PaperlayerView exact revision on arXiv

Source-linked AI summary

Ergodic Capacity Analysis of Amplify-and-Forward MIMO Dual-Hop Systems

Shi Jin, Matthew R. McKay, Caijun Zhong, Kai-Kit Wong

arXiv:0811.4565v1cs.IT

TL;DR

The paper fills the gap in analytical ergodic-capacity results for AF MIMO dual-hop systems with arbitrary finite antenna and relay configurations. It uses finite-dimensional random matrix theory to derive exact and closed-form capacity characterizations, including high-SNR expressions and tight bounds. The results clarify how antenna counts and relay gain affect capacity and connect several asymptotic regimes to conventional MIMO channels.

  • Problem

    Prior AF MIMO dual-hop capacity results were derived mainly with asymptotic methods, leaving a lack of analytical results for arbitrary finite antenna and relaying configurations.

  • Method

    Finite-dimensional random matrix theory is used to derive an exact unordered eigenvalue distribution and random determinant properties for the equivalent AF MIMO dual-hop channel.

  • Results

    The paper derives an exact ergodic-capacity expression, simplified high-SNR formulas, and simplified upper and lower bounds shown to be tight for all SNRs.

  • Takeaways & Limitations

    The analytical results characterize how antenna configuration and relay gain influence capacity and reveal relationships between dual-hop AF relay channels and conventional point-to-point MIMO channels.

  • Takeaways & Limitations

    The analysis assumes channel state information is available at the destination but not at the relay or source terminals.

Abstract

from arXiv · show

This paper presents an analytical characterization of the ergodic capacity of amplify-and-forward (AF) MIMO dual-hop relay channels, assuming that the channel state information is available at the destination terminal only. In contrast to prior results, our expressions apply for arbitrary numbers of antennas and arbitrary relay configurations. We derive an expression for the exact ergodic capacity, simplified closed-form expressions for the high SNR regime, and tight closed-form upper and lower bounds. These results are made possible to employing recent tools from finite-dimensional random matrix theory to derive new closed-form expressions for various statistical properties of the equivalent AF MIMO dual-hop relay channel, such as the distribution of an unordered eigenvalue and certain random determinant properties. Based on the analytical capacity expressions, we investigate the impact of the system and channel characteristics, such as the antenna configuration and the relay power gain. We also demonstrate a number of interesting relationships between the dual-hop AF MIMO relay channel and conventional point-to-point MIMO channels in various asymptotic regimes.

I. INTRODUCTION

The paper addresses the lack of exact finite-dimensional ergodic-capacity results for AF MIMO dual-hop systems by developing analytical expressions valid across antenna and relay configurations. It uses finite-dimensional random matrix theory to obtain exact, high-SNR, and bounded capacity characterizations and examines antenna and relay-gain effects.

  • Research gap: Prior AF MIMO dual-hop capacity results mainly relied on asymptotic methods in which system dimensions grow to infinity.Existing analyses considered large relay arrays, fixed antenna ratios, or other asymptotic regimes.
  • Contributions: The paper derives exact analytical ergodic-capacity results applicable to any finite numbers of MIMO antennas and arbitrary relay configurations.This contrasts with prior results that did not cover arbitrary finite antenna and relaying configurations.
  • Random-matrix analysis: Finite-dimensional random matrix theory provides an exact closed-form unordered eigenvalue distribution for the equivalent cascaded AF MIMO relay channel.The expression uses standard functions and avoids the fixed-point computation required by an earlier asymptotic approximation.
  • Random-matrix analysis: The analysis also derives random determinant properties, including expected characteristic polynomials and expected log-determinants of relevant random matrices.These properties support subsequent capacity expressions for the equivalent relay channel.
  • Capacity characterization: The resulting formulas include simplified high-SNR capacity expressions and tight closed-form upper and lower bounds.The bounds are expressed using standard functions that are easy to compute and are reported as tight for all SNRs.
  • System insights: When source, destination, or relay antennas, or relay gain, becomes large, the dual-hop capacity admits an interpretation through conventional single-hop single-user MIMO capacities.The analysis also shows that AF MIMO dual-hop capacity is upper bounded by the capacity of a SISO AWGN channel.
  • System insights: At high SNR, multiplexing gain equals the minimum source, destination, and relay antenna count, while power offset depends on all three.Increasing destination antennas can significantly improve power offset when the initial destination count is small, with diminishing relative gains as that count increases.

