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The Degrees of Freedom Region and Interference Alignment for the MIMO Interference Channel with Delayed CSI

Chinmay S. Vaze, Mahesh K. Varanasi

arXiv:1101.5809v2cs.IT

TL;DR

The paper studies how to characterize the degrees-of-freedom region of general MIMO interference channels with delayed channel-state information. It derives an outer bound and develops interference-alignment schemes by antenna configuration, showing that the resulting regions coincide for all possible antenna quadruples.

  • Problem

    Characterizing the degrees-of-freedom regions of general MIMO interference channels remains an open question.

  • Method

    The paper derives an outer bound and develops interference-alignment achievability schemes for classes defined by the relative numbers of antennas, using only delayed channel-state information.

  • Results

    The achievable and outer-bound degrees-of-freedom regions coincide for all possible values of the four-terminal antenna tuple.

  • Takeaways & Limitations

    The fundamental degrees-of-freedom region is obtained for the general MIMO interference channel under delayed channel-state information.

Abstract

from arXiv · show

The degrees of freedom (DoF) region of the 2-user multiple-antenna or MIMO (multiple-input, multiple-output) interference channel (IC) is studied under fast fading and the assumption of {\em delayed} channel state information (CSI) wherein all terminals know all (or certain) channel matrices perfectly, but with a delay, and each receiver in addition knows its own incoming channels instantaneously. The general MIMO IC is considered with an arbitrary number of antennas at each of the four terminals. Dividing it into several classes depending on the relation between the numbers of antennas at the four terminals, the fundamental DoF regions are characterized under the delayed CSI assumption for {\em all} possible values of number of antennas at the four terminals. In particular, an outer bound on the DoF region of the general MIMO IC is derived. This bound is then shown to be tight for all MIMO ICs by developing interference alignment based achievability schemes for each class. A comparison of these DoF regions under the delayed CSI assumption is made with those of the idealistic `perfect CSI' assumption where perfect and instantaneous CSI is available at all terminals on the one hand and with the DoF regions of the conservative `no CSI' assumption on the other, where CSI is available at the receivers but not at all at the transmitters.

I. INTRODUCTION

The paper characterizes the DoF region of the general four-antenna-parameter MIMO interference channel with delayed CSI. It derives an outer bound and matching interference-alignment schemes across antenna configurations, then compares delayed CSI with perfect and no CSI.

  • Results: Delayed CSI can outperform no CSI, including the (1, 1.5) DoF pair for the (3, 1, 4, 2) MIMO IC, which lies outside the no-CSI DoF region.The paper establishes this pair as lying on the boundary of the DoF region.
  • Contribution: The study characterizes the DoF region of the general (M1, M2, N1, N2) MIMO IC under delayed CSI.The channel has Mi antennas at transmitter i and Ni antennas at receiver i for i ∈ {1, 2}.
  • Contribution: An outer bound is derived for the general MIMO IC, and interference-alignment schemes achieve it across classes defined by the relative antenna numbers.The resulting DoF regions coincide with the outer bound for all classes.
  • Results: For a class of MIMO ICs, delayed CSI achieves the entire perfect-CSI DoF region even though the no-CSI region is strictly smaller than the perfect-CSI region.This identifies a class where delayed CSI closes the gap to perfect CSI.
  • Comparison: The paper gives a comparative classification according to whether the no-CSI, delayed-CSI, and perfect-CSI DoF regions are strictly contained or equal.The comparison covers the idealized perfect-CSI and conservative no-CSI assumptions.

II. THE CHANNEL MODEL · III. THE DOF REGION OF THE IC WITH DELAYED CSI

The paper specifies a fast-fading MIMO interference channel with delayed CSI and characterizes its DoF region for every antenna configuration, including variants with delayed transmitter or cross-channel CSI.

