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Optimal Power Allocation for Fading Channels in Cognitive Radio Networks: Ergodic Capacity and Outage Capacity

Xin Kang, Ying-Chang Liang, Arumugam Nallanathan, Hari Krishna Garg, Rui Zhang

arXiv:0808.3689v2cs.IT

TL;DR

The paper addresses optimal power allocation for secondary users sharing spectrum with primary users while satisfying transmit- and interference-power constraints. It analyzes ergodic, delay-limited, and outage capacities across peak/average constraints and fading models, finding capacity gains from average constraints and beneficial SU-Tx-to-PU-Rx fading.

  • Problem

    The problem is how to maximize secondary-user fading-channel capacities while allowing coexistence with primary users under regulated interference.

  • Method

    The paper derives optimal SU power allocation strategies for ergodic, delay-limited, and outage capacities under combinations of peak and average constraints.

  • Results

    Average interference constraints are more flexible than peak constraints at the same threshold, yielding higher SU fading-channel capacities.

  • Takeaways & Limitations

    Fading between the SU transmitter and PU receiver can beneficially enhance SU channel capacities across the analyzed fading settings.

Abstract

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A cognitive radio network (CRN) is formed by either allowing the secondary users (SUs) in a secondary communication network (SCN) to opportunistically operate in the frequency bands originally allocated to a primary communication network (PCN) or by allowing SCN to coexist with the primary users (PUs) in PCN as long as the interference caused by SCN to each PU is properly regulated. In this paper, we consider the latter case, known as spectrum sharing, and study the optimal power allocation strategies to achieve the ergodic capacity and the outage capacity of the SU fading channel under different types of power constraints and fading channel models. In particular, besides the interference power constraint at PU, the transmit power constraint of SU is also considered. Since the transmit power and the interference power can be limited either by a peak or an average constraint, various combinations of power constraints are studied. It is shown that there is a capacity gain for SU under the average over the peak transmit/interference power constraint. It is also shown that fading for the channel between SU transmitter and PU receiver is usually a beneficial factor for enhancing the SU channel capacities.

I. INTRODUCTION

The paper studies spectrum sharing in cognitive radio networks, where secondary users coexist with primary users under regulated interference. It derives optimal power allocation for several SU capacities and constraint combinations, finding gains from average constraints and beneficial effects from interference-link fading.

  • Motivation: Spectrum sharing lets secondary users transmit alongside primary users when their interference remains within an acceptable quality-of-service level.This contrasts with opportunistic access, which requires detecting vacant spectrum precisely.
  • Scope: The paper studies ergodic, delay-limited, and outage capacities of secondary-user block-fading channels under spectrum sharing.For block-fading channels, the channel remains constant within each transmission block and may change between blocks.
  • Approach: The authors derive optimal SU power allocation strategies under combinations of peak or average transmit-power and interference-power constraints.The SU transmit-power constraint is considered alongside the interference constraint protecting the primary user.
  • Findings: Average constraints provide SU capacity gains over corresponding peak transmit/interference power constraints.The paper also provides closed-form delay-limited capacity and outage-probability results for Rayleigh, Nakagami, and log-normal fading.
  • Findings: Fading between the SU transmitter and PU receiver can enhance SU channel capacities.The conclusion identifies this interference-link fading effect as beneficial for maximizing capacity.

II. SYSTEM MODEL AND POWER CONSTRAINTS

The system model contains one primary and one secondary user, with flat-fading SU-to-PU and SU-to-SU links and perfect channel state information at the SU transmitter. The paper combines SU transmit-power and PU interference-power limits, each imposed as either peak or average constraints.

  • System model: The model has one PU and one SU, with flat-fading links characterized by instantaneous channel power gains g0 and g1.g0 describes the SU-Tx-to-PU-Rx link, while g1 describes the SU-Tx-to-SU-Rx link.
  • System model: The channel gains are ergodic and stationary, and perfect CSI for both gains is available at the SU transmitter.The receiver noises are modeled as independent circularly symmetric complex Gaussian variables.
  • Transmit-power constraints: The SU transmitter uses instantaneous power P(g0, g1) subject to peak and/or average transmit-power limits.Ppk denotes the peak transmit-power limit and Pav the average transmit-power limit.
  • Interference constraints: PU protection is represented by peak or average received-interference power constraints at the PU receiver.Average interference limits support long-term QoS, whereas peak limits suit instantaneous QoS requirements.
  • Constraint combinations: The analysis combines transmit-power and interference-power constraints into four constraint sets.These combinations cover peak and average limits for each power type.

