Grassmannian Packing Based Aligned Precoder Designs for
Aligned Precoder Designs for Interference Channels based on Chordal Distance Khawla Alnajjar1 , Vaneet Aggarwal2 , Vinay A. Vaishampayan2 , and Xiaodong Wang1 1
Department of Electrical Engineering, Columbia University, New York, NY 10027, USA 2 AT&T Labs - Research, Florham Park, NJ 07932, USA [email protected], [email protected], [email protected], [email protected]
Abstract—In order to manage interference in the K-user interference channel we study optimal interference aligned solutions using a modified version of the chordal distance between the signal and the interference space as a metric. A locally optimal algorithm to optimize this distance is described. The proposed metric and algorithm are validated by an improvement in the probability of error as compared to the baseline aligned solution. Index Terms– Grasmannian packing, Chordal distance, Interference Channel, Precoder.
II. S YSTEM M ODEL A K-user interference channel consists of K transmitters and K receivers. The inputs of the k th transmitter at time i are denoted by Xk [i] ∈ C, k = 1, 2, · · · , K, and the outputs at j th receiver in time i can be written as Yj [i] ∈ C, j = 1, 2, · · · , K. The received signal Yj [i], j = 1, 2, · · · , K is given by Yj [i]
=
Hkj [i]Xk [i] + Zj [i],
(1)
k=1
I. I NTRODUCTION Interference management in wireless networks has been studied extensively [1]-[15]. In [1], interference alignment was proposed as an approach to mitigate interference and the interference aligned approach was shown to be optimal in terms of degrees of freedom. The aligned solution should also maximize an appropriate distance measure between the interference and the signal subspace. This is the subject of this paper. We first propose a metric to study interference management solutions that optimize the distance between the signal and the interference subspace while maintaining the dimensions of the signal subspaces for K-user SISO interference channels. The proposed metric is a modified version of the chordal distance (proposed in [16]) between the signal and the interference spaces. We then propose an algorithm that can be used to find Grasmannian packing for the subspaces that locally optimize the distance between the subspaces. This algorithm locally optimizes the distance using a gradient-descent approach, starting from the aligned solution proposed in [1]. This approach is validated by an improvement in probability of error as compared to the baseline approach in [1]. The rest of the paper is organized as follows. In Section II, we introduce our system model. In Section III, we introduce the baseline interference alignment scheme of [1] and the modified chordal distance between the subspaces. In Section V, we propose our optimization framework for maximizing the distance between the interference and the signal subspaces. In Section VI, we provide a local optimization scheme for the interference channels and provide numerical results to demonstrate the approach. Section VII concludes the paper. Although the theoretical framework is developed for the K user interference channel, our performance results focus the case K = 3.
K X
where Hkj [i] ∈ C is the channel gain associated with each transmitter k and receiver j, and Zj [i] are additive white complex Gaussian random variables of unit variance(CAW GN ). We avoid the degenerate channel conditions of letting all the channel coefficient to be equal or letting the channel coefficient to be zero or infinity. We assumed that channel knowledge is causal and globally available. By using a symbol extension of the original channel for M channel uses, the output at the j th receiver in time-slot (each made up of M time instants) l is given by
Yj [l]
=
K X
Hkj [l]Xk [l] + Zj [l],
k=1
where l is the time slot index (or frequency subcarrier index), Xk [l] is an M × 1 column vector representing the M symbol extension of the transmitted symbol Xk ,
Xk [l]
=
Xk [M (l − 1) + 1] Xk [M (l − 1) + 2] . . . Xk [M l]
(2)
Similarly Yj [l] represents M symbol extension of the output signal Yj at the j th receiver, and Zj [l] represents M symbol extension of Zj . Hkj [l] is a diagonal M × M matrix which represents the M symbol expansion of the channel Hkj from transmitter k to receiver j. In this paper, we consider precoder designs that encode the message linearly to the input symbol. In other words, Xk [l] = Vk [l]dk [l], where Vk [l] is the precoder of size M × Dk and
Hkj [l]
=
Hkj [M (l − 1) + 1] 0 . . . 0
0 Hkj [M (l − 1) + 2] ... 0
dk [l] represents the data streams sent by transmitter k, which is Dk × 1. At receiver j, the received signal is Yj [l]
=
K X
Hkj [l]Vk [l]dk [l] + Zj [l],
(4)
k=1
