An Alternative Encoding of the Golden Code and its Low Complexity Detection

The Golden code is a full-rate full-diversity (FRFD) space-time block code. The encoder of the Golden code takes four complex symbols and generates two pairs of Golden codewords. The encoding of each Golden codeword can be regarded as superposition coding with “complex” power allocation. In this paper, we propose an alternative encoding of the Golden code which can be regarded as superposition coding with “real” power allocation. The Golden code with “complex” power allocation or “real” power allocation is hereinafter referred to as the C-Golden code or R-Golden code, respectively. The R-Golden code also preserves the FRFD property. The R-Golden code can be easily implemented in passband modulation using in-phase and quadrature components compared to the C-Golden code. An equivalent received signal model of the R-Golden code system is constructed, then used to derive a closed-form on the lower bound of the average bit error probability. We further propose a low complexity detection scheme, the fast essentially maximum-likelihood detection with adaptive signal detection subset (FE-ML-ASDS) for the R-Golden code. Both simulation and theoretical results show that both the C-Golden code and the R-Golden code achieve the same error performance. Compared to the fast essentially ML detection with signal detection subset (FE-ML-SDS), at high signal-to-noise ratios, the proposed FE-ML-ASDS further reduces detection complexity by at least 68% for the 16QAM or 64QAM R-Golden code with three receive antennas.


