Low Complexity Decoding of the 4×4 Perfect Space-time Block Code

Procedia Computer Science 32 ( 2014 ) 223 – 228

1877-0509 © 2014 The Authors. Published by Elsevier B.V. Open access under CC BY-NC-ND license. Selection and Peer-review under responsibility of the Program Chairs.

doi: 10.1016/j.procs.2014.05.418

ScienceDirect

The 5th International Conference on Ambient Systems, Networks and Technologies (ANT-2014)

Low Complexity Decoding of the 4ൈ4 Perfect Space-Time Block Code

Elie Amani

a

, Karim Djouani

a,b

and Anish Kurien

a

a_{FSATI/Dept. of Electrical Engineering, Tshwane University of Technology (TUT), Pretoria, 0001, South Africa} b_{Université Paris-Est Créteil (UPEC), 12/LISSI Lab, Créteil, France}

Abstract

The 4ൈ4 perfect space-time block code (STBC) is one type in a family of perfect STBCs that have full rate, full diversity, a non-vanishing constant minimum determinant that improves spectral efficiency, uniform average transmitted energy per antenna, and good shaping. These codes suffer very high complexity maximum likelihood (ML) decoding. The exhaustive ML decoding of the 4ൈ4 perfect STBC for instance has complexity of O(N16_{), with N being the size of the underlying QAM constellation. This paper} suggests a fast decoding algorithm for the 4ൈ4 perfect STBC that reduces decoding complexity from O(N16_{) to O(N}8_{). The} algorithm is based on conditional minimization of the ML decision metric with respect to one subset of symbols given another. This low complexity decoding approach is essentially ML because it suffers no loss in symbol error rate (SER) performance when compared to the exhaustive ML decoding.

© 2014 The Authors. Published by Elsevier B.V.

Selection and peer-review under responsibility of Elhadi M. Shakshuki.

Keywords: Low complexity decoding; maximum likelihood decoding; multiple-input multiple-output (MIMO); perfect space-time block code.

1. Introduction

Space-Time Block Coding is one of the most extensively used MIMO transmission schemes in wireless communications. This scheme allows transmission of redundant signals from numerous antennas at different time slots, and thus mitigates the effects of multipath fading. The first full-rate, full-diversity, orthogonal STBC to be discovered was the Alamouti Code [1] which uses two transmit antennas, achieves rate-one and low complexity ML decoding. Many orthogonal and quasi-orthogonal STBCs have been constructed that allow more antennas at the transmitter [2], [3], [4], [5], [6]. Perfect STBCs were introduced by Oggier et al. [7] to achieve higher data rates and suit next generation wireless communications. These codes outperform the Alamouti Code and all orthogonal and quasi-orthogonal STBCs in that they have full rate, full diversity, a non-vanishing constant minimum determinant, uniform average transmitted energy per antenna, and good shaping [7]. However, they suffer very high decoding © 2014 The Authors. Published by Elsevier B.V. Open access under CC BY-NC-ND license.

complexity which grows exponentially with the number of transmit antennas and the constellation size. Researchers have applied conditional optimization to reduce ML decoding complexity of the 2ൈ2 and 3ൈ3 perfect STBCs and other multiplexed orthogonal designs [8]. This technique is based on optimization of a subset of symbols conditioned on the remaining symbols. It reduces the 2 ൈ 2 perfect STBC (Golden Code [9]) ML decoding complexity from O(N4_{) to O(N}2_{) [10] and the 3ൈ3 perfect STBC ML decoding complexity from O(N}9_{) to O(N}6₎ [11], where N is the QAM or HEX constellation size.

This paper applies conditional optimization to the decoding of the 4ൈ4 perfect STBC. A low complexity algorithm that takes advantage of the structure of the 4ൈ4 perfect STBC and its similarity with the Golden Code is derived. The proposed algorithm reduces ML decoding complexity of the 4ൈ4 perfect STBC from O(N16_{) to O(N}8₎ without loss in SER performance. The rest of the paper is organised as follows. The structure and transmission scheme of the 4ൈ4 perfect STBC are described in section 2. In Section 3, the proposed low complexity decoding algorithm is presented. Section 4 discusses experimental results. Finally, conclusions are provided in section 5. 2. The 4ൈ4 perfect STBC

The 4ൈ4 perfect STBC transmits 16 complex information symbols, from an N-QAM constellation, over four time slots from four antennas. The transmit codewords of the 4ൈ4 perfect STBC are matrices of the form:

