A low computational complexity V-BLAST/STBC detection mechanism in MIMO system
- Jin Hui Chong^{1},
- Chee Kyun Ng^{2, 3}Email author,
- Nor Kamariah Noordin^{3} and
- Borhanuddin Mohd Ali^{3}
https://doi.org/10.1186/s13673-014-0002-1
© Chong et al.; licensee Springer 2014
Received: 12 January 2013
Accepted: 28 January 2014
Published: 18 June 2014
Abstract
The idea of multiple antenna arrays has evolved into multiple-input multiple-output (MIMO) system, which provides transmit and receive diversities. It increases robustness of the effect of multipath fading in wireless channels, besides yielding higher capacity, spectral efficiency and better bit error rate (BER) performance. The spatial diversity gain is obtained by transmitting or receiving multiple copies of a signal through different antennas to combat fading and improves the system BER performance. However, the computational complexity of MIMO system is inevitably increased. Space-time coding (STC) technique such as Alamouti’s space-time block code (STBC) that combines coding, modulation and signal processing has been used to achieve spatial diversity. Vertical Bell Laboratories Layered Space-Time (V-BLAST) uses antenna arrays at both the transmitter and receiver to achieve spatial multiplexing gain. Independent data streams that share both frequency bands and time slots are transmitted from multiple antennas and jointly detected at the receiver. The theoretical capacity of V-BLAST increases linearly with the number of antennas in rich scattering environments. It's well-known that maximization of spatial diversity gain leads to degradation of spatial multiplexing gain or vice versa. In order to achieve spatial multiplexing and diversity gains simultaneously, the V-BLAST/STBC scheme has been introduced. This hybrid scheme increases MIMO system capacity and maintains reliable BER performance at the same time. However, both V-BLAST and STBC layers, in this hybid scheme, assume each other as an interferer. Thus, the symbols must be decoded with a suitable detection mechanism. In this paper, a new low complexity detection mechanism for V-BLAST/STBC scheme based on QR decomposition, denoted as low complexity QR (LC-QR) decomposition, is presented. The performance of the proposed LC-QR decomposition detection mechanism in V-BLAST/STBC transceiver scheme is compared with other detection mechanisms such as ZF, MMSE and QR decomposition. It is shown that the BER performance in V-BLAST/STBC scheme is better than V-BLAST scheme while its system capacity is higher than orthogonal STBC scheme when the LC-QR decomposition detection mechanism is exploited. Moreover, the computational complexity of proposed LC-QR decomposition mechanism is significantly lower than other abovementioned detection mechanisms.
Keywords
MIMO V-BLAST STBC MMSE ZF QR decomposition Spatial diversity gain Spatial multiplexing gainIntroduction
Conventional single-input single-output (SISO) system, which is a wireless communication system with a single antenna at the transmitter and receiver, is vulnerable to multipath fading effect. Multipath is the arrival of the multiple copies of transmitting signal at the receiver through different angles, time delay or differing frequency (Doppler) shifts due to the scattering of electromagnetic waves. Each copy of the transmitted signal will experience differences in attenuation, delay and phase shift while travelling from the transmitter to the receiver. As a result, constructive or destructive interference is experienced at the receiver. The random fluctuation in signal level, known as fading [1, 2], can severely affect the quality and reliability of wireless communication. Strong destructive interference will cause a deep fade and temporary failure of communication due to severe signal power attenuation. Moreover, the constraints posed by limited power, capacity and scarce spectrum make the design of SISO with high data rate and reliability extremely challenging.
The use of multiple antennas at the receiver and transmitter in a wireless network is rapidly superseding SISO to provide higher data rates at longer ranges especially for Long Term Evolution (LTE) systems [3] without consuming extra bandwidth or power. It is also a solution to the capacity limitation of the current wireless systems. The idea of multiple antennas has evolved into multiple-input multiple-output (MIMO) system, which provides transmit and receive diversities. It increases robustness of the effect of multi-path fading in wireless channels, besides yielding higher capacity, spectral efficiency and better bit error rate (BER) performance over conventional SISO systems in multipath fading environments [1, 4]. However, the revolution of SISO to MIMO causes the computation complexity to be increased.
