Fast Maximum-Likelihood Decoder for 4*4 Quasi-Orthogonal Space-Time Block Code

This letter introduces two fast maximum-likelihood (ML) detection methods for 4*4 quasi-orthogonal space-time block code (QOSTBC). The first algorithm with a relatively simple design exploits structure of quadrature amplitude modulation (QAM) constellations to achieve its goal and the second algorithm, though somewhat more complex, can be applied to any arbitrary constellation. Both decoders utilize a novel decomposition technique for ML metric which divides the metric into independent positive parts and a positive interference part. Search spaces of symbols are substantially reduced by employing the independent parts and statistics of noise. Finally, the members of search spaces are successively evaluated until the metric is minimized. Simulation results confirm that the proposed decoder is superior to some of the most recently published methods in terms of complexity level. More specifically, the results verified that application of the new algorithm with 1024-QAM would require reduced computational complexity compared to state-of-the-art solution with 16-QAM.


I. INTRODUCTION UASI-ORTHOGONAL
space-time block codes (QOSTBCs) are noteworthy nowadays due to their desired performances and pairwise detections.However, complexity of pairwise maximum-likelihood (ML) decoder rises drastically when size of constellation is increased.A survey of related literature shows that much research has been underway to develop an approach that reduces complexity of QOSTBC decoder [1]- [7].Authors of [1] employ QR decomposition and sorting to simplify detection of QOSTBC with four transmit antennas.This method achieves ML performance and offers low complexity.QOSTBCs with minimum decoding complexity are proposed in [2] with degraded error performance compared to conventional QOSTBC.A suboptimum decoder is presented in [3] based on sorted QR decomposition and real-valued representation of [2].The decoder unveiled in [3] offers near-ML error A. Ahmadi is with the Department of Electrical Engineering, Shahid Bahonar University of Kerman, Kerman, Iran (e-mail: adel.ahmadi@ieee.org).
S. performance and its complexity is independent of constellation size.In [4], a low complexity ML decoder is introduced on the basis of QR and real valued lattice representation.For rotated QOSTBC with four transmit antennas and quadrature amplitude modulation (QAM) constellation, [5] explores a fast scheme with ML error performance and reduced complexity.In [6], a different structure is proposed for QOSTBCs decoding with three or four transmits antennas.This method achieves a near-ML performance with low complexity decoder.A new 4×4 QOSTBC is reported in [7] that employs precoder and rotated symbols.The complexity of decoder is low and a reasonable suboptimum error performance is obtained when rotation angles are optimized.In [8], a suboptimum fast decoding algorithm is investigated for block diagonal QOSTBC with arbitrary transmit antennas.
The innovative methods developed in this letter decompose the received vector into two pairs of symbols which are detected independently.To do this, the ML metric minimization of each pair is transformed into sum of independent positive parts and an interference part.It should be noted that the independency between parts facilitates detection by helping to considerably limit search spaces of the symbols.The candidates placed in relevant partial search area are gradually evaluated and then transmitted symbols are estimated by computing interference between them.If the search areas are small and no symbol is detected, then they are extended and evaluation is repeated till transmitted symbols are detected.The novel ML metric decomposition studied in this letter boasts two important features, namely; decomposition is not a highly complex process, and more significantly, most of the decomposed parts are independent of each other.The first proposed method utilizes the structure of QAM to further reduce complexity of detection and the second method, which is relatively more complex, can be applied to any arbitrary constellations.Both of these new algorithms offer the desired ML performance.
The rest of the letter is organized as follows.Section II describes a wireless communication system based on 4×4 QOSTBC.The proposed fast ML detection methods are divulged in Section III.This is followed by Section IV which briefly covers simulation set up and compares the results against those of other fast methods.The letter ends with a conclusion in Section V.
where , and the data symbols , , belong to constellation .A space-time communication system that transmits four symbols over four time slots can be equivalently represented for th receive antenna as: , where , and .The equivalent received vector, equivalent additive white Gaussian noise (AWGN) vector and equivalent channel matrices are represented by , and , respectively.The signal-to-noise ratio (SNR) is indicated by , and and denote equivalent received signal and noise at the th time slot of the th receive antenna, respectively.The AWGN has complex Gaussian distribution with zero mean and unit variance.The equivalent channel for the th receive antenna can be defined as: where stands for channel fade between the th transmit antenna and the th receive antenna such that: .
The complete received vectors can be concatenated as: ,

