- Email: [email protected]
Performance analysis of QAM for L-MRC receiver with estimation error over independent Hoyt fading channels
Int. J. Electron. Commun. (AEÜ) 107 (2019) 15–20
Contents lists available at ScienceDirect
International Journal of Electronics and Communications (AEÜ) journal homepage: www.elsevier.com/locate/aeue
Regular paper
Performance analysis of QAM for L-MRC receiver with estimation error over independent Hoyt fading channels Dimpee Das ⇑, Rupaban Subadar 1 Department of Electronics and Communication Engineering, North Eastern Hill University, India
a r t i c l e
i n f o
Article history: Received 30 November 2018 Accepted 2 May 2019
Keywords: Imperfect Channel Estimation Error estimation M-QAM Hoyt fading L-MRC ABER
a b s t r a c t In modern wireless communication, smart receivers are used to overcome the effect of fading and includes the estimation of the channels as a part of its signal processing. From a theoretical point of view these type of receivers offer high grade performance, however from practical point of view the performance depends on the accuracy of the estimation of channels. In real practice, ideal channel estimator is very difficult, which leads to some estimation error in both phase and envelope of the estimated channels information. In this paper, a general expression for average bit error rate (ABER) has been presented for arbitrary branch Maximal Ratio Combining (MRC) while taking an Imperfect Channel Estimation (ICE) into consideration over independent Hoyt fading channels. The expression is formulated for M-ary Quadrature Amplitude Modulation (QAM) scheme. The degradation in the receiver operation due to this error estimation has been analyzed in this paper by numerical evaluation of the obtained expressions. From our analysis we have observed that the envelope estimation error is much critical from the receiver performance point of view. Ó 2019 Elsevier GmbH. All rights reserved.
1. Introduction Multipath propagation of the transmitted signal characterizes wireless communication technology which most commonly uses radio wave propagation due to the phenomena of reflection, diffraction, and scattering. This degrades the performance of wireless communication systems to a great extent, as a constructive and destructive addition of these multiple copies at the receiver leads to different amplitudes and unequal propagation delays, a phenomenon called fading. Ideally, perfect channel estimation is not possible due to the presence of both phase and envelope errors. So an imperfect channel estimation (ICE) must be taken into account during the mathematical modeling of the system. Different fading models have been proposed which describes the statistical nature of the amplitude envelope, of which one is the Hoyt distribution [1] also referred to as Nakagami-q distribution.Whenever the fading conditions of the channel are more severe than that of Rayleigh, the Hoyt distribution is used for modelling the system. It is commonly observed in satellite links subject to strong ionospheric scintillation. In [2], the bit error probability (BEP) performance of narrow band FSK signals is studied over Hoyt fading ⇑ Corresponding author. 1
E-mail address: [email protected] (D. Das). Co-author.
https://doi.org/10.1016/j.aeue.2019.05.005 1434-8411/Ó 2019 Elsevier GmbH. All rights reserved.
channels considering both the Doppler effects and inter symbol interference (ISI). In [3], the performance analysis of signal to noise ratio, outage probability, average symbol error probability, etc. is done over Hoyt-lognormal channel. The performance of generalized selection combining (GSC) over Hoyt fading in terms of the probability density function of the output signal, outage probability etc is done in [4]. In [5], performance analysis of arbitrary branch maximal ratio combining receiver has been done over Hoyt fading channel in terms of outage probability, amount of fading, average bit error rate, etc. The nature of the modulation dramatically influences the performance of the wireless system. The most commonly used digital modulation schemes being Amplitude Shift Keying (ASK), Phase Shift Keying (PSK), Frequency Shift Keying (FSK), Quadrature Amplitude Modulation (QAM). QAM is a combination of ASK and PSK in which an in-phase and quadrature phase signal is amplitude modulated and then summed up. QAM has higher data rates among all techniques. Further, for better performance of a communication system, the effects of fading are combated by using diversity to transmit the signal over multiple channels that experience independent fading and coherently combining them at the receiver. Selection Combining (SC), Maximal Ratio Combining (MRC), and Equal Gain Combining (EGC) are the most commonly used diversity combiners. Out of these, after careful analysis, it is seen that Maximal Ratio Combiner provides optimum performance
16
D. Das, R. Subadar / Int. J. Electron. Commun. (AEÜ) 107 (2019) 15–20
