1
An Iterative Minmax Per-Stream MSE Transceiver Design for MIMO Interference Channel Chiao-En Chen∗ , Member, IEEE
Abstract— This article presents a new MSE-based transceiver design for the MIMO interference channel (MC). The proposed design monotonically decreases the maximum per-stream MSE of all users by alternatively optimizing between a minimum MSE solution for the equalizers and a second- order-cone-programming (SOCP) solution for the precoders. As the system’s error rate performance at high signal-to-noise ratio (SNR) is dominated by the spatial stream with the largest MSE, minimizing the maximum per-stream MSE not only improves the overall system performance but also ensures the fairness in error rate among users. The performance advantage of the proposed algorithm is verified by the numerical simulations. Index Terms— MIMO, interference channel, transceiver, MSE
I. I NTRODUCTION
M
ULTIPLE-input-multiple-output (MIMO) communications have drawn considerable recent interests due to its potential of providing enhanced capacity and link reliability comparing to its single-input-single-output (SISO) counterparts. More recently, the interests on MIMO channels have been gradually migrated from the point-to-point MIMO, MIMO broadcasting channel (BC), to the more complicated MIMO interference channel (IC). Despite the general capacity characterization of the MIMO interference channel is not fully understood to date, recent progress on the Degrees-offreedom (DOF), i.e. the number of interference-free signalingdimensions in the network [1, 2] has inspired many new transceiver and wireless network designs [3]–[5]. An important class of these transceiver designs are based on the technique of interference alignment (IA), which is a signal processing approach aiming at simultaneously aligning the interference in a lower dimensional signal subspace at each receiver such that the desired signal can be received at an essentially interference-free subspace. It has been shown that IA achieves the optimal DOF [3] of the interference channel. Another class of these transceiver designs are based on the mean-square-error (MSE) approach [6, 7]. It has been shown that the MSE-based transceiver design is capable of achieving better error rate performance compared to IA in the low to intermediate SNR region. In [6], the authors proposed two new MSE-based designs, namely, the minimum sum-MSE and the minmax per-user MSE transceivers. The minmax per-user
This work was supported by the National Science Council (NSC), Taiwan, R.O.C. under Grant Number NSC-100-2221-E-194-032-MY2. Chiao-En Chen∗ is with the Department of Electrical Engineering, National Chung Cheng University, Chiayi, Taiwan, R.O.C. (e-mail:
[email protected]). Wei-Ho Chung is with the Research Center for Information Technology Innovation, Academia Sinica, Taipei, Taiwan, R.O.C. (e-mail:
[email protected]).
and Wei-Ho Chung, Member, IEEE
MSE transceiver ensures each user acquire the same sum-MSE and hence improves the fairness among users as compared to the minimum sum-MSE transceiver. In this letter, we propose a new MSE-based transceiver. The motivation of this work is based on the observation that the error rate of a user at sufficiently high SNR is dominated by the spatial stream with the worst SINR (largest MSE). As a result, the error rate performance among users can still be very different even though they have the same peruser MSE if the MSEs of all spatial streams are severely unbalanced. Based on this observation, we proposed a new transceiver design targeted in minimizing the maximum perstream MSE. An iterative algorithm employing alternative optimization between a minimum MSE solution for equalizers and a second-order-cone-programming (SOCP) solution for precoders is presented. The proposed algorithm monotonically decreases the maximum per-stream MSE and is guaranteed to converge to at least a local optimal solution. Notations : Throughout this letter, matrices and vectors are set in boldface, with uppercase letters for matrices and lower case letters for vectors. The superscripts T , H , and −1 denote the transpose, conjugate transpose, and matrix inverse respectively. E{·} is the expectation operator, and IN is the N × N identity matrix. 0a,b and 0N denote the a × b all-zero matrix and the N × 1 all-zero vector, respectively. ⊗ denotes the Kronecker product, and vec(X) is the operator that stacks up all the columns of matrix X into a vector. tr{·} is the trace operator, and k · k denotes the Euclidean-norm. The operator blkdiag{A1 , . . . , AN } builds a block diagonal matrix with the given matrices A1 , . . . , AN on the diagonal. II. S YSTEM D ESCRIPTION We consider linear transceiver design for a K-user MIMO interference channel as shown in Fig. 1. For simplicity, we consider uncoded transmission. The kth transmitter first precodes the Dk × 1 symbol vector dk by a linear precoder Vk , and then transmits the precoded vector xk via its Mk transmit antennas under a power constraint E{kxk k2 } ≤ Pk , for all k = 1, . . . , K. Assuming the interference channel is frequency flat, the channel between the ith transmitter and the jth receiver can be characterized by a Nj ×Mi channel matrix Hji . The chosen {Dk }K k=1 is assumed to meet the feasibility conditions [8] for the maximum achievable DOF. At the receiver side, the kth receiver receives both the desired signal from the kth transmitter as well as the undesired signal (interference) from all transmitters other than k via its Nk receive antennas, for all k = 1, . . . , K. It follows that the
2
Fig. 1. Linear Transceiver architecture for a K-user MIMO interference channel.
