The Dirty MIMO Multiple-Access Channel

The Dirty MIMO Multiple-Access Channel

Abstract

In the scalar dirty multiple-access channel, in addition to Gaussian noise, two additive interference signals are present, each known non-causally to a single transmitter. It was shown by Philosof et al. that for strong interferences, an i.i.d. ensemble of codes does not achieve the capacity region. Rather, a structured-codes approach was presented, that was shown to be optimal in the limit of high signal-to-noise ratios, where the sum-capacity is dictated by the minimal (“bottleneck”) channel gain. In this paper, we consider the multiple-input multiple-output (MIMO) variant of this setting. In order to incorporate structured codes in this case, one can utilize matrix decompositions that transform the channel into effective parallel scalar dirty multiple-access channels. This approach however suffers from a “bottleneck” effect for each effective scalar channel and therefore the achievable rates strongly depend on the chosen decomposition. It is shown that a recently proposed decomposition, where the diagonals of the effective channel matrices are equal up to a scaling factor, is optimal at high signal-to-noise ratios, under an equal rank assumption. This approach is then extended to any number of transmitters. Finally, an application to physical-layer network coding for the MIMO two-way relay channel is presented.

1Introduction

The dirty-paper channel, first introduced by Costa [1], is given by

where is the channel output, is the channel input subject to an average power constraint , is an additive white Gaussian noise (AWGN) of unit power, and is an interference which is known non-causally to the transmitter but not to the receiver.

Costa [1] showed that the capacity of this channel, when the interference is i.i.d. and Gaussian, is equal to that of an interference-free channel , i.e., as if . This result was subsequently extended to ergodic interference in [2] and to arbitrary interference in [3], where to achieve the latter, a structured lattice-based coding scheme was used.

This model serves as an information-theoretic basis for the study of interference cancellation techniques, and was applied to different network communication scenarios; see, e.g., [4].

Its multiple-input multiple-output (MIMO) variant as well as its extension to MIMO broadcast with private messages can be easily treated either directly or via scalar dirty-paper coding (DPC) and an adequate orthogonal matrix decomposition, the most prominent being the singular-value decomposition (SVD) and the QR decomposition (QRD); see, e.g., [5].

Philosof et al. [10] extended the dirty-paper channel to the case of multiple (distributed) transmitters, each transmitter, corresponding to a different user, knowing a different part of the interference:

where and are as before, () is the input of transmitter and is subject to an average power constraint , and is an arbitrary interference sequence which is known non-causally to transmitter but not to the other transmitters nor to the receiver.

This scenario is encountered in practice in cases where, for instance, non-cooperative base stations transmit data (over a multiple-access link) to a common receiver as well as to separate distinct receivers, which serve as interferences known at the transmitters for the communication to the common receiver. This scenario is described in Fig. ? for receivers.

The capacity region of this scenario, termed the dirty multiple-access channel (DMAC) in [10], was shown to be contained (“outer bound”) in the region of all rate tuples satisfying

and to contain (“achievable region”) all rate tuples satisfying1

where . These two regions coincide in the limit of high signal-to-noise ratios (SNRs) —  — thus establishing the capacity region in this limit to be equal to the region of all rate tuples satisfying

That is, the sum-capacity suffers from a bottleneck problem and reduces to the minimum of the individual capacities in this limit, where by the individual capacity of user , we mean the capacity from transmitter to the receiver where all other transmitters are silent. Interestingly, Costa’s random binning technique does not achieve the rate region or the high-SNR region , and structured lattice-based techniques need to be used [10].

The MIMO counterpart of the problem is given by

For simplicity, we assume for now that all vectors are of equal length .2 We further assume, without loss of generality, that the square channel matrices all have unit determinant, since any other value can be absorbed in . The AWGN vector z has i.i.d. unit-variance elements, while the interference vectors s are arbitrary as in the scalar case. The transmitters are subject to average power constraints .

In the high-SNR limit (where all powers satisfy ), the individual capacity of the -th user is given by:3

Thus, similarly to the scalar case , one can expect the high-SNR capacity region to be given by

However, in contrast to the single-user setting , the extension of the scalar DMAC to the MIMO case is not straightforward. As structure is required even in the scalar case , one cannot use a vector random codebook. To overcome this, we suggest to employ parallel scalar schemes, each using the lattice coding technique of [10]. This is in the spirit of the capacity-achieving SVD [12] or QRD [13] based schemes, that were proposed for MIMO communications (motivated by implementation considerations). The total rate is split between multiple scalar codebooks, each one enjoying a channel gain according to the respective diagonal value of the equivalent channel matrix obtained by the channel decomposition.