and is the most preferred among the widely used combiners [8]. Depending on the type of fading model considered, as well as the modulation scheme and diversity combining technique used, each system model displays different performance concerning the widely used analysis parameters such as ABER, outage probability, channel capacity, etc. However, when the practical environment is considered, perfect channel estimation is a challenging job and does not persist in reality. From the above analysis it is seen that MRC is an optimal diversity combining technique and QAM is preferably an effective modulation scheme, while we consider a perfect channel. In [9] estimation error has been considered while obtaining ABER and outage probability expressions for an L-MRC system over Nakagami-m fading channel. In [10], general closed-form expressions of ABER have derived for SpaceFrequency block code orthogonal frequency division multiplexing system (SFBC-OFDM) for both M-ary PSK and M-ary QAM modulation techniques, while taking into account the effect of Imperfect Channel Estimation on the ABER expression. An empirical mode decomposition (EMD) together with adaptive filter (AF), is proposed for estimation in communication channel in OFDM system in [16]. The simulation results depict the improvement in BER performance of this proposed method in comparison to conventional AF and also studies show the efficiency of this method. In [17], two pilot pattern design schemes have been proposed for compressed sensing based channel estimation. Simulation results show that The channel estimation performance using the proposed design scheme pilot patterns precedes in terms of normalized mean square error and bit error rate. Further, in [11], the performance degradation of L-MRC receiver by considering imperfect channel estimation is analyzed regarding ABER and channel capacity, over Hoyt fading channels considering the Binary PSK modulation scheme. In [18], a new approach for the performance analysis of a system under Hoyt fading has been proposed. It shows that squared Hoyt distribution can be constructed form a conditional exponential distribution using which, performance analysis of systems under Hoyt link is simplified and various results for information on Hoyt fading is obtained.In [19], the sensing performance in Nakagami-q fading channel is improved by using energy detection at each cognitive radio (CR) user to investigate performance of cooperative spectrum sensing (CSS). It is found that with increase in Hoyt fading parameter, q, the performance of single CR user based spectrum sensing improves. The performance of MRC diversity receiver over Hoyt fading for M-ary modulation schemes such as 16, 4 and 32 (Phase Shift Keying) PSK has been analyzed in [20]. The PDF based approach has been used to derive the performance measures of MRC receiver over Hoyt and the obtained results have been verified by Monte Carlo simulation. In [21], the BER performance has been studied for different values of receiver antennas of an L-MRC diversity receiver over Hoyt fading channel. The analysis has been done for Binary Phase Shift Keying (BPSK), Quadrature Phase Shift Keying (QPSK) and Differential Binary Phase Shift Keying (DBPSK) modulation schemes. Error probability expressions for higher order branch post detection switch-andexamine combiner (SEC) has been derived for non-coherent Mary Frequency Shift Keying (MFSK) modulation over Hoyt fading channel in [22]. In [23], the error probability of non-coherent Mary frequency shift keying is analyzed. Also Symbol Error Probability (SEP) values are calculated using the closed-form expressions of moment generating functions (MGF) for the switched diversity case along with other performance parameters such as outage probability and average SNR. In [24], the effects of perfect and imperfect channel state information on Cooperative spectrum sensing (CSS) is studied over Hoyt and Rician fading. Performance analysis is done based on various performance analysis parameters and a comparison is made between fusion rules for different fading channels.
From the literature review, it is seen that, ABER performance with channel estimation error has not been reported for L-MRC over Hoyt fading channel considering the M-QAM modulation scheme. This generates a motive to derive the unknown performance measures of QAM for M-MRC over Hoyt fading channel with estimation error. Following is the organization of the paper.The expression of signal-to-noise ratio (SNR) considering error estimation is described in Section 2 using which, the ABER expression of QAM is deduced. In Section 3, the numerical results and in Section 4 the conclusions are presented. 2. System model The Hoyt wireless communication channel is considered over here. Assuming the channel to be slow and frequency nonselective, the low pass complex equivalent of the received signal, over tth symbol interval, can be given as,
rðt Þ ¼ rej/ pðt Þ þ nðtÞ;
ð1Þ
where pðtÞ is the transmitted symbol having energy Ep and nðtÞ is the complex Gaussian noise having zero mean and two sided power spectral density 2N o . Random variable (RV),/, represents the phase. The envelope PDF of Hoyt distributed fading amplitude,rðt Þ is given by [8]
2 2 2 1 þ q2 r ð1þq 2Þ r 1 q4 r 2 4q X pðr Þ ¼ e I0 4q2 X qX
ð2Þ
where X ¼ E r 2 . The Hoyt fading parameter is q 2 ½0; 1 and I0 ð:Þ is the modified Bessel Function of the first kind and zeroth order. In the MRC scheme, the received signals from all diversity antennas are co-phased, multiplied to a weight factor which is proportional to the amplitude of the signal and algebraically added in the MRC combiner. The output SNR of an MRC receiver can be given in [8]