where the desired P signal is Hjj [l]Vj [l]dj [l] and the interference signal is j6=k Hkj [l]Vk [l]dk [l]. The transmit power at transmitter k is E[Xk [l]† Xk [l]] ≤ M Pk 1 . III. P RELIMINARIES In this section, we describe a modified version of the Grassmanian distance metric which will be used throughout this paper. We also describe the interference alignment solution proposed in [1]. A. Chordal Distance The chordal distance is a distance measure between two subspaces. Let W1 , W2 be two k-dimensional subspaces of CM , M ≥ k and F1 , F2 be the corresponding orthogonal projectors. Traditionally chordal or Frobenius distance is used to measure how far away W1 and W2 are from each other. Definition 1. The chordal distance between W1 and W2 is the quantity d(W1 , W2 ) = k − Tr (F1 F2 ), where Tr (A) denotes the trace of matrix A. In this article we will use a slightly different definition of distance which was defined in [16]. Definition 2 ([16]). d(W1 , W2 ) = Tr (F1 ) + Tr (F2 ) − 2Tr (F1 F2 ). The reasons for the change of distance definitions are the following. First, definition 2, unlike definition 1, is applicable for the case when Tr (F1 ) 6= Tr (F2 ). Second, let us consider the linear space formed by linear operators acting in CM with the inner product defined by Tr (A† B), A, B ∈ CM . Then, remembering that for any projection operator F , F F = F and F † = F , we see that definition 2 is the standard definition of the Euclidean distance in a vector space. Third, if Tr (F1 ) = Tr (F2 ) then the distance computed according to definition 2 is just twice the distance computed according to definition 1. The trace of the projection operator gives the dimension of the image subspace spanned by the projection operator. Thus, the modified chordal distance is the sum of the dimension of the two subspaces minus twice the dimension of the space formed by the intersection of the two subspaces. 1 A†
represents complex transpose of A.
... ... . . . ...
0 0 . . . Hkj [M l]
(3)
B. Interference Alignment Interference channels are interference limited rather than noise limited. The authors of [1] proposed an interference management technique called interference alignment. This approach results in K/2 degrees of freedom for a K user interference channel. The design aligns precoders such that all the interference at the receivers is in a space of dimension about half of the whole space leaving the remaining dimensions for the signal of interest. The precoder designs used by [1] achieves the rate tuple n n n+1 , 2n+1 , 2n+1 ) by designing precoders for the 2n + 1 ( 2n+1 channel extension of the original channel. The design of precoders Vk is given as follows (We will omit time index l for brevity). V1 V2 V3
= = =
A
(5)
H23
−1
H13 C
(6)
H32
−1
H12 B,
(7)
where A, B and C are given as given as A = w Tw T2 w ... Tn w B = Tw T2 w ... Tn w C = w Tw ... T(n−1) w,
(8) (9) (10)
where T = H21 H12
−1
H32 H23
−1
H13 H31
−1
,
(11)
and w is an all-one vector. These precoders align the interference at the first receiver such that both the interferers span the same n dimensional subspace giving n + 1 degrees of freedom for the intended signal. At the other two receivers, the interference from the other transmitter (from second or third) lies in a subspace contained within the interference subspace of the first transmitter thus leaving n dimensions for the intended signal. IV. M OTIVATION FOR C HORDAL D ISTANCE Assume that our system must transmit S streams in all, and that data from each stream is modulated onto a subspace of unit dimension. Let N denote the dimension of the space in which communication occurs. The case of interest is when S > N , and as previous work has shown, when precoder vectors are properly designed so that the interference is properly aligned it is possible to guarantee each data stream
an interference-free dimension. Under such conditions, if we denote the chordal distance between the signal and interference at the receiver for stream j by dj , then the effective signal-tointerference ratio (SIR) for stream j, assuming unit variance noise and unit transmit power is sirj = (dj + 2 − N )/2.
(12)
Thus maximizing the minimum per-stream chordal distance is well motivated. V. O PTIMAL P RECODER D ESIGNS In this section, we propose our optimization framework for maximizing the distance between the interference and the signal spaces such that the spaces are aligned and are of proper dimension. The idea is to maximize the minimum distance between the desired and interference spaces. We assume that the precoders V1 , V2 , · · · , VK are used on an M -dimensional space and that the K precoders are of dimension M1 , M2 , · · · , MK , respectively. The distance between the interference space and the signal space at the receiver j ∈ {1, 2, · · · , K} is given by Lj = T r Dj + T r Ij − 2T r Dj Ij ,
(13)
where Dj Ij Cj
+ = Hjj Vj Hjj Vj , + = Cj Cj , and = H1j V1 1j6=1 · · · HKj VK 1j6=K ,
(14) (15) (16)
where A+ denotes the Moore-Penrose pseudoinverse of A. The optimization problem we propose is to maximize min(L1 , L2 , · · · , LK ), where the design of K precoders is a design parameter for a fixed M1 , M2 , · · · , MK , M and the constraint that Tr (Dj ) = Mj . We also focus on aligned solutions such that the dimension of interference is the maximum of the dimension of each interference signal, or Tr (Cj ) = maxk6=j (Mk ). This maximizes the minimum distance between the signal and interfering spaces at each receiver for fixed dimension of the precoders. We note that an aligned solution may not always exist for every value of M1 , · · · , MK , M and channel parameters. It was shown that an aligned solution exists for M1 = n + 1, M2 = · · · = MK = n and M = 2n + 1 for all n > 0 in [1] with probability 1 over the channel gains. Thus, optimization will result in a solution with probability 1 in these configurations. This optimization problem distances the signal and interference space at each receiver such that the interference spaces are aligned at each receiver and that the dimensions of the signal spaces at each receiver are fixed. VI. A N A LGORITHM FOR L OCALLY O PTIMAL A LIGNED S OLUTION In this section, we give a locally optimal solution for the above problem and evaluate this algorithm in terms of error probability.
