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Real Interference Alignment: Exploiting the Potential of Single Antenna Systems

Abolfazl Seyed Motahari, Shahab Oveis Gharan, Mohammad-Ali Maddah-Ali, Amir Keyvan Khandani

arXiv:0908.2282v2cs.IT

TL;DR

The paper addresses how to achieve high degrees of freedom in single-antenna systems without relying on channel variation. It proposes real interference alignment based on fractional signaling dimensions and proves optimal total DOF results for three static channel classes, with stated almost-all-realization scope.

  • Problem

    For K-user Gaussian interference channels, Han-Kobayashi interference management is insufficient, while fast channel variation used by prior alignment schemes is not practically realistic.

  • Method

    The paper proposes real interference alignment, sending multiple streams along designed directions so interfering signals align in fractional dimensions, supported by Diophantine approximation on non-degenerate manifolds.

  • Results

    The scheme characterizes total DOF as K/2 for the K-user GIC, KM/(M+1) for cellular uplink, and KM/(K+M−1) for the K × M X channel, almost surely.

  • Takeaways & Limitations

    Static single-antenna systems can attain the total DOF of these three channels without requiring channel variation over time, frequency, or space.

Abstract

from arXiv · show

In this paper, the available spatial Degrees-Of-Freedoms (DOF) in single antenna systems is exploited. A new coding scheme is proposed in which several data streams having fractional multiplexing gains are sent by transmitters and interfering streams are aligned at receivers. Viewed as a field over rational numbers, a received signal has infinite fractional DOFs, allowing simultaneous interference alignment of any finite number of signals at any finite number of receivers. The coding scheme is backed up by a recent result in the field of Diophantine approximation, which states that the convergence part of the Khintchine-Groshev theorem holds for points on non-degenerate manifolds. The proposed coding scheme is proved to be optimal for three communication channels, namely the Gaussian Interference Channel (GIC), the uplink channel in cellular systems, and the $X$ channel. It is proved that the total DOF of the $K$-user GIC is $\frac{K}{2}$ almost surely, i.e. each user enjoys half of its maximum DOF. Having $K$ cells and $M$ users within each cell in a cellular system, the total DOF of the uplink channel is proved to be $\frac{KM}{M+1}$. Finally, the total DOF of the $X$ channel with $K$ transmitters and $M$ receivers is shown to be $\frac{KM}{K+M-1}$.

I. INTRODUCTION

The paper develops real interference alignment to manage interference in static single-antenna channels, addressing limitations of orthogonal schemes and time-varying approaches. It characterizes the total DOF of the K-user GIC, cellular uplink, and X channel, while clarifying exceptional channel realizations and extensions.

  • Motivation: Orthogonal schemes limit throughput in dense networks, where allowing and managing multi-user interference is optimal.Interference alignment reduces harm by merging dimensions occupied by interfering signals.
  • Motivation: For K-user GICs, Han-Kobayashi interference management is insufficient, motivating interference alignment in the signaling.The paper addresses this challenge for channels with static coefficients and single antennas.
  • Contribution: Real interference alignment achieves total DOF without channel variation over time, frequency, or space in the K-user GIC, cellular uplink, and X channel.The scheme uses fractional dimensions embedded into a single real line, supported by an extension of the Khintchine-Groshev theorem.
  • Main Results: K/2 is the total DOF of the real, time-invariant K-user GIC for almost all channel realizations.If all channel gains are rational, the total DOF is strictly less than K/2.
  • Main Results: KM/(M+1) is the total DOF of a cellular system with K cells and M users per cell for almost all channel realizations.The result concerns the uplink channel, in which users transmit independent messages to their cell’s base station.
  • Main Results: KM/(K+M−1) is the total DOF of the K × M X channel with real, time-invariant coefficients.The theorem states this value for almost all channel realizations.
  • Scope and Extensions: The theorems do not cover infinitely many channel realizations beyond rational gains, although some uncovered cases still achieve the total DOF.For example, the K-user GIC can achieve 1/2 per user when cross gains are rational and direct gains are algebraic irrationals.
  • Scope and Extensions: The scheme also extends the K-user GIC result to MIMO and complex-coefficient channels through virtual-user representations and related coding extensions.Separate encoding and decoding across paired antennas yields the MIMO result, while real and imaginary parts can be paired for complex coefficients.

III. MAIN IDEAS AND BASIC EXAMPLES

The section develops real interference alignment through finite constellations, carefully chosen transmit directions, and number-theoretic minimum-distance guarantees. Its examples show how partial alignment can approach perfect alignment while preserving separability of intended signals.

