Achievable Ergodic Capacity of a MIMO System with a MMSE Receiver

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  • ABSTRACT

    This paper considers the multiple-input multiple-output (MIMO) system with linear minimum mean square error (MMSE) detection under ideal fast fading. For Nt transmit and Nr(≥Nt)receive antennas, we derive the achievable ergodic capacity of MMSE detection exactly. When MMSE detection is considered in a receiver, we introduce a different approach that gives the approximation of a MIMO channel capacity at high signal-to-noise ratio (SNR). The difference between the channel capacity and the achievable capacity of MMSE detection converges to some constant that depends only on the number of antennas. We validate the analytical results by comparing them with Monte Carlo simulated results.


  • KEYWORD

    Capacity , Multiple-Input Multiple-Output (MIMO) , Minimum Mean Square Error (MMSE) , Signal to Interference and Noise Ratio (SINR)

  • I. INTRODUCTION

    The use of multiple antennas at the transmitter and receiver provides a higher capacity than that of a single antenna system [1]. Multiple-input multiple-output (MIMO) communications have been intensively studied over the last few years and are widely considered as suitable for improving performance of modern wireless communications [2].

    Maximum-likelihood (ML) detection gives the maximum receive diversity and has high computational complexity. We can apply linear zero-forcing or minimum mean square error (MMSE) detectors to reduce computational complexity. Unfortunately, linear receivers have a smaller diversity order than that of the ML detector.

    In [3], the authors derived the achievable instantaneous capacity of a MMSE detector for a given channel matrix H. However, we have introduced the achievable ergodic capacity for an uncorrelated channel matrix H. In this paper, we investigate the analytical measurement associated with MMSE detection. We use the probability density function (PDF) of signal-tointerference plus noise ratio (SINR) [4] to obtain a precise value for the achievable capacity of the MMSE detector. We also derive the achievable ergodic capacity of the MMSE detector at high SNR. Finally, we confirm for a small number of antennas that the difference between the channel capacity and the achievable capacity of the MMSE detector converges to some constant that depends on the number of transmit and receive antennas.

    II. SYSTEM DESCRIPTION

    We consider single-user communications and investigate a point-to-point link, where the transmitter is equipped with Nt antennas and the receiver employs Nr(≥Nt) antennas. Suppose that no inter-symbol interference (ISI) exists. Let hij be the complex-valued channel coefficient from transmit antenna j to receive antenna i. If the complex modulated signals are transmitted via the Nt antennas, then the received signal at antenna i can be represented as ,where ni represents additive white Gaussian noise. This relation is easily written in a matrix form as

    image

    where denotes the received complex vector, is the transmitted vector with the power is an independent and identically distributed (i.i.d) complex Gaussian fading channel matrix with unit variance, and is an additive white Gaussian noise with zero mean and variance N0.

    Suppose that the transmitter does not know the channel realization. Minimizing the mean squared error (MSE) between the actually transmitted symbol xi and the output of a linear MMSE detector leads to the filter vector where and hi is the ith column of H. Applying this filter vector into Eq. (1) yields where and the interference-plus-noise term w is defined as The variance of w is computed as The SINR of a linear MMSE detector on the ith spatial stream can be computed as For the uncorrelated channel, the statistical property of SINRi is the same for all 1 ≤ iNt. Then we omit the subscript i of SINRi.

    III. ACHIEVABLE CAPACITY OF MMSE RECEIVER

    We introduce the PDF of SINR at output of linear MMSE detector to calculate the ergodic capacity. We use it for analytical derivation of the ergodic capacity obtained by the MMSE detector. For a high SNR, we will prove that the difference between the channel capacity and the capacity of the MMSE detector converges to the constant that depends only on Nt and d = NrNt + 1.

       1. PDF of SINR

    For small Nr and Nt , we summarize the PDF of SINR [4] as following:

    image

       2. Achievable Capacity

    For the uncorrelated MIMO channel, the ergodic capacity achievable by the MMSE detector can be defined as [3]

    image

    For example, in Nr 2 and Nt = 2 , we compute the integral directly, as follows:

    image

    We can easily check the first term in Eq. (4) by [5, 4.337]:

    image

    where and E1(s) = –Ei(–s) [5]. The second term in Eq. (4) is calculated by integration by parts and [5, 3.352] and then the third one in Eq. (4) is eliminated:

    image

    We can obtain the exact ergodic capacity as

    image

    In a similar way, can be calculated as the following:

    image
    image

    where s = Nt/SNR. Note that is a function of SNR.

    We consider an independent and identically distributed Rayleigh fading channel with Nt. transmit antennas and Nr receive antennas. The simulated capacity performance of the MMSE detector under a typical realization is displayed with respect to SNR. This is validated by the simulation results shown in Fig. 1.

       3. Constant Gap between the Channel Capacity and Achievable Capacity of the MMSE Detector at High SNR

    At a high SNR regime, the capacity of the MMSE detector for a MIMO system can be approximated as

    image

    Up to Nr = 4 and Nt = 4, by integrating Eq. (9), the approximated ergodic capacities of MMSE detector can be computed easily as

    image

    where d = NrNt + 1 and ψ(k) is expressed as [6]

    image

    Note that μ is Euler’s constant. In [6], the authors found the approximation for ergodic channel capacity at high SNR as

    image

    The difference between the MIMO channel capacity and the capacity of the MMSE detector converges to some constant as

    image

    The Eq. (13) shows that the achievable capacity of the MMSE detector has constant capacity degradation. We will validate the capacity degradation by the simulation results shown in Fig. 2.

    IV. CONCLUSION

    In this paper, we have studied the achievable ergodic capacity of a MIMO system with a MMSE detector for a small number of antennas. We have also shown that the difference between the channel capacity and achievable capacity of the MMSE detector converges to some constant at high SNR.

  • 1. Telatar E. 1999 "Capacity of multi-antenna Gaussian channels," [European Transactions on Telecommunications] Vol.10 P.585-595 google doi
  • 2. Foschini G. J. 1996 "Layered space-time architecture for wireless communication in a fading environment when using multielement antennas," [Bell Labs Technical Journal] Vol.1 P.41-59 google doi
  • 3. Zhang X., Kung S. Y. 2003 "Capacity analysis for parallel and sequential MIMO equalizers," [IEEE Transactions on Signal Processing] Vol.51 P.2989-3002 google doi
  • 4. Kim N., Lee Y., Park H. 2008 "Performance analysis of MIMO system with linear MMSE receiver," [IEEE Transactions on Wireless Communications] Vol.7 P.4474-4478 google doi
  • 5. Gradshteyn I. S., Ryzhik I. M. 2007 Table of Integrals, Series, and Products google
  • 6. Oyman O., Nabar E. U., Bolcskei H., Paulraj A. J. 2002 "Tight lower bounds on the ergodic capacity of Rayleigh fading MIMO channels," [in Proceedings of IEEE Global Telecommunications Conference (GLOBECOM)] P.1172-1176 google
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  • [Fig. 1.] The ergodic capacities obtained by the minimum mean square error (MMSE) detector. SNR = signal-to-noise ratio.
    The ergodic capacities obtained by the minimum mean square error (MMSE) detector. SNR = signal-to-noise ratio.
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  • [Fig. 2.] Convergence of the difference between the channel capacity and the capacity obtained by the minimum mean square error (MMSE) detector. SNR = signal-to-noise ratio.
    Convergence of the difference between the channel capacity and the capacity obtained by the minimum mean square error (MMSE) detector. SNR = signal-to-noise ratio.