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## SAIEE Africa Research Journal

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*On-line version* ISSN 1991-1696

*Print version* ISSN 0038-2221

### SAIEE ARJ vol.109 n.3 Observatory, Johannesburg Sep. 2018

**ARTICLE**

**Low-complexity near-ML detection algorithms for NR-STAR-MQAM spatial modulation**

**R. Pillay ^{I}; H. Xu^{II}; N. Pillay^{III}**

^{I}School of Engineering, Department of Electrical, Electronic and Computer Engineering, King George VAvenue, Durban 4041, University of KwaZulu-Matal, South Africa. E-mail: 208514668@stu.ukzn.ac.za

^{II}School of Engineering, Department of Electrical, Electronic and Computer Engineering, King George VAvenue, Durban 4041, University of KwaZulu-Matal, South Africa. E-mail: xuh@ukzn.ac.za

^{III}School of Engineering, Department of Electrical, Electronic and Computer Engineering, King George VAvenue, Durban 4041, University of KwaZulu-Matal, South Africa. E-mail: pillayn@ukzn.ac.za

**ABSTRACT**

In this study, the authors propose two low-complexity near-maximum-likelihood (ML) detection algorithms for spatial modulation (SM) systems, employing the new multiple-ring star-M-ary quadrature amplitude modulation (MR-STAR-MQAM) constellation. The proposed detectors exploit the specific orientation of MR-STAR-MQAM, in order to avoid searching across all constellation points. As a result, the computational complexity is independent of both the constellation size and the number of rings presented in MR-STAR-MQAM. In addition, these detectors are generalized and can be applied to the entire star*-M*QAM family. The Monte Carlo simulation results demonstrate that the proposed detection algorithms achieve the same average bit error rate (ABER) as ML detection for SM but at a much lower computational complexity. For example, in a 4x4, 2R-STAR-16QAM aided SM system, the proposed optimal and sub-optimal detectors achieve an 88.8% and 90.5% reduction in computational complexity, respectively, compared to the ML detector. Furthermore, the simulation results are supported by a closed-form union-bound theoretical ABER expression.

**Keywords**: Low-complexity near-maximum-likelihood detection, multiple-input multiple-output, M-rings star-M-ary quadrature amplitude modulation, spatial modulation.

**1. INTRODUCTION**

Spatial modulation (SM) is a relatively new transmission scheme developed to reduce the cost and complexity of conventional multiple-input multiple-output systems [1]. In SM, only one transmit-antenna is active during each transmission interval, which effectively means that inter-channel interference (ICI) is completely avoided and inter-antenna synchronization (IAS) is not needed [2]. The transmission of information in SM can be described as a simple approach; the transmitter uses the index of the activate antenna, which is determined by the bitstream to be transmitted, as an additional means to convey information in conjunction with the use of conventional amplitude/phase modulation (APM) constellation, i.e., Mary phase shift keying *(M*PSK) or *M*-ary quadrature amplitude modulation *(M*QAM) [3].

At the receiver, the need to correctly detect the transmitted information bitstream is of utmost importance. In [3], Jeganathan *et al. *derived an optimal detection algorithm for SM based on the maximum-likelihood (ML) method. This detector involves the joint estimation of the transmit-antenna index and the transmitted symbol. Although optimal error performance is achieved, the detector imposes a very high search complexity that grows as *N _{T}M, *where

*N*and Μ represent the number of transmit antennas and the modulation order of the employed APM constellation, respectively. This high receiver complexity poses a considerable challenge for practical implementation. In order to address this problem, several researchers have proposed low-complexity detection algorithms for SM [4]-[12]. In [4], the authors proposed a multistage (MS) detection algorithm that operates over two steps. The first step selects the most probable estimates of the active transmit-antenna index based on the modified maximum ratio combiner (MMRC) method. The second step uses the ML method to detect the transmitted symbol. However, the MS detector imposes a high computational complexity for large order APM constellations. In [5], a simple detection algorithm for

