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A Tractable Model for Non-Coherent Joint-Transmission Base Station Cooperation
Ralph Tanbourgi, Sarabjot Singh, Jeffrey G. Andrews, Friedrich K. Jondral
TL;DR
BS cooperation lacks tractable models that capture irregular deployments and interacting cooperation mechanisms without relying on simplistic abstractions or opaque simulations. This paper develops a stochastic-geometry model for non-coherent joint transmission and applies it to SINR, cluster sizing, and intra-cluster scheduling, finding density and path-loss gains but practical limits from cluster size and cell load.
Problem
BS cooperation is difficult to analyze because realistic models must capture multiple cooperating serving BSs and channel-dependent cooperation, while existing alternatives are simplistic or opaque.
Method
The paper models BS locations using a Poisson point process and derives a tractable, general SINR distribution for user-centric non-coherent joint transmission, approximating dependent interference effects with moment-matched Gamma variables.
Results
The analysis finds that SINR improves with uniformly increased BS density and larger path-loss exponent, while imperfect-CSI spectral efficiency saturates around 7 cooperating BSs and scheduling savings can reach 60%.
Takeaways & Limitations
For NC-JT, coordinated scheduling suits lightly loaded cells with generous cooperation activation, whereas frequency reuse can be preferable in moderately loaded cells to save resources and avoid overload.
Takeaways & Limitations
When cell load limits performance, increasing cluster size can reduce per-user throughput, so larger clusters may not be beneficial regardless of backhaul effort.
Abstract
from arXiv · showhide
This paper presents a tractable model for analyzing non-coherent joint transmission base station (BS) cooperation, taking into account the irregular BS deployment typically encountered in practice. Besides cellular-network specific aspects such as BS density, channel fading, average path loss and interference, the model also captures relevant cooperation mechanisms including user-centric BS clustering and channel-dependent cooperation activation. The locations of all BSs are modeled by a Poisson point process. Using tools from stochastic geometry, the signal-to-interference-plus-noise ratio ($\mathtt{SINR}$) distribution with cooperation is precisely characterized in a generality-preserving form. The result is then applied to practical design problems of recent interest. We find that increasing the network-wide BS density improves the $\mathtt{SINR}$, while the gains increase with the path loss exponent. For pilot-based channel estimation, the average spectral efficiency saturates at cluster sizes of around $7$ BSs for typical values, irrespective of backhaul quality. Finally, it is shown that intra-cluster frequency reuse is favorable in moderately loaded cells with generous cooperation activation, while intra-cluster coordinated scheduling may be better in lightly loaded cells with conservative cooperation activation.
I. INTRODUCTION
The paper develops a tractable stochastic-geometry model for non-coherent joint-transmission BS cooperation in irregular cellular networks, addressing complex interactions among clustering, channel-dependent activation, fading, path loss, and interference. It characterizes the SINR distribution and derives design insights about densification, cluster size, and intra-cluster scheduling.
- A. Challenges facing BS Cooperation: BS cooperation is difficult to analyze because simplistic models obscure parameter effects, while system-level simulations are time-consuming and opaque.The challenge includes user-to-BS association, opportunistic channel-dependent cooperation, and jointly characterizing useful signal and interference powers.
- B. Related Work: Multiple cooperating BSs are modeled within a fixed cooperation area, with arbitrary finite-moment fading and channel-dependent cooperation activation.The fixed area reflects constraints such as limited backhaul complexity or cost.
- C. Contributions and Outcomes: The model uses stochastic geometry to characterize the SINR distribution of a typical user served by multiple cooperating BSs under NC-JT.The result has a compact semi-closed form and does not require a particular fading distribution.
- C. Contributions and Outcomes: Increasing the path loss exponent increases cooperation gains, while increasing network-wide BS density in a fixed cooperation region decreases SINR outage probability exponentially.The densification can be random without careful site planning.
- C. Contributions and Outcomes: For typical scenarios with pilot-based MMSE channel estimation, average spectral efficiency stops benefiting from cluster sizes beyond roughly 7 BSs, even with perfect backhaul.Imperfect CSI becomes performance-limiting as cluster size increases.
- C. Contributions and Outcomes: Intra-cluster frequency reuse is favored in moderately loaded cells with generous cooperation activation, whereas coordinated scheduling is favored in lightly loaded cells with conservative activation.Frequency reuse can provide load gains, while coordinated scheduling suits lightly loaded cells.
A. Cooperation and Cluster Model
The model combines user-centric clustering with channel-dependent NC-JT activation, representing signal, intra-cluster, and out-of-cluster interference in a stochastic-geometry framework. A Gamma approximation makes the interference-plus-noise term tractable and matches simulated SINR distributions well across tested settings.
- Cooperation and clustering: User-centric clustering assigns each user a cooperative BS set based on long-term received signal strength, grouping sufficiently close BSs.The cooperative region C defines which BSs may participate for a typical user.
- Signal and interference model: NC-JT combines useful signals non-coherently, while inactive in-cluster BSs contribute interference under frequency reuse or avoid same-resource transmissions under coordinated scheduling.The spectral efficiency is R = log2(1 + SINR), and out-of-cluster BSs are treated as interfering sources for tractability.