  • Coding and decoding: The scheme uses finite integer-based constellations whose received points must remain sufficiently separated for reliable decoding under additive Gaussian noise.Transmitters are subject to power constraints, and the minimum distance of the received constellation is central to the construction.
  • Number-theoretic foundation: The Khintchine-Groshev extension supplies the same minimum-distance lower bound when channel coefficients lie on a non-degenerate manifold.For relations such as b = a^2, the theorem applies on the manifold, with nonsatisfying points having measure zero.
  • Basic examples: In the two-user X channel, two interfering streams are combined into one received direction, leaving two dimensions for intended signals and one for interference.The construction treats the interfering sum as a single effective term; its doubled integer range changes only a constant in the minimum-distance bound.
  • Basic examples: 3 is achievable in total for the two-user X channel, meeting the upper bound through interference alignment.The alignment reduces the power of Q in the minimum-distance expression, enabling higher degrees of freedom.
  • Partial alignment: Single-stream transmission cannot perfectly align three transmitters at two receivers, so the scheme uses multiple streams and channel-dependent directions for partial alignment.The transmit-direction set is chosen to minimize the number of received directions across receivers.
  • Partial alignment: Any finite number of signals can be partially aligned at any finite number of receivers, with alignment efficiency made arbitrarily close to one by increasing the number of directions.The construction therefore approaches perfect alignment while retaining distinct intended directions for decoding.

IV. DIOPHANTINE APPROXIMATION: KHINTCHINE-GROSHEV TYPE THEOREMS

The section develops Khintchine–Groshev results for rational approximation and explains their extension to non-degenerate manifolds, which supports interference-alignment proofs.

  • Khintchine theorem: Khintchine’s theorem characterizes when linear inequalities involving real numbers and integer pairs have infinitely many or finitely many solutions.Its convergence part implies that, for almost all real numbers, only finitely many sufficiently close rational approximations occur.
  • Khintchine–Groshev theorem: The Khintchine–Groshev theorem extends this measure characterization to rational approximation of linear forms with multiple real and integer components.The convergence and divergence conditions determine whether the relevant approximation set has measure zero or full measure.
  • Applications to coding: Earlier results used these approximation tools to establish degrees-of-freedom achievability for the two-user X channel and obtain 4/3 for the three-user GIC.The earlier theorem did not cover related components, preventing a proof of 3/2 for the three-user GIC.
  • Manifold extension: The convergence theorem was extended to non-degenerate manifolds, addressing cases where the parameter vector lies on a lower-dimensional manifold.This extension is needed because such manifolds have zero Lebesgue measure in the surrounding Euclidean space.
  • Applications to coding: Monomials satisfy the theorem’s analytic and linear-independence conditions when they are distinct, making them suitable functions for the coding analysis.With one variable, the resulting family is {1, v, v^2, v^3, …}.

V. CODING SCHEME AND PERFORMANCE ANALYSIS

The coding scheme uses rationally independent transmit directions, finite integer constellations, and channel-dependent monomials to align interference while preserving separability and reliable decoding.

  • Performance analysis: The scheme applies the same encoding and decoding design across transmitters and receivers, with universal parameters selected to satisfy the power constraint.The input constellation is scaled using Q = γP^(2(m+ε)).
  • Encoding scheme: The finite constellation C = (−Q, Q)Z enables feasible alignment, while Q controls constellation cardinality and A controls received minimum distance.The parameters Q and A depend on the reciprocal multiplexing gain m of each stream.
  • Performance analysis: A random codebook over the finite constellation converts the channel into a reliable one, and the error probability analysis yields the target multiplexing gain at high SNR.The theorem states conditions for achieving 1/m DOF per data stream for almost all channel realizations.
  • Encoding scheme: A single antenna can support fractional DOFs because it has infinitely many bases when viewed as a vector space over the rationals.The scheme therefore transmits multiple independent streams along rationally independent real directions.
  • Transmit directions: Channel-dependent monomial directions provide both interference alignment and separability at receivers.Monomials form a non-degenerate manifold, enabling Khintchine–Groshev-based performance analysis.
  • Transmit directions: As the design parameter n grows, alignment efficiency can approach one, allowing finite collections of transmitters to align signals at finite collections of receivers.The efficiency is defined through the ratio of desired and interference direction counts.

A. System Model

The K-user Gaussian interference channel consists of K transmitter–receiver pairs sharing bandwidth, with time-invariant real gains, AWGN, and power-constrained inputs.