_{T}*M*QAM aided SM (MQAM-SM) systems was developed. This algorithm was based on searching partitioned symbol sets for the active transmit-antenna index and transmitted symbol. Then in [6], a distance-based ordered detection (DBD) algorithm was proposed and was aimed at achieving a flexible tradeoff between error performance and computational complexity. In [7], the authors proposed two-sphere decoding algorithms that were tailored for specific SM systems. It was shown that the computational complexity of these algorithms grows linearly with the constellation size. In [8], Rajashekar

*et al.*proposed a hard-limiter based ML detector whose computational complexity is independent of

*M*and only grows with

*N*. However, this detector is specific for SM systems employing the rectangular

_{T}*-M*QAM and square

*-M*QAM constellations. In [9]-[10], a signal vector-based detection (SVD) algorithm was presented. Although the receiver complexity of SVD is very much lower than that of ML, it demonstrated some performance loss in MQAM-SM systems. Meanwhile in [11], the authors proposed a novel detection algorithm specific for MPSK aided SM systems. This detector was shown to achieve the same error performance of ML detection but at a complexity that is independent of M. Recently, the authors of [12] proposed two innovative low-complexity detectors for SM employing the

*M*-ary amplitude-PSK (M-APSK) constellation. The M-APSK constellation belongs to the family of star-MQAM constellations, namely: conventional star-MQAM [13], M-APSK [14] and new dual-ring star-MQAM [15]. This family outperforms the conventional APM constellations in terms of achievable mutual information over peak-power-limited systems [15]-[16]. Hence, it has been widely adopted in most satellite and space communication standards, such as 2nd generation digital video broadcasting system (DVB-S2), internet protocol over satellite (IPoS) and advanced broadcasting systems via satellite (ABS-S) [13]-[16]. Consequently, the motivation and contributions of this paper are:

*1.1. Motivation*

Based on the above background, SM provides many attractive advantages for next generation networks. It is useful to note that single-symbol and multiple-symbol generalised SM (GSM) [17]-[21] address the constraint on the number of required transmit antennas in SM to achieve high spectral efficiency. It was shown in [17] that singlesymbol GSM has an inferior error performance compared to SM, while the drawback of multiple-symbol GSM [18]-[21] is the very high hardware complexity due to multiple radio-frequency chains, ICI and the need for IAS. Hence, this motivates us to investigate SM.

Although the low-complexity detectors of [12] achieve near-ML error performance, these detectors are specific to SM systems employing only the *M*-APSK constellation. Furthermore, the computational complexities of these detectors have been shown to be independent of *M *but linearly dependent on both *N _{T}*and the number of rings (N) presented in the

*M*-APSK constellation. Moreover, these detectors require pre-processing at the receiver, which can easily translate into additional hardware costs.

Despite the merits of the M-APSK constellation [12], [14], the rest of the star*-M*QAM family may prove beneficial for different systems. For instance, in SM systems, the new dual-ring star-16QAM slightly outperforms the popular square-16QAM in terms of error performance [15]. Based on this research, we have extended the new dual-ring star-*M*QAM to multiple-ring star*-M*QAM [22], which is denoted as NR-STAR-MQAM for simplicity. It was shown in [22] that NR-STAR-MQAM utilizes up to 98% less average signal energy, and has fewer amplitude levels and phase differences than its square*-M*QAM counterpart. In addition, this constellation is superior to other multiple-ring constellations [23]-[24], in terms of improving error performance and deriving a closed-form solution.

*1.2. Contributions*

Based on the above motivation, the contributions of this paper are: we propose two low-complexity near-ML detection algorithms for SM systems employing any constellation from the star*-M*QAM family [12]-[15], which is now inclusive of NR-STAR-MQAM [22]. Unlike the detectors in [12], which are specific to *M*-APSK aided SM *(M*-APSK-SM) systems, the proposed detectors are generalized and can be applied to SM employing any multiple-ring constellation. Based on the work in [11]-[12], we achieve the proposed detection algorithms by exploiting the key features of the star*-M*QAM family. Due to the advantages of *N*R-STAR*-M*QAM [22], we investigate the average bit error rate (ABER) of the proposed detectors in SM systems employing said constellation. Furthermore, we show that the computational complexities of the proposed detection algorithms are independent of both the constellation size and the number of rings presented in the *N*R-STAR-*M*QAM constellation.

