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gamma shadowed fading model over indoor off body communication channel
Accepted Manuscript Regular paper Performance Analysis of κ-μ/gamma Shadowed fading model over Indoor off body Communication Channel Hari Shankar, Ankush Kansal PII: DOI: Reference:
S1434-8411(17)32390-7 https://doi.org/10.1016/j.aeue.2018.06.004 AEUE 52362
To appear in:
International Journal of Electronics and Communications
Received Date: Accepted Date:
9 October 2017 1 June 2018
Please cite this article as: H. Shankar, A. Kansal, Performance Analysis of κ-μ/gamma Shadowed fading model over Indoor off body Communication Channel, International Journal of Electronics and Communications (2018), doi: https://doi.org/10.1016/j.aeue.2018.06.004
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Title page with Author Detail
Title: Performance Analysis of κ-µ/gamma Shadowed fading model over Indoor off body Communication Channel
Author Details: (1) Hari Shankar: Research Scholar, Thapar University, India Email id - [email protected] (2) Ankush Kansal: Assistant Professor, Thapar University, India Email id - [email protected]
Performance Analysis of κ-µ/gamma Shadowed fading model over Indoor off body Communication Channel Abstract The κ-µ/gamma distribution has an importance role to model the small scale fading and shadowing over human off body indoor communication channel. This composite fading model has various special cases like κ-µ, Rician, Nakagami-m, Rayleigh, Rayleigh/gamma, Nakagami/gamma and Rice/logormal. In this paper, the expression for bit error rate (BER) using various modulation schemes, average channel capacity (ACC) and outage probability (OP) over κ-µ/gamma shadowed fading channel are derived. All the derived expressions are novel and presented in analytical form. The expression for BER and channel capacity are in form of wellknown Meijer G function, whereas the outage probability expression is obtained from cumulative distribution function (CDF) proposed in previous literature. The derived expressions of BER (BPSK), average channel capacity and outage probability reduces to special cases for validation purpose. The study shows that binary phase shift keying (BPSK) modulation technique has better BER performance as compare to other modulation techniques. Moreover, on increasing α and β while κ and µ kept constant and vice versa, the ACC get increases but below the Additive white Gaussian Noise (AWGN) channel capacity as expected. Also, better outage probability performance is obtained at lowest threshold signal to noise ratio (5dB). Keywords- Body Area Network, composite fading model, shadowing, small scale fading
1. INTRODUCTION In last decades, the composite fading models have been extensively used to model the multipath faded and shadowed environment. The composite fading model is the combination of multipath fading and shadowing. The distribution like Rician, Rayleigh, Nakagami-m and κ-µ are used to model the multipath faded scenarios whereas lognormal, gamma and inverse gamma distribution have been extensively used to model the shadowed environment [1]. The composite fading model can be further divided into two parts, depending on path of obstruction of signal; line of sight (LOS) and multiplicative shadowed fading. The former occurs when the object completely or partially obstruct the dominant LOS signal that causes random fluctuation in dominant signal component. Rice/lognormal [2], Rice-Nakagami [3], κ-µ/Nakagami [4] and κ-µ/lognormal [5], and κ-µ shadowed [6] are the example of this type of composite fading model. On other hand,
the latter occurs when the object obstructs both dominant and multipath signal component that causes the random fluctuations in total signal power. Rice/lognormal [7], Rayleigh/lognormal [8], Rayleigh/gamma (K distribution) [9], Nakagami-m/gamma (KG) [10] and κ-µ/gamma [11], [12] are example of multiplicative shadowed fading model. In body centric communication, the phenomenon like multipath fading, large scale fading and path loss has an important role to characterize the received signal. The body centric communication channel, depending on the position of transmitter and receiver(s), can be defined in three different ways; off body, on body and body to body communicational channel. In off body communication, one or more device is located on human body which communicates with local base station or transceiver. In on body communication, the antennas are positioned on human body so the shadowing occurs due to body part that causes attenuation in the received signal and in body to body communication; the antennas are located on different human bodies. Recently, various literatures have been studied that represent the characteristics of Indoor off body communication channel using different operating frequencies, types of antenna, body worn locations and user movements[12-16]. κ-µ distribution was proposed in [17] that have various special cases like Rice (Nakagami-n), Nakagami-m, Rayleigh and one sided Gaussian distribution. The measurement was performed to validate the proposed distribution in [17]. κ-µ/ Nakagami-m shadowed fading model was proposed in [4] which is applicable to land mobile satellite (LMS) communications and underwater acoustic communications (UAC). In [5], the author proposed κ-µ/lognormal distribution that characterizes the body centric communication channel. LOS Shadowed fading model was proposed in [6] in which resultant dominant component is characterized by Nakagami –m distribution. In [6], the measurements were performed in outdoor environment that agreed with proposed model. The κ-µ/gamma multiplicative shadowed fading model for indoor off body communication was proposed in [12] that was initially proposed in [11]. In [12], the investigations were performed at 5.8 GHz in three different indoor environments i.e. indoor laboratory, open office area and seminar room and in each indoor environment four scenarios were taken out: line of sight, NLOS walking, random and rotational movements. The author further, in [12], derived the expression for PDF (probability density function), cumulative distribution function, moment generating function (MGF), moment and Amount of fading (AF). In [18], the κ-µ /inverse composite fading model was proposed, in which mean signal power