  • Channel model: Each user communicates with its corresponding receiver while all transmitters share the same bandwidth and create interference for other receivers.The received signal at each receiver is the sum of all channel-weighted inputs plus noise.
  • Channel assumptions: The channel gains are real and time invariant, the noise has unit variance, and every transmitter satisfies power constraint P.The gain h_ji denotes the link from transmitter i to receiver j.
  • DOF formulation: The DOF region is the high-SNR shape of the capacity region after scaling by log SNR.The total DOF concerns the sum-rate objective with λ = {1, 1, …, 1}.
  • DOF bound: K/2 is an upper bound on the total DOF, so each user can obtain at most one half of its maximum DOF.The paper’s alignment construction is intended to attain this bound for almost all channel realizations.

B. Three-user Gaussian Interference Channel: DOF = 3

For the three-user GIC, the paper selects channel-dependent transmit directions and proves that the target total DOF is achievable in both algebraic and transcendental cases.

  • DOF result: 3/2 total DOF is achievable for almost all three-user GIC channel realizations.The proof uses an appropriate selection of transmit directions.
  • Standard channel: Every three-user GIC has an equivalent standard channel for DOF analysis, with normalized cross-links and gains G0, G1, G2, and G3.The standard model preserves the relevant DOF characterization.
  • Direction design: The transmit directions are selected from monomials generated by G0, and the proof treats algebraic and transcendental values of G0 separately.Although algebraic G0 has measure zero, the paper analyzes both cases under the theorem’s direction conditions.

1) Case I:

This case selects transmit directions from powers of G0 so interfering signals align while desired directions remain distinguishable over the rationals. The construction achieves the stated three-user DOF for almost all channel realizations.

  • Case I: The basis T = {1, G0, G0^2, . . ., G0^(d−1)} restricts transmit directions while preserving rational independence.The polynomial relation represents higher powers of G0 using this basis with rational coefficients.
  • Case I: Each transmitter sends Li = d data streams using all directions in T.This common direction set makes the interfering signals align at the receivers.
  • Case I: At Receiver 1, signals from Transmitters 2 and 3 align, reducing their received directions to L′1 = d.The received signals at the other receivers have analogous alignment properties.
  • Case I: The construction also satisfies the required receiver conditions, including the corresponding direction-count conditions at all three receivers.The maximum number of received directions is m = 2d.
  • Case I: The special case d = 1 corresponds to rational G0 and was previously shown to achieve the channel’s total DOF.The paper identifies this as a case considered in earlier work.

2) Case II:

This case uses an asymmetric direction assignment: one transmitter sends n+1 streams while the other two send n streams. Pairwise interference aligns at each receiver, yielding an achievable DOF of 3n+1 over 2n+1.

  • Case II: The asymmetric design assigns L1 = n+1 streams to Transmitter 1 and L2 = L3 = n streams to Transmitters 2 and 3.The direction sets use powers of G0 through n for Transmitter 1 and through n−1 for Transmitters 2 and 3.
  • Case II: The resulting DOF 3n+1 over 2n+1 is achievable for every n ∈ N and hence for the three-user GIC almost surely.The paper states the achievability conclusion after applying Theorem 6.
  • Case II: At Receiver 1, Transmitters 2 and 3 align, producing n effective interfering directions.Although 2n directions can arrive, only n are effective.
  • Case II: At Receivers 2 and 3, the corresponding interfering transmitter pairs also align, with n + 1 effective received directions at each receiver.The receiver conditions C2 and C3 hold in both cases.
  • Case II: The maximum total number of received directions is 2n + 1 across the receivers.This count follows from the asymmetric stream allocation and aligned interference.

C. K-user Gaussian Interference Channel: DOF = K

For the K-user GIC, transmit directions are selected as channel-gain monomials that exclude each user’s direct gain, causing all interference to align while desired directions remain distinct. The construction achieves K/2 DOF almost surely.

  • C. K-user Gaussian Interference Channel: DOF = K: Excluding direct gains keeps desired directions distinct from interference, satisfying C2 at each receiver.The paper notes that C3 does not hold in this construction because all received directions are irrational.
  • C. K-user Gaussian Interference Channel: DOF = K: The analysis assumes transcendental channel gains, while algebraic gains can sometimes reduce the number of directions needed to achieve total DOF.The paper states that the transcendental-gain assumption excludes a measure-zero set and does not cover all realizations.
  • C. K-user Gaussian Interference Channel: DOF = K: The direction set Ti uses monomials whose exponent of the intended user’s direct gain is zero and whose other exponents are bounded.This selection gives each transmitter a direction set satisfying condition C1.
  • C. K-user Gaussian Interference Channel: DOF = K: All interfering users align within the common received-direction set Tr at every receiver.Each interferer’s received direction set Tik is a subset of Tr.
  • C. K-user Gaussian Interference Channel: DOF = K: Applying Theorem 6 after verifying the receiver conditions yields the channel’s achievable DOF expression.The derivation combines the desired and interference direction counts.
  • C. K-user Gaussian Interference Channel: DOF = K: K/2 DOF is achievable for the K-user GIC almost surely.The conclusion follows because the construction parameter n can be arbitrarily large.