*Organization: *Section 2 details the NR-STAR-MQAM aided SM (NR-STAR-MQAM-SM) system model and the features of the star*-M*QAM family. Section 3 presents the proposed low-complexity near-ML detection algorithms. In Section 4, the computational complexities of the proposed detection algorithms are formulated. Section 5 demonstrates and discusses the numerical results of the work. Finally, concluding remarks are drawn in Section 6.

*Notation: *Bold lowercase and uppercase letters are used for column vectors and matrices, respectively. [·]^{τ}, |·| and ||-||_{F} represents the transpose, Euclidean norm and Frobenius norm, respectively. operators represent the minimum and maximum of the argument with respect to w, respectively. operator selects the values of w which maximize the argument. E{^{.}} is the mathematical expectation and (^{.})* represents the conjugate of the vector. and represent the real and imaginary part of the complex argument, respectively. represents all possible values of the argument limited to the upper- and lower bounds, and *j *denotes a complex unit. min(^{.}) returns the smallest element in an array. mod(a, n) denotes that a is the computed modulus of n, round(a) rounds the element a to its nearest integer and [a] rounds the element a to the nearest integer greater than or equal to a.

**2. MR-STAR-MQAM-SM SYSTEM MODEL**

*2.1. SM Transmission and ML Detection*

Consider an N_{r}xN_{s}, MR-STAR-MQAM-SM system model shown in Figure 1, where N_{r} and N_{Rl} represent the number of transmit and receive antennas, respectively. The SM mapper assigns a *b *= log_{2}(MN_{r}) bit binary input to both a transmit-antenna index i (1 ≤ i ≤ N_{r}) and a complex-valued transmit symbol s_{?} (1 ≤ *q *≤ M) from the APM symbol set s. Throughout this paper, we consider s to be the MR-STAR-MQAM constellation [22], unless otherwise stated.

The assignment of i and s_{q} are defined by the SM mapping table, which is known at both the transmitter and receiver [3]-[5]. The SM mapper output can be expressed as:

where *χ* is the N_{T} x 1 transmit vector, s_{?} denotes the q^{th }symbol from the MR-STAR*-*MQAM constellation with

The transmit vector defined by (1) has N_{r} - 1 zero entries corresponding to the dormant transmit antennas and a single non-zero entry s_{q} at the i^{th} position corresponding to the active transmit-antenna. After the mapping process is completed, the transmit vector *χ* is transmitted over a Rayleigh, frequency-flat fading channel experiencing N_{fl} -dimensional additive white Gaussian noise . The N_{R} x N_{T} channel gain matrix *Η* is given by *Η* = [h_{1} h_{2} ... h_{NT}], where hj represents a column vector that is defined as . The elements of both *Η* and *η* are assumed to be independent and uniformly distributed complex Gaussian random variables with zero mean and unit variance.

The corresponding N_{R} x 1 received signal vector *y *is given by:

where ρ is the average signal-to-noise ratio (SNR) per receive antenna and the transmit vector *χ* is of the form, *χ* = S_{q}e_{j}, where the vector e_{;} is selected from the N_{T} -dimensional standard basis vectors, e.g., e_{1} = [1,0,... ,0]^{T }[4]. The set of all *χ* is contained in Γ.

At the receiver, the SM detector obtains estimates of the transmit-antenna index and transmitted symbol using the ML detection rule, which is given by:

where and represents the estimates of the transmit-antenna index and transmitted symbol, respectively.

In order to gain a better understanding of the proposed detectors, we first introduce the features of the star-*M*QAM family and then briefly explain the latest low-complexity detectors [12].

