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Exploiting Multi-Antennas for Opportunistic Spectrum Sharing in Cognitive Radio Networks

Rui Zhang, Ying Chang Liang

arXiv:0711.4414v1cs.IT

TL;DR

The paper addresses the tradeoff between secondary throughput and interference protection for primary receivers in cognitive-radio spectrum sharing. It characterizes secondary capacity under transmit- and interference-power constraints, using multi-antenna spatial-spectrum optimization and extensions across channels, time, and frequency. The resulting framework includes optimal convex-optimization designs and easier-to-implement suboptimal algorithms.

  • Problem

    The paper studies how secondary users can maximize throughput while maintaining primary-user QoS under interference-power constraints.

  • Method

    The paper uses multi-antenna transmit spatial-spectrum design, convex optimization, suboptimal SVD-based algorithms, and dual decomposition for multi-channel allocation.

  • Results

    The paper characterizes exact secondary-user capacity under explicit transmit-power and primary-receiver interference-power constraints for arbitrary numbers of primary receivers and antennas.

  • Takeaways & Limitations

    Multi-antennas provide spatial adaptation that complements time- and frequency-domain resource adaptation for opportunistic spectrum sharing.

  • Takeaways & Limitations

    The optimum transmit spatial-spectrum design assumes channel information is available at the secondary transmitter.

Abstract

from arXiv · show

In cognitive radio (CR) networks, there are scenarios where the secondary (lower priority) users intend to communicate with each other by opportunistically utilizing the transmit spectrum originally allocated to the existing primary (higher priority) users. For such a scenario, a secondary user usually has to trade off between two conflicting goals at the same time: one is to maximize its own transmit throughput; and the other is to minimize the amount of interference it produces at each primary receiver. In this paper, we study this fundamental tradeoff from an information-theoretic perspective by characterizing the secondary user's channel capacity under both its own transmit-power constraint as well as a set of interference-power constraints each imposed at one of the primary receivers. In particular, this paper exploits multi-antennas at the secondary transmitter to effectively balance between spatial multiplexing for the secondary transmission and interference avoidance at the primary receivers. Convex optimization techniques are used to design algorithms for the optimal secondary transmit spatial spectrum that achieves the capacity of the secondary transmission. Suboptimal solutions for ease of implementation are also presented and their performances are compared with the optimal solution. Furthermore, algorithms developed for the single-channel transmission are also extended to the case of multi-channel transmission whereby the secondary user is able to achieve opportunistic spectrum sharing via transmit adaptations not only in space, but in time and frequency domains as well.

I. INTRODUCTION

The paper studies how multi-antenna secondary transmitters can balance secondary throughput against interference constraints at higher-priority primary receivers. It formulates capacity-achieving spatial-spectrum designs and develops optimal, suboptimal, and multi-channel resource-allocation algorithms.

  • Opportunistic spectrum sharing lets secondary users reuse primary-allocated spectrum, potentially enhancing spectrum utilization.
  • The central challenge is maximizing secondary throughput while preserving primary-user QoS through interference-power constraints at primary receivers.
  • Multi-antennas add spatial degrees of freedom, enabling the secondary transmitter to trade spatial multiplexing against interference avoidance alongside time and frequency adaptation.
  • The paper formulates capacity-achieving transmit spatial-spectrum design as convex optimization under secondary transmit-power and primary interference-power constraints.
  • For MISO channels, beamforming is optimal and a closed-form beamforming vector is derived for one single-antenna primary receiver.
  • For general MIMO and multi-channel settings, the paper presents D-SVD, P-SVD, hybrid algorithms, and Lagrange dual decomposition for spatial, temporal, and frequency adaptation.

II. SYSTEM MODEL AND PROBLEM FORMULATION

The system models a secondary MIMO link sharing spectrum with K primary receivers under secondary transmit-power and receiver-side interference-power constraints. It formulates capacity maximization over the secondary transmit covariance matrix as a convex optimization problem, while noting imperfect channel knowledge yields capacity upper bounds.

  • Practical scope: When instantaneous channels cannot be estimated perfectly, the paper’s capacity results serve as upper bounds for secondary transmission.The model assumes channel knowledge for resource adaptation, but explicitly identifies imperfect estimation as a practical boundary.
  • Problem formulation: Choosing Γ_k much smaller than the primary receiver noise power φ_k can make the primary capacity loss arbitrarily small.This motivates regulating interference through total receiver-side power constraints.
  • System model: The secondary transmitter maximizes its rate while satisfying its own power limit and total interference-power constraints at K primary receivers.The constraints may involve total interference across each primary receiver’s antennas or individual per-antenna limits.
  • Problem formulation: The optimization maximizes secondary channel capacity subject to power, interference, and positive-semidefinite covariance constraints.Its objective is concave in S and its feasible set is convex, enabling efficient numerical solution for arbitrary secondary antenna counts and primary receiver configurations.

