Multi-Access MIMO Systems with Finite Rate Channel State Feedback

The purpose of this paper is to quantify the effect of the finite rate channel state feedback on the sum rate. The effect of finite rate feedback on single user MIMO systems are well studied. MIMO systems with only one on-beam are considered in [1] and [2] while systems with multiple on-beams are discussed in [3]- [8]. In the recent works [7] and [8], the effect of finite rate feedback is accurately quantified by characterizing the distortion rate function in the Grassmann manifold. For multi-access systems, the throughput capacity region is characterized in [9] with the assumption that each user has only one antenna and the full channel information is available to all users.

To characterize the feedback gain, we propose to control the users jointly. An simple extension of [10] can show that the optimal strategy is to select the covariance matrices of the transmit signals of the users jointly. It is different from the current systems where the base station controls the users individually. The gain of joint control over individual control is analogous to that of vector quantization over scalar quantization. However, it is difficult to either implement or analyze the fully joint control. For simplicity, this paper proposes a suboptimal strategy employing power on/off strategy, where we first choose the onusers jointly and then select the beamforming vectors jointly. The effect of user choice can be analyzed by extreme order statistics. To quantify the effect of beamforming, the composite Grassmann manifold is introduced in this paper. By characterizing the distortion rate function on the composite Grassmann manifold and calculating the logdet function of a random composite Grassmann matrix, a good sum rate approximation is derived. According to the distortion rate function on the composite Grassmann manifold, the loss of finite beamforming decreases exponentially as the feedback bits on beamforming increases.

Assume that there are L R antennas at the base station and N users communicating with the base station. Assume that the user i has L T,i antennas 1 ≤ i ≤ N. In this paper, we let L T,i = L T,j = L T for 1 ≤ i, j ≤ N. The signal transmission model is

where Y ∈ C L R ×1 is the received signal at the base station, H i ∈ C L R ×L T is the channel state matrix for user i, T i is the transmitted Gaussian signal vector for user i and W ∈ C L R ×1 is the additive Gaussian noise vector with zero mean and covariance matrix I L R . In this paper, we assume the Rayleigh fading channel model, i.e., the entries of H i are independent and identically distributed (i.i.d.) circularly symmetric complex Gaussian variables with zero mean and unit variance (CN (0, 1)) and H i ’s are independent for each channel use.

We assume that there exists a common feedback link from the base station to all the users. At the beginning of each channel use, the channel states H i ’s are perfectly estimated at the receiver. A message, which is a function of the channel state, is sent back to all users through a feedback channel. The feedback is error-free and rate limited. The feedback directs the users to choose their Gaussian signal covariance matrices. In multi-access system, users are uncoordinated. It is reasonable to assume that

be the overall transmitted Gaussian signal for all users and Σ E TT † be the overall signal covariance matrix. Then Σ is an NL T × NL T block diagonal matrix whose i th diagonal block is the

Since the variance of the Gaussian noise is normalized, the average power constraint ρ is also the average received signal-to-noise ratio (SNR).

For compositional clarity, this section assembles the useful mathematical results that we derive for later analysis. Due to the space limit, we omit all the proofs.

Let X i = L j=1 |h i,j | 2 where h i,j 1 ≤ j ≤ L, 1 ≤ i ≤ n are i.i.d. circularly symmetric complex Gaussian variables with zero mean and unit variance. Let us rearrange these i.i.d. chi-square random variables

Let n approach infinity, the following theorem gives a formula for E l k=1 X i n-k+1 where l is a fixed positive integer.

Theorem 1: Let X = L j=1 |h j | 2 where h j ∼ CN (0, 1). Denote the distribution function of X by F X (x). Then for any fixed positive integer l,

where a n is the solution of

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