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Achievable Sum Rate of MIMO MMSE Recievers: A General Analytic Framework

Matthew R. McKay, Iain B. Collings, Antonia M. Tulino

arXiv:0903.0666v1cs.ITmath.IT

TL;DR

Closed-form achievable-rate analysis for practical finite-dimensional MIMO systems with linear MMSE receivers is limited. The paper relates MMSE sum rate to ergodic MIMO mutual information and applies that framework to Rayleigh and Rician channels. Its analysis finds rate losses from spatial correlation or line-of-sight at both high and low SNR, with larger losses for MMSE than optimal receivers.

  • Problem

    Closed-form achievable-rate results for MIMO MMSE receivers are scarce, while some existing asymptotic or channel-restricted analyses may not cover practical antenna dimensions and fading models.

    Source: §I · p. 2

  • Method

    The framework relates MIMO MMSE achievable sum rate to ergodic mutual information, then derives expressions for uncorrelated and spatially correlated Rayleigh and uncorrelated Rician channels.

    Source: §I · pp. 3–4

  • Results

    At both high and low SNR, spatial correlation or line-of-sight reduces achievable sum rate for MMSE and optimal receivers, with a more significant loss for MMSE.

    Source: §I · p. 4

  • Takeaways & Limitations

    The analysis identifies increased power offset at high SNR and reduced wideband slope at low SNR as the respective sources of MMSE rate loss.

    Source: §I · p. 4

Abstract

from arXiv · show

This paper investigates the achievable sum rate of multiple-input multiple-output (MIMO) wireless systems employing linear minimum mean-squared error (MMSE) receivers. We present a new analytic framework which unveils an interesting connection between the achievable sum rate with MMSE receivers and the ergodic mutual information achieved with optimal receivers. This simple but powerful result enables the vast prior literature on ergodic MIMO mutual information to be directly applied to the analysis of MMSE receivers. The framework is particularized to various Rayleigh and Rician channel scenarios to yield new exact closed-form expressions for the achievable sum rate, as well as simplified expressions in the asymptotic regimes of high and low signal to noise ratios. These expressions lead to the discovery of key insights into the performance of MIMO MMSE receivers under practical channel conditions.

A General Analytic Framework

The paper develops a general analytic framework for achievable sum rates with linear MMSE receivers by relating them to ergodic MIMO mutual information. It applies this framework to practical channel models where direct SINR-based analysis is limited, under a specified MIMO signal model.

I. INTRODUCTION

  • Few closed-form MMSE rate results cover finite-dimensional systems, and prior approaches often focus on asymptotics or restricted channel models.Large-system results may be less accurate at current MIMO antenna dimensions, while some CDMA-derived results exclude practical correlated Rayleigh and Rician models.Source: §I · pp. 2–3
  • The framework relates linear-MMSE achievable sum rates to ergodic MIMO mutual information, allowing existing mutual-information results to be applied without explicitly characterizing the MMSE output SINR.It yields analytic expressions across a broad class of channel scenarios without requiring a large number of antennas.Source: §I · p. 3, §II · pp. 5–6
  • The framework recovers equivalent expressions for uncorrelated Rayleigh fading and derives new exact and high- and low-SNR expressions for spatially correlated Rayleigh and uncorrelated Rician fading.The paper reports no comparable prior results for the latter two channel scenarios.Source: §I · pp. 3–4

II. SIGNAL MODEL AND ACHIEVABLE SUM RATE

  • The signal model uses spatial multiplexing with independent equal-power Gaussian signaling and assumes the channel is known at the receiver but unknown at the transmitter.The channel is flat-fading in the stated model; frequency-selective fading can be handled by decomposing it into parallel frequency-flat subchannels.Source: §II · pp. 4–5
  • With each linear-filter output decoded independently, achievable sum rate is expressed using the individual output SINRs, whose exact distributions are unavailable in closed form for many practical channels.The text identifies i.i.d. and semi-correlated Rayleigh fading as specific cases where a closed-form distribution is available.Source: §II · pp. 5–6

A. Exact Characterization

The exact characterization links the MMSE achievable sum rate to ergodic mutual-information expectations for optimal receivers. At high SNR, its rate behavior depends on the antenna regime: it saturates when Nr < Nt, while for Nr ≥ Nt it has a non-zero power offset.

