we propose a simplified iterative water-filling (SIWF) algorithm for per-user optimum ... computational complexity for the iterative water-filling process through ...
A Simplified Iterative Water-filling Algorithm for Per-user Power Allocation in Multiuser MMSE-Precoded MIMO Systems Min Lee and Seong Keun Oh School of Electrical and Computer Engineering Ajou University, Korea E-mails: {minishow, oskn}@ajou.ac.kr
Abstract— In this paper, we deal with a per-user power allocation problem in multiuser multiple input multiple output (MU-MIMO) systems based on minimum mean square error (MMSE) precoding. The MMSE-precoding technique provides a reasonable performance due to minimization of a composite interference-plus-noise power through allowing inter-user interference as well as the higher capacity with multiuser spatial multiplexing. The technique also has a reasonable computational complexity with linear transmit processing. The problem of optimizing per-user power allocation under inter-stream interference is not convex. Hence, we invoke some iterative approaches. In this paper, we propose a simplified iterative water-filling (SIWF) algorithm for per-user optimum power allocation in multiuser MMSE-precoded MIMO systems, in order to maximize the downlink sum capacity. This technique can reduce greatly the computational complexity for the iterative water-filling process through accelerating the iteration process, as compared with the existing modified iterative water-filling (MIWF) algorithm [1], without any performance loss. In the proposed algorithm, both the taxation and interference terms are updated at every iteration of the inner loop of iterative water-filling. In addition, per-user power levels at every iteration for the inner loop are normalized so that the total transmit power constraint could be satisfied, prior to the next iteration. From computer simulations and complexity analyses, we show that the proposed algorithm has much lower complexity but the same capacity, as compared with the original MIWF algorithm. Keywords—Multiuser MIMO; MMSE-precoding; Iterative water-filling
I. INTRODUCTION During the last decade, MIMO techniques have drawn a lot of attentions, because of their promising improvement in terms of both spectral efficiency and performance. Among many MIMO issues, MU-MIMO precoding is currently one of hottest topics, since the MU-MIMO precoding exploits synergically the high capacity achievable with MIMO multiplexing and the benefit of space division multiple access (SDMA). It also makes mobile terminals simple, since a lot of complex signal processing tasks are done in advance at the transmitter [2]. So far, various MU-MIMO precoding techniques including zero-forcing (ZF) technique, block diagonalization (BD) technique [3], successive MMSE (SMMSE) techniques [4]-[5], Tomlinson- Harashima precoding (THP) techniques [6], successive optimization THP (SO-THP) technique [7] have been developed. Among them, the MMSE precoding technique [8] has not only the benefits of SDMA, but also provides a reasonable performance when all user terminals are equipped with
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a single receive antenna. The technique has relatively low computational complexity due to linear transmit processing as compared with non-linear precoding techniques. Compared with the ZF technique that enhances the noise, the MMSE technique improves the system performance due to minimizing composite interference-plus-noise power through allowing a certain amount of inter-stream interference especially at low SNR. Proper power allocation can result in significant performance improvement. However, the problem of optimizing per-user power allocation under inter-stream interference is difficult to solve, since it is not convex. Recently, the MIWF algorithm [1] was proposed for per-user power allocation to maximize the sum capacity under inter-stream interference. The algorithm is capable of finding sub-optimal solutions due to a taxation scheme that takes into account the interacting effect of inter-stream interference in an iterative manner. In the MIWF algorithm, while the interference term is updated at every iteration for the inner loop of iterative water-filling, the taxation term is updated at every iteration for an outer loop of iterative water-filling. In addition, per-user power levels at every iteration for the inner loop cannot be satisfied the total power constraint. Hence, it can delay the iterative process to converge into the solutions In this paper, we propose a SIWF algorithm for per-user power allocation in multiuser MMSE-precoded MIMO systems, to maximize the sum capacity. In the proposed algorithm, both the taxation and interference terms are updated at every iteration of the inner loop of iterative water-filling. In addition, per-user power levels at every iteration for the inner loop are normalized so that the total transmit power constraint could be satisfied, prior to the next iteration. This normalization can make the interference and taxation terms computed more accurately. Hence, the above two modifications in the proposed algorithm can reduce greatly the overall computational complexity for the iterative water-filling process as compared with the MIWF algorithm, without any performance loss. We present the system model of the MU-MIMO downlink system with precoding in Section 2, and introduce the simplified iterative water-filling algorithm for the MU-MIMO systems using MMSE precoding in Section 3. After presenting the simulation results in Section 4, we analyze the complexity in Section 5.
