Eigenmode BER based MU-MIMO Scheduling forRate Maximization with Linear Precoding and

Power AllocationKyeongjun Ko, Hyungmin Cho ,and Jungwoo Lee

School of Electrical Engineering and Computer SciencesSeoul National University, Seoul 151-744, Korea

Email: smuff@wmspl.snu.ac.kr, hyungmin.cho@lgericsson.com and junglee@ee.snu.ac.kr

Abstract—In conventional multi-user MIMO systems, Shannoncapacity has been used mostly as the performance measurefor user selection. In this paper, we propose two new MIMOscheduling techniques based on BER instead of capacity as theperformance measure for rate maximization with target BER.One of the key contributions of this paper is to use BER insteadof capacity as the user selection metric, and another is the novelpower allocation techniques considering user scheduling for therate maximization strategy. Simulation results show that theproposed BER based algorithms produce higher rate comparedto the conventional capacity based user selection algorithm withmuch lower computational complexity as well as the schemesmeet target BER.

Index Terms—multiuser MIMO, BER, power allocation,scheduling, block diagonalization

I. INTRODUCTION

When there are many users in a cellular system withmultiple antennas, we need to select a subset of users becausethere is a limit for the number of serviced users. Numerous lowcomplexity multiuser scheduling algorithms were proposedsince the optimal scheduling algorithm which has prohibitivecomputational complexity is impractical [3]–[5].

However, existing multiuser scheduling algorithms weremostly based on capacity as the performance metric. Thecapacity measure is ideal in that it assumes infinite code blocklength, so it is not a practical measure. The bit error rate (BER)measure may be a more practical alternative. Therefore weconsider uncoded BER without channel coding for multiuserMIMO (MU-MIMO) systems. We first formulate a powerallocation strategy to minimize the bit error rate for MU-MIMO systems where we assume the transmitter knows thechannel of all the receivers. The water-filling based powerallocation is optimal in terms of capacity, but optimal powerallocation in terms of BER has not been discussed much inthe literature.

In this paper, we use BER as the performance measure, andpropose BER-based power allocation algorithm for rate max-imization with target BER. We also propose two schedulingalgorithms based on BER with block diagonalization (BD) [1],[2] as precoding sheme. One is BER based algorithm with agreedy approach, and the other is low complexity algorithmwhich assumes user cooperation. We compare the proposed

algorithms with the TDMA system and the existing capacitybased algorithm. Although we use uncoded BER as the userselection measure, the results of this paper is meaningful evenfor the coded BER with channel codes because the relativebehavior of the uncoded BER is expected to be similar to thatof the coded BER.

The rest of the paper is organized as follows. Section IIintroduces the system model. The power allocation algorithmare discussed in Section III. Section IV presents two new mul-tiuser selection algorithms based on BER, and the simulationresults are given in Section V. Finally, conclusions are madein Section VI.

II. SYSTEM MODEL

We consider an MU-MIMO downlink channel with a singlebase station (BS) which has 𝑀 transmit antennas and 𝐾𝑇

users with 𝑁 receive antennas. We assume that the receiversestimate their channels perfectly, and the transmitter knows theexact channel state information (CSIT) of all the receivers.In this paper, we differentiate between candidate users andthe scheduled users. The scheduler (user selection) selects𝐾 streams out of 𝑁𝐾𝑇 candidate streams. The schedulerfunction is denoted by 𝜋(𝑖) = (𝜋𝑠(𝑖), 𝜋𝑢(𝑖)), where 𝜋 :{1, 2, ⋅ ⋅ ⋅ , 𝑁𝐾𝑇 } → {1, 2, ⋅ ⋅ ⋅ , 𝑁} × {1, 2, ⋅ ⋅ ⋅ ,𝐾𝑇 }. 𝜋(𝑖)stands for the 𝑖th selected stream which corresponds to the𝜋𝑠(𝑖)th (1 ≤ 𝜋𝑠(𝑖) ≤ 𝑁) stream of the 𝜋𝑢(𝑖)th (1 ≤𝜋𝑢(𝑖) ≤ 𝐾𝑇 ) user. The scheduled user stream set is definedas 𝒦 = {𝜋(1), 𝜋(2), . . . , 𝜋(𝐾)} where 𝐾 is the number ofsimultaneously selected streams.

