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On Ergodic Sum Capacity of Fading Cognitive Multiple-Access and Broadcast Channels

Rui Zhang, Shuguang Cui, Ying-Chang Liang

arXiv:0806.4468v2cs.IT

TL;DR

The paper addresses how secondary networks can maximize throughput while limiting interference to primary receivers under fading spectrum sharing. It formulates ergodic sum-capacity problems for cognitive MAC and BC channels across mixed long-term and short-term constraints, derives power-control policies and D-TDMA conditions, and finds D-TDMA optimal for the cognitive BC in all considered cases.

  • Problem

    The paper examines ergodic sum capacity and transmission optimality for fading cognitive MAC and BC channels under mixed transmit-power and interference-power constraints.

  • Method

    The paper uses convex capacity optimization, channel-state-based dynamic allocation, and KKT conditions to derive optimal power policies and D-TDMA optimality conditions.

  • Results

    D-TDMA is optimal for the fading cognitive BC under all considered mixed constraints, while for the cognitive MAC it is optimal under LT-TPC with LT-IPC and conditionally optimal in other cases.

  • Takeaways & Limitations

    The results provide constraint-specific power-control policies and characterize when dynamic TDMA achieves ergodic sum capacity in fading cognitive networks.

Abstract

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This paper studies the information-theoretic limits of a secondary or cognitive radio (CR) network under spectrum sharing with an existing primary radio network. In particular, the fading cognitive multiple-access channel (C-MAC) is first studied, where multiple secondary users transmit to the secondary base station (BS) under both individual transmit-power constraints and a set of interference-power constraints each applied at one of the primary receivers. This paper considers the long-term (LT) or the short-term (ST) transmit-power constraint over the fading states at each secondary transmitter, combined with the LT or ST interference-power constraint at each primary receiver. In each case, the optimal power allocation scheme is derived for the secondary users to achieve the ergodic sum capacity of the fading C-MAC, as well as the conditions for the optimality of the dynamic time-division-multiple-access (D-TDMA) scheme in the secondary network. The fading cognitive broadcast channel (C-BC) that models the downlink transmission in the secondary network is then studied under the LT/ST transmit-power constraint at the secondary BS jointly with the LT/ST interference-power constraint at each of the primary receivers. It is shown that D-TDMA is indeed optimal for achieving the ergodic sum capacity of the fading C-BC for all combinations of transmit-power and interference-power constraints.

I. INTRODUCTION

The paper characterizes ergodic sum capacity and optimal dynamic resource allocation for fading cognitive MAC and BC networks sharing spectrum with primary receivers under mixed power constraints.

  • Motivation: Spectrum underlay lets secondary users transmit while primary transmissions are active, provided interference power at each primary receiver stays below a predefined threshold.Dynamic resource allocation uses channel-state information to adjust secondary transmit powers, rates, bandwidths, and antenna beams.
  • System scope: The paper studies SISO fading C-MAC and C-BC networks with K secondary users, M primary receivers, and centralized allocation based on perfect channel-state information.The ergodic sum capacity measures maximum secondary sum-rate averaged over fading states under delay-tolerant traffic.
  • C-MAC: For the C-MAC, the paper derives optimal power-control policies under LT-TPC or ST-TPC combined with LT-IPC or ST-IPC.The four combinations define convex feasible sets, enabling capacity maximization through convex optimization.
  • C-MAC: D-TDMA is optimal for the fading C-MAC under LT-TPC with LT-IPC, but is generally suboptimal under the other three mixed constraint combinations.For LT-TPC with ST-IPC, the number of simultaneously transmitting secondary users need not exceed M + 1 for an optimal solution.
  • C-BC: D-TDMA is optimal for the fading C-BC under all considered combinations of LT/ST transmit-power and interference-power constraints.The corresponding BS power allocations have closed-form solutions resembling single-user water-filling.
  • Evaluation: The paper compares ergodic sum capacities with and without TDMA and optimal power control under different mixed power constraints.It also reports numerical results for both fading cognitive channel types.

III. ERGODIC SUM CAPACITY FOR FADING COGNITIVE MAC

This section formulates ergodic sum-capacity optimization for the fading cognitive multiple-access channel under long-term transmit- and interference-power constraints. Using convex duality and per-fading-state KKT analysis, it establishes when dynamic TDMA is optimal and derives the associated user-selection and power-control rules.

