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The Degrees of Freedom Regions of MIMO Broadcast, Interference, and Cognitive Radio Channels with No CSIT

Chinmay S. Vaze, Mahesh K. Varanasi

arXiv:0909.5424v3cs.IT

TL;DR

The paper characterizes DoF regions for several MIMO broadcast, interference, X, cognitive-radio, and multi-hop networks without CSIT. It obtains exact characterizations in broad cases and identifies when lacking CSIT causes DoF loss.

  • Problem

    Some interference-channel cases have inner and outer bounds that do not coincide, leaving their DoF regions unresolved.

  • Method

    The paper derives DoF-region characterizations and inner and outer bounds for multiple MIMO network classes, using general fading and noise distributions.

  • Results

    The K-user MIMO broadcast-channel DoF region is exactly characterized; two-user MIMO interference-channel and cognitive-radio bounds coincide except for a few cases, and several broader network classes are derived.

  • Takeaways & Limitations

    The resulting DoF regions reveal when a lack of CSIT causes DoF loss and support comparisons with perfect CSIT.

  • Takeaways & Limitations

    The analysis uses fading-channel and additive-noise distributions that subsume i.i.d. Rayleigh and isotropic fading as special cases.

Abstract

from arXiv · show

The degrees of freedom (DoF) regions are characterized for the multiple-input multiple-output (MIMO) broadcast channel (BC), interference channels (IC) (including X and multi-hop interference channels) and the cognitive radio channel (CRC), when there is perfect and no channel state information at the receivers and the transmitter(s) (CSIR and CSIT), respectively. For the K-user MIMO BC, the exact characterization of the DoF region is obtained, which shows that a simple time-division-based transmission scheme is DoF-region optimal. Using the techniques developed for the MIMO BC, the corresponding problems for the two-user MIMO IC and the CRC are addressed. For both of these channels, inner and outer bounds to the DoF region are obtained and are seen to coincide for a vast majority of the relative numbers of antennas at the four terminals, thereby characterizing DoF regions for all but a few cases. Finally, the DoF regions of the $K$-user MIMO IC, the CRC, and X networks are derived for certain classes of these networks, including the one where all transmitters have an equal number of antennas and so do all receivers. The results of this paper are derived for distributions of fading channel matrices and additive noises that are more general than those considered in other simultaneous related works. The DoF regions with and without CSIT are compared and conditions on the relative numbers of antennas at the terminals under which a lack of CSIT does, or does not, result in the loss of DoF are identified, thereby providing, on the one hand, simple and robust communication schemes that don't require CSIT but have the same DoF performance as their previously found CSIT counterparts, and on the other hand, identifying situations where CSI feedback to transmitters would provide gains that are significant enough that even the DoF performance could be improved.

I. INTRODUCTION

The paper studies how lack of CSIT affects DoF regions across several MIMO wireless networks. It exactly characterizes the K-user MIMO BC and derives bounds or exact regions for broad classes of interference, X, cognitive, and multi-hop networks.

  • Scope: The study examines no-CSIT DoF regions for K-user broadcast, two-user interference and cognitive radio channels, plus selected K-user interference, X, cognitive, and multi-hop networks.The considered channels allow arbitrary antenna configurations at four terminals in the two-user cases.
  • MIMO BC: The K-user MIMO BC has an exactly characterized DoF region achieved by simple time sharing.The proof matches a time-division weighted-sum inner bound with an outer bound obtained using genie-aided side information and mutual-information bounds.
  • Generality and CSIT comparison: The results apply to broader fading and noise models, including Rayleigh, Rician, correlated Rayleigh, non-Gaussian noise, isotropic fading, and temporally correlated channel matrices.The paper compares perfect- and no-CSIT DoF regions to identify antenna regimes where CSIT loss does or does not reduce DoF.
  • MIMO IC: For the two-user MIMO IC, inner and outer bounds coincide for all antenna quadruples except when min(M1, N1) > N2 > M2 or the symmetric condition holds.The inner bound uses time sharing and receive zero-forcing, while the outer bound adapts the BC bounding technique.
  • K-user networks: The K-user IC and X channel have identical no-CSIT DoF regions when all transmitters have equal antenna counts and all receivers have equal antenna counts.The paper also derives DoF regions for selected K-user cognitive and multi-hop interference networks.
  • Cognitive radio channels: For the K-user cognitive radio channel, DoF regions are derived when secondary transmitters have at least as many antennas as paired receivers or all transmitters have M and receivers N antennas with N > M.The two-user cognitive channel is characterized except for a specified antenna inequality involving min(N1, M1 + M2), N2, and M2.

