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Dynamic Resource Allocation in Cognitive Radio Networks: A Convex Optimization Perspective

Rui Zhang, Ying-Chang Liang, Shuguang Cui

arXiv:1001.3187v1cs.IT

TL;DR

Cognitive-radio resource allocation must share spectrum while protecting primary transmissions under interference constraints. The paper surveys IT-based spectrum sharing and formulates spatial and space-time-frequency allocation problems, showing that convex optimization and related decompositions provide rigorous, efficient solution methods. It concludes that these results motivate broader interference-management designs for cognitive radio and related multiuser systems.

  • Problem

    Cognitive-radio systems need resource-allocation designs that enable secondary transmission while protecting primary links under interference constraints.

  • Method

    The paper surveys IT-based spectrum sharing and studies spatial and joint space-time-frequency resource-allocation problems using convex optimization, dual decomposition, and related methods.

  • Results

    Convex optimization efficiently solves the convex allocation cases, while dual decomposition splits joint allocation into L convex subproblems; non-convex cases may retain a duality gap.

  • Takeaways & Limitations

    The presented allocation results support interference-management designs for cognitive-radio and related multiuser communication systems.

Abstract

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This article provides an overview of the state-of-art results on communication resource allocation over space, time, and frequency for emerging cognitive radio (CR) wireless networks. Focusing on the interference-power/interference-temperature (IT) constraint approach for CRs to protect primary radio transmissions, many new and challenging problems regarding the design of CR systems are formulated, and some of the corresponding solutions are shown to be obtainable by restructuring some classic results known for traditional (non-CR) wireless networks. It is demonstrated that convex optimization plays an essential role in solving these problems, in a both rigorous and efficient way. Promising research directions on interference management for CR and other related multiuser communication systems are discussed.

I. INTRODUCTION

Cognitive radio networks let secondary users share spectrum allocated to primary users, motivating dynamic spectrum access and resource allocation under primary-protection constraints. This article focuses on interference-temperature spectrum sharing and convex optimization for allocating transmission resources across space, time, and frequency.

  • CRs, or secondary users, communicate over bandwidth originally allocated to primary networks, supporting dynamic spectrum access.
  • OSA allows secondary transmission when primary transmitters are inactive, whereas SS permits simultaneous transmission subject to tolerable interference at primary receivers.
  • The article focuses on IT-based SS because secondary transmitters can protect primary receivers by constraining interference power using estimated or known cross-links.
  • Dynamic resource allocation adapts transmit power, bit-rate, bandwidth, and antenna beams to channel-state information from primary and secondary networks.
  • The article covers spatial optimization and joint space-time-frequency optimization under transmit-power and interference-power constraints.

II. CR NETWORK MODELS

The paper models infrastructure-based and ad hoc cognitive-radio networks with coexisting primary terminals, then formulates transmit-power and interference-power constraints across transmit dimensions. Dynamic allocation is needed because spatial channels vary over time and frequency.

  • CR networks are either infrastructure-based, with an S-BS coordinating transmissions, or ad hoc, with distributed secondary links.
  • Infrastructure-based uplinks use multiple-access channels, while downlinks use broadcast channels that can support multicast or unicast transmission.
  • Ad hoc secondary networks are modeled as interference channels connecting distributed secondary transmitters and receivers.
  • Primary interference may be treated as additional noise at secondary receivers, with total noise modeled as a zero-mean CSCG random vector with identity covariance.
  • Spatial channels are fixed within one transmit dimension but vary across time and frequency, making dynamic scheduling relevant for secondary users.
  • Peak and average transmit-power constraints, together with peak and average interference-power constraints, define the main resource-allocation settings.

III. COGNITIVE BEAMFORMING OPTIMIZATION

For a single transmit dimension, cognitive beamforming optimizes secondary spatial transmission under peak transmit-power and peak interference-power constraints. The paper unifies these constraints through generalized linear transmit covariance constraints.

  • With L = 1, cognitive beamforming maximizes cognitive-radio throughput through spatial transmit optimization under PTPCs and PIPCs.
  • The optimization assumes perfect knowledge of secondary-network channels and channels from secondary transmitters to primary users.
  • PIPCs and PTPCs can be unified as generalized linear transmit covariance constraints.
  • This general constraint form is important because PIPCs extend conventional transmit-optimization problems beyond individual-power and sum-power constraints.

A. Convex Problem Formulation

When the corresponding traditional MIMO transmit-optimization problem is convex, adding linear interference-power constraints preserves convexity. The resulting cognitive-radio problem can therefore be solved efficiently with standard convex optimization techniques.

  • Linear PIPCs preserve convexity when the corresponding traditional MIMO optimization problem is already convex.
  • The CR point-to-point MIMO case is treated as a single active secondary link in a MAC, BC, or IC network.

I + HSHH (P1)

The CR transmit-covariance problem is convex under power and interference constraints, enabling efficient optimal solutions and structural characterizations. Special cases yield beamforming, convex reformulations, water-filling, and comparisons with a suboptimal projection heuristic.