Vertical Bell Laboratories Layered Space-Time (V-BLAST) uses antenna arrays at both the transmitter and receiver to achieve spatial multiplexing gain. Independent data streams that share both frequency bands and time slots are transmitted from multiple antennas and jointly detected at the receiver. The theoretical capacity of V-BLAST increases linearly with the number of antennas in rich scattering environments [5]. The spatial diversity gain is obtained by transmitting or receiving multiple copies of a signal through different antennas. This scheme is designed to combat fading and improves the system BER performance. Space-time coding (STC) technique such as space-time trellis code (STTC) that combines coding, modulation and signal processing has been used to achieve spatial diversity [6]. It achieves maximum diversity and coding gain but the system computational complexity increases exponentially with transmission rate. Alamouti’s space-time block code (STBC) [7] is another technique used to reduce the computational complexity in STTC. It supports linear decoding complexity for maximum-likelihood (ML) decoding. Orthogonal space-time block code (O-STBC), which is a generalization of the Alamouti’s scheme to an arbitrary number of transmit antennas, was introduced in [8].
However, it was shown in [9] that there is a trade-off between spatial diversity gain and spatial multiplexing gain of MIMO systems. For instance, while the V-BLAST scheme increases spatial multiplexing gain, but it does not provide any spatial diversity gain. The V-BLAST scheme is more susceptible to multipath fading and noise compared to Alamouti’s STBC scheme as there is no redundant information. Besides, the error propagation in V-BLAST detection causes BER performance degradation and limits the potential capacity of the V-BLAST scheme [10]. Although Alamouti’s STBC provides full transmit and receive antenna diversity, the maximum code rate of one can be achieved for two transmit antennas only. For more than two antennas, the maximum possible code rate is 3/4 [11], thus Alamouti’s STBC could not satisfy the demand of the desired high system capacity in real time system with good quality of service (QoS) [12, 13]. Therefore, in order to achieve spatial multiplexing and diversity gains simultaneously, the hybrid MIMO system has been introduced [14–16]. One of the hybrid MIMO systems is V-BLAST/STBC scheme. However, this hybrid scheme will further induce inevitably higher computational complexity in designing the system.
The V-BLAST/STBC scheme, which was introduced in [14] is a combination of the Alamouti’s STBC and V-BLAST schemes. A number of research efforts on V-BLAST/STBC scheme have been carried for MIMO system with the goal of maximizing the system capacity and reducing its computational complexity. The V-BLAST/STBC scheme improves the performance of MIMO by combining spatial multiplexing and diversity techniques together [17]. However, the spatially-multiplexed V-BLAST and STBC layers in the V-BLAST/STBC scheme assume each other as an interferer. Therefore, the transmitted symbols must be decoded with well-known detection mechanisms such as zero-forcing (ZF), minimum mean-squared error (MMSE) and QR decomposition which are employed in V-BLAST scheme [18]. Thus, the lowest computation complexity detection mechanism will be preferred. In this paper, a new detection mechanism based on QR decomposition, denoted as low complexity QR (LC-QR) decomposition, is presented. The QR decomposition of A × B channel matrix H is a factorization H = QR, where Q is A × B unitary matrix and R is B × B upper triangular matrix. The computational implementation of QR decomposition is less than ZF and MMSE [19], thus the computational complexity of V-BLAST/STBC scheme can be further reduced by using the proposed LC-QR decomposition detection mechanism.
The performance of V-BLAST/STBC transceiver scheme with proposed LC-QR decomposition mechanism is compared with V-BLAST and Alamouti’s STBC schemes. It is shown that the BER performance of V-BLAST/STBC scheme is better than V-BLAST scheme while the system capacity of V-BLAST/STBC scheme is higher than STBC scheme when the LC-QR decomposition mechanism is exploited. Moreover, the computational complexity of proposed LC-QR decomposition mechanism is significantly lower than traditional ZF, MMSE and QR decomposition detection mechanisms. Since MIMO scheme is considered as the latest multiple access technique for the next generation human computer interaction (HCI) [20] or mobile computing devices, especially in LTE-Advanced system, higher computational complexity inherited with it is inevitable. Any computational complexity reduction mechanism can further reduce the computational cost and power consumption. Therefore, by using the proposed LC-QR decomposition detection mechanism in V-BLAST/STBC MIMO scheme the system performance is not only significantly improved but the computational complexity of the overall system is also significantly reduced.
