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Introduction
The demand for communication systems that effectively exploit the wireless channel's limited capacity (Telatar, 1999) has rapidly grown during the last decade. In recent years, multiple-input, multiple-output (MIMO) systems have emerged as an attractive technique to increase the bit rate without increasing neither power nor bandwidth resources. A MIMO system employs multiple antennas, both at the transmitter and the receiver, adding an extra degree of freedom in the design of communication systems. In particular, two techniques have been developed to take advantage of MIMO systems: Spatial Multiplexing and Diversity Transmission. The first technique aims at increasing the spectral efficiency. One of its main proponents is Sorted QR decomposition (SQRD), which was first introduced in (Wubben et al., 2001). The second technique is aimed at increasing diversity gain. This was achieved by the Space-Time Block Codes (STBC) (Tarokh et al., 1999). The implementation of STBC decoders is relatively easy to carry out, but one drawback is that their spectral efficiency is low. A popular scheme that reaches full diversity and full-rate was proposed by Alamouti in (Alamouti, 1998).
SQRD can attain very high spectral efficiency while maintaining a very low complexity of their receiver. However, because it does not obtain the optimal detection order, and use a scheme of Successive Cancelation Interference (SIC), it suffers great degradation due to the error propagation, in his decision feedback. In (Raza et al., 2004) a scheme called KIL-VBLAST was proposed, whose main goal is diminishing the error propagation between layers, it sacrifices spectral efficiency because it transmits a known symbol in the receiver. In this case, they propose that it belongs at the first sub-stream to detect, with this proposal the total diversity of V-BLAST is increased achieved a better performance, their principal disadvantage is that need hardware resources to transmit known information. In (Jiang et al., 2005), a scheme called GMD-VBLAST was proposed, these scheme uses a joint transceiver in the receiver and the transmitter based on the Geometric Mean Decomposition (GMD) .
It achieves a better performance than V-BLAST, maintaining their high spectral efficiency. However, the complexity of the receiver and transmitter is increased considerably, and it also needs Channel State Information (CSI) to work, therefore their hardware implementation can be complicated and expensive.
An alternative approach, known as hybrid coding (HC) (Mao, Motani, 2005), (Meng, Tuqan, 2007), (Cortez, et al., 2007), (Longoria, et al., 2007), (Bazdresch, et al., 2012) consists in the selection of some of the available transmitting antennas to work in STBC mode, while the remaining ones operate as V-BLAST. In particular, the STBC-VBLAST scheme (Mao, Motani, 2005), (Meng, Tuqan, 2007) is an interesting example of hybrid coding, as it uses n s Alamouti layers and n s spatial layers, increasing spectral efficiency over pure STBC, and its structure allows using an OSIC scheme decoding based on the QR decomposition. The principal disadvantage of (Meng, Tuqan, 2007) is that it has high receiver complexity and for (Mao, Motani, 2005) is that it needs two kinds of decoders in the receiver: STBC decoder and V-BLAST decoder. Also, the spectral efficiency of the system is greatly diminished when the number of Alamouti encoders is increased in the transmitter.
This paper presents a modification to the hybrid coding system proposed in (Mao, Motani, 2005). The scheme consists of only one Alamouti space-time block code unit in the last layer, plus n s antennas operating as V-BLAST in the transmitter. We obtained an equivalent channel matrix using the Linear Dispersion Space-Time Codes (LDCs) techniques proposed in (Hassibi, Hochwald, 2002), with this rearrangement we can achieve the benefits of the schemes described previously (Wubben, et al., 2001), (Raza, et al., 2004).
The receiver is based on scheme proposed in (Wubben, et al., 2001). It refers to this new architecture as ZF-SQRD-LDSTBC, the estimation and detection of the transmitted symbols are carried out of identical way as in SQRD (Wubben, et al., 2001). The Symbols exhibiting diversity gain (Alamouti Layer) are detected first, followed by the ordered spatially-multiplexed symbols.
The scheme proposal increased the total diversity of the system and mainly of the first layer to detect, and since the receiver is based on the scheme proposed in (Wubben, et al., 2001), in this case, an efficient detector with low complexity was proposed, a better performance is obtained, and no CSI is needed in this case. Therefore, an implementation can be carried out in hardware at lower costs. The implementation of the sorted QR decomposition was done using Modified Gram-Schmidt algorithm (MGS) (Golub, Van Loan, 1996). The scheme exploit the special matrix structure that comes from the LDCs representation to improve the lower complexity of the receiver proposed in (Wubben, et al., 2001).
This proposal was compared with the schemes proposed in (Wubben, et al., 2001), (Raza, et al., 2004). The results obtained shown that, at the same spectral efficiency, this proposal outperforms both schemes in terms of achieved Bit Error Rate (BER), with the same number of receiver and transmitter antennas. With respect to the scheme proposed in (Jiang, et al., 2005), this scheme achieves a lower performance but with a less computational complexity. In the next sections, the system model, method and results were presented.
Method
A space-time block code (STBC) is a mapping from a vector of symbols to a space- time code matrix that specifies how symbols are transmitted over the available antennas and time intervals. The STBC-VBLAST linear space-time block code transmits
a. Linear dispersion codes (LDC)
An LDC codeword is defined as a matrix S given by:
Where we have assumed that
b. Channel model
The propagation channel between each transmitting and receiving antenna can be modeled as a Rayleigh narrowband stationary stochastic process. Also this work considers that the channel is scatterer-rich at both the transmitter and receiver ends.