2 N Transmitter

This section defines the K-user MIMO BC channel model, no-CSIT setting, channel and noise assumptions, and DoF as high-SNR rate slopes. It states that the BC DoF region is achieved by time division under the specified model class.

  • Channel Model: The K-user MIMO BC has M transmit antennas, user i has Ni receive antennas, receivers know channel realizations, and the transmitter knows only their distribution.The transmitted signal is independent of the actual channel realizations under no CSIT.
  • Channel Model: The AWGN assumption uses independent, identically distributed zero-mean complex Gaussian noises with identity covariance across time.The channel class D0(M, N̄) includes independent users, identically distributed channel rows, independent channel norms and directions, full rank almost surely, and finite row differential entropy.
  • Generalization: The channel-distribution result extends from D0(M, N̄) with AWGN to a much wider class of MIMO BCs.The included special cases encompass i.i.d. Rayleigh fading and the broader model assumptions stated in the section.
  • Definitions: The DoF region is the set of nonnegative high-SNR multiplexing gains associated with achievable rate tuples relative to log(P).Equivalently, DoF represent simultaneously accessible spatial signaling dimensions per channel use.

B. The DoF Region

The paper characterizes the no-CSIT DoF region of the K-user MIMO BC and proves that simple time division is optimal. It also identifies when no CSIT causes DoF loss and applies the result to related networks.

  • Theorem 1 characterizes the K-user MIMO BC DoF region under no CSIT.The region is established as the fundamental DoF region of the channel.
  • Time division achieves the region, while the converse proves it is also an outer bound.Thus, time division is DoF-region optimal.
  • The same DoF region applies with respective CSIR, where each receiver knows only its own channel matrices.Time division does not require receivers to know other users’ channel matrices, and the outer bound remains valid.
  • With perfect CSIT, the sum-DoF is min(M, P), whereas no CSIT gives min(M, max_i N_i).When M = P and N_i = 1 for all i, the sum-DoF collapses from M to 1.
  • Theorem 1 also supports DoF-region results for certain K-user MIMO interference, X, cognitive interference, and multi-hop interference networks.The multi-hop setting uses multiple relay layers to help transmit independent messages to respective receivers.
  • The proof combines channel enhancement, Fano’s inequality, and a key differential-entropy multiplexing-gain lemma to derive the converse.The lemma is applied successively to bounds on d_1, d_2, and subsequent users.

D. Generalizations of Theorem 1

The paper generalizes the no-CSIT MIMO BC result beyond its initial fading and noise assumptions. The same DoF region extends to broader channel distributions, colored Gaussian noise, correlated fading, and isotropic fading classes.

  • D1(M, N̄) includes distributions with full-rank channel matrices, finite row differential entropy, and exchangeable joint distributions of selected rows.The class strictly contains D0(M, N̄).
  • AWGN is a special case of ACGN, whose receiver noises are temporally i.i.d. complex Gaussian vectors with positive-definite covariance matrices.This broadens the noise model beyond white Gaussian noise.
  • Theorem 2 shows that the no-CSIT DoF region with ACGN and fading in D1(M, N̄) equals Theorem 1’s region.Achievability again follows from time division, while the outer bound is proved separately.
  • Examples: Rician fading belongs to D1(M, N̄), so its no-CSIT DoF region is given by Theorem 2 despite dependence between row norms and directions.The entries are modeled as i.i.d. CN(η, σ^2) variables with |η|^2 + σ^2 < ∞.
  • Examples: Correlated Rayleigh fading with separable correlations is covered by transforming the channel with a unitary or invertible receiver operation, yielding a D1 distribution and possibly colored noise.The resulting DoF region is therefore given by Theorem 2.
  • These general results apply to K ≥ 2 and broader channel and noise distributions than earlier results, but under the more restricted no-CSIT assumption.A prior two-user proof strategy cannot be extended to BCs in D2.