III. ONE SINGLE-ANTENNA PRIMARY RECEIVER

For a single-antenna primary receiver, the interference channel from the secondary transmitter is represented by a vector, reducing the general formulation to a simpler constrained problem.

  • Single-antenna primary receiver: With one single-antenna primary receiver, the secondary-to-primary interference channel is a MISO vector g.The tolerable interference power is represented by γ, and the general problem is simplified accordingly.

A. MISO Secondary User’s Channel

For a MISO secondary channel, the optimal covariance is rank one, so beamforming is optimal. With one primary receiver, the beamforming vector has a closed-form structure combining the interference direction and its null-space projection.

  • Closed-form solution: For the single-antenna primary-receiver case, the paper derives a closed-form optimal transmit covariance through the rank-one beamforming representation.The result applies to the MISO secondary channel under the simplified problem P2.
  • MISO optimality: Rank(S)=1 is optimal for the MISO secondary channel, establishing beamforming as the optimal transmission strategy.The covariance can therefore be written as S=vv†.
  • Beamforming structure: The optimal beamforming vector lies in the span of the primary-interference channel g† and the secondary-channel projection onto the null space of g†.This structure balances useful secondary-channel gain against interference avoidance.
  • Beamforming structure: The beamforming weights are obtained by solving the equivalent two-component constrained problem, with separate cases for inactive and active interference constraints.When the constraint is active, the power allocated toward the interference direction must be regulated by γ.
  • Special cases: When the interference constraint is inactive, the optimal beamformer reduces to the conventional MRC direction for the secondary MISO link.The active-constraint case instead limits the component aligned with the primary receiver’s channel.

B. MIMO Secondary User’s Channel

For a full MIMO secondary channel, interference constraints can make spatial multiplexing optimal while preventing the optimal covariance from diagonalizing the channel. The paper therefore gives exact numerical optimization and lower-complexity SVD-based alternatives.

  • Optimal solution: The optimal MIMO covariance has no closed-form solution in general and must be obtained numerically through the constrained problem.This contrasts with the closed-form MISO result.
  • MIMO optimality: In the MIMO case, Rank(S)>1 can be optimal, implying spatial multiplexing rather than beamforming.The optimal strategy depends on the interference-constrained covariance design.
  • Direct-Channel SVD: Direct-Channel SVD diagonalizes the secondary MIMO channel into parallel sub-channels, simplifying encoding and decoding through independent processing.It is capacity-achieving when the interference constraint is inactive, such as when γ is sufficiently large.
  • MIMO optimality: Under interference constraints, the optimal covariance generally differs from the secondary-channel SVD and does not diagonalize the MIMO channel.Capacity-achieving transmission may consequently require joint coding and decoding across antennas or streams.
  • Suboptimal algorithms: Two lower-complexity closed-form algorithms, Direct-Channel SVD and Projected-Channel SVD, are proposed for MIMO covariance design.The first decomposes the original channel, while the second decomposes its projection into the null space of the interference channel.
  • Power allocation: The SVD-based power allocation uses multi-level water-filling governed by Lagrange multipliers for the transmit-power and interference-power constraints.The interference multiplier is zero when that constraint is inactive; otherwise, it is adjusted by bisection.

2) Projected-Channel SVD (P-SVD):

P-SVD projects the secondary channel into the primary receivers’ interference-free null space, then uses SVD-based precoding and power allocation for secondary transmission. It is optimal under zero interference and asymptotically achieves the optimal spatial multiplexing gain at high transmit power.

  • P-SVD: P-SVD projects the secondary channel into the null space of the primary-channel vector before constructing the precoder.The resulting precoder lies in the projected channel’s right-singular-vector basis.
  • P-SVD: The projection makes the interference power at the primary receiver zero, satisfying any nonnegative interference-power limit.The construction uses S = U_⊥Σ(U_⊥)† and yields gSg† = 0.
  • P-SVD: When the interference constraint is γ = 0, the paper proves that P-SVD is optimal.For a MISO secondary channel, P-SVD is generally suboptimal unless γ = 0.
  • Performance Comparison: As Pt → 0, D-SVD is optimal because the interference constraint becomes inactive, whereas P-SVD can lose rate through channel projection.At low power, the achievable rates use the strongest channel gains of the respective schemes.
  • Performance Comparison: As Pt → ∞, P-SVD approaches the optimal capacity’s spatial multiplexing gain, while D-SVD can saturate and have zero gain.P-SVD’s rate grows linearly with log2 Pt by the projected-channel dimension; D-SVD eventually becomes power-limited by interference constraints.