c ¼ c1 þ c2 þ c3 þ . . . þ cL
ð3Þ
where L implies the total number of branches in the diversity receiver. The square of the Hoyt distribution can be represented by using the Hoyt fading model in [6],
r2k ¼ X 2k þ Y 2k
ð4Þ
where the independent zero mean Gaussian RVs, X k and Y k have variances r2x and r2y respectively.The PDF of the Hoyt RV, r k is given r
in (2) where the Hoyt fading parameter, q ¼ ryx . We assume
r2x ¼ 1
which results in r2y ¼ q2 . The PDF of the output SNR of the MRC receiver is obtained by some mathematical arrangements followed by RV transformation.The PDF of the SNR of MRC receiver, represented in a closed form expression is given in [5]
2 c 1þq 2 L 1þq e 2cq2 L 1 q4 f c ðcÞ ¼ cðL1Þ 1 F 1 ; L; c 2 2 2cq CðLÞ 2cq
ð5Þ
where Cð:Þ is the gamma function and 1 F 1 ð:; :; :Þ is the confluent hypergeometric function [15]. 2.1. Considering Imperfect Channel Estimation (ICE) To analyze the effect of ICE on MRC receivers over Hoyt fading channels an estimated channel vector at the MRC receiver side is h u ðt Þ ¼ rb ej ub1 ðt Þ; . . . ; considered, which if denoted by b r ej b 1
bL ju
rbL e
T
ðtÞ , then at each branch, the channel estimation error can
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D. Das, R. Subadar / Int. J. Electron. Commun. (AEÜ) 107 (2019) 15–20
be considered as, el ðt Þ ¼ rbl ej ubl ðtÞ r l ejul ðtÞ. The model of channel estimation error model for any arbitrary linear channel estimation is given in [12] as, jb u f ;l
juf ;l
ðtÞ ¼ ql br f ;l e
r f ;l e
ðt Þ þ wf ;l ðt Þ
l
l¼1 l
l
plane decision method [13], a new DV is obtained by rotating the e with a plane angle of b which is given as, complex DV G
e jb GðbÞ ¼ R Ge
ð7Þ
where plane angle, b is given as,
p p b¼ þ 2 M
ð8Þ
Here M is the constellation size of the modulation. Therefore, the effective output SNR of an MRC receiver, considering half plane method is given as [12] L X
b cl
ð9Þ
l¼1
jq j where, BðbÞ ¼ l
2
cos2 ðDhl bÞ
1jql j
2
;b cl ¼
No
1 þ q2
!L
e
CðLÞ
2b cq
b c ðL1Þ 1 F 1
L 1 q4 b c ; L; 2 2b c q2
ð10Þ
f bc b c ¼
1 þ q2
!L
b
B1b c
2 c q2
e
BCðLÞ
2b cq 1F1
1þq2
L 1 q4 1 ; L; B b c 2 2b c q2
ðL1Þ
B
ðL1Þ b c
! ð11Þ
In any digital communication system for M-ary modulations, the ABER is obtained by taking the average of the conditional bit error rate corresponding to the modulation over the PDF of the receiver output SNR.
Z
0
1
P ej b c MRC f bc
MRC
pffiffiffiffi 4ð M 1Þ pffiffiffiffi M
M1 2
NBFSK CBFSK/NDBPSK –
– DBPSK MPSK
– – –
– NMFSK – –
–
–
Rect. QAM
–
qffiffiffiffiffiffiffiffiffiffiffiffiffi P ej b c MRC ¼ aQ c MRC bb
b c MRC dbc MRC
ð13Þ
where parameters a and b are given in Table 1 for different modulation schemes of interest. Also, the Q ð:Þ function can be expressed in terms of incomplete gamma function and Eq. (13) can be expressed as,
a 1 bb c P ej b c MRC ¼ pffiffiffiffi C ; MRC 2 2 2 p
ð14Þ
2.3. ABER of L-MRC receiver over Hoyt fading with ICE in M-QAM modulation In this analysis the M-ary QAM modulation schemes for 8, 16 and 32 QAMs is considered. The confluent hypergeometric function 1 F 1 ð:; :; :Þ, in the output SNR PDF expression with ICE for L-MRC given in Eq. (11) is expressed in terms of infinite series using [14, 07.20.02.0001.01]. Eq. (11) can then be given as,
f bc b c ¼
1 þ q2
!L
b
B1b c
2 c q2
e
2b cq
1þq2
B CðLÞ L
1 ðL1Þ X b c
ð12Þ
where P ej b c MRC is the conditional BER corresponding to the modulation scheme used. The expression for conditional BER in case of coherent modulations is given as,
L 2 k
k¼0
1q4
2b c q2 B ðLÞk k!
b c
k ð15Þ
Substituting Eqs. (15) and (14) in Eq. (12) and solving the integral using [15, (6.455,1)], we obtain the final ABER expression as,
Pe ¼
2
a ffiffiffi p
1þq2
p
L
2b cq
B CðLÞ
1 X
L
k¼0
2b c q2 B
c
b b q2 B
L 2 k
1q4
k
2b c q2 B
ðLÞk k! !Lþkþ0:5
qffiffi
b 2
CðL þ k þ 0:5Þ ðL þ k Þ
þ 1 þ q2
2 F 1 1; L þ k þ 0:5; L þ k þ 1;
2.2. ABER of MRC receiver for M-ary modulation with ICE
Pe ¼
1
3 M1
!
According to Eq. (9), we multiply Eq. (10) with the function B followed by a RV transformation which gives,
0:5
The expression for the PDF of SNR for MRC receiver is given in Eq. (5). For jql j ¼ 1, the SNR without and with ICE, that is c and b c are identical. Hence Eq. (5) can be regenerated for b c given in [11]
f bc b c ¼
2sin2 ðp=MÞ
b j ubl 2 rl e
cl þ1
1þq2 b c c q2 2b
a
0:5 1 2
l
of ql . When ICE is considered, either jql j < 1 or Dhl – 0 or both may arise, which causes degradation in receiver performance.
L wf ;l ðt Þ l¼1 denote the independent and identically distributed (iid) equivalent estimation error terms having zero mean and variance r2w . In order to detect the transmitted symbol pðt Þ at the MRC receiver, considering ICE, we take the help of complex decision varie ¼ PL rb ej ubl ðt Þr ðt Þ[7]. By the half able (DV) which is given as G
cMRC ICE ¼ BðbÞ
b
ð6Þ
where ðl ¼ 1; 2; 3; . . . ; LÞ and ‘f ’ is the diffused component. ql ¼ jql jejDhl denotes the normalized estimation correlation coefficient between r ejul ðt Þ and rb ej ubl ðt Þ, where Dh is the phase offset l
Table 1 Values of a and b for some coherent and non-coherent modulations.
