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Degrees of Freedom of the $K$ User $M \times N$ MIMO Interference Channel

Tiangao Gou, Syed A. Jafar

arXiv:0809.0099v1cs.IT

TL;DR

The paper studies total degrees of freedom in K-user MIMO Gaussian interference channels under time-varying, continuously distributed channel coefficients. It develops inner and outer bounds based on interference alignment, characterizes the tight integer-ratio case, and gives constant-channel examples combining alignment with zero forcing.

  • Problem

    The paper asks for the total degrees of freedom of the K-user MIMO Gaussian interference channel and whether interference alignment can improve on zero forcing in some constant-channel settings.

  • Method

    The paper develops inner and outer bounds using interference alignment, including beamforming over multiple symbol extensions, and combines alignment with zero forcing for constant channels.

  • Results

    When R=max(M,N)/min(M,N) is an integer, the total degrees of freedom is min(M,N)K for K≤R and min(M,N)R/(R+1)K for K>R.

  • Takeaways & Limitations

    The bounds are tight for integer R, and alignment combined with zero forcing can achieve more degrees of freedom than zero forcing alone in some constant-channel MIMO interference channels.

Abstract

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We provide innerbound and outerbound for the total number of degrees of freedom of the $K$ user multiple input multiple output (MIMO) Gaussian interference channel with $M$ antennas at each transmitter and $N$ antennas at each receiver if the channel coefficients are time-varying and drawn from a continuous distribution. The bounds are tight when the ratio $\frac{\max(M,N)}{\min(M,N)}=R$ is equal to an integer. For this case, we show that the total number of degrees of freedom is equal to $\min(M,N)K$ if $K \leq R$ and $\min(M,N)\frac{R}{R+1}K$ if $K > R$. Achievability is based on interference alignment. We also provide examples where using interference alignment combined with zero forcing can achieve more degrees of freedom than merely zero forcing for some MIMO interference channels with constant channel coefficients.

I. INTRODUCTION

The introduction frames interference management as a longstanding wireless-design problem and notes that general Gaussian interference-channel capacity characterization remains open, especially with multiple-antenna nodes.

  • Interference management is presented as an important problem in wireless system design.
  • Gaussian interference-channel capacity characterization has been studied from an information-theoretic perspective for more than thirty years.
  • Several inner and outer bounds are known for two-user Gaussian interference channels with single-antenna nodes.
  • The capacity region of the Gaussian interference channel remains open in general.
  • Interference channels with multiple-antenna nodes have also been studied, but the general characterization remains unresolved.

A. Motivating Example

The motivating examples contrast zero forcing with interference alignment in multiuser MISO and SIMO channels. They show why alignment is needed when users cannot independently eliminate all interference, while also identifying feasibility limits for one-to-one alignment in SIMO channels.

  • A. Motivating Example: 2 degrees of freedom are achievable in the 2-user MISO channel with 2 transmit antennas by zero-forcing interference.
  • A. Motivating Example: Only 2 degrees of freedom are achievable by zero forcing in the 3-user MISO channel with 2 antennas per transmitter.
  • A. Motivating Example: The motivating questions ask whether interference alignment can outperform zero forcing and what total degrees of freedom the K-user MIMO channel has.
  • A. Motivating Example: For K > 2, time-varying or frequency-selective channels drawn from a continuous distribution allow each SISO user to achieve 1/2 degree of freedom using interference alignment.
  • A. Motivating Example: For K > 3, each user in the 2-antenna MISO channel can still achieve 2/3 degrees of freedom under the stated channel conditions.
  • A. Motivating Example: One-to-one interference alignment is infeasible in the illustrated SIMO setting because the relevant channel-matrix ranges have null intersection with probability one.

B. Overview of Results

The paper gives inner and outer bounds for the total degrees of freedom of the K-user MIMO Gaussian interference channel, with tight bounds when max(M,N)/min(M,N)=R is an integer. It also studies interference alignment for selected constant-channel MIMO cases, where it can outperform zero forcing.

  • Time-varying channels: The paper provides both achievability and converse bounds for the total degrees of freedom of the K-user MIMO interference channel.The channel has M antennas at each transmitter and N antennas at each receiver.
  • Time-varying channels: min(M,N)K degrees of freedom are achievable when K ≤ R, and the total degrees of freedom is bounded above by min(M,N)K.The bounds are tight when max(M,N)/min(M,N)=R is an integer.
  • Time-varying channels: When K ≤ R, every user can achieve min(M,N) degrees of freedom, matching the interference-free value.This result includes MISO and SIMO interference channels as special cases.
  • Time-varying channels: When K > R, every user achieves a fraction R/(R+1) of the interference-free degrees of freedom, giving R/(R+1)K in the stated tight-bound case.The paper describes this as a per-user loss when the number of users exceeds R.
  • Constant channels: For constant channels, interference alignment achieves additional degrees of freedom in selected R+2-user MIMO configurations, including cases using finite symbol extension or no symbol extension.The paper compares these results with zero forcing, which achieves only RM degrees of freedom in the cited examples.
  • Constant channels: Interference alignment can achieve more degrees of freedom than merely zero forcing in some constant-channel MIMO interference channels.This provides examples answering whether zero forcing can be improved upon.