III. NEW RANDOM MATRIX THEORY RESULTS

The paper develops exact, unified random-matrix results for unordered eigenvalue distributions and random determinant properties in AF MIMO dual-hop analysis. These closed-form results apply across arbitrary finite antenna dimensions and support subsequent capacity derivations.

  • Unordered eigenvalue distribution: A new exact closed-form unordered eigenvalue distribution is derived for the relevant finite-dimensional random matrix.The expression applies to arbitrary finite system dimensions and uses standard functions that can be efficiently evaluated.
  • Unordered eigenvalue distribution: Lemma 1 provides a simpler, computationally efficient unified expression for arbitrary n_s and q.It replaces separate prior expressions for n_s ≤ q and n_s > q.
  • Unordered eigenvalue distribution: Theorem 1 supplies the unconditional marginal p.d.f. of an unordered eigenvalue using cofactors of a matrix whose entries involve standard special functions.The formula includes the modified Bessel function of the second kind and the confluent hypergeometric function.
  • Asymptotic relationships: The exact unordered eigenvalue distribution converges to the Rayleigh-product channel distribution as the relay parameter a grows without bound.The limiting relationship is established using the high-gain form of the distribution.
  • Numerical validation: The exact distribution agrees with Monte Carlo simulations for the (2, 3, 4) system configuration using 100,000 channel realizations.The comparison is presented for the unordered eigenvalue p.d.f.
  • Numerical validation: For small systems, the prior asymptotic eigenvalue p.d.f. has noticeable inaccuracies, whereas exact and asymptotic distributions converge as antenna numbers grow.The inaccuracies are reported for (2, 1, 2) and (4, 2, 4), while convergence is visible for (16, 8, 16).
  • Random determinant properties: New closed-form random determinant properties yield a unified expected determinant and expected log-determinant treatment across the relevant antenna cases.These results are subsequently used to derive tight ergodic-capacity bounds.

IV. ERGODIC CAPACITY ANALYSIS

This section presents analytical expressions for the ergodic capacity of AF MIMO dual-hop systems.

  • The paper presents new analytical expressions for the ergodic capacity of AF MIMO dual-hop systems.

A. Exact Expression for Ergodic Capacity

The exact capacity analysis is validated against simulation and connected to conventional single-hop MIMO capacity in several antenna and relay-gain limits. These relationships show how dual-hop capacity behaves under large-array and high-relay-gain regimes.

  • Exact capacity expression: The exact analytical capacity expressions agree exactly with Monte Carlo simulations for two antenna and relay configurations.The comparison uses the exact capacity derived from the analytical expressions.
  • Asymptotic relationships: As the number of relay antennas grows large, AF MIMO dual-hop capacity becomes one half of a single-hop MIMO capacity with n_s transmit antennas, n_d receive antennas, and average SNR ρ.
  • Asymptotic relationships: As the number of source antennas grows large, AF MIMO dual-hop capacity becomes one half of a single-hop MIMO capacity with n_r transmit antennas, n_d receive antennas, and average SNR ρ.
  • Asymptotic relationships: As the number of destination antennas grows large, AF MIMO dual-hop capacity becomes one half of a single-hop MIMO capacity with n_s transmit antennas, n_r receive antennas, and average SNR ρ.
  • Asymptotic relationships: As the relay power gain α grows large, capacity remains bounded and becomes one half of single-hop MIMO capacity with n_s transmit antennas, q = min(n_r, n_d) receive antennas, and average SNR ρ.
  • Asymptotic relationships: Closed-form expressions for the limiting cases can be obtained from known single-hop MIMO capacity results.

