BER analysis of TDD downlink multiuser MIMO systems with imperfect channel state information
© Zhou et al; licensee Springer. 2011
Received: 23 March 2011
Accepted: 16 November 2011
Published: 16 November 2011
In downlink multiuser multiple-input multiple-output (MU-MIMO) systems, the zero-forcing (ZF) transmission is a simple and effective technique for separating users and data streams of each user at the transmitter side, but its performance depends greatly on the accuracy of the available channel state information (CSI) at the transmitter side. In time division duplex (TDD) systems, the base station estimates CSI based on uplink pilots and then uses it through channel reciprocity to generate the precoding matrix in the downlink transmission. Because of the constraints of the TDD frame structure and the uplink pilot overhead, there inevitably exists CSI delay and channel estimation error between CSI estimation and downlink transmission channel, which degrades system performance significantly. In this article, by characterizing CSI inaccuracies caused by CSI delay and channel estimation error, we develop a novel bit error rate (BER) expression for M-QAM signal in TDD downlink MU-MIMO systems. We find that channel estimation error causes array gain loss while CSI delay causes diversity gain loss. Moreover, CSI delay causes more performance degradation than channel estimation error at high signal-to-noise ratio for time varying channel. Our research is especially valuable for the design of the adaptive modulation and coding scheme as well as the optimization of MU-MIMO systems. Numerical simulations show accurate agreement with the proposed analytical expressions.
Owing to their high spectral efficiency, multiple-input multiple-output (MIMO) wireless antenna systems have been recognized as a key technology for future wireless communication systems such as long-term evolution (LTE), LTE-advanced (LTE-A), WiMax, etc. Multiuser MIMO (MU-MIMO) has become one of the main features in LTE-A systems because of several key advantages over single-user MIMO (SU-MIMO) [1, 2]. There are several kinds of classic transmission methods for downlink MU-MIMO: Dirty Paper Coding (DPC) , Block Diagonalization (BD) [4, 5], and zero-forcing (ZF) (or channel inversion) . Though DPC is optimal and can achieve sum-rate capacity, it is difficult to implement in practical systems because of its high complexity. BD and ZF are suboptimal methods with tolerable performance degradation and their lower complexities make them easier to implement. Furthermore, compared to BD, ZF is an even simpler algorithm which essentially separates multiple data streams from the same user equipment (UE) at transmitter side. So, in this article, ZF transmission method is chosen for the performance analysis of downlink MU-MIMO systems; however, similar analytical methods may be applied to other transmission methods as well.
As is well known, the availability of accurate channel state information (CSI) is very important for downlink MU-MIMO schemes. However, in practice, CSI is always imperfect because of the existence of CSI delay, quantization error, and channel estimation error. This would cause not only self-interference among different data streams of the same user, but also interference among users, severely degrading the performance especially in case of high mobile users or long delay. Hence, it is important to characterize the performance of MU-MIMO system in the presence of imperfect CSI.
Most recent study [7–12] about the impact of imperfect CSI on MU-MIMO focused on the frequency division duplex systems. In , the authors investigated the impact of feedback delay and estimation error on the sum-rate of MU-MIMO systems. In , the authors studied upper and lower bounds on the achievable sum-rate of a correlated/uncorrelated MU-MIMO channel with channel estimation error and feedback delay. The achievable ergodic rates were derived for multi-user MIMO systems with CSI delay and quantization error in [9, 10]. In , the impact of imperfect CSI on sum-rate scaling law was investigated for downlink MU-MIMO systems. In , the authors quantified the impact of channel estimation errors, quantization errors, and outdated quantized CSI on the rate loss of MU-MIMO system.
To the authors' knowledge, the impact of imperfect CSI on MU-MIMO in time division duplex (TDD) systems is almost rarely investigated. In this article, we study the impact of imperfect CSI caused by both CSI delay and channel estimation error on bit error rate (BER) for TDD downlink MU-MIMO ZF systems. In order to clearly indicate the impact of imperfect CSI on MU-MIMO, we only analyze un-coded MU-MIMO systems although channel coding techniques are indispensable in practical systems. In TDD system, the base station (BS) estimates CSI at transmitter side based on the uplink pilots periodically sent by the mobile users. Then, BS uses it through channel reciprocity to generate precoding matrix for the downlink data transmission. Because of the constraints of the TDD frame structure and the uplink pilot overhead, there inevitably exists both CSI delay and channel estimation error between uplink estimated channel (used to generate precoding matrix during downlink transmission) and downlink transmission channel, which degrades the system performance. In this article, using the correlation between the actual channel and the estimated one , as well as the channel's time-correlation , we obtain an expression for post-processing signal-to-interference plus noise ratio (SINR) of each data stream of TDD downlink MU-MIMO systems. Based on the post-processing SINR, we then obtain the expression for average BER of uncoded TDD MU-MIMO ZF systems with M-quadrature amplitude modulation (QAM)-modulated signals. Numerical simulations verify our analysis.
Notation: E(·), (·)H, (·)T, (·)*, and ||·||F denote expectation, Hermitian, transpose, complex conjugation, and Frobenius norm, respectively. I M is the M × M identity matrix. (·)† denotes the right pseudo inversion and (A)†≜AH(AAH)-1. denotes the complex Gaussian distribution with mean vector μ and variance matrix Σ.
2. System model
In TDD MU-MIMO systems, the procedures at the physical layer for the downlink data transmission are as follows:
Step 1: BS obtains the delay estimated version of CSI based on the received uplink pilots at the (m - Md)th symbol interval. Here, , Md denotes the delay in symbol between the uplink channel estimation and downlink data transmission, and the value of Md ranges from 1 to nd for the different downlink data symbol as in Figure 2.
Step 3: each user estimates the downlink channel through the downlink pilots and then detects the received signal.