- Cooperation and clustering: Channel-dependent activation engages a cooperative BS only when its instantaneous received signal strength exceeds threshold T ≥ 0.The threshold can be represented relative to the received power from a hypothetical BS at the cluster edge.
- Interference-plus-noise approximation: The denominator of the SINR is approximated by a Gamma random variable whose shape and scale are obtained through second-order moment matching.This approximation replaces the compound interference-plus-noise term with a tractable distribution.
- Interference-plus-noise approximation: Increasing the cooperative region converts more interfering BSs into cooperative BSs, while the limiting case T = 0 makes interference arise only outside the cluster.As D → ∞, the relevant interference moments tend to zero; as D → 0, the moments do not exist under the singular path-loss law.
- Interference-plus-noise approximation: The Gamma approximation provides a good fit to empirical CDFs over a wide range of system parameters, including the tested path-loss exponents and cooperation-region settings.The comparison uses η = 162 dB, with representative regions averaging approximately 3 and 8 cooperative BSs.
B. Main Result: SINR Distribution
The paper derives bounded and approximated SINR distributions for non-coherent joint transmission using Laplace-transform derivatives, while retaining broad fading-model generality. It also analyzes tightness and path-loss effects, showing that SINR worsens as the path-loss exponent approaches 2.
- Main SINR characterization: Theorem 1 bounds the typical user's SINR CDF using derivatives of the useful-power Laplace transform.The result is stated in a compact form involving Laplace-transform derivatives.
- Main SINR characterization: The formulation remains general because it requires only the useful-power Laplace transform and does not specify the fading PDF.Laplace-transform superposition can also be exploited for independent fading components.
- Effect of path loss: As α approaches 2, P(SINR ≤ β) approaches 1 because interference from many far BSs outweighs the milder path loss of cooperative links.The paper therefore expects SINR to improve with larger α.
- Approximation and tightness: The approximation interpolates between the lower and upper bounds according to the distance of k from its neighboring integers.The bounds arise because the relevant sum is truncated at floor(k) or extended to ceil(k).
- Approximation and tightness: The approximation is remarkably tight, and either bound is exact whenever k is integer-valued.The worst-case gap equals the last summand in the bounding expression.
1) Fixed Number of Cooperating BSs:
For a fixed cooperative-cluster size, conditioning the Poisson deployment on exactly K cooperating BSs yields a binomial point process. The analysis treats useful power and cluster interference as independent, with the resulting discrepancy counted as additional Gamma-approximation error.
- Conditional model: Fixing the number of cooperating BSs to K conditions the Poisson point process and makes the cluster locations follow a binomial point process.This conditional formulation enables comparison with analyses that assume a fixed cluster size.
- Approximation assumption: In the conditional case, useful power P and cluster interference J_C are negatively correlated but are treated as statistically independent.The resulting discrepancy is interpreted as additional inaccuracy of the Gamma approximation.
- Conditional model: Lemma 1 provides the Laplace transform of the useful-power distribution conditioned on exactly K cooperating BSs.The proof is referenced to an appendix.
2) Poisson Number of Cooperating BSs:
The unconditional model treats the number of cooperative BSs inside the cooperation region as Poisson-distributed and derives its interference characterization by deconditioning over cluster size.
- The number of cooperative BSs inside C is modeled as a Poisson random variable, defining the unconditional case.The unconditional result is obtained by deconditioning the conditional Laplace transform on the random cluster size.
- The unconditional interference Laplace transform is given in Lemma 2.
- Increasing BS density λ increases both interference and the likelihood of joint service by multiple cooperative BSs.The cooperative effect appears through decay of the interference Laplace transform.
- As λ →∞, the SINR increases even though denser deployment adds cooperative and interfering BSs simultaneously.This conclusion assumes fixed D, so the average cluster size K also increases.
- The required derivatives can be computed efficiently using Faà di Bruno’s rule and Bell polynomials.
C. Numerical Examples
The numerical examples validate the analytical approximations across fading and rate transformations, then examine imperfect CSI, cluster-size saturation, and scheduling-related trade-offs.
- Numerical validation: The Gamma approximation accurately fits the SINR CDF over a wide range of fading and system parameters.For larger α in the conditional case, accuracy slightly decreases because intra-cluster interference dominates and induces negative correlation.
- Numerical validation: Theorem 1 and Corollary 1 retain their accuracy after transforming SINR into rate R = log2(1 + SINR).
- Imperfect CSI: Pilot-based MMSE channel estimation is modeled through an effective SINR, with estimation resources shared among K cooperating BSs.The factor 1/K accounts for sharing pilot resources among cooperative transmitters, while estimation error accumulates with K.
- Imperfect CSI: K = 7 is the approximate spectral-efficiency saturation point for typical Npilot values.Imperfect CSI has little effect for small K, where out-of-cluster interference limits performance.
- Imperfect CSI: Larger cluster sizes are likely to provide little benefit because NC-JT increases cell load, potentially reducing per-user throughput in high-load scenarios.This conclusion is expected to hold regardless of the engineering effort spent on backhaul.
