO) systems. The optimal user scheduling involves anexhaustive search, which becomes very complex for realistic num-bers of users and transmit antennas. Among various suboptimalbut low complexity algorithms, greedy algorithms with heuristicscheduling metrics have been shown to achieve performance closeto an exhaustive search. Meanwhile, genetic algorithms (GAs)are a rapid, though suboptimal, option of performing a utility(in this case scheduling) metric optimization. In this paper, wepropose and analyze the performance and complexity of greedyand genetic scheduling algorithms for multiuser MIMO systemswith successive zero-forcing precoding. We demonstrate that atlower K, the genetic algorithm performs better than the greedyalgorithm, where K denotes the total number of users requestingservice. For large K, however, the greedy algorithm outperformsthe genetic algorithm. The greedy algorithm also achieves similarsum-rate growth (with K) as the exhaustive search. A detailedcomplexity analysis shows that the order of complexity of thegenetic algorithm is higher than that of the greedy algorithm bya factor equal to K20 , where K0 denotes the maximum numberof simultaneously served multiple-antenna users. Both algorithmsachieve a sum rate very close to the exhaustive search with muchless complexity for a small number of transmit antennas.

I. INTRODUCTIONEfficient user scheduling methods for linearly precoded

multiuser MIMO channels with multiple antennas at the basestation and multiple antennas at each user have captured muchresearch attention recently. It is known that the capacity ofa multiuser MIMO broadcast channel (BC) can be achievedthrough the use of dirty paper coding (DPC) [1], [2]. However,DPC is highly nonlinear and very complex to implement inpractice. Therefore, reduced complexity precoding methodsare of interest to reduce the effect of multiuser interfer-ence (MUI). Such methods include zero-forcing beamforming(ZFB) [3] for systems with single-antenna users, and blockdiagonalization (BD) [4] and successive zero-forcing (SZF)[5] for systems with multiple-antenna users.

In particular, BD is a technique that completely nulls theinterference between users. However, this nulling operationimposes a constraint that the total number of receive antennasbe no larger than the number of transmit antennas. This alsoyields a reduction in the performance and number of users that

Our work made use of the infrastructure and computational resourcesof AICT (Academic Information and Communication Technologies) at theUniversity of Alberta. The authors also gratefully acknowledge fundingfor this research provided by TRLabs, Huawei Technologies, the RohitSharma Professorship, the Natural Sciences and Engineering Research Council(NSERC) of Canada, the Alberta Informatics Circle of Research Excellence(iCORE), and the Alberta Ingenuity Fund.

can be served relative to DPC. In many situations, a completeremoval of MUI is not beneficial. SZF is one example whichdoes not completely null the MUI. In [5] it is shown that withmultiple antenna users, the achievable throughput of SZF ishigher than that of BD in several cases. Additionally, SZF canrelax the transmit/receive antenna constraints and sometimesserve a higher number of users simultaneously compared toBD.

In multiuser systems, typically there are very many usersthat request service simultaneously. The above mentionedantenna constraints, as well as a transmit power constraint,results in the necessity for user scheduling, since all users re-questing service cannot be served simultaneously. The optimalscheduling method is combinatorially complex and involvesan exhaustive search over the user pool in order to findthe selection that maximizes some sort of utility function.Additionally, for successive encoding methods such as SZF,the user encoding order also affects the performance of thesystem, further complicating the choice. Thus, various heuris-tic scheduling methods have been proposed to simplify thescheduling process [3], [6], [7], which perform close to theoptimal exhaustive search.

In this paper, we propose and analyze the performance andcomplexity of greedy and genetic user scheduling algorithmsto maximize the system throughput in a multiuser MIMOsystem with multiple-antenna receivers, in the context of SZF.The first is a low complexity greedy heuristic algorithm foruser selection and ordering, as proposed in [8]. The secondmethod involves the use of genetic algorithms (GAs) [9] forutility function maximization. GAs are known for achievingvery good solutions to optimization problems very quickly.GAs for scheduling remove almost all of the complexity fromthe user selection process and move it to the utility functioncalculation. We compare the performance of a GA similar tothat described in [10], adapted for use with SZF.