3. BER analysis
In this section, we first derive the post-processing SINR under the given , and then derive the average BER based on post-processing SINR.
where 0 ≤ |ρe, k| ≤ 1. Because the SNR of pilots of each user can be different, ρe, kof each user can be different.
where 0 ≤ |ρd, k| ≤ 1. Because each user can have a different mobile velocity, ρd, kof each user can be different.
Because ZF precoding has already separated all data streams at transmitter side, from the receiver's perspective MU-MIMO system has reduced into a lot of parallel "equivalent SISO systems" one of which bears one data stream. Although a more complicated receiver could be used in each "equivalent SISO system" to demodulate the data stream to obtain better performance, the main purpose of this article is to investigate the impact of CSI delay and channel estimation error on MU-MIMO systems, so we use the simple receiver "ZF equalizer" in this article to demodulate each data stream.
where γs, k= Es/N0, kis the pre-processing SNR of downlink data symbol. Note that all data streams to the same user have the same SINR because we do not consider the power allocation strategy for all data streams and each data stream has the equal power.
Based on (13), we below derive the expression for the average BER of TDD MU-MIMO ZF systems with M-QAM modulated signals.
where γ is post-processing SNR.
It is observed from (13) and (18) that pb, k, i[m] is dependent on random matrix , so we need to calculate the expectation of pb, k, i[m] with respect to as follows.
where . So, pb, k, inow depends on the square σ i of each singular value of .
While it is difficult to obtain a closed-form expression for (22), the integral is fairly straightforward to evaluate numerically, at least when min(M, N) is small (in practical communication systems, the number of antennas of BS is at most eight at present), so it is valuable for the design of the adaptive modulation and coding scheme in practical communication systems. Moreover, since the BER expression includes the parameters related to channel conditions (e.g., Doppler frequency shift, uplink pilot SNR, CSI delay length, etc.) and the parameters related to system configurations (e.g., modulation mode, symbol duration, number of BS antennas, number of UE antenna, etc.), it provides the hints for people to optimize the MU-MIMO performance in TDD systems from different perspectives.
The BER function is only determined by three parameters including c k , N, and M. Hence, we can summarize the impact of imperfect CSI as follows.
1. Increase BER
As Md T s Fd, kincreases or γp, kdecreases, |ρ k | decreases, so c k decreases and in turn increases. In other words, system performance degrades when the Doppler shift is high, or when the SNR of pilot symbols is low.
2. Error floor
If CSI is perfect and γs, k→ ∞, then c k → ∞ and . However, if CSI is imperfect and γs, k→ ∞, then and , which means c k approaches an upper-bound and the BER thus exhibits an error floor when γs, kis high, further increases in γs, kgain nothing. This error floor worsens as |ρ k | decreases, i.e., as the channel estimation error or CSI delay of user k increases.
4. Simulation results
f c (GHz)
Md (in symbol)
1, 2, ..., 42
CSI delay in symbol for different downlink data symbol
Number of antennas of each user
Number of antennas of BS
Number of user
Number of antennas of all users
Pre-processing data SNR
Pre-processing pilot SNR
Mobile velocity of user
4, 16, 64
4QAM, 16QAM and 64QAM are used respectively for modulation
The above integral can be evaluated with numerical calculation software, e.g., Matlab (2009a), Mathematica, etc.
1. Without CSI delay and without estimation error: 0 km/h and γp = ∞
There are no CSI delay and no estimation error between CSI and downlink transmission channel. It is the perfect CSI case which is as the comparison baseline for other three cases.
2. Without CSI delay and with estimation error: 0 km/h and γp = γs
There is no CSI delay but exists estimation error between CSI and downlink transmission channel.
3. With CSI delay and without estimation error: 10 km/h and γp = ∞
There exists CSI delay but is no estimation error between CSI and downlink transmission channel.
4. With CSI delay and with estimation error: 10 km/h and γp = γs
There are both CSI delay and estimation error between CSI and downlink transmission channel.
One can see that the simulation curves match the analytical ones very well, demonstrating the correctness of our average BER expression. It is also observed that, the BER increases as the channel estimation error and/or CSI delay (or mobile velocity) increase(s), and an error floor is evident at high SNR for the cases with CSI delay, which agrees with our summary about imperfect CSI impact. Furthermore, one can find that channel estimation error causes array gain loss by comparing the curves of the same mobile velocity but with different pilot SNR while CSI delay causes diversity gain loss by comparing the curves of the same pilot SNR but with different mobile velocities. Moreover, CSI delay causes more performance degradation at high SNR than channel estimation error as the latter diminishes when the SNR is high.
In this article, we have investigated the BER of TDD downlink MU-MIMO ZF systems in the presence of imperfect CSI. By exploiting the correlation between the actual channel and the estimated one as well as channel time-correlation, we have developed the novel BER expression for TDD downlink MU-MIMO systems with M-QAM-modulated signals. Furthermore, we find that CSI delay and channel estimation error degrade system performance and even cause error floor, among which channel estimation error causes array gain loss while CSI delay causes diversity gain loss. At high SNR, CSI delay causes more performance degradation than channel estimation error. Especially, our research is valuable for the design of the adaptive modulation and coding scheme as well as the optimization of MU-MIMO systems. Numerical simulations have verified our theoretical analysis.
This paper was supported jointly by China Middle&Long term project "Next generation wideband wireless communications network"(2010ZX03002-003), National Nature Science Foundation of China (No. 60872017, No. 60832009), important National Science & Technology Specific projects (No. 2010ZX03003-002-03, No. 2011ZX03003-001-03), and Chinese National Programs for high technology research development project (No.2009AA011505), and important National Science & Technology Specific Projects (No. 2011ZX03003-001-03)
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