A New Family of LowComplexity Decodable STBCs for Four Transmit Antennas
Abstract
In this paper we propose a new construction method for rate1 FastGroupDecodable (FGD) SpaceTimeBlock Codes (STBC)s for transmit antennas. We focus on the case of and we show that the new FGD rate1 code has the lowest worstcase decoding complexity among existing comparable STBCs. The coding gain of the new rate1 code is then optimized through constellation stretching and proved to be constant irrespective of the underlying QAM constellation prior to normalization. In a second step, we propose a new rate2 STBC that multiplexes two of our rate1 codes by the means of a unitary matrix. A compromise between rate and complexity is then obtained through puncturing our rate2 code giving rise to a new rate3/2 code. The proposed codes are compared to existing codes in the literature and simulation results show that our rate3/2 code has a lower average decoding complexity while our rate2 code maintains its lower average decoding complexity in the low SNR region at the expense of a small performance loss.
I Introduction
The need for lowcomplexity decodable STBCs is inevitable in the case of highrate communications over MIMO systems employing a number of transmit antennas higher than two. The decoding complexity may be evaluated by different measures, namely the worstcase decoding complexity measure and the average decoding complexity measure. The worstcase decoding complexity is defined as the minimum number of times an exhaustive search decoder has to compute the Maximum Likelihood (ML) metric to optimally estimate the transmitted symbols codeword [1, 2], or equivalently the number of leaf nodes in a search tree if a sphere decoder is employed, whereas the average decoding complexity measure may be numerically evaluated as the average number of visited nodes by a sphere decoder in order to optimally estimate the transmitted symbols codeword [3]. Arguably, the first proposed lowcomplexity rate1 code for the case of four transmit antennas is the QuasiOrthogonal (QO)STBC originally proposed by H. Jafarkhani [4] and later optimized through constellation rotation to provide full diversity [5, 6]. The QOSTBC partially relaxes the orthogonality conditions by allowing two complex symbols to be jointly detected. Subsequently, rate1, fulldiversity QOSTBCs were proposed for an arbitrary number of transmit antennas that subsume the original QOSTBC as a special case [7]. In this general framework, the quasiorthgonality stands for decoupling the transmitted symbols into two groups of the same size. However, STBCs with lower decoding complexity may be obtained through the concept of multigroup decodability laid by the S. Karmakar et al. in [8, 9]. Indeed, the multigroup decodability generalizes the quasiorthogonality by allowing more than two groups of symbols to be decoupled not necessarily with the same size.
However, due to the strict rate limitation imposed by the multigroup decodability, another family of STBCs namely Fast Decodable (FD) STBCs [1] has been proposed. These codes are conditionally multigroup decodable thus enabling the use of the conditional detection technique [10] which in turn significantly reduces the overall decoding complexity. Recently, STBCs that combine the multigroup decodability and the fast decodability namely the FastGroup Decodable (FGD) codes have been proposed [11]. These codes are multigroupgroup decodable such that each group of symbols is fast decodable. The contributions of this paper are summarized in the following:

We propose a novel systematic construction of rate1 FGD STBCs for transmit antennas. The rate1 FGD code for a number of transmit antennas that is not a power of two is obtained by removing the appropriate number of columns from the rate1 FGD code corresponding to the nearest greater number of antennas that is a power of two (e.g. the rate1 FGD STBC for three transmit antennas is obtained by removing a single column from the four transmit antennas rate1 FGD STBC).

We apply our new construction method to the case of four transmit antennas and show that the resulting new 44 rate1 code can be decoded at half the worstcase decoding complexity of the best known rate1 STBC.

