[Paper Review] Interference Alignment and a Noisy Interference Regime for Many-to-One Interference Channels

Viveck R. Cadambe, Syed A. Jafar|ArXiv.org|Dec 15, 2009

Wireless Communication Security Techniques31 references18 citations

TL;DR

This paper identifies a noisy interference regime in discrete memoryless many-to-one interference channels where random coding and treating interference as noise achieve sum-capacity, even with MIMO or finite constellations. The key contribution is a general condition—stochastic degradation of interference to other receivers—under which interference alignment is implicitly achieved via random coding, eliminating the need for structured codes in this regime.

ABSTRACT

We study the capacity of discrete memoryless many-to-one interference channels, i.e., K user interference channels where only one receiver faces interference. For a class of many-to-one interference channels, we identify a noisy interference regime, i.e., a regime where random coding and treating interference as noise achieves sum-capacity. Specializing our results to the Gaussian MIMO many-to-one interference channel, which is a special case of the class of channels considered, we obtain new capacity results. Firstly, we extend the noisy interference regime, previously studied for (many-to-one) interference channels with average power constraints on the inputs, to a more general class of inputs. This more general class includes the practical scenario of inputs being restricted to fixed finite-size constellations such as PSK or QAM. Secondly, we extend noisy interference results previously studied in SISO interference channels with full channel state information (CSI) at all nodes, to MIMO and parallel Gaussian many-to-one interference channels, and to fading Gaussian many-to-one interference channels without CSI at the transmitters. While the many-to-one interference channel requires interference alignment, which in turn requires structured codes in general, we argue that in the noisy interference regime, interference is implicitly aligned by random coding irrespective of the input distribution. As a byproduct of our study, we identify a second class of many-to-one interference channels (albeit deterministic) where random coding is optimal (though interference is not treated as noise). The optimality of random coding in this second class of channels is due to an interference resolvability condition which precludes interference alignment and hence obviates the need of structured codes.

Motivation & Objective

• To characterize a noisy interference regime in discrete memoryless many-to-one interference channels where interference alignment is implicitly achieved via random coding.
• To extend prior noisy interference results from single-antenna to MIMO and parallel interference channels.
• To establish sum-capacity optimality of random coding under average power constraints and finite constellations (e.g., PSK/QAM).
• To analyze fading Gaussian many-to-one interference channels without CSI at transmitters, showing random coding remains optimal.
• To identify a second class of deterministic many-to-one channels where random coding is optimal due to resolvability of multiple interferers, precluding the need for interference alignment.

Proposed method

• Proposes a stochastic degradation condition: the interference-plus-noise seen by the interfered receiver must be a stochastically degraded version of signals at other receivers.
• Uses random coding with i.i.d. Gaussian inputs and treats interference as noise at the single interfered receiver.
• Applies the joint typicality argument and union bound to bound error probability across all possible incorrect message combinations.
• Employs Fourier-Motzkin elimination to eliminate auxiliary rate parameters (Ωi) and derive the achievable rate region.
• Derives a sufficient condition for vanishing error probability: R₁ + ∑ᵢ∈ₛ Ωᵢ ≤ H(Y₁|Uₛᶜ) for all subsets 𝒮 ⊆ 𝒦₁.
• Establishes that in the noisy interference regime, structured codes (e.g., lattice codes) are unnecessary because interference is implicitly aligned by random coding.

Experimental results

Research questions

• RQ1 Under what conditions does random coding with interference treated as noise achieve sum-capacity in many-to-one interference channels?
• RQ2 Can the noisy interference regime be extended beyond single-antenna Gaussian channels to MIMO and parallel interference channels?
• RQ3 Does the optimality of random coding persist when inputs are constrained to finite constellations like PSK or QAM?
• RQ4 Is random coding optimal in fading Gaussian many-to-one interference channels without CSI at transmitters?
• RQ5 Can random coding be optimal in deterministic interference channels where interference alignment is not feasible?

Key findings

• A noisy interference regime exists where random coding and treating interference as noise achieve sum-capacity in a broad class of discrete memoryless many-to-one interference channels.
• The condition for this regime is that the interference-plus-noise at the interfered receiver is stochastically degraded relative to the signals at other receivers.
• For MIMO and parallel interference channels, the noisy interference regime is preserved under the same stochastic degradation condition.
• The result holds even when inputs are restricted to finite constellations (e.g., PSK/QAM), extending beyond average power constraints.
• In fading Gaussian channels without CSI at transmitters, random coding remains optimal in the noisy interference regime.
• A second class of deterministic many-to-one channels is identified where random coding is optimal due to resolvability of multiple interferers, making structured codes unnecessary.