The remainder of this paper is organized as follows. SectionII describes the system model and the pertinent details of SZF.Section III describes the proposed greedy and genetic schedul-ing algorithms. Section IV provides a novel and extensiveanalysis of the algorithms complexity. Simulation results areprovided in Section V, and concluding remarks are given inSection VI.

II. SYSTEM MODELWe consider the downlink of a multiuser MIMO system

with MT transmit antennas and Nk receive antennas ateach of the K multiple-antenna users requesting service. Let

978-1-4244-5626-0/09/$26.00 2009 IEEE

Hk CNkMT denote the downlink channel matrix of thekth user, k = 1, 2, ...,K. We assume a spatially uncorrelatedflat Rayleigh fading channel model, i.e. the elements of Hkare independent and identically distributed complex Gaussianrandom variables with variance of 0.5 per dimension. Weassume that the transmitter has perfect knowledge of thechannel state information of all users (perfect CSIT), andeach user knows its channel perfectly. The data vector ofuser k, sk CNk1, is preprocessed at the transmitter withthe beamforming matrix Wk CMTNk to produce thetransmitted signal vector xk CMT1. The Nk 1 receivedsignal vector of the kth user can be expressed as

yk = HkK

j=1Wjsj + nk (1)

where nk CNk1 denotes zero mean additive white Gaussiannoise with E{nknHk } = 2nIN .

Block diagonalization (BD) designs Wk to pre-eliminatemultiuser interference such that HkWj = 0, k =j and1(j, k)K. This decomposes the multiuser channel intoequivalent single user channels, and the received signal vector(1) becomes yk = HkWksk + nk. In comparison, succes-sive zero-forcing (SZF) precoding does not completely pre-eliminate the multiuser interference.

Due to the successive nature of SZF, the user precodingorder is important for sum-rate maximization. For a given setof users with encoding order , for each user j {1, ...,K}the received signal can be expressed as [5]

y(j) = H(j)(W(j)s(j) +

ijW(i)s(i)) + n(j) (2)

The precoding matrix W(j) is designed such that it lies inthe null space of the aggregate channel Hj1 of the j 1previously precoded users channels:

Hj1 = [HT(1)H

T(2) . . .H

T(j1)]

T (3)SZF of K users channels is possible1 if MT > rank(HK1).

Let us denote the SVD of (3) asHj1 = Uj1j1V

Hj1 = Uj1j1[V

1j1V

0j1]

H (4)where Vj1 CMTMT , and V0j1 holds the MTrank(Hj1)right column vectors defining the null-space basis of Hj1.The precoding matrix of the jth user W(j) is constrained tolie in the subspace defined by V0j1. Hence, the third term in(2) is canceled by the subspace constraint on the design of theprecoding matrices for users i > j. Then, (2) reduces to

y(j) = H(j)(W(j)s(j) +

i> MT . Thenext section provides the details of the proposed algorithms.

III. USER SCHEDULING ALGORITHMSThe optimal user scheduling requires an exhaustive search

through all possible combinations of subsets of simultaneouslyserved users and is thus computationally very complex. Hence,the main objective of this paper is to investigate and analyzethe performance and complexity of reduced-complexity greedyand genetic scheduling algorithms for SZF assuming perfectCSIT. The proposed algorithms are outlined in the following.

A. Greedy AlgorithmLet U = {1, 2, ...,K} denote the set of all users requesting

service, and Ul U denote the possible subset of userssuch that |Ul| K0, l = 1, 2, ...

