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Resource Allocation for Wireless Fading Relay Channels: Max-Min Solution
Yingbin Liang, Venugopal V. Veeravalli, H. Vincent Poor
TL;DR
The paper asks how to allocate resources in wireless fading relay channels when source and relay nodes have separate power constraints. It develops capacity bounds and max-min allocation methods for parallel, Gaussian, full-duplex, and half-duplex relay models. Capacity is established for degraded parallel channels and certain fading channels, while optimal power and channel-resource allocations are characterized across the studied cases.
Problem
Wireless relay networks need resource allocation under separate source and relay power constraints rather than the commonly assumed total power constraint.
Method
The paper analyzes parallel relay channels, connects separate-constraint allocation to minimax two-hypothesis testing, and optimizes power and channel-resource allocations for full- and half-duplex fading models.
Results
Capacity is established for degraded parallel relay channels and certain fading channels, with Gaussian synchronized and asynchronized capacities and optimal source-relay power allocations characterized.
Takeaways & Limitations
The resulting strategies include closed-form or structured allocations, including two-level, orthogonal-division, and iterative water-filling depending on channel statistics and power constraints.
Abstract
from arXiv · showhide
As a basic information-theoretic model for fading relay channels, the parallel relay channel is first studied, for which lower and upper bounds on the capacity are derived. For the parallel relay channel with degraded subchannels, the capacity is established, and is further demonstrated via the Gaussian case, for which the synchronized and asynchronized capacities are obtained. The capacity achieving power allocation at the source and relay nodes among the subchannels is characterized. The fading relay channel is then studied, for which resource allocations that maximize the achievable rates are obtained for both the full-duplex and half-duplex cases. Capacities are established for fading relay channels that satisfy certain conditions.
I. INTRODUCTION
The paper develops capacity and resource-allocation results for parallel and fading wireless relay channels under separate source and relay power constraints. It establishes capacity for degraded parallel channels, characterizes Gaussian synchronized and asynchronized cases, and derives full- and half-duplex fading allocations.
- Motivation and approach: Separate source and relay power constraints make fading-relay resource allocation a max-min problem connected to minimax two-hypothesis testing.The paper applies this connection to obtain max-min optimal allocation strategies.
- Parallel relay channels: The parallel relay channel provides a basic model for fading channels, with partial decode-and-forward lower and cut-set upper capacity bounds.For degraded subchannels, the bounds match and establish capacity; this extends a prior result to multiple subchannels.
- Parallel relay channels: Parallel relay capacity can exceed the sum of independent subchannel capacities because information decoded over one subchannel can be forwarded over another.Thus, the relay can remain useful on reversely degraded subchannels by forwarding information decoded elsewhere.
- Gaussian parallel relay channels: For Gaussian degraded parallel channels, the paper obtains synchronized and asynchronized capacities and characterizes capacity-achieving source and relay power allocations.The asynchronized case has independent source and relay inputs and a closed-form optimal power allocation.
- Fading relay channels: For fading channels, adaptive resource allocation is studied in full-duplex and half-duplex models using channel-state information at transmitters and receivers.Half-duplex analysis jointly optimizes power and, in general, the channel-resource parameter θ across three scenarios.
III. OPTIMAL RESOURCE ALLOCATION FOR GAUSSIAN PARALLEL RELAY CHANNELS WITH DEGRADED SUBCHANNELS
The section develops optimal resource allocations for Gaussian parallel relay channels with degraded subchannels, including synchronized and asynchronized cases.
- The max-min problems in (12) and (14) determine optimal correlation parameters and source-relay power allocations.The synchronized solution is analytic when the optimization is convex, while the asynchronized solution is closed form.
A. Technique to Solve a Class of Max-Min Problem
The paper solves a class of max-min problems by reducing the objective to geometric comparisons between functions and selecting a minimizing parameter.
- A general technique based on geometric properties of V(α) and R(α,t) solves the max-min problem.The technique uses the relationship between a convex curve and the lines defined by R(α,t).
- The technique is related to the minimax detection rule for two-hypothesis testing.
- For fixed α, R(α,t) is a straight line whose endpoint minimum determines the maximization in the max-min problem.
- The solution chooses α* minimizing V(α), after which t(α*) is a max-min rule.The relationship between the resulting quantities falls into three cases illustrated in Fig. 3.
B. Optimal Resource Allocation for Gaussian Parallel Relay Channel: Synchronized Case
The synchronized Gaussian parallel-relay allocation is obtained by applying the max-min technique and KKT conditions to jointly optimize correlation and source-relay powers.
- Proposition 1 is applied to find jointly optimal correlation parameters and source-relay power allocations for the synchronized capacity.
