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MIMO Networks: the Effects of Interference

Marco Chiani, Moe Z. Win, Hyundong Shin

arXiv:0802.0738v3cs.IT

TL;DR

Capacity loss from interference in MIMO networks has received limited analytical study. This paper develops a unified framework for correlated Rayleigh-fading MIMO systems with multiple co-channel interferers, deriving eigenvalue distributions and ergodic mutual information formulas.

  • Problem

    MIMO co-channel interference requires capacity characterization in a general setting, beyond the few existing simulation- or approximation-based studies.

  • Method

    The paper generalizes determinant representations for hypergeometric functions with matrix arguments and derives unified eigenvalue distributions for Wishart matrices and Gaussian quadratic forms.

  • Results

    The resulting analysis characterizes ergodic mutual information for MIMO Rayleigh-fading channels with arbitrary-multiplicity correlation matrices and multiple MIMO interferers.

  • Takeaways & Limitations

    The framework supports analysis of MIMO systems with transmit or receive correlation and multiple interferers.

Abstract

from arXiv · show

Multiple-input/multiple-output (MIMO) systems promise enormous capacity increase and are being considered as one of the key technologies for future wireless networks. However, the decrease in capacity due to the presence of interferers in MIMO networks is not well understood. In this paper, we develop an analytical framework to characterize the capacity of MIMO communication systems in the presence of multiple MIMO co-channel interferers and noise. We consider the situation in which transmitters have no information about the channel and all links undergo Rayleigh fading. We first generalize the known determinant representation of hypergeometric functions with matrix arguments to the case when the argument matrices have eigenvalues of arbitrary multiplicity. This enables the derivation of the distribution of the eigenvalues of Gaussian quadratic forms and Wishart matrices with arbitrary correlation, with application to both single user and multiuser MIMO systems. In particular, we derive the ergodic mutual information for MIMO systems in the presence of multiple MIMO interferers. Our analysis is valid for any number of interferers, each with arbitrary number of antennas having possibly unequal power levels. This framework, therefore, accommodates the study of distributed MIMO systems and accounts for different positions of the MIMO interferers.

I. INTRODUCTION

The paper develops an analytical framework for ergodic MIMO capacity with multiple cochannel MIMO interferers and AWGN under no transmitter CSI, perfect receiver CSI, and frequency-flat Rayleigh fading. It generalizes matrix-argument methods to derive eigenvalue distributions and capacity expressions covering arbitrary correlations, antenna counts, interferer numbers, and unequal powers.

  • MIMO antennas provide high spectral efficiency and link reliability in fading environments.
  • The framework assumes transmitters have no CSI, the receiver has perfect CSI, and all links undergo frequency-flat Rayleigh fading.The analysis considers rich scattering environments and AWGN alongside multiple MIMO cochannel interferers.
  • The paper generalizes determinant representations of hypergeometric functions with matrix arguments to matrices whose eigenvalues have arbitrary multiplicities.This generalization supports subsequent eigenvalue-distribution derivations for complex Gaussian quadratic forms and Wishart matrices with arbitrarily repeated covariance eigenvalues.
  • The resulting capacity expressions cover single-user MIMO correlations and multiple MIMO interferers with arbitrary antenna counts, any interferer number, and possibly unequal power levels.Single-user results allow arbitrary power levels and correlation across transmitting elements or at the receiver; multi-interferer results also accommodate distributed MIMO positions.

II. SYSTEM MODELS

The system model describes a desired MIMO link affected by multiple MIMO co-channel interferers and Gaussian noise under Rayleigh fading. Because interference produces covariance matrices with eigenvalues of arbitrary multiplicity, the analysis develops corresponding distributional tools for mutual-information characterization.

  • Network and channel model: The model comprises a MIMO-(N_T0, N_R) desired link, N_I MIMO co-channel interferers with arbitrary antenna counts, and additive noise.The channel matrices H_k have size N_R×N_Tk, with H_0 representing the desired link.
  • Network and channel model: All links use uncorrelated MIMO Rayleigh fading, with channel entries modeled as i.i.d. zero-mean, unit-variance circularly symmetric complex Gaussian variables.The received power per antenna, P_k, generally differs across users because of transmit power, path-loss, and shadowing.
  • CSI and signaling assumptions: The receiver has perfect CSI, transmitters have no CSI, and each user therefore transmits zero-mean circularly symmetric Gaussian vectors with i.i.d. elements.The transmit power per antenna is P_k/N_Tk, and unequal antenna power levels can be represented by decomposing a transmitter into virtual sub-transmitters.
  • Analytical challenge and approach: Interference generally yields non-identity covariance matrices whose eigenvalues can have multiplicities greater than one, requiring CSU(n_T, n_R, Φ) in a general setting.The next sections derive expressions for matrix-argument hypergeometric functions with non-distinct eigenvalues and joint p.d.f.s for central Wishart matrices and Gaussian quadratic forms with arbitrary covariance.