(KK0

), where |Ul| denotes the

cardinality of set Ul; K0 denotes the maximum number ofusers that can be served simultaneously. Denote the selecteduser subset as Us. In this paper, a simple Frobenius-norm (F-norm) based heuristic scheduling metric is designed as in [8]for the sum-rate maximization objective. Further simplificationof the greedy scheduling algorithm is obtained by using aintermediate user grouping technique as in [6]. Considering(6), the proposed scheduling metric maximizes the signal-to-approximate-interference (from the previously selected users)ratio to maximize the rate of a selected user. Once the usersto be served are selected using the proposed algorithm, theSZF technique described in Section II is used to optimize the

TABLE ISIMPLIFIED GREEDY USER SCHEDULING ALGORITHM FOR SZF

1. i 1; U = {1, 2, ..., K}; Us = {}.Select a user u1 such that u1 = argmax

kUHk2F .

Set Us = Us {u1}; U1 = U\{u1}.2. i i + 1.

Define H(Us)=[HTu1HTu2 ...HTui1 ]T=Uii[V1i V0i ]H .3. [a.] If (|Us| < K0),

Intermediate user grouping:Find Ui =

{k Ui1, k / Us| HkV

1i F

HkF V1i F<

}.

[b.] If (|Ui| = 0),Select a user such that

ui =

arg maxkUi

HkV0i 2F if i = 2,

arg maxkUi

HkV0i 2Fi1j=2

HkV0j2Fotherwise.

Set Us = Us {ui}; Ui = U\{ui}; Go to Step 2.Else exit

channel input covariance matrices for the selected users. Theproposed algorithm is described in Table I.

The algorithm starts by selecting a user with the maximumF-norm of its channel. In Step 3a, an intermediate usergrouping is performed based on a specified threshold to finda subset Ui. If the subset Ui is not empty, the next best user isselected from that subset. Otherwise, the algorithm terminates.The intermediate user grouping technique significantly reducesthe complexity of the regular greedy search algorithm bylimiting the search to Ui as |Ui|

1s is reduced to K0, or a single 0 is randomly toggledto a 1 if there are no 1s. Non-unique order values are setrandomly to a unique value. Iteration: The process of selection and breeding is repeated

until a new set of Np chromosomes is created. This newset then replaces the old population. The whole process thenrepeats for a total of Ng generations.

Our GA also employs elitism, where the best chromosomeC of the past generation is kept in the new one. Duringeach generation, Np2 chromosomes are created through theselection and breeding processes, one is inserted as a copyof C, and one is an additional copy of C, except that theencoding order of two users is swapped at random. Fig. 1shows an example of typical chromosomes and the operationof the proposed genetic scheduling algorithm.

IV. COMPLEXITY ANALYSISIn this section, we compare the complexity of the greedy and

genetic algorithms in terms of the number of flops required. Aflop is a real-value floating point operation; an addition, multi-plication, or division are each 1 flop. A complex-value additionand multiplication take 2 and 6 flops, respectively. In general,most matrix operations require about an equal number ofmultiplications and additions. Thus, we assume that complex-valued operations need 4 times the flops as the real-valuedones. For the analysis, we assume Nk=N,k,K0=MT /N,and that the algorithms schedule the maximum of K0 users.Since K0=MT /N=MT /N+, 0

by Gi=Hi(I +

j =i HjPjHj)1/2

, which involves matrixadditions and multiplications, and an inverse square root. Theblock-diagonal matrix formed by these Gi is waterfilled inorder to obtain covariance matrices Si, which are in turnused to update each Pi for the next iteration. Of all thecalculations, the most complex is the inverse square rootof an MT MT matrix for each of the K0 users duringeach iteration. Thus, the complexity of Step 1 is O(K0M3T )flops, where is the number of iterations required for thealgorithm to converge. From the figures in [11] and from ourown simulations, 3-5 iterations are generally enough for thealgorithm to converge to less than 1% error in the DPC sumrate, which is sufficient for scheduling purposes. Hence, theoverall complexity of Step 1 is O(K0M3T ). Step 2: The DPC covariance matrices j are

determined successively (assuming user 1 isencoded first on the MAC) by calculating Aj=I+Hj(j1

i=1i)HHj , Bj=I+

K0i=j+1

HHi PiHi andj=B

1/2j FjG

Hj A

1/2j PjA

1/2j GjF

Hj B

1/2j . This involves

matrix sums and multiplications, square roots, and an SVDto find Fj and Gj via B1/2j HHj A

