J. Cent. South Univ. (2014) 21: 1347−1352 DOI: 10.1007/s11771-014-2072-7

Performance of M-QAM, M-DPSK and M-PSK with MRC diversity in a Nakagami-m fading channel

Alam S. M. Shamsul Alam, Choi GoangSeog

Department of Information and Communication Engineering,

Chosun University, 309 Pilmoon Daero, Dong, GwangJu, 501-759, Korea

© Central South University Press and Springer-Verlag Berlin Heidelberg 2014

Abstract: The nature of a wireless communication channel is very unpredictable. To design a good communication link, it is required to know the statistical model of the channel accurately. The average symbol error probability (ASER) was analyzed for different modulation schemes. A unified analytical framework was presented to obtain closed-form solutions for calculating the ASER of M-ary differential phase-shift keying (M-DPSK), coherent M-ary phase-shift keying (M-PSK), and quadrature amplitude modulation (QAM) over single or multiple Nakagami-m fading channels. Moreover, the ASER was estimated and evaluated by using the maximal ratio-combining (MRC) diversity technique. Simulation results show that an error rate of the fading channel typically depends on Nakagami parameters (m), space diversity (N), and symbol rate (M). A comparison between M-PSK, M-DPSK, and M-QAM modulation schemes was shown, and the results prove that M-ary QAM (M-QAM) demonstrates better performance compared to M-DPSK and M-PSK under all fading and non-fading conditions. Key words: M-ary differential phase-shift keying (M-DPSK); M-ary phase shift keying (M-PSK); Nakagami-m fading channel; probability density function (PDF); symbol error rate (SER); M-ary quadrature amplitude modulation (M-QAM)

1 Introduction

The theoretical analysis of fading environments is one of the important topics in designing efficient wireless communication systems. This is a long-standing issue and has been the focus of many researchers in recent years. The goal of these studies is the derivation of exact closed-form expressions for key performance measures such as the average symbol error probability (ASER) induced by M-ary modulation schemes over fading channels. These closed-form solution techniques are very useful for investigating channel modeling, and the results alleviate the need for Monte Carlo simulations to get response of the channel. Therefore, by using this type of closed-form solution, it is very easy to calibrate wireless system parameters to optimize the overall performance [1]. Many studies have been performed on M-ary quadrature amplitude modulation (M-QAM) over fading channels with single and multiple receptions. Since its bandwidth efficiency is high, this modulation technique is often used to implement rate-adaptive transmission strategies that are designated to cope with current and future multimedia wireless communications [9]. Moreover, exact closed-form expressions for the

symbol error rate (SER) of general rectangular QAM over Nakagami-m fading channels exist only for square QAM [1]. Furthermore, for M-QAM, the SER of antenna subset diversity (SSD) has been studied, including the case of maximal ratio combining (MRC) on Rayleigh fading channels with imperfect channel estimation (ICE) [2]. In Ref. [3], the response of ICE on the bit error rate of M-QAM was investigated using MRC and equal gain combining (EGC) diversity over a Nakagami-m fading channel.

The analytical methods described in this work were used to calculate the SER of a coherent M-PSK system, an M-DPSK, and an M-QAM over a slow and flat Nakagami-m fading channel. The Nakagami distribution is sometimes considered to be flexible and accurate as the Rayleigh, Rician, and log-normal distributions [4].

Several statistical distributions have been proposed for channel modeling of a fading envelope under short-term and long-term fading conditions. Short-term fading models include the Rayleigh, Weibull, Rice, and Nakagami distributions [5]. TURIN [6] proposed one model that associates each path with a set of three random variables such as the strength, modulation delay, and carrier phase shift [4]. When the delay is unknown, the fading is followed as a Nakagami distribution; this

Foundation item: Project supported by Research Fund Chosun University 2014, Korea Received date: 2013−03−15; Accepted date: 2013−06−30 Corresponding author: Choi GoangSeog; E-mail: gschoigs@chosun.ac.kr

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approach seems to perform better than a Rice distribution [4]. FOTOHABADI and SAID [7] showed that the consumed power in a Nakagami sub-channel is less than that in a Rayleigh sub-channel. KARAGIANNIDIS et al [5] investigated the suitability of a Nakagami distribution to approximate the lognormal distribution. JUANG and PARSLEY [8] presented a method for parameter selecting of a finite-state Markov chain to match a Nakagami-m fading channel. In this work, the performances of different modulation techniques were simulated over a Nakagami-m fading channel. 2 Statistical characteristics of Nakagami-m

fading channel

The use of lognormal distributions in systems is not widespread, and it seems that a complete and satisfactory theoretical model is required for this distribution. Therefore, this lognormal distribution has been developed using parameters (m, Ω), which is known as the Nakagami distribution [4]. The PDF of a received signal under Nakagami fading with parameters m and Ω is given by