 ൌ ଵ൦ šଵ šଶ Œšସ šଵ šଷ šସ šଶ šଷ Œšଷ Œšସ Œšଶ Œšଷ šଵ šଶ Œšସ šଵ ൪ ൅ ଶ൦ šହ š଺ Œš଼ šହ š଻ š଼ š଺ š଻ Œš଻ Œš଼ Œš଺ Œš଻ šହ š଺ Œš଼ šହ ൪ ൅ ଷ൦ šଽ šଵ଴ Œšଵଶ šଽ šଵଵ šଵଶ šଵ଴ šଵଵ Œšଵଵ Œšଵଶ Œšଵ଴ Œšଵଵ šଽ šଵ଴ Œšଵଶ šଽ ൪ ൅ ସ൦ šଵଷ šଵସ Œšଵ଺ šଵଷ šଵହ šଵ଺ šଵସ šଵହ Œšଵହ Œšଵ଺ Œšଵସ Œšଵହ šଵଷ šଵସ Œšଵ଺ šଵଷ ൪         where: ଵൌ ሺͳ െ ͵Œሻସ൅ Œʣଶ ଶൌ ሺͳ െ ͵Œሻʣ ൅ Œʣଷ ଷൌ െŒସ൅ ሺെ͵ ൅ ͶŒሻʣ ൅ ሺͳ െ Œሻʣଷ ସൌ ሺെͳ ൅ Œሻସെ ͵ʣ ൅ ʣଶ൅ ʣଷ ʣ ൌ †‹ƒ‰ሺɅଵǡ Ʌଶǡ Ʌଷǡ Ʌସሻǡ Ʌ୩ൌ ʹ…‘•ሺ ʹ୩_Ɏ ͳͷሻ

Let hlk be the channel gain from transmit antenna l to receive antenna k. Assuming that the channel state

information is available at the receiver, the received signal is given by:

 

r

HX



n

      where: Hൌ ൦ Šଵଵ Šଶଵ Šଵଶ Šଶଶ Šଷଵ Šସଵ Šଷଶ Šସଶ Šଵଷ Šଶଷ Šଵସ Šଶସ Šଷଷ Šସଷ Šଷସ Šସସ

൪ is the channel matrix, X is the codeword in Eq. 1 and n is a matrix of complex Gaussian noise variables with zero-mean and variance ો૛_Ǥ

Let ܛ ൌ ሾšଵ šଶšଷ šସšହ š଺š଻ š଼ሿ and ܋ ൌ ሾšଽ šଵ଴šଵଵ šଵଶšଵଷ šଵସšଵହ šଵ଺ሿ ; Eq. 2 can be

rewritten as:

ܚ ൌ ܛۯ ൅ ܋۰ ൅ ܖ, (3)

ۯ ൌ _» ¼ º « ¬ ª 2 2 2 2 1 1 1 1 F C G H F C G H ۰ ൌ _» ¼ º « ¬ ª 4 4 4 4 3 3 3 3 F C G H F C G H m

H , G_m,C_m and F_m are the induced channel matrices from the four transmit antennas to the 1st_{, 2}nd_{, 3}rd _{and 4}th

receive antenna respectively.

m H ൌ ൦ ˜୫ଵŠଵଵ ˜୫ଶŠଶଵ Œ˜୫ସŠସଵ ˜୫ଵŠଵଵ ˜୫ଷŠଷଵ ˜୫ସŠସଵ ˜୫ଶŠଶଵ ˜୫ଷŠଷଵ Œ˜୫ଷŠଷଵ Œ˜୫ସŠସଵ Œ˜୫ଶŠଶଵ Œ˜୫ଷŠଷଵ ˜୫ଵŠଵଵ ˜୫ଶŠଶଵ Œ˜୫ସŠସଵ ˜୫ଵŠଵଵ ൪ m

G

ൌ ൦ ˜୫ଵŠଵଶ ˜୫ଶŠଶଶ Œ˜୫ସŠସଶ ˜୫ଵŠଵଶ ˜୫ଷŠଷଶ ˜୫ସŠସଶ ˜୫ଶŠଶଶ ˜୫ଷŠଷଶ Œ˜୫ଷŠଷଶ Œ˜୫ସŠସଶ Œ˜୫ଶŠଶଶ Œ˜୫ଷŠଷଶ ˜୫ଵŠଵଶ ˜୫ଶŠଶଶ Œ˜୫ସŠସଶ ˜୫ଵŠଵଶ ൪ m

C

ൌ ൦ ˜୫ଵŠଵଷ ˜୫ଶŠଶଷ Œ˜୫ସŠସଷ ˜୫ଵŠଵଷ ˜୫ଷŠଷଷ ˜୫ସŠସଷ ˜୫ଶŠଶଷ ˜୫ଷŠଷଷ Œ˜୫ଷŠଷଷ Œ˜୫ସŠସଷ Œ˜୫ଶŠଶଷ Œ˜୫ଷŠଷଷ ˜୫ଵŠଵଷ ˜୫ଶŠଶଷ Œ˜୫ସŠସଷ ˜୫ଵŠଵଷ ൪ m

