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Real Interference Alignment with Real Numbers

Abolfazl S. Motahari, Shahab Oveis Gharan, Amir K. Khandani

arXiv:0908.1208v2cs.ITmath.IT

TL;DR

Single-antenna, time-invariant networks need a way to align interference and improve multiplexing gains. This paper uses rational constellation points and irrational transmission directions, with single- and multi-layer designs. The single-layer scheme achieves 4/3 DOF almost surely in the two-user X channel, while the symmetric three-user GIC's achievable DOF is everywhere discontinuous in channel gain.

  • Problem

    The paper asks whether interference alignment and higher multiplexing gains can be achieved in time-invariant networks with single-antenna nodes.

    Source: §I · p. 3

  • Method

    The scheme encodes data with rational constellation points sent along irrational directions, using single-layer or multi-layer constellations and Diophantine approximation results.

    Source: A. Bringing Another Dimension to Life:… · p. 5, §VI · p. 29

  • Results

    The single-layer scheme achieves 4/3 DOF almost surely in the time-invariant two-user X channel, and the multi-layer scheme's achievable DOF for the symmetric three-user GIC is everywhere discontinuous in channel gain.

    Source: §VI · p. 29

  • Takeaways & Limitations

    A time-invariant single-antenna X channel can achieve its total DOF, while the symmetric three-user GIC has different achievable-DOF behavior for rational and irrational gains.

    Source: §VI · p. 29

Abstract

from arXiv · show

A novel coding scheme applicable in networks with single antenna nodes is proposed. This scheme converts a single antenna system to an equivalent Multiple Input Multiple Output (MIMO) system with fractional dimensions. Interference can be aligned along these dimensions and higher Multiplexing gains can be achieved. Tools from the field of Diophantine approximation in number theory are used to show that the proposed coding scheme in fact mimics the traditional schemes used in MIMO systems where each data stream is sent along a direction and alignment happens when several streams arrive at the same direction. Two types of constellation are proposed for the encoding part, namely the single layer constellation and the multi-layer constellation. Using the single layer constellation, the coding scheme is applied to the two-user $X$ channel and the three-user Gaussian Interference Channel (GIC). In case of the two-user $X$ channel, it is proved that the total Degrees-of-Freedom (DOF), i.e. 4/3, of the channel is achievable almost surely. This is the first example in which it is shown that a time invariant single antenna system does not fall short of achieving its total DOF. Using the multi-layer constellation, the coding scheme is applied to the symmetric three-user GIC. Achievable DOFs are derived for all channel gains. As a function of the channel gain, it is observed that the DOF is everywhere discontinuous.

I. INTRODUCTION

The paper extends interference alignment to time-invariant, single-antenna networks by using irrational numbers as transmission directions and rational dimensions. Its single-layer scheme achieves 4/3 DOF almost surely for the two-user X channel, while the multi-layer scheme yields gain-dependent results for the symmetric three-user GIC.

I. INTRODUCTION

  • The paper frames interference management as increasingly important as more users share a medium, and presents alignment as a way to reduce interference at receivers.Earlier one-dimensional schemes had shown results for some classes of time-invariant single-antenna channels; this work broadens the applications of alignment.Source: §I · pp. 2–3

A. Bringing Another Dimension to Life: Rational Dimension

B. Breaking the Ice: Alignment in One dimension

  • The total DOF of 4/3 is achieved almost surely; two independent data streams are sent from each transmitter, and two interfering streams are aligned at each receiver, giving multiplexing gain 1/3 per data stream.Source: B. Breaking the Ice: Alignment in One… · p. 5

C. K-user GICs: Channel Gains May Help

The paper characterizes achievable DOF in terms of channel gains and develops coding and alignment tools for single-dimensional systems. For the symmetric three-user GIC, its multi-layer construction yields an achievable DOF that varies discontinuously with channel gain.

