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Weighted Sum Rate Optimization for Cognitive Radio MIMO Broadcast Channels
Lan Zhang, Yan Xin, Ying-Chang Liang
TL;DR
The paper studies weighted sum-rate maximization for a cognitive-radio MIMO broadcast channel with both sum-power and interference constraints. It transforms the problem, develops BC–MAC duality for the reformulation, and proposes a subgradient-based algorithm. The algorithm is shown to converge to a globally optimal solution.
Problem
The CR MIMO-BC weighted sum-rate problem includes both a sum-power constraint and interference constraints, making it difficult to formulate an efficiently solvable dual problem.
Source: §I · p. 2, §II · p. 5Method
The authors transform the multiconstraint problem into an equivalent form with auxiliary variables, establish BC–MAC duality for fixed variables, and develop a subgradient-based iterative algorithm.
Source: C. Contribution · p. 5, B. CR MIMO BC-MAC Duality · p. 9Results
The proposed algorithm converges to a globally optimal solution for the CR MIMO-BC weighted sum-rate problem.
Source: C. Contribution · p. 5, §VII · p. 20Takeaways & Limitations
The duality extends conventional single-sum-power results to MIMO-BC problems with multiple linear constraints.
Source: C. Contribution · p. 5, B. CR MIMO BC-MAC Duality · p. 11
Abstract
from arXiv · showhide
In this paper, we consider a cognitive radio (CR) network in which the unlicensed (secondary) users are allowed to concurrently access the spectrum allocated to the licensed (primary) users provided that their interference to the primary users (PUs) satisfies certain constraints. We study a weighted sum rate maximization problem for the secondary user (SU) multiple input multiple output (MIMO) broadcast channel (BC), in which the SUs have not only the sum power constraint but also interference constraints. We first transform this multi-constraint maximization problem into its equivalent form, which involves a single constraint with multiple auxiliary variables. Fixing these multiple auxiliary variables, we propose a duality result for the equivalent problem. Our duality result can solve the optimization problem for MIMO-BC with multiple linear constraints, and thus can be viewed as an extension of the conventional results, which rely crucially on a single sum power constraint. Furthermore, we develop an efficient sub-gradient based iterative algorithm to solve the equivalent problem and show that the developed algorithm converges to a globally optimal solution. Simulation results are further provided to corroborate the effectiveness of the proposed algorithm.
I. INTRODUCTION
The paper formulates weighted sum-rate maximization for secondary users in a cognitive-radio MIMO broadcast channel with both sum-power and interference constraints. It develops an equivalent formulation, an extended BC–MAC duality, and algorithms intended to solve the problem globally.
A. System Model and Problem Formulation
- The CR MIMO-BC problem maximizes users’ weighted sum rates subject to both a BS sum-power constraint and an interference constraint protecting the primary user.The channel model assumes channel knowledge at the BS and SUs; obtaining it requires information feedback.Source: §I · p. 2, A. System Model and Problem Formulation · pp. 2–3
B. Related Work
- Conventional BC–MAC duality for a single sum-power constraint does not solve the CR problem’s multiple-constraint case; varying user weights can describe the entire CR MIMO-BC capacity region.Prior approaches described include methods for equally weighted rates, MISO channels, or single-constraint MIMO-BC problems.Source: A. System Model and Problem Formulation · p. 3, B. Related Work · p. 4
C. Contribution
- The paper converts the multi-constraint problem to an equivalent single-constraint problem with auxiliary variables, establishes a generalized BC–MAC duality, and proposes an algorithm that converges globally.The duality applies to arbitrary linear power constraints, extending conventional results limited to a single sum-power constraint.Source: C. Contribution · pp. 4–5
D. Organization and Notation
- The paper is organized around problem reformulation and duality, dual-MAC optimization, covariance mapping, a complete solution, simulations, and conclusions; it also defines its matrix and operator notation.Source: D. Organization and Notation · p. 5
II. EQUIVALENCE AND DUALITY
The section addresses why the CR weighted sum-rate problem is difficult to solve directly and outlines a reformulation and duality approach. The reformulation uses auxiliary variables, which are fixed when establishing the BC–MAC duality.
II. EQUIVALENCE AND DUALITY
- The CR problem is non-convex and difficult to solve directly; its sum-power and interference constraints complicate formulation of an efficiently solvable dual problem.Source: §II · p. 5
- The authors transform the multiconstraint problem into an equivalent problem with a single constraint and multiple auxiliary variables.Source: §II · p. 6
- They develop BC–MAC duality for the reformulated problem when its multiple auxiliary variables are fixed.Source: §II · p. 6
A. An Equivalent MIMO-BC Weighted Sum Rate Problem
Problem 1 and its equivalent formulation have the same optimal solution. The formulation introduces auxiliary dual variables for the interference and sum-power constraints, which are fixed to apply the subsequent duality result.