III. ERGODIC CAPACITY

The paper formulates ergodic-capacity optimization for the SU link under four power-constraint combinations and derives corresponding adaptive power-allocation policies. The policies account for both the SU channel and the interference channel to the PU receiver.

  • Ergodic capacity is obtained by optimizing expected SU-link rate over channel gains under four power-constraint combinations, F1 through F4.
  • Under F1, capacity is maximized by transmitting at the maximum instantaneous power allowed by the combined constraints.
  • When g0 is below a threshold, SU-Tx uses peak power; above it, transmit power decreases inversely with g0.
  • The F2 allocation has a 2D-TCI-like structure involving both the SU channel and the SU-Tx-to-PU-Rx channel, unlike conventional schemes tied to one channel.
  • Under F3, sufficiently severe interference-channel fading or sufficiently large Qpk reduces the allocation to conventional water-filling, while the Qpk cap increases as g0 decreases.

D. Average transmit power constraint and average interference power constraint

The delay-limited-capacity analysis maximizes a constant SU transmission rate across fading blocks under F4, because realistic fading makes the capacity zero under the other constraint combinations. Results are evaluated for Rayleigh and Nakagami fading.

  • Delay-limited capacity is the maximum constant transmission rate achievable over every fading block, making it relevant to delay-sensitive services.
  • The optimization adapts SU-Tx power to maximize this constant rate while keeping PU-Rx interference below its threshold.
  • The analysis focuses on F4 because delay-limited capacity is zero under the other three constraint combinations for realistic fading models.
  • Rayleigh fading: For Rayleigh fading with independent unit-mean channel gains, the delay-limited capacity is zero.
  • Nakagami fading: For Nakagami fading, the channel power gain follows a Gamma distribution, and the resulting capacity depends on the fading parameter and applicable power constraints.

C. Log-normal shadowing

The paper models log-normal fading through independent Gaussian exponents for the SU and interference channels, then derives the associated delay-limited and outage-capacity formulations. Outage capacity is posed as optimizing power allocation for a target outage probability.

  • In log-normal fading, channel gains are modeled as g0=e^X0 and g1=e^X1 with independent zero-mean Gaussian exponents of variance σ2.
  • The ratio g0/g1 is log-normal because Y=X0−X1 is Gaussian with zero mean and variance 2σ2.
  • Delay-limited capacity: The derived log-normal delay-limited capacity has separate forms depending on whether average transmit power is finite or unbounded.
  • Outage capacity: Outage capacity seeks the optimal power allocation maximizing rate for a specified outage probability, equivalently minimizing outage probability for a target rate r0.
  • Peak constraints: For peak transmit and peak interference constraints, the optimal policy resembles truncated channel inversion but uses both g0 and g1, motivating the name 2D-TCI.

B. Peak transmit power constraint and average interference power constraint

For the remaining outage-capacity constraint combinations, the paper derives optimal policies and minimum outage probabilities, including policies identified as two-dimensional truncated channel inversion. It also gives analytical and fading-model-specific evaluations.

  • F2: Under F2, the optimal outage-capacity policy has the same structure as the peak-constraint solution and is therefore 2D-TCI.
  • F2: The F2 minimum outage probability is obtained from the resulting policy, with its dual variable determined by the average interference constraint when active.
  • F3: Under F3, the paper derives an optimal policy and corresponding minimum outage probability under the stated constraint combination.
  • F3: The F3 policy is also 2D-TCI.
  • F4: Under F4, the optimal policy uses two dual variables jointly determined by the average transmit-power and average interference-power constraints.
  • Fading-model evaluation: Minimum outage probabilities are evaluated under Rayleigh, Nakagami, and log-normal fading, using analytical special-function forms where applicable.

2) Average interference power constraint only:

The paper evaluates minimum outage probability under an average interference power constraint across Rayleigh, Nakagami, and log-normal fading models, deriving model-specific expressions and zero-outage conditions.