  • II. THE CHANNEL MODEL: Each transmitter sends an independent message to its paired receiver, while its signal also reaches the unintended receiver as interference.The channel uses transmitter antenna counts M_i, receiver antenna counts N_i, additive Gaussian noise, and per-transmitter power constraint P.
  • II. THE CHANNEL MODEL: The model assumes i.i.d. Rayleigh fading across channel entries and time, with channel and noise realizations independent of each other.All channel entries are zero-mean, unit-variance complex normal random variables.
  • II. THE CHANNEL MODEL: Under delayed CSI, each receiver knows its incoming channel matrices instantaneously, while all other terminals learn them perfectly one time unit later.The paper also defines perfect CSI and no CSI, with their DoF regions satisfying Dno−CSI ⊆Dd−CSI ⊆Dp−CSI.
  • III. THE DOF REGION OF THE IC WITH DELAYED CSI: The delayed-CSI DoF region is bounded by individual-user limits and five weighted-sum constraints, including L1 and L2.The stated bounds include L1 ≡ d1 min(N1 + N2, M1) + d2 min(N2, M1) ≤min(N2, M1 + M2) and its symmetric counterpart L2.
  • III. THE DOF REGION OF THE IC WITH DELAYED CSI: Theorem 1 establishes an outer bound for the MIMO IC with delayed CSI, using point-to-point limits, perfect-CSI bounds, and additional delayed-CSI inequalities.The proof exploits symmetry between L1 and L2 and between L4 and L5, reducing the required arguments to L1 and L4.
  • III. THE DOF REGION OF THE IC WITH DELAYED CSI: Theorem 2 states that the outer bound is tight for all possible values of (M1, M2, N1, N2).Achievability is developed across three main cases subdivided into seven cases, with some configurations satisfying Dno−CSI = Dd−CSI and others requiring new alignment schemes.
  • III. THE DOF REGION OF THE IC WITH DELAYED CSI: With delayed CSI at the transmitters but perfect instantaneous CSI at the receivers, the corresponding DoF region is also characterized by the stated outer bound.The outer-bound inequalities remain applicable because bounds L1 and L4 were derived assuming receivers have perfect and instantaneous CSI.
  • III. THE DOF REGION OF THE IC WITH DELAYED CSI: With delayed CSI only for cross-channel matrices at the transmitters, the same outer bound remains achievable and characterizes Dd−CSI−c.The achievability schemes use only delayed cross-channel CSI at the transmitters, so they apply directly under this reduced-information assumption.

A. Summary of Results · B. Comparison of the DoF Regions with No, Delayed, and Perfect CSI

The paper characterizes delayed-CSI DoF regions across antenna configurations by identifying active outer-bound inequalities, then compares them with no-CSI and perfect-CSI regions. The comparisons show when delayed CSI is unnecessary and when the three regions differ.

  • A. Summary of Results: The delayed-CSI DoF-region summary covers all antenna cases through active bounds that determine the region’s shape.The analysis assumes N1 ≥ N2 without loss of generality, with the opposite ordering obtained by switching users.
  • A. Summary of Results: Under Case A.I, M2 ≥ N1 and bounds L{1,2} are generally active; specifically, only L1 is active under Case A.I.1.These case distinctions determine which inequalities remain essential in the DoF-region description.
  • A. Summary of Results: Under Cases A and B, d1 = min(M1, N1) implies d2 = 0, while under Case A, d2 = min(M2, N2) implies d1 = 0.The second implication does not hold under Case B.
  • B. Comparison of the DoF Regions with No, Delayed, and Perfect CSI: The delayed-CSI region equals the perfect-CSI region in Cases 0, A.I.2, B.0, and B.II.1; only in B.II.1 does it differ from the no-CSI region.The perfect-CSI DoF region is identified with Dd−CSI.
  • B. Comparison of the DoF Regions with No, Delayed, and Perfect CSI: Under Cases 0, A.I.2, and B.0, the no-CSI region equals the delayed-CSI region because perfect-CSI performance is achievable without CSI.Thus delayed CSI provides no additional DoF over no CSI in these cases.
  • B. Comparison of the DoF Regions with No, Delayed, and Perfect CSI: Under Cases A.I.1 and A.II.1, delayed CSI cannot improve the no-CSI DoF region, although the no-CSI and perfect-CSI regions differ.These cases satisfy N1 ≥ M1 > N2 and M2 ≥ N2.