A. Exact Characterization

  • The MMSE achievable sum rate can be expressed through the same expectations used for optimal-receiver ergodic mutual information, enabling established channel results to yield closed forms.This connection avoids explicitly characterizing the output SINR distribution of the linear MMSE filter.Source: A. Exact Characterization · p. 6

B. High SNR Characterization

  • When Nr ≥ Nt and H has i.i.d. entries, Corollary 2 expresses the high-SNR rate using expectations of J for H and for H with one fewer transmit dimension.The stated expression is log2 Nt − EH[J(Nr,Nt,H)] + EH1[J(Nr,Nt−1,H1)].Source: B. High SNR Characterization · p. 8
  • For Nr < Nt, the high-SNR MMSE sum rate converges to a finite asymptote because residual inter-stream interference remains as receiver noise vanishes.The receiver lacks enough degrees of freedom to perfectly cancel interference from the other transmit antennas.Source: B. High SNR Characterization · p. 8
  • For Nr ≥ Nt, the high-SNR MMSE rate has a non-zero power offset and uses the same types of expectations as high-SNR optimal-receiver mutual-information analysis.Existing closed-form mutual-information results can therefore be used to derive MMSE power-offset expressions across a range of fading scenarios.Source: B. High SNR Characterization · pp. 7–8

C. Low SNR Characterization

The low-SNR characterization uses minimum normalized energy per information bit and wideband slope to describe achievable-rate behavior. MMSE receivers attain the optimal minimum required energy per bit, but are suboptimal in the low-SNR regime through a reduced wideband slope.

C. Low SNR Characterization

  • The wideband slope captures second-order behavior of I(snr) as snr approaches zero.The minimum normalized energy per information bit and wideband slope are calculated from I(snr).Source: C. Low SNR Characterization · p. 9
  • At low SNR, the MMSE achievable sum rate has a general characterization in terms of parameters for the low-SNR regime.The paper introduces random-matrix dispersion as a measure used in subsequent derivations.Source: C. Low SNR Characterization · pp. 8–10
  • MMSE receivers are optimal in terms of the minimum required Eb/N0, but are suboptimal at low SNR through a typically reduced wideband slope S0.The wideband slope depends on the random-matrix channel through its dispersion.Source: C. Low SNR Characterization · p. 10
  • For both MMSE and optimal receivers, the minimum required Eb/N0 is independent of transmit-antenna count and varies inversely with receive-antenna count.The text attributes the receive-antenna dependence to increased captured channel energy while total transmit energy is constrained.Source: C. Low SNR Characterization · p. 10

IV. ACHIEVABLE SUM RATE OF MIMO MMSE IN FADING CHANNELS

The paper applies its general framework to practical fading models to obtain explicit MMSE achievable-sum-rate solutions. For i.i.d. Rayleigh fading, it gives an exact closed form that applies at all SNRs and antenna configurations, with analytic results agreeing precisely with simulations in the reported comparison.

IV. ACHIEVABLE SUM RATE OF MIMO MMSE IN FADING CHANNELS

  • The framework is particularized to fading models of practical interest to produce explicit MMSE achievable-sum-rate solutions.The section begins with the canonical i.i.d. Rayleigh case, described as representative of rich-scattering non-line-of-sight environments with sufficiently spaced antennas.Source: §IV · p. 10, A. Uncorrelated Rayleigh Fading · p. 10

1) Exact Analysis:

  • The analytic and Monte Carlo simulated achievable-sum-rate curves agree precisely in the i.i.d. Rayleigh comparison.Fig. 1 compares analysis and simulations for different antenna configurations.Source: Figure 1 · p. 11, 1) Exact Analysis · p. 12
  • For i.i.d. Rayleigh fading, Proposition 1 gives an exact closed-form MMSE achievable sum rate for all SNRs and arbitrary antenna configurations.The result is obtained by applying the ergodic mutual-information expression for i.i.d. Rayleigh MIMO channels to the framework.Source: 1) Exact Analysis · p. 11

2) High SNR Analysis:

The high-SNR analysis characterizes MMSE rate through its power offset and slope, showing how antenna counts affect both. It also derives large-antenna limits for the power offset and its excess.

2) High SNR Analysis:

  • With fixed transmit antennas, adding receive antennas leaves the high-SNR slope unchanged but reduces the power offset and improves achievable rate.The paper attributes this improvement to additional captured power and increased interference-cancellation capability; when receive antennas greatly exceed transmit antennas, the offset tends to −∞ dB.Source: 2) High SNR Analysis · p. 13
  • For equal antenna counts, n = 4 has a steeper high-SNR slope than n = 2, while n = 2 has the smaller power offset.Despite the larger offset, the overall MMSE achievable sum rate increases with n.Source: 2) High SNR Analysis · p. 13
  • The general high-SNR approximation is reported to remain accurate at moderate SNR, including within 20 dB.The section says it is formally valid in the very-high-SNR regime.Source: 2) High SNR Analysis · p. 13
  • In the large-antenna limit, the i.i.d. Rayleigh high-SNR power offset is unbounded when β = 1 but converges and decreases with β for β < 1.The resulting expression agrees with an earlier large-system result for MMSE receivers in randomly spread CDMA systems.Source: 2) High SNR Analysis · p. 14
  • The paper also gives exact and large-antenna characterizations of the excess high-SNR power offset for i.i.d. Rayleigh channels.The large-system expression is reported to agree with a prior result for randomly spread CDMA systems.Source: 2) High SNR Analysis · pp. 14–15

3) Low SNR Analysis:

The low-SNR analysis characterizes MMSE performance through its wideband slope and compares it with optimal reception. The reported comparison indicates a larger relative loss when transmit antennas outnumber receive antennas, while many receive antennas bring MMSE performance closer to optimal.