B. High SNR Capacity Analysis

The high-SNR analysis derives exact slope and power-offset characterizations for arbitrary antenna configurations, then examines how relay gain and antenna additions affect capacity.

  • B. High SNR Capacity Analysis: The high-SNR analysis considers regimes where source and relay powers grow proportionately, including α →∞, ρ →∞ with α/ρ = β.The resulting capacity expression is simplified in this joint high-SNR regime.
  • B. High SNR Capacity Analysis: The high-SNR slope depends only on min(ns, nr, nd), whereas the power offset depends on all three antenna counts.Theorem 4 gives exact characterizations of both parameters for arbitrary source, relay, and destination antenna numbers.
  • B. High SNR Capacity Analysis: The high-SNR analytical approximations converge to their exact capacity curves at quite moderate SNR levels, such as below 20 dB.The paper compares these approximations across antenna configurations and uses α/ρ = 2 in the reported figure context.
  • B. High SNR Capacity Analysis: Increasing β leaves the high-SNR slope unchanged, decreases the power offset, and increases high-SNR ergodic capacity.This follows from the monotonic behavior of the function gl(x) used in the power-offset expression.
  • B. High SNR Capacity Analysis: For ns = nr = 1, adding destination antennas improves ergodic capacity without changing the high-SNR slope, but produces diminishing returns.The power-offset shift approaches 0 dB as the number of destination antennas becomes large.

2) Large Source Power, Fixed Relay Power:

With fixed relay power and increasing source power, the ergodic capacity remains bounded because the relay-destination link limits the system; closed-form upper bounds describe this behavior.

  • 2) Large Source Power, Fixed Relay Power:: With fixed relay gain α and large source power ρ, the ergodic capacity remains bounded as a function of α.This identifies a distinct asymptotic regime from the joint source-and-relay high-SNR analysis.
  • 2) Large Source Power, Fixed Relay Power:: The bounded capacity is attributed to the weakest link in the relay network, here the relay-destination link.The source-power increase therefore does not remove the relay-side bottleneck in this regime.
  • A. Upper Bound: The upper bound is very tight across all SNRs in the configurations examined, and it coincides with exact capacity near ρ ≈ 5 dB.The comparison uses α = 2ρ for the upper-bound evaluation.
  • A. Upper Bound: When there is one relay antenna, large destination-antenna count or relay gain yields an AWGN SISO capacity upper bound.The corresponding lower-bound result uses an AWGN SISO channel with scaled average SNR.

B. Lower Bound

The paper derives a closed-form lower bound for AF MIMO dual-hop ergodic capacity and shows that it remains tight across SNR and relay-gain ranges studied.

  • B. Lower Bound: A new closed-form lower bound is derived for the ergodic capacity of AF MIMO dual-hop systems.Theorem 6 states the lower-bound result, with simplified high-SNR forms obtained subsequently.
  • B. Lower Bound: The lower bound is tight across the entire SNR range examined and coincides with exact capacity around ρ ≈ 15 dB.The comparison covers different system configurations.
  • B. Lower Bound: For one relay antenna, both upper and lower bounds remain quite tight across the relay-gain range considered.Their high-relay-gain asymptotic approximations converge for moderate relay gains, approximately within α ≈ 20 dB.
  • B. Lower Bound: The high-SNR approximations for exact capacity and both bounds are very accurate even at moderate SNR levels.The comparison is shown for the antenna configuration (3, 4, 2).
  • VI. CONCLUSIONS: The paper concludes with exact capacity expressions, simplified high-SNR forms, and bounds validated against numerical simulations under destination-only CSI.The analytical results rely on random-matrix eigenvalue and determinant expressions for the equivalent relay channel.

APPENDIX I

Appendix I derives unordered-eigenvalue density expressions by treating separate antenna-dimension cases and applying joint-density transformations, cofactors, and determinant expansions.