IV. MULTIPLE PRIMARY RECEIVERS

The multiple-primary-receiver case formulates the secondary capacity problem under multiple interference constraints and establishes structural and computational properties of its solutions. For a MISO secondary channel, beamforming remains optimal, while convex reformulations support numerical solution methods.

  • Problem Formulation: The section studies a secondary transmission problem with multiple primary receivers, each possibly having multiple receive antennas.The formulation retains the secondary transmit-power constraint and multiple interference-power constraints.
  • MISO Secondary User’s Channel: Beamforming is optimal when the secondary channel is MISO, and the optimal transmit covariance remains rank one with multiple primary receivers or antennas.The paper also states that the optimal beamforming vector has a closed-form solution in the MISO case.
  • Optimization Methods: The MISO problem can be reformulated as a second-order cone program and solved through convex numerical optimization.Strong duality follows after verifying Slater’s condition, making the dual problem equivalent to the primal problem.
  • Computational Complexity: The interior-point method, SOCP, and SDP have similar variable-count complexity orders, but SOCP has much lower complexity than the interior-point method.The SDP can be preferable to SOCP when K ≪ Mt,s.

B. MIMO Secondary User’s Channel

For a MIMO secondary channel, the paper generally solves the constrained capacity problem with an interior-point method and adapts SVD-based algorithms to multiple interference constraints. The D-SVD adaptation uses unchanged channel precoding but adjusts power allocation through multiple dual variables.

  • MIMO Secondary User’s Channel: For a MIMO secondary channel, the constrained optimization problem generally requires the interior-point method.The paper also modifies two SVD-based solutions for multiple primary receivers and antennas.
  • D-SVD: In the D-SVD adaptation, the precoding matrix remains the right-singular-vector matrix of the secondary channel regardless of primary-receiver count.Power allocation is changed to incorporate all interference-power constraints.
  • D-SVD: Multiple nonnegative Lagrange multipliers are introduced, one for each interference-power constraint.The resulting power assignments are obtained as multi-level water-filling solutions.
  • D-SVD: The multipliers are iteratively updated together with the transmit-power multiplier to satisfy the multiple constraints.The updates use subgradient information for each interference constraint.

2) Projected-Channel SVD (P-SVD):

The hybrid D-SVD/P-SVD algorithm projects the secondary channel into a selected interference subspace, retaining remaining dimensions for transmission. This creates a tunable tradeoff between interference avoidance and spatial multiplexing, with asymptotic optimality results for the component SVD schemes.

  • Hybrid D-SVD/P-SVD: The hybrid algorithm projects the secondary MIMO channel into the null space of a selected subspace of the aggregate primary-channel matrix.Unlike P-SVD, it does not necessarily remove the entire space spanned by the primary channels.
  • Hybrid D-SVD/P-SVD: The hybrid method can operate when Mt,s ≤ Mr,p, a regime where P-SVD is not implementable.This is one stated reason for using the hybrid design.
  • Performance Comparison: For a single-antenna primary receiver, D-SVD is asymptotically optimal as Pt → 0 and P-SVD as Pt → ∞.For multiple primary antennas, the paper conjectures that the hybrid method can outperform both at some moderate transmit powers.
  • Hybrid D-SVD/P-SVD: Selecting the subspace associated with the largest normalized singular values targets the primary-channel directions contributing the most interference.The paper proposes this as a heuristic because the optimal selection rule remains unknown.
  • Hybrid D-SVD/P-SVD: The projected channel is diagonalized into min(Mr,s, Mt,s − b) sub-channels, while residual interference may remain when b < M̂G.Power allocation is then adjusted to satisfy all primary-receiver interference constraints.
  • Interference Diversity: Channel fading, unequal average gains, and correlations can spread the normalized primary-channel singular values, making subspace projection more effective.The paper links this interference-diversity setting to greater flexibility in balancing multiplexing and interference avoidance.

V. MULTI-CHANNEL TRANSMISSION

The paper extends opportunistic spectrum sharing from single-channel transmission to multiple channels across space, time, and frequency. The resulting total-rate optimization remains convex and can be decomposed into independent, similarly structured subproblems.