- Imperfect CSI: The imperfect-CSI approximation closely matches simulation results across K, with a noticeable discrepancy mainly at K = 1.
B. NC-JT Intra-Cluster Scheduling: FR vs. CS
The comparison between intra-cluster frequency reuse and coordinated scheduling exposes a trade-off between radio-resource savings and intra-cluster interference.
- Intra-cluster frequency reuse allows non-participating cooperating BSs to reuse NC-JT resources, whereas coordinated scheduling prohibits those transmissions.
- Switching from coordinated scheduling to frequency reuse is evaluated through the spatially averaged radio-resource saving.
- The resource-saving measure Δ is independent of BS density λ.For exponential fading, the paper gives a closed-form reduction in terms of the activation threshold and path-loss exponent.
- At small T, frequency reuse changes SINR little while saving approximately 12.8% of radio resources; at larger T, savings can reach 60% but worsen SINR.Frequency reuse is favored in moderately loaded cells, whereas coordinated scheduling is favored in lightly loaded cells to mute intra-cluster interference.
V. DISCUSSION AND CONCLUSION
The paper develops a tractable NC-JT cooperation model and derives design insights about density, path loss, cluster size, and intra-cluster coordination.
- Increasing BS density while fixing the cooperation radius improves SINR, with cooperation gains increasing as the path loss exponent grows.
- Average spectral efficiency saturates at a cluster size of around 7 BSs when CSI-R is imperfect.
- Intra-cluster coordinated scheduling should be used in lightly loaded cells with generous channel-dependent cooperation activation.
- Intra-cluster frequency reuse should be used otherwise, including settings with more conservative cooperation activation.
- A more practical model would use RSS-difference-based user-centric clustering so that mainly cell-edge users receive cooperation.
APPENDIX A. Non-Coherent JT: Transmission, Reception and SINR
The appendix derives the NC-JT received signal and SINR by modeling OFDM transmission, channel effects, non-coherent interference, and noise. It then uses fading moments and Campbell’s theorem to obtain tractable parameter expressions.
- Transmission and reception: NC-JT models the received OFDM symbol as the circularly convolved useful signal plus interference and receiver noise.The cyclic prefix induces circular convolution and prevents timely-dispersed multipath from creating inter-symbol interference.
- Transmission and reception: The effective channel includes path loss, fading, and independent timing offsets across cooperating BSs.Independent fading is also assumed because cooperative BSs perform no phase-mismatch correction.
- Frequency-domain model: DFT processing converts timing offsets into frequency-dependent modulation terms on each subcarrier.The modulation term may vary across subcarriers even when fading coefficients remain constant over the coherence bandwidth.
- Frequency-domain model: CSI-R phase correction determines the useful-signal power from the effective channel gain on each subcarrier.
- SINR derivation: Non-coherent superposition of interfering transmissions yields the interference-plus-noise power, which combines with the signal power to form SINR.The normalized transmit-to-noise parameter is defined as η = ρ/σ2.
- Fading approximation: The fading approximation parameters are obtained by matching the first two moments, computed using Campbell’s theorem.The relations are E[˜J] = kθ and Var[˜J] = kθ2.
C. Proof of Theorem 1
The proof derives the SINR CDF by conditioning on the serving-related variable, applying Laplace-transform convolution, and evaluating the resulting contour integrals with residues.
- Conditioning on P and applying the law of total probability produces the starting representation for P(SINR ≤ β).
- A possible jump of the density at P = 0 is separated before applying Laplace-transform arguments.Excluding that jump makes the density strictly continuous and adjusts its Laplace transform by fP(0−).
- The s-convolution theorem combines the Laplace transforms of the relevant distributions because their abscissas of convergence sum to −1/θβ.
- The contour parameter c is chosen in (−1/θβ, 0), while both integrands have a singularity at z0 = −1/θβ.
- Residue-theorem evaluation converts the first contour integral to a residue at z0 and treats the second analogously.The contour is parameterized by z → Re^jφ − c, and dominated convergence controls the limiting integral.
- Integer-valued replacement of k ensures holomorphy for the residue calculation, using either ⌊k⌋ or ⌈k⌉.Laurent and Taylor expansions determine the coefficient needed for the residue.
- Substituting the evaluated integrals back into the conditioned expression yields the theorem’s result.
D. Proof of Lemma 1
The lemma evaluates a stochastic-geometry integral by exploiting the independence of binomial point processes, the distance density, a variable substitution, and partial integration.
- The proof evaluates the target integral using BPP independence, f∥x∥(r) = 2r/D2, the substitution r^-α → t, and partial integration.
E. Proof of Lemma 2
The proof de-conditions on the Poisson cooperation count, derives the interference Laplace transform using the PPP probability generating functional, and evaluates the resulting derivatives through standard integration steps.
- De-conditioning on K, with K = Φ(C) Poisson with mean λπD2, initiates the derivation.
- The PPP probability generating functional yields the interference Laplace transform LP(s) in exponential integral form.
- Tonelli’s theorem, Leibniz’s integration rule, and the substitution r^-α → t justify the derivative calculation and final evaluation at s = −/θβ.