On the Pareto-Optimal Beam Structure and Design for Multi-User MIMO Interference Channels
In this paper, the Pareto-optimal beam structure for multi-user multiple-input multiple-output (MIMO) interference channels is investigated and a necessary condition for any Pareto-optimal transmit signal covariance matrix is presented for the -pair Gaussian interference channel. It is shown that any Pareto-optimal transmit signal covariance matrix at a transmitter should have its column space contained in the union of the eigen-spaces of the channel matrices from the transmitter to all receivers. Based on this necessary condition, an efficient parameterization for the beam search space is proposed. The proposed parameterization is given by the product manifold of a Stiefel manifold and a subset of a hyperplane and enables us to construct a very efficient beam design algorithm by exploiting its rich geometrical structure and existing tools for optimization on Stiefel manifolds. Reduction in the beam search space dimension and computational complexity by the proposed parameterization and the proposed beam design approach is significant when the number of transmit antennas is larger than the sum of the numbers of receive antennas, as in upcoming cellular networks adopting massive MIMO technologies. Numerical results validate the proposed parameterization and the proposed cooperative beam design method based on the parameterization for MIMO interference channels.
Interference channels, multi-input multi-output (MIMO), Pareto-optimality, beamforming, Stiefel manifolds
Multi-user multiple antenna interference channels have gained intensive interest from research communities in recent years because of the significance of proper interference control in current and future wireless networks. One of the break-through results in this area is interference alignment by Cadambe and Jafar , which provides an effective way to achieving maximum degrees-of-freedom (DoF) for MIMO interference channels. However, interference alignment is only DoF optimal, i.e., it is optimal at high signal-to-noise ratio (SNR), whereas in typical cellular networks most receivers experiencing severe interference are located at cell edges and hence operate in the low or intermediate SNR regime. Thus, Jorswieck et al. investigated the multiple antenna interference channel problem from a different perspective based on Pareto-optimality . The framework of Pareto-optimality is especially useful for interference channels since the users in an interference channel basically form a group for negotiation. Under this framework, Jorswieck et al. showed for multiple-input single-output (MISO) interference channels that any Pareto-optimal beam vector at a transmitter is a normalized convex combination of the zero-forcing (ZF) beam vector and the matched-filtering (MF) beam vector in the case of two users and a linear combination of the channel vectors from the transmitter to all receivers in the general case of an arbitrary number of users. Their result and subsequent results by other researchers provide useful parameterizations for the optimal beam search space for efficient cooperative beam design in MISO interference channels [3, 4, 5, 6, 7, 8]. However, not many results for the Pareto-optimal beam structure for MIMO interference channels are available, although there exist some results in limited circumstances [9, 10, 11].
In this paper, we provide a necessary condition for Pareto-optimal beamformers for the -pair Gaussian interference channel,111In the -pair Gaussian channel, we have transmitter-receiver pairs, and every transmitter has transmit antennas and receiver has receive antennas. which can model general MIMO interference channels, and show that any Pareto-optimal transmit signal covariance matrix at a transmitter should have its column space contained in the union of the eigen-spaces of the channel matrices from the transmitter to all receivers. Based on this, we provide an efficient parameterization for the beam search space not missing Pareto-optimality whose dimension is independent of the number of transmit antennas and is determined only by , when . The proposed parameterization is given by the product manifold of a Stiefel manifold and a subset of a hyperplane and enables us to construct a very efficient cooperative beam design algorithm by exploiting its rich geometrical structure and existing tools for optimization on Stiefel manifolds. Reduction in the beam search space dimension and computational complexity by the proposed parameterization and the proposed beam design algorithm is significant, when as in upcoming cellular systems adopting massive MIMO technologies [12, 13]. Furthermore, the proposed beam design algorithm does not need to fix the number of data streams for transmission beforehand and it finds an (locally) optimal DoF for a given finite SNR. This is beneficial because the optimal DoF is not known for a finite SNR in most cases.
Notations and Organization In this paper, we will make use of standard notational conventions. Vectors and matrices are written in boldface with matrices in capitals. All vectors are column vectors. For a matrix , , , , and indicate the Hermitian transpose, 2-norm, trace, and determinant of , respectively. or denotes the element in the -th row and the -th column of . denotes the column space of and denotes the orthogonal complement of . denotes the orthogonal projection of a vector onto a linear subspace . represents the orthogonal projection onto and . For matrices and , means that is positive semi-definite. stands for the identity matrix of size (the subscript is omitted when unnecessary). or denotes the matrix composed of vectors and denotes the diagonal matrix with elements . means that is circularly-symmetric complex Gaussian-distributed with mean vector and covariance matrix . , , and denote the sets of real numbers, non-negative real numbers, and complex numbers, respectively. denotes the -dimensional Euclidean space and denotes the vector space of all complex -tuples. is the set of all matrices with complex elements. For a complex number , denotes the real part of .