I. INTRODUCTION
For real-time applications in the next-generation of wireless communication systems, it is necessary to have high data transmission rate, good wireless link reliability and low complexity detection schemes. Multiplexing is a technique which can be used to increase data transmission rate, while diversity is an approach to improve link reliability. Multipleinput multiple-output (MIMO) techniques are able to offer either multiplexing gain and/or diversity order. But not all MIMO schemes simultaneously provide multiplexing gain and diversity order. There is a trade-off between multiplexing gain and diversity order in MIMO systems [1]. The trade-off between multiplexing gain and diversity order depends on the encoding scheme employed in the MIMO system.
For example, the Alamouti scheme [2], space-time labelling diversity (STLD) [3], and Golden code [4] are all space-time block codes (STBCs), which are all The associate editor coordinating the review of this manuscript and approving it for publication was Wei Feng .
MIMO systems with two transmit antennas. In the above three STBC schemes, the Alamouti scheme is a full-diversity half-rate STBC code with linear maximum-likelihood (ML) detection, the STLD scheme is a labelling and full-diversity half-rate STBC code with O(M 2 ) detection complexity for ML detection, and the Golden code is a full-rate full-diversity (FRFD) STBC with O(M 4 ) detection complexity for ML detection, where M is the modulation order and the code rate is defined as the number of transmitted symbols per antenna use per transmission time slot.
Amongst the above three STBCs with two transmit antennas, Alamouti STBC has been used in modern wireless technologies like Wireless Fidelity (Wi-Fi) IEEE 802.11n [5] and IEEE 802.11ah low power Wi-Fi [6]. However, the Golden code is a competitor of the Alamouti STBC because the Golden code achieves both spatial multiplexing gains and full diversity order. The Golden code has already been incorporated in the IEEE 802.16 (WiMAX) standard [7]. In this paper, we focus on the Golden code with low complexity detection. Very recently, the component-interleaved Golden VOLUME 10, 2022 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ code (CI-Golden code) and the multiple complex symbol Golden code (MCS-Golden code) were proposed in [8] and [9]. Both the CI-Golden code and the MCS-Golden code are also full-rate STBC codes. Compared to the Golden code, both the CI-Golden code and the MCS-Golden code achieve higher diversity order and further improve error performance. The encoder of the Golden code takes four complex symbols and generates four Golden codewords. Let j = √ −1 be a complex number. There are four parameters in the four Golden codewords: α = 1 + jθ,ᾱ = 1 + jθ , θ = 1+ √ 5 2 andθ = 1 − θ. One of the four Golden codewords is αθ . The Golden codeword is rewritten as x s = α 1 x 1 + α 2 x 2 . Borrowing the concept of the superposition coding in the downlink nonorthogonal multiple access (DL-NOMA) system, the encoding of the Golden code in the form of x s = α 1 x 1 + α 2 x 2 can be regarded as superposition coding with ''complex'' power allocation. This is because |α 1 | 2 + |α 2 | 2 = 1, and both α 1 and α 2 are complex, not real. Let α = 1 + jθ = |α|e jϕ . Motivated by real power allocation of superposition coding in DL-NOMA system, we rotate α by an angle −ϕ, and then we have e −jϕ α = |α|.
Similarly, we rotate 1 |α|θ . The alternative encoding β 1 x 1 +β 2 x 2 can be regarded as superposition coding with ''real'' power allocation because |β 1 | 2 + |β 2 | 2 = 1, and both β 1 and β 2 are real, not complex. Similarly other Golden codewords can also be rotated until superposition coding with ''real'' power allocation is achieved. For convenience of discussion, the Golden code with ''complex'' power allocation and the Golden code with ''real'' power allocation are hereinafter referred as C-Golden code and R-Golden code, respectively. Since the average transmit signal power of β 1 x 1 + β 2 x 2 in the R-Golden code is the same as 1 √ 5 α(x 1 + x 2 θ ) in the C-Golden code, the error performance of the R-Golden code system is the same as the C-Golden code system. In Eqs. (3) or (4) of [10], the variances of the equivalent channel models of error performance bound A, also explained that only the amplitudes, not the phases affect the error performance of the proposed system.
In the C-Golden code, one of the Golden codewords is rewritten as , and x s = α 1 x 1 + α 2 x 2 = x I s + jx Q s . Then we have: Similarly, in the R-Golden code one of the Golden codewords is In this paper, this is the key motivation to propose an alternative encoding of the Golden code.
The detection complexity of the optimal ML detection for the R-Golden code is extremely high, which is proportional to O(M 4 ). Thus a low complexity detection algorithm is also important to the applications of the R-Golden code. Based on the structure of the R-Golden code, all detection algorithms for the C-Golden code can be applied to detect the R-Golden code. Reduced complexity detection algorithms for the Golden code have been recently surveyed in [10]. Two new reduced complexity detection algorithms have also been proposed in [10]. These two detection algorithms are the fast essentially ML with signal detection subset (FE-ML-SDS) and the sphere decoding with signal detection subset (SD-SDS). The detection complexity of the FE-ML-SDS is O(2 × M 1 2 ), M 1 ≤ M , which is much smaller than the ML detection complexity O(M 4 ). The average cardinality of the signal set used in SD-SDS is reduced from M 2 to M 1 2 for M -ary quadrature amplitude modulation (M QAM). The detection complexity of the SD-SDS is also greatly reduced compared to the conventional sphere decoding. Very recently, the sphere decoding with sorted signal detection subset (SD-SSDS) has been proposed in [9]. The detection complexity of the SD-SSDS is even smaller than the SD-SDS. In this paper, we only focus on the FE-ML based low complexity detections of the R-Golden code.
The threshold based signal detection (TSD) has been proposed to detect the transmitted symbols in the superposition coded signal for specific power allocation in the DL-NOMA system [11]. The TSD is modified to detect the superposition coded signal with non-specific power allocation. In the R-Golden code, x 1 and x 2 can be also estimated using the modified TSD. This is an important advantage compared to the C-Golden code. We will exploit this advantage in the FE-ML-SDS to further reduce detection complexity.
Given two complex symbols, the FE-ML-SDS algorithm decomposes the ML detection of four complex symbols into two ML detections of the remaining two complex symbols. These two given complex symbols belong to the predefined signal detection subsets. The FE-ML-SDS algorithm implemented two decompositions using the predefined signal detection subsets. In order to further reduce detection 30148 VOLUME 10, 2022 complexity, in this paper, the FE-ML with adaptive signal detection subset (FE-ML-ASDS) is proposed. The FE-ML-ASDS is achieved by aid of the TSD.
Based on the above, the main contributions of this paper are summarized as: • The alternative encoding of the Golden code is proposed in this paper. The alternative encoding of the Golden code is treated as superposition coding with ''real'' power allocation, which is named as R-Golden code. The R-Golden code is also an FRFD STBC. Compared to the C-Golden code, the R-Golden code is easily implemented in passband in terms of • The FE-ML-ASDS is also proposed in this paper. Compared to the FE-ML-SDS, the FE-ML-ASDS can further reduce detection complexity for the R-Golden code.
The remainder of the paper is organized as follows: In Section II, the system model, which includes the Golden codeword, the R-Golden codeword and the R-Golden code system, is presented. In Section III, we present the signal constellation of the R-Golden codewords and the TSD for the R-Golden codewords. The lower bound on the average bit error probability (ABEP) for the R-Golden code system is derived in Section IV. Different detection schemes, which includes QR decomposition based signal detection, fast essentially ML detection, FE-ML-SDS and the proposed FE-ML-ASDS, are described in Section V. In Section VI, the simulation results are demonstrated. Finally, the paper 1 is concluded in Section VII.