A. Algorithm In the following, we describe our algorithm for 3 user interference channel. We chose M1 = n + 1, M2 = M3 = n, M = 2n + 1 for some n > 0. Let w to be (2n + 1) × 1 vector, w11 .. (17) w = . w(2n+1)1 We start with the initial vector w made up of all ones as in [1] which is referred to as the baseline approach. For each choice of w, the precoders are found using (5)-(11). Gradient descent is used to find locally optimal w that maximizes min(L1 , L2 , L3 ) which are described in Equations (12)-(15). The locally optimal solution give an aligned solution that is better in terms of modified chordal distance between the signal and interference spaces than the baseline approach in [1]. We illustrate the algorithm with the following example. Let H11 [1] = 0.3802 + 0.6096i, H11 [2] = −1.2968 + 0.2254i, H11 [3] = −1.5972 − 0.9247i, H21 [1] = −0.3066 + 1.9583i, H21 [2] = 0.2423 − 0.9545i, H21 [3] = 2.5303 + 2.1460i, H31 [1] = 0.5129 − .01449i, H31 [2] = −0.0446 − 0.0878i, H31 [3] = 0.5054 + 1.0534i, H12 [1] = 0.9963 − 0.8538i, H12 [2] = 1.0021 + 0.5072i, H12 [3] = 0.4748 + 1.1528i, H22 [1] = 0.3457 − 0.2146i, H22 [2] = 0.7316 + 0.2078i, H22 [3] = 0.5140 − 0.5567i, H32 [1] = 0.6282 − 0.5724i, H32 [2] = −0.8111 − 2.0819i, H32 [3] = −0.7558 + 1.0171i, H13 [1] = 0.2299 − 1.2102i, H13 [2] = −0.5338 − 0.0723i, H13 [3] = 0.9689 − 0.1707i, H23 [1] = 0.2257 − 0.0212i, H23 [2] = 0.2212 − 0.1166i, H23 [3] = −0.6116 + 0.4439i, H33 [1] = 0.7731 + 0.0547i, H33 [2] = 0.7844 − 0.8585i, H33 [3] = −0.6107 − 0.7874i. Using the approach in [1], the precoders are as follows.
V1
0.0222 = 0.0222 0.0222
0.2259 + 0.1833i −0.4551 + 0.8365i −0.0406 − 0.0730i
V2
0.2520 − 0.8717i = −0.2928 − 0.2089i −0.1954 − 0.0952i
(18)
V3
0.5139 + 0.4436i = −0.1667 − 0.7042i −0.1241 − 0.0041i
(19)
(20)
The gradient descent approach (using 300 iterations for gradient descent with step size as 1/f . 6 for iteration f ) proposed in the paper gives the following precoders.
