An efficient interference alignment (IA) scheme is developed for $K$-user single-input single-output frequency selective fading interference channels. The main idea is to steer the transmit beamforming matrices such that at each receiver the subspace dimensions occupied by interference-free desired streams are asymptotically the same as those occupied by all interferences. Our proposed scheme achieves a higher multiplexing gain at any given number of channel realizations in comparison with the original IA scheme, which is known to achieve the optimal multiplexing gain asymptotically.
As an effective technique for interference management, interference alignment (IA) 1 has been proposed to achieve the optimal multiplexing gain asymptotically in single-input single-output (SISO) multi-user fading interference channel (IC) [2]. Subsequently, the IA scheme has been further studied in an explicit manner, which can be classified into two categories: One is to achieve the IA in signal scale [3]- [5]. Specifically, multi-user interferences at each receiver are aligned based on carefully constructed signal structures. The other is to align interferences in signal space [6], [7]. In this approach, transmit beamforming technique is used to align the multi-user interferences, which are nullified by zero-forcing (ZF) at each receiver.
The main contribution of this paper is to propose an efficient IA scheme in signal space when the number of users is greater than or equal to 3. Specifically, we propose a transmit beamforming design criterion so that a strictly higher multiplexing gain is attained in comparison with the original IA scheme [2] at any given number of channel realizations.
We consider the following channel model:
where M is the number of available frequency selective channel realizations,
is the received (transmitted) signal vector of size M × 1 at the k-th receiver (transmitter), and H [kl] ∈ C M×M is the M × M diagonal channel matrix from the l-th transmitter to the k-th receiver. Here, the (i, i)-th component h [kl] (i) of the channel matrix H [kl] is the channel coefficient in the i-th frequency channel realization. We assume the channel is not time-varying. The noise vector Z [k] ∈ C M is complex Gaussian with zero mean and the covariance of I M , where I M is the identity matrix of size M × M .
We assume the following in this paper.
• All transmitters and receivers are equipped with a single antenna.
• All H [kl] ’s are known in advance at all the transmitters and all the receivers. • The message of the i-th transmitter is independent from that of the j-th transmitter, ∀i, j ∈ K with i = j. • All diagonal components of H [kl] ’s are drawn independent and identically distributed (i.i.d.) from a continuous distribution, and absolute values of all diagonal elements are bounded below by a nonzero minimum value and above by a finite maximum value. Remark 1: In SISO frequency selective fading IC, it is possible to use multiple frequency selective channel instances in a combined manner, which leads to diagonal channel matrices H [kl] ’s, ∀k, l ∈ K.
The multiplexing gain r [8] of the K-user IC is defined as
where R + (SNR) is an achievable sum-rate2 at signal-to-noise ratio (SNR), where SNR is defined as the total power across all transmitters.
The essence of the IA scheme lies in design of transmit beamforming matrices. Basically, interferences are aligned at each receiver using the designed beamforming matrices to maximize the total number of interference-free desired streams.
In the original IA scheme, the following IA conditions were proposed [2].
and
where k] transmit beamforming matrix of the k-th user. Here, d [k] is the number of streams of the k-th user. Note that in (1) we have
3 V [3] ≺ V [1] , T
[k]
2 V [3] ≺ V [1] , •V [3] . T [2] 4 V [3] ≺ V [1] , T
[k]
3 V [3] ≺ V [1] , . . . . . .
K V [3] ≺ V [1] . T
k-1 V [3] ≺ V [1] , T
[k]
k+1 V [3] ≺ V [1] , T
[k] k+2 V [3] ≺ V [1] ,
. . .
K V [3] ≺ V [1] .
where S [k] is a d [k] × 1 vector of streams, and A ≺ B means that the column space of A is included in that of B. Note that (3) means that all the interferences are aligned exactly to occupy the same subspace at receiver 1, while (4) implies that the subspace spanned by the first transmitter’s interferences includes all the subspaces spanned by all the other transmitters’ interferences.
We define T
[k] j ’s as
with j = k. Then, the IA conditions (3) and (4) are equivalently rewritten as in TABLE I. Note that with probability 1 all T
[k] j ’s are full-rank and T
[b]
c for a = c or b = d since all the channel coefficients are assumed to be drawn i.i.d. from a continuous distribution.
Suppose that there exist V [k] ’s (∀k ∈ K) satisfying the IA conditions in TABLE I, where the number of column vectors in V [k] is d [k] . When each transmitter transmits streams using the corresponding transmit beamforming matrix V [k] and each receiver decodes the desired streams by ZF, the multiplexing gain r becomes r = (K -1)d [3] + d [1] d [3] + d [1] ,
where d [2] = d [3] [1] from (3) and (4). Remark 2: When d [1] /d [3] approaches 1, (7) becomes close to the upper bound of K/2 on the optimal multiplexing gain [2], [9].
Remark 3: Note that at the k-th receiver (∀k ∈ K) the ZF is possible due to the fact that effective channel matrix of size (d [3] + d [1] ) × (d [3] + d [1] ) is full-rank with probability 1 [2], where the effective channel matrix is formed by the channel matrices H [kl] ’s and beamforming matrices
In this paper, we propose an efficient beamform
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