The remainder of this paper is organized as follows. The system model is described in Section II. In Section III, a necessary condition and a parameterization for Pareto-optimal transmit beamformers for MIMO interference channels are provided. In Section IV, a beam design algorithm under the obtained parameterization is presented. Numerical results are provided in Section V, followed by conclusions in Section VI.
Ii System Model
In this paper, we consider a Gaussian interference channel with transmitter-receiver pairs, where every transmitter has transmit antennas and receiver has receive antennas. We assume that , , and . Due to interference from the unwanted transmitters, the received signal vector at receiver is given by
where denotes the channel matrix from transmitter to receiver ; is the transmit signal vector at transmitter generated from Gaussian distribution ; and is the additive Gaussian noise vector at receiver with distribution . Here, the transmit signal covariance matrix at transmitter is chosen among the feasible set
where the rank constraint is imposed to guarantee that the number of transmitted data streams is at least one and is less than or equal to the possible maximum for transmitter , . Note that any value of degree-of-freedom (DoF) from one to the maximum is feasible within the feasible set . From here on, we will call the considered MIMO interference channel the -pair Gaussian MIMO interference channel. The considered -pair Gaussian MIMO interference channel model is especially useful for downlink cooperative transmit beamforming in cellular systems. In the cellular downlink case, the transmitters, i.e., basestations can be equipped with many transmit antennas and the number of transmit antennas can be set to be the same in the phase of network design. On the other hand, each receiver, i.e., a mobile station has one or two receive antennas and furthermore the receivers forming a cooperative beamforming group together with the cooperating basestations may not have the same number of antennas. The -pair MIMO interference channel model fits this situation exactly.
Due to the assumption of , the channel matrix is a fat matrix (i.e., the number of its columns is larger than or equal to that of its rows) and its singular value decomposition (SVD) is given by
where is a unitary matrix; is a diagonal matrix composed of the singular values of ; is a submatrix composed of orthonormal column vectors that span the eigen-space of ; and is a submatrix composed of orthonormal column vectors that span the zero-forcing space of . Thus, and . From here on, we shall refer to and as the parallel and vertical spaces of (or simply with slight abuse of notation), respectively. For the purpose of beam design in later sections, we assume that the channel information is known to all the transmitters.
Under the assumption that interference is treated as noise at each receiver, for a given set of transmit signal covariance matrices and a given set of realized channel matrices , the rate of the -th transmitter-receiver pair is given by
for . Then, for the given set of realized channel matrices, the achievable rate region of the MIMO interference channel with interference treated as noise is defined as the union of rate-tuples that can be achieved by all possible combinations of transmit covariance matrices:
The outer boundary of the rate region in the first quadrant is called the Pareto boundary of and it consists of rate-tuples for which the rate of any one user cannot be increased without decreasing the rate of at least one other user.
In the rest of this paper, we shall investigate the Pareto-optimal transmit beam structure for the -pair Gaussian MIMO interference channel and develop an efficient beam design algorithm based on the obtained Pareto-optimal beam structure.
Iii A Necessary Condition for Pareto-Optimality for Transmit Beamforming in MIMO Interference Channels
In this section, we provide a necessary condition for Pareto-optimal transmit covariance matrices for the -pair Gaussian MIMO interference channel, which reveal the structure of Pareto-optimal transmit beamformers. The necessary condition is given in the following theorem.
For the -pair Gaussian MIMO interference channel in which the channel matrices are randomly realized and interference is treated as noise at each receiver, any Pareto-optimal transmit signal covariance matrix at transmitter should satisfy
Proof: First, we consider the case that . Suppose that the matrix has rank . 222When , the condition (6) is trivially satisfied since the channel matrices are randomly realized and thus spans the whole space. Then, there exists an orthonormal basis that spans , i.e.,
Now, suppose that a set of covariance matrices is Pareto-optimal (i.e., it achieves a Pareto boundary point of the achievable rate region ) and that at transmitter . Then, we can express as
where , , and . Here, implies that for some . Let be such an index and let
with . Then, and is positive semi-definite.333The positive semi-definiteness of can be shown as in . First, by the definition of and . For any vector orthogonal to , we have by the positive semi-definiteness of . Since any vector in is contained in . The claim follows. Thus, is a valid transmit signal covariance matrix. Now consider the rate-tuple that is achieved by . Let the interference covariance matrix at receiver be denoted by
Then, with the new set of transmit signal covariance matrices, the rate of the -th transmitter-receiver pair is given by
where step (a) holds because and hence . Similarly, the rate of the -th transmitter-receiver pair () is given by
where step holds again because and hence . Therefore, the rate-tuple does not change by replacing with .