II. SYSTEM MODEL
The key component of the R-Golden code is the encoding of the Golden code. In this section, we firstly present the encoding of the C-Golden code, then describe the proposed alternative encoding of the Golden code, and finally present the R-Golden code system.

A. THE ENCODING OF THE C-GOLDEN CODE
The encoder of the C-Golden code takes four complex input symbols and outputs four super-symbols, which are regarded as Golden codewords in this paper. Suppose that information bits are grouped into four bit streams, M is the signal set of M QAM signals. The C-Golden code transmission matrix is given by [2]: where x c ik , i, k ∈ [1 : 2] are expressed as: 3) [10], shows that x c 12 ∈ C , x c 21 ∈ C and x c 22 ∈ C . For convenience, in this paper, we regard x c ik , i, k ∈ [1 : 2], as the C-Golden codewords.

B. THE ENCODING OF THE R-GOLDEN CODE
The encoder of the R-Golden code also takes four complex input symbols and produces four R-Golden codewords. The derivation of the four R-Golden codewords is based on the C-Golden codewords.
Compared to the implementation of (3.1) to (3.2) in passband, in terms of y = x I cos(2π f c t) + x Q sin(2π f c t) for x = x I + jx Q , the implementation of (4.1) to (4.4) is much simpler. This is the key advantage of the R-Golden code.
From (4.1) to (4.4), it is easily seen that there are two pairs of the R-Golden codewords, (x s 11 , x s 22 ) and (x s 12 , x s 21 ). Let x s 11 ∈ S , where S is the signal set of x s 11 . We also have x s 12 ∈ S . Given x i ∈ M , we also have −x i ∈ M . We further have x s 22 ∈ S and x s 21 ∈ S . Correspondingly, the R-Golden code transmission matrix is written as: In the R-Golden code system, x 1 and x 2 or x 3 and x 4 can be very easily detected using the TSD algorithm if x s ik , VOLUME 10, 2022 i, k ∈ [1 : 2] is known at the receiver. This is an important feature of the R-Golden code, which will be used to further reduce detection complexity of the FE-ML-ASDS. However, in the C-Golden code system, x 1 and x 2 or x 3 and x 4 may not be easily detected even if x c ik , i, k ∈ [1 : 2] is known at the receiver.

C. THE R-GOLDEN CODE SYSTEM MODEL
Consider an R-Golden code system with N t = 2 transmit antennas and N r receive antennas, N r ≥ N t [10]. The received signal in time slot i, i ∈ [1 : 2] is given by: where is the channel gain matrix corresponding to the i th time slot with C N r ×1 column vectors h i,1 and h i,2 . n i ∈ C N r ×1 is the additive white Gaussian noise (AWGN) vector for the i th time slot. The entries of h i,k and n i are independent and identically distributed (i.i.d.) complex Gaussian random variables (RVs) distributed as CN (0,1) and CN (0, 2ε ρ ), respectively. ρ 2 is the average signal-to-noise ratio (SNR) at each receive antenna.
From (6), it is easily seen that each transmit antenna transmits two input symbols in two time slots. Hence, the R-Golden code also is a full-rate code. Further, it is also easily seen that each transmitted input symbol experiences two different fadings in two time slots. Therefore, the R-Golden code also achieves full-diversity order 2N r . So the R-Golden code also is an FRFD STBC.

