The Performance of Successive Interference Cancellation in Random Wireless Networks

Xinchen Zhang; Martin Haenggi

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The Performance of Successive Interference Cancellation in Random Wireless Networks

Xinchen Zhang, Martin Haenggi

arXiv:1402.1557v1Feb 7, 2014cs.IT

TL;DR

The paper asks how effectively SIC can decode multiple users in spatially random wireless networks under general fading and power-law path loss. It develops a unified analytical framework and finds that SIC’s marginal benefit usually falls rapidly with the number of canceled users, while low-rate codes and clustered users make it more valuable. In heterogeneous cellular networks, most contemporary narrow-band SIC gain comes from canceling a single interferer.

Problem
The paper addresses how to quantify SIC performance across spatial user distributions, fading conditions, path loss, coding rates, and heterogeneous cellular networks.
Method
The paper uses a unified PLPF-based framework to analyze successive decoding probabilities, aggregate throughput, and SIC coverage effects in heterogeneous cellular networks.
Results
The probability of successively decoding at least k users decays super-exponentially with k in high-rate settings, while SIC is especially beneficial with very low-rate codes.
Takeaways & Limitations
For contemporary narrow-band systems, most SIC gain is achieved by canceling a single interferer at the receiver.

Abstract

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This paper provides a unified framework to study the performance of successive interference cancellation (SIC) in wireless networks with arbitrary fading distribution and power-law path loss. An analytical characterization of the performance of SIC is given as a function of different system parameters. The results suggest that the marginal benefit of enabling the receiver to successively decode k users diminishes very fast with k, especially in networks of high dimensions and small path loss exponent. On the other hand, SIC is highly beneficial when the users are clustered around the receiver and/or very low-rate codes are used. Also, with multiple packet reception, a lower per-user information rate always results in higher aggregate throughput in interference-limited networks. In contrast, there exists a positive optimal per-user rate that maximizes the aggregate throughput in noisy networks. The analytical results serve as useful tools to understand the potential gain of SIC in heterogeneous cellular networks (HCNs). Using these tools, this paper quantifies the gain of SIC on the coverage probability in HCNs with non-accessible base stations. An interesting observation is that, for contemporary narrow-band systems (e.g., LTE and WiFi), most of the gain of SIC is achieved by canceling a single interferer.

I. INTRODUCTION

The paper develops a unified, receiver-side framework for quantifying SIC in spatially distributed wireless networks and applies it to heterogeneous cellular networks.

The paper analyzes SIC in d-dimensional wireless networks with power-law density functions, arbitrary fading distributions, and parameters including path loss, coding rate, and user distribution.
The framework directly addresses spatial received-power ordering, unlike guard-zone approximations that provide limited insight into SIC beyond canceling one or two interferers.
SIC benefits are characterized through bounds on successive decoding and aggregate throughput, including fading effects and the dependence on per-user information rate in interference-limited and noisy networks.
In interference-limited networks, aggregate throughput decreases with per-user information rate, whereas noisy networks have a positive rate that maximizes aggregate throughput.
The HCN application shows that SIC can improve coverage with overloaded or closed-access base stations, but typical systems obtain most of its gain by canceling one interferer and see little average-throughput benefit.

A. The Power-law Poisson Network with Fading (PPNF)

The PPNF model represents transmitters through a power-law spatial density with fading and reduces received-power analysis to a one-dimensional path loss process.

The PPNF is a Poisson network with fading and density λ(x) = a∥x∥^b, where b controls clustering around the receiver and b = 0 gives a homogeneous network.
Successive decoding orders users by received power, and a user is decoded when stronger users have been decoded and its signal-to-residual-interference ratio meets the threshold.
The model evaluates both the probability of decoding at least k users and the aggregate throughput obtained from all active transmitters.
Mapping the faded spatial network to a path loss process yields a one-dimensional Poisson process whose exponent is β = (d + b)/α.
Scale invariance removes dependence on absolute density, and in interference-limited PPNFs the probability of decoding k users is invariant to fading when E[h^β] < ∞.

IV. BOUNDS ON THE PROBABILITY OF SUCCESSIVE DECODING

The paper derives complementary tractable bounds on the probability of successively decoding k users because the exact joint distribution and resulting integrals are difficult to evaluate.

Exact evaluation of p_k requires the joint distribution of the k strongest received powers and residual interference, which is difficult even for a one-dimensional homogeneous PPP.
The paper therefore develops multiple bounds that are tractable in different parameter regimes and collectively explain how p_k depends on system parameters.
The basic bounds are asymptotically tight as θ → ∞ and remain informative for moderate, realistic decoding thresholds.
The high-rate lower bound p_k ≥ (1 + θ)^−βk(k−1) decays super-exponentially with k, so its predicted marginal SIC benefit diminishes rapidly.
The low-rate lower bound applies for k < 1/θ + 1 and behaves better than the high-rate bound as θ → 0, while higher-moment refinements offer only marginal improvements without closed forms.