received data vector at the kth receiver can be expressed as yk = Hkk xk +
K X
Hki xi + nk
i=1
i=1,i6=k
= Hkk Vk dk +
K X
Hki Vi di + nk ,
Through balancing the MSE of all the spatial streams, the optimal solution to this problem will not only ensure the fairness in error rate among users, but also improve the average error rate of the overall system. Since the optimization problem in (4) is convex over each of K {Vk }K k=1 or {Rk }k=1 but not jointly, a standard technique is to solve the problem iteratively using alternating minimization. Although global convergence is not guaranteed in general, this approach ensures local convergence and often leads to a good suboptimal design when initialized properly. We first start by having {Vk }K k=1 fixed and then optimize over {Rk }K . Let r and v be the ℓk th column of Rk k,ℓ k,ℓ k k k=1 and Vk , respectively. It follows that the MSE for the ℓk th spatial stream of the kth user ! K X H H H [Ek ]ℓk ,ℓk = rk,ℓk Hki Vi Vi Hki rk,ℓk − rH k,ℓk Hkk vk,ℓk H 2 2 − vk,ℓ HH kk rk,ℓk + 1 + σ krk,ℓk k , k 2 = 1 − rH k,ℓk Hkk vk,ℓk
(1)
i=1,i6=k
where nk is the noise vector, modeled as a zero-mean circularly symmetric Gaussian vector with covariance matrix σ 2 INk . Each element of dk is assumed to be generated independently and identically distributed (i.i.d.) with zero-mean and unity power. With this assumption, the power constraint can be equivalently expressed as tr{Vk VkH } ≤ Pk . To suppress the receiver noise and the undesired signal Dk ×Nk (interference), a linear equalizer RH is applied k ∈ C at the kth receiver, resulting in an estimate for dk , given by H ˜ k = RH d k y k = Rk
K X
Hki Vi di + RH k nk .
K X
+
i=1,i6=k
+
Dk X
H tr ViH HH ki rk,ℓk rk,ℓk Hki Vi
ℓ=1,ℓ6=ℓk
(2)
The objective for the design problem considered in this paper is to jointly design {V1 , . . . , VK , R1 , . . . , RK } such that the maximum per-stream MSE is minimized. III. M INIMIZATION THE MAXIMUM P ER -S TREAM MSE ˜ k is given by From (2), the MSE matrix for d H ˜ ˜ Ek = E dk − dk dk − dk ! K X H = RH Hki Vi ViH HH k ki Rk − Rk Hkk Vk −
+ IDk + σ 2 R H k Rk .
min
max
max
K k=1,...,K ℓk =1,...,Dk {Vk }K k=1 ,{Rk }k=1
rH kℓk
=
[Ek ]ℓk ,ℓk
s.t. tr{Vk VkH } ≤ Pk , for all k = 1, . . . , K.
H vk,ℓ HH kk k
K X
Hki Vi ViH HH ki
+
σn2 IDk
i=1
!−1
H H RH k = Vk Hkk
K X
2 Hki Vi ViH HH ki + σn IDk
i=1
!−1
(6)
.
(7)
Alternatively, for fixed {Rk }K k=1 , the following constrained optimization problem has to be solved: max
min
max
k=1,...,K ℓk =1,...,Dk {Vk }K k=1
[Ek ]ℓk ,ℓk
s.t. tr{Vk VkH } ≤ Pk , for all k = 1, . . . , K.