III. SIGNAL CONSTELLATION OF THE R-GOLDEN CODEWORDS
Superposition coded signal constellation has been analyzed in [11]. In this section, we use the approach in [11] to analyze the signal constellation of the R-Golden codewords.
In the R-Golden code system, there are two pairs of four R-Golden codewords (x s 11 , x s 22 ) and (x s 12 , x s 21 ). However, in this section, we only analyze the signal constellation for one pair of R-Golden codeword (x s 11 , x s 22 ). The approach can be used to analyze the signal constellation for another pair of R-Golden codeword. Let . Then x s 11 and x s 22 in (4.1) and (4.4) are rewritten as:   .2) can be rewritten as: As discussed in [11], the superposition coding of (8.1) and (8.
where k s = (k 1 −1)×K +k 2 . It is also noted that z k s s 1 , z k s s 2 ∈ p S . In the R-Golden code system, the encoding functions in (9.1) and (9. and f −1 2 , which are given by:   The TSD is proposed to detect the superposition coded signal with specific power allocation [11]. It is required the superposition coded signal z i . This is an important constraint to detect x p 1 and x p 2 using the TSD. From FIGURE 1 it is easy to know that the constellation in FIGURE 1 does not meet the constraint for the TSD algorithm. For example, FIGURE 1(a) shows z 5 s 1 < z 2 s 1 and FIGURE 1(b) shows z 8 s 2 < z 2 s 2 . In order to apply the TSD to detect both x p 1 and x p 2 , we perform an ascending sort for the superposition coded signal z k s s i , k s ∈ [1 : K 2 ], which is given by: where The argsort (·) operator arranges the K 2 elements in ascending order. i is a vector of the index mapper, i = 1, 2, · · · , K 2 = i (l 1 ), · · · , i (l K 2 ) . Consequently a vector of the inverse index mapper is defined as i −1 = l 1 , l 2 , · · · , l K 2 = i (1) −1 , · · · , i (K 2 ) −1 . In the above example, we have: The signal constellation in terms ofz k s s i , k s ∈ [1 : K 2 ] is shown in FIGURE 2.
We will apply the TSD in the new sequenceZ s i to detect the transmitted x p 1 and x p 2 signals in Section V.

IV. ERROR PERFORMANCE ANALYSIS OF THE R-GOLDEN CODE
In R-Golden code system, four M QAM symbols are transmitted in two time slots using two transmit antennas. The received signal model for the R-Golden code is the same as the C-Golden code. The error performance of the C-Golden code has been analyzed in literature. Appendix B in [10] has proven that the bounded conditional pairwise error probability (PEP) for the C-Golden code at high SNR is equivalent to assume that only one input M QAM symbol is detected with errors, while the remaining three input M QAM symbols are detected correctly. The simulated error performance and the theoretical error performance derived in literature [10] validated that the above assumption works very well for N r ≥ 3 VOLUME 10, 2022 at high SNR for the C-Golden code. In the error performance analysis of the R-Golden code system, we also assume that one input M QAM symbol x 1 , is detected with errors, while the remaining three input M QAM symbols x i , i ∈ [2 : 4], are detected correctly.

V. DETECTION SCHEMES FOR THE R-GOLDEN CODE SYSTEMS
In this section, we discuss the detection schemes for the R-Golden code system. We firstly present the QR decomposition based signal detection, then present the TSD, followed by the FE-ML-SDS, and finally describe the proposed FE-ML-ASDS.

A. QR DECOMPOSITION BASED SIGNAL DETECTION
Based on the complex QR decomposition, H i in (6) is rewritten as: where where R i2 is a zero matrix with (N r − 2) × 2 dimension and R i1 is an upper-triangular matrix with 2 × 2 non-negative real diagonal elements, given by: (6) can be further written as: Now, multiplying both sides of (17) by Q H i , we have: we have: Based on the R-Golden transmission matrix in (5), the equivalent u ij = x s ij + n ij , i, j ∈ [1 : 2] can be derived from (19), which are given by:  : 4]. Based on the superposition encoding in (4.1) to (4.4),x i can be further derived from u 11 and u 12 or u 21 and u 22 . The derivedx i are given by: (21.1) 2) 3) Finally, the estimation of x i is given by: The detection complexity of the QR decomposition based signal detection is very low. However, the error performance is worse compared o the ML detection. In the next subsection, we will present one of the low complexity detection schemes, the TSD.