Unfortunately, for the MIMO DMAC problem, neither the SVD nor the QRD is suitable, i.e., their corresponding achievable rates cannot approach . Applying the SVD is not possible in the MIMO DMAC setting as joint diagonalization with the same orthogonal matrix on one side does not exist in general. Applying the QRD to each of the orthogonal matrices, in contrast, is possible as it requires an orthogonal operation only at the transmitter.4 However, the resulting matrices will have non-equal diagonals in general, corresponding to non-equal SNRs. Specifically, denoting the -th diagonal element of the -th matrix by , the resulting high-SNR sum-rate would be limited to

in this case. As this represents a per-element bottleneck, the rate is in general much lower than .

In this work we make use of a recently proposed joint orthogonal triangularization [17] to remedy the problem, i.e., to transform the per-element bottleneck into a global one as in . Specifically, the decomposition allows to transform two matrices (with equal determinants) into triangular ones with equal diagonals, using the same orthogonal matrix on the left — corresponding to a common operation carried at the receiver — and different orthogonal matrices on the right — corresponding to different operations applied by each of the transmitters. The equal-diagonals property implies that the minimum in is not active and hence the per-element bottleneck problem, incurred in the QRD-based scheme, is replaced by the more favorable vector bottleneck .

The rest of the paper is organized as follows. We start by introducing the channel model in Section 2. We then present the ingredients we use: the matrix decomposition is presented in Section 3, and a structured coding scheme for the single-user “dirty” MIMO channel is presented in Section 4. Our main result, the high-SNR capacity of the two-user MIMO DMAC is given in Section 5, using a structured scheme. We extend this result to the -user case in Section 6. We then demonstrate the usefulness of the proposed technique for MIMO physical-layer network coding in Section 7, by constructing a scheme that achieves capacity in the limit of high SNR for the MIMO two-way relay channel. We conclude the paper in Section 8.

2Problem Statement

The -user MIMO DMAC is given by:

where y is the channel output vector of length ,5 x () is the input vector of transmitter of length and is subject to an average power constraint defined formally in the sequal, z is an AWGN vector with an identity covariance matrix, and s is an interference vector of length which is known non-causally to transmitter but not to the other transmitters nor to the receiver. The interference vector signals s are assumed to be arbitrary sequences. We consider a closed-loop scenario, meaning that the channel matrix is known everywhere and that it satisfies the following properties.

Transmission is carried out in blocks of length . The input signal transmitted by transmitter is given by

where we denote by blocks of at time instants , i.e., , is the conveyed message by this user which is chosen uniformly from , is its transmission rate, and is the encoding function. The input signal x is subject to an average power constraint

The receiver reconstructs the messages from the channel output, using a decoding function :

A rate tuple is said to be achievable if for any , however small, there exist , and , such that the error probability is bounded from the above by :

The capacity region is defined as the closure of all achievable rate tuples.

3Background: Orthogonal Matrix Triangularization

In this section we briefly recall some important matrix decompositions that will be used in the sequel. In Section 3.1 we recall the generalized triangular decomposition (GTD) and some of its important special cases. Joint orthogonal triangularizations of two matrices are discussed in Section 3.2.

3.1Single Matrix Triangularization

Let be a proper matrix (recall Definition ?) of dimensions . A generalized triangular decomposition (GTD) of is given by:

where and are orthogonal matrices of dimensions and , respectively, and is a generalized lower-triangular matrix:

Namely, it has the following structure:

The diagonal entries of always have a unit product.6 Necessary and sufficient conditions for the existence of a GTD for a prescribed diagonal are known, along with explicit constructions of such a decomposition [18].

The following three important special cases of the GTD are well known; all of them are guaranteed to exist for a proper matrix .7

SVD (see, e.g., )

Here, the resulting matrix in is a diagonal matrix, and its diagonal elements are equal to the singular values of the decomposed matrix .

QR Decomposition (see, e.g., )

In this decomposition, the matrix in equals the identity matrix and hence does not depend on the matrix . This decomposition can be constructed by performing Gram–Schmidt orthonormalization on the (ordered) columns of the matrix .

GMD (see )

The diagonal elements of in this decomposition are all equal to the geometric mean of its singular values , which is real and positive.

3.2Joint Matrix Triangularization

Let and be two proper matrices of dimensions and , respectively. A joint triangularization of these two matrices is given by:

where , and are orthogonal matrices of dimensions , and , respectively, and and are generalized lower-triangular matrices of dimensions and , respectively.