(8)
Introducing an auxiliary variable t, the optimization problem (9) can be equivalently expressed as min
{Vk }K k=1 ,t
s.t.
q
t
[Ek ]ℓk ,ℓk ≤ t, for all ℓk = 1, . . . , Dk , k = 1, . . . , K,
tr{Vk VkH } ≤ Pk , for all k = 1, . . . , K.
(4)
,
for all ℓk = 1, . . . , Dk , and k = 1, . . . , K, or more concisely,
(3)
The (ℓk , ℓk )th element of Ek corresponds to the MSE of the ℓk th spatial stream of the kth user, where ℓk = 1, . . . , Dk . Since the symbol error rate of each user at high SNR is dominated by the spatial stream with the largest MSE, we consider the following criterion of minimizing the maximum per-stream MSE among all users:
(5)
is only dependent on rk,ℓk , and hence the optimization problem for each stream is decoupled. Since the problem is convex in rk,ℓk , the minimum point can therefore be obtained by setting the first-order derivative of [Ek ]ℓk ,ℓk over rk,ℓk to zero. This gives the following solution
i=1
i=1 VkH HH kk Rk
H rk,ℓ Hkk vk,ℓ 2 + σ 2 krk,ℓ k2 . k k k
For convenience, we introduce the following notations
(9)
3
(9) can be put in the following form:
H=
H11 H21 .. .
H12 H22 .. .
... ... .. .
H1K H2K .. .
HK1 HK2 . . . HKK V = blkdiag{V1 , V2 , . . . , VK }, i h Θk = 0Nk ×Pk−1 Nj , INk , 0Nk ×PK j=k+1 Nj j=1 i h , Γk = 0Mk ×Pk−1 Mj , IMk , 0Mk ×PK j=k+1 Mj j=1 i h Ξk = 0Dk ×Pk−1 Dj , IDk , 0Dk ×PK j=k+1 Dj j=1
ek,ℓk = [0Tℓk −1 , 1, 0TDk −ℓk ]T ,
min
{Vk }K k=1 ,t
(10)
(11) (12) (13) (14) (15)
for all k = 1, . . . , K, and ℓk = 1, . . . , Dk . With these new notations, (5) can then be rewritten as [Ek ]ℓk ,ℓk 2 H = 1 − rH k,ℓk Θk HΛk VΞk ek,ℓk K X
+
i=1,i6=k
+
Dk X
(16)
o n H H tr Ξi VH Λi HH ΘH k rk,ℓk rk,ℓk Θk HΛi VΞi
jk =1,jk 6=ℓk
2 H 2 2 rk,ℓk Θk HΛk VΞH k ek,jk + σ krk,ℓk k ,
where Λk = ΓH k Γk . By using the properties of Kronecker products [9, p. 30], we further rewrite (16) as [Ek ]ℓk ,ℓk 2 = 1 − eTk,ℓk Ξk ⊗ rH vec(V) k,ℓk Θk HΛk
(17)
K X
2
Ξ i ⊗ rH + vec(V) k,ℓk Θk HΛi i=1,i6=k
+
Dk X
jk =1,jk 6=ℓk
2 T ek,j Ξk ⊗ rH vec(V) k,ℓk Θk HΛk k
+ σ 2 krk,ℓk k2 .
As a result, the minmax per-stream MSE optimization problem
TABLE I S UMMARY OF THE PROPOSED MINMAX PER - STREAM MSE TRANSCEIVER 1: Set n = 0 (the iteration number). Initialize {Vk }K k=1 . 2: Set n = n + 1. −1 P K H H H H 2 3: Set RH i=1 Hki Vi Vi Hki + σn IDk k = Vk Hkk for all k = 1, . . . , K. 4: Update {Vk }K k=1 by solving the SOCP in (18). 5: Repeat step 2-4 until the algorithm converges or reached a predefined number of iterations.
t
i σ krk,ℓk k h
1 − eT Ξ ⊗ rH Θ HΛ vec(V) k
k,ℓk k k,ℓk k h i
H Ξ1 ⊗ rk,ℓk Θk HΛ1 vec(V)
..