1 þ q2 bb c q2 B þ 1 þ q2
! ð16Þ
3. Numerical results and discussion In this section the analytical results of the ABER expression has been presented with proper analysis. In Fig. 1, the ABER of the system is analyzed for 8, 16 and 32 QAM constellations for diversity order, L ¼ 3 and Hoyt parameter q ¼ 0:5. It is seen that while the phase error Dh, is kept constant and the envelope error qis varied, the ABER increases for values lesser than 1. With a slight change in the value of envelope error, q, from 0.9985 to 0.990 and form 0.990 to 0.980, the ABER increases considerably. In Fig. 2, the variation of ABER is shown while considering only the phase error and keeping envelope error q ¼ 1. The ABER decreases as the phase error p to Dh ¼ p and further from Dh ¼ p to changes from Dh ¼ 32 40 40 p . The Hoyt parameter q is kept constant at 0.5 for all the Dh ¼ 56 variations and the 8-QAM constellation is chosen. Also it is seen
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D. Das, R. Subadar / Int. J. Electron. Commun. (AEÜ) 107 (2019) 15–20
Fig. 1. ABER performance of a QAM system in Hoyt channels with ICE.
Fig. 2. ABER performance of a QAM system in Hoyt channels with ICE.
that the ABER decreases for higher values of diversity order, L. In Fig. 3, an analysis of the ABER of the system is done while varying the Hoyt parameter q. Analysis is done for q ¼ 0:5; q ¼ 0:7 and q ¼ 0:7. As expected, the ABER decreases as the value of Hoyt parameter, q is increased. Moreover, it is seen that the ABER decreases for higher diversity orders. In Fig. 4 a comparison of the ABER of a perfect channel is made with an imperfect channel envelope estimation. In this plot the phase error is being considp and we have varied the correlation between ered to be Dh ¼ 32 the actual channel and the estimated channel q from 1 to 0:98 for L ¼ 2 and L ¼ 3. It is seen that the ABER is nearly constant when we are decreasing q to 0:98. From this analysis we can conclude that the accurate envelope estimation is critical to ensure the per-
formance of the receiver. Even with 2% error in the estimation can completely destroy the receiver performance.
4. Conclusions In this paper, the performance analysis of L-MRC over Hoyt fading is done for M-QAM modulation, considering channel estimation. An expression for the ABER has been derived considering both the phase and envelope estimation errors. After the analysis, it can be concluded that the performance of the MRC receiver degrades when an imperfect channel is considered. Also, the performance further degrades as the QAM constellation
D. Das, R. Subadar / Int. J. Electron. Commun. (AEÜ) 107 (2019) 15–20
19
Fig. 3. ABER performance of a QAM system in Hoyt channels with ICE.
Fig. 4. ABER comparison of ICE with perfect CSI in Hoyt channels.
size is increased. Hence, we see that an inaccurate estimator may not yield the results that are expected from the MRC system.
[4]
Conflict of interest [5]
The authors declared that there is no conflict of interest. References [1] Hoyt RS. Probability functions for the modulus and angle of the normal complex variate. Bell Syst Tech J 1947;26:318–59. [2] Hajri Nazih, Youssef Neji. On the performance analysis of fsk using differential detection over Hoyt fading channels. In: 2008 2nd international conference on signals, circuits and systems; January 2009. [3] Kumar Sandeep, Soni Sanjay Kumar, Jain Priyanka. Performance analysis of Hoyt-lognormal composite fading channel. In: 2017 international conference
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on wireless communications, signal processing and networking (WiSPNET); February 2018. Krstic Dragana, Nikolic Petar, Stamenovik Goran, Stefanovic Dusan. Performance analysis of generalised selection combiner in the presence of Hoyt fading. In: 2009 9th international conference on telecommunication in modern satellite, cable and broadcasting services; October 2009. Subadar Rupaban, Sahu PR. Performance of L-MRC receiver over independent Hoyt fading channels. In: 2010 national conference on communications (NCC); January 2010. p. 1–5. Fraidenraich G, L’eveque O, Cioffi JM. On the MIMO channel capacity for the dual and asymptotic cases over Hoyt channels. IEEE Commun Lett 2007;11 (1):31–3. Ma Y, Schober R, Pasupathy. Performance of M-PSK with GSC and EGC with gaussian weighting errors. IEEE Trans Veh Technol 2005;54:149–62. Simon MK, Alouini MS. Digital communication over fading channels. John Wiley and Sons Inc.; 2005. Saikia Bhargabjyoti, Subadar Rupaban. Effect of estimation error on adaptive LMRC receiver over nakagami-m fading channels. Int J Appl Eng Res 2018;13 (15):12077–83.