It turns out that the existence of such a decomposition depends on the diagonal ratios . Necessary and sufficient conditions were given in [17]. Specifically, it was shown that there always exists a decomposition with unit ratios, i.e.,

Such a decomposition is coined the joint equi-diagonal triangularization (JET).8 Technically, the existence of JET is an extension of the existence of the (single-matrix) GMD.

Unfortunately, JET of more than two matrices is in general not possible [23]. Nonetheless, in Section 6 we present a way to overcome this obstacle.

4Background: Single-User MIMO Dirty-Paper Channel

In this section we review the (single-user) MIMO dirty-paper channel, corresponding to setting in :

We suppress the user index of x, s, and in this case.

For an i.i.d. Gaussian interference vector, a straightforward extension of Costa’s random binning scheme achieves the capacity of this channel,

which is, as in the scalar case, equal to the interference-free capacity. In the high-SNR limit, we have the following.

The Costa-style scheme for the MIMO dirty-paper channel suffers from two major drawbacks. First, it requires vector codebooks of dimension , which depend on the specific channel . And second, it does not admit an arbitrary interference. Both can be resolved by using the orthogonal matrix decompositions of Section 3 to reduce the coding task to that of coding for the scalar dirty-paper channel . For each scalar channel, the interference consists of two parts: a linear combination of the elements of the “physical interference” s and a linear combination of the off-diagonal elements of the triangular matrix which also serves as “self interference”. When using the lattice-based scheme of [3], the capacity is achieved even for an arbitrary interference sequence s.

As is well known, the zero-forcing (ZF) DPC scheme approaches capacity for proper channel matrices in the limit of high SNR. This is formally stated as a corollary of Lemma ?.

The ZF MIMO DPC scheme achieves a rate of

where the last equality follows from .

5Two-User MIMO DMAC

In this section we derive outer and inner bounds on the capacity region of the two-user MIMO DMAC . We show that the two coincide for proper channel matrices in the limit of high SNRs.

The following is a straightforward adaptation of the outer bound of [10] for the scalar case to the two-user MIMO setting . It is formally proved in the Appendix.

We next introduce an inner bound that approaches the upper bound in the limit of high SNRs.

We give a constructive proof, employing a scheme that uses the JET of Section 3.2 to translate the two-user MIMO DMAC into parallel SISO DMACs with equal channel gains (corresponding to equal diagonals). As explained in the introduction, this specific choice of decomposition is essential.

We use this scheme for the proof of the theorem.

The proposed scheme achieves any rate pair whose sum-rate is bounded from below by

where is the achievable rate of transmitter () over sub-channel (), follows from , and holds true due to .

By comparing Proposition ? with Theorem ? in the limit of high SNR, (recall Lemma ?), the following corollary follows.

6-User MIMO DMAC

In this section we extend the results obtained in Section 5 to MIMO DMACs with users.

The outer bound is a straightforward extension of the two-user case of Proposition ?.

For an inner bound, we would have liked to use a JET of matrices. As such a decomposition does not exist in general, we present a “workaround”, following [23].

We process jointly channel uses and consider them as one time-extended channel use. The corresponding time-extended channel is

where , , , are the time-extended vectors composed of “physical” (concatenated) output, input, interference and noise vectors, respectively. The corresponding time-extended matrix is a block-diagonal matrix whose blocks are all equal to :

where denotes the Kronecker product operation (see, e.g., [28]). As the following result shows, for such block-diagonal matrices we can achieve equal diagonals, up to edge effects that can be made arbitrarily small by utilizing a sufficient number of time extensions .

The value of in stems from a matrix truncation operation where we omit rows for which triangularization is not guaranteed; see, e.g., [23] for an example of the case , and general .

Since the ratio between and approaches 1 when goes to infinity:

the loss in rate due to the disparity between them goes to zero. This is stated formally in the following theorem.

Fix some large enough . Construct the channel matrices as in , and set , and according to Theorem ?. Now, over consecutive channel uses, apply the natural extension of the scheme of Section 5 to users, replacing with the obtained matrices. As in the proof of Theorem ?, we can attain any rate approaching

As we used the channel times, we need to divide these rates by ; by the proof is complete.

By comparing Proposition ? with Theorem ? in the limit of high SNR, we can extend Corollary ? as follows.

7Application to the MIMO Two-Way Relay Channel

In this section we apply the MIMO DMAC scheme of Section 5 to the MIMO two-way relay channel (TWRC). For Gaussian scalar channels, the similarity between these two settings was previously observed; see, e.g., [4].

