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Interference Alignment for the MIMO Interference Channel with Delayed Local CSIT

Akbar Ghasemi, Abolfazl Seyed Motahari, Amir Keyvan Khandani

arXiv:1102.5673v1cs.ITmath.IT

TL;DR

Because CSIT is hard to obtain in practice, this paper studies interference channels with delayed local CSIT. It develops two-user MIMO achievable DoF regions and examines how sum-DoF compares across CSIT regimes.

  • Problem

    Because CSIT is hard to obtain in practice, the paper considers interference alignment with no or partial CSIT, using degrees of freedom as the performance measure.

    Source: §I · p. 2

  • Method

    The paper develops achievable DoF regions for the two-user MIMO interference channel and studies the K-user MISO interference channel with delayed local CSIT.

    Source: §I · p. 2, D. Discussions · pp. 7–8

  • Results

    For two-user MIMO antenna class C4, full-CSIT sum-DoF can exceed delayed-local-CSIT sum-DoF, which exceeds no-CSIT sum-DoF; in other classes, sum-DoF does not change with CSIT knowledge.

    Source: D. Discussions · pp. 9–10

  • Takeaways & Limitations

    The achievable DoF region is tight for two-user MIMO subclasses C61 and C62.

    Source: D. Discussions · p. 8

Abstract

from arXiv · show

We consider the MIMO (multiple-input multiple-output) Gaussian interference channel with i.i.d. fading across antennas and channel uses and with the delayed local channel state information at the transmitters (CSIT). For the two-user case, achievability results for the degrees of freedom (DoF) region of this channel are provided. We also prove the tightness of our achievable DoF region for some antenna configurations. Interestingly, there are some cases in which the DoF region with delayed local CSIT is identical to the DoF region with perfect CSIT and that is strictly larger than the DoF region with no CSIT. We then consider the $K$-user MISO (multiple-input single-output) IC and show that the degrees of freedom of this channel could be greater than one with delayed local CSIT.

I. INTRODUCTION

The paper studies interference alignment with delayed local CSIT, deriving achievable DoF-region inner bounds for two-user MIMO interference channels and identifying antenna configurations where those bounds are tight. It also bounds the DoF of the K-user MISO channel, including a sum-DoF gain when M ≥ K.

I. INTRODUCTION

  • Because instantaneous perfect CSIT is hard to obtain, the paper studies delayed local CSIT as a partial-CSIT alternative using degrees of freedom as the performance measure.Source: §I · p. 2

A. The Two-User MIMO Gaussian IC

  • The delayed local CSIT model assumes i.i.d. Rayleigh fading, instantaneous perfect own-channel knowledge at each receiver, and perfect own-channel knowledge at each transmitter with one-unit delay.Each transmitter has delayed knowledge of its own channel matrices, rather than global channel knowledge.Source: §I · p. 2

I. INTRODUCTION

  • For the K-user MISO IC, when M ≥ K, delayed local CSIT achieves sum-DoF strictly greater than one, versus one with no CSIT [3].The paper presents this as evidence that delayed local CSIT can improve sum-DoF for K > 2.Source: §I · pp. 2–3

C. Main Results

  • Theorem 1 gives an achievable DoF-region inner bound for the two-user MIMO IC with delayed local CSIT, except for subclass S of C6; the scheme is tight for some antenna configurations.The inner bound applies to all antenna configurations outside the stated subclass.Source: §I · p. 2, C. Main Results · p. 5
  • For subclass S1, Theorem 2 provides an inner bound on the two-user delayed-local-CSIT DoF region.The supplied passages do not include the region’s full defining expression.Source: C. Main Results · p. 6
  • For subclass S2, Theorem 3 states that the DoF region contains a pentagon.The supplied text identifies corner points (0, 0), (M1, 0), and (0, N2), but truncates the remaining coordinates.Source: C. Main Results · p. 6
  • Theorem 4 establishes tightness of the achievable regions from Theorems 1–3 for the displayed antenna configurations d) and e).The conditions are M1 ≤ ∆ < N1 ≤ N2 < N1 + N2 − M1 < M2 and M1 ≤ ∆′ < N1 ≤ N2 < M2 ≤ N1 + N2 − M1.Source: C. Main Results · p. 6
  • For the K-user MISO interference channel with at least K antennas per transmitter and delayed local CSIT, Theorem 5 gives lower and upper bounds on DoF.The numerical bound expressions are truncated in the supplied text.Source: C. Main Results · p. 6

D. Discussions

The achievable two-user DoF region is tight for several antenna configurations, while its relation to perfect- and no-CSIT regions varies by class. The discussion also identifies a CSIT-sensitive exception in sum-DoF and extends the results to the K-user MISO channel.