  • Capacity formulation: The ergodic sum capacity is obtained by maximizing over feasible power-control policies under LT-TPC and LT-IPC constraints.The feasible set is convex because the power constraints are affine, enabling efficient numerical optimization.
  • Dual solution: Zero duality gap makes the primal capacity problem equivalent to minimizing its Lagrange dual function over nonnegative transmit- and interference-power prices.Strict feasibility satisfies Slater’s condition, while the dual function decomposes across fading states.
  • D-TDMA optimality: At each fading state, the optimal solution has at most one secondary user transmitting with positive power, yielding a D-TDMA structure.The active user and its power are determined from the KKT-based per-state solution, followed by optimization of the dual variables.
  • Power control: The optimal transmit power is given by the paper’s KKT-derived allocation rule, with (x)+ defined as max(0, x).The rule is applied to the user selected at each fading state using the optimal dual solutions.
  • User selection: With inactive LT-IPC constraints, the selected user is associated with the largest h_i/λ_i ratio among users.When LT-IPC constraints are active, selection and power additionally depend on interference prices and instantaneous interference-channel gains.

B. Long-Term Transmit-Power and Short-Term Interference-Power Constraints

This section studies the fading cognitive multiple-access channel with LT transmit-power and ST interference-power constraints. The optimization is solved using duality and per-state convex analysis, yielding bounds on the number of simultaneously active users and conditions for D-TDMA optimality.

  • Capacity optimization: Under LT-TPC with ST-IPC, the capacity problem is solved by introducing dual variables for the long-term constraints and decomposing across fading states.The resulting per-state problem is convex but generally lacks a closed-form solution, so standard convex methods can be used.
  • Active-user structure: At most M + 1 secondary users can transmit with strictly positive power at any fading state.Thus, the maximum number of active users depends on the number M of primary receivers or interference-power constraints.
  • Special cases: For one primary receiver, the general active-user bound permits at most two simultaneously transmitting secondary users, so D-TDMA may be close to optimal.This is a structural observation rather than a universal D-TDMA optimality result for the full LT-TPC/ST-IPC setting.
  • D-TDMA conditions: D-TDMA is optimal at a fading state if and only if one transmitting user satisfies one of the theorem’s two condition sets involving channel and interference gains.The first set applies when the user with the largest h_i/λ_i satisfies all ST-IPC constraints; otherwise, the second set uses K − 1 inequalities.
  • Special cases: When only ST-IPC constraints are active, the LT-TPC prices can be set to zero, simplifying the D-TDMA conditions to inequalities involving h_jg_im′ − h_ig_jm′.With a single primary receiver, D-TDMA is optimal under this constraint-only case.

C. Short-Term Transmit-Power and Long-Term Interference-Power Constraints

Under ST-TPC with LT-IPC, the paper formulates ergodic sum-capacity optimization and derives closed-form power allocation. D-TDMA is optimal only when a stated channel-dependent condition holds.

  • Optimization formulation: The ergodic sum capacity is formulated as an optimization problem under ST-TPC with LT-IPC and solved using Lagrange duality.The dual function is minimized over nonnegative interference-constraint multipliers, with per-state maximization problems.
  • Closed-form allocation: The KKT-based solution orders users by channel-related quantities and identifies the active transmitting-user set.The active set consists of the leading users under the relevant permutation, with its size determined by a threshold condition.
  • Closed-form allocation: At most one active user transmits below its ST power constraint; every other active user transmits at maximum power.This structural property follows directly from the closed-form solution.
  • D-TDMA optimality: D-TDMA is optimal at a fading state if and only if user π(1) satisfies the theorem’s channel-dependent condition.When the condition holds, user π(1) is selected and its optimal transmit power is given by the corresponding closed-form rule.
  • D-TDMA optimality: Without LT-IPC, all users transmit at their maximum ST powers, so D-TDMA cannot be optimal in this special case.This follows because the interference multipliers become zero and the D-TDMA condition is never satisfied.

D. Short-Term Transmit-Power and Interference-Power Constraints

With both ST-TPC and ST-IPC, the fading C-MAC problem separates across fading states. The resulting convex per-state problem generally lacks a closed-form solution, while D-TDMA has explicit optimality conditions.

  • Optimization formulation: The ergodic sum capacity is obtained by solving an optimization problem with both ST-TPC and ST-IPC.Because all constraints are short-term, the problem decomposes into independent subproblems for individual fading states.
  • Optimization formulation: Each per-state rate-maximization problem is convex but generally has no closed-form solution.The paper suggests interior-point or Lagrange-duality methods for solving it.
  • D-TDMA optimality: D-TDMA is optimal at a fading state if and only if one transmitting user satisfies both stated conditions relative to every other user.The comparison uses each other user’s most restrictive primary-receiver interference channel.
  • D-TDMA optimality: When the D-TDMA conditions hold, the selected user’s optimal transmit power follows the corresponding closed-form expression.The theorem identifies the active user and its power rule together.

IV. ERGODIC SUM CAPACITY FOR FADING COGNITIVE BC

The paper formulates ergodic sum-capacity problems for the fading cognitive broadcast channel under four mixed transmit- and interference-power constraint cases. It proves D-TDMA optimality in every case.