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[10] Singh Daljeet, Joshi Hem Dutt. Performance analysis of SFBC-OFDM system with channel estimation error over generalized fading channels. Trans Emerg Telecommun Technol 2018. [11] Saikia Bhagarbjyoti, Subadar Rupaban. L-MRC receiver with estimation error over hoyt fading channels. Int J Eng Technol 2017;9(4). [12] Ma Y, Schober R, Pasupathy S. Effect of channel estimation error on MRC diversity in rician fading channels. IEEE Trans Veh Tech 2005;54:2137–42. [13] Jeong Woojin, Lee Jaeyoon, Yoon Dongweon. New BER expression of MPSK. IEEE Trans Veh Tech 2011;60(4):1916–24. [14] http://functions.wolfram.com/07.20.02.0001.01. [15] Gradshteyn IS, Ryzhik IM. Table of integrals, series, and products,. 6th ed. San Diego, CA: Academic; 1994. [16] Hari Krishna E, Sivani K, Ashoka Reddy K. On the use of EMD based adaptive filtering for OFDM channel estimation. Int J Electron Commun 2018;83. [17] Xiao Shenyang, Jin Zhigang, Su Yishan. Pilot design for sparse MIMO-OFDM channel estimation with generalized shift invariance property. Int J Electron Commun 2018;96. [18] Romero-Jerez Juan M, Javier Lopez-Martinez F. A new framework for the performance analysis of wireless communications under Hoyt (Nakagami-q) fading. IEEE Trans Inform Theory 2017;63.
[19] Nallagonda Srinivas, Chandra Aniruddha, Dhar Roy Sanjay, Kundu Sumit. Performance of cooperative spectrum sensing in hoyt fading channel under hard decision fusion rules. Comput Dev Commun (CODEC); 2012. [20] Kour Rupender, Mannan Manasvi. MRC receiver in correlated hoyt fading channel with multiple modulation technique. Int J Eng Res Technol 2017;6. [21] Sweeti, Kaur Kiranpreet, KamaljitKaur. Performance analysis of L-MRC receiver over nakagami q (Hoyt) flat fading channels. Int J Adv Res Electron Commun Eng 2016;5. [22] Chandra Aniruddha, Bose Chayanika, Bose Manas K. Performance of order-L post detection switch and examine combiner in Hoyt fading. IEICE Commun Exp 2012;1. [23] Chandra Aniruddha, Bose Chayanika, Bose Manas K. Performance of noncoherent MFSK with selection and switched diversity over hoyt fading channel. Wirel Pers Commun 2013;68. [24] Nallagonda Srinivas, Chandra Aniruddha, Dhar Roy Sanjay, Kundu Sumit. The effects of channel knowledge on cooperative spectrum sensing in Nakagami-n/ q fading channels. Wirel Netw; 1–13.
Contents lists available at ScienceDirect
International Journal of Electronics and Communications (AEÜ) journal homepage: www.elsevier.com/locate/aeue
Regular paper
Performance analysis of QAM for L-MRC receiver with estimation error over independent Hoyt fading channels Dimpee Das ⇑, Rupaban Subadar 1 Department of Electronics and Communication Engineering, North Eastern Hill University, India
a r t i c l e
i n f o
Article history: Received 30 November 2018 Accepted 2 May 2019
Keywords: Imperfect Channel Estimation Error estimation M-QAM Hoyt fading L-MRC ABER
a b s t r a c t In modern wireless communication, smart receivers are used to overcome the effect of fading and includes the estimation of the channels as a part of its signal processing. From a theoretical point of view these type of receivers offer high grade performance, however from practical point of view the performance depends on the accuracy of the estimation of channels. In real practice, ideal channel estimator is very difficult, which leads to some estimation error in both phase and envelope of the estimated channels information. In this paper, a general expression for average bit error rate (ABER) has been presented for arbitrary branch Maximal Ratio Combining (MRC) while taking an Imperfect Channel Estimation (ICE) into consideration over independent Hoyt fading channels. The expression is formulated for M-ary Quadrature Amplitude Modulation (QAM) scheme. The degradation in the receiver operation due to this error estimation has been analyzed in this paper by numerical evaluation of the obtained expressions. From our analysis we have observed that the envelope estimation error is much critical from the receiver performance point of view. Ó 2019 Elsevier GmbH. All rights reserved.
1. Introduction Multipath propagation of the transmitted signal characterizes wireless communication technology which most commonly uses radio wave propagation due to the phenomena of reflection, diffraction, and scattering. This degrades the performance of wireless communication systems to a great extent, as a constructive and destructive addition of these multiple copies at the receiver leads to different amplitudes and unequal propagation delays, a phenomenon called fading. Ideally, perfect channel estimation is not possible due to the presence of both phase and envelope errors. So an imperfect channel estimation (ICE) must be taken into account during the mathematical modeling of the system. Different fading models have been proposed which describes the statistical nature of the amplitude envelope, of which one is the Hoyt distribution [1] also referred to as Nakagami-q distribution.Whenever the fading conditions of the channel are more severe than that of Rayleigh, the Hoyt distribution is used for modelling the system. It is commonly observed in satellite links subject to strong ionospheric scintillation. In [2], the bit error probability (BEP) performance of narrow band FSK signals is studied over Hoyt fading ⇑ Corresponding author. 1
E-mail address: [email protected] (D. Das). Co-author.
https://doi.org/10.1016/j.aeue.2019.05.005 1434-8411/Ó 2019 Elsevier GmbH. All rights reserved.