As in the scalar DMAC setting , structured schemes based on lattice coding are known to be optimal in the limit of high SNR for the scalar TWRC [29]. This suggests in turn that, like in the MIMO DMAC setting, using the JET in conjunction with scalar lattice codes can achieve a similar result in the MIMO TWRC setting.

We start by introducing the MIMO TWRC model.

7.1Channel Model

The TWRC consists of two terminals and a relay. We define the channel model as follows. Transmission takes place in two phases, each one, without loss of generality, consisting of channel uses. At each time instant in the first phase, terminal () transmits a signal and the relay receives according to some memoryless multiple-access channel (MAC) . At each time instant in the second phase, the relay transmits a signal and terminal () receives according to some memoryless broadcast (BC) channel . Before transmission begins, terminal possesses an independent message of rate , unknown to the other nodes; at the end of the two transmission phases, each terminal should be able to decode, with arbitrarily low error probability, the message of the other terminal. The closure of all achievable rate pairs is the capacity region of the network.

In the Gaussian MIMO setting, terminal () has transmit antennas and the relay has receive antennas, during the MAC phase (see also Fig. ?):

where the channel matrices and the vectors z are defined as in . We assume that the channel matrices are proper.9 We denote by () the input covariance matrix used by terminal during transmission.

We shall concentrate on the symmetric setting:

The exact nature of the BC channel is not material in the context of this work. We characterize it by its common-message capacity .


Before treating the MIMO setting, we start by reviewing the structured physical network coding (PNC) approach for the scalar setting, in which and are replaced with scalar channel gains and .

7.2PNC for the Scalar Gaussian TWRC

In the scalar case, reduces to

In this case, the min-cut max-flow theorem [32] reduces to the minimum between the individual capacities over the MAC link and the common-message capacity of the BC link (see, e.g., [30], [34]):

In the (structured) PNC approach [29], both terminals transmit codewords generated from the same lattice code. Due to the linearity property of the lattice code, the sum of the two codewords is a valid lattice codeword. This sum is decoded at the relay and sent to the terminals. Each terminal then recovers the sum codeword and subtracts from it its own lattice codeword, to obtain the codeword transmitted by the other terminal. The rate achievable using this scheme is given by [29]:

As is shown in [30], the PNC rate is within half a bit from the cut-set bound and approaches the cut-set bound in the limit of high SNR. We next extend the asymptotic optimality of the PNC scheme to the MIMO setting.

We note that, in the scalar case, other strategies can also be used rather than PNC, yielding better performance at different parameter regimes. We address these strategies in Section 7.4 along with their extension to the MIMO setting.

7.3PNC for the Gaussian MIMO TWRC

The cut-set bound, in this case, is given by

where

are the individual capacities of the MIMO links, and the maximization is carried over all subject to the input power constraint .

The PNC approach can be extended to work for the MIMO case by applying the JET of Section 3.1 to the channel matrices and . The off-diagonal elements of the resulting triangular matrices are treated as interferences known at the transmitters. Specifically, the following rate is achievable:10

To see this, use the scheme for MIMO DMAC of Section 5 with the DMAC codes replaced with PNC ones: over each sub-channel, each of the users transmits a codeword from the (same) lattice, and the relay (who takes the role of the receiver in the scheme for the MIMO DMAC) recovers the sum codeword (modulo the coarse lattice) as in scalar PNC and continues as in the standard scalar PNC scheme. The achievable symmetric-rate is equal to the sum-rate achievable by the MIMO DMAC scheme , which is equal, in turn, to .

The achievability of implies that when the channel is “MAC-limited” (the bottleneck being the first phase), the capacity in the high SNR regime is achieved by applying the JET to the channel matrices and using PNC over the resulting scalar channels.

7.4Extensions and Comparisons

A different approach to the MIMO TWRC was proposed by Yang et al. [35]. In this approach, parallel scalar TWRCs are obtained via a joint matrix decomposition, but instead of the JET, the generalized singular value decomposition (GSVD) [36] is used, which may be seen as a different choice of joint matrix triangularization ; also, rather than DPC, successive interference cancellation (SIC) is used.

We next compare the approaches, followed by a numerical example; the reader is referred to [34] for further details.

  1. High-SNR optimality.

    In the symmetric case , both approaches achieve the high-SNR optimal rate .

  2. Finite-SNR improvements.

    At finite SNR, one can easily improve upon to get a slightly better rate, see Remark ?. As we show in [34] using the technique of [37], the finite-SNR rate of the GSVD-based scheme can also be slightly improved.11 Analytically comparing the finite-SNR performance of the schemes after the improvement is difficult, although numerical evidence suggests that the JET-based scheme has better finite-SNR performance in many cases (e.g., Example ?).