II. SYSTEM MODEL The block diagram of MU-MIMO downlink system is illustrated in Fig. 1. In what follows, we assume channel state in-
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formation (CSI) knowledge at both the base station (BS) and mobile stations (MSs) for downlink precoding. M and K are used to represent the number of transmit antennas and receive antennas, i.e. there are K users quipped with a single antenna.
P
F
h1
r1
K
ri = h i ∑ fk
nK hK
2 pi hi ,i max ∑ log 2 1 + K 2 i =1 σ n + ∑ p j hi , j j =1( j ≠ i ) K
(2)
fK ] ,
s.t.
0 , pK
T
nK ] , T
(8)
2
≤ PT
i = 1,…, K ,
(9)
where hi , j is an effective channel gain experienced by the signal of the j-th user that interferes with the i-th user, and PT denotes the total power constraint. Then, the effective channel matrix can be expressed as
(3)
H eff
(4)
h1,1 h = H ⋅ F = 2,1 h K ,1
h1,2 h2,2 hK ,2
h1, K h2, K . hK , K
(10)
Solving the Lagrangian optimization problem of (9), we obtain the following solution as: 2 pi hi ,i L ({ pi } , µ ) = ∑ log 2 1 + K 2 i =1 σ n + ∑ p j hi , j j =1( j ≠ i )
(5)
K
where si is a signal transmitted to the i -th user. The noise vector n is decomposed as n = [ n1
i
pi ≥ 0
where pi is a square root power allocated the signal of the i-th user. The transmitted signal vector s can be decomposed as sK ] ,
K
∑p i =1
where fi denotes an M×1 column vector of per-user precoding vector for the i -th user. The power allocation matrix is made up of square root power allocated users’ signals as
s = [ s1
,K .
In this section, we present the SIWF algorithm for multiuser MMSE-precoded MIMO systems with inter-stream interference. To find an effective algorithm of optimizing per-user power allocation under inter-stream interference, we invoke some iterative approaches to solve the Karush-Kuhn-Tucker (KKT) system of the non-convex optimization problem. When the inter-stream interference is regarded as noise, the K-user optimization problem to maximize the sum capacity can be expressed as
(1)
where hi (i=1,2,…,K) denotes an 1×M per-user MISO channel vector made up of channel gains from transmit antennas to receive antenna for the i -th user. The precoding matrix F again consists of combining per-user precoding vectors as
p 1 P= 0
i = 1, 2,
III. SIMPLIFIED ITERATIVE WATER-FILLING ALGORITHM
where r = (r1,r2,…,rK) is a column vector of received signals at K MSs. H is a K×M composite channel matrix. F is an M×K precoding matrix, P is a K×K diagonal matrix for power allocation, s is an 1×K column vector for transmit signals, and n is an 1×K column vector for zero-mean additive Gaussian noises at MSs. Here, H is the MU-MIMO downlink channel matrix made up of per-user row channel vectors as follows:
f = [ f1
(7)
rK
T
h1 H = , h K
pk sk + ni ,
k =1
Fig. 1. MU-MIMO downlink system with precoding.
The received signal vector at MSs is r = HFPs + n ,
0 s1 n1 + . pK sK nK
Hence, the received signal of the i -th user can then be expressed as
n1
s
p 1 fK ] 0
r1 h1 = [ f1 rK hK
(6)
2
K −µ ∑ pi − PT . i =1
where ni is an additive white Gaussian noise at the mobile receiver of the i -th user. From (2) through (6), we can rewrite (1) as
(11)
By taking the derivative of the above with respect to pi, we can obtain the KKT system of the optimization problem:
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1 hi ,i
2
K
2
∑
pi +
= ti + µ ,
p j hi , j + σ n2
obtained from (16). Once the optimum ({pi }, µ) is obtained, we may then iterate the water-filling loop over total users, taking into account updating the taxation and interference terms at every iteration for the inner loop of iterative water-filling, until the process converges. Finally, repeat the process until the entire KKT system is solved. The algorithm is summarized as follows:
(12)
j =1( j ≠ i )
where ti is defined as follow [9]-[10]: K
ti =
2
p j h j, j
∑
j =1( j ≠ i )
p j h j, j
2
K
∑
+
pk h j , k
2
⋅ + σ n2
Algorithm Consider a K-user system. Assume the total power constraint PT. The power allocation algorithm works as follows:
k =1( k ≠ j ) 2
h j ,i K
∑
pk h j , k
2
.