C. The Upper Bound

The paper develops upper bounds for successive decoding probabilities, including combined and SMUD-specific bounds, and establishes their behavior across decoding thresholds. The results show rapidly diminishing benefits from decoding additional users and extend exact strongest-user results to arbitrary fading and nonuniform distributions.

For θ > 1, the combined upper bound decays with the number of successively decoded users, so added SIC capability has rapidly diminishing gain.
The SMUD upper bound is much tighter than previous bounds in the sequential multi-user decoding regime.SMUD is defined by decoding threshold θ ≥ 1.
In the SMUD regime, multiple packet reception requires SIC, whereas for θ < 1 it is possible through parallel decoding without SIC.
The SMUD bounds exploit the fact that at most one k-user set can satisfy the required received-power condition, preventing overcounting for θ ≥ 1.
The strongest-user probability result holds for arbitrary fading, including non-fading, in d-dimensional networks with nonuniform user distributions.
The closed-form strongest-user estimate agrees closely with simulation for θ > −4 dB.

F. Comparison of the Bounds

The paper compares complementary bounds for decoding probability, mean decoded-user count, and aggregate throughput across network clustering and threshold regimes. SMUD bounds are strongest for θ ≥ 1, while low-rate and other bounds are needed at small thresholds.

SMUD bounds are generally tighter than combined bounds and are the best estimates when θ ≥ 1.
For θ ≪ 1, SMUD bounds become uninformative, requiring complementary bounds such as the low-rate lower bound.The SMUD upper bound can exceed one in this regime.
The more clustered the network, the more useful SIC becomes, and clustering can increase aggregate throughput.Smaller b corresponds to transmitters being more clustered around the receiver.
The expected number of decoded users increases without bound as θ decreases, confirming SIC’s particular benefit for low-rate applications.Examples include node discovery and cell search.
The low-rate lower bound captures the small-θ asymptotics that the ordinary lower bound misses, while the SMUD upper bound is especially tight for θ > 1.
In the interference-limited network, aggregate throughput approaches a nonzero limit as θ → 0 and the asymptotic bound is numerically tight.

C. A Laplace Transform-based Approximation

The paper introduces a Laplace-transform approximation for aggregate throughput and extends its analysis to noisy networks. In noiseless networks the approximation is accurate near low thresholds, whereas noise forces throughput toward zero and creates a positive optimal threshold.

The Laplace-transform approximation is asymptotically accurate as θ becomes small and provides a tractable alternative to numerical inversion.
Most noiseless-network analytical bounds can be adapted to noise, but the resulting bounds generally lack closed-form expressions.
In noisy networks, aggregate throughput goes to zero as θ → 0.The residual interference is increasingly canceled, leaving noise to dominate throughput.
There exists at least one positive θ that maximizes aggregate throughput in a noisy network.This optimum balances interference and noise.
In noisy networks, fading reduces the probability of decoding k users, the mean decoded-user count, and aggregate throughput.
Independent random power control cannot increase network throughput in a noisy Poisson-powered path-loss field.

VII. APPLICATION IN HETEROGENEOUS CELLULAR NETWORKS

The paper applies its SIC framework to K-tier heterogeneous cellular networks with accessible and non-accessible base stations. A marked Poisson model and PLPF mapping reduce tier-specific powers, fading, and access probabilities to common analytical parameters.

SIC is analyzed as a tool for mitigating interference within and across HCN tiers, where receivers connect to one accessible base station.
HCNs are modeled as K-tier marked Poisson networks whose base stations carry independent accessibility marks.
Non-accessible base stations model closed or overloaded cells and can act as pure interferers to typical users.
The PLPF representation combines the K tiers into a marked inhomogeneous PPP with intensity measure Λ([0,r]) = Zπr^β and Bernoulli marks with P(tξ = 1) = η.
Different tier transmit powers, fading distributions, and access probabilities are subsumed by the parameters Z and η.

B. The Coverage Probability

The section defines HCN coverage as successful decoding of at least one accessible base station, possibly after canceling non-accessible interferers. It then characterizes coverage with infinite SIC through a condition involving successive cancellation and accessible base stations.

Coverage is the probability that a typical UE connects successfully to at least one accessible base station after possibly canceling non-accessible base stations.
Without SIC in the interference-limited model, coverage is formulated using the strongest accessible base station and the SIR threshold θ.
When θ ≥ 1, a UE without SIC cannot decode any base station weaker than the strongest one, so coverage depends on decoding the strongest base station and its accessibility.
With infinite SIC, coverage occurs when some accessible base station can be decoded after successively canceling a suitable number of stronger non-accessible base stations.
Proposition 13 expresses the coverage probability with SIC using the probability that an accessible base station remains decodable after cancellation.