- Case 1: In case 1, both source and relay allocations have water-filling forms, with the source allocation using equivalent noise levels.The optimal correlation parameter indicates that coherent combining is not needed in this case.
- Case 2: Case 2 never occurs because the KKT conditions force the relevant source powers to zero, making the required condition unsatisfiable.
- For the nonconvex fixed-α* optimization, KKT conditions provide only a necessary condition, so brute-force search may be required.
- Jointly optimizing correlated inputs and power allocations may be too complex to implement, motivating study of the simpler asynchronized case.Independent inputs are optimal in case 1.
C. Optimal Resource Allocation for Gaussian Parallel Relay Channel: Asynchronized Case
The asynchronized Gaussian parallel relay channel admits three optimal power-allocation structures, selected by power-constraint conditions and obtained through water-filling or iterative optimization.
- The max-min optimization over source and relay power allocations is solved by KKT-based analysis.The solution is organized into three possible structures.
- The relay allocation in two-level water-filling is first obtained by treating its effective term as an equivalent noise level.The resulting relay allocation is then combined with source allocation under the separate power constraints.
- Case 1: Two-level water-filling applies when the relay power exceeds the source-dependent threshold PR,u(P).The threshold is determined by an equality condition.
- Case 2: Orthogonal division water-filling allocates positive power to either the source or relay on each subchannel.This structure follows from the relevant KKT equation and applies when its associated condition is satisfied.
- Case 3: Iterative water-filling is used for the remaining case and is obtained by alternating source and relay updates.The iteration converges to an optimal allocation, with α* selected by an equalizer condition.
IV. FADING FULL-DUPLEX RELAY CHANNELS
The fading full-duplex relay channel is analyzed under separate source and relay power constraints, with adaptive allocations maximizing capacity bounds in the asynchronized setting.
- The fading full-duplex model adds multiplicative stationary, ergodic fading and additive white Gaussian noise to each relay-channel link.The source, relay, and destination form a three-terminal channel with full-duplex relay operation.
- Separate source and relay power constraints replace the earlier sum-power constraint, and optimal allocations for both capacity bounds are derived.The paper also identifies when the bounds match and capacity is established.
- The lower-bound allocation has the same three structures as the Gaussian parallel-relay solution.The structures are summarized using source- and relay-power constraints over fading states.
- In general, the upper and lower bounds do not match, but matching conditions yield the asynchronized capacity.The matching condition depends on channel statistics and the separate power constraints.
- When the matching condition holds, the capacity-achieving allocation uses orthogonal time-division water-filling.This regime essentially requires relay power to be small relative to source power.
V. FADING HALF-DUPLEX RELAY CHANNELS
The fading half-duplex relay model separates source transmission and relay transmission into orthogonal channels and optimizes both resource sharing and adaptive power allocation.
- The source transmits to the relay and destination on one channel, while the relay transmits to the destination on an orthogonal channel.The model represents these links with separate channel resources.
- The resource-allocation parameter θ controls the time and bandwidth division between the two channels.The channel model retains the fading and Gaussian-noise assumptions used for the full-duplex case.
- With channel state information available at transmitters and receiver, source and relay powers can adapt to instantaneous channel states.The goal is joint optimization of θ and the two power allocations.
- Three scenarios vary the flexibility of θ: fixed θ = 1/2, a common θ across states, and state-dependent θ.The third scenario permits the most general channel-resource and power allocation.
A. Scenario I: Fixed θ = 1/2
For Scenario I, equal resource sharing leads to a max-min power-allocation problem with three cases whose behavior depends on the relative source and relay power constraints.
- Scenario I fixes the two-channel resource allocation at θ = 1/2 and uses an achievable-rate formulation.The rate is optimized over source and relay power allocations.
- The relay power allocation always has a water-filling form based only on the relay-to-destination fading gain h2.The source allocation generally depends on both h1 and h3 and is not generally water-filling.
- Achievable rates increase with relay power in cases 2 and 3 but saturate in case 1.In case 1, relay power is sufficient to forward all information decoded at the relay, leaving the source-to-relay link rate-limiting.
- When relay power is small relative to source power, the optimal allocation falls into case 2; when relay power is large, it falls into case 1.The transition boundaries are given by the threshold functions PR,u(P) and PR,l(P).
- Beyond PR,u(P), additional relay power is not useful for decode-and-forward under Scenario I.The solid threshold therefore identifies the relay powers providing the best decode-and-forward rates for corresponding source powers.
B. Scenario II: Same θ for All Channel States
Scenario II jointly optimizes a single channel-resource parameter θ and state-dependent power allocations under separate source and relay constraints. The bounds coincide under a stated condition, establishing capacity; otherwise, the gap remains small in the reported example.
