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Uplink Macro Diversity of Limited Backhaul Cellular Network

Amichai Sanderovich, Oren Somekh, H. Vincent Poor, Shlomo Shamai

arXiv:0805.4620v1cs.IT

TL;DR

The paper addresses uplink joint multicell processing with finite-capacity backhaul, rather than the infinite-capacity links assumed in much prior work. It derives compress-and-forward and local-decoding achievable rates for symmetric Wyner-type Gaussian and fading models, and finds that both schemes can approach the upper bound in complementary backhaul regimes.

  • Problem

    Prior multicell-processing analyses mostly assume reliable backhaul links with infinite capacity, whereas this work studies finite-capacity links between cell-sites and the central processor.

  • Method

    The paper derives achievable rates for oblivious distributed Wyner-Ziv compress-and-forward and partial local decoding in Wyner and soft-handoff models.

  • Results

    Both schemes approach the upper bound when backhaul capacity is either low relative to the unlimited rate or high relative to it, for Gaussian and fading channels.

  • Takeaways & Limitations

    Local decoding can attain the limited-backhaul upper bound below a threshold, while its benefit is marginal when interference is small.

Abstract

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In this work new achievable rates are derived, for the uplink channel of a cellular network with joint multicell processing, where unlike previous results, the ideal backhaul network has finite capacity per-cell. Namely, the cell sites are linked to the central joint processor via lossless links with finite capacity. The cellular network is abstracted by symmetric models, which render analytical treatment plausible. For this idealistic model family, achievable rates are presented for cell-sites that use compress-and-forward schemes combined with local decoding, for both Gaussian and fading channels. The rates are given in closed form for the classical Wyner model and the soft-handover model. These rates are then demonstrated to be rather close to the optimal unlimited backhaul joint processing rates, already for modest backhaul capacities, supporting the potential gain offered by the joint multicell processing approach. Particular attention is also given to the low-SNR characterization of these rates through which the effect of the limited backhaul network is explicitly revealed. In addition, the rate at which the backhaul capacity should scale in order to maintain the original high-SNR characterization of an unlimited backhaul capacity system is found.

I. INTRODUCTION

The paper studies uplink joint multicell processing when cell-sites connect to a central processor through finite-capacity backhaul links. It develops achievable schemes for symmetric Wyner-type cellular models under Gaussian and fading channels, including oblivious compression and partial local decoding.

  • Motivation: Joint multicell processing is analyzed because it has been identified as a tool for enhancing cellular-system performance.The models retain cellular interference structure while facilitating analytical treatment.
  • Motivation: Finite-capacity backhaul is introduced to replace the idealized infinite-capacity links commonly assumed in multicell-processing analyses.The cell-sites remain connected by reliable, error-free links to a remote central processor.
  • Approach: The paper evaluates an oblivious scheme using distributed Wyner-Ziv compress-and-forward and a partial local-decoding scheme that splits user messages.The oblivious scheme forwards compressed received signals for joint decoding, while the second scheme also decodes part of each message locally at the relevant base stations.
  • Models: The analysis uses circular Wyner and circular soft-handoff models with infinitely many cells, covering both non-fading Gaussian and flat Rayleigh fading channels.In the Wyner model each user is received by three base stations; in the soft-handoff model each user is received by its local and left-neighboring base stations.
  • Contributions: The paper characterizes the effects of limited backhaul at low SNR and determines how backhaul capacity must scale with SNR to preserve unlimited-backhaul high-SNR behavior.It provides closed-form low-SNR parameters for both limited and unlimited backhaul versions.

A. Gaussian Channels, No Fading (nf)

This section reviews unlimited-backhaul Wyner and soft-handoff benchmarks for Gaussian and fading cellular uplinks. It also records low-SNR characterizations and contrasts fading with non-fading transmission schemes.