1/2j =FjjG

Hj . As with

step 1, the most complex operation is the inverse squareroot of the MT MT matrices Bj for K0 users. Thus, thecomplexity of Step 2 is also O(K0M3T ). Step 3: To convert the DPC matrices j to SZF covariance

matrices Qi, for each user j, the null space basis vectorsV0j for the aggregate channel matrix of the previousj1 encoded users are found through an SVD or a QRdecomposition; V01=I. For users 1 through K01, theSZF matrices are found as Qj=V0j V0

H

j jV0j V

0H

j . Forthe final user K0, first a temporary covariance matrixQK0 is found by waterfilling over an effective channelmatrix Heff=(I+HK0(

K01j=1

Qj)HHK0)

12HK0V0K0V

0H

K0

with power constraint PK01j=1

Tr(Qj), thenQK0=V

0K0V

0H

K0QK0V0K0V

0H

K0 . Over all users, the calculationof the null space vectors is O

(K20M

2TNK30MT N2+K40N3

),

while the matrix multiplications for the Qj matrices areO(K0M3T ). With K0=MT /N, the above terms are allabout O(K0M3T ), which is therefore the overall complexityof Step 3. Step 4: In calculating the sum rate (7), each user requires

2 determinant calculations (except for the first, where anidentity matrix is in the denominator). The sum of Qjmatrices is updated once per user at a cost of 2M2T flops. Withthe sum calculated, each determinant value requires a total of8M2TN + 8MTN

2 +N + 8/3N3 + 6N flops for the matrixmultiplications and the determinant value calculation. Lastly,2 flops are required per user to multiply and divide all thereal determinant values together. Thus, the total complexityof Step 4 is (2K0 1)(8M2T N + 8MT N2 + 8/3N3 + 7N) +2K0M

2T + 2K0, which is O(K0M2T N) O(M3T ).

Thus, one GA metric calculation requires O(K0M3T ) flops. TheGA calculates this metric NpNg times. We use Np=5K0and Ng=K/2 in our simulations. Thus, the entire schedul-ing process is O(KK20M3T ). Hence, the genetic algorithm is

10 20 30 40 50 60 70 80 90 1006.5

7

8

9

9.8

Ave

rage

sum

rate

(bits/

s/Hz)

3 10 20 30 40 50 60 70 80 90 100

11

12

13

14

15

Number of users ( K )

Exhaustive searchProposed genetic algorithmProposed greedy algorithm

(a) SNR = 5 dB

(b) SNR = 10 dB

Fig. 2. Performance of exhaustive search, proposed greedy and geneticscheduling algorithms; MT = 4, Nk = 2,K0 = 2; SNR = 5 and 10 dB.

more complex than the greedy algorithm by a factor of K20 .However, it should be noted that the greedy algorithm willstill have to calculate (7) once (i.e. perform the equivalent ofone GA metric calculation) to find the transmit covariancematrices, at a cost of O(K0M3T ) flops. whereas, the GAhas already performed that calculation. In comparison, anexhaustive search to find the user selection and ordering thatmaximizes (8) will be O (K0!(KK0

)K0M

3T

) O(KK0K0M3T ).

V. SIMULATION RESULTSIn this section, we present simulation results demonstrating

the performance of our proposed algorithms. A performancecomparison of the optimal exhaustive search, proposed greedyalgorithm and proposed genetic algorithm (GA) for SZF arepresented. For the greedy algorithm, the optimal correlationthreshold (that maximizes the sum rate) is determined as in[6] through simulation. It has been observed that the optimumuser grouping threshold decreases with increasing K [6]. Forexample, a threshold in the range of 0.45 to 0.375 has beenobserved to be optimal for K = 10 to 100. The optimal caseexhaustively searches through all possible user combinationsand orders.