III. HYPERGEOMETRIC FUNCTIONS WITH MATRIX ARGUMENTS HAVING ARBITRARY

This section extends determinant representations of hypergeometric functions of two matrix arguments beyond the distinct-eigenvalue case. The extension uses continuous limits with derivative columns and scaling factors to handle repeated eigenvalues, including multiple coincident groups and zero eigenvalues.

  • Motivation: The existing determinant representation is simpler than the zonal-polynomial series but requires all eigenvalues of both argument matrices to be distinct.The series form is difficult to manage for further analysis, whereas the determinant form uses matrices whose entries are scalar-argument hypergeometric functions.
  • Generalization: The paper generalizes this representation to matrices whose eigenvalues are not necessarily distinct.This generalization is enabled by a lemma giving the continuous extension of determinant-like expressions at coincident arguments.
  • Continuous extension: For an eigenvalue of multiplicity L, the continuous extension removes zero denominator factors, replaces corresponding columns with successive derivatives, and divides by Γ(L)(L).The procedure applies when L coincident arguments occur and assumes the underlying functions have derivatives through order at least m − 1 near the arguments.
  • Generalized determinant form: Each repeated eigenvalue w of multiplicity L > 1 produces L derivative-based columns with the proper scaling factor Γ(L)(L), and the rule extends to multiple coincident groups.For exponential entries, the columns are generated from λ_i^(L−1)e^(λ_iw), …, e^(λ_iw).
  • Zero eigenvalues: The generalized formulas also cover repeated zero eigenvalues through corollaries derived from the repeated-eigenvalue representations.The zero-eigenvalue cases are obtained by setting the repeated argument to zero in the corresponding generalized formulas.

IV. GAUSSIAN QUADRATIC FORMS WITH COVARIANCE MATRIX HAVING EIGENVALUES OF ARBITRARY MULTIPLICITY We now derive the joint p.d.f. of the eigenvalues for Gaussian quadratic forms and central

This section presents Lemma 6, which gives the joint eigenvalue density for Gaussian quadratic forms with positive-definite covariance matrices whose eigenvalues may have arbitrary multiplicities. The result covers central Wishart and pseudo-Wishart cases with arbitrary one-sided correlation and supports marginal eigenvalue distributions.

  • Lemma 6: Lemma 6 gives the joint p.d.f. of the ordered non-zero eigenvalues of W = HΦH† for complex Gaussian H and positive-definite Φ.H has zero-mean, unit-variance i.i.d. entries.
  • Lemma 6: The density accommodates arbitrary eigenvalue multiplicities in the covariance structure through the distinct eigenvalues of Φ−1 and their corresponding multiplicities.The formulation uses the Vandermonde matrix and multiplicity-dependent terms.
  • Applicability: Lemma 6 provides a compact general joint distribution for central Wishart matrices when p ≥ n and central pseudo-Wishart or quadratic forms when n ≥ p.The same lemma applies in both dimensional regimes, with the stated simplifications for each case.
  • Applicability: The result applies to arbitrary one-sided correlation matrices with not-necessarily distinct eigenvalues.This extends the joint eigenvalue characterization beyond distinct-eigenvalue covariance cases.
  • Marginal distributions: Using Lemma 6 and prior results, the analysis also derives marginal distributions for individual eigenvalues or arbitrary subsets of eigenvalues.The marginalization extends the joint characterization to selected eigenvalue statistics.

V. ERGODIC MUTUAL INFORMATION OF A SINGLE-USER MIMO SYSTEM

This section develops a unified analysis of ergodic mutual information for single-user Rayleigh-fading MIMO systems with arbitrary transmit power/correlation and receive correlation. It expresses the analysis through eigenvalues of a Gaussian quadratic form and evaluates the resulting nested integral using earlier results.