  • Wyner Model: The Wyner model’s per-cell sum-rate capacity is reviewed together with its low-SNR characterization.The cited results provide the benchmark used later for limited-backhaul analysis.
  • Transmission protocols: For non-fading channels, both intra-cell TDMA and wide-band protocols can achieve the unlimited-backhaul Wyner rate under a total cell-power constraint.The result is stated for the asymptotic Wyner model and fixed total cell power.
  • Soft-Handoff Model: The soft-handoff model likewise has reviewed per-cell sum-rate and low-SNR characterizations, with both intra-cell TDMA and wide-band protocols capacity achieving under a total cell-power constraint.The soft-handoff setup is treated as a complementary unlimited-backhaul benchmark.
  • Fading channels: In flat Rayleigh fading, intra-cell TDMA is no longer optimal, while the wide-band protocol is capacity achieving.For large K, its per-cell sum-rate has a tight upper bound corresponding to a single-user non-fading link with an additional multicell gain.

C. Upper Bound

The section develops upper bounds and achievable rates for symmetric Wyner and soft-handoff uplinks with limited backhaul, including oblivious cell-sites and local decoding. For infinitely many cells, the central achievable-rate result is obtained by optimizing a compression parameter and approaches unlimited-backhaul joint processing as capacity grows.

  • Upper bound: The cut-set-like bound limits per-cell sum-rate using cuts between the central processor and cell-sites and between cell-sites and mobile stations.The second cut yields the per-cell sum-rate of the corresponding unlimited-backhaul setup, while accounting for the absence of mobile-station cooperation.
  • Achievable schemes: Oblivious cell-sites compress their received signals and forward them to the remote central processor for joint message decoding.The general achievable rate region uses compression parameters r_j = I(Y_j; U_j|X_N), constrained by the link capacities.
  • Gaussian channels: Intra-cell TDMA is optimal for achievable throughput in the considered non-fading homogeneous model.The result applies to both the Wyner and soft-handoff models with equal-rate users and equal-capacity links.
  • Gaussian channels: For Gaussian channels, complex Gaussian signaling reduces the achievable-rate expression to a form involving the channel transfer matrix and optimized compression parameters.In the symmetric case, equal capacities and concavity imply an invariant compression parameter r_j = r, with the sum-rate inequality dominant.
  • Gaussian channels: For α > 0 and N →∞, Proposition IV.3 gives a central achievable rate for circular models with equal limited capacities and oblivious cell-sites.The rate is expressed through F(r*) with r* determined by the implicit equation F(r*) = C − r*; monotonicity permits numerical solution.
  • Gaussian channels: When C →∞, the optimized rate reduces to the per-cell sum-rate capacity of optimal joint processing with unlimited backhaul.For finite C, the implicit equation defining r* remains numerically tractable because F(r) is monotonic for the symmetric models.

1) Low-SNR Characterization:

The low-SNR analysis characterizes how limited backhaul changes achievable rates through minimum energy per bit and low-SNR slope. It also identifies conditions under which the limited-backhaul characterization approaches the unlimited-backhaul one, while the high-SNR analysis shows a fixed-capacity ceiling.

  • Low-SNR characterization: The low-SNR characterization uses a Taylor expansion of the per-cell sum-rate and yields minimum energy per bit and low-SNR slope parameters.The expansion is derived for the asymptotic achievable rate in the low-SNR regime P ≪ 1.
  • Low-SNR characterization: With increasing backhaul capacity, the limited-backhaul low-SNR characterization coincides with that of the unlimited-backhaul channel.The low-SNR parameters are expressed in a form that makes the backhaul effect explicit.
  • Low-SNR characterization: Allocating at least C ≈3.2 bits/sec/Hz to backhaul keeps the minimum required energy within 0.5 dB of the unlimited-backhaul value.This quantitative statement concerns the limited channel relative to the unlimited-backhaul baseline.
  • High-SNR characterization: For fixed backhaul capacity C and increasing SNR P, the oblivious-scheme rate converges to C and has zero multiplexing gain.Thus, the high-SNR rate remains finite when C is fixed.

2) High-SNR characterization:

The paper derives high-SNR conditions under which limited-backhaul systems preserve the characterization of their unlimited-backhaul counterparts. The required backhaul scaling and the low-SNR effects of finite capacity are made explicit.

