Achievable Ergodic Capacity of a MIMO System with a MMSE Receiver
 Author: Kim Jae Hong, Kim Nam Shik, Song Bong Seop
 Organization: NESSLAB, Daejeon, Korea.; NESSLAB, Daejeon, Korea.; NESSLAB, Daejeon, Korea.
 Publish: Journal of electromagnetic engineering and science Volume 14, Issue4, p349~352, Dec 2014

ABSTRACT
This paper considers the multipleinput multipleoutput (MIMO) system with linear minimum mean square error (MMSE) detection under ideal fast fading. For
N_{t} transmit andN_{r} (≥N_{t} )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 signaltonoise 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 , MultipleInput MultipleOutput (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]. Multipleinput multipleoutput (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].
Maximumlikelihood (ML) detection gives the maximum receive diversity and has high computational complexity. We can apply linear zeroforcing 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 matrixH. In this paper, we investigate the analytical measurement associated with MMSE detection. We use the probability density function (PDF) of signaltointerference 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 singleuser communications and investigate a pointtopoint link, where the transmitter is equipped with
N_{t} antennas and the receiver employsN_{r} (≥N_{t} ) antennas. Suppose that no intersymbol interference (ISI) exists. Leth_{ij} be the complexvalued channel coefficient from transmit antennaj to receive antennai. If the complex modulated signals are transmitted via theN_{t} antennas, then the received signal at antennai can be represented as ,wheren_{i} represents additive white Gaussian noise. This relation is easily written in a matrix form aswhere 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
N _{0}.Suppose that the transmitter does not know the channel realization. Minimizing the mean squared error (MSE) between the actually transmitted symbol
x_{i} and the output of a linear MMSE detector leads to the filter vector where andh _{i} is thei th column ofH . Applying this filter vector into Eq. (1) yields where and the interferenceplusnoise termw is defined as The variance ofw 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 SINR_{i} is the same for all 1 ≤i ≤N_{t} . Then we omit the subscripti of SINR_{i}.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
N_{t} and d =N_{r} –N_{t} + 1.1. PDF of SINR
For small
N_{r} andN_{t} , we summarize the PDF of SINR [4] as following:2. Achievable Capacity
For the uncorrelated MIMO channel, the ergodic capacity achievable by the MMSE detector can be defined as [3]
For example, in
N_{r} 2 andN_{t} = 2 , we compute the integral directly, as follows:We can easily check the first term in Eq. (4) by [5, 4.337]:
where and
E _{1}(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:We can obtain the exact ergodic capacity as
In a similar way, can be calculated as the following:
where
s =N_{t} /SNR . Note that is a function of SNR.We consider an independent and identically distributed Rayleigh fading channel with
N_{t} . transmit antennas andN_{r} 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
Up to
N_{r} = 4 andN_{t} = 4, by integrating Eq. (9), the approximated ergodic capacities of MMSE detector can be computed easily aswhere
d =N_{r} –N_{t} + 1 and ψ(k ) is expressed as [6]Note that μ is Euler’s constant. In [6], the authors found the approximation for ergodic channel capacity at high SNR as
The difference between the MIMO channel capacity and the capacity of the MMSE detector converges to some constant as
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.

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[Fig. 1.] The ergodic capacities obtained by the minimum mean square error (MMSE) detector. SNR = signaltonoise 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 = signaltonoise ratio.