We propose a new rate2 STBC through multiplexing two of the new rate1 codes by the means of a unitary matrix and numerical optimization. We then propose a significant reduction of the worstdecoding complexity at the expense of a rate loss by puncturing the rate2 code to obtain a new rate3/2 code.
We compare the proposed codes to existing STBCs in the literature and found through numerical simulations that our rate3/2 code has a significantly lower average decoding complexity while our rate2 code is decoded with a lower average decoding complexity at low SNR region. Performance simulations show that this reduction in the decoding complexity comes at the expense of a small performance loss.
The rest of the paper is organized as follows: The system model is defined and the families of lowcomplexity STBCs are outlined in Section II. In Section III we propose our scheme for the rate1 FGD codes construction for the case of transmit antennas, and then the FGD code construction method is applied to the case of four transmit antennas giving rise to a new 44 rate1 STBC. In Section IV, the rate of the proposed code is increased through multiplexing and numerical optimization. Numerical results are provided in Section V, and we conclude the paper in Section VI.
Notations:
Hereafter, small letters, bold small letters and bold capital letters will designate scalars, vectors and matrices, respectively. If is a matrix, then , and denote the hermitian and the transpose of , respectively. We define the as the operator which, when applied to a matrix, transforms it into a vector by simply concatenating vertically the columns of the corresponding matrix. The operator is the Kronecker product and the operator returns 1 if its scalar input is and 1 otherwise. The operator rounds its argument to the nearest integer. The operator concatenates vertically the real and imaginary parts of its argument. means modulo .
Ii Preliminaries
We define the MIMO channel inputoutput relation as:
(1) 
where is the number of channel uses, is the number of receive antennas, is the number of transmit antennas, is the received signal matrix, is the code matrix, is the channel matrix with entries , and is the noise matrix with entries . In the case of Linear Dispersion (LD) codes [14], a STBC that encodes real symbols is expressed as a linear combination of the transmitted symbols as:
(2) 
with and the are complex matrices called dispersion or weight matrices that are required to be linearly independent over . The MIMO channel model can then be expressed in a useful manner by using (2) as:
(3) 
Applying the operator to the above equation we obtain:
(4) 
where is the identity matrix. If , and designate the ’th column of the received signal matrix , the channel matrix and the noise matrix respectively, then equation (4) can be written in matrix form as:
(5) 
Thus we have:
(6) 
A real system of equations is be obtained by applying
the operator to the (6):
(7) 
where , and . Assuming that , the QR decomposition of yields:
(8) 
where ,, , is a real upper triangular matrix and is a null matrix. Accordingly, the ML estimate may be expressed as:
(9) 
where is the vector space spanned by information vector . Noting that multiplying a column vector by a unitary matrix does not alter its norm, the above reduces to:
(10) 
where .
In the following, we will briefly review the known families of lowcomplexity STBCs and the structures of their corresponding matrices that enable a simplified ML detection.
Iia Multigroup decodable codes
Multigroup decodable STBCs are designed to significantly reduce the worstcase decoding complexity by allowing separate detection of disjoint groups of symbols without any loss of performance. This is achieved iff the ML metric can be expressed as a sum of terms depending on disjoint groups of symbols.
Definition 1.
For instance if a STBC that encodes real symbols is group decodable, its worstcase decoding complexity order can be reduced from to with being the size of the used square QAM constellation. The worstcase decoding complexity order can be further reduced to if the conditional detection with hard slicer is employed. In the special case of orthogonal STBCs, the worstcase decoding complexity is as the PAM slicers need only a fixed number of arithmetic operations irrespectively of the square QAM constellation size.
IiB Fast decodable codes
A STBC is said to be fast decodable if it is conditionally multigroup decodable.
Definition 2.
A STBC that encodes real symbols is said to be FD if its weight matrices are such that:
(12) 
where is the set of weight matrices associated to the ’th group of symbols.
In this case the conditional detection may be used to significantly reduce the worstcase decoding complexity. The first step consists of evaluating the ML estimate of conditioned on a given value of the rest of the symbols that we may note by . In the second step, the receiver will have to minimize the ML metric only over all the possible values of . For instance, if a STBC that encodes real symbols is FD, its corresponding worstcase decoding complexity order for square QAM constellations is reduced from to . If the FD code is in fact conditionally orthogonal, the worstcase decoding complexity order is reduced to .
IiC Fast group decodable codes
A STBC is said to be fast group decodable if it is multigroup decodable such that each group is fast decodable.
Definition 3.
A STBC that encodes real symbols is said to be FGD if its weight matrices are such that:
(13) 
and that the weight matrices within each group are such that:
(14) 
where (resp. ) denotes the set of weight matrices that constitute the ’th group (resp. the number of inner groups) within the ’th group of symbols .
For instance, if a STBC that encodes real symbols is FGD, its corresponding worstcase decoding complexity order for square QAM constellations is reduced from to . Similarly, if each group is conditionally orthogonal, the worstcase decoding complexity order is equal to .
(17) 
(20)  
(21) 
Iii The proposed FGD scheme
Let the set denote the weight matrices of the square orthogonal STBC for transmit antennas [15]. The proofs of the following propositions are omitted due to space limitations.
Proposition 1.
Proposition 2.
The rate of the proposed family of FGD codes is equal to one complex symbol per channel use.
The weight matrices of our new FGD construction method for four and eight transmit antennas are listed in Table VI.
Tx  

