Improved QR Decomposition-Based SIC Detection Algorithm for MIMO System

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INTRODUCTION
Spatial Multiplexing Multiple-Input Multiple-Output (MIMO) systems with rich scattering wireless channels can provide enormous capacity improvements without increasing the bandwidth or transmitted power.Channel capacity linearly increases with the number of antennas.In MIMO system, a serial data stream is converted to parallel, with each data symbol transmitted on a separate antenna.In order to decode symbols corrupted by inter-antenna interference, efficient signal detection algorithms for MIMO systems have attracted much interest in recent years.Many detection techniques had been proposed such as zero forcing linear detection, minimum mean-square error linear detector, Maximum Likelihood (ML) detection, Bell-Labs layered space-time detection, etc.
Among these schemes, ML detection is the optimized detection algorithm when considering the performance of Bit-Error Rate (BER), but the computational complexity of ML detection increases exponentially with the number of transmit-antennas and the order of modulation which makes it difficult to use ML detection in practical system.Linear detection algorithms have low computation complexity comparing with ML detection, but the performance is also lower than that of ML detection.A lot of efforts have been put into the search of detection algorithms achieving ML or near-ML performance with lower complexity.
The popular reduced-complexity alternative technique is the V-BLAST detection algorithm and QR decomposition-based successive interference cancellation detection algorithm.For these detection algorithms, incorrect symbol detection in the early layers will create errors in the following layers, so they are easy to suffer from the error propagation problem.The following layers depend highly on the results of the first detected layers (Foschini and Gans, 1998;Telatar, 1999;Zheng et al., 2011;Seethaler and Bolcskei, 2010;Zhang et al., 2012;Chen and Sheen, 2011;Chen and Yu, 2007;Chen et al., 2007;Xu et al., 2008).
In order to improve the detection performance, the first detected layers should be cancelled perfectly.Some improved detection algorithms have been proposed in the literatures.Two kinds of detection method are combined together, or different searching methods like tree search are used in MIMO detection.Ordered QR decomposition of channel matrix can be utilized to reduce the complexity of detection at the cost of lowing performance.Inspired by this idea, we introduce a new sorted method which can obtain the best detection order according to the rule of SNR without calculations of pseudo-inverses (Wolniansky et al., 1998;Golden and Foschini, 1999;Chang et al., 2010;Kyeong et al., 2008).
On the other hand, due to the error propagation, the overall performance is determined by the first layers to be decoded, we proposed an improved method to increase the validity of the first layers detection (Lin and Wu, 2011;Myung-Sun et al., 2009;Chang and Chung, 2012;Goldberger and Leshem, 2011;Studer and Bolcskei, 2010).After performing improved sorted QR decomposition of the channel matrix, the ML detection of symbols with length L is performed and the partial accumulated metrics are calculated and ordered, which gives a sequence set of the first L layers, from the sequence, we select the first T sequences from the set with the smallest partial accumulated metrics, then successive interference cancellation algorithm is used to search the left layers (Jia et al., 2008;Kang et al., 2008;Wübben et al., 2001).Finally, the minimum accumulated metric symbol is put out as the transmitted symbols.The proposed scheme provides 2 trade-off parameters with better performance and low complexity.

SYSTEM MODEL AND ORIGINAL ALGORITHM
Description of system model: Here we consider the un-coded MIMO system consisting of M transmitted antennas at the transmitter and N antennas at the receiver (M≤N ).The wireless channel is assumed to be quasi-static, so that the channel remains constant during a block of certain length and changes to an independent realization for the next block.The received signal vector can be represented as: where,  n : The noise symbols (.) H : Vector complex conjugate transposition the symbols x m (t) and the noise n n (t) are mutually uncorrelated, zero-mean random processes with variances E{|x m (t)| 2 } = 1 and E{|n n (t) 2 |} = σ 2 .The element h ij of H(t) represents channel gains between the j th transmitter and the i t receiver antennas at the discrete time t.As the channel has slowly time varying flat fading and the channel state information H is known perfectly in the receiver, for brevity of notation the discrete time index t is abandoned in the subsequent consideration so the received signal vector can be described as: Maximum likelihood detection algorithm: For detecting transmitted symbols from the received signals, Maximum Likelihood (ML) detection of the transmitted signal can be formulated as finding: ML detection is an exhaustive searching which performs searching over the whole alphabet that the computational complexity is very high.It's very difficult to put ML detection into a practical system.

QR decomposition based Sucssicive Interference Cancellation (QR-SIC):
With perfect state information of channel H, the ordered QR-decomposition of channel matrix is obtained firstly, H = QR, here Q is a N×M unitary matrix, Q H Q = I, R is M×M upper triangular matrix.After left-multiplying received signal by: where, y and n are 1 M × and n have the equal statistical properties.
So the ML detection problem can be reformulated as: where, R i, j : The (i, j) th component of R : The absolute value The vector x depends on the constellation size of modulated signal and the number of transmit antennas.