B. THE THRESHOLD BASED SIGNAL DETECTION
In the previous subsection, the transmitted M QAM symbols x i are estimated based on two time slot received signals. Alternatively, based on the received signal y 1 or y 2 in each time slot, the transmitted M QAM symbols x i can be estimated by use of the TSD.
The thresholds play a key role in the TSD algorithm. Let t k th represent these thresholds in FIGURE 2 (a) and (b), where k ∈ [1 : K 2 − 1]. These thresholds t k th are easily derived as: The thresholds t k th , k ∈ [1 : K 2 − 1] are also shown in FIGURES 2 (a) and (b). Now we apply the TSD in [11] to the signal constellation in FIGURE 2 (a) and (b) to detect the transmitted signals. Since the TSD is not directly applied in FIGURE 1, we regard the signal detection scheme as a modified TSD.
As an example, let can also be estimated. In the above modified TSD, letx t 1 i be the estimated M QAM symbols from u 11 and u 12 andx t 2 i be the estimated M QAM symbols from u 21 and u 22 . It is easily seen that botĥ x t 1 i andx t 2 i are based on one time slot received signal. It is also easily seen thatx i in (22) are based on two time slot received signals. The error performance of the above estimatedx i is better thanx t 1 i orx t 2 i . However, the error performance of the estimatedx i is still far from the optimal detection.
In the subsequent subsections, we will discuss another two low complexity detection schemes, the FE-ML-SDS and the proposed FE-ML-ASDS. The two low complexity detection schemes can achieve the optimal error performance.

C. FAST ESSENTIALLY ML DETECTION (FE-ML)
The FE-ML was proposed in [7], which reduces the detection complexity from O(M 4 ) to O(M 2 ). For convenient discussion of the FE-ML detection, we rewrite (6) as: where Y = y 1 y 2 Ignoring noise N in (25), the pair X 13 of symbols can be estimated, given the pair X 24 of symbols: Alternatively, the pair X 24 of symbols can be estimated, given the pair X 13 of symbols: The FE-ML detection is shown in Algorithm 1.
The FE-ML-SDS algorithm firstly performs the coarse estimation of x i given in (22) based on QR decomposition based detection, finds signal detection subset (x i , δ) for eachx i , replaces M with (x i , δ) in the FE-ML algorithm (Algorithm 1) and lastly conducts the final estimation of x i .
As an example, we consider x i , i ∈ [1 : 4] being 16QAM symbols. Let δ = 4. It is easily found that the average cardinality of (x i , δ) is 4. The detection complexity is proportional to O(2×4 2 ), which is much smaller than O(2×16 2 ).
In general, the detection complexity is proportional to O(2 × L 2 i ), where L i is the average cardinality of (x i , δ). If the average cardinality of the signal detection subset is reduced, the overall detection complexity can be further reduced. In the next subsection, an adaptive signal detection subset is proposed to further reduce the average cardinality of the signal detection subset.  where the argsort (·) operator arranges the L i elements from most probable to least probable.ĩ = ĩ 1 , · · · ,ĩ L i . Let τ 3 , τ 2 and τ 1 be positive integers with τ 3 > τ 2 > τ 1 . Ifx t 1 i =x t 2 i = z k i andx i = z k i , the probability of transmitting symbol x i = z k i is very large. In this case, we use small cardinality τ 1 to construct the signal detection subset for x i .
In this case, we use a little large cardinality τ 2 to construct the signal detection subset for x i . The last case isx t 1 i =x t 2 i . In this case, we use large τ 3 to construct the signal detection subset for x i . The implementation of ASDS is given in Algorithm 2.