Now, construct another transmit signal covariance matrix as
where satisfies while for all . Such exists almost surely in (i.e., ) for randomly realized channel matrices, because the event has measure zero.444The dimension of is and the dimension of is at most which is strictly less than by the assumption . The probability that a randomly realized subspace of is contained in another randomly realized subspace of with dimension strictly less than is zero. Here, is chosen so that (this is possible since . See (10).) and
Thus, is a valid transmit signal covariance matrix. Now consider the rate-tuple that is achieved by . Here, we define
Then, the rate of the -th transmitter-receiver pair receiver () is given by
where step holds by the construction of and step (d) holds by (13). On the other hand, the rate of the -th transmitter-receiver pair with is given by
Next, suppose that but . Then, by the same argument as before, there almost surely exists such that and for all , when . Let
where is chosen to be so that . Then, the rate of the -th transmitter-receiver pair () does not change by the same argument as in (17) and the rate of the -th transmitter-receiver pair strictly increases by the same argument as in (18). Thus, in the case of , each transmitter should use full power for Pareto optimality.
Under the same conditions as in Theorem 1, we have
Proof: First, consider the difference:
Thus, . This implies that the ordered eigenvalues of majorize those of . That is, let be the -th largest eigenvalue of and let be the -th largest eigenvalue of . Then,
Next, consider the difference of the traces of the two matrices:
by the construction of satisfying . By (21), (22) and the fact that the trace of a matrix is the sum of its eigenvalues, there exists at least one eigenvalue that is strictly larger than . Therefore, we have
since the determinant of a matrix is the product of its eigenvalues and both the matrices are strictly positive-definite due to the added identity matrix in , i.e., . Finally, (20) follows by the monotonicity of logarithm.
Theorem 1 states that the column space of any Pareto-optimal transmit signal covariance matrix at transmitter should be contained in the union of the parallel spaces of the channels from transmitter to all receivers. In the case that for all , the parallel space is simply the 1-dimensional linear subspace spanned by the matched filtering vector. Thus, this result in Theorem 1 can be regarded as a generalization of the result in the MISO interference channel by Jorswieck et al.  to general MIMO interference channels described by the -pair interference channel model.
Iii-a The Symmetric -User Case
In this subsection, we consider the symmetric two-user case and present another representation for Pareto-optimal transmit signal covariance matrices in this case.
In the two-user case in which the number of receive antennas is the same and , any Pareto-optimal transmit signal covariance matrix at transmitter should satisfy
and , where .
Proof: The proof is by showing the equivalence of the two subspaces:
Any vector in of the right-hand side (RHS) of (24) can be expressed as
for some , whereas any vector in of the left-hand side (LHS) of (24) can be expressed as
for some . Eq. (26) can be rewritten as
Furthermore, is invertible almost surely.555 and are the parallel spaces of and , respectively. The event that is non-invertible requires that is contained in a strict subspace of with dimension less than determined by . Such an event has measure zero for randomly realized channel matrices. Thus, there exists an isomorphism between and given by
Thus, the two subspaces are equivalent, i.e., . Since by Theorem 1, the claim follows.
As in the MISO case , the Pareto-optimal beam space is contained in the union of the self-parallel space of and the vertical or zero-forcing space of the channel to the other user in the two-user symmetric MIMO case.
Iii-B Parameterization for the Pareto-Optimal Beam Structure in MIMO Interference Channels
Theorem 1 provides a necessary condition for Pareto-optimal transmit signal covariance matrices for the -pair Gaussian MIMO interference channel with interference treated as noise. Based on Theorem 1, in this section, we develop a concrete parameterization for Pareto-optimal transmit signal covariance matrices for the -pair Gaussian MIMO interference channel for construction of a very efficient beam design algorithm in the next section. Here, we mainly focus on the case of , although the parameterization result here can be applied to the case of .
Since , any Pareto-optimal transmit signal covariance matrix at transmitter can be expressed as
where is a positive semi-definite matrix with rank less than or equal to . Note that is a matrix and it has full column rank almost surely for randomly realized channels.666The full column rank assumption is not necessary. In fact, the complexity of the beam design problem is reduced when the matrix does not have full column rank. This step will be explained in Algorithm 1 in Section IV. Let the (skinny) QR factorization of be
where is a matrix with orthonormal columns and is a upper triangular matrix. With the QR factorization, the Pareto-optimality subspace condition (6) can be rewritten as
where is a positive semi-definite matrix with rank less than or equal to . Since is Hermitian, i.e., self-adjoint, by the spectral theorem, it has the spectral decomposition given by
where is a matrix with orthonormal columns, i.e., and is a