F

ൌ ൦ ˜୫ଵŠଵସ ˜୫ଶŠଶସ Œ˜୫ସŠସସ ˜୫ଵŠଵସ ˜୫ଷŠଷସ ˜୫ସŠସସ ˜୫ଶŠଶସ ˜୫ଷŠଷସ Œ˜୫ଷŠଷସ Œ˜୫ସŠସସ Œ˜୫ଶŠଶସ Œ˜୫ଷŠଷସ ˜୫ଵŠଵସ ˜୫ଶŠଶସ Œ˜୫ସŠସସ ˜୫ଵŠଵସ ൪

ܚ ൌ  ሾ”ଵ ”ଶሿ contains two signal vectors and each vector is a combinations of signals received from a group of two

antennas during the four transmission time slots. Thus,

”ଵൌ  ሾ”ଵଵ ”ଵଶ”ଵଷ ”ଵସ”ଶଵ ”ଶଶ”ଶଷ ”ଶସሿ and ”ଶൌ  ሾ”ଷଵ ”ଷଶ”ଷଷ ”ଷସ”ସଵ ”ସଶ”ସଷ ”ସସሿ,

where rlk is the received signal at antenna l in time slot k.

The likelihood function is:

’ሺܚȁܛǡ ܋ሻ ן ‡š’(െ ଵ

ଶો૛ԡܚ െ ܛۯ െ ܋۰ԡ

ଶ_{), (4)}

where ԡǤ ԡ denotes Euclidian norm and the ML solution corresponding to Eq. 4 is

ሺܛොǡ ܋Ƹሻ ൌ  ƒ”‰‹ୱǡୡ஫ୗఴԡܚ െ ܛۯ െ ܋۰ԡଶ (5)

The low complexity algorithm in the next section is derived by maximizing the likelihood function in Eq. 4 with respect to ܋given ܛ or vice-versa to obtain the new likelihood function as:

’ሺܚȁܛǡ ܋ሻ ן ‡š’ ቀെ ଵ

ଶો૛ሺܚ െ ܛۯሻሺ۷૚૟െ ۰

ା_ሺ۰۰ା_ሻିଵ_{۰ሻ ൈ ሺܚ െ ܛۯሻ}ା_ቁ

ൈ ‡š’ ቀെ_ଶોଵ૛ሺ܋ െ ܋෤ሺܛሻሻ۰۰ାሺ܋ െ ܋෤ሺܛሻሻାቁ, (6)

3. Low Complexity Decoding Algorithm

The conditional optimization algorithm divides the symbols to be decoded into two subsets. Then one subset is chosen to be conditioned on the other i.e. estimated based on the other then quantized. This avoids exhaustive search on the constellation because the subset that is conditioned on the other is not chosen on the constellation during decoding but rather estimated based on the subset that is chosen on the constellation. There are two possible decoding solutions. The best alternative is the one that will result in the most accurate quantization (rounding off to the nearest constellation points). The best choice is made by comparing the determinants of covariance matrices ۯۯା _{and ۰۰}ା_{. The largest determinant will determine the solution to be chosen. Thus, the low complexity}

algorithm can be summarized as follows: If ݀݁ݐሺۯۯା_{ሻ ൒ ݀݁ݐሺ۰۰}ା_ሻ:

ƒ Step 1: Choose ܋ ൌ ሾšଽ šଵ଴šଵଵ šଵଶšଵଷ šଵସšଵହ šଵ଺ሿ where šଽ, šଵ଴, … šଵ଺ arefrom the

N-QAM constellation ܁.

ƒ Step 2: Given c, calculate s as : ܛ෤ሺ܋ሻ ൌ ሺܚ െ ܋۰ሻۯା_ሺۯۯା_ሻିଵ

Quantize ܛ෤ሺ܋ሻ:

ܛොሺ܋ሻ ൌ ൫ܛ෤ሺ܋ሻ൯ ؠ ቀ൫ܠ෤૚ሺ܋ሻ൯ǡ ൫ܠ෤૛ሺ܋ሻ൯ǡ ൫ܠ෤૜ሺ܋ሻ൯ǡ ൫ܠ෤૝ሺ܋ሻ൯ǡ ൫ܠ෤૞ሺ܋ሻ൯ǡ ൫ܠ෤૟ሺ܋ሻ൯ǡ ൫ܠ෤ૠሺ܋ሻ൯ǡ ൫ܠ෤ૡሺ܋ሻ൯ቁ,

with  being the quantizer for the N-QAM constellation. ƒ Step 3: Calculate the decision metric:

ୢൌ ԡܚ െ ܛොሺ܋ሻۯ െ ܋۰ԡଶ

ƒ Step 4: Repeat steps 1 to 3 until all possibilities to choose ܋ are exhausted (N8 _{possibilities). Identify the} smallest decision metric ୢ. Look for the value of ܋ that corresponds to it. Estimate the associated symbol

set and round it off to the nearest constellation points. The values ܛොand ܋Ƹobtained are the estimated solutions: ܋Ƹ ൌ  ƒ”‰‹ ୡ஫ୗఴ ԡܚ െ ܛොሺ܋ሻۯ െ ܋۰ԡ ଶ_ ܛො ൌ ൫ܛ෤ሺ܋Ƹሻ൯ If ݀݁ݐሺ۰۰ା_{ሻ ൒ ݀݁ݐሺۯۯ}ା_ሻ:

ƒ Step 1: Choose ܛ ൌ ሾšଵ šଶšଷ šସšହ š଺š଻ š଼ሿ where š_ଵ, šଶ, … š଼ arefrom the N-QAM

constellation ܁.

ƒ Step 2: Given s, calculate c as : ܋෤ሺܛሻ ൌ ሺܚ െ ܛۯሻ۰ା_ሺ۰۰ା_ሻିଵ

Quantize ܋෤ሺܛሻ:

܋Ƹሺܛሻ ൌ ൫܋෤ሺܛሻ൯ ؠ ቀ൫ܠ෤ૢሺ܋ሻ൯ǡ ൫ܠ෤૚૙ሺ܋ሻ൯ǡ ൫ܠ෤૚૚ሺ܋ሻ൯ǡ ൫ܠ෤૚૛ሺ܋ሻ൯ǡ ൫ܠ෤૚૜ሺ܋ሻ൯ǡ ൫ܠ෤૚૝ሺ܋ሻ൯ǡ ൫ܠ෤૚૞ሺ܋ሻ൯ǡ ൫ܠ෤૚૟ሺ܋ሻ൯ቁ,

with  being the quantizer for the N-QAM constellation. ƒ Step 3: Calculate the decision metric:

ୢൌ ԡܚ െ ܛۯ െ ܋Ƹሺܛሻ۰ԡଶ

ƒ Step 4: Repeat steps 1 to 3 until all possibilities to choose ܛ are exhausted (N8 _{possibilities). Identify the} smallest decision metric ୢ. Look for the value of ܛ that corresponds to it. Estimate the associated symbol

set and round it off to the nearest constellation points. The values ܛොand ܋Ƹobtained are the estimated solutions: ܛො ൌ  ƒ”‰‹ ୱ஫ୗఴ ԡܚ െ ܛۯ െ ܋Ƹሺܛሻ۰ԡ ଶ_ ܋Ƹ ൌ ൫܋෤ሺܛොሻ൯

It is clearly observed that the complexity of the algorithm above is of O(N8_{) because there are at most N}8 possibilities to choose a subset of 8 symbols from a constellation of N points, meaning N8 _{evaluations of the decision} metric are performed. For an exhaustive ML algorithm on the other hand, 16 symbols are chosen from an N-point constellation which results in N16 _{possible choices or a complexity of O(N}16_).

4. Experimental Results

Fig. 1. (a) SER performance of Exhaustive ML decoding of the 4x4 perfect STBC (QPSK); (b) SER performance of Optimized ML decoding of the 4x4 perfect STBC (QPSK).

The performed simulations assume perfect knowledge of channel information at the receiver. The SER performance values are averaged over 103 _{channel realizations obtained by simulation of slow Rayleigh fading}

channels. Figure 1 (a) shows the SER performance of the exhaustive ML decoding of the 4ൈ4 perfect STBC with QPSK as modulation scheme. In Fig. 1 (b), the SER performance of the optimized ML decoding of the 4ൈ4 perfect STBC is displayed. It can be observed that the two performance curves are almost the same which implies that there is no loss in SER performance incurred by the optimized algorithm. Thus, the optimized algorithm has essentially ML performance

5. Conclusions

This paper has presented a low complexity algorithm for decoding the 4ൈ4 perfect STBC. The algorithm is based on the conditional optimization technique. The low complexity decoding algorithm is derived by exploiting the structure of the 4ൈ4 perfect STBC and its similarity with the Golden Code, to which conditional optimization decoding has been applied in recent literature. The suggested algorithm has worst-case complexity of O(N8_{) and} retains essentially ML performance. In comparison, the exhaustive ML decoder has worst-case complexity of O(N16_). Although the reduced complexity algorithm is still a challenge to implement in real-time practical systems, this complexity reduction represents a significant improvement.

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