(13)
Initialize pi(0) = PT / K , n = 0 .
+ σ n2
Repeat (outer loop) n = n +1 Repeat (inner loop)
k =1( k ≠ j )
Solving the optimization problem (9) can now be thought of as solving the KKT system (12)-(13) along with the power constraint and positivity constraints on pi, and the equality in (12) needs to be replaced by less than or equal to, when pi = 0. The ti term stands for the effect of interference caused the i-th user towards other users and is called as the taxation term. To solve this non-convex optimization problem, we fix both the ti term and the interference term from other users, denoted by Ii , where Ii =
K
∑
j =1( j ≠ i )
p j hi , j
2
.
pi ← pi( n −1) ti =
pi +
p j hi , j
j =1( j ≠ i )
hi ,i
∑
2
2
+
2
p j hi, j
2
1 I +σ2 Obtain µ via bisection on PT = ∑ − i 2n i =1 µ + t i hi , i K
σ n2 hi ,i
2
1 = . µ + ti
I +σ 2 1 pi = − i 2n hi ,i µ + ti
I +σ 2 1 − i 2n pi = µ + ti hi ,i
i =1
pi( n ) ← pi
(15)
Until pi( n ) − pi( n −1) < ε
IV. SIMULATION RESULTS We performed computer simulations to evaluate the performance of the proposed algorithm and to compare it with that of the MIWF algorithm [1] for MU-MIMO downlink system with MMSE precoding. The channel H was assumed to be spatially white and flat Rayleigh fading. In order to describe the antenna configuration of the MU-MIMO systems, we use the notation M R1 , , M RK × M T . In all simulations for obtaining
(16)
{
+
.
+
Until pi converges.
where x+ := max(x,0). Here, µ is essentially the inverse of the water-filling level, but modified by ti term. Next, we get I +σ 2 1 − i 2n PT = ∑ i =1 µ + ti hi ,i
+
K
+
.
Normalize to satisfy PT = ∑ pi
Fixing ti and Ii, the iterative water-filling algorithm can find the optimal pi. The first step is to recognize that given µ, pi can be obtained from (15) as follows:
K
h j ,i
K 2 2 2 K 2 pk h j , k + σ n2 p j h j , j + ∑ pk h j ,k + σ n k =∑ = ≠ ≠ 1( ) 1( ) k k j k j
K
∑
2
j =1( j ≠ i )
(14)
Here, one of our main ideas is to update both the ti and Ii terms according to the newly obtained per-user power levels pi (i=1,…,K) at every iteration for the inner loop of iterative water-filling. The other is to normalize the newly obtained pi so that the total power constraint could be satisfied, prior to the next iteration. These make the algorithm accelerate the iteration process. Then, we repeat the process until convergence. This strategy is numerically efficient because (12) is essentially a modified water-filling step, which can be rewritten as: K
∑
j =1( j ≠i )
Ii =
p j hj, j
K
(17)
With ti and Ii fixed, this is now an equation of a single variable µ. Further, the right-hand side of (17) is a monotonic function of µ. Thus, (17) can be solved through only a one-dimensional search (e.g. using bisection). After µ has been found, pi can be
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}
the ergodic sum capacity, we considered {1,1,1,1}×4 configuration. In addition, we used the receive signal-to-noise ratio defined as SNR = PT / σ n2 [7]. Fig. 2 shows the ergodic sum capacity of the SIWF and MIWF algorithms as a function of SNR. From Fig. 2, we see that both algorithms have equal performance, since both of them use the iterative water-filling algorithm based on the same taxation scheme in order to solve an optimization problem.
are updated for every inner loop in the proposed SIWF algorithm. We summarize the number of multiplications required for the bisection process, the normalization process, interference updating, and taxation updating in the Table 1.
{1,1,1,1} X 4
Ergodic sum capacity [bits/Hz]
18
MIWF SIWF
16 14
Table 1. Number of multiplications for four basic operations in an iterative water-filling algorithm.