The rest of the paper is organized as follows. In Section System models, an overview of the hybrid V-BLAST/STBC system model is presented with its traditional ZF, MMSE and QR decomposition decoder mechanisms in the sub-sections. Then, the proposed new LC-QR decomposition detection mechanism is introduced in Section LC-QR Decomposition Mechanism. In Section The Computational Complexity of LC-QR Decomposition Compared with ZF, MMSE and QR Decomposition Detection Mechanisms, the computational complexity comparison of LC-QR decomposition with other detection mechanisms is discussed. The system capacity and probability of error in V-BLAST/STBC scheme with LC-QR decomposition are examined in Sections System Capacity of V-BLAST/STBC Scheme with LC-QR Decomposition and Probability of Error in V-BLAST/STBC Scheme with LC-QR Decomposition respectively. Section Performance Evaluation of LC-QR Decomposition in V-BLAST/STBC Scheme illustrates the system performance of proposed LC-QR decomposition in V-BLAST/STBC scheme. Finally, this paper concludes in Section Conclusions.
System models
The V-BLAST/STBC scheme, which was introduced in [14, 17], is a combination of the Alamouti’s STBC and V-BLAST schemes. It provides spatial diversity gain for high priority data and spatial multiplexing gain for low priority data simultaneously by partitioning a single data stream into two parallel sub-streams according to the data priority. The high priority data (e.g. frame header, I-frame, P-frame) is assigned to the STBC layer for extra protection while low priority data (e.g. B-frame, best-effort data) is sent to V-BLAST layer with higher capacity. Since high priority data is more important than low priority data, the corruption of high priority data will severely affect the real time service quality. For instance, error of any missing data in the first enhancement of P-frame layer is propagated to the subsequent P-frames, and it significantly degrades the perceived MPEG video quality. In contrast, any data loss in the B-frame layer affects only the corresponding frame, as it is not referred by other frames for decoding.
From (5), it can be easily seen that the obtained system is equivalent to a spatial multiplexing scheme. The spatially multiplexed V-BLAST and STBC layers in the V-BLAST/STBC scheme assume each other as interferer. Therefore, the transmitted symbols can be decoded with well-known detection techniques such as ZF, MMSE and QR decomposition which are employed in V-BLAST scheme [18]. The ZF and MMSE techniques involve the computation of Moore-Penrose pseudo-inverse of a matrix with cubic computational complexity. Beside MMSE and ZF, QR decomposition is also a common signal processing technique for MIMO detection [21]. The QR decomposition of A × B channel matrix H is a factorization H = QR, where Q is A × B unitary matrix and R is B × B upper triangular matrix. The computational implementation of QR decomposition is less than ZF and MMSE [19], thus the computational complexity of V-BLAST/STBC scheme can be reduced using QR decomposition. The brief overview of ZF, MMSE and QR decomposition decoder are presented in following sub-sections.
Zero-forcing (ZF) decoder
In an orthogonal channel matrix, ZF is identical to ML. However, in general ZF leads to noise amplification, which is especially observed in systems with the same number of transmit and receive antennas.
Minimum mean squared error (MMSE) decoder
QR decomposition decoder
LC-QR decomposition mechanism
Case A: N receive antenna is greater than or equal to M transmit antenna (N ≥ M)
The five steps of the LC-QR decomposition mechanism for case A (N ≥ M) are described as follows.
Step 1: Application of QR decomposition to the channel matrix H
Step 2: Calculate the first estimated symbols of STBC layer with STBC 2 × 2 decoder
Step 3: Decode the data symbols of V-BLAST layer based on the first estimated symbols of STBC layer
Step 4: Interference cancellation
Step 5: Decode the data symbols of STBC layer from the new modified received matrix with STBC 2 × N decoder
Case B: N receive antenna is less than M transmit antenna by 1 (N = M − 1)
The five steps of the LC-QR mechanism for case B (N = M − 1) are described as follows.