channels considering both the Doppler effects and inter symbol interference (ISI). In [3], the performance analysis of signal to noise ratio, outage probability, average symbol error probability, etc. is done over Hoyt-lognormal channel. The performance of generalized selection combining (GSC) over Hoyt fading in terms of the probability density function of the output signal, outage probability etc is done in [4]. In [5], performance analysis of arbitrary branch maximal ratio combining receiver has been done over Hoyt fading channel in terms of outage probability, amount of fading, average bit error rate, etc. The nature of the modulation dramatically influences the performance of the wireless system. The most commonly used digital modulation schemes being Amplitude Shift Keying (ASK), Phase Shift Keying (PSK), Frequency Shift Keying (FSK), Quadrature Amplitude Modulation (QAM). QAM is a combination of ASK and PSK in which an in-phase and quadrature phase signal is amplitude modulated and then summed up. QAM has higher data rates among all techniques. Further, for better performance of a communication system, the effects of fading are combated by using diversity to transmit the signal over multiple channels that experience independent fading and coherently combining them at the receiver. Selection Combining (SC), Maximal Ratio Combining (MRC), and Equal Gain Combining (EGC) are the most commonly used diversity combiners. Out of these, after careful analysis, it is seen that Maximal Ratio Combiner provides optimum performance
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D. Das, R. Subadar / Int. J. Electron. Commun. (AEÜ) 107 (2019) 15–20
and is the most preferred among the widely used combiners [8]. Depending on the type of fading model considered, as well as the modulation scheme and diversity combining technique used, each system model displays different performance concerning the widely used analysis parameters such as ABER, outage probability, channel capacity, etc. However, when the practical environment is considered, perfect channel estimation is a challenging job and does not persist in reality. From the above analysis it is seen that MRC is an optimal diversity combining technique and QAM is preferably an effective modulation scheme, while we consider a perfect channel. In [9] estimation error has been considered while obtaining ABER and outage probability expressions for an L-MRC system over Nakagami-m fading channel. In [10], general closed-form expressions of ABER have derived for SpaceFrequency block code orthogonal frequency division multiplexing system (SFBC-OFDM) for both M-ary PSK and M-ary QAM modulation techniques, while taking into account the effect of Imperfect Channel Estimation on the ABER expression. An empirical mode decomposition (EMD) together with adaptive filter (AF), is proposed for estimation in communication channel in OFDM system in [16]. The simulation results depict the improvement in BER performance of this proposed method in comparison to conventional AF and also studies show the efficiency of this method. In [17], two pilot pattern design schemes have been proposed for compressed sensing based channel estimation. Simulation results show that The channel estimation performance using the proposed design scheme pilot patterns precedes in terms of normalized mean square error and bit error rate. Further, in [11], the performance degradation of L-MRC receiver by considering imperfect channel estimation is analyzed regarding ABER and channel capacity, over Hoyt fading channels considering the Binary PSK modulation scheme. In [18], a new approach for the performance analysis of a system under Hoyt fading has been proposed. It shows that squared Hoyt distribution can be constructed form a conditional exponential distribution using which, performance analysis of systems under Hoyt link is simplified and various results for information on Hoyt fading is obtained.In [19], the sensing performance in Nakagami-q fading channel is improved by using energy detection at each cognitive radio (CR) user to investigate performance of cooperative spectrum sensing (CSS). It is found that with increase in Hoyt fading parameter, q, the performance of single CR user based spectrum sensing improves. The performance of MRC diversity receiver over Hoyt fading for M-ary modulation schemes such as 16, 4 and 32 (Phase Shift Keying) PSK has been analyzed in [20]. The PDF based approach has been used to derive the performance measures of MRC receiver over Hoyt and the obtained results have been verified by Monte Carlo simulation. In [21], the BER performance has been studied for different values of receiver antennas of an L-MRC diversity receiver over Hoyt fading channel. The analysis has been done for Binary Phase Shift Keying (BPSK), Quadrature Phase Shift Keying (QPSK) and Differential Binary Phase Shift Keying (DBPSK) modulation schemes. Error probability expressions for higher order branch post detection switch-andexamine combiner (SEC) has been derived for non-coherent Mary Frequency Shift Keying (MFSK) modulation over Hoyt fading channel in [22]. In [23], the error probability of non-coherent Mary frequency shift keying is analyzed. Also Symbol Error Probability (SEP) values are calculated using the closed-form expressions of moment generating functions (MGF) for the switched diversity case along with other performance parameters such as outage probability and average SNR. In [24], the effects of perfect and imperfect channel state information on Cooperative spectrum sensing (CSS) is studied over Hoyt and Rician fading. Performance analysis is done based on various performance analysis parameters and a comparison is made between fusion rules for different fading channels.