follows inverse gamma distribution. In [18], the measured values, performed in body centric communication channel, were fitted to theoretical results. Bit error rate (BER), channel capacity, outage probability has an important role in performance analysis over wireless communication system. In [12], the author derived the expression for probability density function (PDF), cumulative distribution function (CDF), moment generating function (MGF) and Amount of fading (AF) of κ-µ/gamma shadowing fading model. However, the performance analysis of digital communication system over κµ/gamma shadowed fading channel did not further elaborate therein by authors and never reported in previous literature to the best of author’s knowledge. So, the main work of this paper is to derive the expression for BER using different modulation schemes, channel capacity and outage probability over κ-µ/gamma indoor off body communication channel. All derived expressions are presented in infinite series form. The formulas for BER and channel capacity are presented in term of well-known Meijer G function whereas, the expression for outage probability is given in term of hyper geometric function that can be easily implemented in most standard software packages like Maple and Mathematica. The numerical result of average BER (Binary Phase Shift Keying), average channel capacity and outage probability of κ-µ/gamma shadowed fading model are compared (for validation purpose) with ABER (BPSK), ACC and OP of other fading/shadowed fading model, respectively. The work of the paper is organized as follows. In Section-2, PDF of κ-µ/gamma shadowed fading model is presented with its numerous special cases. The expression for ABER, average channel capacity and outage probability are derived in Section-3. Further, in Secrtion-4, the numerical results of ABER, ACC and OP are presented and compared with its special cases. Finally, the paper is summarized with some concluding remark in Section-5. 2. κ-µ/GAMMA SHADOWED FADING MODEL Let X is the envelope of κ-µ /gamma shadowed fading model. The pdf of this fading model is [12],
f X ( x) j 0
4
3 j 2
j
k (1 k ) j
2
x j 1
j!exp( k )( )( j )
j 2
(1 k ) K j 2 x
(1)
where, Km(.) is the modified Bessel function of second kind with m order, x is the envelope of received signal. Parameter µ and κ are related to the κ-µ distribution. µ is the multipath clustering. κ is the ratio of total power of dominant components to the total power of scattered waves. Moreover, the parameter α and β are related to Gamma distribution. α and β is the shape and scale parameter, respectively. κ-µ/gamma model has four parameters κ, µ, α and β. So (1) can be reducing to the various small scale fading model for different value of κ, µ, α and β. For example, as α→∞, β =1.6, the PDF of this model coincides with κ-µ, Rician, Nakagami-m, and Rayleigh with different values of κ and µ. By setting κ = 4.2 and µ = 1.3, it coincides with κ-µ distribution [12, 17]. As, κ = K (Rician Factor) = 3.2 and µ = 1, it approximates the Rician distribution [1, 12]. By putting κ→0 and µ=1, κ-µ/gamma shadowed fading distribution reduce to Rayleigh distribution. In similar way, the Nakagami- m model can be obtain by setting κ→0 with µ parameter equal to m parameter [1,12]. Moreover this composite fading model reduces to other shadowed fading distribution like Rayleigh/gamma [9, 12], Nakagami-m/gamma [10, 12] and Rice-lognormal distribution [2, 12] as special case. For example as κ→0, µ = 1, α = 7.8 and β = 0.7, it coincides with Rayleigh/gamma composite fading distribution. Likewise with κ→0, α = 2.4 and β = 1.5, (1) reduces to Nakagami-m/gamma with µ parameter equals to m parameter. Also, by setting κ = K (Rician Factor) = 2.4, µ = 1, α = 43.9 and β = 0.05, it coincides with Rice Lognormal distribution. The shape parameter (σ) and log-scale parameter (u) of lognormal shadowing can be obtained from α and β parameters of Gamma distribution. Let us consider a signal s, with Es energy symbol, is transmitted over a κ-µ/gamma shadowed faded channel. The received signal z can be defined as, z = s X + n, where n is AWGN (Additive White Gaussian Noise) with one sided power spectral density (psd) of N0, X denotes the small scale fading/shadowing envelope which is κ-µ/gamma distributed. The instantaneous SNR per symbol is defined as γ = X2 Es/N0. So the average SNR per symbol becomes γ = Ω Es/N0, where Ω = E[X2], is the mean squared value of κ-µ/gamma shadowed faded envelope. Hence, the PDF of instantaneous SNR per symbol can be defined as [19], 2 f ( ) f Z
It is noted that Ω = E[X2] can be obtained by putting n=2 in [12, eq. (14)], which is given as
(2)
E[ X 2 ]
exp( k )( 1)( 1)1 F1 ( 1; ; k ) ( )( ) (1 k )
(3)
By using (1) and (2), the expression for PDF of signal to noise ratio (SNR) can be written as,
f ( ) j 0
2
3 j 2
j 2
j j!exp( k )( )( j ) 2 k j (1 k )
j 2
j 2 2
(1 k ) K j 2
d
(4)
In the next section, we will derive the expression of average BER for different modulation schemes, average channel capacity and outage probability over κ-µ/gamma shadowed faded channel using (4).
3. PERFORMANCE ANALYSIS 3.1 Average Bit Error Rate ABER of any modulation technique over shadowed faded channel disturbed with AWGN is calculated by averaging the conditional BER for AWGN channels over PDF of instantaneous SNR. The expression for average BER is defined as [19],
PE ( E ) P ( E | ) f ( )d
(5)
0
where, P(E|γ) is the conditional error probability for AWGN channel and fγ(γ) is the SNR PDF. The conditional bit error rate in an AWGN channel can be written as [19],
Pb ( E | )
(b, a ) 2(b)
(6)
where, a = 1 and b = 0.5 for BPSK, a = 0.5, b = 0.5 for BFSK, a = 0.5, b = 1 for Differentially coherent BPSK (DPSK) and a = 1, b = 1 for non-coherent BFSK (NCFSK). Γ(w, x) is the complementary incomplete gamma function. Now, substituting (4) and (6) in (5), the average BER becomes,
(b, a ) PE 2(b) j 0 0
2
3 j 2
k j (1 k )
j 2
j!exp( k )( )( j )
j 2
j 2
j 2 2
(1 k ) K j 2
d
(7)
d
(8)
By taking the constant term outside the integral, the above equation becomes, 1 PE 2(b) j 0
2
3 j 2
k j (1 k )
j 2
j!exp( k )( )( j )
j 2
j 2
(b, a ) 0
j 2 2
(1 k ) K j 2
The incomplete gamma function can be represented in term the of Meijer G function as [1],
1 (b, a ) G12, 2,0 a b 0
(9)
so, (8) becomes 1 PE 2(b) j 0
2
3 j 2
j 2
j 2 j!exp( k )( )( j ) k j (1 k )
j 2
(1 k ) 1 2 j 2 2 G a K j 0 b 0 2, 0 1, 2
d
(10)
After taking the substitution and by using Table of Integral Series and Product [20], (10) can be represented as, 1 PE 2(b) j 0
2
3 j
j
j
2
2
j (1 k ) 2 j!exp( k )( )( j ) 2
k j (1 k )
a 1, j 1,1 G32,,22 (1 k ) b,0
(11)
The above equation is the expression for average BER over κ-µ/ gamma shadowed faded channel. The above expression coincides with BER expression of small scale fading models like κ-µ, Rician, Nakagami–m and Rayleigh as a special case and also coincides with BER expression of composite shadowed fading model like Rayleigh-Gamma, Nakagami-m-Gamma and Rice lognormal.
3.2 Average Channel Capacity Let us consider, a κ-µ/ gamma shadowed faded channel has a bandwidth W. According to Shannon, the average channel capacity is given by [1, eq. (4.200)],
C W log 2 (1 ) f ( )d
(12)
0
where, fγ(γ) is the PDF of instantaneous SNR per symbol γ as given in (4). The normalized channel capacity can be given by,
C C / W log 2 (1 ) f ( )d
(13)
0
By putting the expression of (4) in (13) we get,
C log 2 (1 ) 0
j 0
2
3 j
j
j
2
2
j 2 j!exp( k )( )( j ) 2
k j (1 k )
j 2 2
(1 k ) K j 2
The function ln (1+γ) can be expressed in term of Meijer G function as [1, eq. (4.279)],
d (14)
1,1 ln(1 ) G21,,22 1,0
(15)
Hence, (14) becomes, 1 C ln( 2) j 0
2
3 j
j
j
2
2
j j!exp( k )( )( j ) 2 2
k j (1 k )
(1 k ) 1,1 2 j 2 1, 2 G K j 2 2 , 2 0 1,0
d
(16)
By using substitution property of integration and using Table of Integral Series [20], the above equation can be written as, 1 C ln( 2) j 0
3 j
j
j
2
2
j (1 k ) 2 j!exp( k )( )( j ) 2
k j (1 k )
1, j 1,1,1 G41,,42 (1 k ) 1,0
(17)
This is the expression for normalized ACC over κ-µ/gamma composite fading channel. In the next subsection, we will derive the expression for outage probability using cumulative distribution function proposed in previous literature.