III. FAST MAXIMUM-LIKELIHOOD DECODER
The ML decoder should minimize the following norm in order to estimate the transmitted symbols: .
The above minimization can be rewritten as: , (7) where is defined as: .
The matrix can be decomposed into: , where , , .
Further, the 2×2 identity matrix is represented by and , , , and .
By employing ( 9)-( 11) and doing some math operations, we are able to simplify as: , . ( On the other hand, can be computed with fewer multiplications by utilizing the above relations, yielding .(15) By substituting (12) into , we obtain , ( Based on ( 16), the ML decoder can detect two pairs and independently: , . ( 19) For the remaining part of this section, we focus on detection of noting that the other pair can be detectable by applying a similar approach.
The ML metric of (18) can be expanded as: , (20) where denotes the real part of .Based on (13) and ( 14), and therefore we can rewrite minimization of (20) as minimization of sum of three positive parts: , (21) where and stands for the signum function.In (21), the first part and the second part are independent of each other which helps to reduce the search space, and the purpose of the third part is to present interference between the other parts and thus lead to detection of and .
In subsection A, a fast ML decoder is presented for QAM constellations and in subsection B a fast ML method is introduced that deals with arbitrary constellations.For the proposed algorithms, algorithm initialization and detection complexity are covered in subsections C and D, respectively.

A. Fast ML Detection for QAM Constellations
In this subsection, we investigate detection of QAM constellations.For the sake of simplicity and clarity, let consider a square -QAM constellation but the solution can also be straightforwardly extended to any rectangular QAM constellations.Under this scenario, the real and imaginary parts of symbols belong to .
Two independent parts of (21) can be expanded to a summation of their real and imaginary parts: , where and stand for the real and imaginary parts of , respectively.The decoder searches within to find the best choices (i.e. and ) for and , when .By analyzing (22) and assuming that the minimum of ( 21) is smaller than , we have: , (23) , (24) where .The above inequalities help us to discard inappropriate members of by comparing them with the real and imaginary parts of and .Therefore, the decoder initially selects certain members of , which are located within intervals and , when .Then, the selected positions are evaluated step-bystep to extract and which minimize the ML metric.
The proposed algorithm that detects for an -QAM can be summarized as follows: Step Step 11: Go back to Step 6.
Step 12: If and are not still obtained, increase the values of and and repeat Steps 3 to 12; Otherwise, the final result of detection is .In the following subsection, a fast ML detector is examined for arbitrary constellations.

B. Fast ML Detection of Arbitrary Constellations
For arbitrary constellation with points, we cannot exploit lattice structure and therefore decoding calls for primary form of (20) and ( 21).The proposed technique can be summarized as follows: Step 1: Compute , and by employing (10)-( 15) and adjust to an appropriate value, then define , , and .
Step 3: Select a new complex value of from , whose real and imaginary parts are located within and , respectively.If there isn't any new point, go to Step 11.
Step 4: Set , and .If , go to the previous step and select another point.
Step 5: From , select a new complex value of , whose real and imaginary parts are located within and , respectively.If there isn't any new point, go to Step 3.
Step 6: Set and .If , go to the previous step and select another point. Step

C. Algorithm Initialization
The proposed methods need to begin with an appropriate initial value for which should be suitably selected to avoid unnecessary complexity.From the definition (8), can be rewritten as: , (25) where is the transmitted vector and is complex Gaussian noise vector: .
The mean of is and its covariance can be simplified as: , where and .If the ML decoder correctly identifies the transmitted symbols, then minimum of ML has occurred for transmitted symbols.Under this condition, the part of minimum metric becomes equal to for .Most of the times, the absolute of is smaller than its standard deviation multiplied by four and therefore is an appropriate choice for initializing.
On the other hand, can also be expressed as where is the minimum distance between two distinct constellation points.In this letter, is used for simulation because it involves lower level of complexity.

D. Complexity of proposed methods
The decoder follows a procedure that can be divided into pre-computation and search stages.The initial value of variables and candidate points are obtained by engaging about relational operator, real additions, 6), 16-QAM ML of (6), 64-QAM ML of (6), 256-QAM ML of (6), 1024-QAM Prop.MLs, 16-QAM Prop.MLs, 64-QAM Prop.MLs, 256-QAM Prop.MLs, 1024-QAM Talebi is with the Department of Electrical Engineering, Shahid Bahonar University of Kerman, Kerman, Iran and the Advanced Communications siamak.talebi@uk.ac.ir).