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SEP performance of triangular QAM with MRC spatial diversity over fading channels
EURASIP Journal on Wireless Communications and Networking volume 2016, Article number: 5 (2016)
Abstract
This paper presents the mathematical model for symbol error probability of triangular quadrature amplitude modulation in a singleinput multioutput environment. The symbol error probability performance is evaluated over fading channels namely Rayleigh, Nakagamim, Nakagamin, and Nakagamiq. The maximalratio combining technique is considered as spatial diversity algorithm and unified momentgeneratingfunctionbased approach is applied to derive the results. The multiple channels considered are independent but not necessarily identically distributed. The results presented are valid for slow and frequency nonselective fading channels only. The symbol error probability expressions obtained contain single integrals with finite limits and integrand composed of elementary functions which help us evaluate our analytical expressions numerically. We also compare these expressions with the error performances obtained through computer simulation, which show excellent agreement. In addition, an example has been simulated to validate our derived mathematical expressions.
Introduction
An efficient signal constellation has always been an active research area since 1960s for the purpose of wired and wireless communication. Quadrature amplitude modulation (QAM) has become the dominant modulation scheme in terms of power and bandwidth efficiency. It was first suggested by C. R. Cahn in 1960 [1]. Since then, many developments have been made in the geometry of QAM. The constellation was named QAM. Hancock and Lucky [2] expanded the work of Cahn. They suggested constellations with signal points taken on concentric circles, with outer ring having more points than on the inner ring. Idea behind this was to remove errors due to phase shift. Later in 1962, Campopiano and Glazer [3] introduced a wellorganized structure of evenbit signal constellation, which is presently known as square QAM (SQAM). In 1960s and 1970s, progressive research on the structure of a potent 16ary constellation had been carried out. Since Square QAM retains a high minimum distance between constellation points and has simple detection technique, plenty of the digital communication systems using high modulation orders with 16 or greater number of constellation points have been utilizing the SQAM. Little literature on constellation designs defeating the SQAM in terms of good transmission capability is available since its birth. In [4], a honeycomblike architecture of constellation whose signal points are taken at the origin and on the first and the second concentric hexagon has been advised. The authors named this constellation as Honeycomb Signal Set. In 1989, Shinjiro Oshita et al. [5] proposed another structure which had hexagonal packing; it was analyzed and named as triangularshaped signal set (TSSS). In spite of the structures of constellations suggested in [4, 5] which give improved performance over the SQAM, they are not practicable because of the increase in detection complexity at the receiver end.
Recently, in 2007, SungJoon Park [6] presented a novel symmetrical structure named as triangular quadrature amplitude modulation (TQAM) in which vertices of the equilateral triangles were taken as constellation signal points. TQAM has a proven efficiency against SQAM in terms of error probability performance and detection complexity in [6, 7]. TQAM uses even number of bits to represent a signal point in the constellation. The key reason behind the efficiency of TQAM constellation is its compact geometry. In 2010, symbol error probability (SEP) of TQAM was evaluated by K. Cho, J. Lee, and D. Yoon for additive white Gaussian noise (AWGN) channel [8] and also, an approximation of SEP expression for TQAM was derived for AWGN and fading channels by T. T. DUY and H. Y. KONG [9], in which maximalratio combining (MRC) was used to analyze TQAM over Rayleigh fading channel with multipath reception. In [10], θQAM was introduced to incorporate SEP of SQAM and TQAM in a single analytical expression over AWGN and Nakagamim channels; however, J. Lee et al. [11] proved their work incorrect for higher modulation order and presented their own equations for exact SEP and bit error probability (BEP) over AWGN channel and also paved way for exact BEP over Rayleigh, Rician, and Nakagamim channels. However, the SEP expressions provided in [10] are valid only for modulation order 16. In 2012, SungJoon Park analyzed the SEP performance of TQAM in AWGN channel using an approximate expression for the SEP [7]. Though an exact generalized mathematical SEP expression in the presence of AWGN channel is provided in [11], the mathematical model we provide in this article can be implemented easily when dealing with TQAM not only in AWGN, but also this model is extended to Rayleigh, Nakagamim, Nakagamin, and Nakagamiq channels. To the best of author’s knowledge, generalized SEP expressions for fading channels incorporating diversity reception have not been presented before in [7–11].
In this paper, TQAM has been analyzed in singleinput multioutput (SIMO) environment with spatial diversity. We use spatial diversity to mitigate fading. Diversity combining is the most powerful way to cater the aftermaths of multipath fading. These combining techniques were introduced by Brennan [12]. Diversity is available whenever multiple, independently fading channels link the transmitter and receiver. Such multiple channels naturally occur in multiinput multioutput (MIMO) applications for which the transmitter or receiver use an antenna array. These diversity combining techniques are actually the operations performed on an antenna array. The idea behind using diversity reception scheme is that, as the signal paths are independent, all of them have very low probability to experience deep fades simultaneously. Thus, we transmit same signal over independently fading paths in diversity reception scheme. In this article, at receiving end these paths are then combined using MRC algorithm that the fading amount of the combined received signal decreases and consequently its signaltonoise ratio (SNR) improves. MRC was first proposed by Kahn [13]. In MRC, if there are total L antennas at the receiving end, the received signals from all of the L branches are weighted according to their individual SNR and then summed together to provide single output.