A. An Equivalent MIMO-BC Weighted Sum Rate Problem
- Problem 2 shares the same optimal solution as the original multiconstraint problem.Source: A. An Equivalent MIMO-BC Weighted Sum… · p. 6
- The equivalent formulation uses auxiliary dual variables for the interference constraint and the sum-power constraint.Source: A. An Equivalent MIMO-BC Weighted Sum… · p. 6
- Fixing those variables reduces Problem 2 to a simplified form that can be solved using the paper’s duality result.Source: A. An Equivalent MIMO-BC Weighted Sum… · p. 6
B. CR MIMO BC-MAC Duality
With the auxiliary variables fixed, the paper establishes a duality between the constrained MIMO-BC and a dual MIMO-MAC. The result extends conventional BC-MAC duality beyond a single sum-power constraint.
B. CR MIMO BC-MAC Duality
- The dual MAC problem provides a route to solving the corresponding constrained BC problem for fixed auxiliary variables.The section describes the dual MAC as a weighted sum-rate problem with noise covariance determined by those variables.Source: B. CR MIMO BC-MAC Duality · pp. 6–7
- For fixed qt and qu, the dual MIMO-MAC and primal MIMO-BC are dual under their SINR constraints, with equal optimal weighted sum rates.The dual MAC noise covariance depends on qt and qu rather than being the identity matrix.Source: B. CR MIMO BC-MAC Duality · pp. 7, 9
- For fixed qt and qu, the constrained BC and dual MAC have the same achievable SINR and rate regions.The rate-region equality follows from equality of the SINR regions and the corresponding achievable rates.Source: B. CR MIMO BC-MAC Duality · pp. 10–11
- The additional interference constraint prevents applying conventional duality results that assume only a single sum-power constraint.The paper presents its duality as an extension to multiple linear constraints.Source: B. CR MIMO BC-MAC Duality · pp. 6, 11
III. DUAL MAC WEIGHTED SUM RATE MAXIMIZATION PROBLEM
The paper develops DIPA to solve the weighted sum-rate problem for the dual MIMO-MAC. It specifies decoding-order choices and establishes convergence to an optimal set of MAC covariance matrices.
III. DUAL MAC WEIGHTED SUM RATE MAXIMIZATION PROBLEM
- Unlike the equally weighted case, weighted sum-rate maximization generally depends on the MAC decoding order and lacks a water-filling solution structure.The paper therefore does not use a water-filling-based algorithm for DIPA.Source: §III · pp. 11, 12, 15
- The optimal user decoding order is descending in user weights, whereas the decoding order among streams within a user is arbitrary.The primal BC encoding order reverses the MAC decoding order.Source: §III · p. 12
- DIPA solves the dual-MAC weighted sum-rate problem through sequential covariance updates and a subgradient-based search for the shared multiplier.The multiplier is updated along a subgradient direction because the convex dual objective need not be differentiable.Source: §III · pp. 13–14
- The DIPA algorithm converges to an optimal set of dual-MAC transmit signal covariance matrices.Its inner loop updates user covariance matrices by gradient ascent, while the outer loop searches for the Lagrangian multiplier.Source: §III · p. 14
IV. MAC-TO-BC COVARIANCE MATRIX MAPPING
The section maps the optimal covariance matrices from the dual MAC to covariance matrices for the BC. This supplies the BC solution using the dual MAC optimization.
IV. MAC-TO-BC COVARIANCE MATRIX MAPPING
- The proposed mapping addresses a gap in the earlier algorithm, which handled single-sum-power sum-rate maximization but not weighted sum rate under multiple constraints.The mapping computes BC covariance matrices from the dual MAC covariance matrices.Source: §IV · p. 15
- The mapping derives MAC beamforming vectors by eigenvalue decomposition, obtains receiver vectors with MMSE, and computes BC stream powers using duality.For the BC, powers are computed in reverse encoding order relative to MAC decoding order.Source: §IV · pp. 15–16
- The MAC-to-BC covariance matrix mapping obtains optimal BC covariance matrices for Problem 3 by solving Problem 4.The paper notes that an idea behind this mapping had appeared previously, but without an explicit algorithm.Source: §IV · p. 16
V. A COMPLETE SOLUTION TO THE CR MIMO-BC WEIGHTED SUM RATE PROBLEM
The solution minimizes the dual objective over auxiliary variables using subgradient updates, with each iteration solving a dual MAC problem and mapping its solution back to the BC. The resulting SIPA algorithm converges globally, and the approach extends to multiple primary users and arbitrary linear power constraints.