Then in a block fading channel, the system model of theMU-MIMO downlink is given by

y𝜋(𝑖) =√

𝑃𝜋(𝑖)W𝐻𝜋(𝑖)H𝜋𝑢(𝑖)V𝜋(𝑖)s𝜋(𝑖)

+√

𝑃𝜋(𝑖)W𝐻𝜋(𝑖)H𝜋𝑢(𝑖)

𝐾∑𝑘=1,𝑘 ∕=𝑖

V𝜋(𝑘)s𝜋(𝑘) +W𝐻𝜋(𝑖)n𝜋𝑢(𝑖)

(1)

where H𝜋𝑢(𝑖) is the 𝑁 × 𝑀 channel matrix of the user𝜋𝑢(𝑖) which is the user index corresponding to the 𝑖th stream,the elements of H𝜋𝑢(𝑖) are independent identically distributed(i.i.d.) complex Gaussian with zero mean and unit variance,

2012 IEEE Wireless Communications and Networking Conference: PHY and Fundamentals

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y𝜋(𝑖) is the received symbol of the user 𝜋𝑢(𝑖), n𝜋𝑢(𝑖) is the𝑁×1 complex white Gaussian noise vector for the user 𝜋𝑢(𝑖)with E{n𝜋𝑢(𝑖)n

𝐻𝜋𝑢(𝑖)

} = I𝑁 , s𝜋(𝑖) is the scalar symbol for thestream 𝜋(𝑖) with E[∣s𝜋(𝑖)∣2]≤ 1, and 𝑃𝜋(𝑖) is the allocatedpower for the stream 𝜋(𝑖) with

∑𝐾𝑙=1 𝑃𝜋(𝑖) = 𝑃 . Note that

the basic scheduling unit of the system model in (1) is not auser, but a stream. When the number of receive antennas is𝑁 , the number of serviced streams for a user may be fewerthan 𝑁 .

We now find the final precoding vector V𝜋(𝑖) and thereceiver combining vector W𝜋(𝑖). Let us denote the BDprecoding matrix of the user 𝜋𝑢(𝑖) by T𝜋𝑢(𝑖), which satisfies

H𝜋𝑢(𝑖)T𝜋𝑢(𝑘) = 0 (𝜋𝑢(𝑖) ∕= 𝜋𝑢(𝑘), 𝜋𝑢(𝑖) ∈ 𝒦, 𝜋𝑢(𝑘) ∈ 𝒦)(2)

where T𝜋𝑢(𝑖) is an 𝑀× (𝑀−∑𝐾𝑗=1,𝑗 ∕=𝑖 ∣Ω𝜋𝑢(𝑗)∣) matrix, and

Ω𝜋𝑢(𝑗) is the set of allocated streams for 𝜋𝑢(𝑗). We assumeeach receiver feeds back the raw channel H𝑙 (1 ≤ 𝑙 ≤ 𝐾𝑇 ) tothe basestation, and H𝑙 is decomposed into U𝑙S𝑙E

𝐻𝑙 by SVD

at the basestation. After the user and the stream selection at thebasestation, the precoding matrix T𝜋𝑢(𝑖) for U𝐻

Ω𝜋𝑢(𝑖)H𝜋𝑢(𝑖)

of the user 𝜋𝑢(𝑖) is calculated where UΩ𝜋𝑢(𝑖)is an 𝑁 ×

∣Ω𝜋𝑢(𝑖)∣ matrix which is obtained from U𝜋𝑢(𝑖) by taking thecolumns corresponding to Ω𝜋𝑢(𝑖). In order to use eigenmodetransmission, the effective channel U𝐻