0 ,exp)(

2)( 212

aa

Ω

ma

Ω

m

mGaP m

m

where G(m) is the Gamma function, a is strength of received signal, Ω=Ea2, and m≥1/2. The Nakagami distribution becomes a Rayleigh distribution if m=1. A Nakagami distribution (m, Ω) can approximate a Rice distribution [4] with m≠1. For a given distribution, the amount of fading (F) is defined as F=vara2/[Ea2]2. For Nakagami distribution, F is F=1/m and 0≤F≤2 [4]. If m=∞, then F=0, which corresponds to the case of no fading. The case of m=1, F=1 corresponds to “Rayleigh fading”. The case of m=1/2, F=2 represents the severest fading assumed by Nakagami fading. 3 SER derivation of M-PSK over Nakagami-

m fading channels

It can be assumed that there are N diversity channels carrying the same information-bearing signal. Each channel is modeled as a frequency non-selective slowly Nakagami-m fading channel corrupted by an additive white Gaussian noise (AWGN) process. In fading, the instantaneous signal-to-noise ratio (SNR) (γb) becomes a random variable. If we assume that the channel is identical, the PDF of (γb) at the output of the maximum ratio combiner (MRC) can be written as [9]

)(/)exp()( b1

bbe mNGΓ

mN

Γ

mNP mN

mN

(1)

where G(·) is the gamma function, m is the Nakagami fading parameter (which is assumed to be identical for all channels), and Γ is the average SNR associated with each symbol (which is related to Гk as Г=NГk). If the PDF of (γb) is known, then the average SER in the fading channel can be calculated as follows:

0 bbSbeser d)()( PPP (2)

The SER of M-PSK is given in the exponential form as follows [5]:

d]sec)π

(sinexp[π

1)(

)π

2

π(

2

π

22bbS

M

MP (3)

Equation (4) represents the integral relationship

described in Ref. [10]:

0

1 1)(de

kuxk

ukGxX (4)

By substituting Eqs. (1) and (3) into Eq. (2) and

applying relationship of Eq. (4), the average SER over a Nakagami fading channel with N-order diversity is

d)/]sec)π

(([sin1π

1)(

π

2

π

2

π

22bser

mN

M mNΓM

P

(5)

Equation (5) represents the average SER of a coherent M-PSK over Nakagami-m fading channels. If m=∞, Eq. (5) represents Eq. (3), which is the SER of an M-PSK over an AWGN channel. 4 SER derivation of M-DPSK over

Nakagami-m fading channels

It is shown in Ref. [11] that the probability of the symbol error of M-DPSK over an AWGN channel is given by

2

π

2

π

bbS /cos)π

cos(1expπ2

)π

sin()(

MMP

dcos)π

cos(1

M (6)

Substituting the values from Eqs. (6) and (1) into

Eq. (2) and applying the relation of Eq. (4), the average symbol error probability of M-DPSK over a Nakagami fading channel is

2

π

2

π

bser //]cos)π

cos(1[1π2

)/πsin()(

mN

mNM

ΓM

P

d)cos)π

cos(1(

M (7)

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5 Performance of M-QAM over Nakagami- m fading channels

In this section, the ASER of a coherent M-QAM over a Nakagami-m fading channel is derived by using the N-th order MRC reception. In this fading environment, we assume that there are N diversity channels that are carrying the same transmitted signal. Therefore, each path or channel can be modeled as frequency non-selective slowly Nakagami-m fading channel that is corrupted by an AWGN process. For a QAM demodulator, it can be shown that the probability of symbol error caused by an AWGN channel depends on the instantaneous value of the SNR (γ) associated with each symbol [12]. The symbol error rate (SER) of M-QAM over an AWGN channel can be written as [13]

||4)( b2

bbS baQbQaP (8)

where ,1

1M

a )1(

3

0

S

MN

Eb and

0

Sb N

E .

By applying exponential bounds for the Q function, Eq. (8) can be written as [14]

)1(2/)exp(2/)2/exp(/ bbS ababnaP

12

b

1

1b )/exp()/exp(

n

nii

n

ii SbSb (9)

where Si=2sin2θi and )4(π/ nii .