IV. PROOF OF THEOREM 1: L1 IS AN OUTER-BOUND · A. Proof of Bound L1 · B. Proof of Lemma 1

The section proves that L1 is an outer bound for the delayed-CSI MIMO interference channel by combining an entropy inequality with an enhanced-channel argument. The proof of Lemma 1 uses statistical equivalence among receive-antenna outputs and bounds residual noise terms within o(log2 P).

  • IV. PROOF OF THEOREM 1: L1 IS AN OUTER-BOUND: The proof introduces m1 = min(M1, N1 + N2) and m2 = min(M1, N2) in the key entropy inequality of Lemma 1.These quantities determine the receive-signal dimensions relevant to the DoF analysis.
  • IV. PROOF OF THEOREM 1: L1 IS AN OUTER-BOUND: L1 is obtained by applying Lemma 1 to an outer-bound argument for the delayed-CSI MIMO interference channel.The paper explicitly identifies proving L1 as the aim and states that the bound follows from the lemma.
  • A. Proof of Bound L1: The capacity region is enlarged by giving both receivers instantaneous knowledge of all channel matrices, non-causal knowledge of M2 to R1, and instantaneous access to Y2(t).Because these assumptions enhance the channel, any resulting rate upper bounds also apply to the original delayed-CSI model.
  • A. Proof of Bound L1: If d1 DoF are achieved for user 1, interference occupies at least m2/m1 times d1 DoF at receiver 2, regardless of the achievability scheme.This lower bound on interference yields the upper bound L1 after combining the rate inequalities.
  • B. Proof of Lemma 1: Lemma 1 is proved through auxiliary lemmas and a corollary that compare differential entropies of the signals received across the two receivers.Although the receivers have N1 and N2 dimensions, only the first m1 − m2 and m2 entries are relevant to the DoF analysis.
  • B. Proof of Lemma 1: Channel inversion constructs the relevant received signals from m1 receive antennas with probability 1, reducing the associated conditional term to the differential entropy of noise variables.The resulting per-time noise terms satisfy qt = o(log2 P), independently of n.
  • B. Proof of Lemma 1: Statistical equivalence shows that, under the stated conditioning, signals received at different antennas provide equal information about M1 when transmitter 2 is silent.This property relates the differential entropies at R1 and R2 and is identified as the important point of the proof.
  • B. Proof of Lemma 1: The final inequalities combine lower and upper differential-entropy bounds from the auxiliary lemmas, with sums or differences of o(log2 P) terms remaining o(log2 P).The second inequality follows because conditioned received-signal entropy equals the entropy of noise terms, of order n · o(log2 P).

C. Comments on the Proof of Lemma 1

The proof of Lemma 1 rests on statistical equivalence between two receive-antenna outputs under delayed CSI. This equivalence depends on conditioning only on past and contemporaneous outputs, unlike the broader no-CSI case, and also supports an alternative outer-bound derivation.

  • Proof of Lemma 1: The equality in (15) follows because, conditioned on the relevant variables, X1(t) is independent of i.i.d. channel vectors H2i1(t) and H2j1(t).The two receive antennas therefore provide equal information about message M2, making statistical equivalence the key step in proving Lemma 1.
  • Proof of Lemma 1: The differential-entropy equality holds when conditioning includes past channel outputs, selected present outputs, channel matrices, and message M2.The conditioning set excludes future channel outputs.
  • Proof of Lemma 1: Conditioning on future channel outputs may invalidate equality in (15), because delayed CSI lets future transmissions depend on present outputs and channel vectors.The vectors H2i1(t) and H2j1(t) may then no longer be identically distributed.
  • Comparison with no CSI: Under no CSI, statistical equivalence holds more generally because channel inputs are independent of all channel matrices, regardless of the conditioning variables.Thus, compared with delayed CSI, no CSI permits equal differential entropies under any common conditioning set.
  • Implications for outer bounds: The argument used for bound L1 can derive the MIMO broadcast-channel outer bound in [22] without invoking the physically-degraded broadcast-channel feedback result in.Earlier delayed-CSI MISO and MIMO broadcast-channel outer bounds used that result and were tight in some special cases.