4  
8  
A new rate1 FGD STBC for four transmit antennas
According to Table VI, the proposed rate1 STBC in the case of four transmit antennas denoted may be expressed as:
(16) 
According to Definition 3, the proposed code is a FGD STBC with and such as . Therefore, the worstcase decoding complexity order is . However, the coding gain of is equal to zero, in order to achieve the fulldiversity, we resort to the constellation stretching [12] rather than the constellation rotation technique, otherwise the orthogonal symbols inside each group will be entangled together which in turns will destroy the FGD structure of the proposed code and causes a significant increase in the decoding complexity.
The full diversity code matrix takes the form of (17) where and is chosen to provide a high coding gain. The term is added to normalize the average transmitted power per antenna per time slot.
Proposition 3.
Taking , ensures the NVD property for the proposed code with a coding gain equal to 1.
Iv The proposed rate2 code
The proposed rate2 code denoted is simply obtained by multiplexing two rate1 codes by means of a unitary matrix. Mathematically speaking, the rate2 STBC is expressed as:
where and are chosen in order to maximize the coding gain. It was numerically verified for QPSK constellation that taking and maximizes the coding gain which is equal to 1. To decode the proposed code, the receiver evaluates the QR decomposition of the real equivalent channel matrix (7). The corresponding uppertriangular matrix takes the form:
(18) 
where has no special structure, is an upper triangular matrix and takes the form:
(19) 
in which indicates a possible nonzero position. For each value of , the decoder scans independently all possible values of and , and assigns to them the corresponding 6 ML estimates of the rest of symbols via hard slicers according to (20)(21). where:
A rate3/2 code that we will denote may be easily obtained by puncturing the rate2 proposed code and may be expressed as:
V Numerical and simulation results
In this section, we compare our proposed codes to comparable lowcomplexity STBCs existing in the literature in terms of worstcase decoding complexity, average decoding complexity and Bit Error Rate (BER) performance over quasistatic Rayleigh fading channels. One can notice from Table VIII that the worstcase decoding complexity of the proposed rate3/2 code is half that of the punctured rate3/2 code in [16] and is reduced by a factor of w.r.t to the worstcase decoding complexity of the rate3/2 code in [17]. Moreover, the worstcase decoding complexity of our rate2 code is half that of the code in [16].
Code  Square QAM 

decoding complexity  
The proposed rate3/2 code  
The punctured P.SrinathS.Rajan rate3/2 code [16]  
The S.Sirianunpiboon et al. code [17]  
The proposed rate2 code  
The rate2 P.SrinathS.Rajan code [16] 
Simulations are carried out in a quasistatic Rayleigh fading channel in the presence of AWGN and 2 receive antennas for our rate3/2 and rate2 codes. The ML detection is performed via a depthfirst tree traversal with infinite initial radius SD. The radius is updated whenever a leaf node is reached and sibling nodes are visited according to the simplified SchnorrEuchner enumeration [18].
From Fig. 1, one can notice that the proposed rate3/2 code gains about 0.4 dB w.r.t to S.Sirianunpiboon et al. code [17] while it loses about 0.5 dB w.r.t to the punctured P.SrinathS.Rajan code [16] at BER. From Fig. 2, it can be noticed that the average complexity of our rate3/2 code is significantly less than that of the punctured P.SrinathS.Rajan code [16] and roughly equal to that of S.Sirianunpiboon et al. code [17]. From Fig. 3, one can notice that the proposed rate2 code loses about 0.8 dB w.r.t the P.SrinathS.Rajan code [16] at BER. However, from Fig. 4 one can notice that our proposed code maintains its lower average decoding complexity in the low SNR region.
Vi Conclusion
In the present paper we have proposed a systematic approach for the construction of rate1 FGD codes for an arbitrary number of transmit antennas. This approach when applied to the special case of four transmit antennas results in a new rate1 FGD STBC that has the smallest worstcase decoding complexity among existing comparable lowcomplexity STBCs. The coding gain of the proposed FGD rate1 code was then optimized through constellation stretching. Next we managed to increase the rate to 2 by multiplexing two rate1 codes through a unitary matrix. A compromise between complexity and throughput may be achieved through puncturing the proposed rate2 code which results in a new lowcomplexity rate3/2 code. The worstcase decoding complexity of the proposed codes is lower than their STBC counterparts in the literature. Simulations results show that the proposed rate3/2 code offers better performance that the S.Sirianunpiboon et al. code [17] but loses about 0.5 dB w.r.t the punctured P.SrinathS.Rajan code [16] at BER. The proposed rate2 code loses about 0.8 dB w.r.t the P.SrinathS.Rajan code [16] at BER. In terms of average decoding complexity, we found that the proposed rate3/2 code has a lower average decoding complexity while the proposed rate2 code maintains its lower average decoding complexity in the low SNR region.
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