III. THE TWO-USER MIMO IC

This section studies the no-CSIT DoF region of a two-user MIMO interference channel with distributed transmitters and arbitrary antenna counts at its four terminals.

  • The problem is to characterize the no-CSIT DoF region of the two-user MIMO interference channel.The setting allows an arbitrary number of antennas at each of the four nodes.

A. Channel Model

The two-user MIMO interference channel has two transmitter-receiver pairs, with each transmitter carrying only its paired receiver’s message while also interfering at the unintended receiver.

  • Transmitters 1 and 2 have M_1 and M_2 antennas, while receivers 1 and 2 have N_1 and N_2 antennas.Each transmitter communicates with its corresponding receiver.
  • Each transmitted signal is received both as a desired signal at its paired receiver and as interference at the unintended receiver.The channel model includes additive noises and channel matrices linking both transmitters to both receivers.

Z Mˆ 22 H Transmitter 2 Receiver 2

The two-user MIMO IC is analyzed under perfect CSIR and no CSIT, yielding inner and outer DoF bounds and conditions for their agreement or separation from perfect-CSIT performance.

  • System model: The model has two transmitter–receiver pairs, with channel matrices known instantaneously at receivers but only statistically at transmitters.Transmit signals are independent of messages, channel matrices, and additive noises, under equal power constraints and AWGN.
  • Inner and outer bounds: The no-CSIT inner and outer bounds coincide for every antenna configuration except min(N1, M1) > N2 > M2 or its symmetric counterpart.The exceptional cases are identified by the relative antenna dimensions at the four terminals.
  • DoF separability: For i.i.d. fast fading, the region in equation (22) remains the DoF region, so the two-user MIMO IC is DoF-separable.The paper addresses whether time-varying fading enlarges the region and answers negatively.
  • Comparison with perfect CSIT: The no-CSIT DoF region equals the perfect-CSIT region if and only if N1 ≥ N2 ≥ M1 or N2 ≥ N1 ≥ M2.In these regimes, CSI-independent robust schemes attain the perfect-CSIT DoF region.
  • Comparison with perfect CSIT: When those antenna conditions fail, perfect CSIT strictly enlarges the DoF region; for M1 = M2 = 2N and N1 = N2 = N, sum-DoF is 2N versus N.The comparison is between perfect CSIT and no CSIT.

D. Proof of the Inner-Bound

The inner-bound proof constructs corner points by allocating streams without CSIT and letting receivers zero-force interference, then extends achievability to the full region by time sharing.

  • Corner-point achievability: Without CSIT, transmitters cannot use cross-channel null spaces, so streams from one transmitter generally create interference at the unintended receiver.This motivates receiver-side interference suppression rather than transmitter beamforming.
  • Corner-point achievability: To achieve d1 = min(M1, N1), receiver 1 zero-forces interference while receiver 2 decodes its useful signal from the remaining dimensions.The resulting stream allocation establishes corner point P1.
  • Corner-point achievability: The achievable second-user DoF at P1 is determined by receiver dimensions after accounting for desired and interference streams.The construction yields d2 = min{N2, min(M1,N1)+min(M2,(N1−M1)+)}−min(N2,N1,M1).
  • Region completion: The full inner region follows from achieving the two corner points and taking their time-sharing convex combinations.The proof also establishes achievability with the corresponding CSIR.
  • Outer-bound strategy: The outer-bound proof enhances the channel, applies Fano’s inequality, and reduces part of the analysis to a broadcast channel with M1 transmit antennas.A statistical-equivalence lemma then controls weighted entropy terms before the weighted-sum bound is completed.