D. Discussions

  • In one antenna class, delayed-local-CSIT achieves a region strictly smaller than perfect-CSIT and no larger than no-CSIT.The discussion identifies this as the only class with that comparison.Source: D. Discussions · p. 7
  • The achievable DoF region is an exact characterization for class C1, subclasses of C3 and C4, and subclasses C61 and C62 of C6.For C61 and C62, the achievable region equals the perfect-CSIT region.Source: D. Discussions · pp. 8–9
  • For class C4, sum-DoF is strictly ordered as full CSIT, delayed local CSIT, then no CSIT; outside C4, it does not change with CSIT knowledge.When min(M1, M2) ≥ N1 + N2, the no-CSIT sum-DoF is max(N1, N2), while delayed local CSIT has a larger value.Source: D. Discussions · pp. 9–10
  • Figure 3 presents the DoF region for the two-user MIMO IC with M1 = M2 = 2 and N1 = N2 = 1 under different assumptions about CSIT information.Source: Figure 3 · p. 10
  • For the K-user MISO IC with M ≥ K antennas per transmitter, the extended scheme achieves sum-DoF strictly greater than one.The paper also develops an upper bound for this setting.Source: D. Discussions · p. 10

III. PROOF OF THEOREM 1

The proof illustrates delayed local CSIT with a two-user channel where each transmitter has two antennas and each receiver one. The scheme achieves (2/3, 2/3) and yields a DoF region strictly larger than with no CSIT, but smaller than with perfect CSIT.

A. An illustrative example

  • Delayed local CSIT gives a DoF region strictly larger than the no-CSIT region and strictly smaller than the perfect-CSIT region in this example.The no-CSIT region is the time-division region, while the perfect-CSIT region is the unit square.Source: A. An illustrative example · p. 10
  • After one channel use, each transmitter knows the other receiver’s interference term and the scheme delivers both interference terms to each receiver for decoding.Each transmitter obtains the other user’s interference term with a unit delay; the delivery phase uses time division.Source: A. An illustrative example · p. 11
  • The DoF pair (2/3, 2/3) is achieved when each transmitter sends two independent symbols over three channel uses.Time sharing with the points (1, 0) and (0, 1) extends the achievable region.Source: A. An illustrative example · p. 11
  • The achievable region is DoF-optimal for this example because it matches the outer bound obtained by allowing the transmitters to cooperate.The resulting broadcast-channel region with delayed CSIT equals the achievable region.Source: A. An illustrative example · p. 11

B. Proof of Achievability for General setting

The general achievability scheme uses two transmission phases to manage interference under delayed local CSIT. Counting equations is not always enough to ensure decoding; the rank conditions must also hold.

B. Proof of Achievability for General setting

  • In phase one, each transmitter sends independent symbols intended for its corresponding receiver over its allocated channel uses.The phase durations may differ between transmitters.Source: B. Proof of Achievability for General… · p. 11
  • In phase two, no new information is sent; transmitters send linear combinations of interference previously observed at non-intended receivers.Delayed local channel information gives a transmitter access to its generated interference terms in later slots.Source: B. Proof of Achievability for General… · p. 12
  • Having at least as many equations as unknowns does not always guarantee decoding, so achievability also requires checking the rank of each receiver's coefficient matrix.The scheme evaluates the rank of P[i] for receiver i and checks rank conditions for the candidate corner points.Source: B. Proof of Achievability for General… · pp. 13–14
  • For M1=2, M2=6, N1=3, and N2=4, the proposed phase durations satisfy the equation-count inequalities, yet receiver two cannot resolve its intended symbols.Its equations form an under-determined subsystem, demonstrating that conditions (6) and (7) alone are insufficient in general.Source: B. Proof of Achievability for General… · p. 13
  • The proof checks the rank conditions for corner points across classes C3, C4, C5, C61, C62, and C63.The corresponding class-specific arguments are assigned to appendices B through F.Source: B. Proof of Achievability for General… · p. 14

IV. PROOF OF THEOREM 2

For subclass S of class C6, the proof uses two-phase transmission schemes to establish achievability of corner points. The decoding argument checks that receivers can resolve the intended symbols and interference terms.