C. K-user GICs: Channel Gains May Help

  • If cross-links arriving at every receiver have rational dimension m or less, the paper states that total DOF K/(m+1) is achievable.The case m = 1 is identified as collapsing to the result of Etkin and Ordentlich.Source: C. K-user GICs: Channel Gains May Help · p. 6

D. Strange Behavior: Discontinuity of DOF

  • For the symmetric three-user GIC, achievable DOF is everywhere discontinuous in channel gain; irrational gains attain 3/2, while rational gains remain below available upper bounds despite carryover.The paper relates rational-gain DOF roughly to the maximum of the gain's numerator and denominator.Source: D. Strange Behavior: Discontinuity of… · p. 6

III. CODING SCHEME

  • The coding scheme is proposed for real, additive, time-invariant channels with additive white Gaussian noise and a transmitter power constraint.The paper says the scheme is not generally optimal, but provides tools for interference alignment.Source: §III · p. 6, B. Received Signal and Interference… · p. 8

A. Encoding

  • The proposed multi-layer constellation supports multiple data layers and is more useful when channel gains are rational; carryover across levels packs points efficiently on the real line.The simpler single-layer constellation selects all integers in an interval, whereas the multi-layer version selects a subset.Source: D. Strange Behavior: Discontinuity of… · p. 6, A. Encoding · p. 7
  • The encoder combines streams as a linear combination with real-number directions, while rationally independent directions preserve a unique mapping from constellation points to messages.Each stream's direction acts like its transmission direction in a multiple-antenna analogy.Source: A. Encoding · p. 8

B. Received Signal and Interference Alignment

  • Interference alignment in one dimension occurs when a receiver observes a rational combination of two streams; rational dimension counts the number of rationally independent directions needed to represent gains.The paper treats that rational dimension as the effective dimension occupied by interference at the receiver.Source: B. Received Signal and Interference… · p. 9

C. Decoding

The decoding analysis turns the received constellation into a discrete output, then maps it back to the intended stream under a zero-noise decodability condition. It bounds error probability to establish achievable rates and motivate the single-layer constellation's use across channels.

C. Decoding

  • The receiver hard-decodes to the nearest received-constellation point, applies the many-to-one map, and then uses a joint-typical decoder on the resulting symbols.The analysis bounds the probability of decoding a different input symbol to lower-bound achievable rate.Source: C. Decoding · p. 10
  • Property Γ requires a many-to-one map from the received constellation to the intended input constellation, enabling zero-error decoding without additive noise.The paper assumes this property for all received constellations and notes that satisfying it at every receiver usually requires careful transmit-constellation design.Source: C. Decoding · p. 10

IV. SINGLE LAYER CONSTELLATION

  • The single-layer constellation is used for all streams, and the decoder analysis is intended to apply to both the X channel and interference channels.The paper describes this constellation as simple but sufficient for alignment that increases throughput.Source: §IV · p. 11

A. Peformance Analysis: The Khintchine-Groshev Theorem

The Khintchine-Groshev theorem bounds how closely linear forms can approach zero, which lets the paper bound the minimum distance in a received constellation. Under the stated regularity conditions, the decoder can reliably recover a data stream with multiplexing gain 1.

A. Peformance Analysis: The Khintchine-Groshev Theorem

  • Rational independence of the received constellation coefficients gives each point a unique representation and satisfies Property Γ.The paper assumes this independence, and states that it holds almost surely in a random environment.Source: A. Peformance Analysis: The… · p. 11
  • For almost all real coefficient tuples, the convergent Khintchine-Groshev result gives a positive lower bound on the relevant linear forms.This is the theorem component the paper uses in its minimum-distance analysis.Source: A. Peformance Analysis: The… · p. 13
  • The Khintchine-Groshev theorem supplies a lower bound on received-constellation minimum distance, enabling analysis of hard-decoding error.The received point is expressed as a linear form in the desired stream and integer-valued interference components.Source: A. Peformance Analysis: The… · pp. 12–13
  • A data stream is reliably decodable with multiplexing gain 1 when the stated regularity conditions hold.The theorem's final two conditions hold almost surely.Source: A. Peformance Analysis: The… · p. 14

B. Two-user X channel: DOF = 43 is Achievable Almost Surely

The two-user X channel uses transmitter-side linear combinations so that undesired streams arrive with matching coefficients and align at each receiver. The paper shows that the time-invariant single-antenna channel achieves total DOF 4/3.

