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Relay Selection and Power Allocation in Cooperative Cellular Networks

Sachin Kadloor, Raviraj Adve

arXiv:0903.2426v1cs.ITmath.IT

TL;DR

In multiuser relay networks, assigning each flow to a relay and allocating relay power jointly is combinatorial, while prior work has not jointly addressed these choices in cellular networks. This paper develops convex bounds and a heuristic, finding that selection is usually inherent in the solution and that the bounds are extremely tight; simulated relaying gains remain context-dependent and must be weighed against added costs.

  • Problem

    In multiuser cellular networks, joint relay assignment and power allocation have not been addressed together, even though each relay must divide limited power among its supported users.

    Source: §I · p. 2

  • Method

    The paper formulates convex upper bounds for sum rate and minimum user rate, analyzes when selection is inherent, and derives a heuristic for relay assignment and power allocation.

    Source: §I · p. 2, §V · p. 19

  • Results

    For practical user counts, the upper bound is indistinguishable from the true solution, and the heuristic’s lower bound is extremely tight.

    Source: §V · p. 19

  • Takeaways & Limitations

    Simulations indicate significant but context-dependent gains from relaying, leaving open whether they compensate for additional infrastructure costs.

    Source: §I · p. 2, E. Discussion · p. 19

    The results are optimistic because they assume relays can always decode and transmitters have the information needed for optimal decisions.

    Source: E. Discussion · p. 19

Abstract

from arXiv · show

We consider a system with a single base station communicating with multiple users over orthogonal channels while being assisted by multiple relays. Several recent works have suggested that, in such a scenario, selection, i.e., a single relay helping the source, is the best relaying option in terms of the resulting complexity and overhead. However, in a multiuser setting, optimal relay assignment is a combinatorial problem. In this paper, we formulate a related convex optimization problem that provides an extremely tight upper bound on performance and show that selection is, almost always, inherent in the solution. We also provide a heuristic to find a close-to-optimal relay assignment and power allocation across users supported by a single relay. Simulation results using realistic channel models demonstrate the efficacy of the proposed schemes, but also raise the question as to whether the gains from relaying are worth the additional costs.

I. INTRODUCTION

Relay selection reduces the overhead of cooperative wireless systems, but assigning relays across multiple users while allocating their limited power is combinatorial. The paper develops convex bounds and a heuristic for this joint problem, then studies a cellular system in which users communicate with help from dedicated relays.

I. INTRODUCTION

  • Selection cooperation lets a single best relay help the source, avoiding the bandwidth inefficiency of having all relays transmit on orthogonal bands and the synchronization demands of distributed space-time codes.Prior work indicates that this can achieve most cooperative-diversity benefits with minimum overhead.Source: §I · p. 1
  • With multiple information flows, each relay must divide its power among supported users, so the best relay for one flow may not be best overall and assignment becomes combinatorial.Prior approaches approximate assignment but do not address resource allocation.Source: §I · p. 2
  • The paper jointly considers relay assignment and power allocation for maximizing sum rate or minimum user rate, using convex upper bounds whose tightness is analyzed and a simplified tight lower bound.Its OFDMA analogy assigns relays to users, but each relay has its own power constraint.Source: §I · pp. 2–3

II. SYSTEM MODEL

  • The system has one base station, multiple users on orthogonal channels, and dedicated decode-and-forward relays; communication uses two time slots and relay power is divided among supported users.The model uses the same codebook at source and relays; the rate expression accounts for both relay and user decoding.Source: §II · pp. 3–4

III. PROBLEM FORMULATION AND SOLUTION

The paper formulates sum-rate and max-min-rate relay assignment and power allocation as combinatorial problems, then relaxes selection to obtain tractable convex upper bounds. KKT analysis shows that optimal power allocation is usually sparse across relays, motivating a heuristic whose lower bound is reported as indistinguishable from the upper bound.