  • Optimal covariance optimization: (P1) is a convex optimization problem because its objective is concave and its constraints define a convex set over S.It can therefore be efficiently solved using an interior-point method.
  • MISO specialization: For the MISO CR channel, capacity-optimal transmit covariance has Rank(S) = 1, so transmission can be represented by a precoding vector.The resulting vector problem is non-convex but can be converted into SOCP or solved through a convex SDP dual; a closed-form solution exists for one single-antenna PU.
  • Optimal covariance optimization: Strong duality permits solving (P1) equivalently through its Lagrange dual, with dual variables updated using subgradients.The subgradients include P − Tr(S⋆) for the transmit-power constraint and Γj − Tr(...) for interference constraints.
  • Optimal covariance optimization: An auxiliary covariance transformation reduces the problem to standard point-to-point MIMO capacity optimization with a single sum-power constraint.The solution uses SVD and the standard water-filling allocation σi = (1/ln 2 − 1/θi^2)+.
  • Heuristic projection: Partial channel projection is a generally suboptimal heuristic that projects away interference-relevant subspaces and allocates power over the resulting parallel channels.Its power allocation satisfies J + 1 linear constraints through a generalized water-filling algorithm and works for any N and Dj’s.
  • Numerical comparison: The convex-optimal covariance solution produces notable achievable-rate gains over partial-projection solutions for different b values under PTPC and PIPCs.The comparison uses M = N = 4, J = 2, D1 = D2 = 1, and Γ1 = Γ2 = 0.1, with rate plotted against SU PTPC P.
  • CR MIMO-MAC: For CR MIMO-MAC transmission, weighted sum-rate maximization with individual PTPCs and joint PIPCs remains convex and is solvable by interior-point or dual iterative algorithms.This formulation assumes optimal non-linear multiuser detection at the secondary base station.

B. Non-Convex Problem Formulation

The article examines non-convex CR MIMO optimization under transmit and interference constraints, using duality and iterative methods to obtain tractable solutions or feasible approximations. It covers MIMO-BC, MISO-BC, and MIMO-IC formulations.

  • CR MIMO-BC: CR MIMO-BC weighted sum-rate maximization is non-convex, so standard Lagrange duality does not directly solve it.The formulation includes a transmit-power constraint and multiple interference-power constraints.
  • CR MIMO-BC: Combining the MIMO-BC constraints into one generalized linear transmit covariance constraint enables a generalized BC-MAC duality approach.The resulting dual MIMO-MAC has a corresponding sum-power constraint and the same achievable rate region.
  • CR MIMO-BC: The dual MIMO-MAC weighted sum-rate problem is convex and can be efficiently solved, after which covariance transformation recovers the corresponding MIMO-BC solution.The constraint multipliers are updated iteratively, with the ellipsoid method using subgradients of the dual function.
  • CR MISO-BC: For CR MISO-BC SINR balancing, a fixed SINR value yields an SOCP feasibility problem, and bisection searches for the optimal SINR.A beamforming duality converts the problem to a dual SIMO-MAC where an efficient iterative algorithm can be applied.
  • CR MIMO-IC: CR MIMO-IC weighted sum-rate maximization remains non-convex and lacks efficient globally optimal algorithms for multiple users.Decomposing each primary interference constraint across secondary transmitters makes a decentralized iterative approach feasible, but it changes the constraint handling.

IV. JOINT SPACE-TIME-FREQUENCY DRA OPTIMIZATION

Joint space-time-frequency DRA extends CR optimization across parallel transmit dimensions under average transmit and interference constraints. Lagrange dual decomposition separates the problem by dimension, with exactness depending on utility concavity or asymptotic time-sharing conditions.

  • Problem formulation: Joint DRA optimizes transmit covariance matrices across L time or frequency dimensions under average transmit and interference-power constraints.The formulation assumes channel knowledge and a separable utility across dimensions.
  • Dual decomposition: Introducing dual variables for the average constraints produces a convex dual problem solvable by the ellipsoid method when each inner maximization is tractable.The inner maximization decomposes into L parallel subproblems with the same structure.
  • Concave utilities: For concave per-dimension utilities, the primal problem is convex, the duality gap is zero, and dual decomposition solves equivalent problems.Each decomposed subproblem is also convex in this case.
  • Non-concave utilities: For non-concave utilities, the primal and subproblems are non-convex, so the dual optimum generally provides only an upper bound.This case includes weighted sum-rate optimization for CR MIMO-BC and MIMO-IC.
  • Non-concave utilities: Time-sharing conditions can make the duality gap vanish asymptotically as L →∞, allowing dual decomposition to remain applicable under continuity considerations.The paper identifies this result for the class of joint DRA problems considered.

A. TDMA/FDMA Constrained DRA: When Is It Optimal?

TDMA/FDMA scheduling is practically attractive because it assigns one user per transmit dimension, but its constrained optimization is generally non-convex. Existing results establish optimality in several fading MAC and BC settings, while the CR IC case remains open.