  • High-SNR scaling: For high SNR, replacing P by P(1 − 2^-r⋆) relates the limited-backhaul system to an unlimited-backhaul system with equivalent high-SNR characteristics.This relation is used when P(1 − 2^-r⋆) is very large.
  • High-SNR scaling: A larger gap between backhaul capacity C and rate F accelerates convergence to the unlimited-backhaul high-SNR behavior.The paper characterizes this gap through Θ(P).
  • High-SNR scaling: C(P) = S∞log2 P + Θ(P), with Θ(P) →∞ as P →∞, suffices to preserve the unlimited-backhaul high-SNR characterization.The exact growth of Θ(P) may be arbitrarily slow, such as log2 log2 P.
  • Rate characterization: The Wyner and soft-handoff models admit closed-form expressions for the achievable rate using their respective known unlimited-backhaul results.The soft-handoff result is stated for oblivious base stations with equal limited backhaul capacities.
  • Asymptotic behavior: For the soft-handoff model, the achievable rate approaches the cut-set bound when either backhaul capacity C or transmit power P increases while the other remains fixed.The corresponding low-SNR penalty is an increased minimum energy per bit and a reduced low-SNR slope, with both effects diminishing as C increases.

B. Fading Channels

The fading-channel analysis extends the limited-backhaul achievable-rate results to Wyner and soft-handoff models. It provides bounds and asymptotic characterizations, while exact rates remain available only in selected cases.

  • Wyner model: The fading Wyner model has a per-cell achievable ergodic sum-rate for the WB protocol with equal limited capacities.The result is obtained by extending the Gaussian-channel analysis with conditioning and expectations over the fading channel.
  • Wyner model: For large K, the constant-r⋆ lower bound is tight, with a very small gap expected already at K = 2.The argument uses channel ergodicity and the six received signals available at each cell-site when K = 2.
  • Wyner model: For the Wyner model, limited backhaul increases minimum energy per bit and decreases the low-SNR rate slope.These are the same low-SNR penalties identified for the non-fading limited-backhaul setting.
  • Soft-handoff model: For the soft-handoff model, the fading result replaces the Wyner transfer matrix with the soft-handoff matrix, while closed-form unlimited-backhaul expressions remain available only in special cases.The analysis includes intra-cell TDMA and WB protocols.
  • Soft-handoff model: For the soft-handoff WB upper bound, C →∞ recovers the unlimited setup, P →∞ with fixed C reaches C, and C scaling like log2 P achieves multiplexing gain 1.The rate itself may require numerical solution in the relevant fading special case.
  • Soft-handoff model: The soft-handoff fading upper bound is tight for K ≫1.This follows from the asymptotic expression used for the rate bound.

V. CELL-SITES WITH DECODING

The paper introduces a rate-splitting scheme that combines local cell-site decoding with central joint decoding under limited backhaul. It characterizes achievable rates, low-SNR behavior, and when local decoding is beneficial.

  • Scheme: Each user splits its message into a central-decoded part and a locally decoded part, allocating powers βP and (1 − β)P.The central-decoded message can interfere with local decoding at the relevant cell-site.
  • Scheme: Forwarding decoded information consumes backhaul capacity that would otherwise support compression, yielding the separate-decoding rate Rsd(C).Cell-sites may decode only their local message or also neighboring interfering messages.
  • Special case: For α = 0, the rate-splitting scheme is optimal because each cell-site can decode at the same rate as the central processor.With no inter-cell interference, local decoding does not incur the corresponding interference penalty.
  • Time-sharing: Time-sharing between local decoding and central decoding can improve the achievable rate because Rsd(C) is not generally concave in C.Numerical calculations favor the two extreme approaches over simultaneous mixed decoding.
  • Low-SNR characterization: The hybrid scheme’s low-SNR parameters can be expressed through the low-SNR parameters of its local-decoding and oblivious components.When λo = 0, or C ≥ r̃m, they coincide with those of the oblivious scheme.
  • High-SNR characterization: With fixed finite C and α > 0, partial local decoding loses multiplexing gain at high SNR and cannot attain the unrestricted joint-processing high-SNR parameters.The analysis therefore focuses on backhaul capacity that increases with SNR.