The main idea of this exponential bound is to choose the arbitrary values of n. By increasing n (i.e., the number of θ), the bound tends to the exact value [14].

By substituting the values from Eqs. (9) and (1) into Eq. (2), and applying Eq. (4), the average symbol error probability of M-QAM over a Nakagami-m fading channel is

mNmN

mN

bΓa

mN

bΓ

n

aP 1

221

2

1)( bSER

1

12sin2

1)1(n

i

mN

imN

bΓa

12

2sin21

n

ni

mN

imN

bΓ

(10)

Equation (10) shows the ASER of coherent square

M-QAM over a Nakagami-m fading channel, where Г shows the average SNR in dB, N represents the number of channels, and m and M denote Nakagami fading parameter and modulation index, respectively.

6 Evaluation of M-PSK over a Nakagami-m

fading channel

Equation (5) shows the exact numerical represent of the SER of M-PSK modulation technique over a Nakagami-m fading channel. Here, Γ represents the average SNR in dB and N is the number of available channels. The SER over a Nakagami-m fading channel can be plotted by varying the channel numbers (N) and the number of symbols M for PSK modulation.

m=1 represents a Rayleigh fading channel and m>>1 represents an AWGN channel. Interestingly, it can be shown that the increasing value of m coverts the channel from Nakagami to AWGN. Figure 1 depicts the ASER of BPSK (M=2) modulation, where the number of channels is one (N=1) and m is varied from 0.5 to 106 (theoretically, infinity). From Fig. 1, we note that performance decreases when m=0.5, and when m is gradually increased, the channel converts to a non-fading channel such as AWGN. However, the overall performance of the fading channel is not especially good because it is plotted for N=1 (no diversity). Figure 2

Fig. 1 SER of BPSK for N=1 (without diversity) over different

fading channels

Fig. 2 SER of BPSK for N=2, 12 (with diversity) over different

channels

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shows the SER for BPSK modulation with multiple channel numbers (N=2, 12). The modulation is plotted for the same conditions as those represented in Fig. 1, except diversity.

After applying the diversity techniques, Fig. 2 shows that it requires less SNR to achieve low SER. If the fading effects are gradually reduced by changing the value of N, then the fading channel converts to the non-fading AWGN channel. For example, from Figs. 1 and 2, it can be shown that for a Nakagami channel (m=2), the SER is almost 10−3 at 15 dB SNR without the diversity condition. When diversity is applied, the SER is reduced to about 10−5.

Figure 3 is simulated for M-PSK (M=4, 16, 64) modulation technique using multiple diversity. For a severely fading channel (m=0.5), Fig. 3 shows that when the number of symbols is low (M=4), the performance is high. As the fading parameter m improves from 0.5 to 1, there is significant improvement in the response curve, for the case without diversity conditions. Similarly, a low symbol value of PSK shows good performance. This is due to the fact that if the phase variation of the carrier signal is low, then the phases of the carrier are less corrupted by the channel. Comparing the different curves in Fig. 3, we note that the vulnerability increases with M. For example, at a given SNR, the SER increases the least for M=4 and increases the most for M=64.

Fig. 3 SER of (M=4, 16, 64) PSK for N=1 (without diversity)

and N=12 (with diversity) over a Nakagami (m=0.5) channel

7 Evaluation of M-DPSK over a Nakagami- m fading channel

Equation (7) shows the exact numerical presentation

of the SER for M-DPSK over a Nakagami-m fading channel.

Figure 4 depicts the simulation results for differential binary phase shift keying (DBPSK) over different fading channels. It is plotted without the

Fig. 4 SER of DBPSK over AWGN, Rayleigh and Nakagami-m

channels

diversity condition except m=0.5 because diversity is applied only for the Nakagami parameter m=0.5. From Fig. 4, we note that when diversity is applied, the fading channels show a good result compared to the case without the diversity condition. Figure 5 shows the performances for M=4, 16, and 64 D-PSK with and without diversity. Here, M=4 D-PSK shows very good result compared to other values of M. Now, if we compare the result between M-PSK and M-DPSK over a Nakagami fading channel, then M-PSK is an attractive communication technique and shows very good performance compared to the M-DPSK modulation technique. Figure 6 shows simulation results for M=4, 16, and 64 differential phase shift keying (DPSK) and PSK over a Nakagami fading channel. From aforementioned simulation curves, we suggest that there should be a trade-off between fading parameter m and diversity N. If a channel exhibits strong fading effects for a low value of m, then it is wise to increase diversity. Similarly, if there is an AWGN channel (m=∞), then it is not necessary to apply diversity. Therefore, in summary, the number of