From the literature review, it is seen that, ABER performance with channel estimation error has not been reported for L-MRC over Hoyt fading channel considering the M-QAM modulation scheme. This generates a motive to derive the unknown performance measures of QAM for M-MRC over Hoyt fading channel with estimation error. Following is the organization of the paper.The expression of signal-to-noise ratio (SNR) considering error estimation is described in Section 2 using which, the ABER expression of QAM is deduced. In Section 3, the numerical results and in Section 4 the conclusions are presented. 2. System model The Hoyt wireless communication channel is considered over here. Assuming the channel to be slow and frequency nonselective, the low pass complex equivalent of the received signal, over tth symbol interval, can be given as,
rðt Þ ¼ rej/ pðt Þ þ nðtÞ;
ð1Þ
where pðtÞ is the transmitted symbol having energy Ep and nðtÞ is the complex Gaussian noise having zero mean and two sided power spectral density 2N o . Random variable (RV),/, represents the phase. The envelope PDF of Hoyt distributed fading amplitude,rðt Þ is given by [8]
2 2 2 1 þ q2 r ð1þq 2Þ r 1 q4 r 2 4q X pðr Þ ¼ e I0 4q2 X qX
ð2Þ
where X ¼ E r 2 . The Hoyt fading parameter is q 2 ½0; 1 and I0 ð:Þ is the modified Bessel Function of the first kind and zeroth order. In the MRC scheme, the received signals from all diversity antennas are co-phased, multiplied to a weight factor which is proportional to the amplitude of the signal and algebraically added in the MRC combiner. The output SNR of an MRC receiver can be given in [8]
c ¼ c1 þ c2 þ c3 þ . . . þ cL
ð3Þ
where L implies the total number of branches in the diversity receiver. The square of the Hoyt distribution can be represented by using the Hoyt fading model in [6],
r2k ¼ X 2k þ Y 2k
ð4Þ
where the independent zero mean Gaussian RVs, X k and Y k have variances r2x and r2y respectively.The PDF of the Hoyt RV, r k is given r
in (2) where the Hoyt fading parameter, q ¼ ryx . We assume
r2x ¼ 1
which results in r2y ¼ q2 . The PDF of the output SNR of the MRC receiver is obtained by some mathematical arrangements followed by RV transformation.The PDF of the SNR of MRC receiver, represented in a closed form expression is given in [5]
2 c 1þq 2 L 1þq e 2cq2 L 1 q4 f c ðcÞ ¼ cðL1Þ 1 F 1 ; L; c 2 2 2cq CðLÞ 2cq
ð5Þ
where Cð:Þ is the gamma function and 1 F 1 ð:; :; :Þ is the confluent hypergeometric function [15]. 2.1. Considering Imperfect Channel Estimation (ICE) To analyze the effect of ICE on MRC receivers over Hoyt fading channels an estimated channel vector at the MRC receiver side is h u ðt Þ ¼ rb ej ub1 ðt Þ; . . . ; considered, which if denoted by b r ej b 1
bL ju
rbL e
T
ðtÞ , then at each branch, the channel estimation error can
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D. Das, R. Subadar / Int. J. Electron. Commun. (AEÜ) 107 (2019) 15–20
be considered as, el ðt Þ ¼ rbl ej ubl ðtÞ r l ejul ðtÞ. The model of channel estimation error model for any arbitrary linear channel estimation is given in [12] as, jb u f ;l
juf ;l
ðtÞ ¼ ql br f ;l e
r f ;l e
ðt Þ þ wf ;l ðt Þ
l
l¼1 l
l
plane decision method [13], a new DV is obtained by rotating the e with a plane angle of b which is given as, complex DV G
e jb GðbÞ ¼ R Ge
ð7Þ
where plane angle, b is given as,
p p b¼ þ 2 M
ð8Þ
Here M is the constellation size of the modulation. Therefore, the effective output SNR of an MRC receiver, considering half plane method is given as [12] L X
b cl
ð9Þ
l¼1
jq j where, BðbÞ ¼ l
2
cos2 ðDhl bÞ
1jql j
2
;b cl ¼
No
1 þ q2
!L
e
CðLÞ
2b cq
b c ðL1Þ 1 F 1
L 1 q4 b c ; L; 2 2b c q2
ð10Þ
f bc b c ¼
1 þ q2
!L
b
B1b c
2 c q2
e
BCðLÞ
2b cq 1F1
1þq2
L 1 q4 1 ; L; B b c 2 2b c q2
ðL1Þ
B
ðL1Þ b c
! ð11Þ
In any digital communication system for M-ary modulations, the ABER is obtained by taking the average of the conditional bit error rate corresponding to the modulation over the PDF of the receiver output SNR.
Z
0
1
P ej b c MRC f bc
MRC
pffiffiffiffi 4ð M 1Þ pffiffiffiffi M
M1 2
NBFSK CBFSK/NDBPSK –
– DBPSK MPSK
– – –
– NMFSK – –
–
–
Rect. QAM
–
qffiffiffiffiffiffiffiffiffiffiffiffiffi P ej b c MRC ¼ aQ c MRC bb
b c MRC dbc MRC
ð13Þ
where parameters a and b are given in Table 1 for different modulation schemes of interest. Also, the Q ð:Þ function can be expressed in terms of incomplete gamma function and Eq. (13) can be expressed as,
a 1 bb c P ej b c MRC ¼ pffiffiffiffi C ; MRC 2 2 2 p
ð14Þ
2.3. ABER of L-MRC receiver over Hoyt fading with ICE in M-QAM modulation In this analysis the M-ary QAM modulation schemes for 8, 16 and 32 QAMs is considered. The confluent hypergeometric function 1 F 1 ð:; :; :Þ, in the output SNR PDF expression with ICE for L-MRC given in Eq. (11) is expressed in terms of infinite series using [14, 07.20.02.0001.01]. Eq. (11) can then be given as,
f bc b c ¼
1 þ q2
!L
b
B1b c
2 c q2
e
2b cq
1þq2
B CðLÞ L
1 ðL1Þ X b c
ð12Þ
where P ej b c MRC is the conditional BER corresponding to the modulation scheme used. The expression for conditional BER in case of coherent modulations is given as,
L 2 k
k¼0
1q4
2b c q2 B ðLÞk k!