Ω𝜋𝑢(𝑖)H𝜋𝑢(𝑖)T𝜋𝑢(𝑖) is

then decomposed into LΩ𝜋𝑢(𝑖)Σ𝜋𝑢(𝑖)R

𝐻𝜋𝑢(𝑖)

by SVD whereLΩ𝜋𝑢(𝑖)

is a ∣Ω𝜋𝑢(𝑖)∣ × ∣Ω𝜋𝑢(𝑖)∣ matrix, Σ𝜋𝑢(𝑖) is a ∣Ω𝜋𝑢(𝑖)∣ ×(𝑀 − ∑𝐾

𝑗=1,𝑗 ∕=𝑖 ∣Ω𝜋𝑢(𝑗)∣) matrix, and R𝐻𝜋𝑢(𝑖)

is an (𝑀 −∑𝐾𝑗=1,𝑗 ∕=𝑖 ∣Ω𝜋𝑢(𝑗)∣)×(𝑀−∑𝐾

𝑗=1,𝑗 ∕=𝑖 ∣Ω𝜋𝑢(𝑗)∣) matrix. There-fore, the final precoding vector V𝜋(𝑖) at the transmitter, andthe receive combining vector W𝜋(𝑖) at the receiver for thestream 𝜋(𝑖) are given by

V𝜋(𝑖) = {T𝜋𝑢(𝑖)R𝜋𝑢(𝑖)}(:,𝜋𝑠(𝑖)),

W𝜋(𝑖) = {UΩ𝜋𝑢(𝑖)LΩ𝜋𝑢(𝑖)

}(:,𝜋𝑠(𝑖)) (3)

where {A}(:,𝑖) is the 𝑖th column of A.Because V𝜋(𝑖) eliminates both the inter-user interferences

and the inter-stream interferences, (1) can be rewritten as

y𝜋(𝑖) =√

𝑃𝜋(𝑖)W𝐻𝜋(𝑖)H𝜋𝑢(𝑖)V𝜋(𝑖)s𝜋(𝑖) +W𝐻

𝜋(𝑖)n𝜋𝑢(𝑖). (4)

III. POWER ALLOCATION ALGORITHMS

To simplify the notation, we substitute 𝜋(𝑖) for 𝑖 from nowon. By V𝑖 and W𝑖, the MU-MIMO system is decomposedinto multiple-stream eigenmodes. Thus, (4) is rewritten by

y𝑖 =√

𝑃𝑖𝜆𝑖s𝑖 + n̂𝑖 (5)

where y𝑖 is the receive symbol of the 𝑖th selected stream,√𝜆𝑖

is the diagonal element of Σ𝑖 corresponded to 𝜋(𝑖), and n̂𝑖 isW𝐻

𝜋(𝑖)n𝜋𝑢(𝑖). An 𝑁 ×𝑀 MU-MIMO channel is decomposedinto 𝐾 independent SISO channels as in (5) with W𝑖 and V𝑖.

In this paper, we consider the case where the data rate ismaximized under the target BER constraint. This correspondsto the case where there is a maximum tolerable BER for eachuser as the quality of service (QoS) requirement. Note that we

TABLE IREQUIRED

√POWER WHEN THE TARGET BER IS 10−2 .

BPSK QPSK 16 QAM 64 QAM 256 QAM

𝐹𝑖 1.6450 2.0538 4.9471 9.6715 18.6082

use uncoded BER or symbol error rate (SER) without channelcoding to simplify the derivation.

The SER for 𝐿-ary QAM is written by

𝑔𝑠 = 1− (1− 𝑝𝑠)2 (6)

where 𝑝𝑠 is the SER of a√𝐿-ary PAM with one-half the

average power in each quadrature dimension of the equivalentQAM system. By appropriately modifying the probability oferror for 𝐿-ary PAM, we obtain

𝑝𝑠 = 2

(1− 1√

𝐿

)𝑄

(√3

𝐿− 1

𝐸𝑠

𝑁0

). (7)

The BER 𝑔𝑏 can be obtained approximately with the assump-tion of 𝑔𝑠 ≪ 1 as

𝑔𝑏 ≈ 𝑔𝑠log2 𝐿

. (8)

The power (or SNR) for the target BER (SER) 𝑔𝑡𝑏 can befound by (6)–(8), BPSK and QPSK BER function in [7]. For𝐿-ary QAM, we can obtain 𝐸𝑠/𝑁0 by using (6)–(8) as follows.