III. PROBLEM FORMULATION AND SOLUTION

  • The formulation targets both total sum rate and the minimum user rate, while each relay must divide its limited power among its assigned users.This joint assignment and power-allocation problem is combinatorial because each user must be assigned to a relay.Source: §III · p. 4

A. Max sum rate

  • The analysis assumes the base station knows the parameters defining the optimization problem; how it obtains that information is outside the paper’s scope.The model also assumes the source-relay-destination channel is the rate bottleneck.Source: A. Max sum rate · p. 5, §III · p. 5
  • Exhaustive search requires testing all J^K relay assignments and solving J water-filling problems for each, making it impractical for realistic J and K.The selection constraint is the source of the difficult integer-programming structure.Source: A. Max sum rate · pp. 5–6
  • Ignoring the single-relay selection constraint yields a convex optimization problem and an upper bound on the sum rate achievable under selection.The remaining power constraints make the relaxed problem tractable for practical values of K and J.Source: A. Max sum rate · pp. 5–6
  • For two relays, continuous independent channel powers imply that at most one user can receive nonzero power from both relays at an optimum, so all other users satisfy selection.The three-relay analysis likewise finds at most two users with multirelay power; generally, at most J−1 users can have such allocations.Source: A. Max sum rate · pp. 7–8
  • For the general J-relay case, when K is much larger than J−1, a large fraction of users are guaranteed to receive power from only one relay.This sparsity in the power-allocation matrix motivates enforcing selection in a heuristic.Source: A. Max sum rate · pp. 8–9
  • The heuristic assigns users receiving power from multiple relays to a relay allocating them maximum power, then uses per-relay water-filling or the existing allocation to form a lower bound.Tied maximum-power relays may be chosen arbitrarily.Source: A. Max sum rate · p. 9
  • The upper and lower bounds are reported as indistinguishable, so the paper avoids a second optimization to refine the lower bound.The conclusion characterizes the resulting heuristic as efficient and its lower bound as extremely tight.Source: A. Max sum rate · p. 9

B. Max sum rate with a minimum rate constraint

With a minimum rate guaranteed to each user, the relaxed optimization still yields a solution in which almost all users receive power from a single relay. The authors use this structure to construct a relay-assignment heuristic, while noting rare feasibility cases where the bounds are not meaningful.

B. Max sum rate with a minimum rate constraint

  • The formulation guarantees each user a minimum rate while requiring each user to be helped by only one relay.The displayed selection constraint requires αjkαlk = 0 for distinct relays j and l.Source: B. Max sum rate with a minimum rate… · pp. 9–10
  • At most J −1 of the J-tuples of relay powers have multiple non-zero entries, and the minimum-rate constraint does not alter this property.This result is stated for the max-sum-rate formulation with a minimum-rate constraint.Source: B. Max sum rate with a minimum rate… · p. 10
  • The heuristic assigns users receiving power from multiple relays to the relay that allocates the most power.The resulting assignment is followed by J water-filling problems to meet each user's minimum-rate requirement.Source: B. Max sum rate with a minimum rate… · p. 11
  • The bounds are not meaningful in rare cases when the simplified problem is feasible but the original problem or a water-filling problem is infeasible.The passage identifies these feasibility cases as occurring very rarely.Source: B. Max sum rate with a minimum rate… · p. 11

C. Max-min rate

The max-min formulation seeks to maximize the rate guaranteed to every user, and its relaxed problem is convex after reformulation. Its solution has at most J −1 users with power from multiple relays; this structure supports a relay-assignment heuristic.