V1
0.0300 0.0210 = 0.0478
0.3051 + 0.2475i −0.4301 + 0.7905i −0.0873 − 0.1569i
(21)
Baseline scheme with ZF Baseline scheme with MMSE Proposed scheme with ZF Proposed scheme with MMSE
0
−1
10
1 0.9 0.8
Baseline scheme with ZF Proposed scheme with ZF
0.7 0.6
CDF
Symbol error rate (SER)
10
−2
10
0.5 0.4
−3
0.3
10
0.2 0.1
−4
10 −10
0
10
20
30
40
50
0
Energy power level (SNR(dB))
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
The maximum minimum distance
Fig. 1. The SER performance for baseline scheme and our proposed scheme Fig. 2. CDF Plot of the Grasmannian distance achieved with the baseline scheme and our proposed scheme using zero forcing
V2
0.2512 − 0.8689i = −0.2042 − 0.1457i −0.3101 − 0.1511i
V3
0.5910 + 0.5100i = −0.1341 − 0.5665i −0.2272 − 0.0076i
(22)
The second decoder is the MMSE equalizer, where the estimated signal at receiver j is given as
(23)
B. Numerical Results In this subsection, we illustrate the performance of the proposed locally optimal algorithm. We assume n = 1 for illustration. The first transmitter sends two data streams and the other transmitters send only one data stream in three time instances. A quasi-static flat Rayleigh fading channel model with QAM input modulation is used. The results assume coherence time of 3000 symbols and the numerical results are averaged over 300 coherence blocks. We used two decoders to illustrate the approach. The first decoder is the Zero Forcing (ZF) equalizer which is described as follows. Let HHi = (24) H1i V1 H2i V2 H3i V3 ,
+
γi = HHi Yi ,
=
3 X
H H Hji Vi Vi Hji
!−1 +
σn2 I
Yj ,
i=1
With CAWGN noise environment in 3000 experiments, the average probability of symbol error for each receivers in the baseline approach are 0.3382, 0.0550 and 0.0040 respectively, which all decrease to 0.3012, 0.0330 and .0003 respectively using the proposed algorithm. Our proposed algorithm increases the minimum modified chrodal distance from 1.8225 to 2.2836. This shows an improvement in the distance as well as error probabilities using our approach.
for i = 1, 2, 3. Further, let
χj
H H Vj Hjj
(25)
for i = 1, 2, 3. Then, the estimated signal at first receiver is the first two elements of γ1 , at the second receiver is the third element of γ2 , and at the third receiver is the fourth element of γ3 .
where σn2 is the noise power and I is the identity matrix. From Figure 1, we see that the symbol error rate (SER) for the proposed scheme is lower than the baseline approach. In Figure 2, we see the cumulative distribution function (CDF) of the modified chordal distance of the baseline approach and our approach. We note that the distances do increase significantly using the optimization. In the baseline approach, distances are smaller than 1.1 with almost 60% probability while we have only 20% probability that the distance is that small. This increase of distance in our approach results in an improvement in the error probabilities. VII. C ONCLUSION In this paper, we formulated the interference management schemes for interference channels that maximize the distance between the interference and the signal subspaces at each receiver. We also propose a locally optimal interference aligned solution which is validated by a corresponding improvement in error probabilities. In this paper, we considered distance between the signal space and the interference space. We can also consider distance considering each dimension of the signal as the signal space and all the rest spaces of the signal and interference as the interference space. Doing this, we can maximize the minimum of all the distances of each dimension of signal space with rest dimensions of the signal and the interference. However, we did not consider this measure in this paper. An interesting open question is to find the optimal precoder designs for given channel gains that result in the best possible distance between the interference and the signal spaces at each receiver.
R EFERENCES [1] V. R. Cadambe and S. A. Jafar, “Interference alignment and the degrees of freedom for the K-user interference channel,” IEEE Transactions on Information Theory, vol. 54, no. 8, pp. 3425-3441, Aug. 2008. [2] V. S. Annapureddy, A. El Gamal, and V. V. Veeravalli, “Degrees of freedom of the K-user interference channel with transmitter cooperation,” in Proc. IEEE International Symposium on Information Theory, pp. 385-389, Austin, Jun 2010. [3] H. Maleki, S. A. Jafar, and S. Shamai, “Retrospective Interference Alignment,” in Proc. IEEE International Symposium on Information Theory, pp. 2756-2760, St. Petersburg, Jul-Aug 2011. [4] C. Wang, S. A. Jafar, S. Shamai, and M. Wigger, “Interference, Cooperation and Connectivity - A Degrees of Freedom Perspective,” in Proc. IEEE International Symposium on Information Theory, pp. 430-434, St. Petersburg, Jul-Aug 2011. [5] I. Santamaria, O. Gonzalez, R. W. Heath, Jr., and S. W. Peters, “Maximum Sum-Rate Interference Alignment Algorithms for MIMO Channels,” in Proc. IEEE Global Telecommunications Conference, pp. 1-6, Dec. 2010. [6] O. El Ayach and R. W. Heath, Jr., “Interference Alignment with