IV. PROOF OF THEOREM 2

  • For subclass S of class C6, corner point T5 is achievable using W=M2−N1 channel uses split into phases of N2−N1 and M2−N2 slots.Transmitter two sends previously observed interference terms in phase two; receiver two's equations must be linearly independent.Source: §IV · pp. 14–15
  • For subclass C62, Theorem 1 establishes achievability of corner point T7.The proof invokes this result to complete the corresponding corner-point argument.Source: §IV · p. 14
  • For subclass S1, corner point T8 is achieved with W=N1−λ channel uses, split into N1−M1 and M1−λ slots.The decoding check establishes that receiver two can resolve its desired symbols using equations from both phases.Source: §IV · pp. 15–16

V. PROOF OF THEOREM 3

The proof establishes achievability for selected corner points and compares the achievable DoF region with outer bounds. The comparisons show that tightness depends on the antenna configuration.

V. PROOF OF THEOREM 3

  • Theorem 3's proof establishes achievability of corner points T7 and T9; for T9, the two-phase scheme lets receiver two resolve its desired symbols.For T9, phase one lasts N2−M1 slots and phase two lasts N1 slots.Source: §V · p. 16, §1 · p. 17

VI. PROOF OF THEOREM 4

  • For another class, the intersection of broadcast and perfect-CSIT outer bounds is tighter than either bound, while the achievable region is strictly smaller.The text states that no specific configuration in this class makes the achievable region coincide with the outer bound.Source: §VI · p. 18
  • For some antenna configurations with min(M1,M2)≥N1+N2, the achievable region is tight and lies strictly between the no-CSIT and perfect-CSIT regions.The proof also identifies configurations in which delayed CSIT matches both regions.Source: §VI · p. 19
  • For subclasses C61 and C62, the perfect-CSIT outer bound is tighter than the broadcast outer bound and coincides with the achievable DoF region.Source: §VI · p. 19

VII. PROOF OF THEOREM 5

The proof bounds the sum-DoF of the K-user MISO interference channel with delayed local CSIT and shows it can exceed one. It uses a two-phase scheme to achieve a sum-DoF result when each transmitter has at least K antennas.

VII. PROOF OF THEOREM 5

  • The proof also establishes an upper bound on sum-DoF by allowing cooperation among K−1 transmitters and their corresponding receivers, reducing the problem to a two-user MIMO IC.The resulting channel's sum-DoF upper-bounds that of the original K-user channel.Source: §VII · p. 21
  • With delayed local CSIT and M≥K antennas per transmitter, the scheme achieves a sum-DoF greater than one for the K-user MISO interference channel.Phase one sends information symbols; phase two retransmits interference terms using delayed local CSIT, allowing receivers to resolve their intended symbols.Source: §VII · pp. 21–22

VIII. CONCLUSION

  • The conclusion reports lower and upper bounds on sum-DoF for the K-user MISO Gaussian IC with delayed local CSIT when M≥K.It also summarizes new two-user MIMO Gaussian IC DoF-region results and tightness in some cases.Source: §VIII · p. 22

APPENDIX A

Appendix A derives rank P[i] by modeling receiver i’s equations as a coefficient matrix, partitioning it into phase-related blocks, and combining their ranks.

APPENDIX A

  • P[i] is the coefficient matrix for receiver i’s NiW equations in the transmitted symbols and effective interference vector.The interference vector is formed from the other transmitter’s channel and precoding quantities.Source: DERIVATION OF RANK OF MATRIX P[i] · p. 22
  • The matrix is partitioned into six random submatrices, with separate blocks corresponding to information symbols and effective-interference quantities across transmission phases.The block definitions distinguish first-phase transmissions from later retransmissions of interference quantities.Source: DERIVATION OF RANK OF MATRIX P[i] · pp. 22–24
  • Independent random matrices multiply to a matrix whose rank is the minimum of their ranks almost surely.The appendix applies this fact to the block rank calculations, including a diagonal channel block that has the stated rank almost surely.Source: DERIVATION OF RANK OF MATRIX P[i] · pp. 23–24
  • rank P[i] is computed as r[i]1 + r[i]2 + r[i]3, the sum of the ranks of its three vertically concatenated blocks.The appendix defines the intermediate rank terms and identifies their sum as the desired result.Source: DERIVATION OF RANK OF MATRIX P[i] · p. 26

APPENDIX B

Appendix B verifies the rank conditions for class C3 by checking the receiver equations under the two orderings of the transmission durations.