C. Max-min rate

  • The max-min objective seeks to maximize the minimum user rate, with optimal power allocation giving each user the same data rate.The formulation also requires single-relay assistance per user and satisfies relay power constraints.Source: C. Max-min rate · p. 11
  • Ignoring the selection constraint leaves a concave optimization problem, and the non-differentiable objective is reformulated with differentiable objective and constraint functions.The reformulation uses that the logarithm is monotonically increasing, so maximizing the minimum of logarithms is equivalent to maximizing the minimum of their arguments.Source: C. Max-min rate · pp. 11–12
  • For J relays, at most J −1 users have power allocated from more than one relay in the relaxed max-min solution.For two relays, the solution obtained by ignoring the selection constraint comes quite close to the original solution.Source: C. Max-min rate · p. 13
  • The heuristic assigns each user receiving power from multiple relays to a relay allocating maximum power, then solves J max-min power-allocation problems.If multiple relays allocate the same maximum power, the user is assigned to any one of them arbitrarily.Source: C. Max-min rate · pp. 13–14

D. Independent codebooks at the relays

The paper extends its optimization approach to independent source and relay codebooks. Relaxing the selection constraint gives a concave maximization problem whose solution supplies an upper bound, while a heuristic supplies a lower bound.

D. Independent codebooks at the relays

  • Independent Gaussian codebooks at the source and relays yield a sum-rate optimization problem with a single-relay selection constraint.Independent codebooks result in higher rates, but decoding the source and relay messages is significantly more complex than with repetition coding.Source: D. Independent codebooks at the relays · p. 14
  • Relaxing the selection constraint gives a concave maximization problem whose solution provides an upper bound on sum rate.The heuristic constructed from this relaxation also serves as a lower bound.Source: D. Independent codebooks at the relays · p. 14
  • A max-min optimization problem can also be formulated for the independent-codebook case.Source: D. Independent codebooks at the relays · p. 14

IV. NUMERICAL RESULTS AND DISCUSSION

The simulations compare relay-assisted cellular networks with SISO and MISO alternatives while holding other system parameters constant. The section also presents parameters for the COST-231 model.

IV. NUMERICAL RESULTS AND DISCUSSION

  • The simulation compares a SISO baseline without relaying, a single-antenna base station with J relays, and a MISO system with J + 1 base-station antennas.Each user has one antenna in all three cases.Source: §IV · p. 14
  • Number of users, total power, and bandwidth remain constant across the compared systems.Source: §IV · p. 14
  • The simulations aim to verify the assumption in Eqn. (6) and assess the tightness of the bounds; Table I is titled as listing COST-231 model parameters.Source: §IV · p. 14, Table I · p. 15

A. Channel Model

The simulations use the COST-231 channel model, with distinct fading and loss assumptions for BS-relay, BS-user, and relay-user links.

A. Channel Model

  • COST-231 models path loss, log-normal large-scale fading, and Rician LoS or Rayleigh non-LoS small-scale fading.BS-relay channels are Rician, while BS-destination and relay-destination channels are Rayleigh; BS-user and relay-user links also include rooftop-to-street diffraction loss.Source: A. Channel Model · p. 15
  • Channel-power attenuation is 20dB/km through 657 meters and 38dB/km at greater distances.The paper describes these parameter-specific attenuation rates as conservative, since the Rician LoS component is expected to attenuate more slowly than non-LoS components.Source: A. Channel Model · p. 15
  • Each user receives an orthogonal 200kHz channel, with noise power set to -120dBm.The chosen system parameters are listed in Table I.Source: A. Channel Model · p. 15

B. Decoding at the relays

The tractable formulation assumes successful relay decoding in the first time slot and limits rate by the compound source-relay-destination channel capacity; simulations check where this assumption holds.

B. Decoding at the relays

  • The formulation assumes relays always decode the BS message successfully in the first time slot, with rate limited by compound source-relay-destination capacity.This is the assumption represented by Eqn. (6).Source: B. Decoding at the relays · p. 15
  • Eqn. (6) holds whenever users are farther than 300m from the BS, covering locations with relatively weak BS channels.The authors characterize the reported percentages as conservative.Source: B. Decoding at the relays · pp. 15–16
  • The check uses 3×10^6 random user locations and Table II reports the percentage satisfying Eqn. (6).Locations are grouped into annular rings of 100 meters.Source: B. Decoding at the relays · p. 15, Table II · p. 16

C. Tightness of the bounds

The upper and lower bounds are extremely tight for both sum-rate and max-min objectives, consistent with the heuristic's near-selection behavior.