Step 1: Application of QR decomposition to the channel matrix H
Step 2: Calculate the first estimated symbols of the STBC layer with STBC 2 × 1 decoder
Step 3: Decode the data symbols of V-BLAST layer based on the first estimated symbols of STBC layer
Step 4: Interference Cancellation
Step 5: Decode the data symbols of STBC layer from the new modified received matrix with STBC 2 × (M − 1) decoder
The computational complexity of LC-QR decomposition compared with ZF, MMSE and QR decomposition detection mechanisms
The channel matrix H with dimension N × M is used to analyze the computational complexity of the LC-QR decomposition. The equivalent channel matrix $\overrightarrow{\mathit{H}}$ with dimension 2 N × 2(M − 1) is used to analyze the computational complexity of ZF, MMSE and QR decomposition. It is observed that there are zeros in channel matrix $\overrightarrow{\mathit{H}}$, therefore the multiplication and addition with zero are not taken into account in ZF and MMSE complexity calculation. The detail of computational complexity of each mechanism is presented in the following sub-sections.
Zero-Forcing (ZF) with $\overrightarrow{\mathit{H}}$
From (7), the process to calculate the ZF equalizer filter matrix W is divided into four steps.
Step 1 involves multiplication of ${\overrightarrow{\mathit{H}}}^{\mathit{H}}$ with $\overrightarrow{\mathit{H}}$, it requires 8 N(M − 1)^{2} multiplications and 4(2 N − 1)(M − 1)^{2} additions, so the number of complex arithmetic operations in step 1 is 4(4 N − 1)(M − 1)^{2}.
Step 2 involves Gaussian elimination matrix inversion of ${\overrightarrow{\mathit{H}}}^{\mathit{H}}\overrightarrow{\mathit{H}}$. According to [23], the computational complexity of Gaussian elimination matrix inversion with dimension a × b matrix is O(ab^{2}). Since the dimension of matrix ${\overrightarrow{\mathit{H}}}^{\mathit{H}}\overrightarrow{\mathit{H}}$ is 2(M − 1) × 2(M − 1), thus the number of complex arithmetic operations in step 2 is 8(M − 1)^{3}.
Step 3 performs multiplication of ${\left({\overrightarrow{\mathit{H}}}^{\mathit{H}}\overrightarrow{\mathit{H}}\right)}^{-1}$ with ${\overrightarrow{\mathit{H}}}^{\mathit{H}}$, it requires 8 N(M − 1)^{2} multiplications and 8 N(M − 1)^{2} − 4 N(M − 1) additions, then the number of complex arithmetic operations in step 3 is 16 N(M − 1)^{2} − 4 N(M − 1).
In step 4, the decoded symbols with ZF mechanism are calculated by multiplying equalizer filter matrix W with the receive vector V and it needs 4 N(M − 1) multiplications and 2(2 N − 1)(M − 1) additions. So, the number of complex arithmetic operations in step 4 is 2 (4 N − 1) (M − 1).
Finally, the total complex arithmetic operation of ZF V-BLAST/STBC is 8(M − 1)^{3} + 4(8 N − 1)(M − 1)^{2} + 2(2 N − 1)(M − 1).
Minimum mean-squared error (MMSE) with $\overrightarrow{\mathit{H}}$
From (9), the process to calculate the MMSE equalizer filter matrix D is divided into five steps.
Step 1 involves multiplication of ${\overrightarrow{\mathit{H}}}^{\mathit{H}}$ with $\overrightarrow{\mathit{H}}$, it requires 8 N(M − 1)^{2} multiplications and 4(2 N − 1)(M − 1)^{2} additions, so the number of complex arithmetic operations in step 1 is 4(4 N − 1)(M − 1)^{2}.
Step 2 performs the addition of ${\overrightarrow{\mathit{H}}}^{\mathit{H}}\overrightarrow{\mathit{H}}$. with σ^{2}I_{2(M − 1)}, it requires I_{2(M − 1)} additions, and then the number of complex arithmetic operations in step 2 is I_{2(M − 1)}.