Fig. 5 SER of (M=4, 16, 64) DPSK for N=1 (without diversity)

and N=12 (with diversity) over a Nakagami fading channel

(m=5)

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Fig. 6 SER of (M=4, 16, 64) for D-PSK and M-PSK over a

Nakagami fading channel

symbols (M) and the SNR are significant parameters in the design of an efficient communication system. It is not always possible to transmit data at a high rate. Depending on the channel response, designers will assign the symbol rate by observing the error responses. 8 Evaluation of M-QAM over a Nakagami-m

fading channel

Equation (10) depicts the performance of coherent square M-QAM over a Nakagami-m fading channel. As previously described, the type of channel depends on the values of m. If m=∞, then the fading channel becomes a non-fading channel such as an AWGN channel. Similarly, for m=1, the channel shows a Rayleigh fading response.

Figure 7 shows the probability of symbol error for different values of m, and the response curves show the gradual conversion of a fading channel to a non-fading environment. This was simulated for a 16-QAM modulation technique by applying different values of m over a single Nakagami fading channel. Here, we see that a small change in m creates a large modification in

Fig. 7 SER of 16-QAM over a single Nakagami-m fading

channel

performance. The QAM modulation technique for different

symbol values (M=4, 16, and 64) is plotted in Fig. 8. It is shown that the performance is highly sensitive to the value of symbol M.

Fig. 8 SER of (M=4, 16, 64) QAM over a single Nakagami-m

fading channel

In practice, for a high value of M, the phase and

amplitude of the carrier carry more symbols. Thus, a greater SNR value is required for a higher-order QAM compared to a low symbol QAM in order to maintain the same BER. For this reason, 4-QAM with fading parameter m=5 produces a better result than that of 64-QAM. Similar behavior also occurs for the fading parameter m=2, which is shown in Fig. 8.

Figure 9 represents the simulation results for 16-QAM modulations for multiple Nakagami-m fading channels using the MRC technique. It is shown that for a fixed fading parameter, the response curve strongly depends on the number of channels. For example, for m=0.5, the performance of multiple channels (N=4) shows very good results compared to single channel transmission. Thus, by increasing space diversity, the

Fig. 9 SER of 16-QAM over a multiple Nakagami-m fading

channel

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effect of fading is noticeably decreased and the channel converts to an AWGN channel, as shown in Fig. 9.

Figure 10 shows a very important correlation among these three modulation techniques. It is simulated to obtain a comparison between the DPSK, PSK, and QAM modulation schemes. As previously stated, the behaviors of the fading (m) and diversity (N) parameters are the same for these three modulation techniques. Thus, for fixed values of m and N, DPSK, PSK, and QAM are plotted in Fig. 10. It is shown that the response of 16- QAM is better than the responses of the other modulation techniques. This behavior is explained as follows. The distance between two consecutive constellation points for 16 PSK and QAM can be written as d16-PSK=

)π/16sin(2 SE and 10/2 SQAM-16 Ed . So, the ratio of the two distances is given by /10/2 SEX

62.1)]π/16sin(2[ S E and XdB=4.19 dB.

Fig. 10 SER of 16 DPSK, PSK, and QAM over a Nakagami-m

fading channel

The distance between two constellation points of

16-QAM is 1.62 times greater than that of the 16-PSK modulation technique. As shown in Fig. 10, for the same SER, 16-QAM requires an SNR that is 4.16 dB less than that of PSK, as marked by an arrow in Fig. 10. Therefore, 64-QAM shows very good performance compared to the 64-PSK and D-PSK modulation techniques. 9 Conclusions

The average SEP of M-DPSK, M-PSK, and QAM over a Nakagami fading channel with arbitrary fading index m has been derived. In this derivation, the approximation bounds are assumed in the analysis. The averaging is performed by using the probability density function method for calculating the error probabilities over fading channels. QAM and PSK modulations are analyzed in detail for different fading channels using space diversity. The simulation results are sufficiently general to allow for an arbitrary fading parameter in the Nakagami model and a diversity parameter for multiple

channel environments. For this reason, our results can provide engineers with useful information regarding the evaluation and design of radio communication links over Nakagami fading channels. Acknowledgements

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(Edited by YANG Bing)