12 10
Function
Number of multiplications
8
Bisection
4K + 1
Normalization
K
Taxation update Interference update
3K 2 − K or 2K with interference term
6 4 2 0
0
5
10
15
20
SNR [dB]
V. COMPLEXITY ANALYSES In this section, we analyzed the overall computational complexity of the proposed SIWF algorithm and compared the complexity with those of the MIWF algorithm and the MIWF algorithm based on per-iteration power normalization (PIPN) [12]. For simplicity, we assumed that the number of transmit antennas is equal to the number of users equipped with a single receive antenna, that is, K. To count operations for computational complexity, we take account of the number of iterations for the outer loop of iterative water-filling, since the computational complexity of iterative water-filling depends heavily on the number of iterations of the outer loop. We take account of only the number of multiplications. In counting the number of multiplications, we first consider the multiplications of four basic operations such as bisection for finding the water-filling level, normalization, and updating of taxation and interference terms. These complexities can be used to compute the overall computational complexity in combination with the number of outer loop iterations, inner loop iterations per outer loop, and bisection loop iterations per inner loop. Let K be the number of users who are positive-power allocated in an inner loop of iterative water-filling. The bisection method is used to find the water-filling level through iterative way. It requires 4 K + 1 multiplications for every bisection [11]. The normalization term is considered only for the MIWF algorithm based on PIPN and the proposed algorithm. This makes a water-filling solution satisfy the total power constraint, and requires K multiplications for every inner loop of iterative water-filing. The taxation and interference terms, respectively, require 3K 2 − K and K 2 − K multiplications from (13) and (14) for every inner loop of iterative water-filing. However, if the interference term is calculated in advance, the taxation term requires only 2K 2 multiplication. While the taxation term is updated for every outer loop, and the interference term is updated for every inner loop in the MIWF algorithm, both terms
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K2 − K
We analyzed the overall computational complexity by computing the total number of multiplications required for the number of outer loop iterations, inner loop iterations per outer loop, and bisection loop iterations per inner loop. We set SNR of 3dB. We counted the average number of multiplications from 10000 independent channel realizations. Fig. 3 shows the average number of iterations for the outer loop in the SIWF and MIWF algorithms as a function of the number of transmit antennas (or users). From Fig. 3, we see that the proposed algorithm requires 20% lower iterations for the outer loop than the MIWF algorithm, except for two transmit antennas (or users). We also see that only the normalization can reduce the number of iterations of the MIWF algorithm by about 8% as compared with the original MIWF algorithm.
10
Avergae number of iterations
Fig. 2. Ergodic sum capacity of the SIWF and MIWF algorithms as a function of SNR.
2
8
6
4
2
0
MIWF MIWF with PIPN SIWF 2
3
4
5
6
7
Number of Tx antennas (or users)
Fig. 3 Average number of iterations of the SIWF and MIWF algorithms.
Fig. 4 shows the average number of multiplications for obtaining the solutions in the SIWF and MIWF algorithms as a function of the number of transmit antennas (or users). From Fig. 4, we see that the proposed algorithm requires fewer multiplications than the MIWF algorithm by about 28%. We also
see that only the normalization can reduce the number of multiplications of the MIWF algorithm by 11% as compared with the original MIWF algorithm.
[3]
[4] 22000
MIWF MIWF with PIPN SIWF
Avergae number of multiplications
20000 18000
[5]
16000 14000 12000
[6]
10000 8000 6000
[7]
4000 2000 0
2
3
4
5
6
7
Number of Tx antennas (or users)
[8]
Fig. 4. Average number of multiplications of the SIWF and MIWF algorithms. [9]
VI. CONCLUSIONS In this paper, we proposed the SIWF algorithm for per-user power allocation suitable for the use in multiuser MMSE-precoded MIMO systems, in order to maximize the downlink sum capacity. This technique can reduce greatly the computational complexity for the iterative process through accelerating the iteration process, as compared with the existing MIWF algorithm, without performance degradation. In the proposed algorithm, both the taxation and interference terms are updated at every iteration of the inner loop of iterative water-filling. In addition, per-user power levels at every iteration for the inner loop are normalized so that the total transmit power constraint could be satisfied, prior to the next iteration. From simulation results and complexity analyses, the proposed algorithm has much lower complexity as compared with the MIWF algorithm, while achieving the same ergodic sum capacity.
[10]
[11]
[12]
ACKNOWLEDGEMENT This research was supported by the MIC(Ministry of Information and Communication), Korea, under the ITRC(Information Technology Research Center) support program supervised by the IITA(Institute of Information Technology Advancement) (IITA-2008-(C1090-0801-0003))
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