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From Spectrum Pooling to Space Pooling: Opportunistic Interference Alignment in MIMO Cognitive Networks

S. M. Perlaza, N. Fawaz, S. Lasaulce, M. Debbah

arXiv:0907.1255v2cs.IT

TL;DR

The paper addresses how a secondary MIMO link can coexist with a primary link when unused spectrum is unavailable. It aligns secondary interference with spatial dimensions unused by the primary, develops processing and power-allocation schemes, and shows that the secondary rate can be of the same order as the primary rate under appropriate SNR and antenna conditions.

  • Problem

    The paper asks whether a secondary MIMO link can transmit without reducing a primary link’s capacity when spectrum white-spaces are absent.

  • Method

    The paper aligns secondary interference with primary-link spatial dimensions left unused by water-filling and optimizes secondary processing and power allocation.

  • Results

    The secondary link can achieve transmission rates of the same order as the primary link, depending on the systems’ SNRs and antenna numbers.

  • Takeaways & Limitations

    Unused spatial directions can be analytically characterized and reused by a secondary link while avoiding interference with the primary link’s reserved dimensions.

Abstract

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We describe a non-cooperative interference alignment (IA) technique which allows an opportunistic multiple input multiple output (MIMO) link (secondary) to harmlessly coexist with another MIMO link (primary) in the same frequency band. Assuming perfect channel knowledge at the primary receiver and transmitter, capacity is achieved by transmiting along the spatial directions (SD) associated with the singular values of its channel matrix using a water-filling power allocation (PA) scheme. Often, power limitations lead the primary transmitter to leave some of its SD unused. Here, it is shown that the opportunistic link can transmit its own data if it is possible to align the interference produced on the primary link with such unused SDs. We provide both a processing scheme to perform IA and a PA scheme which maximizes the transmission rate of the opportunistic link. The asymptotes of the achievable transmission rates of the opportunistic link are obtained in the regime of large numbers of antennas. Using this result, it is shown that depending on the signal-to-noise ratio and the number of transmit and receive antennas of the primary and opportunistic links, both systems can achieve transmission rates of the same order.

I. INTRODUCTION

The paper studies opportunistic MIMO coexistence without cooperation, using interference alignment to let a secondary link exploit spatial dimensions left unused by a primary link. It formulates the MIMO interference-channel setting and motivates transmit opportunities created by the primary link’s capacity-achieving transmission.

  • Motivation: Cognitive radio traditionally relies on spectrum white-spaces, but dense networks may offer few or short-lived opportunities for secondary transmission.Interference alignment provides an alternative when unused spectrum is unavailable.
  • System model: The system model contains two non-cooperating MIMO links with private messages operating simultaneously in the same frequency band.Transmitters send only to their respective receivers, and the secondary link must preserve the primary link’s single-user capacity.
  • Proposed approach: The proposed opportunistic interference-alignment scheme processes the secondary signal so its interference occupies unused primary dimensions rather than reserved ones.The paper analyzes feasibility, signal processing, and achievable opportunistic transmission rates.
  • Primary transmission: The primary link uses channel singular directions and water-filling power allocation to achieve capacity, potentially leaving some transmit dimensions unused.These dimensions arise from the equivalent parallel-channel representation of the primary MIMO link.

B. Transmit Opportunities

The primary link leaves spatial dimensions unused, creating transmit opportunities that the secondary link can exploit through interference alignment without reducing the primary link's single-user rate.

  • B. Transmit Opportunities: The primary equivalent channel separates used dimensions from unused receive dimensions, called secondary transmit opportunities.The unused dimensions contain no primary signal and can be exploited by the secondary link.
  • B. Transmit Opportunities: The number of opportunities S is determined by the channel eigenvalues, water level, and number m1 of primary reserved dimensions.The number of opportunities remains constant over the channel coherence time under the stated assumption.
  • C. Pre-processing Matrix: The IA condition ensures the primary link retains its equivalent single-user transmission rate, while V2 and P2 are subsequently optimized for the secondary rate.The preprocessing matrix satisfies IA independently of the secondary power-allocation matrix.
  • C. Pre-processing Matrix: The secondary transmitter chooses V2 so its interference lies in the null space of the primary cross-interference channel restricted to used dimensions.This aligns secondary interference with the primary receiver dimensions left unused by the primary link.
  • C. Pre-processing Matrix: OIA provides at least as many usable secondary transmit dimensions as zero-forcing beamforming and therefore outperforms it in this sense.The comparison follows from Ker(˜H) being contained in Ker(˜H1), yielding L2,BF ≤ L2.

D. Post-processing Matrix

After interference alignment protects the primary link, the secondary receiver processes colored interference-plus-noise to preserve and evaluate its achievable mutual information.