Analog CSI Feedback,” in Proc. Military Communications Conference, pp. 16441648, Oct-Nov. 2010. [7] O. El Ayach, S. W. Peters, and R. W. Heath Jr., “The Feasibility of Interference Alignment of Measured MIMO-OFDM Channels,” IEEE Transactions on Vehicular Technology,vol. 59. No. 9, Nov. 2010. [8] S. W. Peters and R. W. Heath, Jr., “Interference Alignment Via Alternating Minimization,” in Proc. IEEE International Conference on Acoustics, Speech and Signal Processing, pp. 2445-2448, Apr 2009. [9] K. S. Gomadam, V. R. Cadambe, and S. A. Jafar, “Approaching the Capacity of Wireless Networks through Distributed Interference Alignment,” in Proc. IEEE Global Telecommunications Conference, pp. 1-6, Nov-Dec 2008. [10] J. Lee and W. Choi, “Opportunistic Interference Aligned User Selection in Multiuser MIMO Interference Channels,” in Proc. IEEE Global Telecommunications Conference, pp. 1-5, Dec 2010. [11] H. Yu, J. Park, Y. Sung and H. Lee, “A least Squares Approach to Joint Beam Design for Interference Alignment in Multiuser Interference Channels,” in Proc. IEEE 10th Workshop on Signal Processing Advances in Wireless Communications, pp. 593-597, Jun 2009. [12] H. Sung, K. Lee and S. Park, and I. Lee, “An Iterative Precoder Optimization Method for K-user Interference Channel Systems,” in Proc. IEEE Global Telecommunications Conference, pp. 1-5, Nov-Dec 2009. [13] H. Shen, B. Li, M. Tao, and Yi Luo, “The New Interference Alignment Scheme for the MIMO Interference Channel,” in Proc. IEEE Wireless Communications and Networking Conference, pp. 1-6, Apr 2010. [14] H. Shen, B. Li, M. Tao, and X. Wang, “MSE-Based Transceiver Designs for the MIMO Interference Channel,” IEEE Transactions On Wireless Communications, vol.9, no.11, Nov 2010. [15] H. Shen and B. Li, “The Novel Iterative Interference Alignment Scheme for the SISO Interference Channel,” in Proc. IEEE 71st Vehicular Technology Conference, pp. 1-5, May 2010. [16] V. Aggarwal, A. Ashikhmin, and A. R. Calderbank, “A Grassmannian Packing Based on the Nordstrom-Robinson Code,” in Proc. IEEE Information Theory Workshop, pp. 1-5, Chengdu, Oct 2006.
Department of Electrical Engineering, Columbia University, New York, NY 10027, USA 2 AT&T Labs - Research, Florham Park, NJ 07932, USA [email protected], [email protected], [email protected], [email protected]
Abstract—In order to manage interference in the K-user interference channel we study optimal interference aligned solutions using a modified version of the chordal distance between the signal and the interference space as a metric. A locally optimal algorithm to optimize this distance is described. The proposed metric and algorithm are validated by an improvement in the probability of error as compared to the baseline aligned solution. Index Terms– Grasmannian packing, Chordal distance, Interference Channel, Precoder.
II. S YSTEM M ODEL A K-user interference channel consists of K transmitters and K receivers. The inputs of the k th transmitter at time i are denoted by Xk [i] ∈ C, k = 1, 2, · · · , K, and the outputs at j th receiver in time i can be written as Yj [i] ∈ C, j = 1, 2, · · · , K. The received signal Yj [i], j = 1, 2, · · · , K is given by Yj [i]
=
Hkj [i]Xk [i] + Zj [i],
(1)
k=1
I. I NTRODUCTION Interference management in wireless networks has been studied extensively [1]-[15]. In [1], interference alignment was proposed as an approach to mitigate interference and the interference aligned approach was shown to be optimal in terms of degrees of freedom. The aligned solution should also maximize an appropriate distance measure between the interference and the signal subspace. This is the subject of this paper. We first propose a metric to study interference management solutions that optimize the distance between the signal and the interference subspace while maintaining the dimensions of the signal subspaces for K-user SISO interference channels. The proposed metric is a modified version of the chordal distance (proposed in [16]) between the signal and the interference spaces. We then propose an algorithm that can be used to find Grasmannian packing for the subspaces that locally optimize the distance between the subspaces. This algorithm locally optimizes the distance using a gradient-descent approach, starting from the aligned solution proposed in [1]. This approach is validated by an improvement in probability of error as compared to the baseline approach in [1]. The rest of the paper is organized as follows. In Section II, we introduce our system model. In Section III, we introduce the baseline interference alignment scheme of [1] and the modified chordal distance between the subspaces. In Section V, we propose our optimization framework for maximizing the distance between the interference and the signal subspaces. In Section VI, we provide a local optimization scheme for the interference channels and provide numerical results to demonstrate the approach. Section VII concludes the paper. Although the theoretical framework is developed for the K user interference channel, our performance results focus the case K = 3.
