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

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

\authorblockNJun Zhang, Marios Kountouris, Jeffrey G. Andrews and Robert W. Heath Jr. \authorblockAWireless Networking and Communications Group
Department of Electrical and Computer Engineering
The University of Texas at Austin, Austin, TX 78712-0240
Email: {jzhang2, mkountouris, jandrews, rheath}@ece.utexas.edu
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.

I Introduction

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, 3, 4, 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 multi-user (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.

Ii System Model

We consider a MIMO broadcast channel with antennas at the transmitter and single-antenna receivers. Each time slot, the transmitter determines the number of users to be served, denoted as the transmission mode M, . Eigen-beamforming is applied for the SU mode (), which is optimal with perfect CSIT. ZF precoding is used for the MU mode (), 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 -th user in mode at time is given as

(1)

where is the channel vector for the -th user, and is the normalized complex additive Gaussian noise, . and are the transmit signal and precoding vector for the -th user. The transmit power constraint is , and we assume equal power allocation among different users. As the noise is normalized, is also the average SNR.

To assist the analysis, we assume that the channel is well modeled as a spatially white Gaussian channel, with entries . 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.

Ii-a CSI Delay Model

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

(2)

where is the channel error vector, with i.i.d. entries , and it is uncorrelated with 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 with Doppler spread [14], where is the symbol duration and is the zero-th order Bessel function of the first kind. The variance of the error vector is . The value is the normalized Doppler frequency. Note that this model can also be used to model the estimation or prediction error, which makes our results more general.

Ii-B Channel Quantization

The channel direction information is fed back using a quantization codebook known at both the transmitter and receiver. The quantization is chosen from a codebook of unit norm vectors of size . Each user uses a different codebook to avoid the same quantization vector. The codebook for user is . Each user quantizes its channel direction to the closest codeword, and the closeness is measured by the inner product. Therefore, the index of channel for user is

(3)

where is the channel direction. Random vector quantization (RVQ) is used to facilitate the analysis, where each quantization vector is independently chosen from the isotropic distribution on the -dimensional unit sphere. The codebook based on RVQ is asymptotically optimal in probability as , , with [15].

Ii-C Multi-mode Transmission

In this paper, we consider a homogeneous network where all the users have the same average SNR, mobility, delay and feedback bits. The transmitter will determine how many users to serve, i.e. the active mode , and then users are selected from the total users randomly or with round-robin scheduling. The mode selection is based on the information including average SNR, normalized Doppler frequency, and the quantization codebook size. Only the selected users need to feed back their instantaneous channel information. Such user scheduling is suitable for the system with scheduling independent of the channel status, such as round-robin or the one based on the queue length, or the small system with user number roughly equal to the transmit antennas, such as in a cooperative communication network.

Iii Throughput Analysis and Mode Selection

In this section, we derive the achievable ergodic rate for each mode, based on which the active mode can be selected. Both perfect and imperfect CSIT systems are investigated.

Iii-a Perfect CSIT

Iii-A1 SU Mode (Eigen-beamforming),

With perfect CSIT, the beamforming (BF) vector is the channel direction vector, . The ergodic achievable rate is

(4)

where is the complementary incomplete gamma function. The BF system provides a transmit array gain as .

Iii-A2 MU Mode (Zero-forcing),

The received SINR for the -th user in a linear precoding MU-MIMO system of mode is given by

(5)

Denote , and the pseudo-inverse of as . The precoding vector for the -th user is obtained by normalizing the -th column of . Therefore, , i.e. there is no inter-user interference and each user gets an equivalent interference-free channel. The SINR for the -th user becomes

(6)

Due to the isotropic nature of i.i.d. Rayleigh fading, such orthogonality constraints to precancel inter-user interference consume degrees of freedom at the transmitter. As a result, the effective channel gain of each parallel channel is a chi-square random variable with degrees of freedom [16], i.e. . Therefore, the channel for each user is equivalent to a diversity channel with order and effective SNR . The achievable rate for the -th user in mode is

(7)

where is given in (III-A1). The achievable sum rate for the ZF system of mode is

(8)

When , this reduces to (III-A1).

Iii-A3 Mode Selection

From (III-A2), the system in mode provides a spatial multiplexing gain of and an array gain of for each user. As increases, the achievable spatial multiplexing gain increases but the array gain decreases. Therefore, there is a tradeoff between the achievable array gain and the spatial multiplexing gain. From (III-A2), the mode that achieves a higher throughput for the given average SNR can be determined as

(9)

Iii-B Imperfect CSIT

In this section, we consider a system with both delay and channel quantization. As it is difficult to derive the exact achievable rate for such a system, we provide accurate closed-form approximations for mode selection.

Iii-B1 SU Mode (Eigen-Beamforming)

With delay and channel quantization, the BF vector is based on the delayed version of the quantized channel direction, .