𝐸𝑠

𝑁0= 𝑄−1

⎛⎝2−√4− 4𝑔𝑡𝑏 log2 𝐿

4(1− 1√

𝐿

)⎞⎠

2

(𝐿− 1)

3. (9)

In fact, it is very difficult to find the optimal power alloca-tion for rate maximization with a target BER constraint. Thus,we propose a simple power allocation scheme for rate max-imization with a target BER constraint, where the allocatedpower for each stream is proportional to the minimum SNR(𝐹 2

𝑖 ) required for the target BER given the modulation order.Note that the 𝐹 2

𝑖 which is the same as 𝐸𝑠/𝑁0 is calculatedby (9). If we assume that 𝐹𝑖 is given and there are 𝐾 streamsin a system, the proposed power allocation for the 𝐾 streamssatisfies

𝑃1𝜆1

𝐹 21 𝑛1

=𝑃2𝜆2

𝐹 22 𝑛2

= ⋅ ⋅ ⋅ = 𝑃𝐾𝜆𝐾

𝐹 2𝐾𝑛𝐾

= 𝛽 (10)

where 𝑛𝑖 is the noise power of the 𝑖th user, and 𝛽 is a constantdetermined by the total available power. Note that 𝐹𝑖 can bepre-calculated for each modulation order, and it is listed inTable I for the target BER of 10−2.

The power allocation algorithm matches effective SNR𝑃𝑖𝜆𝑖/𝑛𝑖 based on the ratio of 𝐹 2

𝑖 for each stream by (10).A higher modulation order is allocated to the user with largereffective SNR, and a smaller modulation order is allocatedto the user with smaller effective SNR by (10). The powerallocated for each user is given by

𝑃𝑖 =𝐹 2𝑖 𝑛𝑖

𝜆𝑖𝛽. (11)

143

From the power constraint, we have

𝐾∑𝑚=1

𝑃𝑚 = 𝛽

𝐾∑𝑚=1

𝐹 2𝑚𝑛𝑚

𝜆𝑚= 𝑃. (12)

We then have

𝛽 =𝑃∑𝐾

𝑚=1𝐹 2

𝑚𝑛𝑚

𝜆𝑚

. (13)

Thus, from (11) and (13), the allocated power for each streamis given by

𝑃𝑖 =𝑃∑𝐾

𝑚=1𝐹 2

𝑚𝑛𝑚

𝜆𝑚

⋅ 𝐹2𝑖 𝑛𝑖

𝜆𝑖. (14)

In terms of assigning the modulation order to each user, itis desirable to assign higher modulation order to the channelwith higher channel power (eigenvalue). Thus, the modulationorder is assigned by

𝜆1

𝑛1≥ 𝜆2

𝑛2≥ ⋅ ⋅ ⋅ ≥ 𝜆𝐾

𝑛𝐾=⇒ 𝐹1 ≥ 𝐹2 ≥ ⋅ ⋅ ⋅ ≥ 𝐹𝐾 . (15)

IV. MULTIUSER MIMO SCHEDULINGALGORITHMS BASED ON BER

In this section, we propose two scheduling algorithms basedon BER, which achieve low complexity.

A. BER based Scheduling Algorithm

Algorithm 1 summarizes the BER based Multiuser MIMOscheduling algorithm. Target BER is applied to each stream,and the basic unit of the proposed scheduling scheme is astream by eigen decomposition.