. i h
H Ξk−1 ⊗ rk,ℓk Θk HΛk−1 vec(V)
i h
vec(V) Ξk+1 ⊗ rH
k,ℓk Θk HΛk+1
..
.
i h s.t. H ΞK ⊗ rk,ℓk Θk HΛK vec(V)
i h
vec(V) eTk,1 Ξk ⊗ rH
k,ℓk Θk HΛk
..
.
h i
eTk,ℓk −1 Ξk ⊗ rH Θk HΛk vec(V) k,ℓ k
h i
vec(V) Θ HΛ eTk,ℓk +1 Ξk ⊗ rH k k k,ℓk
. .
h . i
eTk,K Ξk ⊗ rH Θ HΛ vec(V) k k k,ℓk for all ℓk = 1, . . . , Dk , and k = 1, . . . , K, tr{Vk VkH } ≤ Pk , for all k = 1, . . . , K.
≤ t,
(18)
The above optimization problem is a Second-Order Cone Programming (SOCP) problem [10], and hence can be solved efficiently using many existing optimization tools such as MOSEK [11] or SeDuMi [12]. The proposed minmax per-stream MSE algorithm can be described as follows. The algorithm is first initialized by some precoding matrices {Vk }K k=1 . The algorithm then iterates K between (7) and (18), fixing {Vk }K k=1 and {Rk }k=1 to the most updated value respectively until the stopping criterion is met. Since the proposed algorithm monotonically decreases the maximum per-stream MSE and the per-stream MSE is bounded below, it is clear that the proposed algorithm is guaranteed to converge to a local minimum within finite number of iterations. Like most of the non-convex problems, appropriate initialization points are required to ensure a reasonable suboptimal solution. Several reasonable choices such as right singular matrices and IA solutions [3] have been proposed in [6]. Optimal initialization method that guarantees global optimal solution is a very challenging problem and will be left to our future work. A summary of the proposed algorithm is shown in Table I. It is worth noting that the proposed minmax per-stream MSE transceiver has a similar structure as the minmax per-user MSE algorithm in [6] in the sense that both algorithms alternatively optimize between (7) and an SOCP problem. However, the minmax per-stream MSE criterion results in more complicated and also larger number of SOCP inequality constraints, and therefore the proposed minmax per-stream MSE algorithm is expected to require higher complexity comparing to the minmax per-user MSE algorithm.
4
MSE of user 1, stream 1 MSE of user 1, stream 2 MSE of user 2, stream 1 MSE of user 2, stream 2 MSE of user 3, stream 1 MSE of user 3, stream 2 sum MSE of user 1 sum MSE of user 2 sum MSE of user 3 total sum MSE
2.5
−1
10
1.5
1
Bit error rate (BER)
2
−2
10
IA min sum−MSE (iter=4) minmax per−user MSE (iter=4) minmax per−stream MSE (iter=4) min sum−MSE (iter=8) minmax per−user MSE (iter=8) minmax per−stream MSE (iter=8) min sum−MSE (iter=16) minmax per−user MSE (iter=16) minmax per−stream MSE (iter=16)
−3
10 0.5
1
Fig. 2.
2 Scheme index
3
0
The MSE distribution of different MSE-based transceiver designs.