*2.2. Star-MQAM Family of Constellations*

The star-MQAM family is composed of M concentric rings of PSK. In addition, the phase differences between adjacent symbols on the same ring are equal [12]-[15], [22]. The set of constellation symbols s are given as follows:

where *n _{l }*such that , r

_{l}such that r

_{1}< r

_{2}< ... < r

_{n}and Ө; represents the number of symbols, the radius and the initial phase shift of the l

^{th}ring, respectively.

The configuration to denote the number of symbols on each ring is given by n = (n_{1}, n_{2}, ..., n_{N}). In Figure 2, the conventional star-16QAM [13], 16-APSK [14] and the new class of dual-ring star-16QAM [15] constellations are shown, respectively. Note, these dual-ring constellations are the foundation of any multiple-ring constellation that can be both Gray mapped and theoretically validated. The multiple-ring constellations in [23]-[24], do not inherit the features of the dual-ring constellations in Figure 2 and hence, not viable options to both improve and evaluate the ABER performance of SM systems. Nevertheless, the proposed detectors can still be applied to these irregular constellations [23]-[24].

Table 1 presents the different number of amplitude and phase types of the conventional star-16QAM, 16-APSK, new dual-ring star-16QAM and the popular square-16QAM constellation [22]-[23]. It can be seen that the star-16QAM family has fewer amplitudes and phase types than the square-16QAM constellation. This indicates that the star-MQAM family has a stronger ability to resist errors than its square-MQAM counterpart [22]-[23]. For example, square-16QAM has 3 different amplitude and 12 different phase types, whereas conventional star-16QAM has 2 different amplitude and 8 different phase types.

Compared to the multiple-ring constellations in [23]-[24], MR-STAR-MQAM is superior in terms of matching with Gray mapping, error performance and being able to easily derive a closed-form solution [22]. Hence, we investigate the proposed detectors in MR-STAR-MQAM-SM systems. The multiple-ring constellations sizes of interest are the 2R-STAR-16QAM, 4R-STAR-64QAM and 8R-STAR-256QAM. These constellations were designed, based on achieving almost the same Euclidean distance as the square-16QAM, square-64QAM and square-256QAM constellations, respectively, by using as few rings as possible [22]-[23].

*2.3. Latest Low-Complexity near-ML Detection Algorithms*

The authors of [12] developed two low-complexity detection algorithms for SM systems employing the *M-*APSK constellation (Figure 2(b)). The first detection algorithm achieves optimal-ML performance and consists of two steps. The first step lies in identifying the most probable ring in the *M*-APSK constellation to which the receive signal vector belongs to, and is defined as:

In order to achieve the desired outcome of the first step, the receiver is required to perform a few preprocessing tasks that create a threshold decision rule. The computational complexity of this rule may grow depending on the number of rings presented in the *M-*APSK constellation. Let be the estimated ring at the end of the first step.

Then, step 2 involves jointly estimating the transmit-antenna index and transmitted symbol. At this point, the search space of the transmitted symbol is reduced from *M *to the number of symbols that lie on the ring and is given by:

Once (6) has been determined, the most probable symbol on the *1 ^{th }*ring can be determined using [12, Eq. (26)]. Then by using [12, Eq. (27)] the estimated transmit-antenna index can be determined.

The second low-complexity detection algorithm in [12] achieves sub-optimal ML performance and also consists of two steps. The first step selects the most probable transmit-antenna indices, based on the MMRC [4]-[5] criterion and is described as:

The second step jointly estimates the transmit-antenna index from the candidates and the transmitted symbol using the optimal-ML detection algorithm.

Despite the ingenuity of the above-mentioned algorithms, there are two notable disadvantages. Firstly, these algorithms are only suited for SM systems employing the M-APSK constellation. Secondly, the pre-processing tasks at the receiver may add to overall hardware cost of the system.

Hence, to alleviate these disadvantages, we formulate two near-ML low-complexity detection algorithms, which are presented in the next section.

**3. PROPOSED DETECTION ALGORITHMS**

In this section, we propose two near-ML low-complexity detectors for NR-STAR-MQAM-SM systems. Unlike the latest low-complexity detectors [12], which are only suited for SM employing the M-APSK constellation, the two proposed detectors are generalized and can be applied to SM employing any constellation from the star-MQAM family. Note that any multiple-ring constellations, despite both its structure and orientation, belongs to the star-MQAM family.