V-BLAST:
The classical V-BLAST detection algorithm is proposed by researchers of BELL Labs.It can be summarized as 4 steps: nulling, slicing, canceling and recurrence: 1; ; arg min ( ) ; ; arg min ( ) ; 1 where, (G i ) j = The k th i row of G i In this detection algorithm, at each time instant, instead of jointly decoding the signals from all the transmit antennas, V-BLAST decodes the "strongest" signal firstly, then cancels the effect of this strongest transmit signal from each of the received signals and then proceeds to decode the "strongest" of the remaining transmit signal and so on.It is the optimum detection order, but there needs to calculate pseudoinverse in each iteration, so the calculating complexity is much higher than that of QR-SIC.

Proposed algorithm: Improved Sorted QR decomposition (ISQR):
The detection order in V-BLAST becomes important to all the overall performance of the system.The optimal detection order is from the strongest to the weakest signal since it can minimize the error propagation from one step of detection to the next.
Estimated covariance matrix is written as follows: The SNR of i th layer can be described: where, , [.] From here, we notice that optimized ordered rule of SNR is equivalent to the order of row 2-norm of R.
In decomposition of the channel matrix, the layer of minimal ||(R) i || 2 should be put first and order of ||(R) i :|| 2 should be sorted from minimum to maximum.Therefore, the order rule can satisfies the lowest layer of maximum SNR.
So after performing QR decomposition of H, the column order of R is rearranged by the sort of 2-norm of R. The permutation matrix is named P. Then the QR decomposition is performed with the re-arranged R .Now, the optimum detection ordered can be obtained through first QR decomposition, re-sorted and second QR decomposition and without calculating pseudoinverses of the channel matrix H.The propose algorithm (named as ISQR) can use the modified Gram-Schmidt method, householder transformation and Givens transformation to further reduce calculating complexity.
Improved scheme to decrease the error of the first detection layers: Due to interference nulling and error propagation, the performance of V-BLAST is far from that of optimal ML detection scheme.We propose an improved algorithm combined the ML detection and SIC by setting 2 adjustable parameters and the trade-off between performance and complexity can be adjusted by setting the 2 parameters at different values.
In order to improve the detection performance of the first detected layers, referencing the idea of ML detection, we set the first parameter L and L layers (these layer's numbers are M, M-1,… L+1, respectively) are performed exhaustive search in the step after improved sorted QR decomposition.Then calculating the partial accumulated metrics as: Ω L Ω = The set of modulation constellation Ω = The number of constellation points Now we get |Ω| L available candidate paths.Here we need to sort these sequences with the partial accumulated metrics.Without loss of generality that the symbol with lower index has smaller metrics in: d(1)≤d( 2)≤…≤d(λ).
After these preparations, we set another parameter T, then T candidate signal sequences with smaller partial accumulated metrics d(1)≤d( 2)≤…≤d(T)) are selected and the searching for left layers from M-L, M-L-1,…, 1 can be performed by SIC algorithm according to the ordered set mentioned above.
Then we get L candidate signal sequences with length M. The accumulated metrics are calculating for all the L candidates as: 2 1 ˆ= arg min( ) At last, the candidate signal sequence with the smallest accumulated metrics is quantization and decision as the transmitted signal.

SIMULATION EXPERIMENTS
In the computer simulation, we consider a MIMO antenna system.Channel is assumed to be independently Rayleigh faded and quasi-static.Under correlated channels, ZF criteria are used at the receiver side.Following in the previous discussion, the performance is measured in term of BER for a frame of 1000 bits from QPSK constellations averaged over 100 frames.We compare detection performance of proposed sorting method with QR-SIC and V-BLAST.
The BER curves are verified in Fig. 1 and 2.
Figure 1 shows the performance of various detection algorithms with M = 4, N = 4, antennas MIMO system.As expected, the ISQR algorithm's performance near that of V-BLAST and outperform OSIC algorithm.Comparing with V-BLAST, its complexity is much lower.
Figure 2 shows the performance with M = 6, N = 8, antennas MIMO system.The advantage of the proposed algorithm becomes obviously with the increasing number of antennas.
Figure 4 shows the BER performance in case of L = 2, T = 2; L = 4, T = 2; and L = 4, T = 4 with 16-QAM.When L at the same value, the performance better with T increasing.With the increasing of the 2 parameter, the detection performance is close to that of ML detection and the calculating complexity is much lower than ML detection.The trade-off performance can be adjusted by setting L and T at different value.The bigger of value, the performance more close to ML detection with increasing complexity.

Calculating complexity analysis:
Comparing with V-BLAST and QR-SIC, ISQR algorithm needs 2 times QR decomposition and the second QR decomposition is performed with upper triangular matrix which has many 0 elements, so its complexity is still O(M) 3 , a little higher than QR-SIC and much lower than that of V-BLAST, which complexity is O(M) 4 .
Using ISQR in SIC, the proposed algorithm performs P layers ML detection, calculating and sorting the partial accumulated metrics, selecting L partial sequences with smaller partial accumulated metrics to do SIC detection, so its complexity will change with parameters P and L. The detail can be summarized as: (H H H) -1 H H denotes the Moore-Penrose pseudo-inverse w ki = The weight vector which is chosen depending on the criterion of ZF or MMSE