  • System model: The framework covers single-user MIMO systems with arbitrary transmit power levels, transmit correlation, and receive correlation, including non-distinct correlation-matrix eigenvalues.The stated analysis admits arbitrary power levels/correlation among transmitting elements and arbitrary receiver correlation.
  • Capacity formulations: For a Rayleigh-fading MIMO-(nT, nR) channel, the function (28) expresses ergodic mutual information under either receiver-side or transmitter-side correlation.The two listed cases respectively use ΨR = I with transmit covariance Q, or ΨT = I with receiver correlation ΨR and equal power allocation.
  • Capacity formulations: With no receiver correlation, the mutual information is CSU(nT, nR, Φ) for Φ = (1/σ2)ΨTQ; with no transmit correlation and equal allocation, capacity is CSU(nR, nT, Φ).The second case uses Q = P/nTI and Φ = (P/nTσ2)ΨR.
  • Eigenvalue representation: The mutual-information calculation uses the ordered non-zero eigenvalues λ1 ≥ λ2 ≥ . . . ≥ λnmin > 0 of W = HΦH†, where nmin = min(n, p).The eigenvalue density is integrated over the ordered domain defined for these eigenvalues.
  • Eigenvalue representation: The nested integral in (29), containing log(1 + xi), can be evaluated using results from previous sections.The integration uses the joint p.d.f. of the ordered eigenvalues and the ordered integration domain.

Appendix II, leading to the following Theorem. · Appendix II, the multiple integral in (29) reduces to (30).

Theorem 1 gives an exact, unified expression for ergodic mutual information in correlated MIMO Rayleigh channels with receiver CSI, including arbitrary eigenvalue multiplicities. The appendix also shows that the resulting integrals admit standard numerical evaluation and further simplification.

  • Appendix II, leading to the following Theorem.: Theorem 1 characterizes ergodic mutual information for MIMO Rayleigh fading channels with CSI only at the receiver.
  • Appendix II, leading to the following Theorem.: The result allows one-sided correlation matrices whose eigenvalues have arbitrary multiplicities.
  • Appendix II, leading to the following Theorem.: The theorem’s matrix representation uses integrals involving powers, exponential factors, and log(1 + x), with parameters defined through the distinct eigenvalues of Φ−1 and their multiplicities.
  • Appendix II, leading to the following Theorem.: The derivation uses the joint probability density function of the ordered eigenvalues of W, whose matrix elements are real functions of the eigenvalues.
  • Appendix II, the multiple integral in (29) reduces to (30).: The integral in (31) can be evaluated with standard numerical techniques and further simplified using identities involving the incomplete gamma function.
  • Appendix II, the multiple integral in (29) reduces to (30).: Theorem 1 provides exact mutual information uniformly for nR ≥ nT and nT ≥ nR, with arbitrary transmitter or receiver correlation and without Monte Carlo evaluation.
  • Appendix II, the multiple integral in (29) reduces to (30).: Applying the results from Sections III–V enables unified analyses of MIMO systems with arbitrary covariance matrices, including multiuser and distributed MIMO systems.

VI. NUMERICAL RESULTS

The numerical results validate the framework across unequal-power MIMO, relay, and co-channel-interference settings. They show that interference changes the preferred antenna configuration, while receiver CSI and available spatial degrees of freedom determine achievable capacity.

  • Unequal-power single-user MIMO: Capacity decreases as transmit-power imbalance increases from Δ=0 to Δ=1, consistent with majorization-theory analysis.For the MIMO-(6, 3) Rayleigh channel, Δ=0 and Δ=1 correspond to equal power across 6 and 3 transmitting antennas, respectively.
  • MIMO relay networks: Theorem 1 yields the exact relay-network upper bound, whereas Jensen’s inequality can be overly optimistic.The example uses a source with 4 antennas, five 2-antenna relays, and relay received-power weights {1, 2, 5, 10, 20}.
  • Receiver degrees of freedom: For large interference power and NR > NT1, capacity approaches the single-user MIMO-(NT0, NR − NT1) floor; for NR ≤ NT1, capacity approaches zero at small SIR.The floor reflects using NT1 receiver degrees of freedom to null interference, while insufficient receiver degrees of freedom prevent mitigating all interferers and preserving NT0 useful streams.
  • Antenna configuration under interference: With one or two same-sized interferers, minimum transmit-antenna configurations are best at small SIR, whereas maximum configurations are best at large SIR.The Gaussian approximation incorrectly favors the largest antenna count for all conditions; among networks using MIMO-(n, n), larger n achieves higher mutual information for every SIR and SNR.

VII. CONCLUSION

The paper develops generalized matrix-function and eigenvalue-distribution results that enable unified MIMO analyses, including closed-form ergodic mutual information with multiple interferers.

  • The study addresses MIMO communication systems operating with multiple MIMO interferers and noise.
  • The authors generalize determinant representations for hypergeometric functions with matrix arguments to argument matrices with eigenvalues of arbitrary multiplicities.
  • They derive a unified joint p.d.f. for eigenvalues of central Wishart matrices and Gaussian quadratic forms, allowing arbitrary multiplicities among covariance-matrix eigenvalues.
  • These results enable analysis of MIMO scenarios with transmit or receive correlation matrices whose eigenvalues have arbitrary multiplicities.
  • The framework yields closed-form ergodic mutual information for MIMO systems in the presence of multiple MIMO interferers.