P SIC

The paper uses bounds and approximations for successive-decoding probabilities to estimate HCN coverage and throughput under infinite SIC. SIC substantially improves coverage at low SIR thresholds, but its average-throughput benefit can be marginal.

Bounds and approximations provide reasonably good estimates of SIC coverage across the full SIR-threshold range.
SIC yields significant coverage gains when the SIR threshold lies between −10 dB and −5 dB, with the gain depending on η.
The absolute coverage gain from SIC is larger for larger path loss exponent α.
SIC is especially useful for coverage with low-rate codes, but its average-throughput benefit can be marginal.
For the evaluated throughput setting, throughput is maximized at θ about 5 dB, where SIC is not very useful.
Most of the SIC gain at such thresholds can be obtained by canceling a very small number of non-accessible base stations.

P SIC

Finite SIC capability is analyzed through coverage bounds that become tight rapidly as the cancellation capability grows. In contemporary narrow-band HCN settings, most SIC coverage gain comes from canceling only one non-accessible base station.

The finite-SIC lower bound converges to the true coverage probability at least exponentially fast as the capability n increases.
For θ = 2 dB, the n = 2 and n = 10 coverage curves almost completely overlap.
For n = 1, the finite-SIC upper bound is tight, and the bound and simulation curves overlap.
The absolute coverage gain is largest when η is close to 1/2 and smaller when η is close to 0 or 1.
For θ ≥ 1 and β = 1/2, most SIC coverage gain is achieved by canceling a single non-accessible base station.
The difference between infinite SIC and finite cancellation capability decays super-exponentially with n, so additional coverage mainly comes from canceling a small number of non-accessible base stations.
Decoding many base stations becomes more useful as β or θ approaches zero, including clustered networks, low-dimensional networks, and very low-rate codes.

P SIC

The paper develops a unified analysis of SIC and shows that its additional benefit usually diminishes rapidly with cancellation capability, while depending strongly on network geometry, path loss, code rate, and noise. In HCNs, SIC can improve coverage substantially, but contemporary narrow-band systems obtain most of that gain by canceling one interferer.

SIC gain decreases as the path loss exponent decreases because received powers become more comparable and leave less exploitable structure.The associated coverage probability also decreases as the path loss exponent decreases.
The framework analyzes SIC in arbitrary-fading, power-law wireless networks and applies the results to coverage and throughput in heterogeneous cellular networks.The analysis uses a unified PLPF-based framework and counts information rates from all decodable transmitters for aggregate throughput.
The marginal gain of adding cancellation capability diminishes very quickly, especially with high dimension, small path loss exponent, or high-rate codes.The probability of successively decoding at least k users is described as decaying super-exponentially with k under high-rate codes.
SIC is especially beneficial with very low-rate codes, clustered transmitters, or low-dimensional networks such as d = 1.The framework also connects clustering to network-design choices such as distance-dependent access control or power control.
In interference-limited networks, aggregate throughput decreases with per-user information rate, whereas noisy networks have at least one positive rate maximizing aggregate throughput.The paper also reports that fading harms SIC performance in noisy networks.
Most SIC gain in uniform 2-d HCNs comes from canceling a single interferer at the UE side.This conclusion is reported for realistic contemporary narrow-band cellular-system parameters.

APPENDIX A

The appendix develops analytical bounds and probability expressions for sequential multi-user decoding and successive interference cancellation using path-loss and interference models. It also establishes structural properties of decodable user sets and decoding limits when θ ≥ 1.

Order-statistics arguments transform the relevant random variables into uniform distributions to evaluate decoding probabilities and establish successive bounds.
The appendix derives lower bounds on the probability of successively decoding at least k users using conditional interference analysis and Markov's inequality.
The analysis models interference through Poisson path-loss processes, induced exponential fading, Laplace transforms, and probability generating functionals.
For θ ≥ 1, at most one user can be decoded without SIC, even after arbitrarily many cancellation operations.
For θ ≥ 1, multiple packet reception cannot be achieved through parallel decoding, motivating sequential multi-user decoding.

APPENDIX D

The appendix extends the analytical framework to heterogeneous cellular networks and characterizes coverage through accessible and non-accessible base-station marks. It also derives bounds and equality conditions for SIC decoding events.

The first accessible base station index M is geometrically distributed and independent of the path-loss process.
Coverage probability is expressed through the probability of decoding the M-th strongest base station when it is the strongest accessible base station.
The appendix proves equivalence between the indicator functions defining SIC decoding events and accessible-base-station decoding events.
When θ ≥ 1, decoding events beyond the allowed number of sequentially decoded users have probability zero, yielding equality in the corresponding bound.