  • The minimum outage probability is evaluated under different fading models subject to an average interference power constraint.
  • Rayleigh fading: For Rayleigh fading, zero-outage capacity is zero because zero outage requires ω∗ to approach infinity, which occurs only when r0 = 0.
  • Nakagami fading: For Nakagami fading with m = 2, the outage probability has a closed-form expression, and zero outage yields the delay-limited capacity.
  • Log-normal fading: For log-normal fading, ω∗ is determined by the average interference constraint, and zero-outage probability is achieved when ω∗ approaches infinity.

VI. SIMULATION RESULTS

Simulations evaluate the proposed power-allocation strategies for ergodic, delay-limited, and outage capacities under multiple power constraints and fading models. Results show that constraint type and SU–PU channel fading substantially affect achievable capacity and outage probability.

  • The simulations evaluate SU capacities under spectrum sharing using the proposed power-allocation strategies.
  • Ergodic capacity: When peak transmit power is small, it limits ergodic-capacity performance; when sufficiently large relative to interference power, the constraint effects change.
  • Ergodic capacity: Fading in the SU-Tx–PU-Rx channel can increase ergodic capacity, while the outcome depends on which channel is fading.
  • Average interference power constraints provide larger ergodic capacity and smaller outage probability than peak interference constraints at the same power value.
  • Delay-limited capacity: Rayleigh fading has zero delay-limited capacity regardless of Qav, whereas Nakagami and log-normal capacities increase with Qav before saturation from Pav.
  • Outage capacity: For outage capacity, fading can reduce outage probability at small Qpk, but when Qpk equals Ppk, fading may be more restrictive than AWGN.
  • Outage capacity: At sufficiently large Qpk, Ppk becomes dominant and fading and AWGN have the same outage probability.
  • Outage capacity: Analytical and simulation outage results match well, and Nakagami and log-normal outage probabilities fall sharply beyond a Qav threshold.

VII. CONCLUSIONS

The paper studies optimal power allocation for SU ergodic, delay-limited, and outage capacities under combinations of peak and average transmit and interference constraints. Average interference constraints are more flexible, and SU–PU channel fading can improve capacity.

  • The paper studies optimal power allocation for ergodic, delay-limited, and outage capacities under combinations of peak and average constraints.
  • Average interference power constraints are more flexible than peak constraints at the same threshold for maximizing SU fading-channel capacities.
  • The effects of different fading statistics on achievable SU capacities are analyzed.
  • Fading in the channel between SU-Tx and PU-Rx can be beneficial for maximizing SU capacity.

APPENDIX A

The appendix formulates the constrained power-allocation problem through Lagrangian duality and solves its per-fading-state subproblems using convexity and KKT conditions.

  • The average-interference constraint introduces a nonnegative dual variable λ into the Lagrangian formulation.
  • The feasible set restricts power allocation to 0 ≤ P(g0, g1) ≤ Ppk, and the dual function is formed over this set.
  • The convex problem has zero duality gap, so solving the dual problem is equivalent to solving the original problem.
  • For fixed λ, dual decomposition separates the problem into subproblems for individual fading states.
  • The subproblem is convex, and its optimal solution must satisfy the KKT conditions involving the dual variables μ and ν.

APPENDIX B

Appendix B proves the optimal power-allocation structure for the outage-capacity problem under an average interference-power constraint. It uses dual decomposition over fading states and determines the multiplier by enforcing the average-interference constraint.

  • Proof strategy: The proof rewrites the optimization using an indicator function and a partial Lagrangian for the average interference-power constraint.The Lagrangian combines the outage indicator with λ(E{g0P(g0,g1)} − Qav).
  • Proof strategy: For fixed λ, dual decomposition separates the problem into similar subproblems for each fading state.Each subproblem minimizes the Lagrangian over power allocations bounded by 0 ≤ P(g0,g1) ≤ Ppk.
  • Optimal structure: The per-state minimization allocates zero power when the outage indicator is one; otherwise, the optimal allocation follows the structure of (26).The zero-power case has minimum objective value 1, while the alternative is selected when the relevant conditions are simultaneously satisfied.
  • Constraint determination: When λ = 0, the resulting strategy reduces to truncated channel inversion, which applies only when the average interference constraint is slack.This completes the contradiction argument and establishes the stated theorem.
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