3) Low SNR Analysis:

  • The paper derives an i.i.d. Rayleigh wideband slope for MMSE reception and a large-antenna limit for its ratio to the optimal wideband slope.The MMSE wideband slope is reported to increase with both Nr and Nt.Source: 3) Low SNR Analysis · pp. 15–16
  • The low-SNR linear approximations are accurate over a moderate range of N0 values, especially for MMSE reception.The comparison is for Nr = Nt = 3 and includes exact low-SNR curves.Source: 3) Low SNR Analysis · p. 15
  • When transmit antennas exceed receive antennas, MMSE receivers take a significant low-SNR hit relative to optimal receivers, attributed primarily to limited interference suppression.The relative loss approaches its lower bound as transmit antennas grow with the receive count fixed; the upper bound is achieved for one transmit antenna.Source: 3) Low SNR Analysis · pp. 15–16
  • As receive antennas greatly exceed transmit antennas, linear MMSE receivers approach optimal performance at low SNR.The text links this behavior to additional captured received power and enhanced interference suppression.Source: 3) Low SNR Analysis · p. 16

B. Correlated Rayleigh Fading

For spatially correlated Rayleigh fading, the paper introduces an analytical treatment of MMSE achievable sum rate and provides exact results for semi-correlated channels. The treatment draws on a general mutual-information result and its closed forms are checked against simulation.

B. Correlated Rayleigh Fading

  • The paper presents what it identifies as the first analytical investigation of MMSE achievable sum rate under spatial correlation.It uses a separable correlation model, with positive-definite receive and transmit correlation matrices.Source: B. Correlated Rayleigh Fading · p. 17

1) Exact Analysis:

  • The exact analysis focuses on correlation at the transmitter or receiver, not both, although the approach can extend to joint correlation with more cumbersome expressions.The paper says the more general model is treated in its asymptotic-SNR subsections.Source: 1) Exact Analysis · p. 17
  • A new ergodic mutual-information result for semi-correlated Rayleigh fading allows correlation at either link end without restrictions on system dimensions.The paper contrasts this with prior analyses that treated cases separately depending on which end had fewer or more antennas.Source: 1) Exact Analysis · pp. 17–18
  • For semi-correlated Rayleigh fading, Propositions 4 and 5 give exact closed-form MMSE achievable sum rates for receive- and transmit-correlated cases.The results apply to arbitrary transmit and receive antenna counts.Source: 1) Exact Analysis · p. 18
  • For transmit correlation, the closed-form analysis agrees precisely with Monte Carlo simulations and shows rate degradation as correlation increases, especially at high SNR.The reported simulation comparison uses Nt = 3, Nr = 5, and different correlation coefficients.Source: 1) Exact Analysis · p. 19, Figure 3 · p. 19

2) High SNR Analysis:

At high SNR, transmit and receive correlation affect the MMSE power offset through separate functions. The analysis shows rate penalties from correlation and compares the MMSE excess offset with that of optimal reception.

2) High SNR Analysis:

  • At high SNR, transmit and receive correlation are decoupled in the power offset, through separate functions f(·) and g(·).The expression for the effect of receive correlation depends on its eigenvalues and on both Nr and Nt, whereas the transmit-correlation effect depends only on Nt.Source: 2) High SNR Analysis · pp. 20–21
  • Spatial correlation does not affect the high-SNR slope, but transmit correlation reduces MMSE achievable sum rate through an increased power offset.The high-SNR approximation is reported to converge to the exact rate at moderate SNR levels.Source: 2) High SNR Analysis · p. 21
  • For the equal-antenna scenarios shown, the SNR penalty increases with transmit or receive correlation and is more severe for transmit correlation, especially at small n.The comparison uses the exponential correlation model and varies correlation coefficient ρ.Source: Figure 4 · p. 20, 2) High SNR Analysis · p. 21
  • For equal transmit and receive antenna counts, the optimal-receiver high-SNR power offset has a simplified expression involving the determinants of both correlation matrices.The special case is stated for Nr = Nt = n.Source: 2) High SNR Analysis · pp. 21–22
  • In the high-SNR regime, MMSE receivers incur a more significant rate loss from transmit correlation than optimal MIMO receivers.The paper states that the excess offset due to transmit correlation is non-negative.Source: 2) High SNR Analysis · p. 22

3) Low SNR Analysis:

The low-SNR analysis characterizes the wideband slope of MMSE receivers under spatial correlation and compares its effects with optimal MIMO receivers. Both receive and transmit correlation reduce the achievable sum rate through a reduced wideband slope.