3.3 Outage Probability (Pout) The probability of outage (Pout) is defined as the probability that the instantaneous SNR per symbol γ falls below the specified threshold SNR per symbol γ th. The outage probability can be obtained by taking integration of PDF of instantaneous SNR with limits of zero to γth which is nothing but cumulative distribution function of specified threshold SNR (γ th). Hence it can be written as [1, eq. (4.154)],
Pout Pr { 0 }
th
f ( )d F (
0
)
(18)
0
where, fγ(γ) is PDF of instantaneous SNR per symbol(γ), γth is specified SNR threshold value and Fγ(γ) is the CDF of instantaneous SNR per symbol. Instantaneous SNR per symbol is defined as γ =X2 Es/N0 and we know that the average SNR per symbol can be obtained as, =E[γ] = E[x2]* Es/N0=Ω* Es/N0, where Ω is the Mean square value of x as given in (3), hence γ = X2/ Ω. So the CDF of γ i.e. Fγ(γ)=Fz((γ Ω/ )1/2). Therefore the outage probability can be express as, Pout FX th
From [12, eq. (8)], the outage probability can be evaluated as,
(19)
(1 ) th (1 ) th 1 F2 ( ; j 1,1 ; j 0 j! exp( ) sin(( j ) ) (1 j )( j )(1 ) j
Pout
j
j
th (1 ) th 1 F2 ( j;1 j ,1 j; j j!exp( ) sin(( j ) ) (1 j )( )(1 j )
j 2 j (1 ) j
j 0
(20)
This expression represents the outage probability of κ-µ/gamma shadowed fading model. In the next section, we will present the numerical results of average BER, ACC and OP of κ-µ/gamma distribution and compare it with existing model proposed in previous literature.
4. NUMERICAL ANALYSIS AND DISCUSSION In this section, the derived expression for ABER, ACC and OP are numerically evaluated and these performance matrices are compared with performance matrices of existing model like Rician,
Rayleigh,
κ-µ,
Nakagami-m,
Rayleigh/gamma,
Nakagami-m/gamma
and
Rice/lognormal. Fig.1 and Fig.2 shows the numerical result of derived expression of ABER as given in (11) over BPSK modulation scheme for κ-µ/gamma composite fading channel. Fig.1 depicts the plot of ABER as the function average SNR for different parameter κ, µ, α and β. We have used coherent BPSK modulation scheme with modulation parameter a=1 and b=0.5 to plot the curve. The continuous line shows the ABER of κ-µ/gamma composite fading model for different values of function parameter. The results of ABER over κ-µ/gamma channel are compared with result of ABER over small scale fading channel model like κ-µ, Rician, Nakagami-m and Rayleigh. With κ = 4.2, µ = 1.3, α = 100 and β = 1.6, it reduces to BER expression of κ-µ model (square line) [21, eq. (12)] and for κ = 3.2, µ = 1.0, α = 100 and β =1.6, the result of (11) overlapped to BER of Rician model (cross) having same Rice factor (K) as κ [22, eq. (2.10)]. Similarly for κ→0, α = 100 and β =1.6, the result of (11) coincides with BER of Nakagami-m distribution (Asterisk) having m parameter equal to µ parameter [1, eq. (4.138)]. Moreover, for κ→0, µ = 1.0, α = 100 and β = 1.6, the expression for BER of κ-µ/gamma shadowed model agree with BER of Rayleigh model (circle) [1, eq. (4.134)]. From the fig.1, it is observed that on increasing the value of κ, the error rate drop get increases. The result shows that the curve for κ = 3.2 (Rician case) has sharp BER than for k→0 (Rayleigh case).
Fig.1 Average BER verses γ for BPSK over κ-µ/gamma shadowed fading channel (continuous line) with special cases: BER of κ-µ (square), Nakagami-m (Asterisk), Rician (cross) and Rayleigh (circle). The values of α and β are 100 and 1.6, respectively. Fig.2 shows the plot of ABER of BPSK modulation scheme for different parameter values. The results of BER of κ-µ/gamma shadowed model are compared with BER of other composite fading models like Rayleigh/gamma, Nakagami-m/gamma and Rice/lognormal to validate the proposed model. In Fig.2, solid line shows the BER of κ-µ/gamma shadowed fading model for different four parameter values, κ, µ, α and β. For κ→0, µ=1, α=7.8 and β=0.7, (11) agree with BER of Rayleigh/gamma composite model (with m=1 and N=1) [1, eq. (4.152)]. Similarly, for κ→0, α=2.4 and β=1.5, (11) overlapped with BER of Nakagami-m /gamma having m parameter equal to µ parameter [1, eq. (4.152)]. Moreover, for µ=1, α=43.9 and β=0.05, (11) agree with BER of Rice/lognormal having same Rician factor (K) as κ [23, eq. (6)]. We have used shape parameter σ = 0.15 and log-scale parameter u=0.7 of lognormal shadowing to plot the BER of Rice lognormal shadowed fading model. The value of these two parameters can be estimated from parameter α and β [12]. From Fig.2, it is observed that Nakagami-m based composite model has better BER performance as compared to Rayleigh based composite model.
Fig.2 Average BER verses γ for BPSK over κ-µ/gamma shadowed fading channel (continuous line) with its special cases: BER of Rayleigh/gamma (circle), Rice/lognormal (cross) and Nakagami-m/gamma (square).
Fig.3 Average BER verses γ for different modulation scheme over κ-µ/gamma composite fading channel. BPSK (Solid line), BFSK (Dot), DPSK (Dash), NCFSK (Dash dot).
Fig.3 depicts the analytical results of average BER under different modulation schemes like coherent BPSK, coherent BFSK, differentially coherent BPSK (DPSK) and non-coherent BFSK (NCFSK). Various modulation schemes depend on the modulation specific parameters a and b as discussed in previous section. From the fig.3, it is observed that on increasing the κ and µ parameter while α and β kept constant, the error rates drop over any modulation schemes get increases. Similarly, on increasing the shape (α) and scale parameter (β) while κ and µ kept constant, the error rates drop get also increases. Result shows that the bit error probability of BPSK at fixed average SNR is lowest as compare to other modulation schemes (BFSK, DPSK or NCFSK).
Fig.4 Average channel capacity verses γ over κ-µ/gamma shadowed fading channel with its special cases: ACC of of κ-µ (cross), Rician (square), Nakagami-m (Asterisk), and Rayleigh (circle). The values of α and β are 100 and 1.6, respectively. Fig. 4 shows the plot of ACC per unit bandwidth (W) verses average SNR over κ-µ/gamma fading channel (continuous line). As we know that as α→∞, the κ-µ/gamma shadowed fading model reduces to small scale fading model like κ-µ, Rician, Nakagami-m and Rayleigh model. For κ=4.2, µ=1.3, the capacity of κ-µ/gamma shadowed model reduces to capacity of κ-µ fading model [24, eq. (6)] and for κ=3.2, µ=1, (17) reduces to capacity of Rician fading having Kfactor same as κ [1, eq. (4.210)]. Similarly with κ→0, µ=1.4, (17) reduce to capacity of
Nakagami-m fading model with same Nakagami parameter (m) as µ parameter [1, eq. (4.207)] and for κ→0, µ=1, (17) reduces to capacity of Rayleigh model [1, eq. (4.208)]. Fig.5 depicts the analytical results of average channel capacity over κ-µ/gamma shadowed fading channel (continuous line) overlapped with result of ACC of existing model like Rayleigh/gamma (cross), Nakagami-m/gamma (square). As κ→0, µ=1.4, α=7.8 and β=0.7, (17) coincides with capacity of Rayleigh/gamma model with m=1 [1, eq. (4.214)]. Similarly with κ→0, µ = 1.4, α = 2.4 and β = 1.5, (17) reduce to channel capacity of Nakagami-m/gamma shadowed fading model [1, eq. (4.214)].