B. Two-user X channel: DOF = 43 is Achievable Almost Surely

  • The result concerns a two-transmitter, two-receiver X channel with constant channel gains and a power constraint P.Each transmitter sends one stream intended for each receiver.Source: B. Two-user X channel: DOF = 43 is… · pp. 14–15
  • 4/3 total DOF is achievable for the time-invariant two-user X channel, meeting the stated single-antenna upper bound.Each of the four data streams has multiplexing gain 1/3.Source: B. Two-user X channel: DOF = 43 is… · p. 15
  • The transmitters combine their desired and undesired streams so that the undesired streams arrive with the same coefficients at a receiver and can be treated as one stream.At Receiver 1, V1 and V2 align, while Receiver 1 decodes U1 and U2 separately.Source: B. Two-user X channel: DOF = 43 is… · p. 15

3. Similarly, Receiver 2 can decode U2 which

The receiver-side decoding result completes the two-user X-channel achievability argument: both receivers decode their intended streams, yielding total DOF 4/3.

3. Similarly, Receiver 2 can decode U2 which

  • 4/3 total DOF follows when both receivers decode their intended streams with multiplexing gain 1/3.Receiver 1 decodes U1 and Receiver 2 decodes U2.Source: §3 · p. 16

C. K-user Gaussian Interference Channel: Special Cases

For the K-user Gaussian interference channel, the paper establishes a single-stream DOF result conditioned on the rational dimension of cross gains. It notes that multiple streams can yield higher DOF, while single-stream transmission simplifies the analysis.

C. K-user Gaussian Interference Channel: Special Cases

  • Single-stream transmission is not generally optimal, since the paper states that multiple streams can obtain higher DOF.The single-stream case is considered for its simpler analysis.Source: C. K-user Gaussian Interference… · p. 17
  • K/(m+1) DOF is achievable almost surely when each receiver's cross gains have rational dimension at most m.The theorem uses single-data-stream transmission at each user.Source: C. K-user Gaussian Interference… · pp. 17–18
  • Each transmitter sends one data stream using a single-layer constellation, and the receiver groups interference through the rational-dimension representation of its cross gains.The resulting received signal has one desired-stream coefficient and m integer-valued interference sums.Source: C. K-user Gaussian Interference… · p. 17

D. Three-user Gaussian Interference Channel: DOF = 43 is Achievable Almost Surely

The three-user GIC achieves 4/3 total DOF almost surely under the proposed scheme. The proof aligns interference while preserving the channel’s DOF through standardization.

D. Three-user Gaussian Interference Channel: DOF = 43 is Achievable Almost Surely

  • A standard three-user channel can represent every three-user GIC without changing its DOF.Linear operations standardize the gains, powers, and noise; the resulting channel’s capacity region is bounded by two channels with the same high-power DOF.Source: D. Three-user Gaussian Interference… · pp. 18–19
  • The cited result states that a total DOF of 4/3 is achievable almost surely for the three-user GIC, and describes four data streams with multiplexing gain 1 each.Source: D. Three-user Gaussian Interference… · p. 20
  • The asymmetric encoding assigns two data streams to User 1 and one each to Users 2 and 3.Receivers decode their desired streams while treating aligned combinations of other users’ streams as interference.Source: D. Three-user Gaussian Interference… · p. 20

V. MULTI-LAYER CONSTELLATION

The multi-layer constellation is applied to the symmetric three-user GIC. Its construction aligns interference, and it achieves 3/2 DOF for irrational channel gains.