II. SYSTEM MODEL

The system is a K-user MIMO Gaussian interference channel with M transmit antennas and N receive antennas per user, modeled through time-varying channel matrices and additive Gaussian noise.

  • The channel has K transmitters and K receivers, with independent message Wi from transmitter i intended for receiver i.
  • Each receiver output is the sum of signals from all transmitters transformed by channel matrices plus additive noise.
  • Each transmitter uses an M × 1 input vector, while each receiver observes an N × 1 output vector.
  • Channel coefficients are independently drawn from a continuous distribution, and their values vary at every channel use.
  • Perfect knowledge of all channel coefficients is available to every transmitter and receiver, under a total power constraint ρ.
  • Spatial degrees of freedom are defined from the high-SNR growth rate of the sum capacity CΣ(ρ).

III. OUTERBOUND ON THE DEGREES OF FREEDOM FOR THE K USER MIMO INTERFERENCE CHANNEL

The paper derives an outerbound on the total degrees of freedom by combining antenna limitations with cooperation-based bounds on user subsets. The resulting bound applies to both time-varying and constant nonzero channel coefficients.

  • The converse does not require time variation and holds for constant nonzero channel coefficients as well.
  • Theorem 1 bounds total degrees of freedom by K min(M, N) when K ≤ R.
  • For K > R, the outerbound is max(M, N)R K/(R + 1), with R determined by the antenna ratio.
  • Cooperating among R transmitters and their corresponding receivers reduces the R + 1-user problem to a two-user MIMO interference channel.
  • The resulting two-user channel has max(M, N) degrees of freedom, so the original R + 1-user channel cannot exceed that value.
  • Every selected set of R + 1 users satisfies a sum constraint no greater than max(M, N).

3. Next we present a detailed

The paper constructs interference-alignment schemes for SIMO and general MIMO channels, then matches the outerbound when the antenna ratio is an integer. The schemes align interference into reduced subspaces while preserving desired-signal independence.

  • SIMO achievable scheme: The construction uses symbol extensions, beamforming vectors, and precoding matrices selected so required subspace containments hold.
  • SIMO achievable scheme: The SIMO scheme aligns interference from one transmitter within the subspace generated by interference from other transmitters at each receiver.
  • SIMO achievable scheme: Desired signal vectors remain linearly independent of interference vectors with probability one, enabling zero-forcing decoding.
  • SIMO achievable scheme: For K > R + 1, the SIMO channel achieves R/(R + 1)K degrees of freedom per orthogonal time dimension.
  • MIMO achievable scheme: The MIMO innerbound achieves min(M, N)K degrees of freedom when K ≤ R and R/(R + 1) min(M, N)K when K > R.
  • Tightness: When max(M, N)/min(M, N) is an integer, the innerbound and outerbound match, yielding the exact total degrees of freedom.

V. ACHIEVABLE DEGREES OF FREEDOM FOR THE MIMO INTERFERENCE CHANNEL WITH CONSTANT

For constant channel coefficients, the paper gives examples where interference alignment combined with zero forcing exceeds zero forcing alone. In a 4-user 4×8 channel, the scheme achieves 9 degrees of freedom without channel extension.

  • Example 1: 9 degrees of freedom are achieved on that channel by combining interference alignment with zero forcing, without channel extension.
  • Example 1: Users 1, 2, and 3 each achieve 2 degrees of freedom, while user 4 achieves 3, totaling 9.
  • Example 1: The construction aligns one interference vector at each receiver so zero forcing leaves the desired streams decodable.
  • Example 2: In the 2×4 constant-coefficient channel, a 2-symbol extension achieves 9 degrees of freedom over the extension, or 4 1/2 degrees of freedom per channel use.
  • Example 2: The extension-based construction cannot reuse Example 1 because the extension channel matrix has block-diagonal structure.

VI. CONCLUSION

The paper studies degrees of freedom in K-user MIMO Gaussian interference channels and finds interference alignment optimal under an integer antenna-ratio condition. It also examines constant-channel settings where combining interference alignment with zero forcing can outperform zero forcing alone.