Fig.5 Average channel capacity verses γ over κ-µ/gamma shadowed fading channel (continuous line) with its special cases: ACC of Rayleigh/gamma (cross), Nakagami-m/gamma (square). Fig.6 depicts the plot of ACC per unit bandwidth verses average SNR over κ-µ/gamma composite fading model for different values of four parameters κ, µ, α and β. The capacity of AWGN channel (continuous line) is more as compare to κ-µ/gamma model as expected. From fig. 6, it is observed that, as the κ, µ parameter increases while α and β kept constant and vice versa, the channel capacity gets increases.
Fig.6 Average channel capacity verses γ over κ-µ/gamma composite fading channel for different values of κ, µ, α, and β.
Fig.7 utage Probability verses γ over κ-µ/gamma shadowed fading channel (continuous line) with its special cases: Outage Probability of Rician (cross), Nakagami-m (circle), Rayleigh (Asterisk), Rayleigh/ gamma (solid square), Nakagami-m/gamma (square). The fixed threshold SNR (γth) = 10dB.
Fig. 7 depicts the numerical results of outage probability of κ-µ/gamma distribution (continuous line) as the function of average signal to noise ratio (SNR). The outage probability of this shadowed fading are compared with outage probability of other existing model like Rician, Nakagami-m, Rayleigh, Rayleigh/ gamma and Nakagami-m/gamma. As α=100, β=1.6, κ→0, (21) reduce to outage probability of Nakagami-m distribution with m-parameter equal to µ parameter [1,eq. (4.157)], as κ→0, µ = 1,α=100, β=1.6, (21) reduce to outage probability of Rayleigh model [1, Table (4.4)], as µ=1, α=100, β=1.6 , (21) reduce to outage Probability of Rician model with K parameter equals to κ parameter [19, Table 9.5], As κ→0, u=1, α = 7.8, β=0.7, (21) reduce to outage probability of Rayleigh/gamma[1, Table(4.4)] and as κ→0, α = 2.4, β=1.5, (21) reduce to P of Nakagami-m/gamma model (Generalized K ) [1,Table (4.4)].
Fig.8 utage Probability verses γ over κ-µ/gamma shadowed fading channel for different values of parameters with different threshold SNR (γth = 5, 10 and 15 dB). Fig.8 illustrates the numerical result of outage probability of κ-µ/gamma composite fading model as the function average SNR with three different threshold SNR values 5, 10 and 15 dB. We have plot three cluster curves by using different κ, µ, α, β parameters for 5 to 15dB threshold SNR values. From the figure, it is observed that on increasing the values of threshold SNR, the probability of outage get increases at any fixed average SNR values. For example, with 40dB
average SNR, the OP is high for 15dB threshold SNR as compared to 5dB threshold SNR. Moreover on increasing the κ, µ parameter for fixed values of α, β and γth, the OP performance get increases. Similarly, on increasing α, β parameter with fixed κ, µ and γth, the OP performance also get increases as illustrated in fig.8.
CONCLUSION In this paper, we have derived the analytical expression for average bit error rate (ABER) using different modulation scheme, average channel capacity (ACC) and outage probability of wireless communication system over κ-µ/gamma shadowed fading channel. This shadowed fading model is suitable for indoor off body communication channel and have various special cases that is it can be reduce to small scale fading model like κ-µ, Rician, Nakagami-m, Rayleigh, and shadowed fading model like Rayleigh/gamma, Nakagami-m/gamma and Rice/lognormal. The expression for ABER and ACC are presented in term of Meijer G function while OP expression is presented in term of hypergeometric function that can be easily implemented in most commonly available software package like Maple and Mathematica. Further, we have presented the numerical result of ABER (BPSK), ACC and
P of κ-µ/gamma shadowed fading model and
compared it with the ABER (BPSK), ACC and OP, respectively, of existing model like κ-µ, Rician, Nakagami-m, Rayleigh Rayleigh/gamma, Nakagami-m/gamma and Rice/lognormal. The study shows that error probability for BPSK modulation scheme is minimum as compared to other modulation schemes like BFSK, DPSK and NCFSK at fixed average SNR for any parameter values. For example at 40dB average SNR for κ =1, µ =1, α =2, β =2, the BER for BPSK, BFSK, DPSK and NCFSK is 3.51×10-5,6.99×10-5,6.35×10-5 and 13.1×10-5, respectively that shows BPSK modulation schemes has better BER performance. Moreover, as κ and µ increases while α and β kept constant and vice versa, the average channel capacity gets increases but below the AWGN channel capacity as expected. Further, on increasing the values of κ and µ (0.5 to 1.5) while α and β kept constant with any threshold SNR values ranges between 5 and 15dB at 60dB average SNR, the probability of outage gets decreases in the range of 10 -2 to 10-5 approximately that improve the system performance. References [1] Shankar PM. Fading and Shadowing for Wireless system. New York: Springer; 2005.
[2] Loo C. A statistical model for a land mobile satellite link. IEEE Trans Veh Technol 1985; 34(3):122–27. [3] Abdi A, Lau WC, Alouini MS, Kaveh M. A new simple model for land mobile satellite channels: First- and second-order statistics. IEEE Trans Wirel Commun 2003; 2(3): 519–28. [4] Paris JF. Statistical characterization of κ–μ shadowed fading. IEEE Trans Veh Technol 2014; 63(2): 518–26. [5] Cotton SL. A statistical model for shadowed body-centric communications channels: Theory and validation. IEEE Trans Antennas Propag 2014; 62(3): 1416–24. [6] Cotton SL. Human body shadowing in cellular device-to-device communications: Channel modeling using the shadowed κ–μ fading model. IEEE J. Sel. Areas Commun 2015; 33(1): 111–19. [7] Corazza GE, Vatalaro F. A statistical model for land mobile satellite channels and its application to nongeostationary orbit systems. IEEE Trans Veh Technol 1994; 43(3): 73842. [8] Suzuki H. A statistical model for urban radio propagation. IEEE Trans Commun 1997; 25(7): 673–80. [9] Abdi A, Kaveh M. K distribution: An appropriate substitute for Rayleigh-lognormal distribution in fading-shadowing wireless channels. IET Electron Lett 1998; 34(9): 851–52. [10] Shankar P.M. Error rates in generalized shadowed fading channels. Wirel Pers Commun 2004; 3: 233–38. [11] Sofotasios PC, Freear S. On the κ–μ/gamma composite distribution: A generalized multipath/shadowing fading model. In: Proc. SBMO/IEEE MTT-S Int Microw Optoelectron Conf (IM C’11). Oct. /Nov. 2011. p. 390–94. [12] Yoo SK, Cotton SL, Sofotasios PC, Freear S. Shadowed Fading in Indoor Off-Body Communication Channels: A Statistical Characterization Using the κ-µ/gamma Composite Fading Model. IEEE Trans Wirel Commun 2016; 15(8):5231-44. [13] Ambroziak SJ, Correia LM, Katulski RJ, Mackowiak M, Oliveira C, Sadowski J, Turbic K. An Off-Body Channel Model for Body Area Networks in Indoor Environments. IEEE Transactions on antennas and propagation 2016; 64(9): 4022-35.