Algorithm 1: BER based Multiuser MIMO Scheduling Al-gorithm

1) Initialization

𝒯 = {(𝑗, 𝑘)∣1 ≤ 𝑗 ≤ 𝑁, 1 ≤ 𝑘 ≤ 𝐾𝑇 }; 𝒮 = ∅;where 𝑗 is the stream index and 𝑘 is the user index.𝑝 = argmax(𝑗,𝑘)∈𝒯 𝜆𝑟𝑎𝑤

𝑗,𝑘 , where 𝜆𝑟𝑎𝑤𝑗,𝑘 is the

square of singular value of {U𝑘}𝐻(:,𝑗)H𝑘 (H𝑘 =

U𝑘S𝑘E𝐻𝑘 ).

𝒮 = 𝒮 + {𝑝}; , 𝒯 = 𝒯 − {𝑝};∙ 𝑄1

𝐵 = 𝐺𝐵(𝒮)∙ Ψ1

𝐵 = Φ𝐵(𝒮)∘ 𝑄1

𝑅 = 𝐺𝑅(𝒮)∘ Ψ1

𝑅 = Φ𝑅(𝒮)2) Loop

FOR 𝑖 = 2 to 𝑀

FOR each (𝑗, 𝑘) ∈ 𝒯Let 𝒟𝑗,𝑘 = 𝒮 + {(𝑗, 𝑘)}.Find the precoding matrix T𝑚 for{U𝑚}𝐻Ω𝑚

H𝑚 where Ω𝑚 = {𝑙∣(𝑙,𝑚) ∈ 𝒟𝑗,𝑘}Find the precoding vector V(𝑙,𝑚) and thereceiver combining vector W(𝑙,𝑚) with{U𝑚}𝐻Ω𝑚

H𝑚T𝑚

where V(𝑙,𝑚) and W(𝑙,𝑚) are the final precod-ing vector and the receiver combining vectorfor the 𝑙th stream of the 𝑚th user, respectively.

Find the 𝜆𝑙,𝑚 of W𝐻(𝑙,𝑚)H𝑚V(𝑙,𝑚) for

(𝑙,𝑚) ∈ 𝒟𝑗,𝑘.∙ 𝜂𝒟𝑗,𝑘

= 𝐺𝐵(𝒟𝑗,𝑘)∘ 𝜇𝒟𝑗,𝑘

= 𝐺𝑅(𝒟𝑗,𝑘)

END FOR∙ 𝑝 = argmin(𝑗,𝑘)∈𝒯 𝜂𝒟𝑗,𝑘

∘ 𝑝 = argmax(𝑗,𝑘)∈𝒯 𝜇𝒟𝑗,𝑘

𝒮 = 𝒮 + {𝑝}; , 𝒯 = 𝒯 − {𝑝};∙ 𝑄𝑖

𝐵 = 𝐺𝐵(𝒮), ∙ Ψ𝑖𝐵 = Φ𝐵(𝒮)

∘ 𝑄𝑖𝑅 = 𝐺𝑅(𝒮), ∘ Ψ𝑖

𝑅 = Φ𝑅(𝒮)END FOR

3) USER SELECTION

∙ 𝑖∗ = argmin𝑖 𝑄𝑖𝐵

∘ 𝑖∗ = argmax𝑖 𝑄𝑖𝑅

Finally selected user set: 𝕊 = 𝒮(1 : 𝑖∗).∙ modulation order pair of 𝕊: Ψ𝑖∗

𝐵

∘ modulation order pair of 𝕊: Ψ𝑖∗𝑅

At first, the total stream set 𝒯 = {(𝑗, 𝑘)∣1 ≤ 𝑗 ≤ 𝑁, 1 ≤𝑘 ≤ 𝐾𝑇 } is defined, and we consider the raw channel H𝑘

(1 ≤ 𝑘 ≤ 𝐾𝑇 ) in Step I since there is only one user. Therefore,the transmitter proceeds scheduling with {U𝑘}𝐻(:,𝑗)H𝑘 foreigenmode (H𝑘 = U𝑘S𝑘E

𝐻𝑘 by SVD).