APPENDIX I PROOFS · A. Proof of Lemma 2

The proof establishes Lemma 2 by induction for K = 1 and coincident arguments, using Taylor expansion and determinant properties. It concludes that the resulting expression retains the form of (15), with extensions to different K and additional coincident-argument groups stated as straightforward.

  • A. Proof of Lemma 2: The proof first restricts without loss of generality to K = 1 and proceeds by induction.The base case is L = 1, where (15) coincides with (14).
  • A. Proof of Lemma 2: The induction step assumes (15) for L and shows it also holds for L + 1.This establishes the recursive structure needed for the proof.
  • A. Proof of Lemma 2: When w1 = w2 = · · · = wL, the product Q contains exactly L factors with value ǫ ≜ wL − wL+1.The proof rewrites wL+1 as wL − ǫ to analyze coincident arguments.
  • A. Proof of Lemma 2: Taylor expansion is applied to the resulting functions, retaining terms while denoting omitted contributions by O(ǫ).The determinant is linear in the entries of any one column when the other columns are fixed.
  • A. Proof of Lemma 2: Determinants associated with n = 0, . . . , L −1 vanish because they contain coincident columns.As ǫ → 0, only the term of grade L remains.
  • A. Proof of Lemma 2: After simplifying and cyclically reordering the first L + 1 columns, the matrix again has the form of (15).This completes the induction for w1 = · · · = wL.
  • A. Proof of Lemma 2: The proof states that extending the result to different K and more groups of coincident arguments is straightforward.This is the final generalization stated after the induction argument.

B. Proof of Lemma 5.

The proof of Lemma 5 uses derivatives of scalar-argument hypergeometric functions and applies the resulting shifted-parameter expression in Lemma 2 and equation (10).

  • The proof begins by expressing derivatives of the hypergeometric function with scalar arguments.
  • The derivative expression contains products of rising-factorial terms for the numerator parameters.
  • The denominator contains the corresponding products of rising-factorial terms for the lower parameters.
  • This derivative result is then used with Lemma 2 and equation (10).
  • The resulting hypergeometric function has all numerator and denominator parameters shifted by n.

C. Proof of Lemma 6 … AN IDENTITY ON MULTIPLE INTEGRALS INVOLVING DETERMINANTS

The appendix proves Lemma 6 for eigenvalue distributions of Gaussian quadratic forms, extending covariance results from distinct to arbitrary eigenvalue multiplicities. It also states an identity for ordered multiple integrals involving determinants.

  • C. Proof of Lemma 6: The proof addresses Gaussian quadratic forms and random matrices with covariance structures represented by Φ.
  • C. Proof of Lemma 6: Lemma 6 concerns the eigenvalue distribution of W = HΦH†, where H has uncorrelated Gaussian entries and Φ is positive definite.Φ represents the channel covariance matrix.
  • 1) Correlation on the shortest side - distinct eigenvalues:: For distinct covariance eigenvalues, prior results covered both n ≥ p and p ≥ n cases, assuming unit eigenvalue multiplicity.
  • 3) Generalization to covariance matrix with arbitrary eigenvalues:: Lemma 2 generalizes the formulas to arbitrary eigenvalue multiplicities by replacing rows of G(x, µ) with successive eigenvalue derivatives and normalization factors.Derivative orders are assigned according to each eigenvalue’s multiplicity and row position.
  • 1) Correlation on the shortest side - distinct eigenvalues:: The shortest-side derivation gives the joint p.d.f. of the ordered eigenvalues of W when Φ has distinct eigenvalues and n ≥ p.
  • 2) Correlation on the largest side - distinct eigenvalues:: For the largest-side case, the appendix derives the joint p.d.f. of W’s ordered eigenvalues using Corollary 1, rather than the previous literature’s approach.This case is stated for n ≥ p and distinct eigenvalues of Φ^-1.
  • 2) Correlation on the largest side - distinct eigenvalues:: The distinct-eigenvalue formulas are limited to covariance matrices whose eigenvalues all have multiplicity one.
  • AN IDENTITY ON MULTIPLE INTEGRALS INVOLVING DETERMINANTS: Theorem 2 states an identity for integrals of determinants involving an arbitrary p × p matrix and two arbitrary functions over the ordered domain Dord.The proof follows the same steps as [6, Theorem 3].
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