*3.1. Proposed Low-Complexity Optimal Near-ML Detection Algorithm*

The proposed optimal near-ML detection algorithm for SM systems operates over two steps. The first step involves detecting the two rings closest to (5) and lastly, the second step involves jointly estimating the transmitted symbol and the transmit-antenna index.

**Step 1: **The first step is aimed at determining the two rings closest to Zj. This is achieved using, step 3 to step 7, in Algorithm 1. Note the estimates of the two closest rings are stored in t.

Once the two most probable rings have been determined, the search space of the second step has been reduced to the symbols that lie on the estimated two rings and is given by [11]-[12]:

where with .

**Step 2: **The ML-based criterion of (3) can be split-up into an inner- and outer optimization problem that can be rewritten as:

For a given value of *i, *the inner optimization problem of (9) can be written as and is equally described as [11]:

where , respectively.

Then (10) can be further expressed as:

The equivalent of (11) can be directly expressed as:

The function of (12) is to determine the transmitted symbol by comparing the angles of each symbol in s_{t}to the angle of the receive signal vector z_{j}. However, this will not work when s_{t}contains symbols with different amplitude levels. This is because there exists a scenario where a symbol will be chosen based on (12) but another symbol will lie closer to Z_{i} based on Euclidean distance. Note, the latter part of the scenario will obtain the correct result. Hence, we establish a relationship between the orientation of each ring and Z_{i}. Thus for an initial phase shift of , the relationship can be described as:

where

In order to accommodate for rings that have an initial phase shift equal to , we formulate a simple expression:

Thus from (13) and/or (14), we can estimate the most probable symbol on each ring as follows [12]:

By using (10) and (15), we can correctly estimate the transmitted symbol, which is given as:

After obtaining the correct transmitted symbol , we estimate the transmit-antenna index by substituting in (9), which is given by:

In order to reduce the complexity of (17), the authors of [8] considered that:

From (17) and (18), we obtain that:

For the optimization problem of (19), does not need to be computed again, which further reduces the computational complexity [8].

Hence the low-complexity optimal near-ML detection algorithm is summarized in Algorithm 1:

*3.2. Proposed Low-Complexity Sub-Optimal Near-ML Detection Algorithm*

In this subsection, we propose a low-complexity detection algorithm which achieves sub-optimal error performance.

This algorithm matches the ABER of SM systems using the MS detection algorithm [4]. However, unlike the MS detection algorithm, the proposed sub-optimal detector is suitable for SM systems that employ large order APM constellations. The proposed low-complexity sub-optimal detection algorithm operates over the following two steps, which are summarized as follows:

**Step 1: **The first step is aimed at reducing the transmit-antenna search space, which is efficiently achieved by using (7).

**Step 2: **After reducing the transmit-antenna search space to possible candidates, we then apply Algorithm 1, which is discussed in the previous subsection. Note that z_{; }does not need to be calculated in Algorithm 2, since the computation of (7) is the same as (5). The sub-optimal ML detection algorithm is summarized in Algorithm 2:

Hence, the proposed sub-optimal ML detection algorithm is a viable option for SM systems that have a large *N _{T}*,

*M*or both.

**4. COMPUTATIONAL COMPLEXITY ANALYSIS**

In this section, we present the computational complexity at the receiver, in terms of the total number of floating point operations (flops) [22] required for both the proposed suboptimal detector (PSOD) and proposed optimal detector (POD). Thereafter, we compare these detectors to a few other notable SM detectors, i.e., the MMRC detection [4], SVD [9], DBD [6], MS detection [4] and ML-based detection [3]. We assume results required for future computations can be stored, therefore redundant computation is neglected and full channel knowledge is known at the receiver for all the detection algorithms.

*4.1. Formulation of Computational Complexity*

**ML Detection: **It is straightforward from (3) that it can be rewritten as and that the order of complexity grows as *N _{T}M. *For a given , the complexity imposed by the inner term is equal to

*6N*

_{R}_{ }flops. Hence, the total computational complexity imposed is .