To get a good approximation for the achievable rate for the SU mode, we first make the following approximation on the instantaneous received SNR

(10)

i.e. we remove , which is normally insignificant compared to . Eq. (10) is equivalent for a limited feedback system. From [17], the achievable rate of the limited feedback BF system is given by

(11)

where is the -th order exponential integral. So the achievable rate of the BF system with both delay and channel quantization can be approximated as

(12)

Iii-B2 MU Mode (Zero-Forcing)

For the MU mode with imperfect CSIT, the precoding vectors are designed based on the quantized channel directions with delay, which achieve . The SINR for the -th user in mode is given as

(13)

Following Theorem 3 in [11], the average residual inter-user interference for the -th user is

(14)
Remark 1

The residual interference depends on delay, codebook size, , and . It increases with delay, and decreases with the codebook size. At high SNR, it makes the system interference-limited. With other parameters fixed, residual interference increases as increases, and mode selection will take this into consideration.

To approximate the achievable ergodic rate, we first approximate the signal term as

(15)

where step (a) removes , which is normally very small. Step (b) approximates the actual channel direction by the quantized version, which is justified for small quantization error. As is independent from other users, similar to the system with perfect CSIT, .

(16)
(17)
(18)

The received SINR for the -th user can then be approximated as in (16), on the top of the next page. The approximation for the denominator comes from removing the terms with both and . For the interference terms in (16), as , , and they are independent, is an exponential random variable with mean . The following lemma in [12] provides the distribution of the term .

Lemma 1

Based on the quantization cell approximation [5], the interference term due to quantization, , can be approximated as an exponential random variable with mean .

Remark 2

The residual interference terms due to both delay and quantization are exponential random variables, which means the delay and quantization error have equivalent effects but with different means.

Based on the distributions of the interference terms, we can get the following theorem on the achievable rate for the MU mode.

Theorem 1

The achievable ergodic rate for the -th user in mode () with both delay and channel quantization can be approximated by (17), where , , , , is the following integral

(19)

and

with .

{proof}

This approximation is derived by assuming the interference terms are pair-wise independent, and also independent of the signal term. Then the CDF of the SINR can be derived, following which the achievable rate can be calculated. A closed form expression can be derived for , which is not provided due to space limitation.

As a special case, with only delay, following (III-B2) and (16) the received SINR for the -th user is approximated as

(20)

for which the achievable ergodic rate is provided as follows.

Corollary 1

The achievable ergodic rate for the -th user in the delayed system of mode () can be approximated by (18), where , , , and is the integral in (19).

From Lemma 1, the effects of delay and channel quantization are equivalent, so the result in this corollary also applies for the limited feedback system, replacing by .

Iii-C Mode Selection

Based on (12) and (17), the active mode in the system with both delay and channel quantization is selected according to

(21)

where .

Remark 3

Considering (III-B2), (III-B2), and (17), the mode is now related to the residual interference, the transmit array gain, and the spatial multiplexing gain. The multi-mode transmission is to balance between these effects to improve the system throughput.

Iv Numerical Results: Verification of Analysis and Key Observations

Fig. 1: Comparison of simulations and approximations for different , , mobility km/hr, msec, .

We first verify the derived approximations for the achievable rates in Fig. 1. We see that the approximation is very accurate at low to medium SNRs. At high SNR, the approximation becomes a lower bound, and the accuracy decreases as increases. Interestingly, we see that the mode always provides a higher throughput than the full MU mode . This is due to the fact that the full mode has the highest level of residual interference while provides no array gain. Therefore, it is desirable to do spatial division multiplexing for fewer than users, which provides addition array gain and reduces the residual interference for each user. We see that the proposed multi-mode transmission provides a significant throughput gain over the SU-MIMO system (). In addition, it is able to provide a throughput gain around bps/Hz over the dual-mode switching [11, 12] at medium SNR.

(a) Different , .
(b) Different , , msec., and km/hr.
Fig. 2: Operating regions for different modes with both CSI delay and channel quantization, . The mode means that there are active users. In this plot, ‘’ is for , ‘’ is for , ‘’ is for , and ‘’ is for .

From (21), we can determine the active mode for a given scenario. Accordingly, the operating regions for different modes can be plotted for different system parameters. Fig. 2(a) and Fig. 2(b) show the operating regions for different and different , respectively. There are several key observations.

  1. For the given and , the SU mode () will be active at both low and high SNRs, due to its array gain and the robustness to imperfect CSIT, respectively.

  2. For higher MU modes to be active, needs to be small while needs to be large. Specifically, to get the mode activated, we need with as in Fig. 2(a), and need with km/hr, GHz, and msec as in Fig. 2(b).

  3. The highest mode is not active with the considered parameters. We find that it is possible to be active only when and is large enough.

V Conclusions

In this paper, we derive the achievable ergodic rates for the MIMO broadcast channel with imperfect CSIT. Due to residual inter-user interference, it is not desirable to serve too many users with spatial division multiplexing. Based on the derived rates, a multi-mode transmission strategy is proposed that adaptively adjusts the number of active users. It is based on the channel statistics information, and only the selected users need to feed back the instantaneous channel information.

References

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