Ergodic and Outage Capacity of Narrowband MIMO Gaussian Channels

This paper presents exact results on the capacity of narrowband multiple-input-multiple-output (MIMO) Gaussian channels when perfect channel state information (CSI) is assumed at the receiver, but not always at the transmitter. Expressions for the MIMO channel capacity over ergodic flat fading channel, and formulas for the variance of the instantaneous capacity are derived. The cumulative distribution function (CDF) of the instantaneous channel capacity is also obtained using a Gaussian approximation. This CDF specifies the probability of not achieving a certain level of capacity (outage capacity) of the non-ergodic MIMO channels. I. I NTRODUCTION More recently, the application of multiple antennas at both the transmitter and the receiver configurations has attracted considerable interest both within academia and industry as a mean of providing significant capacity gains over conventional single antenna array based solutions [1], [2], [3]. This configuration is the so-called Multiple-input-multipleoutput (MIMO) system. This novel approach offers a unique solution for increasing demand for high performance next generation wireless communications. As demonstrated by Bell Laboratories [4], the spectral efficiencies is up to 42 bps/Hz, a spectacular increase compared to currently achievable spectral efficiencies of 2-3 bps/Hz in wireless mobile and LAN system. Therefore, study and analysis the capacity of MIMO channels using information theory would provide a deeper understanding of the configuration, and realization of the fundamental capacity limit of this approach. Channel capacity is defined as the maximum rate at which data can be transmitted at an arbitrary small error probability [5]. The capacity of MIMO channels has been well studied for the i.i. d. Rayleigh fading scenario [2], [3], and an exact formula for the ergodic (or mean) channel capacity was obtained in [3] assuming the receiver has the perfect channel state information (CSI). Few years later, a closed-form expression for the i.i. d. Rayleigh MIMO capacity with equal power allocation to each of the transmit antennas was derived [6], [7]. The capacity variance of this MIMO channel was then derived in [8], and [8] proposed that the MIMO channel capacity can be accurately approximated by a Gaussian random variable when the CSI is not available at the transmitter (equal power allocation). The implication of this Gaussian approximation is that only the capacity mean and variance are needed to get an accurate approximation to the capacity cumulative distribution function (CDF). It specifies the probability of not achieving a certain capacity, defined as the outage capacity probability . It was shown by simulations that the Gaussian approximation technique is very accurate even for a small number of antenna elements, and with a wide range of signal-to-noise ratio (SNR) [8], [9]. More recently, researchers are working on calculation of MIMO channel capacity under different scenarios: Correlation fading [10], Moment Generating Function (MGF) approach for independent Rician fading [11] and so on. The remainder of this paper is organized as follows. The next section we review some relevant results from linear algebra and numerical analysis. We present the MIMO system model in section III. The capacity of MIMO systems with fixed and deterministic coefficients will be studied in section IV. In section V, we discuss the capacity of MIMO systems with i.i. d. random channel coefficients for fast and block Rayleigh fading channel model. In section VI, we discuss the outage capacity of MIMO systems for slow i.i. d. Rayleigh fading model. In section VII some conclusions are presented. II. PRELIMINARIES We begin by establishing the notation and by reviewing the results from linear algebra and numerical analysis that we will employ. • For any matrixA, A† denote the Hermitian (transpose conjugate) ofA. Recall from linear algebra that an n×n matrix is Hermitian ifA = A† • The Matrix is nonnegative definiteif x†Ax ≥ 0 for any n× 1 complex vectorx. • An n× n matrix U is unitary, if U†U = In whereIn is the n× n identity matrix. • An eigenvectorv of ann×n matrix A corresponding to eigenvalueλ is a n × 1 vector of unit length such that vA = λv for some complex number λ. The vector space spanned by the eigenvectors of A corresponding to the non-zero eigenvalue has dimension r0, wherer0 is the rank of A. • The eigenvalues of a Hermitian matrix are real. • The eigenvalues of a nonnegative definite Hermitian matrix are nonnegative. • Given an×n Hermitian MatrixA, there exists a unitary matrixU and a real diagonal matrix D such thatU†AU = D. The columns ofU are an orthonormal basis of the ndimensional complex space, Cn, given by eigenvectors of A. The diagonal elements of D are the eigenvalues λi, i = 1, 2, · · · , n of A.