PaperlayerView exact revision on arXiv

Source-linked AI summary

Cognitive Beamforming Made Practical: Effective Interference Channel and Learning-Throughput Tradeoff

Rui Zhang, Feifei Gao, Ying-Chang Liang

arXiv:0809.2148v3cs.ITmath.IT

TL;DR

Cognitive beamforming must balance CR performance against interference to concurrent PR transmissions, while conventional designs assume knowledge of CR-to-PR channels. This paper learns an effective interference channel from observed PR signals and designs beamforming from it. It reports a conditional capacity gain over zero-forcing and derives a learning-time choice for finite, imperfect estimation.

  • Problem

    Earlier cognitive-beamforming designs assume the CR transmitter knows the channels from it to PR terminals, although obtaining those channels may require PR-side estimation and feedback.

    Source: §II · p. 3

  • Method

    The CR estimates an effective interference channel from observed PR transmissions and uses it for beamforming, including a two-phase protocol for finite learning time.

    Source: §I · pp. 1–2

  • Results

    When multi-antenna PRs use only a subspace of available spatial dimensions, EIC-based beamforming gains capacity over conventional zero-forcing even with exact CR-to-PR channel knowledge.

    Source: §I · p. 2

  • Takeaways & Limitations

    For finite learning time, the paper derives a learning-time choice that maximizes CR effective throughput subject to CR transmit-power and PR interference-power constraints.

    Source: §I · p. 2

    Imperfect EIC estimates from finite learning time cause leakage interference at PR terminals.

    Source: §I · p. 2

Abstract

from arXiv · show

This paper studies the transmit strategy for a secondary link or the so-called cognitive radio (CR) link under opportunistic spectrum sharing with an existing primary radio (PR) link. It is assumed that the CR transmitter is equipped with multi-antennas, whereby transmit precoding and power control can be jointly deployed to balance between avoiding interference at the PR terminals and optimizing performance of the CR link. This operation is named as cognitive beamforming (CB). Unlike prior study on CB that assumes perfect knowledge of the channels over which the CR transmitter interferes with the PR terminals, this paper proposes a practical CB scheme utilizing a new idea of effective interference channel (EIC), which can be efficiently estimated at the CR transmitter from its observed PR signals. Somehow surprisingly, this paper shows that the learning-based CB scheme with the EIC improves the CR channel capacity against the conventional scheme even with the exact CR-to-PR channel knowledge, when the PR link is equipped with multi-antennas but only communicates over a subspace of the total available spatial dimensions. Moreover, this paper presents algorithms for the CR to estimate the EIC over a finite learning time. Due to channel estimation errors, the proposed CB scheme causes leakage interference at the PR terminals, which leads to an interesting learning-throughput tradeoff phenomenon for the CR, pertinent to its time allocation between channel learning and data transmission. This paper derives the optimal channel learning time to maximize the effective throughput of the CR link, subject to the CR transmit power constraint and the interference power constraints for the PR terminals.

I. INTRODUCTION

The paper proposes cognitive beamforming that uses an effective interference channel (EIC) learned from observed primary-radio signals, rather than requiring prior CR-to-PR channel knowledge. It presents opportunistic spatial sharing and studies both ideal learning and finite-time learning, including the resulting throughput tradeoff.

I. INTRODUCTION

  • The EIC-based scheme needs no prior CR-to-PR channel knowledge and can be estimated at the CR transmitter by observing PR transmissions.This avoids the overhead of having PR terminals estimate those channels and feed them back to the CR.Source: §I · pp. 1–2
  • Opportunistic spatial sharing lets the CR transmit alongside PRs over detected available spatial dimensions, using PR-transmission side information to improve CR spectral efficiency over underlay.The paper contrasts this with interweave, which transmits only when PR transmissions are off.Source: §I · pp. 1–2
  • When a multi-antenna PR communicates over only part of its available spatial dimensions, EIC-based beamforming gains capacity over conventional zero-forcing even with exact CR-to-PR channel knowledge.The paper attributes the gain to side information about PR transmit dimensions extracted from the EIC.Source: §I · p. 2
  • With finite learning time, EIC estimation errors cause leakage interference at PR terminals, creating a tradeoff between PR protection and CR throughput.The paper formulates an optimal learning-time choice to maximize CR effective throughput under CR transmit-power and PR interference-power constraints.Source: §I · p. 2

II. SYSTEM MODEL

The system model considers one multi-antenna CR link sharing a frequency band with one PR link. It motivates learning-based beamforming by the difficulty of obtaining actual CR-to-PR channels and proposes the EIC to address implementation issues.