The performance of Mary QAM with space diversity in various fading channels has been analyzed in [9, 14–24], on which light, is shredded in terms of comparison in Section 5. Recently in 2010, Xichun Zhang et al. [25] utilized the MGFbased approach to evaluate the performance of cross QAM over fading channels, and later in 2013, their work was extended by Hua Yu et al. [26] to SIMO systems with MRC reception. Here, based on the momentgenerating function (MGF) method, average SEP of Mary TQAM is analyzed, whereas, MRC is used as the spatial diversity technique at the receiving end. The fading channels considered in this article are Rayleigh, Nakagamim, Nakagamin (Rice), and Nakagamiq (Hoyt). The results presented are valid for slow flat fading channels only. Moreover, we consider a coherent general order TQAM signal assuming perfect channel estimation. We are using finite integral form of Gaussian Qfunction and the unified MGFbased approach to reach the final expressions. SEP closedform expression provided here consists of single finite range integrals, and the integrand is composed of elementary functions which provide easy numerical evaluation. Moreover, the mathematical expressions are valid for general modulation order TQAM and are accurate and elementary enough that it becomes conducive and fast to quantify the SEP performance of TQAM with MRC.
The remaining article is organized in six sections. In Section 2, we show the SEP expression for AWGN channel. In Section 3, we present the channel model for MRC spatial diversity. In Section 4, we pursue the derivation of SEP expression for Mary TQAM with MRC. Section 5 presents numerical results with brief discussion. An example has been simulated in Section 6, which validates our results derived in Section 4, whereas, conclusion is given in Section 7.
SEP of TQAM over AWGN channel
From Fig. 1, it is observed that unlike SQAM, in TQAM, all the nearest neighbors to any signal point in constellation are equidistant. This is the key reason that SEP expressions provided here give exact fit over the simulation curves. Point P is taken at Euclidean distance d/2 from the origin at an angle of 60°, where d is the length of one side of the triangle, or we can say distance between any two adjacent signal points is d. Mary TQAM is an evenbit representation of constellation points, i.e., M = 22^{m} where m = 2, 3, 4, …, and M is the modulation order of the constellation.
In Fig. 2, dots p _{ j } represents jth signal point, where j = 1, 2, 3, …, M and the lines show decision boundaries for 16TQAM. Now here, four of the innermost signal points have hexagonal decision regions which resemble like honeycomb. The decision boundary lines for all the signal points are drawn based on the CampopianoGlazer construction rule ([27]: Article 9.9.2). For 64TQAM, [7: Fig. 1b] is referred. We assume here that all the signal points are equally probable and Re(p _{ j }) and Im(p _{ j }) are the real and imaginary values of signal point p _{ j }, respectively. Average energy per symbol E _{ s } for Mary TQAM can be evaluated as:
In (2), upper limit of the summation is always a positive integer. Solving summation of polynomial expressions using Faulhaber’s formula ([28]: p. 106):
This leads us to minimum Euclidean distance d expression,
since symbol’s signaltonoise ratio γ can be written as:
where σ ^{2} = N _{ O }/2 is the variance of Gaussian probability distribution function and β is the normalized least distance between adjacent symbols. Thus:
The SEP expressions in this section and Section 4 are derived in terms of β.
The main reason for using the MGFbased approach is to utilize the technique of writing Gaussian Qfunction with finite integration limits. Now in this constellation, there are five types of signal points based on the number of neighbors each point has, and unlike SQAM, for this particular constellation, each symbol has all the neighbors as the nearest neighbors. Table 1 tells us about the number of nearest neighbors for Mary TQAM, where S _{ N } means number of signal points having N nearest neighbors.
Probability of correct symbol reception P _{ C, N } for a symbol having N nearest neighbors is written as:
where Q(β) is the Gaussian Qfunction and to evaluate it numerically, [29]: equation (9)] is used, which is its finite limit integral form. The exact probability of correct receiving symbol for Mary TQAM is given as:
since P _{ e }(β) = 1 − P _{ C }(β):
Equation (15) gives us the exact SEP of Mary TQAM in the presence of AWGN channel. In Section 4, using (15), we evaluate the SEP over various fading channels with MRC reception after presenting channel model in Section 3.
MRC spatial diversity channel models
In SIMO system, the signal is transmitted over L diversity paths where each copy of the signal struggles through individual fading amplitude. The multipath receiver uses the algorithm of MRC diversity reception scheme to increase the SNR of the combined received signal by decreasing the SEP. For MRC, SNR of the combined output signal γ _{∑} at the receiver is expressed as ([30]: equation (5.98)):
where γ _{ i } is the instantaneous received SNR per symbol and α _{ i } is the instantaneous fading amplitude at the ith diversity path. SEP evaluation of TQAM over fading channels while using MRC diversity receiver demands knowledge of probability density function (pdf) of SNR γ _{∑} of the combined signal at output. Before that, we take a look at the pdf of the instantaneous received SNR per symbol γ _{ i } of the ith diversity path over Rayleigh, Nakagamim, Nakagamin and Nakagamiq channels respectively, provided from [31] as:
where \( \overline{\gamma_i}=E\left[{\gamma_i}^2\right] \) is the average received SNR per symbol. E[.] is the expectation operator and I _{ 0 } (.) is the modified Bessel function of the first kind and zero order. Using (18)–(21) along with (22), the MGFs of Rayleigh, Nakagamim, Nakagamin, and Nakagamiq fading channels are respectively given in [31] as:
From (23) to (26), we can observe that these are MGFs of the individual ith diversity path. As we are considering multiple channels which are independent but not necessarily identical, then the MGF of the γ _{∑} is written as the product of the individual MGFs of γ _{ i }:
Equation (27) gives us the MGF of the SNR of the combined signal at the output of the receiver.
SEP of TQAM with MRC spatial diversity
Now we attend our main objective, i.e., to find the SEP expression for Mary TQAM in MRC diversity scheme. This is achieved by averaging the SEP formula (15) over the pdf of the SNR \( {p}_{\gamma_{\sum }}\left(\gamma \right) \) of the combined signal at the output.
P _{ e }(γ) is the exact SEP for Mary TQAM in AWGN channel as provided in (15), whereas, from (6) we know that β is a function of γ.
Ignoring the higher order terms Q ^{i}, i > 4, the average symbol error probability is approximately given as:
In (30), we remove the infinite upper limit in the integrals by using a few mathematical tools. From [32: equation (2)], we can write \( Q\left(\beta \right)=2{Q}_a\left(\beta, \raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{1ex}{$2$}\right.\right) \) and from [33: equation (12)], we can write \( {Q}^2\left(\beta \right)=2{Q}_a\left(\beta, \raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{1ex}{$4$}\right.\right) \), both for β ≥ 0. Where,
The form in (31) simplifies the evaluation of SEP performance over fading channels. For higher powers of the Gaussian Qfunction, we refer to [31: Article 4.1.4]. The following relationships are applied:
Still we have not eliminated the infinite upper limit in the integrals of (30). We have only simplified the evaluation of higher powers of the Gaussian Qfunction yet. Now, if
then from relationships provided in (6) and (32), we can write
This integral can be expressed in terms of MGF of γ (22). Using (31), we get
Now (40) is applicable to 1st and 2nd power of the Gaussian Qfunction. For 3rd and 4th power of the Gaussian Qfunction, we define I _{ Q3 }(D, φ) and I _{ Q4}(D, φ) as:
Now (30) can be expressed as:
The difference in (30) and (43) is that we have overcome the infinite limits of integration. The SEP expression (43) is used to measure performance of Mary TQAM in diversity systems using MRC technique. On substituting MGF from (27) in (40), we get following four integrals with finite limits for Rayleigh, Nakagamim, Nakagamin, and Nakagamiq fading mediums, respectively as:
Similarly, on substituting MGF from (27) in (41), we get following four integrals with finite limits for Rayleigh, Nakagamim, Nakagamin, and Nakagamiq fading mediums, respectively as:
Treating (42) in similar fashion with (27), we get I _{ Q4}(D, φ) for the stated fading mediums.
Since the four integrals from (44) to (55) are finite range, single integrals and integrand composed of elementary functions only; using (43), the average SEP of general modulation order TQAM with MRC spatial diversity over fading channels can be conveniently assessed through numerical integration methods.
Numerical results and discussion
Here, we verify our analytical formulas using computer simulations. In Fig. 3, (15) is compared with the SEP approximation provided in ([7]: equation (5)). Figure 3 shows the SEP of 16TQAM and 64TQAM against SNR in AWGN channel, and it is observed that our expression for exact SEP (15) completely agrees with the simulation curve. Figures 4 and 5 show symbol error rate performance of 16TQAM and 64TQAM over Rayleigh fading channel using MRC diversity scheme, respectively. In Fig. 4, comparison has been made with the SQAM results provided in [15]. We investigate the effect of m, the Nakagamim fading parameter (m ≥ 0.5), in Figs. 6 and 7, which displays the SEP performance of M = 16 and M = 64 for Nakagamim channel against SNR with MRC reception, where m = 2, 4. Figure 6 shows comparison with SQAM [16]. Figures 8 and 9 explain the SEP performance in Nakagamin channel, where K = n ^{2} and various values of K considered are K = 1 dB, 7 dB. Here, n is the Nakagamin fading parameter, which ranges from 0 to ∞. For comparison with SQAM, we apply the results of [17] in Fig. 8. Similarly from Figs. 10 and 11, we confirm analytical expression for Nakagamiq fading channel derived in (43) along with (47), (51), and (55) for q = 0, 0.3, where q is the Nakagamiq fading parameter (0 ≤ q ≤ 1). Comparison with SQAM [22] is shown in Fig. 10 for Nakagamiq medium.
Table 2 provides power gains achieved because of using antenna array instead of single antenna in MRC scenario. We observe quite significant power gain, as L increases. The highest gain is obtained by going from single antenna to twobranch diversity. Now, as the diversity paths are increased from two to three, the power gain lessens as it was for going from one to two, generally as we keep on increasing L, the power gain diminishes. The gains for Nakagamim, m = 1, and Nakagamiq, q = 1, are same as for the Rayleigh fading channel. It is clearly observed that the system performance improves as the diversity order increases. The results illustrate the advantage of diversity as a means for combating the fading phenomena.
Example
To further validate the analytical SEP expressions (43) to (55) for Mary TQAM in MRC, we simulate an example. We take five diversity branches, each with different type of channel fading. The experiment is performed over both 16TQAM and 64TQAM, respectively. Following are the channel fadings being considered over different diversity branches:

First branch: Rayleigh fading

Second branch: Nakagamim fading (m = 2)

Third branch: Nakagamim fading (m = 4)

Fourth branch: Nakagamiq fading (q = 0)

Fifth branch: Nakagamiq fading (q = 0.3)
Now to evaluate the analytical expression for this experiment, we use (43); however, the integrals are evaluated using (56), (57), and (58) as below:
The simulation result for this example is shown in Fig. 12. A good fit of SEP simulation curve over the theoretical curve adds to the validity of our analytical expressions derived in Section 4.
Conclusion
In this article, the SEP performance of TQAM with MRC spatial diversity over independent but not necessarily identical multibranch fading channels, including Rayleigh, Nakagamim, Nakagamin, and Nakagamiq channels, have been evaluated based on the unified MGFbased approach. The SEP expressions are simple and accurate and can be applied to any evenbit general modulation order TQAM. These SEP expressions consist of single integrals with finite limits and integrand composed of elementary functions only, which can be accurately evaluated numerically. The simulation results, along with the example, also confirm the efficacy of numerical expressions obtained for the abovementioned channels. So, by choosing only the modulation order of the constellation and the diversity order of MRC, we can study the impact of diversity reception, which removes the need for Monte Carlo simulations to optimize any wireless system parameters using TQAM.
Abbreviations
 dB:

deciBel
 MGF:

momentgenerating function
 MIMO:

multiinput multioutput
 MRC:

maximalratio combining
 pdf:

probability density function
 QAM:

quadrature amplitude modulation
 SEP:

symbol error probability
 SIMO:

singleinput multioutput
 SNR:

signaltonoise ratio
 SQAM:

square quadrature amplitude modulation
 TQAM:

triangular quadrature amplitude modulation
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The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper.
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Qureshi, F.H., Sheikh, S.A., Khan, Q.U. et al. SEP performance of triangular QAM with MRC spatial diversity over fading channels. J Wireless Com Network 2016, 5 (2016). https://doi.org/10.1186/s1363801505112
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Keywords
 Symbol error probability
 Triangular quadrature amplitude modulation
 Maximalratio combining
 AWGN
 Momentgenerating function
 Fading channel
 Rayleigh
 Nakagamim
 Nakagamiq