F. Discussion of the Cases where the Inner and Outer Bounds Do Not Coincide

The paper examines exceptional antenna regimes where the stated IC bounds differ, then explains related CRC results and identifies when cognition or CSIT changes the DoF region.

  • IC exceptions: The exceptional IC regime min(M1,N1) > N2 > M2 requires interference suppression that would exploit a cross-channel null space, unavailable without CSIT.The discussion concerns why the inner and outer bounds do not initially coincide there.
  • IC exceptions: Later work showed the IC outer bound is loose in these cases and that the no-CSIT DoF region equals the inner bound.The passage attributes this result to reference [37].
  • IC exceptions: Blind interference alignment can achieve the exceptional point without CSIT under staggered block fading, outside Theorem 5’s model.The construction uses channel null spaces that remain constant during the coherence period.
  • Cognitive radio channel: For the CRC, the cognitive transmitter noncausally knows the primary message and can aid primary transmission while sending its own message.The CRC otherwise has the same input-output relationship as the MIMO IC.
  • Cognitive radio channel: The CRC inner and outer bounds coincide except when min(N1,M1+M2) > N2 > M2.The exact region is therefore characterized outside this antenna regime.
  • Cognitive radio channel: Cognition is useful under no CSIT only when N1 > M1; when N1 ≤ M1, the IC and CRC inner bounds remain equal.With perfect CSIT, the CRC region can nevertheless be strictly larger in the latter case.

C. Proof of the Inner-Bound

The CRC inner-bound proof uses its message-sharing structure to derive corner points and handles the outer bound through broadcast-channel reductions for the two receiver-ordering cases.

  • Inner-bound achievability: Because the cognitive transmitter knows the primary message, the primary receiver can achieve up to d1 = min(N1, M1 + M2).Both transmitters can contribute to the primary message under the CRC model.
  • Inner-bound achievability: At the maximum primary DoF point, the proof sets the cognitive-receiver DoF to d2 = 0 because the primary receiver’s signal space is fully used.Depending on antenna dimensions, either the cognitive transmitter uses all M2 streams or the primary receiver uses all N1 dimensions.
  • Inner-bound achievability: The second corner point matches the corresponding IC corner point when d2 = min(M2, N2).The proof then bounds the primary-user DoF achievable alongside the cognitive user’s maximum DoF.
  • Outer-bound proof: The outer-bound proof treats N1 ≥ N2 and N2 > N1 separately, applying Fano’s inequality and channel-enhancement arguments analogous to those for the IC.For N2 > N1, conditioning on the primary message reduces the relevant step to a broadcast channel with the cognitive transmitter.
  • Outer-bound proof: When conditioning on the primary message, the cognitive transmitter’s signal remains message-dependent, so the proof pools both transmitters into a BC with M1 + M2 antennas.This reduction supports the required entropy inequality and outer bound.

E. Discussion of the Case where the Inner and Outer Bounds Do Not Coincide

For the K-user MIMO IC, the paper gives an outer bound and fully characterizes the no-CSIT DoF region under specified antenna conditions. It also discusses a CRC case where inner and outer bounds remain separated and conjectures that exploiting a channel null space may be necessary to attain intermediate points.

  • CRC bounds do not coincide: The CRC example with (M1, N1, M2, N2) = (3, 5, 2, 4) satisfies min(N1, M1 + M2) > N2 > M2.Its weighted-sum outer bound passes through (2.5, 2), while the inner bound passes through (2, 2).
  • CRC bounds do not coincide: Achieving points on the segment beyond P2 is conjectured to require exploiting the channel matrix’s null space at the second receiver.The discussion identifies this as a possible mechanism for closing the gap between the bounds.
  • K-user MIMO IC: For the K-user MIMO IC, an overall BC outer bound follows by allowing all transmitters to cooperate perfectly.The resulting broadcast channel has total transmit antennas Mtot, yielding per-user limits based on min(Mi, Ni) and the broadcast-channel constraints.
  • K-user MIMO IC: The no-CSIT DoF region of the K-user MIMO IC is exactly characterized when the stated antenna conditions of Theorem 9 hold.The proof uses time division in one case and receive zero-forcing with time sharing in another.
  • Implications of no CSIT: The theorem completely characterizes the equal-antenna class, with equal numbers of antennas at all transmitters and equal numbers at all receivers.In the equal-antenna SISO specialization, perfect CSIT can achieve K/2 sum-DoF, whereas no CSIT limits the sum-DoF to 1.
  • Implications of no CSIT: When M = N, the comparison reports a sum-DoF collapse from MK with perfect CSIT to M without CSIT.This identifies a setting where transmitter CSI can improve even the DoF performance.