  • Capacity formulation: The C-BC ergodic sum capacities under the four mixed constraint cases are expressed as optimization problems with affine power constraints.The formulations parallel the C-MAC problems while using the secondary BS’s state-dependent transmit power.
  • D-TDMA optimality: D-TDMA is optimal across all fading states in Cases I–IV for achieving the ergodic sum capacity of the fading C-BC.Thus, the result covers every considered combination of LT/ST transmit-power and interference-power constraints.
  • D-TDMA optimality: At each fading state, the BS selects the user with the largest h_i for transmission.The theorem also gives the BS power-assignment rule for each constraint case using the relevant dual variables.
  • Relation to prior results: The result extends the traditional fading SISO-BC conclusion to the cognitive setting with primary-receiver interference constraints.The traditional result already establishes D-TDMA optimality regardless of the BS’s LT- or ST-TPC.

V. NUMERICAL EXAMPLES

Numerical examples evaluate ergodic sum capacity under four mixed transmit- and interference-power constraints, compare dynamic and fixed resource allocation, and examine TDMA effects in fading C-MAC and C-BC networks.

  • Simulation setup: The simulations use symmetric fading channels, 10,000 randomly generated channel power-gain vectors, identical transmit-power constraints across secondary users, and unit interference-power constraints.The four cases combine LT or ST transmit-power constraints with LT or ST interference-power constraints.
  • A. Effects of LT/ST TPC/IPC on Ergodic Sum Capacity: For the fading C-MAC, Case I has the largest ergodic sum capacity and Case IV the smallest for every secondary-user transmit-power constraint P.As P increases, capacity eventually saturates because the interference-power constraint becomes dominant.
  • A. Effects of LT/ST TPC/IPC on Ergodic Sum Capacity: For the fading C-BC, LT interference-power constraints yield much larger capacities than ST interference-power constraints at large BS transmit power.With one BS transmitter and multiple primary receivers, ST interference constraints can limit BS power more stringently than in the corresponding C-MAC example.
  • B. Fading C-MAC With (w/) vs. Without (w/o) TDMA Constraint: Removing the explicit TDMA constraint increases achievable C-MAC ergodic sum capacity in Cases II–IV for both K = 2, M = 1 and K = 4, M = 2.The TDMA constraint limits dynamic resource-allocation flexibility.
  • B. Fading C-MAC With (w/) vs. Without (w/o) TDMA Constraint: The gap between C-MAC capacities with and without TDMA diminishes as P becomes sufficiently large because interference-power constraints eventually become the only active constraints.This aligns with the reported optimality of D-TDMA under the remaining active interference-power cases.
  • C. Dynamic vs. Fixed Resource Allocation: Dynamic resource allocation achieves substantial throughput gains over fixed resource allocation in the C-MAC and increasingly significant gains in the C-BC as the number of secondary users grows.At K = 20 in the C-BC, DRA capacity is 2.75 times FRA for M = 1 and 3.83 times FRA for M = 4.
  • VI. CONCLUDING REMARKS: The paper characterizes achievable ergodic sum capacity and presents optimal dynamic resource-allocation schemes for fading C-MAC and C-BC networks under mixed LT/ST constraints.The techniques are also stated to extend to settings with all LT/ST constraints, prioritized users, and parallel Gaussian channels.

APPENDIX I PROOF OF LEMMA 3.1

The proof uses KKT conditions and independence arguments to characterize sparse optimal transmission and interference-multiplier patterns. It establishes when at most one secondary user is active and derives conditions for Theorem 3.2.

  • Single active user: At most one user can have a strictly positive optimal power value under the stated KKT equalities.The relevant equality would otherwise hold with zero probability because the channel gains are independent while the multipliers are fixed.
  • Feasible active sets: With multiple users having positive power, the KKT equations imply that their number cannot exceed M + 1.There are M dual variables but only |J| − 1 independent equations, requiring M ≥ |J| − 1.
  • Multiplier cases: The proof distinguishes the cases in which all interference multipliers are zero or exactly one multiplier is strictly positive.More than one strictly positive multiplier is ruled out through contradictory KKT implications.
  • Theorem 3.2 conditions: The first and second sets of Theorem 3.2 conditions are obtained from the corresponding zero-multiplier and single-positive-multiplier cases.Sufficiency follows because strict convexity makes the KKT conditions necessary and sufficient for the unique primal and dual optimum.

APPENDIX V PROOF OF LEMMA 3.4

The supplied passage states that Lemma 3.4 follows directly from the two preceding inequalities.

  • Proof transition: Lemma 3.4 follows from the two inequalities established immediately beforehand.The passage provides the proof transition but does not state the lemma’s substantive claim.
  • Proof transition: The argument is presented as an immediate consequence rather than through a separate optimization derivation.No additional assumptions or quantitative result are specified in the supplied passage.
  • Proof transition: The passage marks completion of the preceding inequality-based step in the proof.It does not identify the variables or conditions appearing in those inequalities.