  • A. TDMA/FDMA Constrained DRA: When Is It Optimal?: TDMA/FDMA schedules only one user for transmission at each time or frequency dimension, supporting implementation simplicity.The resulting utility enforces a single-user contribution at each dimension.
  • A. TDMA/FDMA Constrained DRA: When Is It Optimal?: The TDMA/FDMA utility is non-concave, making the associated resource-allocation problem non-convex.When the number of transmit dimensions is large, approximate time-sharing permits dual decomposition to solve the problem.
  • A. TDMA/FDMA Constrained DRA: When Is It Optimal?: TDMA is optimal for ergodic or long-term sum-capacity in traditional fading MACs under average transmit-power constraints.The cited result concerns single-antenna fading MACs with user average transmit-power constraints over time.
  • A. TDMA/FDMA Constrained DRA: When Is It Optimal?: The same TDMA optimality extends to fading CR MACs and CR BCs under average transmit- and interference-power constraints.KKT conditions also characterize optimality under combined peak and average transmit/interference constraints.
  • A. TDMA/FDMA Constrained DRA: When Is It Optimal?: Whether TDMA/FDMA is optimal for CR interference channels under additional peak or average interference constraints remains an open question.The unresolved issue is the performance gap relative to DRA allowing multiple users per dimension.

B. Peak vs. Average Interference Power Constraints: A New Interference Diversity

Comparing peak and average interference constraints reveals interference diversity: varying interference across transmit dimensions can reduce primary capacity loss, and can outperform constant interference under equal average power.

  • B. Peak vs. Average Interference Power Constraints: A New Interference Diversity: Average interference-power constraints are more flexible for secondary DRA than peak interference-power constraints under the same threshold.The primary-network impact is less obvious because average constraints permit interference variation across dimensions.
  • B. Peak vs. Average Interference Power Constraints: A New Interference Diversity: For a single-antenna fading primary link, average interference-power constraints outperform peak constraints under the same average power threshold.This result holds for both ergodic and outage primary capacities, with or without primary power control.
  • B. Peak vs. Average Interference Power Constraints: A New Interference Diversity: With equal average interference, fluctuating instantaneous interference across fading states yields larger primary ergodic capacity than constant interference.The comparison follows from the stated capacity expression and Jensen’s inequality.
  • B. Peak vs. Average Interference Power Constraints: A New Interference Diversity: The cited results identify interference diversity, where randomized secondary interference powers can be more advantageous than deterministic powers across space, time, or frequency.The criterion is minimizing resulting primary-network capacity losses.

C. Beyond Interference Temperature: Exploiting Primary Link Performance Margins

The article considers replacing conventional interference-power constraints with a primary ergodic-capacity constraint that is more directly tied to primary performance. This can improve both primary and secondary rates, but makes secondary power allocation non-convex.

  • C. Beyond Interference Temperature: Exploiting Primary Link Performance Margins: A primary ergodic-capacity constraint directly imposes a minimum threshold on the primary link’s ergodic capacity.The threshold is denoted by C̄_p.
  • C. Beyond Interference Temperature: Exploiting Primary Link Performance Margins: The primary ergodic-capacity constraint is more directly related to primary transmission performance than the conventional average interference-power constraint.Its improved alignment with the primary link comes at the cost of a harder optimization problem.
  • C. Beyond Interference Temperature: Exploiting Primary Link Performance Margins: The new constraint is generally non-convex in secondary transmit power, unlike the convex average interference-power constraint.Consequently, secondary power allocation becomes more challenging.
  • C. Beyond Interference Temperature: Exploiting Primary Link Performance Margins: The primary ergodic-capacity constraint achieves notable rate improvements for both primary and secondary links over the conventional average interference-power constraint.The comparison is reported for the optimal power-allocation rules studied in the cited work.

V. CONCLUSIONS AND DIRECTIONS FOR FUTURE WORK

The article presents dynamic resource allocation as central to cognitive-radio transmit optimization under primary protection and highlights convex optimization as a rigorous, efficient solution framework. It also identifies imperfect channel knowledge and active interference-temperature control as important future directions.

  • V. CONCLUSIONS AND DIRECTIONS FOR FUTURE WORK: Dynamic resource allocation exploits cognition about primary and secondary networks while enforcing required primary protection.The article surveys new design problems spanning cognitive-radio transmit optimization.
  • V. CONCLUSIONS AND DIRECTIONS FOR FUTURE WORK: Convex optimization techniques play a key role in solving the article’s cognitive-radio resource-allocation problems rigorously and efficiently.The conclusion describes an extensive set of challenging and distinctive DRA design problems.
  • V. CONCLUSIONS AND DIRECTIONS FOR FUTURE WORK: Robust cognitive beamforming is needed because secondary-to-primary channel estimates may contain errors.Required channel-state information is difficult to obtain across separately operated primary and secondary networks.
  • V. CONCLUSIONS AND DIRECTIONS FOR FUTURE WORK: Active interference-temperature control sets interference-temperature levels across coexisting links to improve spectrum-sharing throughput.The article presents this as a relatively new paradigm for interference management in cognitive-radio and related multiuser systems.
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