II. SYSTEM MODEL

  • The model has one multi-antenna CR transmitter and one CR receiver coexisting with two PR terminals, which operate in TDD with reciprocal channels.The PR terminals treat CR interference as additional noise; the CR also treats PR interference at its receiver as noise.Source: §II · pp. 2–3
  • Unlike earlier designs that assume perfect knowledge of the CR-to-PR channels, the proposed approach learns channel information by observing PR signals and using reciprocity.The paper notes that requiring PRs to estimate these channels and feed them back may be difficult when the systems belong to different operators.Source: §II · p. 3
  • The CR can observe effective channels GH_j A_j rather than actual channels G_j when PR transmit beamformers A_j are unknown, and separate estimation can require synchronization.The paper also identifies uncertainty about controlling interference when PR receive beamformers differ from their transmit beamformers.Source: §II · p. 3
  • The EIC is proposed to resolve the effective-channel and synchronization issues without critical assumptions or prior knowledge of PR signal processing.The remaining issue—analyzing the effect of CR interference on PR transmissions—is addressed for ideal and imperfect EIC estimates in later sections.Source: §II · p. 3

III. EFFECTIVE INTERFERENCE CHANNEL

The CR estimates an aggregate effective interference channel from observations of PR transmissions. The resulting EIC supports beamforming design without separately estimating the two PR-to-CR channels or synchronizing with each PR transmit direction.

III. EFFECTIVE INTERFERENCE CHANNEL

  • Before data transmission, the CR listens for PR signals over N symbol periods to obtain observations for channel learning.The received-signal model includes the active PR transmission and additive noise.Source: §III · p. 3
  • The model treats the two PR transmission directions as disjoint in time, with independent encoded signals and additive noise.PR activity probabilities satisfy α1 + α2 ≤ 1; silent intervals may occur between alternating TDD transmissions.Source: §III · p. 3
  • The CR's sample covariance estimates the PR-signal covariance from noisy observations; with infinitely many samples, the sample covariance converges almost surely to that covariance plus noise power.For finite N, the PR-signal covariance must be estimated from the sample covariance.Source: §III · p. 4
  • Using channel reciprocity, the paper defines the EIC from the aggregate effective channels and studies beamforming based on it instead of the actual CR-to-PR channels.This avoids separating the two PR-to-CR channels and removes the need to synchronize with each PR transmit direction; the impact of CR interference on PR transmissions is then analyzed for ideal and finite learning.Source: §III · p. 4

IV. PERFECT CHANNEL LEARNING

With perfect EIC learning, the CR can null its effective interference channel to protect PR transmissions when the stated subspace conditions hold. The EIC-based design can also retain more CR degrees of freedom than P-SVD when the PRs use only part of their available spatial dimensions.

IV. PERFECT CHANNEL LEARNING

  • The EIC-based beamformer lies in the null space of G_eff, leaving at most Mt − (d1 + d2) CR spatial degrees of freedom when d1 + d2 < Mt.The remaining CR covariance design is equivalent to precoding over the auxiliary channel H U under the CR power constraint.Source: §IV · pp. 4–5
  • When Span(AH_j G_j) contains Span(B_j G_j) for both PRs, enforcing G_eff A_CR = 0 eliminates adverse effects on PR transmissions.The result assumes perfect EIC learning; if the conditions fail, the scheme can still impair PR performance.Source: §IV · pp. 4–5
  • For spatial multiplexing, the stated subspace condition holds regardless of B_j; for eigenmode transmission, it holds when both PR directions use the same number of streams.The eigenmode result includes beamforming as the one-stream special case.Source: §IV · p. 4
  • Fig. 2 compares CR capacity for the proposed CB and P-SVD schemes in a setup with M1 = M2 = 2, d1 = d2 = 1, Mt = 5, and Mr = 3.The supplied figure caption and experiment description identify the comparison and setup, but do not state the plotted outcome.Source: Figure 2 · p. 5, §IV · p. 5
  • When d1 + d2 < Mt but M1 + M2 ≥ Mt, the EIC scheme can have positive CR DoF while P-SVD has zero DoF.The proposed DoF bound is min((Mt − d1 − d2)+, Mr), versus min((Mt − M1 − M2)+, Mr) for P-SVD; the proposed scheme exploits PR signal-dimension side information.Source: §IV · p. 5

A. Estimation of Geff

The EIC estimate depends on the estimated PR-signal covariance, Q_s. The section describes two estimators that differ in their noise-power treatment and in how they estimate the covariance rank.