Step 3 involves Gaussian elimination matrix inversion of $\left({\overrightarrow{\mathit{H}}}^{\mathit{H}}\overrightarrow{\mathit{H}}+{\mathit{\sigma}}^{2}{\mathit{I}}_{2\left(\mathit{M}-1\right)}\right)$. Since the dimension of matrix $\left({\overrightarrow{\mathit{H}}}^{\mathit{H}}\overrightarrow{\mathit{H}}+{\mathit{\sigma}}^{2}{\mathit{I}}_{2\left(\mathit{M}-1\right)}\right)$ is 2(M − 1) × 2(M − 1), thus the number of complex arithmetic operations in step 3 is 8(M − 1)^{3}[23].
Step 4 performs multiplication of ${\left({\overrightarrow{\mathit{H}}}^{\mathit{H}}\overrightarrow{\mathit{H}}+{\mathit{\sigma}}^{2}{\mathit{I}}_{2\left(\mathit{M}-1\right)}\right)}^{-1}$ with ${\overrightarrow{\mathit{H}}}^{\mathit{H}}$, it requires 8 N(M − 1)^{2} multiplications and 8 N(M − 1)^{2} − 4 N(M − 1) additions, then the number of complex arithmetic operations in step 4 is 16 N(M − 1)^{2} − 4 N(M − 1).
In step 5, the decoded symbols with MMSE mechanism are calculated by multiplying equalizer filter matrix D with the receive vector V and it needs 4 N(M − 1) multiplications and 2(2 N − 1)(M − 1) additions. So, the number of complex arithmetic operations in step 5 is 2 (4 N − 1) (M − 1).
Finally, the total complex arithmetic operation of MMSE V-BLAST/STBC is 8(M − 1)^{3} + 4(8 N − 1)(M − 1)^{2} + 4 N(M − 1).
Conventional QR decomposition with $\overrightarrow{\mathit{H}}$
Step 1 involves application of QR decomposition to the channel matrix $\overrightarrow{\mathit{H}}$ and multiplication of ${\overrightarrow{\mathit{Q}}}^{\mathit{H}}$ with V. According to [23], the computational complexity of QR decomposition by Householder reflection with size a × b matrix is O(ab^{2} − b^{3}/3). Since the dimension of channel matrix $\overrightarrow{\mathit{H}}$ is 2 N × 2(M − 1), thus the number of complex arithmetic operations for the QR decomposition of $\overrightarrow{\mathit{H}}$ being 8 N(M − 1)^{2} − (8/3)(M − 1)^{3}. Besides, for the multiplication of ${\overrightarrow{\mathit{Q}}}^{\mathit{H}}$ with V, it needs 2 N × 2 (M − 1) multiplications and (2 N − 1) × 2(M − 1) additions. Therefore, the number of complex arithmetic operations in step 1 is 8 N (M − 1)^{2} − (8/3)(M − 1)^{3} + 2(4 N − 1) × (M − 1).
Step 2 decodes the transmitted symbols with backward substitution with cancellation; it requires (M − 2) × (2 M − 3) additions, 2(M − 1) − 1 subtractions, (2 M − 3) × (M − 1) multiplications and 2(M − 1) divisions.
Finally, the total complex arithmetic operation of conventional QR decomposition is 8 N(M − 1)^{2} − (8/3)(M − 1)^{3} + (2 M − 3)^{2} + 2(M − 1)(4 N + 1) − 1.
LC-QR decomposition with H
Step 1 of LC-QR decomposition involves application of QR decomposition to the channel matrix H and multiplication of Q^{ H } with y. Since the dimension of channel matrix H is N × M, thus the number of complex arithmetic operations for the QR decomposition of H is NM^{2} − M^{3}/ 3. Besides, for the multiplication of Q^{ H } with you, it needs N × M multiplications and (N − 1) × M additions. Therefore, the number of complex arithmetic operations in step 1 is NM^{2} − M^{3}/ 3 + M(2 N − 1).
Step 2 of LC-QR decomposition calculates the first estimate symbols of the STBC layer with STBC 2 × 2 decoder. It needs six multiplications, three additions and one subtraction. So, the number of complex arithmetic operations in step 2 is ten.
Step 3 of the LC-QR decomposition decodes the data symbols of V-BLAST layer based on the first estimate symbols of STBC layer, so it requires (M − 2)(M − 1) additions, 2(M − 2) subtractions, (M − 2)(M + 1) multiplications and 2(M − 2) divisions. Therefore, the number of complex arithmetic operations in step 3 is 2 (M − 2) (M + 2).