  • D. Post-processing Matrix: Once V2 performs IA, the secondary receiver still experiences co-channel interference from the primary transmitter.IA removes harmful interference at the primary link but does not eliminate interference at the secondary receiver.
  • D. Post-processing Matrix: The combined primary interference and receiver noise are modeled as colored Gaussian noise with covariance matrix Q2.This covariance enters the secondary mutual-information expression.
  • D. Post-processing Matrix: A whitening filter D2 = Q2^-1/2 achieves equality by removing the effect of the secondary receiver's colored noise covariance.The resulting post-processing preserves the mutual information between the transmitted symbols and received signal.

E. Power Allocation Matrix Optimization

The secondary transmitter optimizes its input covariance after preprocessing and post-processing are fixed, with uniform allocation serving as a high-SNR reference and water-filling giving the optimal allocation.

  • E. Power Allocation Matrix Optimization: The optimization selects P2 to maximize the opportunistic link's achievable transmission rate after V2 and D2 are fixed.The problem is formulated in terms of the secondary input covariance matrix.
  • E. Power Allocation Matrix Optimization: Uniform power allocation spreads total transmit power across the identified transmit opportunities and approaches optimal allocation at high SNR.It can also reduce information requirements and transmitter complexity.
  • E. Power Allocation Matrix Optimization: The optimized covariance becomes diagonal in the singular-vector coordinates of the effective secondary channel.Choosing a diagonal transformed covariance maximizes the relevant transmission-rate expression.
  • E. Power Allocation Matrix Optimization: Water-filling determines the transformed powers, with its water level selected to saturate the secondary transmit-power constraint.The resulting diagonal powers define the optimal power-allocation matrix.

IV. ASYMPTOTIC PERFORMANCE OF THE SECONDARY LINK

The asymptotic analysis characterizes the secondary link's transmit opportunities and usable dimensions when antenna numbers grow proportionally, using limiting eigenvalue distributions and water-filling behavior.

  • IV. ASYMPTOTIC PERFORMANCE OF THE SECONDARY LINK: The regime of large numbers of antennas studies secondary performance while antenna dimensions grow according to fixed positive ratios αij.This regime supports deterministic asymptotic characterizations.
  • IV. ASYMPTOTIC PERFORMANCE OF THE SECONDARY LINK: The analysis focuses on S, the number of transmit opportunities, and L2, the number of secondary transmit dimensions available without harming the primary link.L2 also equals the number of independent symbols the secondary system can transmit simultaneously.
  • IV. ASYMPTOTIC PERFORMANCE OF THE SECONDARY LINK: The asymptotic primary water level is uniquely determined by maximum power and receiver noise power rather than a specific channel realization.This yields a deterministic basis for computing asymptotic primary occupancy.
  • IV. ASYMPTOTIC PERFORMANCE OF THE SECONDARY LINK: In the large-antenna regime, empirical eigenvalue distributions converge almost surely to deterministic limiting laws, including the Marčenko-Pastur law.These limits provide the spectral quantities used to derive asymptotic water levels and dimensions.
  • IV. ASYMPTOTIC PERFORMANCE OF THE SECONDARY LINK: Higher primary-link SNR lowers the number of transmit opportunities, but practical SNR values still permit a non-zero number of opportunities.The asymptotic expressions determine both the opportunity count and the secondary usable-dimension count.

B. Asymptotic Transmission Rate of the Opportunistic Link

In the large-antenna regime, the opportunistic transmission rate per antenna converges almost surely to an expression characterized by limiting eigenvalue distributions and Stieltjes transforms. The result applies under compact-support assumptions and is exact for power allocations independent of the noise variance, while optimal allocation has an additional low- and medium-SNR dependence.

  • Asymptotic rate: The opportunistic rate per antenna converges almost surely in the large-antenna regime under limiting-eigenvalue-distribution assumptions.The limiting distributions of P1 and V2P2V2^H are assumed to have compact support.
  • Asymptotic rate: The asymptotic rate is obtained through Stieltjes transforms satisfying fixed-point equations with a unique solution on the non-positive real axis.The transforms correspond to the limiting eigenvalue distributions of the relevant matrices.
  • Power allocation: The asymptotic result holds for any power allocation matrix independent of the noise variance.Uniform power allocation satisfies this condition, as does optimal allocation in the high-SNR regime.
  • Power allocation: At low and medium SNRs, optimal power allocation generally depends on the noise variance because it is obtained by water-filling.The resulting technical problem is left as an extension of the work.