PROOF OF ACHIEVABILITY FOR CLASS C3

  • For one duration ordering, the rank requirement is met at receiver one with r[1]1 + r[1]2 + r[1]3 = N1W.The proof also checks receiver two’s rank requirement for this ordering.Source: PROOF OF ACHIEVABILITY FOR CLASS C3 · p. 27
  • The receiver-two checks use the class assumptions and inequalities comparing the weighted transmission durations with N2(W − W1) or N2(W − W2).The proof states these inequalities follow from its preceding conditions.Source: PROOF OF ACHIEVABILITY FOR CLASS C3 · pp. 27–28
  • For the other duration ordering, receiver one’s rank sum equals N1W and receiver two’s rank condition is also satisfied.The proof checks the corresponding receiver-one equality and receiver-two condition under the stated antenna assumptions.Source: PROOF OF ACHIEVABILITY FOR CLASS C3 · p. 28

APPENDIX C

Appendix C verifies the rank conditions for class C4 using selected transmission durations, concluding that the conditions hold at both receivers.

APPENDIX C

  • For class C4, the proof chooses W = M′1M′2 − N1N2 and W2 = N2(M′1 − N1) to satisfy the rank conditions.These duration choices are presented as the construction for the achievable point under consideration.Source: PROOF OF ACHIEVABILITY FOR CLASS C4 · p. 28
  • The proof checks an inequality involving N1W2 + N2W1 and N2(W − Wimax) to establish the required rank bound.It states that the inequality proved is stronger than the one needed.Source: PROOF OF ACHIEVABILITY FOR CLASS C4 · p. 29
  • With these checks, the rank condition is met at both receivers, completing the class C4 proof.The proof gives r[i]1 + r[i]2 + r[i]3 = NiW for i = 1, 2.Source: PROOF OF ACHIEVABILITY FOR CLASS C4 · p. 29

APPENDIX D

Appendix D verifies rank conditions for class C5 and presents constructions for point T4 in subclasses C61 and C62.

PROOF OF ACHIEVABILITY FOR CLASS C5

  • For class C5, the proof chooses W1 = N1 and W2 = N1 − M1, then verifies the rank condition for each receiver.The receiver-one rank sum equals N1^2; the proof separately establishes receiver two’s condition.Source: PROOF OF ACHIEVABILITY FOR CLASS C5 · pp. 29–30

APPENDIX E

APPENDIX F

Appendix F verifies achievability constructions by checking the rank conditions at both receivers. It gives distinct choices of W, W1, and W2 for points T5 and T6.

APPENDIX F

  • For point T5, W = M1(M2 − N1), W1 = N1(M2 − N2), and W2 = M1(N2 − N1) satisfy the rank conditions at both receivers.The proof states that the rank condition is met for receiver one and receiver two.Source: Proof of Achievability for point T5 · pp. 32–33
  • The rank checks use antenna assumptions and inequalities involving W, W1, and W2, including M1 < N1 < N2 < M2 and W − W1 < W2 ≤ W1.These conditions are invoked in proving the relevant inequalities for the rank checks.Source: Proof of Achievability for point T5 · p. 32
  • For point T6, the construction W = W1 = N1 and W2 = N1 − M1 is used to satisfy the rank conditions.The proof considers W1 > W2 and sets imax = 1 and imin = 2 when substituting in the rank conditions.Source: Proof of Achievability for point T5 · p. 34

1. On the other hand M1W1 +

This proof continuation verifies rank conditions by treating separate cases and bounding the receiver-two expression under an antenna constraint.

1. On the other hand M1W1 +

  • For receiver one, the passage states that the rank condition is verified and concludes the proof.Source: §1 · pp. 34–35
  • The proof distinguishes cases after establishing N1 < N2 < M2 and uses M1 > ∆′ in a subsequent bound.The excerpt also gives r[2]1 + r[2]2 = M1^2 + N1(N1 − M1) + N2(N1 − M1).Source: §1 · p. 35
  • The receiver-two bound uses M2 < N1 + N2 − M1 to show M2W2 + min(M1, N2)W1 < r[2]1 + r[2]2.The passage states that the proof is complete for this case.Source: §1 · p. 35
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