𝐺𝑅(𝒮) is a function that computes the maximum sumof modulation orders for given 𝒮 exhaustively using (15)with the target BER constraint, which corresponds to therate maximization case with target BER. To compute 𝐺𝑅(𝒮),we consider all possible combinations of modulation orderthat satisfy (15) for selected users, i.e., from (1, ⋅ ⋅ ⋅ , 1) to(8, ⋅ ⋅ ⋅ , 8). We can then find 𝛽 corresponding to the modula-tion set by (13), and the effective channel gain of each streamby (10).

After the allocated power of each stream is determined with𝛽, we just need to check if 𝛽 > 1. If 𝛽 > 1, the set ofmodulation order satisfies the target BER constraint. With theabove method, 𝐺𝑅(𝒮) finds the maximum sum of modulationorders among all possible modulation sets to satisfy the targetBER constraint for given 𝒮. 𝑄𝑖

𝑅 is the storage variable for thevalue of 𝐺𝑅(𝒮) in the 𝑖th iteration. But the stream set thatproduces the maximum sum of modulation orders may not beunique. In this case, the stream set that has the maximum 𝛽 isselected. The higher 𝛽 we have, the lower BER we get. Ψ𝑖

𝑅

is the storage variable for the modulation order set computedby Φ𝑅(𝒮) which corresponds to 𝐺𝑅(𝒮) in the 𝑖th iteration.

From the 2nd step, the precoding matrix based on BD formultiple users is computed. Suppose 𝒟𝑗,𝑘 is the union of theset of previously selected streams 𝒮 and the candidate stream(𝑗, 𝑘). The precoding vector V(𝑙,𝑚) and the receiver combiningmatrix W(𝑙,𝑚) are found with the (𝑙,𝑚) element of 𝒟𝑗,𝑘. Wecan then find 𝜆𝑙,𝑚, which is W𝐻

(𝑙,𝑚)H𝑚V(𝑙,𝑚) for (𝑙,𝑚) ∈𝒟𝑗,𝑘, and determine 𝐺𝑅(𝒟𝑗,𝑘). When 𝜇𝒟𝑗,𝑘

= 𝐺𝑅(𝒟𝑗,𝑘), thestream which has maximum 𝜇𝒟𝑗,𝑘

is selected. At the end ofthe 𝑀 th iteration, the served user set 𝕊 and the correspondingmodulation order set are determined.

144

B. Low Complexity BER based Scheduling Algorithm

Algorithm 2 shows the details of the low complexity BERbased MU-MIMO scheduling algorithm, which has lowercomplexity compared to Algorithm 1.

Algorithm 2: Low Complexity Multiuser MIMO Schedul-ing Algorithm

1) Initialization

𝒯 = {(𝑗, 𝑘)∣1 ≤ 𝑗 ≤ 𝑁, 1 ≤ 𝑘 ≤ 𝐾𝑇 }, 𝒮 =∅, A = ∅, X = ∅, Y = I𝑀FOR all (𝑙,𝑚) ∈ 𝒯

B(𝑙,𝑚) = {U𝑚}𝐻(:,𝑙)H𝑚

X(𝑙,𝑚) = I𝑀 −B𝐻(𝑙,𝑚)(B(𝑙,𝑚)B

𝐻(𝑙,𝑚))

−1B(𝑙,𝑚)

END FOR

2) Loop

FOR 𝑖 = 1 to 𝑀

FOR all (𝑙,𝑚) ∈ 𝒯Γ(𝑙,𝑚) = 𝑡𝑟((AX(𝑙,𝑚)A

𝐻)−1) +𝑡𝑟((B(𝑙,𝑚)YB𝐻

(𝑙,𝑚))−1)

END FOR(𝑙∗,𝑚∗) = argmin(𝑙,𝑚)∈𝒯 Γ(𝑙,𝑚)