[14] Petrillo L, Mavridis T, Sarrazin J, Delai AB, Doncker PD. Wideband Off-Body Measurements and Channel Modeling at 60 GHz. IEEE Antennas and wireless propagation letters 2017; 16: 1088-91. [15] Yoo SK, Cotton SL, Chun YJ, Scanlon WJ, Conway GA. Channel Characteristics of Dynamic Off-Body Communications at 60 GHz under Line-of-Sight (LOS) and Non-LOS Conditions. IEEE Antennas and Wireless Propagation Letters 2017; 16: 1553-56. [16] Yoo SK, Cotton SL. A Statistical Characterization of Shadowed Fading in Indoor Off-Body Communications Channels at 5.8 GHz. In: Proceeding European Conference on Antennas and Propagation (EuCAP’15). April 2015. [17] Yacoub MD. The κ–μ distribution and the η−μ distribution. IEEE Antennas Propag. Mag. 2007; 49(1): 68–81. [18] Yoo SK, Cotton SL, Sofotasios PC, Matthaiou M, Valkama M, Karagiannidis GK. The κ µ/inverse gamma fading model. In: Proceedings of IEEE Personal Indoor and Mobile Radio Communication (PIMRC’15). September 2015.p. 527–31. [19] Simon MK, Alouini MS. Digital Communication over Fading Channels. New York: John Wiley & Sons; 2005. [20] Gradshteyn IS, Ryzhik IM. Table of Integrals, Series, and Products. Academic Press, New York; 2007. [21] Costa DB da, Yacoub MD. Moment Generating Functions of Generalized Fading Distributions and Applications. IEEE Commun Lett 2008; 12(2):112-14. [22] Deergha RK. Performance of Digital Communication over Fading Channels. In: Channel Coding Techniques for Wireless Communications. New Delhi: Springer; 2015. [23] Vatalaro F, Corazza GE. Probability of Error and Outage in a Rice-Lognormal Channel for Terrestrial and Satellite Personal Communications. IEEE Trans Commun 1996; 44(8): 92124. [24] Costa DB da, Yacoub MD. Average Channel Capacity for Generalized Fading Scenarios. IEEE Commun Lett 2007; 11(12): 949-51.
S1434-8411(17)32390-7 https://doi.org/10.1016/j.aeue.2018.06.004 AEUE 52362
To appear in:
International Journal of Electronics and Communications
Received Date: Accepted Date:
9 October 2017 1 June 2018
Please cite this article as: H. Shankar, A. Kansal, Performance Analysis of κ-μ/gamma Shadowed fading model over Indoor off body Communication Channel, International Journal of Electronics and Communications (2018), doi: https://doi.org/10.1016/j.aeue.2018.06.004
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Title page with Author Detail
Title: Performance Analysis of κ-µ/gamma Shadowed fading model over Indoor off body Communication Channel
Author Details: (1) Hari Shankar: Research Scholar, Thapar University, India Email id - [email protected] (2) Ankush Kansal: Assistant Professor, Thapar University, India Email id - [email protected]
Performance Analysis of κ-µ/gamma Shadowed fading model over Indoor off body Communication Channel Abstract The κ-µ/gamma distribution has an importance role to model the small scale fading and shadowing over human off body indoor communication channel. This composite fading model has various special cases like κ-µ, Rician, Nakagami-m, Rayleigh, Rayleigh/gamma, Nakagami/gamma and Rice/logormal. In this paper, the expression for bit error rate (BER) using various modulation schemes, average channel capacity (ACC) and outage probability (OP) over κ-µ/gamma shadowed fading channel are derived. All the derived expressions are novel and presented in analytical form. The expression for BER and channel capacity are in form of wellknown Meijer G function, whereas the outage probability expression is obtained from cumulative distribution function (CDF) proposed in previous literature. The derived expressions of BER (BPSK), average channel capacity and outage probability reduces to special cases for validation purpose. The study shows that binary phase shift keying (BPSK) modulation technique has better BER performance as compare to other modulation techniques. Moreover, on increasing α and β while κ and µ kept constant and vice versa, the ACC get increases but below the Additive white Gaussian Noise (AWGN) channel capacity as expected. Also, better outage probability performance is obtained at lowest threshold signal to noise ratio (5dB). Keywords- Body Area Network, composite fading model, shadowing, small scale fading
1. INTRODUCTION In last decades, the composite fading models have been extensively used to model the multipath faded and shadowed environment. The composite fading model is the combination of multipath fading and shadowing. The distribution like Rician, Rayleigh, Nakagami-m and κ-µ are used to model the multipath faded scenarios whereas lognormal, gamma and inverse gamma distribution have been extensively used to model the shadowed environment [1]. The composite fading model can be further divided into two parts, depending on path of obstruction of signal; line of sight (LOS) and multiplicative shadowed fading. The former occurs when the object completely or partially obstruct the dominant LOS signal that causes random fluctuation in dominant signal component. Rice/lognormal [2], Rice-Nakagami [3], κ-µ/Nakagami [4] and κ-µ/lognormal [5], and κ-µ shadowed [6] are the example of this type of composite fading model. On other hand,
the latter occurs when the object obstructs both dominant and multipath signal component that causes the random fluctuations in total signal power. Rice/lognormal [7], Rayleigh/lognormal [8], Rayleigh/gamma (K distribution) [9], Nakagami-m/gamma (KG) [10] and κ-µ/gamma [11], [12] are example of multiplicative shadowed fading model. In body centric communication, the phenomenon like multipath fading, large scale fading and path loss has an important role to characterize the received signal. The body centric communication channel, depending on the position of transmitter and receiver(s), can be defined in three different ways; off body, on body and body to body communicational channel. In off body communication, one or more device is located on human body which communicates with local base station or transceiver. In on body communication, the antennas are positioned on human body so the shadowing occurs due to body part that causes attenuation in the received signal and in body to body communication; the antennas are located on different human bodies. Recently, various literatures have been studied that represent the characteristics of Indoor off body communication channel using different operating frequencies, types of antenna, body worn locations and user movements[12-16]. κ-µ distribution was proposed in [17] that have various special cases like Rice (Nakagami-n), Nakagami-m, Rayleigh and one sided Gaussian distribution. The measurement was performed to validate the proposed distribution in [17]. κ-µ/ Nakagami-m shadowed fading model was proposed in [4] which is applicable to land mobile satellite (LMS) communications and underwater acoustic communications (UAC). In [5], the author proposed κ-µ/lognormal distribution that characterizes the body centric communication channel. LOS Shadowed fading model was proposed in [6] in which resultant dominant component is characterized by Nakagami –m distribution. In [6], the measurements were performed in outdoor environment that agreed with proposed model. The κ-µ/gamma multiplicative shadowed fading model for indoor off body communication was proposed in [12] that was initially proposed in [11]. In [12], the investigations were performed at 5.8 GHz in three different indoor environments i.e. indoor laboratory, open office area and seminar room and in each indoor environment four scenarios were taken out: line of sight, NLOS walking, random and rotational movements. The author further, in [12], derived the expression for PDF (probability density function), cumulative distribution function, moment generating function (MGF), moment and Amount of fading (AF). In [18], the κ-µ /inverse composite fading model was proposed, in which mean signal power