**Latest low-complexity ML Detection Algorithms: **In [12], two detectors, termed PHD and PMML were proposed specifically for *M*-APSK-SM systems. The complexity imposed by both PHD and PMML are flops and flops, respectively. Note, that the complexity of PHD and PMML matches the complexity of the PSOD and the POD, respectively, only for *M*-APSK-SM systems.

**Proposed optimal-ML Detection: **It can be seen that Algorithm 1 grows as *N _{T}*. The complexity imposed by (5) is

*6N*flops [11]-[12]. The computation of calculating the most probable symbol on each ring is equal to a total of 8 flops, according to (15). Then the estimated transmitted symbol is obtained by (16) which amounts to a total of 4 flops. Finally, the computation of (19) which estimates the transmit-antenna index incurs 5 flops, since has already been computed while obtaining (5). Hence, the total number of flops imposed is equal to

_{R}+ 2**Proposed sub-optimal ML Detection: **The computation of the first step, which estimates the most probable transmit antenna(s) is equal to *6N _{R}N_{T} + 4N_{T}* flops [4]. Thereafter, we apply Algorithm 1 but with two slight modifications. Firstly,

*N*needs to be replaced by

_{T}*N.*Secondly (5), does not need to be computed because it is the same as (7). Thus the complexity imposed by Algorithm 1 is equal to 12 flops for step 8 to step 11 and 5 flops for estimating the transmit-antenna index. Hence, the total computational complexity imposed is .

It is worth mentioning that the complexity of Algorithm 1 may be further reduced for SM systems employing the conventional star*-M*QAM constellation (Figure 2(a)). This is because the Euclidean distance between any two neighboring constellation points on adjacent rings are equal. This means that step 3 to step 7 in Algorithm 1 can be easily modified to find the single closest ring to (5). Then, step 8 to step 11 in Algorithm 1 will reduce to 6 flops. Hence, the total flops imposed for SM systems employing the conventional star*-M*QAM constellation [13] or any multiple-ring constellation adopting this specific orientation [23] is . Note that this algorithm will achieve optimal-ML error performance.

*4.2. Evaluation of Computational Complexity*

In this subsection, we evaluate the computational complexity at the receiver, in terms of the total number of flops, as a function of constellation size M, number of receive antennas *N _{R}*and number of transmit antennas

*N*. The computational complexity of the proposed detectors as well as the closest competing detectors are summarized in Table 2.

_{T}

From Figure 3(a), a *4x4 *SM system is considered with the size of *M *changing according to the 2R-STAR-16QAM, 4R-STAR-64QAM and 8R-STAR-256QAM constellations. It can be seen that the complexity of the proposed detectors does not grow with *M.*

In Figure 3(b), we show the computational complexities by changing *N _{R}*when

*N*and

_{T}*M*are fixed at 4 and 16 (2R- STAR-16QAM), respectively. The results show that the proposed detectors offer much reduced complexities than the other notable detectors.

Finally, in Figure 3(c), we consider the computational complexity when *N _{R} = *4 and Μ = 16 (2R-STAR-16QAM), while varying

*N*Though the computational complexity of the MMRC detector is lower than the proposed detectors when

_{T}.*N*is large, it has a non-negligible performance loss, which can be seen in Figure 5 -Figure 7.

_{T}

In the next section, we present the Monte Carlo simulation results in terms of ABER performance for the proposed near-ML detectors in NR-STAR-MQAM-SM systems.

**5. NUMERICAL ANALYSIS**

The aim of this section is to evaluate and compare the ABER performance of the proposed detectors, which was presented in Section 3, to a few other low-complexity SM detectors, namely, MMRC detection [4], SVD [9], DBD [6], MS detection [4] and the benchmark ML-based detection [3]. As mentioned earlier, the proposed suboptimal detector and proposed optimal detector are dubbed PSOD and POD, respectively, for ease of use. The simulation results are investigated over i.i.d. Rayleigh frequency-flat fading channels. Here the ABER is plotted against the average SNR at each receive antenna. As stated earlier, the spectral efficiency of SM is given as log_{2}(N_{T}M) bits per channel use (bpcu).