Algorithm 2 Implementation of Adaptive Signal Detection Subset
, and three predetermined cadinalities:

VI. SIMULATION RESULTS
In this section, we will present simulation results for the R-Golden code systems in frequency-flat Rayleigh fading with AWGN as described in Section II. It is assumed that the CSI is fully known at the receiver. As per the discussion in [10], the R-Golden code system is valid for N r ≥ 2.
However, in this section, we only consider N r > 2 to enable comparison with the theoretical ABEP (14) derived in Section III. Based on TABLE 3 in [10], δ = 16 and δ = 20 are set for simulations of 16QAM and 64QAM Golden code with N r = 3, respectively; and δ = 4 is set for simulations of both 16QAM and 64QAM Golden code with N r = 4.
Let L i be the average cardinality of (x i , δ). τ k for the FE-ML-ASDS are tabulated in TABLE 2.
In the following two subsections, we firstly present the simulated detection complexity and then present the simulated bit error rate (BER).

A. DETECTION COMPLEXITY ANALYSIS
The detection complexity of the FE-ML-SDS has been analyzed in [10]. Similar to the discussion of detection complexity in [10], the metric of the detection complexity discussed  Let Com SDS and Com ASDS be the detection complexity of the FE-ML-SDS and the FE-ML-ASDS, respectively. Now we define the percentage of complexity reduction for the FE-ML-ASDS compared to the FE-ML-SDS as: From the simulated detection complexity in FIGURES 3 and 4 it is observed that: 1) The detection complexity of the FE-ML-SDS is almost constant because only one δ is used to construct symbol detection subsets.  2) The detection complexity of the proposed FE-ML-ASDS decreases as SNR increases. From Algorithm 2 it is seen that the lower end of the detection complexity of the FE-ML-ASDS is (τ 1 ) 2 . 3) For N r = 3, β = 78.6 for 64QAM at 28.5 dB and β = 68.7 for 16QAM at 22.5 dB. 4) For N r = 4, β = 63.5 for 64QAM at 25.5 dB and β = 67.8 for 16QAM at 19.5 dB.

B. BIT ERROR RATE ANALYSIS FOR R-GOLDEN CODE
In this paper, both the FE-ML-SDS and the FE-ML-ASDS are applied in the 16QAM and 64QAM R-Golden code with N r = 3 and N r = 4. Again, we set the simulation parameters for the R-Golden code according to TABLE 1 in this paper. All BER simulation results of the two detection algorithms are shown in FIGURES 5 and 6. All theoretical results based on (14) are also shown in FIGURES 5 and 6. For comparison, we also simulated the 16QAM and 64QAM C-Golden codes.
The BER simulation results of the 16QAM and 64QAM C-Golden code are also shown in FIGURES 5 and 6.
From the results in FIGURES 5 and 6, it is observed that: 1) Both the C-Golden code and the R-Golden code achieve the same error performance. This is because the average transmit signal power of β 1 x 1 + β 2 x 2 in the R-Golden code is the same as 1 √ 5 α(x 1 + x 2 θ ) in the C-Golden code.
2) As the number of receive antennas increases, the simulated BER draws closer to the theoretical bound. 3) For N r = 3, the proposed FE-ML-ASDS detection algorithm achieves the same error performance of the FE-ML-SDS detection algorithm until a BER of 2 × 10 −5 . However the detection complexity of the FE-ML-ASDS is much lower than the FE-ML-SDS at high SNR. This is because the proposed FE-ML-ASDS has a large probability to correctly estimate the transmitted symbols with small cardinality of the SDS at high SNR.

VII. CONCLUSION
In this paper, we proposed an alternative encoding of the Golden code. The alternative encoding of the Golden code is regarded as superposition coding with ''real'' power allocation. Compared to the C-Golden code, it is comparatively easy to implement the R-Golden code in passband modulation. The lower bound on the ABEP for the R-Golden code system was also derived. Furthermore, the FE-ML with adaptive SDS was proposed in this paper, to further reduce the detection complexity. Simulation and theoretical results validated that both the C-Golden code and the R-Golden code achieve the same error performance. For the R-Golden code system with N r = 3, the detection complexity of the FE-ML-ASDS is at least 68 percent lower than the FE-ML-SDS at high SNR.