Fig. 2 shows the performance for MT = 4. It is observedthat the proposed algorithms perform very close to the ex-haustive search. For small K (e.g. K < 15 at SNR = 5dB, and K < 30 at SNR = 10 dB in this example) the GAoutperforms the greedy algorithm, whereas for large K thegreedy algorithm outperforms the GA. At SNR = 10 dB, theperformance of the greedy algorithm and GA has been foundto be close to each other (with the greedy algorithm performingmarginally better than the GA at larger K); hence the plotsin Fig. 2b are not clearly distinguishable. The performance ofboth algorithms is less than 0.8 bits/s/Hz inferior than optimal,achieving about 95 98% of the sum rate of an exhaustivesearch.

Similar results are observed for MT = 8 in Fig. 3. Acrossover of the performance curves of the greedy algorithmand GA can be seen from these figures. The reason for thisas follows. When scheduling the maximum number of userssimultaneously, the cancelation of MUI becomes a significantfactor. It is often best to schedule less than the maximum

4 10 20 30 40 50 60 70 80 90 10010

11

12

13

14

15Ave

rage

sum

rate

(bits/

s/Hz)

4 10 20 30 40 50 60 70 80 90 10016

18

20

22

24

Number of users (K)

Exhaustive searchProposed genetic algorithmProposed greedy algorithm(b) SNR = 10 dB

(a) SNR = 5 dB

Fig. 3. Performance of exhaustive search, proposed greedy and geneticscheduling algorithms; MT = 8, Nk = 2,K0 = 4; SNR = 5 and 10 dB.

servable number of users in order to maximize the throughputfor small K. The greedy algorithm is biased towards schedul-ing the maximum number of users, whereas the GA is not.Consequently, the performance of the greedy algorithm suffersat small K. On the other hand, as K increases, the likelihoodof users having near-orthogonal channels increases as a resultof multiuser diversity, making it more likely to be optimalto schedule K0 users. For example, at 10 dB, our exhaustivesearch results indicate that at K=8, it is optimal to schedule 3users instead of 4 about 61% of the time, whereas at K=20,this drops to about 6.5%. For similar reasons, it becomes morelikely at higher K that the user with the best channel should bescheduled, which the GA does not guarantee. Furthermore, it isobserved that the proposed greedy algorithm achieves similarsum-rate growth versus K as the exhaustive search; the sumrate of any beamforming (including SZF) grows as log(logK)[15]. A plot of the greedy scheduling results vs. log(logK)(not included due to lack of space) is indeed linear. The GAdoes not keep up with the growth rate of the exhaustive searchat higher MT . Reasons for this and ways to compensate arediscussed in [10], but these are beyond the scope of this paper.

We also note that since SZF can serve more users at onceand can handle more interference than block diagonalization(BD), it is also optimal to schedule all K0 users at com-paratively lower values of K for SZF than for BD. Hence,the crossover in performance is observed for SZF, whereasno such crossover was observed up to K=100 in our relatedscheduling work in the context of BD [16]. The performanceof both proposed algorithms is still quite close to that of anexhaustive search, though not quite as close as for MT=4.Full exhaustive search results are not available for larger K,due to the combinatorially increasing complexity.

VI. CONCLUSIONSWe have proposed and analyzed low complexity greedy

and genetic user scheduling algorithms for linearly precodedmultiuser MIMO downlink. The proposed algorithms are muchless complex, but perform close to the highly complex optimalexhaustive search. We demonstrate that at lower K, thegenetic algorithm performs better than the greedy algorithm,but at large K the greedy algorithm outperforms the geneticalgorithm. The greedy algorithm achieves similar sum-rate

growth with K as the exhaustive search, whereas the geneticalgorithm does not for larger MT . A detailed complexityanalysis of the proposed algorithms has also been presented,and it shows that the genetic algorithm is more complex thanthe greedy algorithm by an order of K20 , where K0 denotes themaximum number of simultaneously served users. The currentwork has shown that the proposed suboptimal algorithms canpotentially be used to improve the performance of fourthgeneration wireless systems. However, in order to understandthe practicality of these scheduling algorithms, fairness shouldalso be incorporated into the scheduling metric, and the impactof imperfect channel knowledge or correlation in the channelon system performance should be investigated. Consideringfairness must first entail optimizing the transmit covariancematrices for a weighted sum-rate, which to the best of ourknowledge, has not yet been considered in the literature.