B. The K-User CRC

The K-user CRC and X channel DoF regions are characterized under no CSIT for several antenna configurations, with exact results in important classes and broad conclusions about CSIT loss.

  • The K-User CRC: The K-user MIMO CRC is defined as a K-user MIMO interference channel with the first transmitter-receiver pair cognitive.
  • The K-User CRC: An outer bound for the K-user CRC constrains the primary DoF by min(Mtot, N1) and each other user by min(Mi, Ni).
  • The K-User CRC: The outer bound is tight in specified cases, and the resulting region can be achieved by time division.
  • The K-User CRC: The characterization is complete when all transmitters and receivers have equal antenna counts.
  • The MIMO X Channel: For the no-CSIT MIMO X channel with equal transmitter and receiver antenna counts, the DoF region is characterized exactly.
  • Conclusion: The paper establishes no-CSIT DoF regions for broad classes of MIMO interference, X, cognitive, and multi-hop networks under general fading assumptions.
  • Conclusion: Comparisons identify antenna configurations where lacking CSIT causes DoF loss and where CSI-independent schemes match CSI-dependent performance.
  • Conclusion: The CRC DoF region with partial CSIT remains an open direction beyond the no-CSIT results.

APPENDIX A

The appendix studies multi-hop interference channels when relays lack channel knowledge. It shows that relay CSI is decisive for achieving the larger perfect-CSI DoF in the two-hop case and extends the result more broadly.

  • APPENDIX A: The appendix considers an n-hop interference network with K transmitter-receiver pairs separated by n−1 relay layers.
  • Two-Hop IC: For a two-hop IC, receivers know both hop channel matrices perfectly and instantaneously, while relays know only their distributions.
  • Two-Hop IC: The two-hop IC DoF region is given independently of the relays’ knowledge of the channel matrices under the stated assumptions.
  • Proof: Time division achieves the region, while a broadcast-channel outer bound supplies d1 + d2 ≤ M.
  • Implication: Perfect CSI at transmitters cannot improve the DoF region unless relays know the final-hop channel matrices.
  • Implication: For M = 1, achieving 2 sum-DoF depends critically on relays having perfect knowledge of the final-hop channel matrices.
  • Generalization: The argument extends to n-hop K-user networks whose last relay layer lacks knowledge of the last-hop channel matrices.

THE TWO-USER MIMO IC IS SEPARABLE IN THE DOF SENSE

The section proves DoF separability for the two-user MIMO interference channel under broad fading conditions by deriving the required sum-DoF bounds through statistical and matrix arguments.

  • DoF Separability: The perfect-CSIT DoF region of the two-user IC with i.i.d. fast fading is established as the starting point for the separability proof.
  • Proof Strategy: It suffices to prove the sum-DoF bound d1 + d2 ≤ max(N1, M2), treating separately N1 ≥ M2 and the remaining case.
  • Proof Strategy: When N1 ≥ M2, the proof obtains d1 + d2 ≤ N1 using Fano’s inequality and statistical-equivalence arguments.
  • Proof Strategy: The remaining case M2 > N1 is handled using the corresponding corollary to establish the required sum-DoF bound.
  • General Fading: For general fading distributions, auxiliary transformed outputs preserve the inequalities needed for Steps I and II, after which Step III completes the proof.
  • General Fading: For the D1 case, statistical equivalence of the constructed outputs validates the key inequality and completes that case.
  • General Fading: For the D2 case, singular-value-based constructions ensure the required diagonal inequalities and statistical equivalence.
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