APPENDIX VI PROOF OF LEMMA 3.5

The proof of Lemma 3.5 characterizes the optimal active-user set through KKT-derived inequalities and identifies the corresponding optimal solution structure.

  • Optimal solution: Problem 3.6 has at most one user indexed by i with 0 < p_i*.The active user is indexed as i = π(|I|), and the transmitting users’ optimal sum-power satisfies the stated bound in the lemma.
  • Optimal solution: If two users had strictly positive powers below P_ST, their KKT equalities would hold simultaneously with zero probability.Independence of the users’ channel gains yields the contradiction underlying the single-positive-power result.
  • Case analysis: Lemma 6.1 restricts the optimal powers to one of two solution sets.The supplied passage introduces this dichotomy without reproducing the sets themselves.
  • Active-user count: The optimal number of active users |I| is the largest x satisfying inequality (76).The proof verifies the inequality for the last active user and its failure for the next user, establishing that (76) determines |I|.

APPENDIX VII PROOF OF THEOREM 3.4

Theorem 3.4 is proved with KKT optimality conditions, showing a restricted pattern for positive interference multipliers and deriving channel conditions for a transmitting user.

  • KKT framework: The proof uses KKT conditions for Problem 3.8, with dual variables associated with its power and interference constraints.The KKT conditions provide the basis for both necessity and sufficiency in the theorem’s proof.
  • Multiplier structure: At most one interference multiplier can be strictly positive at a time.Having two positive multipliers leads to contradictory implications involving the interference-channel gains and positive transmit power.
  • Multiplier structure: There is one and only one strictly positive interference multiplier in the considered transmitting-user case.The proof rules out both multiple positive multipliers and the possibility that all multipliers are zero.
  • Sufficiency: The theorem’s sufficiency direction follows because strict convexity makes the KKT conditions necessary and sufficient for the unique optimum.This establishes the converse after the channel and multiplier conditions are derived.

TDMA CONSTRAINT

Under the TDMA constraint, only one secondary user transmits in each fading state, and an optimization over user selection and power control determines the ergodic sum capacity. The resulting TDMA solutions coincide with the unconstrained problem's solutions, with zero duality gap established for the considered LT-TPC/LT-IPC case.

  • TDMA formulation: TDMA restricts each fading state to one transmitting secondary user selected by the mapping Π(α).The mapping Π(α) assigns the transmitting user for each channel realization α.
  • Optimization approach: For a fixed user-selection function, capacity maximization over feasible powers is convex, whereas optimizing the selection function is not necessarily convex.The analysis therefore uses a dual formulation to solve the selection and power-control problem.
  • Optimization approach: The dual minimization over the nonnegative power- and interference-constraint multipliers is a convex optimization solvable by the ellipsoid method.The per-fading-state maximization is solved separately for fixed dual variables.
  • Optimal user selection: For each candidate user, the optimal nonnegative transmit power is obtained first, after which the user maximizing the resulting objective is selected.This reduces the per-state TDMA decision to comparing users after their power solutions are substituted into the objective.
  • Optimality of TDMA: The TDMA-constrained problem has the same solution set as the corresponding unconstrained problem, because the unconstrained optimum is already TDMA-based.For Problem 8.1, the duality gap is zero and the constrained and unconstrained formulations share the same solutions.

B. Long-Term Transmit-Power and Short-Term Interference-Power Constraints

The paper treats mixed long-term and short-term transmit- and interference-power constraints by formulating separate TDMA-constrained optimization problems. Lagrange duality yields the corresponding power-control and user-selection policies, while fully short-term constraints permit statewise decomposition.

  • LT transmit-power and ST interference-power constraints: With LT transmit-power and ST interference-power constraints, the TDMA ergodic sum capacity is posed as a constrained optimization problem.The problem introduces dual variables for the long-term transmit-power constraints.
  • LT transmit-power and ST interference-power constraints: The LT-TPC/ST-IPC problem has zero duality gap because it satisfies the time-sharing conditions.The optimal transmitting user at each fading state maximizes a dual-adjusted expression among all users.
  • ST transmit-power and LT interference-power constraints: With ST transmit-power and LT interference-power constraints, Lagrange duality similarly determines the optimal power-control policy and per-state user selection.The policy uses optimal dual solutions for the long-term interference constraints obtained by the ellipsoid method.
  • ST transmit-power and ST interference-power constraints: Under ST transmit-power and ST interference-power constraints, all constraints separate across fading states, decomposing the problem into independent per-state subproblems.The optimal user is selected by maximizing the specified expression among users in each fading state.
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