- Open Access
Adaptive selection of antenna grouping and beamforming for MIMO systems
© Kim et al; licensee Springer. 2011
- Received: 27 November 2010
- Accepted: 2 November 2011
- Published: 2 November 2011
Antenna grouping algorithms are hybrids of transmit beamforming and spatial multiplexing. With antenna grouping, we can achieve combining gain through transmit beamforming, and high spectral efficiency through spatial multiplexing. In an independent identically distributed channel, the antenna grouping method has better bit error rate (BER) performance than the beamforming method. However, if the channel is correlated, then the BER performance of antenna grouping degrades. In that case, it is better to use beamforming instead of antenna grouping. In this article, we investigate the mode selection algorithms which select between the antenna grouping and the beamforming modes. By selecting a suitable mode for a given channel, we can achieve more robustness of the system performance. We introduce several mode selection criteria as well as a low complexity criterion which is derived from a low complexity antenna grouping algorithm. Simulation results show that the proposed mode selection algorithm performs better than the antenna grouping and the beamforming modes in various channel conditions.
- Singular Value Decomposition
- Channel State Information
- Mode Selection
- Minimum Euclidean Distance
- Spatial Multiplex
Multiple-input and multiple-output (MIMO) systems have been investigated extensively for their high spectral efficiency and reliable transmission of data [1, 2] over single-input and single-output (SISO) systems. Through multiple transmit antennas, we can transmit several independent data streams by spatial multiplexing mode. We can also send only one data stream by transmit beamforming or diversity modes. With spatial multiplexing, we can achieve high spectral efficiency, but the reliability of data transmission gets worse especially when there is a correlation between antennas. On the other hand, we can obtain combining gain (SNR gain) by sacrificing spectral efficiency in the beamforming mode.a
We assume an MIMO system which has N t transmit antennas and N r receive antennas. The availability of channel state information (CSI) at the transmitter helps to make the system more efficient . Beamforming is one of the strategies which use the CSI at the transmit side. By singular value decomposition (SVD), it divides MIMO channel into min(N t , N r ) SISO channels and transmits one data stream through the best SISO channel. It increases the received signal-to-noise ratio (SNR), and improves the reliability. Especially in a highly correlated channel, beamforming is the best transmit strategy for the bit error rate (BER) performance. But transmission of only one stream can make beamforming inefficient with respect to spectral efficiency. When the bit per channel use (BPCU) is fixed, the modulation order of beamforming tends to be higher than that of spatial multiplexing, and the BER performance will be degraded in an independent identically distributed (IID) channel. The eigenmode transmission also uses SVD to find precoding matrix. In the eigenmode transmission, min(N t , N r ) streams can be transmitted with adequate power allocation. To maximize the capacity, water-filling-based power allocation is optimal, while inverse water-filling minimizes the mean square error . General multi-mode precoding [5–7] can also be used, and it adapts the number of transmission streams to minimize the BER or maximize the capacity. In multi-mode precoding systems, each instantaneous channel prefers a particular mode.
Antenna grouping is a combination of beamforming and spatial multiplexing . We also introduced some antenna grouping criteria . When N t is larger than the N r , N t transmit antennas can be partitioned into N r groups. The antennas in each group are used for beamforming, and an independent data stream is transmitted in each group. In short, antenna grouping transmits N r independent data streams through partitioned beamforming. In this algorithm, we can improve the BER performance by achieving combining gain through beamforming, and multiplexing gain through spatial multiplexing. We assume that SVD is performed at the receiver instead of the transmitter, so that we need to feedback beamforming vector(s) or right unitary matrix of SVD instead of full CSI. Feedback information in antenna grouping is a beamforming vector (N t × 1 vector) plus additional antenna grouping information while required feedback information in eigenmode transmission is an N t × N r matrix. As the antenna correlation increases, the BER performance of antenna grouping gets worse, and beamforming is the best strategy as mentioned earlier. In case of an ill-conditioned channel (i.e., the condition number of a matrix is large), the BER performance of antenna grouping may not be better than that of beamforming because we cannot send N r streams through an ill-conditioned channel. In average, an ill-conditioned channel occurs more frequently in a correlated channel. This is why beamforming is the best strategy in a highly correlated channel.
To overcome performance degradation in an ill-conditioned or a correlated channel, we propose to use mode selection for each instantaneous channel. We only consider beamforming and antenna grouping as the two possible modes in the mode selection algorithm to limit the feedback information and the complexity. Eigenmode transmission requires an N t × N r matrix feedback which is much more than beamforming and antenna grouping. Multi-mode precoding systems are complex because it considers all possible numbers of data transmission streams. The rest of this article is organized as follows. In Section 2, we provide the system model. We review the antenna grouping algorithms and introduce mode selection criteria in Sections 3 and 4. The proposed antenna grouping algorithm and the mode selection criterion are presented in Sections 3.5 and 4.5, respectively. Section 5 provides simulation results, and conclusions are given in Section 6.
where R and T are receiver and transmitter antenna correlation matrix, respectively. , the (i,j)th element of H w , is modeled as an independent and identical complex Gaussian distribution with 0 mean and unit variance. When the channel has IID Rayleigh fading, R and T in (1) is an identity matrix I. The noise n is an AWGN vector with variance of 's.