K X
where Hkj [i] ∈ C is the channel gain associated with each transmitter k and receiver j, and Zj [i] are additive white complex Gaussian random variables of unit variance(CAW GN ). We avoid the degenerate channel conditions of letting all the channel coefficient to be equal or letting the channel coefficient to be zero or infinity. We assumed that channel knowledge is causal and globally available. By using a symbol extension of the original channel for M channel uses, the output at the j th receiver in time-slot (each made up of M time instants) l is given by
Yj [l]
=
K X
Hkj [l]Xk [l] + Zj [l],
k=1
where l is the time slot index (or frequency subcarrier index), Xk [l] is an M × 1 column vector representing the M symbol extension of the transmitted symbol Xk ,
Xk [l]
=
Xk [M (l − 1) + 1] Xk [M (l − 1) + 2] . . . Xk [M l]
(2)
Similarly Yj [l] represents M symbol extension of the output signal Yj at the j th receiver, and Zj [l] represents M symbol extension of Zj . Hkj [l] is a diagonal M × M matrix which represents the M symbol expansion of the channel Hkj from transmitter k to receiver j. In this paper, we consider precoder designs that encode the message linearly to the input symbol. In other words, Xk [l] = Vk [l]dk [l], where Vk [l] is the precoder of size M × Dk and
Hkj [l]
=
Hkj [M (l − 1) + 1] 0 . . . 0
0 Hkj [M (l − 1) + 2] ... 0
dk [l] represents the data streams sent by transmitter k, which is Dk × 1. At receiver j, the received signal is Yj [l]
=
K X
Hkj [l]Vk [l]dk [l] + Zj [l],
(4)
k=1
where the desired P signal is Hjj [l]Vj [l]dj [l] and the interference signal is j6=k Hkj [l]Vk [l]dk [l]. The transmit power at transmitter k is E[Xk [l]† Xk [l]] ≤ M Pk 1 . III. P RELIMINARIES In this section, we describe a modified version of the Grassmanian distance metric which will be used throughout this paper. We also describe the interference alignment solution proposed in [1]. A. Chordal Distance The chordal distance is a distance measure between two subspaces. Let W1 , W2 be two k-dimensional subspaces of CM , M ≥ k and F1 , F2 be the corresponding orthogonal projectors. Traditionally chordal or Frobenius distance is used to measure how far away W1 and W2 are from each other. Definition 1. The chordal distance between W1 and W2 is the quantity d(W1 , W2 ) = k − Tr (F1 F2 ), where Tr (A) denotes the trace of matrix A. In this article we will use a slightly different definition of distance which was defined in [16]. Definition 2 ([16]). d(W1 , W2 ) = Tr (F1 ) + Tr (F2 ) − 2Tr (F1 F2 ). The reasons for the change of distance definitions are the following. First, definition 2, unlike definition 1, is applicable for the case when Tr (F1 ) 6= Tr (F2 ). Second, let us consider the linear space formed by linear operators acting in CM with the inner product defined by Tr (A† B), A, B ∈ CM . Then, remembering that for any projection operator F , F F = F and F † = F , we see that definition 2 is the standard definition of the Euclidean distance in a vector space. Third, if Tr (F1 ) = Tr (F2 ) then the distance computed according to definition 2 is just twice the distance computed according to definition 1. The trace of the projection operator gives the dimension of the image subspace spanned by the projection operator. Thus, the modified chordal distance is the sum of the dimension of the two subspaces minus twice the dimension of the space formed by the intersection of the two subspaces. 1 A†
represents complex transpose of A.
... ... . . . ...
0 0 . . . Hkj [M l]
(3)
B. Interference Alignment Interference channels are interference limited rather than noise limited. The authors of [1] proposed an interference management technique called interference alignment. This approach results in K/2 degrees of freedom for a K user interference channel. The design aligns precoders such that all the interference at the receivers is in a space of dimension about half of the whole space leaving the remaining dimensions for the signal of interest. The precoder designs used by [1] achieves the rate tuple n n n+1 , 2n+1 , 2n+1 ) by designing precoders for the 2n + 1 ( 2n+1 channel extension of the original channel. The design of precoders Vk is given as follows (We will omit time index l for brevity). V1 V2 V3
= = =
A
(5)
H23
−1
H13 C
(6)
H32
−1
H12 B,
(7)
where A, B and C are given as given as A = w Tw T2 w ... Tn w B = Tw T2 w ... Tn w C = w Tw ... T(n−1) w,
(8) (9) (10)
where T = H21 H12
−1
H32 H23
−1
H13 H31
−1
,
(11)
and w is an all-one vector. These precoders align the interference at the first receiver such that both the interferers span the same n dimensional subspace giving n + 1 degrees of freedom for the intended signal. At the other two receivers, the interference from the other transmitter (from second or third) lies in a subspace contained within the interference subspace of the first transmitter thus leaving n dimensions for the intended signal. IV. M OTIVATION FOR C HORDAL D ISTANCE Assume that our system must transmit S streams in all, and that data from each stream is modulated onto a subspace of unit dimension. Let N denote the dimension of the space in which communication occurs. The case of interest is when S > N , and as previous work has shown, when precoder vectors are properly designed so that the interference is properly aligned it is possible to guarantee each data stream
an interference-free dimension. Under such conditions, if we denote the chordal distance between the signal and interference at the receiver for stream j by dj , then the effective signal-tointerference ratio (SIR) for stream j, assuming unit variance noise and unit transmit power is sirj = (dj + 2 − N )/2.