IV. S IMULATION R ESULTS This section presents the simulation results of the proposed minmax per-stream MSE transceiver in comparison of several existing transceiver designs. For fair comparison, all the iterative transceivers use right singular matrices as initialization. A 3-user MIMO interference channel with M1 = M2 = M3 = 4 and N1 = N2 = N3 = 4 is simulated. This channel is known to have 6 degrees of freedom and can be achieved using IA [5] with D1 = D2 = D3 = 2. The MIMO channel is assumed to be frequency flat and Rayleigh faded with each element of the channel matrix generated as i.i.d. circularly-symmetric complex Gaussian of zero-mean and unit-variance. All the data symbols are assumed to be modulated using 16-QAM. Fig. 2 presents the MSE distributions of three different MSE-based schemes, with scheme 1 corresponds to the minimum sum-MSE design [6], scheme 2 corresponds to the minmax per-user MSE design [6], and scheme 3 corresponds to our proposed minmax per-stream MSE design. The simulation result is obtained from one typical realization of the random PK channel, simulated under SNR = σ12 k=1 Pk = 5 dB. It is observed that the minimum sum-MSE design achieves the lowest sum-MSE, but can be unbalanced in both the perstream and per-user MSE. The minmax per-user MSE design improves the fairness among users in the sense of equal peruser MSE, but can still be unbalanced in the per-stream MSE, and hence results in unequal error rate among users. On the contrary, our proposed minmax per-stream MSE design ensures the same per-stream MSE, or equivalently the same error rate among all users at the expense of only slightly increased total sum-MSE and per-user MSE. Fig. 3 shows the average bit error rate performance of several transceiver designs. It can be observed from the figure that the closed-form IA solution [3] has the worst error rate performance comparing to all MSE-based approaches. The proposed minmax perstream MSE design starts to outperform the other MSE-based designs for sufficiently high SNR. At BER=10−3 , the proposed minmax per-stream MSE design provides roughly 0.8 dB and 2.9 dB gain with 16 iterations as compared to the minmax minmax per-user MSE design and minimum sum-MSE design, respectively. More performance gain can be acquired with increased number of iterations.
−4
10
0
5
10
15 Eb/N0 (dB)
20
25
30
Fig. 3. The average bit error rate performance of different transceiver designs.
V. C ONCLUSION A new minmax per-stream MSE transceiver design for MIMO interference channel is proposed. The proposed algorithm iteratively reduces the maximum per-stream MSE by employing alternatively optimization between the MMSE solution for equalizers and a second-order-cone-programming (SOCP) solution for precoders. Compared to other existing designs, the proposed algorithm ensures fairness among users and improves the overall error rate for sufficiently high SNR at the expense of increased complexity as verified by the computer simulations. R EFERENCES [1] S. A. Jafar and M. Fakereddin, “Degrees of freedom for the MIMO interference channel,” IEEE Trans. Inf. Theory, vol. 53, no. 7, pp. 2637– 2642, Jul. 2007. [2] S. A. Jafar and S. Shamai, “Degrees of freedom region for the MIMO X channel,” IEEE Trans. Inf. Theory, vol. 54, no. 1, pp. 151–170, Jan. 2008. [3] V. R. Cadambe and S. A. Jafar, “Interference alignment and the degrees of freedom for the k user interference channel,” IEEE Trans. Inf. Theory, vol. 54, no. 8, pp. 3425–3441, Aug. 2008. [4] H. Yu and Y. Sung, “Least squares approach to joint beam design for interference alignment in multiuser multi-input multi-output interference channels,” IEEE Trans. Signal Process., vol. 58, no. 9, pp. 4960–4966, Sep. 2010. [5] K. S. Gomadam, V. R. Cadambe, and S. A. Jafar, “A distributed numerical approach to interference alignment and applications to wireless interference networks,” IEEE Trans. Inf. Theory, vol. 57, no. 6, pp. 3309– 3322, Jun. 2011. [6] H. Shen, B. Li, M. Tao, and X. Wang, “MSE-based transceiver designs for the MIMO interference channel,” IEEE Trans. Wireless Commun., vol. 9, no. 11, pp. 3480–3489, Nov. 2010. [7] J. Shin, J. Moon, and J. Ahn, “Weighted-sum-rate-maximizing linear transceiver filters for the k-user mimo interference channel,” in Proc. IEEE Conf. Global Commun. (GLOBECOM), Houston, Texas, USA, Dec. 2011. [8] C. M. Yetis, T. Gou, S. A. Jafar, and A. H. Kayran, “On feasibility of interference alignment in MIMO interference networks,” IEEE Trans. Signal Process., vol. 58, no. 9, pp. 4771–4782, Sep. 2010. [9] J. R. Magnus and H. Neudecker, Matrix differential calculus with applications in statistics and econometrics. John Wiley and Sons, 2002. [10] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press, 2004. [11] MOSEK: Solutions through mathematical optimization. [Online]. Available: http://www.mosek.com [12] J. F. Sturm, “Using SeDuMi 1.02, a MATLAB tool for optimization over symmetric cones,” Optimization Methods and Software, vol. 11-12, pp. 625–653, 1999.