follows inverse gamma distribution. In [18], the measured values, performed in body centric communication channel, were fitted to theoretical results. Bit error rate (BER), channel capacity, outage probability has an important role in performance analysis over wireless communication system. In [12], the author derived the expression for probability density function (PDF), cumulative distribution function (CDF), moment generating function (MGF) and Amount of fading (AF) of κ-µ/gamma shadowing fading model. However, the performance analysis of digital communication system over κµ/gamma shadowed fading channel did not further elaborate therein by authors and never reported in previous literature to the best of author’s knowledge. So, the main work of this paper is to derive the expression for BER using different modulation schemes, channel capacity and outage probability over κ-µ/gamma indoor off body communication channel. All derived expressions are presented in infinite series form. The formulas for BER and channel capacity are presented in term of well-known Meijer G function whereas, the expression for outage probability is given in term of hyper geometric function that can be easily implemented in most standard software packages like Maple and Mathematica. The numerical result of average BER (Binary Phase Shift Keying), average channel capacity and outage probability of κ-µ/gamma shadowed fading model are compared (for validation purpose) with ABER (BPSK), ACC and OP of other fading/shadowed fading model, respectively. The work of the paper is organized as follows. In Section-2, PDF of κ-µ/gamma shadowed fading model is presented with its numerous special cases. The expression for ABER, average channel capacity and outage probability are derived in Section-3. Further, in Secrtion-4, the numerical results of ABER, ACC and OP are presented and compared with its special cases. Finally, the paper is summarized with some concluding remark in Section-5. 2. κ-µ/GAMMA SHADOWED FADING MODEL Let X is the envelope of κ-µ /gamma shadowed fading model. The pdf of this fading model is [12],
f X ( x) j 0
4
3 j 2
j
k (1 k ) j
2
x j 1
j!exp( k )( )( j )
j 2
(1 k ) K j 2 x
(1)
where, Km(.) is the modified Bessel function of second kind with m order, x is the envelope of received signal. Parameter µ and κ are related to the κ-µ distribution. µ is the multipath clustering. κ is the ratio of total power of dominant components to the total power of scattered waves. Moreover, the parameter α and β are related to Gamma distribution. α and β is the shape and scale parameter, respectively. κ-µ/gamma model has four parameters κ, µ, α and β. So (1) can be reducing to the various small scale fading model for different value of κ, µ, α and β. For example, as α→∞, β =1.6, the PDF of this model coincides with κ-µ, Rician, Nakagami-m, and Rayleigh with different values of κ and µ. By setting κ = 4.2 and µ = 1.3, it coincides with κ-µ distribution [12, 17]. As, κ = K (Rician Factor) = 3.2 and µ = 1, it approximates the Rician distribution [1, 12]. By putting κ→0 and µ=1, κ-µ/gamma shadowed fading distribution reduce to Rayleigh distribution. In similar way, the Nakagami- m model can be obtain by setting κ→0 with µ parameter equal to m parameter [1,12]. Moreover this composite fading model reduces to other shadowed fading distribution like Rayleigh/gamma [9, 12], Nakagami-m/gamma [10, 12] and Rice-lognormal distribution [2, 12] as special case. For example as κ→0, µ = 1, α = 7.8 and β = 0.7, it coincides with Rayleigh/gamma composite fading distribution. Likewise with κ→0, α = 2.4 and β = 1.5, (1) reduces to Nakagami-m/gamma with µ parameter equals to m parameter. Also, by setting κ = K (Rician Factor) = 2.4, µ = 1, α = 43.9 and β = 0.05, it coincides with Rice Lognormal distribution. The shape parameter (σ) and log-scale parameter (u) of lognormal shadowing can be obtained from α and β parameters of Gamma distribution. Let us consider a signal s, with Es energy symbol, is transmitted over a κ-µ/gamma shadowed faded channel. The received signal z can be defined as, z = s X + n, where n is AWGN (Additive White Gaussian Noise) with one sided power spectral density (psd) of N0, X denotes the small scale fading/shadowing envelope which is κ-µ/gamma distributed. The instantaneous SNR per symbol is defined as γ = X2 Es/N0. So the average SNR per symbol becomes γ = Ω Es/N0, where Ω = E[X2], is the mean squared value of κ-µ/gamma shadowed faded envelope. Hence, the PDF of instantaneous SNR per symbol can be defined as [19], 2 f ( ) f Z
It is noted that Ω = E[X2] can be obtained by putting n=2 in [12, eq. (14)], which is given as
(2)
E[ X 2 ]
exp( k )( 1)( 1)1 F1 ( 1; ; k ) ( )( ) (1 k )
(3)
By using (1) and (2), the expression for PDF of signal to noise ratio (SNR) can be written as,
f ( ) j 0
2
3 j 2
j 2
j j!exp( k )( )( j ) 2 k j (1 k )
j 2
j 2 2
(1 k ) K j 2
d
(4)
In the next section, we will derive the expression of average BER for different modulation schemes, average channel capacity and outage probability over κ-µ/gamma shadowed faded channel using (4).
3. PERFORMANCE ANALYSIS 3.1 Average Bit Error Rate ABER of any modulation technique over shadowed faded channel disturbed with AWGN is calculated by averaging the conditional BER for AWGN channels over PDF of instantaneous SNR. The expression for average BER is defined as [19],
PE ( E ) P ( E | ) f ( )d
(5)
0
where, P(E|γ) is the conditional error probability for AWGN channel and fγ(γ) is the SNR PDF. The conditional bit error rate in an AWGN channel can be written as [19],
Pb ( E | )
(b, a ) 2(b)
(6)
where, a = 1 and b = 0.5 for BPSK, a = 0.5, b = 0.5 for BFSK, a = 0.5, b = 1 for Differentially coherent BPSK (DPSK) and a = 1, b = 1 for non-coherent BFSK (NCFSK). Γ(w, x) is the complementary incomplete gamma function. Now, substituting (4) and (6) in (5), the average BER becomes,
(b, a ) PE 2(b) j 0 0
2
3 j 2
k j (1 k )
j 2
j!exp( k )( )( j )
j 2
j 2
j 2 2
(1 k ) K j 2
d
(7)
d
(8)
By taking the constant term outside the integral, the above equation becomes, 1 PE 2(b) j 0
2
3 j 2
k j (1 k )
j 2
j!exp( k )( )( j )
j 2
j 2
(b, a ) 0
j 2 2
(1 k ) K j 2
The incomplete gamma function can be represented in term the of Meijer G function as [1],
1 (b, a ) G12, 2,0 a b 0
(9)
so, (8) becomes 1 PE 2(b) j 0
2
3 j 2
j 2
j 2 j!exp( k )( )( j ) k j (1 k )
j 2
(1 k ) 1 2 j 2 2 G a K j 0 b 0 2, 0 1, 2
d
(10)
After taking the substitution and by using Table of Integral Series and Product [20], (10) can be represented as, 1 PE 2(b) j 0
2
3 j
j
j
2
2
j (1 k ) 2 j!exp( k )( )( j ) 2
k j (1 k )
a 1, j 1,1 G32,,22 (1 k ) b,0
(11)
The above equation is the expression for average BER over κ-µ/ gamma shadowed faded channel. The above expression coincides with BER expression of small scale fading models like κ-µ, Rician, Nakagami–m and Rayleigh as a special case and also coincides with BER expression of composite shadowed fading model like Rayleigh-Gamma, Nakagami-m-Gamma and Rice lognormal.