- Resource allocation: Scenario II fixes θ across channel states while adapting only source and relay powers to instantaneous channel conditions.The common θ simplifies system design compared with state-dependent resource allocation.
- Optimization: The paper derives an achievable-rate lower bound, a cut-set upper bound, and optimal joint channel-resource and power allocations for the lower bound.The lower-bound allocation is characterized through cases and iterative procedures involving θ and power variables.
- Capacity condition: Under condition (94), the lower and upper bounds match, so Scenario II capacity is achieved by the iteratively obtained allocation.The capacity refers to the largest rate over all allowed common θ values and power-allocation rules.
- Numerical results: When relay power is below 4 dB in the reported Rayleigh example, the optimized lower and upper bounds match and determine capacity.The gap remains small even when relay power is large.
- Numerical results: Scenario II lacks the saturating case present in Scenario I, so its achievable rate continues increasing beyond Scenario I’s saturation point.The reported comparison attributes this behavior to jointly optimizing θ and power allocation.
- Numerical results: The optimal θ is nonmonotonic in relay power: it decreases at low relay power and increases once relay power is sufficiently large.The allocation shifts more channel resource toward the relay-to-destination link at low relay power.
C. Scenario III: θ Changes with Channel States
Scenario III allows θ to vary with each channel-state realization, jointly with adaptive power allocation. This improves achievable rates over fixed-resource scenarios but requires channel knowledge across all transmission links and creates a high-dimensional optimization problem.
- Scenario definition: Scenario III permits θ(h) to change with channel-state realizations, whereas Scenario II uses one θ for all states.Power allocations remain jointly optimized with the state-dependent resource allocation.
- Practical scope: Scenario III requires each node to know channel realizations on all transmission links, making system design more complex and less practical than Scenario II.The paper includes this scenario mainly for completeness.
- Optimization: The paper derives achievable-rate and cut-set bounds and characterizes iterative resource allocations that maximize the lower bound.The upper-bound optimization uses similar steps, while the lower and upper bounds generally do not match.
- Capacity condition: Under condition (110), the lower and upper bounds match, establishing the capacity for Scenario III.The capacity is the largest rate over state-dependent θ(h) and power-allocation rules.
- Numerical results: Scenario III achieves larger rates than Scenario II because θ(h) can adapt to instantaneous channel-state information.The comparison is reported alongside Scenario I and the direct source-to-destination link.
- Computational scope: The Rayleigh-fading optimization is high dimensional, particularly in Scenario III, although the paper reports fast convergence for its analytical-structure-based algorithm.Standard convex programming techniques may converge slowly, while the reported algorithm takes only a few iterations.
VI. CONCLUDING REMARKS
The paper develops capacity bounds and resource allocations for parallel and fading relay channels under separate source and relay power constraints. It also provides a technique for the resulting max-min optimization problems.
- Conclusions: Capacity theorems are established for degraded parallel relay channels, their Gaussian case, and full- and half-duplex fading relay channels under stated conditions.The parallel-channel lower and upper bounds match for degraded subchannels.
- Conclusions: The resource-allocation analysis covers Gaussian parallel relay channels and fading relay channels under both full-duplex and half-duplex models.The formulation uses separate source and relay power constraints rather than a total constraint.
- Conclusions: The paper treats the resource-allocation problem as a max-min problem and provides a technique for solving such problems.The technique is motivated by max-min forms arising in decode-and-forward relay-channel achievable rates.
(78) FOR SCENARIO I
The appendix derives the Scenario I lower-bound resource allocation using a compact max-min formulation and KKT conditions. The resulting optimal power allocation is organized into three cases.
- Max-min formulation: The two terms in the Scenario I minimization are expressed in a compact max-min form before deriving the allocation.This formulation supports the subsequent KKT-based analysis.
- KKT derivation: The optimal power allocation is derived from KKT conditions and must satisfy the corresponding case conditions.The appendix explicitly invokes KKT conditions for the allocation and associated constraints.
(89) FOR SCENARIO II
Scenario II’s max-min resource-allocation problem is reduced to convex optimization and KKT conditions, with the solution characterized through cases and iterative algorithms.
- For fixed α, maximizing the relevant rate function is a convex programming problem because its Hessian is negative semidefinite.
- The max-min problem is analyzed by applying Proposition 1 and considering three resource-allocation cases.
- The optimal resource allocation and power allocation in each case are obtained from the KKT conditions.
- The Scenario II proof includes case-specific conditions, allows α∗ = 0 in case 3, and omits detailed proofs for cases 2 and 3 in the related analysis.
- The KKT conditions constrain the optimal θ through boundary and interior derivative conditions, while ∂θ has at most one root on 0 ≤ θ ≤ 1.
- The iterative algorithms converge to KKT solutions, which achieve the optimum because the corresponding rate functions are concave.