4) High-SNR Characterization:

The high-SNR analysis of local decoding specializes the general scheme to the Wyner model. It identifies concrete decoding strategies and shows how interference affects the usefulness of local processing.

  • Wyner model: Wyner local decoding can decode all three received messages, only the strongest message, or only adjacent-cell signals.These alternatives produce the local-decoding rate used in the hybrid analysis.
  • Wyner model: For the Wyner model, substituting the local-decoding characterization into the hybrid expressions yields the achievable rate.The low-SNR parameters are obtained by combining the local-decoding and oblivious characterizations.
  • Low-SNR characterization: The low-SNR decoding time ratio is λo = 1 − C/rm.Here rm is determined using the low-SNR optimization and fixed-point characterization.
  • Low-SNR characterization: The threshold r̃m decreases as the intra-cell interference factor α increases, so local decoding becomes beneficial below a lower backhaul threshold.At α = 0, r̃m = ∞ and local decoding alone is optimal for any C; at α = 0.2, r̃m ≈ 2.15 bits.

2) The Soft-Handoff Model:

For the soft-handoff model, the paper extends the limited-backhaul achievable-rate analysis to fading and large-user regimes, including low-SNR characterizations. The resulting expressions parallel those for the Wyner model, with the soft-handoff model’s distinct power gain.

  • Fading channels: Fading-channel achievable rates are obtained by repeating the limited-backhaul construction with fading-specific rate expressions.The construction covers finite-user and large-user regimes.
  • Large-user regime: For large K with fixed total cell SNR P, the analysis uses the strong law of large numbers to reduce the fading expressions to simpler forms.The resulting formulation uses F(r) = Rrf−lk(P(1−2−r)).
  • Low-SNR regime: The soft-handoff low-SNR characterization combines the low-SNR parameters of local decoding and oblivious processing.For finite K, the threshold parameter is rm = max{C, ˜rm}.
  • Implications: The soft-handoff analysis reports conclusions similar to those established for the corresponding non-fading channels.The same type of low-SNR threshold behavior is identified for the fading setting.
  • Computational caveat: The fading expressions include integrations over hypergeometric functions, but these integrals are omitted because they are numerically unstable, especially for large K.The expressions are noted as mathematically rewritable but computationally unstable.

C. The Soft-Handoff Model - Fading Channels

The soft-handoff fading analysis evaluates finite- and large-user achievable rates under limited backhaul and compares local decoding and oblivious processing numerically. Across the reported examples, local decoding helps particularly when backhaul capacity is small or interference is limited.

  • Rate construction: Finite-user soft-handoff fading rates are constructed for several cases, including compact suboptimal rates, upper bounds, and TDMA with Rayleigh fading.The construction replaces the Wyner transfer matrix with the soft-handoff matrix where appropriate.
  • Large-user regime: For large K with fixed total cell SNR P, the Wyner array power gain 1 + 2α2 is replaced by the soft-handoff array power gain 1 + α2.The large-user fading expressions otherwise follow the corresponding Wyner derivation.
  • Numerical results: At P = 10 dB, limited-backhaul rates for C = 3 and C = 6 bits/channel use reveal backhaul degradation, while local decoding benefits low-interference conditions.The interference threshold for local-decoding benefit decreases as C increases.
  • Numerical results: For C = 3, local decoding reaches the limited-backhaul upper bound below an interference threshold, whereas oblivious processing does not reach it at finite C.At C = 6, the local-decoding equality range reduces to α = 0.
  • Fading channels: With Rayleigh fading, rates generally increase with the number of users per cell K, and the qualitative observations resemble those for Gaussian channels.The comparison is reported for C = 3 and C = 6.
  • SNR dependence: With C = 6 and α = 0.15, oblivious and local-decoding rates are close because interference is small, while oblivious processing approaches the upper bound at low and high SNR extremes.The same behavior is observed for fading channels.
  • Backhaul dependence: For Gaussian and fading channels, both schemes approach the upper bound when C is either much below or much above the unlimited rate; local decoding reaches C below a threshold.This behavior is shown as a function of backhaul capacity.
  • Low-SNR validation: Low-SNR approximations broadly match the numerical results, although the approximated low-SNR slope is somewhat optimistic and local decoding helps when C is small.The numerical study uses C = 2, 4, and 6 bits in the Wyner Gaussian setup.