*5.1. Evaluation and Comparison of the ABER Performance*

The variance, which is normalized to one, is given as , where *NF *and *s _{APM}, *represents the normalizing factor and symbol set of the employed APM constellation, respectively. Here, we denote the symbol set for the 2R-STAR-16QAM, 4R-STAR-64QAM and 8R-STAR-256QAM constellations as $

_{2R*16}, S

_{4R*64}and S

_{8R*256}, respectively.

In Figure 4, we compare the ABER of the detection algorithms presented in [12], namely PHD and PMML along with ML, PSOD and POD in 32-APSK-SM. Note that the ML detector is used for benchmarking purposes. The 32-APSK-SM system achieves a spectral efficiency of 7 bpcu and has radii = (1,2.5,4) with , where S_{32-APSK} represents the symbol set of 32-APSK. It can be seen that the ABER performance of PHD and PMML closely matches the ABER performance of PSOD and POD, respectively.

The following analysis is performed in SM employing the 2R-STAR-16QAM, 4R-STAR-64QAM and 8R-STAR-256QAM constellations with *N _{T} = *4 and

*N*2. In order to support the Monte Carlo simulation results, a closed-form union-bound theoretical ABER is provided for

_{R}=*M*R-STAR

*-M*QAM [22].

In Figure 5, the 2R-STAR- 16QAM-SM system has radii = (1,1.8) with and achieves a spectral efficiency of 6 bpcu. Note, the optimum radii for *M*R-STAR*-M*QAM-SM is efficiently calculated using the optimum ring radii algorithm in [22]. It can be seen that the POD matches very closely with the benchmark ML detector and that the theoretical expression, which was derived in [22], confirms this observation. Although, the PSOD has a slight deviation from the ML detector after 12 dB, it still has a superior error performance than the other detectors. At an ABER of 10^{-4}, the POD achieves an SNR gain of approximately 1 db compared to both the PSOD and MS detection. Also, when compared to DBD and both MMRC detection and SVD, the POD achieves an SNR gain of 2 dB and 15 dB, respectively.

In Figure 6, the spectral efficiency is increased to 8 bpcu by employing the 4R-STAR-64QAM constellation in SM. The optimal radii for this system is (1,1.5,2.0,2.5) with . These simulation results, exhibit analogous ABER performances compared to Figure 5. For example, at an ABER of 10^{-4}, the proposed detectors achieve an SNR gain of approximately 3 dB compared to DBD and 11 dB compared to both the MMRC detection and SVD.

In Figure 7, similar behaviour to Figure 5 and Figure 6 are demonstrated by the proposed detectors. The 8R-STAR-256QAM-SM system has an optimal radii of (1,1.3,1.6,1.9,2.2,2.5,2.8,3.1) with . Here the MS detector, ML detector. PSOD and POD exhibit approximately the same error performance down to an ABER of 10^{-4}. At the same ABER, the proposed detectors achieve SNR gains of 4 dB and 6 dB compared to DBD and both the MMRC detector and SVD. Hence, these proposed detectors may prove to be a feasible alternative to the other notable detectors due to its superior ABER performance and lower computational complexity.

**6. CONCLUSION**

In this paper, we have proposed two low-complexity near-ML detection algorithms for SM systems suitable for employing any constellation from the star*-M*QAM family, more specifically the *M*R-STAR*-M*QAM constellation. We have provided a detailed computational complexity comparison of the proposed detectors and existing notable detectors. It has been shown that the computational complexity imposed by the proposed detectors are independent of both the size of the symbol set and the number of rings presented in the *M*R-STAR*-M*QAM constellation. This effectively means that the complexity is only dependent on the number of transmit antennas in the system. Hence, the proposed near-ML detectors are capable of offering near-ML error performance but at a much reduced computational complexity. Furthermore, the ABER results have been supported by a closed-form union-bound theoretical solution.

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