VII. CELLULAR SYSTEMS: UPLINK

In the cellular uplink, users within each cell transmit to their serving base station, and transmit directions are designed so non-intended-cell signals align at every base station. The construction achieves KM/(M+1) total DOF.

  • VII. CELLULAR SYSTEMS: UPLINK: The uplink has K cells with M users per cell, where users transmit independent messages to their in-cell base station.The section considers uplink operation rather than downlink broadcasting.
  • VII. CELLULAR SYSTEMS: UPLINK: An upper bound is obtained by allowing users within each cell to cooperate, converting the uplink into a MISO K-user GIC.This cooperation-based channel provides the comparison used for the DOF upper bound.
  • VII. CELLULAR SYSTEMS: UPLINK: Each user in Cell k selects directions from channel-gain monomials whose exponent pattern excludes gains associated with the intended cell.The resulting set Tkm satisfies condition C1.
  • VII. CELLULAR SYSTEMS: UPLINK: All signals from non-intended cells align at every base station within a common interference direction set T.The inclusion Ti ⊆ T follows from the bounded channel-gain exponents in the selected transmit directions.
  • VII. CELLULAR SYSTEMS: UPLINK: The total number of received directions is obtained by combining the desired in-cell directions with the aligned inter-cell interference directions.The construction satisfies C1 and C2 at all base stations.
  • VII. CELLULAR SYSTEMS: UPLINK: KM/(M+1) total DOF is achievable for the cellular uplink.The result is concluded by allowing the design parameter n to grow arbitrarily large.

VIII. K × M X CHANNEL

The K × M X channel uses real interference alignment to place interfering messages into shared signal directions while preserving desired dimensions. This achieves the channel’s DOF upper bound, including KM/(K+M−1) in general and K^2/(2K−1) when M = K.

  • VIII. K × M X CHANNEL: The model assumes real, time-invariant channel gains, unit-variance AWGN, and transmitters subject to power constraint P.The DOF region is evaluated in the high-SNR regime, with message rates normalized by log SNR.
  • VIII. K × M X CHANNEL: KM/(K+M−1) DOF is achievable for the K × M X channel, matching the stated upper bound.The construction assigns each transmitter M signal directions and aligns interference at receivers.
  • VIII. K × M X CHANNEL: Interference from messages intended for other receivers is aligned across receivers using monomial signal directions built from channel-coefficient products.For Receiver 1, the relevant coefficients form H1, and the direction set T1 contains (n + 1)^((M−1)K) monomials.
  • VIII. K × M X CHANNEL: Each receiver combines desired signals with aligned interference and additive noise, producing a finite set of received directions for decoding.At Receiver 1, the interference contributes (M−1)(n + 1)^((M−1)K) directions, while desired signals use Kn^(M−1)(n + 1)^((M−1)(K−1)) directions.
  • VIII. K × M X CHANNEL: K^2/(2K−1) is the total DOF when the numbers of transmitters and receivers are equal, M = K.The paper notes that X-channel and GIC DOF behave similarly as transmitter and receiver counts increase.

IX. CONCLUSION

The paper applies real interference alignment to three static wireless channels and shows that single-antenna systems can exploit fractional signal dimensions like multiple-antenna systems. The result relies on a coding scheme supported by Diophantine approximation and attains the total DOF of the considered systems.

  • IX. CONCLUSION: Real interference alignment attains the total DOF of the K-user GIC, cellular uplink, and K × M X channel.These are the three static channels considered in the paper.
  • IX. CONCLUSION: Single-antenna systems can use signal directions for data transmission and reception similarly to multiple-antenna systems.The scheme embeds several fractional dimensions into a single real line.
  • IX. CONCLUSION: The coding scheme is supported by an extension of the Khintchine-Groshev theorem to non-degenerate manifolds.This Diophantine-approximation result provides the stated theoretical basis for the alignment construction.
  • IX. CONCLUSION: The same main result also achieves the total DOF in MIMO and complex-channel cases.The paper describes these extensions as simple applications of its main result.
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