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vol. 29, no. 3, pp. 439441, May 1983.[2] S. Vishwanath, N. Jindal, and A. Goldsmith, Duality, achievable rates,

and sum-rate capacity of Gaussian MIMO broadcast channels, IEEETrans. Inf. Theory, vol. 49, no. 10, pp. 26582668, Oct. 2003.

[3] T. Yoo and A. Goldsmith, On the optimality of multiantenna broad-cast scheduling using zero-forcing beamforming, IEEE J. Sel. AreasCommun., vol. 24, no. 3, pp. 528541, March 2006.

[4] Q. H. Spencer, A. L. Swindlehurst, and M. Haardt, Zero forcing meth-ods for downlink spatial multiplexing in multiuser MIMO channels,IEEE Trans. Signal Process., vol. 52, no. 2, pp. 461471, Feb. 2004.

[5] A. D. Dabbagh and D. J. Love, Precoding for multiple antenna Gaussianbroadcast channels with successive zero-forcing, IEEE Trans. SignalProcess., vol. 55, no. 7, pp. 38373850, July 2007.

[6] S. Sigdel and W. A. Krzymien, Simplified fair scheduling and antennaselection algorithms for multiuser MIMO orthogonal space-divisionmultiplexing downlink, IEEE Trans. Veh. Technol., vol. 58, no. 3, pp.13291344, March 2009.

[7] Z. Shen, R. Chen, J. G. Andrews, J. R. W. Heath, and B. L. Evans, Lowcomplexity user selection algorithms for multiuser MIMO systems withblock diagonalization, IEEE Trans. Signal Process., vol. 54, no. 9, pp.36583663, Sept. 2006.

[8] S. Sigdel and W. A. Krzymien, Efficient user selection and orderingalgorithms for successive zero-forcing precoding for multiuser MIMOdownlink, in Proc. IEEE VTC09-Spring, April 2009, pp. 16.

[9] J. H. Holland, Adaptation in Natural and Artificial Systems, 1st ed. AnnArbor, MI: Univ. of Michigan Press, 1975.

[10] R. C. Elliott and W. A. Krzymien, Downlink scheduling via geneticalgorithms for multiuser single-carrier and multicarrier MIMO systemswith dirty paper coding, IEEE Trans. Veh. Technol., vol. 58, no. 7, pp.32473262, Sept. 2009.

[11] N. Jindal, W. Rhee, S. Vishwanath, S. A. Jafar, and A. Goldsmith,Sum power iterative water-filling for multi-antenna Gaussian broadcastchannels, IEEE Trans. Inf. Theory, vol. 51, no. 4, pp. 1570 1580,April 2005.

[12] R. C. Elliott and W. A. Krzymien, On the convergence of geneticscheduling algorithms for downlink transmission in multi-user MIMOsystems, in Proc. Int. Symp. Wireless Pers. Multimedia Commun.(WPMC08), Sept. 2008.

[13] G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed.Baltimore, MD: The John Hopkins Univ. Press, 1996.

[14] C.-H. Guo and N. J. Higham, A Schur-Newton method for the matrixpth root and its inverse, SIAM J. Matrix Anal. Appl, vol. 28, no. 3, pp.788804, 2006.

[15] M. Sharif and B. Hassibi, A comparison of time-sharing, DPC, andbeamforming for MIMO broadcast channels with many users, IEEETrans. Commun., vol. 55, no. 1, pp. 1115, Jan 2007.

[16] S. Sigdel, R. C. Elliott, W. A. Krzymien, and M. Al-Shalash, Greedyand genetic user scheduling algorithms for multiuser MIMO systemswith block diagonalization, accepted for publication in Proc. IEEEVTC09-Fall, Anchorage, AK, USA, Sept. 2009.

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