A. The Number S of Transmit Opportunities

The number of transmit opportunities depends on the primary link’s SNR and antenna ratio, and these opportunities determine when opportunistic transmission is feasible. OIA can retain more usable transmit dimensions than ZFBF and can achieve comparable or superior rates depending on antenna configuration and SNR.

  • The number of transmit opportunities: At practical primary-link SNRs of 10–20 dB, the number of transmit opportunities is non-zero.The number of opportunities is non-increasing with primary-link SNR but increases with the antenna ratio α11 = M1/N1.
  • The number of transmit opportunities: When N1 > M1, opportunistic communication remains feasible independently of the primary-link SNR; when α11 ≤ 1, feasibility depends on SNR.The theoretical result also matches finite-antenna simulations for N1 = 10.
  • Comparison between OIA and ZFBF: ZFBF may be infeasible when M2 ≤ N1 and H12 is full column rank, whereas OIA can still transmit with a non-null preprocessing matrix.OIA remains possible in the identified cases involving the rank of the reduced channel matrix.
  • Comparison between OIA and ZFBF: OIA has at least as many usable transmit dimensions as ZFBF, with the difference arising from additional transmit opportunities.The comparison is especially relevant when primary and secondary receivers are close, where ZFBF experiences power reduction.
  • Comparison between OIA and ZFBF: For Nt > Nr, OIA outperforms ZFBF at small antenna counts, while their high-SNR performance becomes nearly identical.At high SNR, the number of transmit opportunities approaches Nt − Nr, matching the number of directions ZFBF can avoid.
  • Opportunistic transmission rate: For Nr = Nt, optimal power allocation matters most with more antennas and low primary-link SNR, while opportunistic rate decreases as primary-link SNR increases.When Nr < Nt, opportunistic transmission requires additional rank conditions and reaches zero at a lower primary-link SNR.
  • Asymptotic transmission rate: The secondary link can achieve transmission rates of the same order as the primary link depending on both links’ SNRs.The asymptotic secondary rates converge rapidly even with a small number of antennas.

APPENDIX A PROOF OF LEMMA 1

The proof establishes the IA condition by constraining the secondary preprocessing matrix to the null space of the reduced primary-channel matrix. This construction confines secondary interference to the primary link’s unused spatial directions.

  • Proof of Lemma 1: A secondary preprocessing matrix satisfying ˜H1V2 = 0_(N1−S)×L2 satisfies the interference-alignment condition.The constraint forces the relevant interference components to vanish.
  • Proof of Lemma 1: The primary channel is decomposed using a sorted singular value decomposition into unitary matrices and a diagonal singular-value matrix.The decomposition separates used and unused spatial directions for the proof.
  • Proof of Lemma 1: The interference-plus-noise covariance is partitioned into block matrices, allowing the IA condition to be enforced through selected submatrix constraints.The proof sets R1 = 0 and R2 = 0 while leaving R3 unrestricted.

APPENDIX B DEFINITIONS

The appendix introduces empirical and limiting eigenvalue distributions together with associated transforms used in the large-system analysis. These definitions support the asymptotic characterization of the opportunistic transmission rate.

  • Definitions: The empirical eigenvalue distribution describes the eigenvalues of a finite square random matrix, while its limiting distribution applies as matrix size grows.The associated probability density functions are defined for the empirical and limiting distributions.
  • Definitions: The Stieltjes transform is introduced as a transform associated with an eigenvalue distribution.The appendix also relates the Υ-transform to the Stieltjes transform.

APPENDIX C PROOF OF PROPOSITION 5

The appendix derives the asymptotic transmission rate of the opportunistic link through successive transforms of random matrix eigenvalue distributions, then integrates the resulting rate derivative.

  • The proof proceeds in four steps: express the rate derivative through Stieltjes transforms, obtain G_M1(z), obtain G_M(z), and integrate the derivative.These steps organize the derivation of the asymptotic opportunistic rate.
  • M and M1 are Hermitian Gramian matrices whose empirical eigenvalue distributions converge almost surely to compactly supported asymptotic distributions.Their eigenvalue decompositions provide the matrices used in the transform-based derivation.
  • The transform of M1 is obtained using a random-matrix theorem, yielding a fixed-point equation with a unique solution for z ∈ R−.The construction uses the asymptotic distribution of the primary covariance-related matrix and the Gaussian interference matrix.
  • The transform of M is derived from S- and Υ-transforms of its component matrices and likewise yields a fixed-point equation with a unique solution.The derivation relates G_M(z) to the transform of V_2P_2V_2^H and the distributions of the constituent matrices.
  • The asymptotic opportunistic rate is obtained by integrating the final expression for its rate derivative.When the relevant asymptotic signal-to-noise parameter tends to infinity, reliable communication is impossible and the asymptotic rate is zero.
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