Step 4 of LC-QR decomposition involves interference cancellation, thus it requires 2 N(M − 2) multiplications, 2 N(M − 3) additions and 2 N subtractions. Therefore, the number of complex arithmetic operations in step 4 is 4 N (M − 2).
In Step 5, it decodes the data symbols of the STBC layer from the new modified received matrix s with STBC 2 × N decoder, it requires 4 N multiplications, N + 2(N − 1) additions and N subtractions. Therefore, the number of complex arithmetic operations in step 5 is 2 (4 N − 1).
Finally, the total complex arithmetic operation of LC-QR is NM^{2} − M^{3}/ 3 + (M + 2)(2 N − 1) + 2(M − 2)(M + 2) + 4 N (M − 1) + 10.
Comparison of number of complex arithmetic operations for ZF, MMSE, QR decomposition and LC-QR decomposition detection mechanisms
Complex arithmetic operation | ZF with$\overrightarrow{\mathit{H}}$ | MMSE with$\overrightarrow{\mathit{H}}$ | QR decomposition with$\overrightarrow{\mathit{H}}$ | LC-QR with H |
---|---|---|---|---|
Addition | 4(4 N − 1) × (M − 1)^{2} − 2(M − 1) | 4(4 N − 1) × (M − 1)^{2} | 2(2 N − 1) × (M − 1) + (M − 2) × (2 M − 3) | N(3 M − 2) + (M − 3) × (M − 1) |
Subtraction | - | - | 2 M − 3 | 2(M − 2) + 2 N + 1 |
Multiplication | 4 N(M − 1) × (4 M − 3) | 4 N(M − 1) × (4 M − 3) | (M − 1) × (4 N + 2 M − 3) | M^{2} + 3MN − M + 4 |
Division | - | - | 2(M − 1) | 2(M − 2) |
Householder reflection | - | - | 8 N(M − 1)^{2} − (8/3)(M − 1)^{3} | NM^{2} − M^{3}/ 3 |
Gaussian elimination | 8(M − 1)^{3} | 8(M − 1)^{3} | - | - |
Total | 8(M − 1)^{3} + 4(8 N − 1)(M − 1)^{2} + 2(2 N − 1)(M − 1) | 8(M − 1)^{3} + 4(8 N − 1)(M − 1)^{2} + 4 N(M − 1) | 8 N(M − 1)^{2} − (8/3)(M − 1)^{3} + (2 M − 3)^{2} + 2(M − 1)(4 N + 1) − 1 | NM^{2} − M^{3}/ 3 + (M + 2)(2 N − 1) + 2(M − 2)(M + 2) + 4 N (M − 1) + 10 |
Reduction of computational complexity for ZF, MMSE and QR decomposition compared to LC-QR decomposition detection mechanisms
Complex arithmetic operation | Reduction of computational complexity compared to ZF and MMSE with$\overrightarrow{\mathit{H}}$(%) | Reduction of computational complexity compared to QR decomposition with$\overrightarrow{\mathit{H}}$(%) | ||||||
---|---|---|---|---|---|---|---|---|
M = 4 | M = 5 | M = 4 | M = 5 | |||||
N = 4 | N = 5 | N = 5 | N = 6 | N = 4 | N = 5 | N = 5 | N = 6 | |
Addition | 93.2 | 92.2 | 94.0 | 94.1 | 17.3 | 17.2 | 21.5 | 21.1 |
Subtraction | - | - | - | - | - | - | - | - |
Multiplication | 89.7 | 90.3 | 92.7 | 93.0 | - | - | 8.3 | 8.1 |
Division | - | - | - | - | 33.3 | 33.3 | 25.0 | 25.0 |
Householder reflection | - | - | - | - | 80.6 | 79.9 | 82.1 | 81.8 |
Gaussian elimination | - | - | - | - | - | - | - | - |
Total | 87.9 | 87.7 | 90.9 | 90.7 | 51.5 | 53.0 | 59.3 | 60.5 |
System capacity of V-BLAST/STBC scheme with LC-QR decomposition
Probability of error in V-BLAST/STBC scheme with LC-QR decomposition
Analytical model of V-BLAST layer
The instantaneous post-detection SNR is determined by the ${\mathrm{R}}_{\mathit{i}\phantom{\rule{0.25em}{0ex}}\mathit{i}}^{2}$. It is shown in [28] that the diversity gain of i-th detected sub-data stream layer of V-BLAST scheme is (N − i + 1). Thus, ${\mathrm{R}}_{\mathit{i}\phantom{\rule{0.25em}{0ex}}\mathit{i}}^{2}$ is a chi-square distribution with the degree of freedom 2(N − i + 1), which is denoted as ${\mathit{\chi}}_{2\left(\mathit{N}-\mathit{i}+1\right)}^{2}$. It can be observed that the when the i is larger, the diversity gain of the i-th layer becomes smaller. Consequently, the largest or M-th layer limits the overall performance of V-BLAST scheme at high SNR. Therefore, the overall diversity gain of V-BLAST scheme is N − M + 1 [29].