(12)
Thus maximizing the minimum per-stream chordal distance is well motivated. V. O PTIMAL P RECODER D ESIGNS In this section, we propose our optimization framework for maximizing the distance between the interference and the signal spaces such that the spaces are aligned and are of proper dimension. The idea is to maximize the minimum distance between the desired and interference spaces. We assume that the precoders V1 , V2 , · · · , VK are used on an M -dimensional space and that the K precoders are of dimension M1 , M2 , · · · , MK , respectively. The distance between the interference space and the signal space at the receiver j ∈ {1, 2, · · · , K} is given by Lj = T r Dj + T r Ij − 2T r Dj Ij ,
(13)
where Dj Ij Cj
+ = Hjj Vj Hjj Vj , + = Cj Cj , and = H1j V1 1j6=1 · · · HKj VK 1j6=K ,
(14) (15) (16)
where A+ denotes the Moore-Penrose pseudoinverse of A. The optimization problem we propose is to maximize min(L1 , L2 , · · · , LK ), where the design of K precoders is a design parameter for a fixed M1 , M2 , · · · , MK , M and the constraint that Tr (Dj ) = Mj . We also focus on aligned solutions such that the dimension of interference is the maximum of the dimension of each interference signal, or Tr (Cj ) = maxk6=j (Mk ). This maximizes the minimum distance between the signal and interfering spaces at each receiver for fixed dimension of the precoders. We note that an aligned solution may not always exist for every value of M1 , · · · , MK , M and channel parameters. It was shown that an aligned solution exists for M1 = n + 1, M2 = · · · = MK = n and M = 2n + 1 for all n > 0 in [1] with probability 1 over the channel gains. Thus, optimization will result in a solution with probability 1 in these configurations. This optimization problem distances the signal and interference space at each receiver such that the interference spaces are aligned at each receiver and that the dimensions of the signal spaces at each receiver are fixed. VI. A N A LGORITHM FOR L OCALLY O PTIMAL A LIGNED S OLUTION In this section, we give a locally optimal solution for the above problem and evaluate this algorithm in terms of error probability.
A. Algorithm In the following, we describe our algorithm for 3 user interference channel. We chose M1 = n + 1, M2 = M3 = n, M = 2n + 1 for some n > 0. Let w to be (2n + 1) × 1 vector, w11 .. (17) w = . w(2n+1)1 We start with the initial vector w made up of all ones as in [1] which is referred to as the baseline approach. For each choice of w, the precoders are found using (5)-(11). Gradient descent is used to find locally optimal w that maximizes min(L1 , L2 , L3 ) which are described in Equations (12)-(15). The locally optimal solution give an aligned solution that is better in terms of modified chordal distance between the signal and interference spaces than the baseline approach in [1]. We illustrate the algorithm with the following example. Let H11 [1] = 0.3802 + 0.6096i, H11 [2] = −1.2968 + 0.2254i, H11 [3] = −1.5972 − 0.9247i, H21 [1] = −0.3066 + 1.9583i, H21 [2] = 0.2423 − 0.9545i, H21 [3] = 2.5303 + 2.1460i, H31 [1] = 0.5129 − .01449i, H31 [2] = −0.0446 − 0.0878i, H31 [3] = 0.5054 + 1.0534i, H12 [1] = 0.9963 − 0.8538i, H12 [2] = 1.0021 + 0.5072i, H12 [3] = 0.4748 + 1.1528i, H22 [1] = 0.3457 − 0.2146i, H22 [2] = 0.7316 + 0.2078i, H22 [3] = 0.5140 − 0.5567i, H32 [1] = 0.6282 − 0.5724i, H32 [2] = −0.8111 − 2.0819i, H32 [3] = −0.7558 + 1.0171i, H13 [1] = 0.2299 − 1.2102i, H13 [2] = −0.5338 − 0.0723i, H13 [3] = 0.9689 − 0.1707i, H23 [1] = 0.2257 − 0.0212i, H23 [2] = 0.2212 − 0.1166i, H23 [3] = −0.6116 + 0.4439i, H33 [1] = 0.7731 + 0.0547i, H33 [2] = 0.7844 − 0.8585i, H33 [3] = −0.6107 − 0.7874i. Using the approach in [1], the precoders are as follows.