𝒮 = 𝒮 + {(𝑙∗,𝑚∗)}, 𝒯 = 𝒯 − {(𝑙∗,𝑚∗)}∙ 𝑄𝑖

𝐵 = 𝐺𝐵(𝒮)∙ Ψ𝑖

𝐵 = Φ𝐵(𝒮)∘ 𝑄𝑖

𝑅 = 𝐺𝑅(𝒮)∘ Ψ𝑖

𝑅 = Φ𝑅(𝒮)A = [A; {U𝑚}𝐻(:,𝑙∗)H𝑚∗ ]

Y = I𝑀 −A𝐻(AA𝐻)−1A

END FOR

3) USER SELECTION

∙ 𝑖∗ = argmin𝑖 𝑄𝑖𝐵

∘ 𝑖∗ = argmax𝑖 𝑄𝑖𝑅

Finally selected user set: 𝕊 = 𝒮(1 : 𝑖∗)∙ modulation order pair of 𝕊: Ψ𝑖∗

𝐵

∘ modulation order pair of 𝕊: Ψ𝑖∗𝑅

The low complexity BER based scheduling algorithm does notfind the singular value of the effective channel of each user.Instead, it calculates the sum of the inverse squared singularvalues of the combined channel matrix of the selected usersthat cooperate with each other. For an 𝑛×𝑛 matrix C, 𝑡𝑟(C) =∑𝑛

𝑖=1 𝛾𝑖 where 𝛾𝑖 is an eigenvalue of C. For an 𝑛×𝑚 matrixT, the pseudo inverse of T is defined by T† = T𝐻(TT𝐻)−1,and it satisfies (T†)𝐻T† = {(TT𝐻)−1}𝐻TT𝐻(TT𝐻)−1 =(TT𝐻)−1. Thus, the squared singular values of T† are eigen-value of (TT𝐻)−1 [9]. By the way, the singular values of Tare the inverse singular values of T† [9].

From [8], blockwise matrix inversion is given by[C GZ Q

]−1

=[(C−GQ−1Z)−1 −(C−GQ−1Z)−1GQ−1

−Q−1Z(C−GQ−1Z)−1 (Q− ZC−1G)−1

](16)

where C and Q are square matrices. Suppose that A is thechannel matrix of the previously selected users, and B is thethe channel matrix of the newly added user. A and B are alsodefined in Algorithm 2. We do not find the singular values of[AB

]directly, but the sum of inverse squared singular values

of the matrix to reduce the complexity. Let us define J by

𝐽 =

[AB

]⋅ [A𝐻 B𝐻

]=

[AA𝐻 AB𝐻

BA𝐻 BB𝐻

]. (17)

Therefore, 𝑡𝑟(J−1) is the same as the sum of inverse squared

singular values of

[AB

]. By computing J−1 with (16) and (17),

we have

(C−GQ−1Z)−1 = (A(I𝑀−B𝐻(BB𝐻)−1B)A𝐻)−1 (18)

(Q− ZC−1G)−1 = (B(I𝑀 −A𝐻(AA𝐻)−1A)B𝐻)−1.(19)

We also have

𝑡𝑟(J−1) = 𝑡𝑟

([C GZ Q

]−1)

= 𝑡𝑟((C−GQ−1Z)−1) + 𝑡𝑟((Q− ZC−1G)−1)

= 𝑡𝑟((A(I𝑀 −B𝐻(BB𝐻)−1B)A𝐻)−1) +

𝑡𝑟((B(I𝑀 −A𝐻(AA𝐻)−1A)B𝐻)−1). (20)

As for the rate maximization strategy, the user set whichmaximizes 𝛽 needs to be selected for given 𝐹𝑖’s. From (13),we can upperbound 𝛽 as

𝛽 ≤ 1

𝐹 2⋅ 𝑃∑𝐾

𝑚=11

𝜆𝑚

(21)

where 𝐹 = min𝑚 𝐹𝑚. Note that the minimization of∑𝐾𝑚=1

1𝜆𝑚

approximately leads to the maximization of theupperbound for 𝛽. In order to maximize (13), we need tominimize 𝑡𝑟((AXA𝐻)−1) + 𝑡𝑟((BYB𝐻)−1) where X =(I𝑀 − B𝐻(BB𝐻)−1B) and Y = (I𝑀 − A𝐻(AA𝐻)−1A)in (20).