The MIMO channel can be modeled as a random matrix H of size
c. ZF-SQRD-LDSTBC Transmitter
ZF-SQRD-LDSTBC employs
The received signal is represented by the
Where N is a matrix of complex Gaussian random noise variables with zero mean and independent real and imaginary parts with variance
d. ZF-SQRD-LDSTBC Receiver
Under the assumption that the Channel State Information (CSI) is perfectly known by the receiver and in presence of Gaussian noise, the detection and decoding process of the transmitted signal vector S, at the
In Equation 3, and from now on, the subscript indicate the correspondent transmitter or receiver antenna, and the super-indices indicate the correspondent block emission time t. Where S is defined as:
And its respective blocks have the structure indicated as follows:
And
In Equations 5 and 6, the matrix at the left indicates the symbol arrangement in the antennas and the notation introduced in the right matrix has been elaborated to simplify the explanation of the detection algorithm. Note that the matrix
The system Equation 3 can be reformulated as a LDSTBC code according to (Hassibi, Hochwald, 2002). The resultant expression is
The Equation 7 can be reformulated in compact form as:
Where
Where
For
Where each element of Equation 11 is given by:
For
A matrix structure similar to Equation 11 is shown in (Longoria, et al., 2007). The reformulation of Equation 8 of the system Equation 3, lead to the next rearrangement of matrix S
Where
And
The reformulation of Equation 3 as Equation 8 allows to consider the MIMO system as an equivalent version with nsym transmit antennas, ignoring any distinction between STBC block and V-BLAST layers.
In this way, we propose a modified and optimized SQRD that can be applied directly over
where
e. ZF-SQRD-LDSTBC as Linear dispersion space-time code
The transmission matrix
The dispersion matrices Aand B for m=2 and
For
Where
And for the Almaouti layer
f. Modified SQRD
For the detection of the nsym transmitted symbols, is necessary to calculate the QR decomposition of
By taking advantage of the structure of
G. norm(x) est calculate
In order to decrease the complexity of the QR decomposition and because the elements of have a distribution Gaussian, we can use an estimator in the calculation of the energy of H to obtain the detection order, such as was described in (Kim, et al., 2006). We have found that the performance is the same using the estimator or the exact equation, therefore a meaningful saving of flops is achieved. The equation to calculate the exact energy of a column vector x is
The estimator that we used to calculate the energy of a column vector x is
The operations required in the estimator, no multiplications are needed but sums, therefore a saving in complexity is achieved without loss of performance in our proposed scheme.
Simulations Results
A. BER Performance
To demonstrate the advantages of the proposed scheme, we have performed several simulations to compare the BER performance of different MIMO systems under the mentioned conditions, employing 16-QAM and 32-QAM modulation schemes. Throughout this paper, the block length is considered fixed to
Table II Spatial Code Rate
| SCHEME | DATA RATE |
|---|---|
| SQRD 4x4 16-QAM | 16 |
| KIL-BLAST 4x4 16-QAM | 12 |
| GMD-VBLAST 4x4 16-QAM | 16 |
| ZF-SQRD-LDSTBC 4x4 16-QAM | 12 |
| ZF-SQRD-LDSTBC 4x4 32-QAM | 15 |
As can be seen from Figure 2, the proposal improves around 7.5dB and 10.5dB with respect to that of SQRD when 32-QAM and 16-QAM are used, respectively for a
B. Complexity analysis
To compare the complexity for the different schemes, we only have considered the number of arithmetical operations necessary to obtain the QR decomposition. Besides, we have assigned 1 flop per sum or multiplication of real numbers executed for the algorithms, and for the square-root and divisions we took the number of flops assigned in (Bazdresch, et al., 2012), which are 8 flops per division of real numbers, and 30 flops for the square root. The results for schemes of different size are presented in Table III.
Table III Complexity comparison
| SCHEME | Number of additions/multiplications of real numbers |
|---|---|
| SQRD 4x4 | 320/628 |
| SQRD 6x6 | 1152/1728 |
| KIL-BLAST 4x4 | 272/600 |
| KIL-BLAST 6x6 | 972/1548 |
| ZF-SQRD-LDSTBC 4x4 | 240/539 |
| ZF-SQRD-LDSTBC 6x6 | 984/1471 |
As might be seen, our proposal improves around twenty percent with respect to the scheme proposed in (Wubben, et. al., 2001). This improvement is mainly due to the use of the estimator for the energy in the order detection. With respect to KIL-BLAST, the complexity to calculate the QR decomposition is the same to our proposal.
Conclusions
In this work we have presented the ZF-SQRD-LDSTBC hybrid space-time codes, along with a receiver algorithm with very low complexity. This receiver, is based on the theory of linear dispersion codes (Hassibi, Hochwald, 2002), it might be implemented in hardware with CORDIC using Givens rotations. The results showed that the BER performance of these codes is better than other purely spatial codes with the same spatial code rate that have been proposed recently (A. Raza, et. al, 2004), including without feedback required. We also have proposed a method to transmit with the same power all the symbols.