  • Multi-channel model: Multi-channel transmission adapts secondary power, rate, and spatial resources across OFDM tones or time-domain fading blocks.Each sub-channel may represent an OFDM tone or a fading block, with primary activity detected across space, time, and frequency.
  • Optimization problem: The optimization maximizes total secondary transmit rate under one total-power constraint and per-sub-channel interference-power constraints.The interference-power threshold is assumed identical across sub-channels in the stated formulation.
  • Solution method: Lagrange dual decomposition converts the multi-channel problem into N independent subproblems with identical structure.A common dual variable regulates power allocation across sub-channels, and the same computational routine can be applied to each subproblem.
  • Solution method: Strong duality holds, so solving the dual problem yields the primal optimum without a duality gap.The dual function is maximized over the spatial spectra and then minimized over the nonnegative power multiplier.
  • Implementation: For MISO channels, SOCP or SDP can solve the convex problem, whereas MIMO channels use an interior-point method because spatial multiplexing is optimal.The multi-channel algorithm can also use D-SVD or P-SVD approximations independently on each sub-channel.

VI. SIMULATION RESULTS

Simulations show that interference constraints reduce secondary capacity, especially at high SNR, while multiple antennas and adaptive SVD-based designs recover substantial rate. Across multiple primary receivers, the hybrid method benefits from interference diversity.

  • Single primary receiver: Interference-power constraints reduce secondary capacity for both SISO and MISO links, with the strongest effect in the high-SNR regime.The simulations use γ = 0.01, Mr,s = 1, and compare Mt,s = 1 with Mt,s = 4.
  • Single primary receiver: Adding transmit antennas produces substantial capacity improvement because spatial adaptation can avoid interference while increasing secondary SNR.The secondary transmitter adapts its spatial spectrum away from the primary receiver’s interfering direction.
  • SVD-based algorithms: With Mt,s = Mr,s = 2, D-SVD is close to interference-constrained capacity at low SNR, whereas P-SVD is close at high SNR.At high SNR, optimal capacity and P-SVD rate scale with log2 SNR by min(Mr,s, Mt,s − 1) = 1, while D-SVD saturates.
  • Multiple primary receivers: As the number K of primary receivers increases, all proposed rates decrease because more interference-power constraints are added.The hybrid method and D-SVD converge for K much larger than Mt,s, but the convergence is very slow.
  • Multiple primary receivers: Dynamic selection of b gives the hybrid D-SVD/P-SVD substantial rate improvements over D-SVD and P-SVD by exploiting interference diversity.For two primary receivers, the preferred algorithm changes from D-SVD to hybrid b = 1 and finally P-SVD as SNR increases.

D. Multi-Channel Transmission

The paper extends its algorithms to broadband multi-tone transmission, allowing adaptation across frequency and time as well as space. Per-tone SVD selection exploits frequency-selective fading and approaches the capacity more closely.

  • Multi-tone model: The broadband model uses N = 64 OFDM tones with a single-antenna primary receiver and Mt,s = Mr,s = 2.The interference threshold is γ = 0.1 at every tone, with four equal-energy independent consecutive multipath delays.
  • Multi-tone model: The secondary user allocates variable transmit rate, power, and spatial spectrum independently across OFDM tones according to the sub-channel channels.This realizes transmit adaptation in the frequency domain while extending the broader space-time-frequency framework.
  • Algorithm and result: The per-tone suboptimal algorithm selects D-SVD or P-SVD separately at each tone and is compared with constrained and unconstrained capacity.Its selection exploits the diversity of the frequency-selective channels.
  • Algorithm and result: Frequency-selective fading makes the per-tone algorithm’s rate gap from actual capacity smaller than in the corresponding single-channel case.The comparison specifically concerns transmission with and without interference-power constraints.
  • Extensions: For multiple primary receivers or antennas, the paper proposes hybrid D-SVD/P-SVD designs to exploit interference diversity.The paper also identifies generalized interference constraints and joint primary-secondary adaptation as extensions.

APPENDIX I

The appendix formulates primary-user capacity loss caused by secondary interference and establishes structural properties of optimal secondary transmission. It supports beamforming for MISO channels and explains why constrained MIMO capacity generally requires non-diagonalizing designs.

  • Primary capacity loss: The secondary transmitter’s interference at primary receiver k has covariance Q_k = G_k S G_k†, where S is the secondary transmit covariance matrix.Primary capacity loss due to secondary transmission is expressed as the difference between capacities with and without that interference.
  • MISO structure: For MISO secondary channels, the optimal transmit covariance has rank one, establishing beamforming as the optimal strategy.The rank-one result follows from the KKT conditions under either the secondary power or interference constraint being active.
  • MISO structure: The optimal MISO beamforming direction favors the component orthogonal to the primary-interference direction because it avoids interference and improves secondary SNR.The appendix states that the orthogonal component does not increase interference power and helps the secondary receiver’s SNR.
  • P-SVD special case: When the interference threshold is zero, P-SVD is optimal because its projected transmit covariance satisfies the zero-interference constraint.The proof uses the nullspace projection associated with the primary channel.
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