3) Low SNR Analysis:

  • The low-SNR analysis gives the wideband slope for transmit- and receive-correlated Rayleigh channels.Proposition 7 states the result, while the receive-correlation-only case admits a simpler form.Source: 3) Low SNR Analysis · pp. 22–23
  • Receive correlation reduces the MMSE achievable sum rate at low SNR through a reduced wideband slope, with bounds corresponding to uncorrelated and fully correlated antennas.The loss is greater than for optimal MIMO receivers, and the ratio of their wideband slopes decreases with receive correlation.Source: 3) Low SNR Analysis · p. 23
  • The ratio of MMSE to optimal-receiver wideband slopes has the same limiting behavior as for uncorrelated Rayleigh channels.Source: 3) Low SNR Analysis · p. 23
  • Transmit correlation likewise reduces the low-SNR MMSE achievable sum rate by reducing the wideband slope.Source: 3) Low SNR Analysis · p. 24

C. Uncorrelated Rician Fading

For uncorrelated Rician channels with a rank-1 specular component, the paper derives an exact MMSE sum-rate expression from optimal-receiver mutual-information results and analyzes high-SNR power offsets. At high SNR, line-of-sight increases the MMSE power offset, with its effect determined by the Rician K-factor rather than the specular component's geometry.

C. Uncorrelated Rician Fading

  • The Rician analysis assumes uncorrelated channels with a rank-1 specular component, parameterized by K and the specular component's arrival and departure angles.The paper identifies this as the first analytical investigation of achievable sum rates for MIMO MMSE receivers in Rician fading.Source: C. Uncorrelated Rician Fading · p. 24

1) Exact Analysis:

  • The exact MMSE achievable sum rate for uncorrelated Rician fading can be obtained by evaluating the framework using exact optimal-receiver mutual-information results.The paper omits the expression to avoid cumbersome notation.Source: 1) Exact Analysis · p. 24

2) High SNR Analysis:

  • The high-SNR power offset increases monotonically with K, so line-of-sight reduces the MMSE achievable sum rate.Its impact depends on the specular component's relative strength, not its angle-of-arrival or angle-of-departure geometry.Source: 2) High SNR Analysis · pp. 24–25
  • The excess power offset h2(K) increases with K, indicating a greater line-of-sight-related loss for MMSE than for optimal MIMO receivers.Figure 5 presents h1(K) and h2(K) for Nr = Nt = 2.Source: Figure 5 · p. 25, 2) High SNR Analysis · p. 26

3) Low SNR Analysis:

For uncorrelated Rician fading, the low-SNR analysis characterizes the MMSE wideband slope and its dependence on the specular component's strength. The concluding remarks report that spatial correlation and line-of-sight reduce MMSE sum rate at both SNR extremes, with greater losses than for optimal receivers.

3) Low SNR Analysis:

  • The Rician wideband slope for MMSE receivers decreases monotonically with K, so line-of-sight damages achievable sum rate at low SNR.The effect depends on the specular component's relative strength, not its geometry.Source: 3) Low SNR Analysis · p. 26
  • The ratio of MMSE to optimal-receiver wideband slopes decreases with K, with its upper bound approached for Rayleigh fading and lower bound for a deterministic channel.Source: 3) Low SNR Analysis · p. 27

V. Concluding Remarks

  • The framework yields exact closed-form sum-rate expressions and high- and low-SNR approximations for correlated and uncorrelated Rayleigh channels and uncorrelated Rician channels.The conclusion reports greater MMSE rate losses than for optimal receivers from spatial correlation or line-of-sight at both SNR extremes.Source: §V · p. 27
  • The authors state that many results apply almost verbatim to multiple-access and MIMO broadcast scenarios and that the framework extends to interference and amplify-and-forward relaying.Source: §V · pp. 27–28

APPENDIX I

The appendices provide derivation steps for the main results, use log-determinant derivatives and channel normalization in further analysis, and present an eigenvalue-density route for evaluating MIMO mutual information.

APPENDIX I

  • Appendix I uses substitutions and a matrix-minor property in deriving a result.Source: Appendix I · p. 28

APPENDIX II

  • Appendix II uses derivatives of a log determinant and channel power normalization to obtain subsequent results.Source: APPENDIX II · pp. 28–29

APPENDIX III

  • Appendix III expresses MIMO mutual information using the density of a nonzero eigenvalue and cites a unified density expression for arbitrary antenna counts.The result is obtained by substitution and integration.Source: APPENDIX III · p. 29
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