A. Estimation of G_eff

  • The EIC estimate depends solely on the estimated signal covariance Q_s, so estimating Q_s provides the basis for estimating G_eff.The section presents maximum-likelihood estimates of Q_s and its eigenspaces after estimating the noise power and rank.Source: A. Estimation of Geff · pp. 6–7
  • The MDL rank estimator uses the geometric and arithmetic means of the last Mt − k eigenvalues of the observed covariance matrix.The section describes its bias correction term and then obtains the estimated signal and noise eigenspaces from the corresponding columns of the covariance eigendecomposition.Source: A. Estimation of Geff · pp. 6–7
  • The two EIC estimators share a similar structure but differ in the noise-power term and in how they estimate rank(Q_s).The available noise-power knowledge at the CR determines which estimation assumptions apply.Source: A. Estimation of Geff · pp. 6–7

B. Effective Leakage Interference

Finite learning leaves residual interference at the PR terminals, quantified as noise-normalized effective leakage power. Its bounds depend on CR power, learning time, and PR parameters, with rank underestimation posing a separate risk.

B. Effective Leakage Interference

  • Underestimating the effective rank can place estimated noise-subspace vectors inside the PR signal subspace, greatly increasing PR interference; the analysis assumes the rank is correctly estimated.The authors leave detailed study of rank-error probabilities and threshold selection for separate work.Source: B. Effective Leakage Interference · p. 7
  • The effective leakage interference power is the interference power normalized by the noise power after both are processed by the PR receive beamformer.Its upper bound is proportional to CR transmit power and inversely proportional to learning length N, α_j, and PR transmit power P_j.Source: B. Effective Leakage Interference · p. 7
  • The leakage measure is invariant to scalar multiplication of G_j, so the CR protects a PR equally despite location-dependent signal attenuation.The paper notes that a finite upper bound applies when α_j > 0; when α_j = 0, that PR is not transmitting.Source: B. Effective Leakage Interference · p. 7
  • With fixed N and CR power, the leakage upper bound decreases as α_j or P_j increases, a property the authors identify as useful for distributing leakage among PRs.The bound is practically meaningful when α_j is a non-negligible positive number.Source: B. Effective Leakage Interference · p. 7
  • Numerical results show inverse leakage power growing linearly with N at both high and low PR SNR, while the theoretical and numerical results match well only at high SNR.At low SNR, the authors attribute the mismatch to an inaccurate first-order approximation.Source: B. Effective Leakage Interference · p. 7

C. Optimal Learning Time

The section formulates CR learning time as a throughput optimization under transmit-power and PR leakage constraints. It accounts for the learning overhead and uses a slowly varying-subspace approximation to make the optimization tractable.

C. Optimal Learning Time

  • The CR chooses learning time τ to maximize effective throughput over block duration T, subject to its power constraint and leakage limits at both PR terminals.The throughput includes a factor (T − τ)/T for the time spent learning rather than transmitting data.Source: C. Optimal Learning Time · p. 8
  • The CR covariance trace is bounded by P_CR under peak power, while average-power operation can allocate the block's total power to the data phase.The section distinguishes the resulting cases through J, including J = min(P_CR, γτ) under a peak constraint.Source: C. Optimal Learning Time · p. 8
  • The interference constraints can be converted into Tr(C_CR) ≤ γτ, with γ set from the stricter of the two PR-specific limits.The calibration also uses a margin for analytical errors and depends on prior knowledge of the margins, leakage threshold, symbol period, and observed PR signal power.Source: C. Optimal Learning Time · p. 8
  • Because the estimated noise subspace changes with τ, the analysis ignores that dependence after noting its estimation error decreases on the order of O(1/√τ).The paper says it verifies this approximation through simulation.Source: C. Optimal Learning Time · p. 8
  • The covariance optimization is transformed using an eigendecomposition of the estimated subspace and yields a diagonal optimal covariance in that basis.The optimal diagonal entries are constrained by 0 ≤ x_i ≤ J, and the CR covariance is recovered from that diagonal matrix.Source: C. Optimal Learning Time · p. 8

1) Peak transmit power constraint:

Under a peak CR transmit-power constraint, the learning-time optimization is divided into regimes, with the optimal time determined by the relevant objective branch. The paper characterizes the branch behavior and illustrates the resulting optimum in Fig. 5.