V. PROOF OF THEOREM 1: L4 IS AN OUTER-BOUND … VI. PROOF OF THEOREM 2: CASES 0 AND A.I

The proof derives the L4 outer bound using enhanced receiver CSI, Fano-based rate bounds, and a unitary transformation, then establishes tight delayed-CSI regions for Cases 0 and A.I. Case A.I is resolved by identifying active bounds and a three-phase interference-alignment scheme that delivers missing linear combinations without additional interference.

  • V. PROOF OF THEOREM 1: L4 IS AN OUTER-BOUND: The L4 outer bound is obtained by upper-bounding the two users’ achievable rates under instantaneous CSI at both receivers.The proof applies Fano’s inequality and bounds the resulting terms through two lemmas.
  • V. PROOF OF THEOREM 1: L4 IS AN OUTER-BOUND: A unitary matrix U12(t) is constructed from the singular-value decomposition of H12(t) so the last N1 − M2 rows of U12(t)H12(t) are zero.This transformation isolates the entries affected by X2(t) while preserving mutual information.
  • V. PROOF OF THEOREM 1: L4 IS AN OUTER-BOUND: The two lemma-based inequalities are combined with equations (21), (22), and (26) to produce the desired bound L4.The final algebra substitutes a lower bound on f(t2) into equation (19), yielding the stated inequality.
  • A. Proof of Lemma 6: Lemma 6 bounds the transformed entropy terms using a retained-row matrix, a noise covariance dominated by the identity, and the receive-antenna DoF limit.The proof invokes positive-semidefinite covariance ordering and conditional independence before substituting the resulting lower bound.
  • B. Proof of Lemma 7: Lemma 7 establishes the required entropy equality by conditioning on M2 and H(n), applying translation invariance, and using the distributional equivalence H′(t) ∼ H(t).The argument relies on U12(t) being a deterministic function of H12(t) and independent of H11(t).
  • VI. PROOF OF THEOREM 2: CASES 0 AND A.I: Under Case 0, Dp−CSI = Dd−CSI, so bound L3 is active; the same equality follows through Dno−CSI = Dp−CSI.The case assumes N1 ≥ N2 ≥ M1.
  • VI. PROOF OF THEOREM 2: CASES 0 AND A.I: Under Case A.I, bounds L{1,2} are active, with three outcomes determined by M1 ≤ N1, M1 > N1 and N2 = M2, or M1 > N1 and N2 < M2.These cases yield respectively Dno−CSI = Dd−CSI ⊂ Dp−CSI; equality among all three regions; or Dno−CSI ⊂ Dd−CSI ⊂ Dp−CSI.
  • VI. PROOF OF THEOREM 2: CASES 0 AND A.I: The Case A.I.3 achievability scheme uses three phases to deliver each receiver’s remaining linear combinations, after which each receiver obtains one linearly independent combination per desired symbol.In the final phase, retransmitted combinations align with previously observed interference, causing no additional interference while enabling decoding.

VII. PROOF OF THEOREM 2: CASE A.II

For Case A.II, the delayed-CSI outer bound is tight, with the active bounds and resulting DoF-region comparisons determined by whether N1 is at least M1. A two-phase interference-alignment scheme achieves the critical intersection point when N1 < M1.