V1
0.0222 = 0.0222 0.0222
0.2259 + 0.1833i −0.4551 + 0.8365i −0.0406 − 0.0730i
V2
0.2520 − 0.8717i = −0.2928 − 0.2089i −0.1954 − 0.0952i
(18)
V3
0.5139 + 0.4436i = −0.1667 − 0.7042i −0.1241 − 0.0041i
(19)
(20)
The gradient descent approach (using 300 iterations for gradient descent with step size as 1/f . 6 for iteration f ) proposed in the paper gives the following precoders.
V1
0.0300 0.0210 = 0.0478
0.3051 + 0.2475i −0.4301 + 0.7905i −0.0873 − 0.1569i
(21)
Baseline scheme with ZF Baseline scheme with MMSE Proposed scheme with ZF Proposed scheme with MMSE
0
−1
10
1 0.9 0.8
Baseline scheme with ZF Proposed scheme with ZF
0.7 0.6
CDF
Symbol error rate (SER)
10
−2
10
0.5 0.4
−3
0.3
10
0.2 0.1
−4
10 −10
0
10
20
30
40
50
0
Energy power level (SNR(dB))
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
The maximum minimum distance
Fig. 1. The SER performance for baseline scheme and our proposed scheme Fig. 2. CDF Plot of the Grasmannian distance achieved with the baseline scheme and our proposed scheme using zero forcing
V2
0.2512 − 0.8689i = −0.2042 − 0.1457i −0.3101 − 0.1511i
V3
0.5910 + 0.5100i = −0.1341 − 0.5665i −0.2272 − 0.0076i
(22)
The second decoder is the MMSE equalizer, where the estimated signal at receiver j is given as
(23)
B. Numerical Results In this subsection, we illustrate the performance of the proposed locally optimal algorithm. We assume n = 1 for illustration. The first transmitter sends two data streams and the other transmitters send only one data stream in three time instances. A quasi-static flat Rayleigh fading channel model with QAM input modulation is used. The results assume coherence time of 3000 symbols and the numerical results are averaged over 300 coherence blocks. We used two decoders to illustrate the approach. The first decoder is the Zero Forcing (ZF) equalizer which is described as follows. Let HHi = (24) H1i V1 H2i V2 H3i V3 ,
+
γi = HHi Yi ,
=
3 X
H H Hji Vi Vi Hji
!−1 +
σn2 I
Yj ,
i=1
With CAWGN noise environment in 3000 experiments, the average probability of symbol error for each receivers in the baseline approach are 0.3382, 0.0550 and 0.0040 respectively, which all decrease to 0.3012, 0.0330 and .0003 respectively using the proposed algorithm. Our proposed algorithm increases the minimum modified chrodal distance from 1.8225 to 2.2836. This shows an improvement in the distance as well as error probabilities using our approach.
for i = 1, 2, 3. Further, let
χj
H H Vj Hjj
(25)
for i = 1, 2, 3. Then, the estimated signal at first receiver is the first two elements of γ1 , at the second receiver is the third element of γ2 , and at the third receiver is the fourth element of γ3 .
where σn2 is the noise power and I is the identity matrix. From Figure 1, we see that the symbol error rate (SER) for the proposed scheme is lower than the baseline approach. In Figure 2, we see the cumulative distribution function (CDF) of the modified chordal distance of the baseline approach and our approach. We note that the distances do increase significantly using the optimization. In the baseline approach, distances are smaller than 1.1 with almost 60% probability while we have only 20% probability that the distance is that small. This increase of distance in our approach results in an improvement in the error probabilities. VII. C ONCLUSION In this paper, we formulated the interference management schemes for interference channels that maximize the distance between the interference and the signal subspaces at each receiver. We also propose a locally optimal interference aligned solution which is validated by a corresponding improvement in error probabilities. In this paper, we considered distance between the signal space and the interference space. We can also consider distance considering each dimension of the signal as the signal space and all the rest spaces of the signal and interference as the interference space. Doing this, we can maximize the minimum of all the distances of each dimension of signal space with rest dimensions of the signal and the interference. However, we did not consider this measure in this paper. An interesting open question is to find the optimal precoder designs for given channel gains that result in the best possible distance between the interference and the signal spaces at each receiver.
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