  • APPENDIX I: The derivation treats separately the cases q < ns and q ≥ ns.This case split organizes the subsequent eigenvalue-density calculations.
  • APPENDIX I: Cofactor expansions, term-by-term integration, Laplace expansion, and a final substitution produce the desired closed-form results.The derivation also uses the relation β = λ/(1 + aλ) to complete a density transformation.
  • APPENDIX I: For q ≥ ns, the joint density of unordered eigenvalues is used to obtain the density of a single unordered eigenvalue.The single-eigenvalue density follows by integrating the joint density over the remaining eigenvalues.
  • APPENDIX I: The appendix derives the joint density of transformed diagonal-matrix variables using a vector transformation and its Jacobian.The transformation maps α_i variables to β_i variables before yielding the joint density of L.

C. Proof of Theorem 1

The proof derives unconditional unordered-eigenvalue distributions from conditional densities and joint eigenvalue distributions, treating q < ns and q ≥ ns separately. Determinant expansions and closed-form integral evaluations complete the result.

  • The proof first re-expresses the conditional unordered eigenvalue p.d.f. and then removes conditioning to obtain the unconditional p.d.f.
  • The integral transformation t = x/(1 − ax) enables evaluation through standard integral identities.
  • The resulting integrals involve the confluent hypergeometric function of the second kind U(·, ·, ·).
  • Applying determinant expansions, closed-form integration results, and factor extraction yields the desired expressions.
  • The proof handles q < ns and q ≥ ns separately using ordered joint eigenvalue densities over the corresponding ordered integration regions.

E. Proof of Lemma 4

The proof of Lemma 4 evaluates log-determinant-related quantities by conditioning on L, using eigenvalue densities and moment-generating functions, then simplifying determinant sums.

  • The proof treats q < ns and q ≥ ns separately when evaluating the relevant log-determinant quantity.
  • The moment-generating function of ln det is used to obtain the required expectations and matrix entries.
  • Multilinearity of determinants and permutation expansions reduce the determinant sums to the final result.
  • For q ≥ ns, applying the joint eigenvalue density and a cited lemma yields a closed-form evaluation.
  • For q < ns, the calculation uses the joint eigenvalue density and integrates over the ordered eigenvalue region.

F. Proof of Theorem 3

The proof removes conditioning on L to derive capacity expressions and then obtains high-SNR behavior by separating q < ns from q ≥ ns. It reports the high-SNR slope and power offset through theorem-based substitutions.

  • The proof removes conditioning on L using Lemma 2 and evaluates the resulting integrals, including a separate treatment when q = s.
  • As nr → ∞, the capacity expression is simplified using the Law of Large Numbers and the identity associated with the channel model.
  • The large-system derivation also uses an equivalent capacity representation and applies the Law of Large Numbers before further substitutions.
  • The high-SNR analysis separates q < ns and q ≥ ns and derives the corresponding slope and power-offset expressions.
  • 2 bit/s/Hz (3dB) is given as the high-SNR slope in the reported case.

D. Proof of Corollary 7

The proof establishes asymptotic results by applying confluent-hypergeometric-function properties and lower-bound expressions across antenna and SNR limits. Separate cases are considered for q < ns and q ≥ ns.

  • The proof uses properties and asymptotic expansions of the confluent hypergeometric function U to obtain the stated limiting results.
  • Both relevant expressions converge to the same limit as nd → ∞, and taking α → ∞ yields the subsequent result.
  • The q < ns and q ≥ ns cases are handled separately when deriving the lower bound on ergodic capacity.
  • For ns → ∞, the digamma function is approximated before substitution into the lower-bound expression.
  • For nd → ∞, special-function identities and asymptotic expansions produce the corresponding limiting expression.
  • For α → ∞, recurrence relations and series expansions for the exponential integral and digamma function simplify the lower-bound expression.
Loading 0811.4565v1…