C. Tightness of the bounds

  • For sum rate, the upper and heuristic lower bounds are indistinguishable; users are rarely allocated power from multiple relays.The heuristic is the paper's final joint selection and power-allocation solution; the simulation sets average total transmit power-to-noise ratio to 30dB.Source: C. Tightness of the bounds · p. 16
  • Exact maximum sum rate was checked only for J = 2 and K = 1–8, because each exhaustive search requires JK water-filling problems.The paper says larger J values are infeasible for this exhaustive search.Source: C. Tightness of the bounds · p. 16
  • For max-min rate, the bounds are extremely tight, with slight differences arising when a user receives power from two relays.This simulation uses a 20dB average total transmit power-to-noise ratio.Source: C. Tightness of the bounds · p. 16
  • The section omits a corresponding plot for max-sum rate with a rate constraint.Source: C. Tightness of the bounds · p. 17

D. Results for a cellular network

In the cellular simulations, MISO has higher average user rates than relaying, but relaying has higher outage rates than MISO under max-min allocation.

D. Results for a cellular network

  • The evaluated users occupy the outer annulus from rcell/2 to rcell, where performance is measured as the cell-size increase enabled by relaying.Users are uniformly distributed at a constant density of (30/π) per square kilometer.Source: D. Results for a cellular network · p. 17
  • The comparison considers multiuser SISO networks with single-antenna base stations, multiuser MISO networks with five-transmit-antenna base stations, and single-antenna base stations assisted by four relays.In the relay-assisted system, four relays are positioned on a ring of radius 0.4rcell.Source: D. Results for a cellular network · p. 17
  • MISO provides higher average user rates than the relay-assisted system in the sum-rate comparison.The comparison uses lower-bound rates, with equal BS power in the first slot and each relay using one fourth of available power in the second slot so total power matches across systems.Source: D. Results for a cellular network · pp. 17–18
  • Relaying provides higher outage rates than MISO for the 10% and 1% outage comparisons.This reversal is reported for max-min power allocation; the paper also notes that BS-user communication is more susceptible to channel fluctuations.Source: D. Results for a cellular network · p. 18

E. Discussion

The discussion finds that relaying can extend service compared with SISO, but the rate metric and deployment costs shape whether those gains are worthwhile. The conclusion presents a joint selection-and-power-allocation framework with tight bounds and an effective heuristic.

E. Discussion

  • MISO provides higher average data rates than relaying, while the relay system has higher outage rates than MISO.The discussion attributes the average-rate result partly to the half-bandwidth loss from cooperative transmission.Source: E. Discussion · p. 18
  • Relay paths help users in heavily shadowed regions, whereas max-sum-rate allocation favors strong-channel users and MISO has higher data rates.The discussion says the half-bandwidth loss of cooperative transmission outweighs its diversity benefit for average-rate performance.Source: E. Discussion · p. 18
  • Cellular geometry and rapid channel-power deterioration limit effective diversity because most users have good channels to only a small subset of relays.The stated order J + 1 diversity applies when a user has the same average channel to all J relays.Source: E. Discussion · p. 18
  • Relaying offers substantial gains over SISO and can expand the effectively serviced area, but those gains must justify relay infrastructure costs.If a relay costs about as much as a base station, the simulated cell-radius improvements do not justify the additional cost.Source: E. Discussion · pp. 18–19
  • The examples are limited, and the reported gains are optimistic because relays are assumed always able to decode and transmitters have the information needed for optimal decisions.A complete financial cost-benefit analysis is also beyond the paper's scope.Source: E. Discussion · p. 19

V. CONCLUSION

  • The conclusion frames the contribution as joint relay selection and power allocation, using tight convex upper bounds and a close heuristic for practical user counts.The heuristic assigns a user receiving power from multiple relays to the relay allocating the most power.Source: §V · p. 19
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