V. A COMPLETE SOLUTION TO THE CR MIMO-BC WEIGHTED SUM RATE PROBLEM
- The subgradient method reaches an ǫ-suboptimal point in finitely many steps, and a square-summable but not summable diminishing step size guarantees convergence to the optimal value.The error ǫ decreases with the step size.Source: §V · p. 17
- Each SIPA iteration solves the dual MAC with DIPA, maps the solution to the BC, and updates auxiliary variables by a subgradient step.The update searches for optimal qt and qu because the dual objective is not necessarily differentiable.Source: §V · pp. 16–17
- SIPA converges to the globally optimal solution of Problem 1.Its proof uses the fixed-auxiliary-variable BC-MAC duality and the convergence of DIPA to its optimal solution.Source: §V · pp. 17–18
- For multiple PUs, the formulation assigns an auxiliary variable qt,j to each PU and SIPA can be modified to solve the resulting problem.The multiple interference constraints can also represent per-antenna power constraints or other arbitrary linear power constraints.Source: §V · p. 19
VI. SIMULATION RESULTS
Simulations test DIPA convergence and examine SIPA’s convergence, constraint satisfaction, and achievable rates under varying power and PU placement. They show convergence in the tested cases and identify rate trade-offs associated with interference constraints and PU distance.
VI. SIMULATION RESULTS
- The simulations assume a path-loss exponent of 4, identity noise covariance, and Pt = 0 dB, with l1 = l2 except where stated otherwise.The channel matrix elements are modeled as zero-mean, unit-variance CSCG random variables.Source: §VI · p. 19
- In the single-SU case, DIPA converges within several iterations to the water-filling optimum; with K = 20, it also reaches its optimum within several iterations.The single-SU test uses Nt = 4, Nr = 4, and Pu = 10 dB; the K = 20 test uses Pu = 10 dB.Source: §VI · pp. 19–20
- At convergence, SIPA’s sum power and PU interference approach their limits of Pu = 13 dB and Pt = 0 dB, respectively.The tested network has K = 5 SUs, two PUs, and step sizes t = 0.1 and t = 0.01; step size affects accuracy and convergence speed.Source: §VI · p. 20
- At low sum power, achievable sum rates with and without a PU are close, while at high sum power the no-PU case is much higher.The paper attributes the difference to the additional constraint reducing system degrees of freedom.Source: §VI · p. 20
- As the PU moves farther from the BS, achievable SU sum rate increases, and the PU’s influence falls to zero beyond a certain l2/l1 threshold.The comparison varies l2/l1 from 1 to 12 at BS sum-power constraints of 15 dB and 20 dB.Source: §VI · p. 20
VII. CONCLUSIONS
The paper develops a BC-MAC duality for multiple linear constraints and uses it to construct an algorithm for CR MIMO-BC weighted sum rate maximization. The algorithm converges to a globally optimal solution.
VII. CONCLUSIONS
- The new BC-MAC duality extends existing duality results beyond sum-power or per-antenna constraints, enabling an algorithm for CR MIMO-BC weighted sum rate maximization that converges globally.Source: §VII · p. 20
APPENDIX
The appendix establishes that a constraint is tight at the fixed-auxiliary-variable optimum and proves that Problems 1 and 2 share an optimal solution.
A. Lemma 4 and its proof:
- For fixed qt and qu, the maximum weighted sum rate is attained when constraint (6) holds with equality.The proof uses the DPC encoding order: the first encoded user’s signal does not affect the other users’ rates, so unused constraint slack could improve that user’s rate.Source: A. Lemma 4 and its proof · p. 21
B. Proof of Proposition 1:
- The cited conclusion states that the optimal solutions for Problem 2 and Problem 1 are the same.Source: B. Proof of Proposition 1 · p. 23
C. Proof of Lemma 1:
The proof shows that data streams within each secondary user can be encoded in any order without changing the optimal solution.
C. Proof of Lemma 1:
- The signal from each secondary user is divided into several data streams.Source: C. Proof of Lemma 1 · p. 23
- Any encoding order for a secondary user's streams achieves the optimal solution because those streams share the same weight.Source: C. Proof of Lemma 1 · p. 23
- The proof relies on the result that equally weighted MAC sum-rate optimization is independent of decoding order.Source: C. Proof of Lemma 1 · p. 23
D. Proof of Lemma 2:
The proofs establish subgradient inequalities for the dual objective using optimal covariance matrices at the selected auxiliary-variable values.
D. Proof of Lemma 2:
- For Lemma 2, the proof uses optimal covariance matrices at two multiplier values to establish the subgradient inequality for g(λ).Source: D. Proof of Lemma 2 · p. 24
- In both proofs, the covariance matrices are optimal for their respective fixed multiplier or auxiliary-variable values.Source: D. Proof of Lemma 2 · p. 24, E. Proof of Lemma 3 · pp. 24–25
E. Proof of Lemma 3:
- For Lemma 3, the proof establishes the corresponding subgradient inequality for g(qt, qu) over feasible auxiliary-variable vectors.Source: E. Proof of Lemma 3 · pp. 24–25