3.2 Average Channel Capacity Let us consider, a κ-µ/ gamma shadowed faded channel has a bandwidth W. According to Shannon, the average channel capacity is given by [1, eq. (4.200)],
C W log 2 (1 ) f ( )d
(12)
0
where, fγ(γ) is the PDF of instantaneous SNR per symbol γ as given in (4). The normalized channel capacity can be given by,
C C / W log 2 (1 ) f ( )d
(13)
0
By putting the expression of (4) in (13) we get,
C log 2 (1 ) 0
j 0
2
3 j
j
j
2
2
j 2 j!exp( k )( )( j ) 2
k j (1 k )
j 2 2
(1 k ) K j 2
The function ln (1+γ) can be expressed in term of Meijer G function as [1, eq. (4.279)],
d (14)
1,1 ln(1 ) G21,,22 1,0
(15)
Hence, (14) becomes, 1 C ln( 2) j 0
2
3 j
j
j
2
2
j j!exp( k )( )( j ) 2 2
k j (1 k )
(1 k ) 1,1 2 j 2 1, 2 G K j 2 2 , 2 0 1,0
d
(16)
By using substitution property of integration and using Table of Integral Series [20], the above equation can be written as, 1 C ln( 2) j 0
3 j
j
j
2
2
j (1 k ) 2 j!exp( k )( )( j ) 2
k j (1 k )
1, j 1,1,1 G41,,42 (1 k ) 1,0
(17)
This is the expression for normalized ACC over κ-µ/gamma composite fading channel. In the next subsection, we will derive the expression for outage probability using cumulative distribution function proposed in previous literature.
3.3 Outage Probability (Pout) The probability of outage (Pout) is defined as the probability that the instantaneous SNR per symbol γ falls below the specified threshold SNR per symbol γ th. The outage probability can be obtained by taking integration of PDF of instantaneous SNR with limits of zero to γth which is nothing but cumulative distribution function of specified threshold SNR (γ th). Hence it can be written as [1, eq. (4.154)],
Pout Pr { 0 }
th
f ( )d F (
0
)
(18)
0
where, fγ(γ) is PDF of instantaneous SNR per symbol(γ), γth is specified SNR threshold value and Fγ(γ) is the CDF of instantaneous SNR per symbol. Instantaneous SNR per symbol is defined as γ =X2 Es/N0 and we know that the average SNR per symbol can be obtained as, =E[γ] = E[x2]* Es/N0=Ω* Es/N0, where Ω is the Mean square value of x as given in (3), hence γ = X2/ Ω. So the CDF of γ i.e. Fγ(γ)=Fz((γ Ω/ )1/2). Therefore the outage probability can be express as, Pout FX th
From [12, eq. (8)], the outage probability can be evaluated as,
(19)
(1 ) th (1 ) th 1 F2 ( ; j 1,1 ; j 0 j! exp( ) sin(( j ) ) (1 j )( j )(1 ) j
Pout
j
j
th (1 ) th 1 F2 ( j;1 j ,1 j; j j!exp( ) sin(( j ) ) (1 j )( )(1 j )
j 2 j (1 ) j
j 0
(20)
This expression represents the outage probability of κ-µ/gamma shadowed fading model. In the next section, we will present the numerical results of average BER, ACC and OP of κ-µ/gamma distribution and compare it with existing model proposed in previous literature.
4. NUMERICAL ANALYSIS AND DISCUSSION In this section, the derived expression for ABER, ACC and OP are numerically evaluated and these performance matrices are compared with performance matrices of existing model like Rician,
Rayleigh,
κ-µ,
Nakagami-m,
Rayleigh/gamma,
Nakagami-m/gamma
and
Rice/lognormal. Fig.1 and Fig.2 shows the numerical result of derived expression of ABER as given in (11) over BPSK modulation scheme for κ-µ/gamma composite fading channel. Fig.1 depicts the plot of ABER as the function average SNR for different parameter κ, µ, α and β. We have used coherent BPSK modulation scheme with modulation parameter a=1 and b=0.5 to plot the curve. The continuous line shows the ABER of κ-µ/gamma composite fading model for different values of function parameter. The results of ABER over κ-µ/gamma channel are compared with result of ABER over small scale fading channel model like κ-µ, Rician, Nakagami-m and Rayleigh. With κ = 4.2, µ = 1.3, α = 100 and β = 1.6, it reduces to BER expression of κ-µ model (square line) [21, eq. (12)] and for κ = 3.2, µ = 1.0, α = 100 and β =1.6, the result of (11) overlapped to BER of Rician model (cross) having same Rice factor (K) as κ [22, eq. (2.10)]. Similarly for κ→0, α = 100 and β =1.6, the result of (11) coincides with BER of Nakagami-m distribution (Asterisk) having m parameter equal to µ parameter [1, eq. (4.138)]. Moreover, for κ→0, µ = 1.0, α = 100 and β = 1.6, the expression for BER of κ-µ/gamma shadowed model agree with BER of Rayleigh model (circle) [1, eq. (4.134)]. From the fig.1, it is observed that on increasing the value of κ, the error rate drop get increases. The result shows that the curve for κ = 3.2 (Rician case) has sharp BER than for k→0 (Rayleigh case).
Fig.1 Average BER verses γ for BPSK over κ-µ/gamma shadowed fading channel (continuous line) with special cases: BER of κ-µ (square), Nakagami-m (Asterisk), Rician (cross) and Rayleigh (circle). The values of α and β are 100 and 1.6, respectively. Fig.2 shows the plot of ABER of BPSK modulation scheme for different parameter values. The results of BER of κ-µ/gamma shadowed model are compared with BER of other composite fading models like Rayleigh/gamma, Nakagami-m/gamma and Rice/lognormal to validate the proposed model. In Fig.2, solid line shows the BER of κ-µ/gamma shadowed fading model for different four parameter values, κ, µ, α and β. For κ→0, µ=1, α=7.8 and β=0.7, (11) agree with BER of Rayleigh/gamma composite model (with m=1 and N=1) [1, eq. (4.152)]. Similarly, for κ→0, α=2.4 and β=1.5, (11) overlapped with BER of Nakagami-m /gamma having m parameter equal to µ parameter [1, eq. (4.152)]. Moreover, for µ=1, α=43.9 and β=0.05, (11) agree with BER of Rice/lognormal having same Rician factor (K) as κ [23, eq. (6)]. We have used shape parameter σ = 0.15 and log-scale parameter u=0.7 of lognormal shadowing to plot the BER of Rice lognormal shadowed fading model. The value of these two parameters can be estimated from parameter α and β [12]. From Fig.2, it is observed that Nakagami-m based composite model has better BER performance as compared to Rayleigh based composite model.
Fig.2 Average BER verses γ for BPSK over κ-µ/gamma shadowed fading channel (continuous line) with its special cases: BER of Rayleigh/gamma (circle), Rice/lognormal (cross) and Nakagami-m/gamma (square).
Fig.3 Average BER verses γ for different modulation scheme over κ-µ/gamma composite fading channel. BPSK (Solid line), BFSK (Dot), DPSK (Dash), NCFSK (Dash dot).