  • Lemma 10: The outer bound is tight in Case A.II, with bounds L{1,3} active.The proof also shows that L2 is implied by L3.
  • Lemma 10: When N1 ≥M1, L1 is active and Dno−CSI = Dd−CSI ⊂Dp−CSI.In this subcase, the delayed-CSI outer bound coincides with the no-CSI region.
  • Lemma 10: When N1 < M1, L{1,3} is active and Dno−CSI ⊂Dd−CSI ⊂Dp−CSI.Both bounds are strictly active, and achieving their intersection point P1,3 suffices to achieve the outer bound.
  • Achievability scheme for point P1,3: In Phase Two, R2 subtracts known interference and recovers its symbols by channel inversion, while R1 zero-forces T2’s interference and decodes its remaining data.Exactly N1 transmit antennas are active during this phase, enabling the stated recovery steps.

VIII. PROOF OF THEOREM 2: CASES B.0 AND B.I

Case B.0 yields equality between the delayed-CSI and perfect-CSI DoF regions, making bound L3 active. For Case B.I, bound L1 is active, and a two-phase interference-alignment scheme achieves the relevant outer-bound point by retransmitting previously caused interference.

  • Case B.0: Under Case B.0, Dd−CSI = Dp−CSI, which implies that bound L3 is active.The proof uses the condition N1 = N2 > M2.
  • Case B.I: Under Case B.I, bound L1 is active and determines the outer-bound Dd−CSI.The proof notes that L3 implies L2 and, since N2 < M1, L1 implies L3.
  • Case B.I: Achievability reduces to the intersection point of the single-user d2 bound and L1, because time sharing then achieves the entire outer-bound.The scheme targets point Po2,1 over N2 time slots.
  • Case B.I: The scheme uses two phases: Phase One creates interference-dependent linear combinations, while Phase Two retransmits them as T2 continues sending M2 new data symbols per slot.Phase One lasts t1 = N2 − M2 time slots; Phase Two takes the remaining M2 time slots.
  • Case B.I: Retransmitted interference lets R2 decode both its new and Phase-One desired symbols, while R1 obtains one interference-free linear combination per desired symbol.During Phase Two, t1 + M2 = N2 transmit antennas are used, enabling both receivers to recover the transmit signals almost surely via channel inversion.
  • Case B.I: The alignment works because T1 retransmits interference already caused at R2, recovering R1’s lost combinations without adding interference at R2.This also enables R2 to learn the interference needed to decode its earlier desired data.

IX. PROOF OF THEOREM 2: CASE B.II

For Case B.II, the proof identifies which outer bounds are active in each antenna regime and constructs interference-alignment schemes that achieve the resulting outer-bound DoF regions.

  • Lemma 13: Under Case B.II, L{1,3} are active, with Dno−CSI ⊂ Dd−CSI ⊂ Dp−CSI when M1 = M′1 and M2 > m.The outer-bound structure depends on the antenna conditions and CSI assumptions.
  • Case B.II.1: When M2 ≤ m, L3 is active and Dd−CSI outer = Dp−CSI.This is the Case B.II.1 regime.
  • Case B.II.2: In Case B.II.2, bounds L{1,3} are active, and separate schemes establish achievability of points Po2,1 and P1,3.The proof treats the two corner points separately to exhaust the outer-bound.
  • Case B.II.1: Over M′1 time slots, the Case B.II.1 scheme achieves M′1(N1 −M2) and M′1M2 DoF for the two users, respectively.The construction uses two phases to deliver data and retransmit interference symbols needed by the receivers.
  • Case B.II.2: Each phase provides one interference-free LC per desired data symbol, enabling both receivers to decode their desired symbols.This decoding conclusion is stated for the achievability construction in the later Case B.II.2 argument.

1. We will design this coding scheme such

The IA-Scheme uses a two-phase transmission strategy: an initial phase sends data while creating interference, and a second phase retransmits grouped symbols so both receivers can decode. Under Lemma 16’s partition conditions, the scheme achieves the intended DoF tuple for the specified antenna configuration.