V. SIMULATION RESULTS

In this section, we compared the performances of theproposed algorithms with TDMA and the capacity basedalgorithm [4] when 𝑀 = 4, 𝑁 = 2, and used 100,000 i.i.d.channel realizations for Monte Carlo simulations. A single useris selected by exhaustive search with 𝐺𝑅(𝒮), and Φ𝑅(𝒮) inthe TDMA system. Note that one user instead of a stream isserved at a time in the TDMA system.

Water-filling based power allocation is used for the capacitybased algorithm. In the rate maximization strategy with a targetBER constraint, the power of each stream is allocated with thewater-filling scheme, and the modulation order for each streamis found by using the minimum required power in Table I. Ifthe target BER for a stream cannot be satisfied, the lowestmodulation order (BPSK) is used for the stream in the capacitybased algorithm, which implies that the target BER may not

145

0 5 10 15 200

5

10

15

20

25

SNR

Su

m−R

ate

BER basedLow complexity BER basedTDMACapacity based

Fig. 1. Sum-rate comparison of various scheduling algorithms for the ratemaximization strategy with the target BER of 10−2.

0 5 10 15 2010

−3

10−2

10−1

SNR

BE

R

BER basedLow complexity BER basedTDMACapacity based

Fig. 2. BER comparison of various scheduling algorithms for the ratemaximization strategy with the target BER of 10−2.

be satisfied for the capacity based algorithm especially at lowSNR.

Fig. 1 shows the average BPCU comparison between theproposed rate maximization algorithms, TDMA, and the ca-pacity based scheduling algorithm under the target BER of10−2. It is observed that the two proposed algorithms achievelarger sum-rate than TDMA and the capacity based schedulingalgorithm, and that the proposed algorithms have similar sum-rate performance. The gap between the proposed algorithmsand TDMA increases as SNR increases, and the proposedalgorithms have gain of more than 5 dB at the SNR of 20dB.

Fig. 2 checks whether the proposed power allocation algo-rithm satisfies the target BER constraint for the simulationsof Fig. 1, which uses the rate maximization strategy withtarget BER. It is observed that the BER of all the algorithms

except the capacity based scheduling algorithm satisfy thetarget BER constraint. The capacity based algorithm does notmeet the target BER requirement at low SNR. It is because thetransmitter selects the users first with the capacity measure,and allocates resources to the selected users regardless ofthe target BER requirement. It is also because the lowestmodulation order (BPSK) is used if no modulation ordersatisfies the target BER requirement. On the other hand, theproposed BER based scheduling algorithms always satisfythe target BER constraint, which is critical in meeting theBER QoS requirement. This suggests that the proposed BERbased algorithms appear to be more suitable for practicalimplementation in terms of QoS requirements.

VI. CONCLUSIONS

In this paper, we proposed MU-MIMO scheduling algo-rithms based on BER instead of capacity as the performancemeasure. Key contributions of this paper include the usageof BER instead of capacity as the user selection metric, andthe power allocation technique for MIMO scheduling withadaptive modulation. We proposed novel power allocationtechnique for the BER based multiuser MIMO schedulingalgorithms with BD for rate maximization with target BERconstraint. We also proposed the low complexity BER basedscheduling algorithm, which reduces the computational com-plexity of the original BER based algorithm. Simulation resultsshow the proposed scheduling algorithms have higher rate aswell as it achieves target BER compared to TDMA and thecapacity based scheduling algorithm.

ACKNOWLEDGMENT

This research was supported in part by Basic ScienceResearch Program (2010-0013397) and Mid-career ResearcherProgram (2010-0027155) through the NRF funded by theMEST, Seoul R&BD Program (JP091007, 0423-20090051),the INMAC, and BK21.

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