1) Peak transmit power constraint:

2) Average transmit power constraint:

With average CR transmit power, the optimal learning time depends on the relation between PCR/γ and T/4 and, in one regime, on two roots of a power-balance equation. The paper summarizes the solution and illustrates it in Fig. 6.

2) Average transmit power constraint:

  • When PCR/γ < T/4, the equation T/(T − τ)PCR = γτ has two roots, τl and τu, that divide the learning-time regions used in the optimization.Where J = γτ, the objective is maximized over the corresponding interval; between the roots, the maximum occurs at the balancing value described in the passage.Source: 2) Average transmit power constraint · p. 9
  • Because g3(τ) decreases over [0, T), its maximum on the stated interval occurs at τl.The paper's summarized solution for the average-power case is illustrated in Fig. 6.Source: 2) Average transmit power constraint · pp. 9–10, Figure 6 · p. 10

VI. NUMERICAL RESULTS

The numerical study examines CR throughput and optimal learning time under a peak transmit-power constraint. It reports close agreement between theoretical and estimator-based throughput curves, throughput saturation at high CR SNR, and learning-time trends with SNR and leakage tolerance.

VI. NUMERICAL RESULTS

  • The simulation uses fixed Rayleigh-fading channel realizations, T = 1,000 normalized symbol periods, and a peak CR transmit-power constraint.The system parameters are Mt = 6, Mr = 3, M1 = 4, M2 = 2, with d1 = d2 = 2 and PR SNR set to 20 dB.Source: §VI · p. 10
  • Theoretical and estimator-based CR-throughput results almost merge; maximum throughput increases with γ as greater leakage tolerance reduces optimal learning time and relaxes the CR power restriction.Fig. 7 plots CR throughput versus learning time for γ = 0.2 and 0.6, with PCR fixed at 100.Source: §VI · p. 10, Figure 7 · p. 10
  • Beyond a CR-SNR threshold for each γ, maximum CR throughput cannot improve because the interference-power constraint becomes dominant; higher γ increases maximum throughput.Fig. 8 presents theoretical maximum-throughput curves versus CR SNR.Source: Figure 8 · p. 10, §VI · p. 11
  • At low CR SNR, optimal τ reaches the assumed minimum of 10; at high SNR it stops increasing, and lower γ yields larger optimal τ.Fig. 9 shows the theoretical optimal learning time versus CR SNR.Source: §VI · p. 11, Figure 9 · p. 11

VII. CONCLUDING REMARKS

The paper proposes a practical cognitive-beamforming scheme based on an effective interference channel learned from received PR signals. With finite learning samples, it finds an optimal learning time for maximizing CR throughput and notes that the approach extends to multiple PR links.

VII. CONCLUDING REMARKS

  • The effective interference channel can be learned at the CR transmitter through blind or semiblind estimation of received PR signals and used to design practical cognitive beamforming.The scheme is intended to minimize the effect of CR-generated interference on PR transmissions.Source: §VII · p. 11
  • With finite channel-learning samples, an optimal learning time exists for maximizing CR link throughput.Source: §VII · p. 11
  • The developed results extend to multiple PR links because the EIC measures the space spanned by coexisting PR signals as a whole.The approach therefore works whether those signals come from one PR link or multiple PR links.Source: §VII · p. 11

APPENDIX I PROOF OF LEMMA 5.1

The appendices bound average interference power under finite-sample and noise effects, and establish that f(z) is increasing, continuous, differentiable, and concave. These properties support the later analysis of the learning-time objective.

APPENDIX I PROOF OF LEMMA 5.1

s )†ZHUCCRU HZY †

APPENDIX II PROOF OF LEMMA 5.2

A. Continuity

  • f(z) is continuous at all points, including the boundaries between its intervals.Source: A. Continuity · p. 12

B. Differentiability

C. Concavity

  • f(z) is concave, as shown using an equivalent min-max optimization with zero duality gap.For total power z, 1/μ(z) is the corresponding water level.Source: C. Concavity · p. 12
Loading 0809.2148v3…