  • The study investigates degrees of freedom with M antennas at each transmitter and N antennas at each receiver.
  • The motivation is the potential benefit of interference alignment for wireless networks.
  • Interference alignment is optimal for the K-user M × N MIMO Gaussian interference channel when max(M,N)/min(M,N) is an integer and channel coefficients vary over time.The channel coefficients are drawn from a continuous distribution.
  • For constant channel coefficients, interference alignment is combined with zero forcing to explore achievable degrees of freedom.
  • Examples show that this combination can achieve more degrees of freedom than merely zero forcing in some MIMO interference channels.

APPENDIX I

The appendix constructs an interference-alignment scheme over symbol extensions for the time-varying MIMO channel. Alignment confines interference while desired signals remain linearly independent, enabling zero-forcing decoding and the stated degrees-of-freedom result.

  • The construction achieves a total of (R + 1)R(n + 1)Γ + (K − R − 1)RnΓ degrees of freedom over µn = (R + 1)(n + 1)Γ symbol extensions.
  • Taking the supremum over arbitrary n establishes a total of RK degrees of freedom.
  • Users 1 through R + 1 achieve R(n + 1)Γ degrees of freedom over the extended channel, while each remaining user achieves RnΓ.
  • Interference vectors are aligned into lower-dimensional subspaces while desired signal vectors remain separable from interference.This leaves interference-free dimensions for decoding.
  • Desired signal vectors are linearly independent of interference vectors with probability one when the relevant beamforming-vector entries are nonzero.
  • Zero forcing decodes the desired streams after alignment makes the desired and interference vectors independent.

CONSTANT CHANNEL COEFFICIENTS

For constant channel coefficients, the paper gives achievable degrees-of-freedom results for R + 2-user MIMO interference channels. Channel extension can improve on zero forcing alone, including when M < R + 2.

  • The section studies the R + 2-user MIMO Gaussian interference channel with M antennas at each transmitter and RM antennas at each receiver.The channel coefficients are constant.
  • Without channel extension, an achievable result is given for the constant-channel setting.
  • For R = 2, a four-user channel with M antennas at each transmitter and 2M antennas at each receiver achieves 2M + ⌊2M/7⌋ degrees of freedom using interference alignment.
  • Zero forcing achieves only 2M degrees of freedom in that example, so the combined scheme is better when M > 3.
  • Allowing channel extension can provide more degrees of freedom than zero forcing even when M < R + 2.
  • For R = 2 and M = 2, interference alignment provides 1/2 more degrees of freedom.

A. Proof of Theorem 4

Theorem 4 aligns selected interference vectors within the spans of other interference vectors, reducing interference dimensions at each receiver. Random remaining beamforming vectors and independent direct channels preserve desired-signal independence, enabling zero-forcing decoding.

  • A. Proof of Theorem 4: Each user achieves d_i degrees of freedom, for a total of RM + ⌊RM/(R + 2)⌋ degrees of freedom.
  • A. Proof of Theorem 4: Each receiver must keep enough interference-free dimensions to decode its desired streams by zero forcing.
  • A. Proof of Theorem 4: The construction aligns selected interference ranges inside subspaces spanned by other interference vectors at receivers 1 through R + 2.
  • A. Proof of Theorem 4: The alignment conditions are expressed through linear transformations involving the channel matrices and transmit beamforming matrices.
  • A. Proof of Theorem 4: The remaining beamforming vectors can be chosen randomly according to a continuous distribution.
  • A. Proof of Theorem 4: Because direct channel matrices do not appear in the alignment equations, multiplying by them independently transforms desired signals and preserves linear independence with probability one.

B. Proof of Theorem 5

The proof constructs an interference-alignment scheme on a symbol-extension channel, then uses zero forcing to decode desired streams. The construction achieves the stated degrees of freedom after translating the extension-channel result back to the original channel.

  • Extension-channel construction: The scheme uses a ⌈R+2/M⌉-symbol extension, with each transmitter sending independently encoded streams through a precoding matrix.The overbar notation denotes the symbol extension, and the transmitted vectors are formed from the precoding matrices and encoded streams.
  • Extension-channel construction: Interference alignment restricts the interference dimension at each receiver so desired streams can be decoded by zero forcing.The interference space must fit within the available dimension, requiring one interference vector to align within the span of the others.
  • Alignment conditions: The alignment conditions are imposed through linear-span equations across receivers, including receivers 1, 2 through R+1, and R+2.The equations specify which transformed interference vector lies in the span of the remaining interference vectors at each receiver.
  • Alignment conditions: Randomly choosing one initial beamforming vector and solving the remaining vectors satisfies the construction under the stated nonzero-entry condition.The vectors are chosen independently of the direct channel matrices, so desired signal vectors are linearly independent of interference vectors almost surely.
  • Degrees-of-freedom result: Each receiver decodes its message by zero forcing the interference, achieving d_i degrees of freedom per user on the extension channel.The resulting total degrees of freedom are then normalized from the symbol-extension channel to the original channel.
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