Analytical model of STBC layer
It can be seen from (51) that the received i-th sub-data stream is composed of STBC layer symbols, AWGN noise and the potential propagation error from V-BLAST layer. The equivalent noise is the combination of the last two parts.
Since h_{i,1} and h_{i,2} are circularly symmetric, i.i.d. Gaussian random variables with zero-mean and unit variance, the γ_{ S } has a chi-square distribution with 2 × 2 × N degrees of freedom.
From (19) and (20), the STBC layer of V-BLAST/STBC scheme is decoded with STBC 2 × N decoder after the V-BLAST layer is decoded. Therefore, the BER performance of V-BLAST layer dominates the BER performance of STBC layer. The statistical properties of BER performance of STBC layer are said to be conditional.
Analytical model of V-BLAST/STBC scheme
Performance evaluation of LC-QR decomposition in V-BLAST/STBC scheme
Symbol rate of various 4 × 4 MIMO schemes
MIMO scheme | Symbol rate (symbol/symbol transmission period) |
---|---|
ZF V-BLAST 4 × 4 | 4 |
O-STBC 4 × 4 | 3/4 |
V-BLAST/STBC 4 × 4 | 3 |
It can be concluded that Figures 4 and 5 present the tradeoffs among ZF V-BLAST, O-STBC and V-BLAST/STBC with LC-QR decomposition in 4 × 4 MIMO system. The O-STBC achieves the best BER performance, but the system capacity is the lowest among the considered schemes. Meanwhile, the system capacity of ZF V-BLAST with 10% outage capacity is the highest for SNR above 37 dB, but the BER performance is the worst among the considered schemes. On the other hand, the system capacity of V-BLAST/STBC with LC-QR decomposition is close to MIMO and better than ZF V-BLAST for SNR below 37 dB. Moreover, the BER performance of V-BLAST/STBC with LC-QR decomposition is significantly better than ZF V-BLAST as V-BLAST/STBC achieves spatial multiplexing and diversity gain simultaneously.
It is clear that V-BLAST layer performance dominates over final decisions of STBC layer in (25) and (26). With increasing SNR, the probability of error in decoding the V-BLAST layer is reduced, the probability of correct decoding is increased at the STBC layer. As the V-BLAST layer transmits four data symbols while STBC layer transmits two data symbols over two consecutive symbol transmission periods, thus a better BER performance of V-BLAST layer with LC-QR decomposition leads to overall V-BLAST/STBC system performance improvement.
Conclusions
In this paper, it is illustrated that V-BLAST/STBC scheme, which achieves spatial multiplexing and diversity gains simultaneously, increases system capacity and maintains reliable BER performance to accommodate the ever growing demand for real time system with tolerably lower QoS. It is also shown that the system capacity of LC-QR decomposition V-BLAST/STBC scheme is close to the ideal MIMO system and better than ZF V-BLAST for SNR below 37 dB. Moreover, the BER performance of LC-QR decomposition V-BLAST/STBC is significantly better than ZF-VBLAST. The LC-QR decomposition mechanism has also significantly reduced the arithmetic operation complexity and remains a satisfactory BER performance compared to ZF, MMSE and QR decomposition mechanisms. The reduction of computational complexity in V-BLAST/STBC MIMO scheme will see a significant reduction in computational cost and power consumption for next generation MIMO mobile computing devices.
Declarations
Authors’ Affiliations
References
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