  • Two-phase IA-Scheme: The scheme takes T time slots and consists of two phases.Phase One occupies t1 slots, while Phase Two occupies the remaining t2 slots.
  • Two-phase IA-Scheme: In Phase One, T1 and T2 transmit their data symbols, after which R1 decodes its symbols while R2 requires the relevant interference values.The receivers observe desired symbols mixed with interference, motivating the retransmission strategy in the next phase.
  • Partitioning lemma: Lemma 16 partitions the symbol set into t2 disjoint subsets of cardinality at most N1, with bounds of M2 and N2 on specified symbol classes.The partition supports simultaneous retransmission while controlling the number of data and unknown-interference symbols per slot.
  • Two-phase IA-Scheme: In Phase Two, T1 retransmits interfering symbols and T2 transmits data symbols from each selected subset, allowing R1 to recover interference and decode.At most N1 symbols are transmitted per slot, enabling channel inversion and one interference-free linear combination per data symbol at R1.
  • Decoding and achievability: R2 subtracts known interference and recovers its desired data along with unknown interfering symbols, so both receivers decode all intended symbols by the end of T time slots.The construction is achievable whenever Lemma 16 holds for the selected parameters.

X. PROOF OF THEOREM 2: CASE B.III

In Case B.III, defined by 1 > N1 > N2 > M2 > m, the outer-bound structure depends on whether M1 is at least N1 + N2 −m. Bounds L{3,4} are active when M1 ≥N1 + N2 −m, whereas bounds L{1,3,4} are active when M1 < N1 + N2 −m.

  • Case B.III is defined by 1 > N1 > N2 > M2 > m.
  • The lemma identifies bounds L{1,3,4} as active for the outer-bound Dd−CSI in Case B.III.
  • M1 ≥N1 + N2 −m makes bounds L{3,4} active.
  • M1 < N1 + N2 −m makes bounds L{1,3,4} active.
  • The two conditions correspond to distinct outer-bound shapes: Case B.III.1 activates L{3,4}, while Case B.III.2 activates L{1,3,4}.These shapes are shown in Fig. 6(a) and Fig. 6(b), respectively.

A. Case B.III.1: Condition 1 holds and M1 ≥N1 + N2 −m

In Case B.III.1, the entire outer bound is achievable by time sharing once points Po2,4 and P3,4 are established. Po2,4 follows from the earlier Case B.II.2 construction, while P3,4 is achieved using IA-Scheme under specified parameters.

  • A. Case B.III.1: Condition 1 holds and M1 ≥N1 + N2 −m: The entire outer bound can be achieved via time sharing provided Po2,4 and P3,4 are achievable.The remainder of the section establishes these two points.
  • A. Case B.III.1: Condition 1 holds and M1 ≥N1 + N2 −m: Po2,4 is achievable because the corresponding MIMO IC belongs to Case B.II.2, where Po2,1 was already established.The earlier construction therefore implies achievability of Po2,4 in Case B.III.1.
  • A. Case B.III.1: Condition 1 holds and M1 ≥N1 + N2 −m: P3,4 remains invariant when M1 exceeds N1 + N2, so the proposed scheme may use at most N1 + N2 antennas at R1.This permits assuming without loss of generality that M1 ≤N1 + N2.
  • A. Case B.III.1: Condition 1 holds and M1 ≥N1 + N2 −m: IA-Scheme is instantiated with T = M′, t1 = N1−M2, t2 = N2, d⋆1 = N1M′1−N22, and d⋆2 = N22.The parameter choice ensures the antenna constraints mt ≤M1 and nt ≤M2 for all t in the specified range.
  • A. Case B.III.1: Condition 1 holds and M1 ≥N1 + N2 −m: The chosen parameters yield P3,4, and Lemma 16 verifies the required inequalities, establishing that P3,4 is achievable.The proof uses nt = N1 + N2 −mt and derives n′ = N2M2 before verifying inequalities (a), (b), and (c).

B. Case B.III.2: Condition 1 holds and M1 < N1 + N2 −m · XI. CONCLUSION

For Case B.III.2, the entire outer bound is achievable through time sharing once three corner points are established. Overall, the paper characterizes the delayed-CSI DoF region for every antenna configuration using statistical-equivalence outer bounds and delayed-CSIT interference-alignment schemes.