b c
k ð15Þ
Substituting Eqs. (15) and (14) in Eq. (12) and solving the integral using [15, (6.455,1)], we obtain the final ABER expression as,
Pe ¼
2
a ffiffiffi p
1þq2
p
L
2b cq
B CðLÞ
1 X
L
k¼0
2b c q2 B
c
b b q2 B
L 2 k
1q4
k
2b c q2 B
ðLÞk k! !Lþkþ0:5
qffiffi
b 2
CðL þ k þ 0:5Þ ðL þ k Þ
þ 1 þ q2
2 F 1 1; L þ k þ 0:5; L þ k þ 1;
2.2. ABER of MRC receiver for M-ary modulation with ICE
Pe ¼
1
3 M1
!
According to Eq. (9), we multiply Eq. (10) with the function B followed by a RV transformation which gives,
0:5
The expression for the PDF of SNR for MRC receiver is given in Eq. (5). For jql j ¼ 1, the SNR without and with ICE, that is c and b c are identical. Hence Eq. (5) can be regenerated for b c given in [11]
f bc b c ¼
2sin2 ðp=MÞ
b j ubl 2 rl e
cl þ1
1þq2 b c c q2 2b
a
0:5 1 2
l
of ql . When ICE is considered, either jql j < 1 or Dhl – 0 or both may arise, which causes degradation in receiver performance.
L wf ;l ðt Þ l¼1 denote the independent and identically distributed (iid) equivalent estimation error terms having zero mean and variance r2w . In order to detect the transmitted symbol pðt Þ at the MRC receiver, considering ICE, we take the help of complex decision varie ¼ PL rb ej ubl ðt Þr ðt Þ[7]. By the half able (DV) which is given as G
cMRC ICE ¼ BðbÞ
b
ð6Þ
where ðl ¼ 1; 2; 3; . . . ; LÞ and ‘f ’ is the diffused component. ql ¼ jql jejDhl denotes the normalized estimation correlation coefficient between r ejul ðt Þ and rb ej ubl ðt Þ, where Dh is the phase offset l
Table 1 Values of a and b for some coherent and non-coherent modulations.
1 þ q2 bb c q2 B þ 1 þ q2
! ð16Þ
3. Numerical results and discussion In this section the analytical results of the ABER expression has been presented with proper analysis. In Fig. 1, the ABER of the system is analyzed for 8, 16 and 32 QAM constellations for diversity order, L ¼ 3 and Hoyt parameter q ¼ 0:5. It is seen that while the phase error Dh, is kept constant and the envelope error qis varied, the ABER increases for values lesser than 1. With a slight change in the value of envelope error, q, from 0.9985 to 0.990 and form 0.990 to 0.980, the ABER increases considerably. In Fig. 2, the variation of ABER is shown while considering only the phase error and keeping envelope error q ¼ 1. The ABER decreases as the phase error p to Dh ¼ p and further from Dh ¼ p to changes from Dh ¼ 32 40 40 p . The Hoyt parameter q is kept constant at 0.5 for all the Dh ¼ 56 variations and the 8-QAM constellation is chosen. Also it is seen
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D. Das, R. Subadar / Int. J. Electron. Commun. (AEÜ) 107 (2019) 15–20
Fig. 1. ABER performance of a QAM system in Hoyt channels with ICE.
Fig. 2. ABER performance of a QAM system in Hoyt channels with ICE.
that the ABER decreases for higher values of diversity order, L. In Fig. 3, an analysis of the ABER of the system is done while varying the Hoyt parameter q. Analysis is done for q ¼ 0:5; q ¼ 0:7 and q ¼ 0:7. As expected, the ABER decreases as the value of Hoyt parameter, q is increased. Moreover, it is seen that the ABER decreases for higher diversity orders. In Fig. 4 a comparison of the ABER of a perfect channel is made with an imperfect channel envelope estimation. In this plot the phase error is being considp and we have varied the correlation between ered to be Dh ¼ 32 the actual channel and the estimated channel q from 1 to 0:98 for L ¼ 2 and L ¼ 3. It is seen that the ABER is nearly constant when we are decreasing q to 0:98. From this analysis we can conclude that the accurate envelope estimation is critical to ensure the per-
formance of the receiver. Even with 2% error in the estimation can completely destroy the receiver performance.
4. Conclusions In this paper, the performance analysis of L-MRC over Hoyt fading is done for M-QAM modulation, considering channel estimation. An expression for the ABER has been derived considering both the phase and envelope estimation errors. After the analysis, it can be concluded that the performance of the MRC receiver degrades when an imperfect channel is considered. Also, the performance further degrades as the QAM constellation
D. Das, R. Subadar / Int. J. Electron. Commun. (AEÜ) 107 (2019) 15–20
19
Fig. 3. ABER performance of a QAM system in Hoyt channels with ICE.
Fig. 4. ABER comparison of ICE with perfect CSI in Hoyt channels.
size is increased. Hence, we see that an inaccurate estimator may not yield the results that are expected from the MRC system.
[4]
Conflict of interest [5]
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