APPENDIX I PROOF OUTLINE OF PROPOSITION IV.1

The appendix proof outline establishes an achievable rate region for lossless finite-backhaul links using random codebooks, compression indices, joint typicality, and separate error-event bounds.

  • Proof setup: The proof invokes a generalized Markov lemma to guarantee jointly typical compressed codewords with high probability.The lemma is attributed to an earlier result and applies to independently generated compression vectors.
  • Code construction: The construction generates user codewords and cell-site compression codebooks, partitioning compression vectors into sets indexed by the backhaul messages.The compression indexing depends on the channel realization H.
  • Encoding and decoding: Each cell-site sends a backhaul index through its lossless link, and the destination jointly identifies compressed vectors and user messages.Correct decoding requires a unique jointly typical tuple.
  • Error analysis: The error analysis separates E1, failure to find a jointly typical compressed vector, from E2, an incorrect message decision.Both probabilities are driven arbitrarily small for sufficiently large block length under the stated rate conditions.
  • Error conditions: The covering condition ˆRj > I(Uj; Yj|H) ensures that E1 becomes arbitrarily unlikely as block length grows.This condition controls whether the cell-site can find a suitable compression codeword.
  • Error conditions: Bounding E2 over incorrect message and compression-index subsets yields the achievable rate region RN,K for all relevant subsets.The appendix states that the resulting region completes the proof.

APPENDIX II PROOF OF LEMMA IV.4

The appendices derive soft-handoff limited-backhaul expressions and connect asymptotic multicell analysis to stationary inter-symbol-interference methods. The soft-handoff fixed-point calculation reduces to a quadratic equation with an explicitly selected valid solution.

  • Asymptotic proof technique: The asymptotic proof treats the multicell system by analogy with a stationary ISI channel and uses a relationship between MMSE and mutual information.The argument focuses on sets of consecutive indices in the large-system limit.
  • Asymptotic proof technique: For consecutive-index sets, stationarity and existence of the relevant limit support the asymptotic argument, which is completed by integration.The proof explicitly invokes channel ergodicity.
  • Soft-handoff rate derivation: The soft-handoff limited-backhaul per-cell sum-rate is obtained using the same general proposition as the Wyner setup.Its fixed-point function uses the unlimited soft-handoff rate as an input.
  • Fixed-point calculation: The soft-handoff fixed-point equation becomes a quadratic because the function F(r∗) has an explicit simple form.The quadratic contains the soft-handoff power-gain term 1 + α2.
  • Fixed-point calculation: Choosing the positive quadratic-root branch gives a valid r∗ smaller than the backhaul capacity C for all P, C, and α.The achievable rate is then C − r∗.

FADING CHANNELS, EQUATION (4-34)

The derivation constructs an achievable rate by time-sharing local decoding with oblivious processing, optimizing the mixing parameter and an auxiliary rate variable under the backhaul constraint.

  • Replacing F(r*) with an upper bound F ub(r) provides a valid solution that also upper-bounds F(r*).
  • Solving the resulting fixed-point equation yields a quadratic equation for the relevant rate parameter.
  • The selected positive-root solution is valid and remains below the backhaul capacity C for all P, C, and α.
  • The achievable rate is derived by time-sharing between local decoding and oblivious processing followed by remote joint decoding.The local-decoding point is (t, t), while the oblivious-processing curve is (F(r)+r, F(r)).
  • The time-sharing weight λ and auxiliary rate r′ satisfy two linear constraints fixing the backhaul usage C and decoded rate Rdec.
  • Optimizing over r′ is restricted by 0 ≤ λ ≤ 1, which requires r′ to exceed r*.
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