The only difference between (5) and (7) is dimension of matrices. The dimensions of W AG and x AG are N t × N r and N r × 1, respectively. But the dimension of w B is N t × 1, and x B is a scalar. The H AG is N r × N r matrix, whereas h B is N r × 1 vector.
There are several antenna grouping techniques, which were introduced in .
Sum capacity of sub-channels (Algorithm A1)
Minimum euclidean distance of received constellations (Algorithm A2)
where X is the set of all possible transmitted vector x AG . We consider all possible effective channel H AG 's in (5). We calculate the minimum Euclidean distance of receive constellation for every possible H AG , and find the optimal sub-channel H i 's and the optimal W AG that maximize (10).
Minimum singular value of effective channel (Algorithm A3)
A MIMO channel can be decomposed into multiple SISO channels by SVD, and the received SNR is proportional to the squared singular value of a channel. The BER performance is thus dominated by the minimum singular value. We find the minimum singular value of each H AG , and pick the best H i 's and W AG which maximize the minimum singular value of H AG .
Effective channel capacity (Algorithm A4)
We can then select the grouping and the precoding matrix which maximize (11).
Based on normalized instantaneous channel correlation matrix (Algorithm A5)
In (13), if the amplitude of r13 is large, then it means that the first and the third columns of H are more correlated than the other pairs. This can be interpreted as the correlation between the transmit antennas 1 and 3 is large.
For simplicity, we consider only the antenna grouping where the size of each group is 2, which we call (2, 2) grouping. The possible antenna grouping cases are (1,2//3,4), (1,3//2,4) and (1,4//2,3). We then compare (|A| + |F|), (|B| + |E|), and (|C| + |D|). If (|A| + |F|) is the largest, it means that the correlation between transmit antennas 1 and 2 and between transmit antennas 3 and 4 is larger than the others so we group (1, 2) and (3, 4) together which are denoted by (1,2//3,4). Similarly, if (|B|+|E|) is the largest, then we use the grouping of (1,3//2,4). If (|C|+|D|) is the maximum, then we use the grouping of (1,4//2,3). The advantage of this algorithm is that it reduces the search complexity significantly. This antenna grouping algorithm can be extended to any MIMO system where N t is an integer multiple of N r . The BER performance of this algorithm is compared to other criteria in , and it is very close to others.
As we compared each possible group with a certain criterion in antenna grouping, we can compare two modes (antenna grouping and beamforming) with a similar criterion. In this section, we provide several mode selection criteria similar to those of antenna grouping.
Minimum Euclidean distance of received constellations (Algorithm M1)
where S is the set of all possible transmitted signals x B , λmax(H) is the maximum singular value of H, and dmin,bis the minimum Euclidean distance of the transmit beamforming constellation. The second equality is because s i 's are scalars in beamforming. If (10) is larger than (15), then we select the antenna grouping mode, and vice versa.
Range of minimum distance (Algorithm M2)
where λmin(H AG ), λmax(H AG ) are minimum and maximum singular values of H AG . dmin ,agis the minimum Euclidean distance of the transmit constellation in antenna grouping. In (15), we can easily calculate the minimum Euclidean distance of received beamforming constellation. As in , we compare and . If the former is larger than the latter, we select the antenna grouping mode, and vice versa.
Effective channel capacity (Algorithm M3)
If C AG is larger than C B , then we select the antenna grouping mode, and vice versa.
Condition number of channel matrix (Algorithm M4)
Based on NICCM (Algorithm M5)
Figure 4 shows the BER performance of mode selection algorithms with the A2 antenna grouping algorithm in highly correlated channels where the angle of departure (AOD) is 45°, and the angular spread (AS) is 6°. In this simulation, the BPCU is 8. We use 16-QAM in the antenna grouping and the eigenmode transmission methods, and 256-QAM in the beamforming mode. In the high SNR region, the M1 algorithm has the best BER performance, and the others have similar performance. The proposed algorithm (M5) has performance similar to the M2 and the M4 algorithms while its complexity is lower.
In an MIMO system with more transmit antennas than receive antennas, we can improve the BER performance by antenna grouping which is a hybrid form of transmit beamforming and spatial multiplexing. But antenna grouping is not always the best strategy. Using mode selection techniques, we can get robust performance irrespective of channel variation. We proposed mode selection techniques between transmit beamforming and antenna grouping for a given channel. If the channel is not ill conditioned, then we can get multiplexing gain by selecting the antenna grouping mode. When the channel is ill conditioned, we can prevent BER degradation by selecting the transmit beamforming mode. In this article, we introduce several mode selection criteria which are similar to the criteria of antenna grouping, and propose a low complexity mode selection criterion. Simulation results show that the proposed mode selection algorithm performs better than the antenna grouping and the transmit beamforming methods in various channel conditions.
a In this article, the beamforming mode refers to the transmit beamforming mode.
This research was supported in part by the Basic Science Research Program (KRF-2008-314-D00287, 2010-0013397), the Mid-career Researcher Program (2010-0027155) through the NRF funded by the MEST, Seoul R&BD Program (JP091007, 0423-20090051), the INMAC, and the BK21.
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