V. MULTI-LAYER CONSTELLATION

  • 3/2 DOF is achievable for the symmetric three-user GIC when the channel gain is irrational.The section derives achievable DOF across channel gains, with this result covering the irrational case.Source: §V · p. 21
  • The constellation uses integer digits in base W, with a digits per layer and L layers.Its points are distinct when a < W, giving a constellation size of a^L and a maximum per-stream rate bounded by L log a.Source: §V · p. 21
  • The multi-layer constellation has a DC component that must be removed at every transmitter, but removal does not affect DOF.The remark says removing it duplicates the achievable rate.Source: §V · p. 21
  • Aligned interference and Property Γ let the receiver distinguish the desired symbol from the interference.The interference is formed by aligned contributions from the other transmitters; Property Γ enables extraction of both the desired symbol and the interference vector.Source: §V · p. 22

A. Rational Channel Gains

For rational gains in the symmetric three-user GIC, the section establishes achievable DOF using multi-layer constellations. The achievable value remains below the upper bound.

A. Rational Channel Gains

  • Theorem 7 gives an achievable DOF for the symmetric three-user GIC with rational channel gain.The proof partitions rational gains into three subsets and analyzes the system in each.Source: A. Rational Channel Gains · p. 23
  • The chosen parameters make the error probability decay exponentially with P, yielding the achievable DOF as ε becomes arbitrarily small.The passage states P_e ≤ exp(−γP^(2ε)) for a constant γ.Source: A. Rational Channel Gains · p. 23
  • The achievable DOF remains below the 3/2 upper bound, although the required relation between W and a is quadratic in all cases.The section identifies the W and a relation in Table I while noting that W = a^2, which would touch the bound, is not possible.Source: A. Rational Channel Gains · p. 24
  • Property Γ holds for all cases listed in Table I, supporting the distinctness needed in the received constellation.The proof establishes this property by induction on L.Source: A. Rational Channel Gains · p. 24

B. Irrational Channel Gains

For irrational gains in the symmetric three-user GIC, the paper proves that 3/2 total DOF is achievable. The proof uses Diophantine approximation to control received-constellation distances.

B. Irrational Channel Gains

  • The irrational-gain proof uses Hurwitz approximation and a lower bound on the received constellation’s minimum distance.Hurwitz’s theorem supplies infinitely many rational approximants to an irrational gain, and the result relies on Diophantine approximation.Source: B. Irrational Channel Gains · pp. 26–27
  • 3/2 total DOF is achievable for the symmetric three-user GIC for all irrational channel gains.Theorem 9 states the result for all irrational gains.Source: B. Irrational Channel Gains · p. 26
  • The paper notes that the result extends to the symmetric K-user GIC.The extension is stated as an achievable-DOF result, without a value specified in these passages.Source: B. Irrational Channel Gains · p. 26

VI. CONCLUSION

The paper’s rational-point constellations, directed by irrational numbers, use Diophantine approximation to achieve the total DOF of several channels. Single- and multi-layer designs yield distinct results for the two-user X channel and symmetric three-user GIC.

VI. CONCLUSION

  • The proposed scheme modulates rational-point constellations and directs them with irrational numbers, using Diophantine approximation tools to establish DOF results.The encoding considers both single-layer and multi-layer constellations.Source: §VI · p. 29
  • The single-layer design achieves DOF 4/3 in the time-invariant two-user X channel and three-user GIC, meeting the upper bound for the X channel.This is the first shown example of a time-invariant single-antenna system achieving its total DOF.Source: §VI · p. 29
  • For the symmetric three-user GIC, the multi-layer design’s achievable DOF is everywhere discontinuous in channel gain and meets the 3/2 upper bound for irrational gains.For rational gains, even allowing carry-over from multiple layers leaves a gap to available upper bounds.Source: §VI · p. 29

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