  • B. Case B.III.2: Condition 1 holds and M1 < N1 + N2 −m: The entire outer bound can be achieved via time sharing if points Po2,4, P1,4, and P1,3 are achievable.This establishes the case’s achievability strategy by reducing the region to three corner points.
  • B. Case B.III.2: Condition 1 holds and M1 < N1 + N2 −m: Point Po2,4 is achievable using the same manner as in the previous case.The passage identifies this point’s scheme by reference to the preceding construction.
  • B. Case B.III.2: Condition 1 holds and M1 < N1 + N2 −m: For P1,3, the condition 2 = N1 + N2 −M1 < M2 permits use of the IA−Scheme.The construction specifies T = M1 −N2, t1 = N1 −N2, and t2 = T = t1 = (M1 −N1).
  • B. Case B.III.2: Condition 1 holds and M1 < N1 + N2 −m: The P1,3 scheme sets d⋆ 1 = M1(N1 −N2) and d⋆ 2 = N2(M1 −N1), with mt = M1 for t ∈[1 : t1].The required lemma for this construction is proved in Appendix D-A.
  • XI. CONCLUSION: The paper obtains inner and outer bounds that coincide for every antenna tuple (M1, M2, N1, N2).The outer bound uses statistical equivalence of channel outputs, while the inner bound uses interference-alignment achievability schemes.
  • XI. CONCLUSION: The interference-alignment schemes require only delayed CSIT.Thus, the stated inner-bound constructions operate under delayed channel-state information at the transmitters.

APPENDIX A LEMMAS USEFUL FOR DETERMINING THE SHAPE OF THE DOF REGION … B. For Point P1,4

The appendices establish lemmas and constructive partition arguments used to characterize the delayed-CSI DoF region. They distinguish antenna configurations and verify the inequalities and subset-capacity conditions required by the achievability proofs.

  • APPENDIX A LEMMAS USEFUL FOR DETERMINING THE SHAPE OF THE DOF REGION: In Case A, d2 = N2 forces d1 = 0, whereas in Case B, d2 = M2 permits d1 = N2 − M2 even without CSI.These conclusions provide the two key endpoint properties established by Lemma 19.
  • APPENDIX A LEMMAS USEFUL FOR DETERMINING THE SHAPE OF THE DOF REGION: Lemma 20 relates Condition 1 to the inequality 1 > N1 > N2 > M2 > m and derives M′1 − N1 = N2 − M2.The appendix proves both directions by rewriting the condition and its implied inequalities.
  • APPENDIX B: Appendix B constructs a partition algorithm that repeatedly selects distinct elements, removes them from further consideration, and terminates at j = t2.Because selected elements are never reused, the resulting subsets are disjoint and partition SR2−known with the required properties.
  • APPENDIX C: Appendix C proves the required partition for Case B.II.2 by splitting on M2(N1 − N2) ≥ N2(M1 − N1) versus the strict reverse inequality.In the first case n′ = 0; in the second, n′ > 0 and new data symbols must be transmitted during Phase Two.
  • APPENDIX C: In Case B.II.2, subsets of size at most N1 accommodate the required elements, including N1t2 − N2t2 = (N1 − N2)t2 remaining positions.The construction assigns each element to exactly one subset and thereby obtains the required partition P.
  • A. For Point P1,3: Appendix D states that when n′ = 0, the proof from Appendix C-A applies unchanged, while n′ > 0 requires proving inequality (a), M2t2 ≥ n′.The contradiction argument uses the Case B.III.2 condition M1 < N1 + N2 − m; the remaining inequalities are described as straightforward.
  • B. For Point P1,4: For Point P1,4, the proof derives N2t2 − |SDS| and bounds |SIS−R2known| = (N2 − 2)(M′1 − N1) by M′2(N1 − N2).Assuming the bound fails contradicts the Case B.III.2 condition M1 < N1 + N2 − m.
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