Fig.3 depicts the analytical results of average BER under different modulation schemes like coherent BPSK, coherent BFSK, differentially coherent BPSK (DPSK) and non-coherent BFSK (NCFSK). Various modulation schemes depend on the modulation specific parameters a and b as discussed in previous section. From the fig.3, it is observed that on increasing the κ and µ parameter while α and β kept constant, the error rates drop over any modulation schemes get increases. Similarly, on increasing the shape (α) and scale parameter (β) while κ and µ kept constant, the error rates drop get also increases. Result shows that the bit error probability of BPSK at fixed average SNR is lowest as compare to other modulation schemes (BFSK, DPSK or NCFSK).
Fig.4 Average channel capacity verses γ over κ-µ/gamma shadowed fading channel with its special cases: ACC of of κ-µ (cross), Rician (square), Nakagami-m (Asterisk), and Rayleigh (circle). The values of α and β are 100 and 1.6, respectively. Fig. 4 shows the plot of ACC per unit bandwidth (W) verses average SNR over κ-µ/gamma fading channel (continuous line). As we know that as α→∞, the κ-µ/gamma shadowed fading model reduces to small scale fading model like κ-µ, Rician, Nakagami-m and Rayleigh model. For κ=4.2, µ=1.3, the capacity of κ-µ/gamma shadowed model reduces to capacity of κ-µ fading model [24, eq. (6)] and for κ=3.2, µ=1, (17) reduces to capacity of Rician fading having Kfactor same as κ [1, eq. (4.210)]. Similarly with κ→0, µ=1.4, (17) reduce to capacity of
Nakagami-m fading model with same Nakagami parameter (m) as µ parameter [1, eq. (4.207)] and for κ→0, µ=1, (17) reduces to capacity of Rayleigh model [1, eq. (4.208)]. Fig.5 depicts the analytical results of average channel capacity over κ-µ/gamma shadowed fading channel (continuous line) overlapped with result of ACC of existing model like Rayleigh/gamma (cross), Nakagami-m/gamma (square). As κ→0, µ=1.4, α=7.8 and β=0.7, (17) coincides with capacity of Rayleigh/gamma model with m=1 [1, eq. (4.214)]. Similarly with κ→0, µ = 1.4, α = 2.4 and β = 1.5, (17) reduce to channel capacity of Nakagami-m/gamma shadowed fading model [1, eq. (4.214)].
Fig.5 Average channel capacity verses γ over κ-µ/gamma shadowed fading channel (continuous line) with its special cases: ACC of Rayleigh/gamma (cross), Nakagami-m/gamma (square). Fig.6 depicts the plot of ACC per unit bandwidth verses average SNR over κ-µ/gamma composite fading model for different values of four parameters κ, µ, α and β. The capacity of AWGN channel (continuous line) is more as compare to κ-µ/gamma model as expected. From fig. 6, it is observed that, as the κ, µ parameter increases while α and β kept constant and vice versa, the channel capacity gets increases.
Fig.6 Average channel capacity verses γ over κ-µ/gamma composite fading channel for different values of κ, µ, α, and β.
Fig.7 utage Probability verses γ over κ-µ/gamma shadowed fading channel (continuous line) with its special cases: Outage Probability of Rician (cross), Nakagami-m (circle), Rayleigh (Asterisk), Rayleigh/ gamma (solid square), Nakagami-m/gamma (square). The fixed threshold SNR (γth) = 10dB.
Fig. 7 depicts the numerical results of outage probability of κ-µ/gamma distribution (continuous line) as the function of average signal to noise ratio (SNR). The outage probability of this shadowed fading are compared with outage probability of other existing model like Rician, Nakagami-m, Rayleigh, Rayleigh/ gamma and Nakagami-m/gamma. As α=100, β=1.6, κ→0, (21) reduce to outage probability of Nakagami-m distribution with m-parameter equal to µ parameter [1,eq. (4.157)], as κ→0, µ = 1,α=100, β=1.6, (21) reduce to outage probability of Rayleigh model [1, Table (4.4)], as µ=1, α=100, β=1.6 , (21) reduce to outage Probability of Rician model with K parameter equals to κ parameter [19, Table 9.5], As κ→0, u=1, α = 7.8, β=0.7, (21) reduce to outage probability of Rayleigh/gamma[1, Table(4.4)] and as κ→0, α = 2.4, β=1.5, (21) reduce to P of Nakagami-m/gamma model (Generalized K ) [1,Table (4.4)].
Fig.8 utage Probability verses γ over κ-µ/gamma shadowed fading channel for different values of parameters with different threshold SNR (γth = 5, 10 and 15 dB). Fig.8 illustrates the numerical result of outage probability of κ-µ/gamma composite fading model as the function average SNR with three different threshold SNR values 5, 10 and 15 dB. We have plot three cluster curves by using different κ, µ, α, β parameters for 5 to 15dB threshold SNR values. From the figure, it is observed that on increasing the values of threshold SNR, the probability of outage get increases at any fixed average SNR values. For example, with 40dB
average SNR, the OP is high for 15dB threshold SNR as compared to 5dB threshold SNR. Moreover on increasing the κ, µ parameter for fixed values of α, β and γth, the OP performance get increases. Similarly, on increasing α, β parameter with fixed κ, µ and γth, the OP performance also get increases as illustrated in fig.8.
CONCLUSION In this paper, we have derived the analytical expression for average bit error rate (ABER) using different modulation scheme, average channel capacity (ACC) and outage probability of wireless communication system over κ-µ/gamma shadowed fading channel. This shadowed fading model is suitable for indoor off body communication channel and have various special cases that is it can be reduce to small scale fading model like κ-µ, Rician, Nakagami-m, Rayleigh, and shadowed fading model like Rayleigh/gamma, Nakagami-m/gamma and Rice/lognormal. The expression for ABER and ACC are presented in term of Meijer G function while OP expression is presented in term of hypergeometric function that can be easily implemented in most commonly available software package like Maple and Mathematica. Further, we have presented the numerical result of ABER (BPSK), ACC and
P of κ-µ/gamma shadowed fading model and
compared it with the ABER (BPSK), ACC and OP, respectively, of existing model like κ-µ, Rician, Nakagami-m, Rayleigh Rayleigh/gamma, Nakagami-m/gamma and Rice/lognormal. The study shows that error probability for BPSK modulation scheme is minimum as compared to other modulation schemes like BFSK, DPSK and NCFSK at fixed average SNR for any parameter values. For example at 40dB average SNR for κ =1, µ =1, α =2, β =2, the BER for BPSK, BFSK, DPSK and NCFSK is 3.51×10-5,6.99×10-5,6.35×10-5 and 13.1×10-5, respectively that shows BPSK modulation schemes has better BER performance. Moreover, as κ and µ increases while α and β kept constant and vice versa, the average channel capacity gets increases but below the AWGN channel capacity as expected. Further, on increasing the values of κ and µ (0.5 to 1.5) while α and β kept constant with any threshold SNR values ranges between 5 and 15dB at 60dB average SNR, the probability of outage gets decreases in the range of 10 -2 to 10-5 approximately that improve the system performance. References [1] Shankar PM. Fading and Shadowing for Wireless system. New York: Springer; 2005.
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