Achievable Throughput of Multi-mode Multiuser MIMO with Imperfect CSI Constraints

📝 Abstract

For the multiple-input multiple-output (MIMO) broadcast channel with imperfect channel state information (CSI), neither the capacity nor the optimal transmission technique have been fully discovered. In this paper, we derive achievable ergodic rates for a MIMO fading broadcast channel when CSI is delayed and quantized. It is shown that we should not support too many users with spatial division multiplexing due to the residual inter-user interference caused by imperfect CSI. Based on the derived achievable rates, we propose a multi-mode transmission strategy to maximize the throughput, which adaptively adjusts the number of active users based on the channel statistics information.

💡 Analysis

For the multiple-input multiple-output (MIMO) broadcast channel with imperfect channel state information (CSI), neither the capacity nor the optimal transmission technique have been fully discovered. In this paper, we derive achievable ergodic rates for a MIMO fading broadcast channel when CSI is delayed and quantized. It is shown that we should not support too many users with spatial division multiplexing due to the residual inter-user interference caused by imperfect CSI. Based on the derived achievable rates, we propose a multi-mode transmission strategy to maximize the throughput, which adaptively adjusts the number of active users based on the channel statistics information.

📄 Content

For the multiple-input multiple-output (MIMO) broadcast channel, channel state information at the transmitter (CSIT) is required to separate the spatial channels for different users and achieve the full spatial multiplexing gain. CSIT, however, is difficult to get and is never perfect. Neither the capacity nor the optimal transmission technique have been fully discovered. Linear precoding combined with limited feedback [1] is a practical option, which has drawn lots of interest recently [2]- [5]. The main finding is that the full spatial multiplexing gain can be obtained with carefully designed feedback strategy and sufficient feedback bits that grow linearly with signal-to-noise ratio (SNR) (in dB) and the number of transmit antennas.

In most systems, the number of feedback bits per user is fixed. In addition, there are other CSIT imperfections, such as estimation error and feedback delay. All of these make the system throughput limited by the residual inter-user interference at high SNR [6]. A simple approach to solve this problem is to adaptively switch between the single-user (SU) and multiuser (MU) modes, as the SU mode does not suffer from the residual interference at high SNR. SU/MU mode switching algorithms for the random beamforming system were proposed in [7], [8], where each user feeds back its preferred mode and the channel quality information. Mode switching for systems with zero-forcing (ZF) precoding and limited feedback was investigated in [9], [10], where the switching is performed during the scheduling process with properly designed channel quality information feedback.

The above mentioned SU/MU mode switching algorithms are based on instantaneous CSIT, and require feedback from each user in each time slot. In [11], [12], a SU/MU mode switching algorithm was proposed for the system with delayed and quantized CSIT. The mode switching is based on the statistics of the channel information, including the average SNR, the normalized Doppler frequency, and the codebook size, which are easily available at the transmitter. But it only switches between the SU mode and the full MU mode that serves the maximum number of users that can be supported, i.e. it is a dual-mode switching strategy.

In this paper, we consider a MIMO broadcast channel with delayed and quantized CSIT, with the amount of delay and the size of the quantization codebook fixed. We derive an achievable ergodic rate for each transmission mode, denoting the number of users served by spatial division multiplexing. It is shown that the number of active users is related to the transmit array gain, spatial multiplexing gain, and the residual inter-user interference. The full MU mode normally should not be activated, as it suffers from the highest interference while provides no array gain. A multi-mode transmission strategy is proposed to adaptively select the active mode to maximize the throughput.

We consider a MIMO broadcast channel with N t antennas at the transmitter and U single-antenna receivers. Each time slot, the transmitter determines the number of users to be served, denoted as the transmission mode M, 1 ≤ M ≤ N t . Eigenbeamforming is applied for the SU mode (M = 1), which is optimal with perfect CSIT. ZF precoding is used for the MU mode (1 < M ≤ N t ), as it is possible to derive closed-form results due to its simple structure, and it is optimal among the set of all linear precoders at asymptotically high SNR [13]. The discrete-time complex baseband received signal at the uth user in mode M at time n is given as

where h u [n] is the channel vector for the u-th user, and z u [n] is the normalized complex additive Gaussian noise, z u [n] ∼ CN (0, 1). x u [n] and f u [n] are the transmit signal and precoding vector for the u-th user. The transmit power constraint is E M u=1 |x u [n]| 2 = P , and we assume equal power allocation among different users. As the noise is normalized, P is also the average SNR.

To assist the analysis, we assume that the channel h u [n] is well modeled as a spatially white Gaussian channel, with entries h i [n] ∼ CN (0, 1). We assume perfect CSI at the receiver and the transmitter obtains CSI through limited feedback. In addition, there is delay in the available CSIT. The models for delay and limited feedback are presented as follows.

We consider a stationary ergodic Gauss-Markov block fading regular process (or auto regressive model of order 1), where the channel stays constant for a symbol duration and changes from symbol to symbol according to

where e[n] is the channel error vector, with i.i.d. entries e i [n] ∼ CN (0, ǫ 2 e ), and it is uncorrelated with h[n -1] and i.i.d. in time. We assume the CSI delay is of one symbol. For the numerical analysis, the classical Clarke’s isotropic scattering model will be used as an example, for which the correlation coefficient is ρ = J 0 (2πf d T s ) with Doppler spread f d [14], where T s is the symbol duration and J 0 (•) is the zeroth order Bessel fu

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