  3. Use of any strategy.

    For the scalar case, we note the following approaches, as an alternative to physical PNC. In the decode-and-forward (DF) approach, the relay decodes both messages with sum-rate . Instead of forwarding both messages, it can use a network-coding approach [39] and sum them bitwise modulo-2 (“xor them”). Then, each terminal can xor out its own message to obtain the desired one. In the compress-and-forward (CF) approach, the noisy sum of the signals transmitted by the sources is quantized at the relay, using remote Wyner–Ziv coding [40], with each terminal using its transmitted signal as decoder side-information. The CF and DF strategies both have an advantage over physical PNC for some parameters. Further, [41] presents various layered combinations of the CF and DF approaches, which yield some improvement over the two. Constructing an optimized scheme via time-sharing between these approaches is pursued in [34].

    Turning our attention to the MIMO setting, since the JET approach uses DPC, any strategy can be used over the subchannels; the decoder for each subchannel will receive an input signal as if this were the only channel. In contrast, the GSVD approach uses SIC, where the task of canceling inter-channel interference is left to the relay. In order to cancel out interference, the relay thus needs to decode; this prohibits the use of CF.

  4. Asymmetric case.

    The GSVD-based approach is suitable for any ratio of rates and powers between the users, and in fact it is optimal in the high-SNR limit for any such ratio. Since the JET-based PNC scheme uses the MIMO DMAC scheme of Section 5 for transmission during the MAC phase, it is limited by the minimum of the powers of the two terminals and cannot leverage excess power in case the transmit powers are different.12 To take advantage of the excess power in case of unequal transmit powers for this scheme, one can superimpose a DF strategy for the stronger user on top of the symmetric JET-based PNC scheme, as was proposed in [43].

Figure 1: Performance of the proposed strategies for {\bf H}_1 = \mathrm{diag}(1/4, 4), {\bf H}_2 = \mathrm{diag}(4, 1/4) and \mathrm{C}_\text{common}= 20 \text{bits}.
Figure 1: Performance of the proposed strategies for , and .

8Discussion: General Channel Matrices

In this paper we restricted attention to full rank channel matrices having more columns than rows. In this case, the column spaces of both matrices are equal. Indeed, the scheme and inner bound of Sections Section 5 and Section 6 can be extended to work for the general case as well; this requires, however, introducing an output projection at the receiver, which transforms the channel matrices to effective proper ones. Since all interferences need to be canceled out for the recovery of the transmitted messages, it seems that such a scheme would be optimal in the limit of large transmit powers . Unfortunately, the upper bound of Proposition ?, which is equal to the maximal individual capacity, is not tight in the non-proper matrix case, which calls for further research.

The proof of Corollary ? is a simple adaptation of the proof of the outer bound for the scalar case of [10].

Take s and s to be i.i.d. Gaussian with zero mean and scaled-identity covariance matrix . Further assume that both users wish to transmit the same common message , and denote the rate of this message by . Clearly, the supremum over all achievable rates bounds from the above the sum-capacity of the two-user MIMO DMAC.

By applying Fano’s inequality, we have

where as the error probability goes to zero and . By retracing (120)–(125) of [10] we attain

By recalling that s0 and using the Cauchy–Schwarz inequality, we have

where for .

By substituting in and the outcome in , and taking , we attain

By switching roles between the users, the following upper bound holds

and the desired result follows.

Footnotes

  1. In addition to , other inner bounds which are tighter in certain cases are derived in [10].
  2. We shall depart from this assumption later and treat the more general case of full-rank channel matrices where the number of receive antennas is larger or equal to that of the transmit antennas of each of the transmitters.
  3. The optimal covariance matrix in the limit of high SNR is white; see Lemma ? in the sequel.
  4. More precisely, RQ decompositions need to be applied to the channel matrices in this case.
  5. All vectors in this paper are assumed column vectors.
  6. Since is a proper matrix it is full rank by definition; thus, all the diagonal values of , , are non-zero.
  7. See [23] for a geometrical interpretation of these decompositions.
  8. See [23] for a geometrical interpretation of the JET.
  9. For a treatment of the case of non-proper channel matrices, see [31].
  10. Indeed, the same was proven using a different approach [35], see Section 7.4 in the sequel for a comparison.
  11. [38] also introduces an improvement to the GSVD-based scheme for certain special cases, but it is subsumed by the improvement in [34].
  12. This statement is precise when the SNRs are high and the interferences are strong. Otherwise, some improvements are possible [10].

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