The effects of scatterers, fluctuation parameter and propagation clusters significantly affect the performance of κ- μ shadowed fading channel. On the other hand, opportunistic relaying is an efficient technique to improve the performance of fading channels reducing the effects of aforementioned parameters. Motivated by these issues, in this paper, a secure wireless multicasting scenario through κ- μ shadowed fading channel is considered in the presence of multiple eavesdroppers with opportunistic relaying. The main purpose of this paper is to ensure the security level in wireless multicasting compensating the loss of security due to the effects of power ratio between dominant and scattered waves, fluctuation parameter, and the number of propagation clusters, multicast users and eavesdroppers, by opportunistic relaying technique. The closed-form analytical expressions are derived for the probability of non-zero secrecy multicast capacity (PNSMC) and the secure outage probability for multicasting (SOPM) to understand the insight of the effects of above parameters. The results show that the loss of security in multicasting through κ- μ shadowed fading channel can be significantly enhanced using opportunistic relaying technique by compensating the effects of scatterers, fluctuation parameter, and the number of propagation clusters, multicast users and eavesdroppers.
The statistical characterization of fading is an important issue in wireless communication system. The most promising method, for improving wireless links by reducing the impacts of fading, scatterers, fluctuations, etc., is cooperative diversity with the best relay selection [
Recently, zero-forcing (ZF) multiplexing system was used in [
Wyner’s wiretap model was used in [
Based on the aforementioned scenario available in the literature and motivated by the importance of security in multicasting, the authors in this work investigated a secure wireless multicasting scenario using κ-μ shadowed fading channels when multiple eavesdroppers are present. Authors developed a mathematical model to ensure that security in κ-μ shadowed fading channels with opportunistic relaying system. We can summarize the major contributions of this paper below.
● At first, based on the probability density function (PDF) of κ-μ shadowed fading channels [
● Then, the effects of power ratio between dominant and scattered waves, fluctuation parameter, and the number of propagation clusters, multicast users, eavesdroppers and relays on the PNSMC and the SOPM are investigated.
● Finally, the derived analytical expressions are justified via Monte-Carlo simulation.
The remaining part of this paper has been arranged as follows. Section II explains the system model and section III narrates the problem formulation. The equations for the PNSMC and the SOPM have been derived respectively in Section IV and V. Section VI describes the numerical results. In the end, the conclusions of this paper are represented in Section VII.
A secure wireless multicasting communication scenario shown in
channels experience independent κ-μ shadowed fading. κ and μ denote two physical fading parameters of this system. κreflects a real positive number representing the power ratio between dominant and scattered-wave components and μ reflects a positive real number representing the effective number of propagation clusters.
This section deals with the derivation of the closed-form analytical expressions for the PDFs of multicast channels and eavesdropper’s channels.
Let γ denotes the SNR of each sub-channel of multicast channel. Then, the PDF of γ denoted by f γ ( γ ) for the κ-μ shadowed fading channels is given by [
f γ ( γ ) = μ μ m m ( 1 + k ) μ Γ ( μ ) ( m + μ k ) m γ ¯ ( γ γ ¯ ) μ − 1 e − μ ( 1 + k ) γ γ ¯ F 1 1 ( m , μ ; μ 2 k ( 1 + k ) μ k + m γ γ ¯ ) , (1)
where F 1 1 ( .,. ; . ) is the confluent hypergeometric function which is defined as
F 1 1 ( p , q ; r ) = Γ ( q + 1 ) Γ ( p + 1 ) ∑ n = 0 ∞ Γ ( p + n ) Γ ( q + n ) r n n ! . Substituting this formula in Equation (1), f γ ( γ ) can be found as
f γ ( γ ) = ∑ n = 0 ∞ A 1 A 4 γ n + μ − 1 e − A 2 γ , (2)
where A 1 = μ μ m m ( 1 + k ) μ Γ ( μ ) ( m + μ k ) m γ ¯ μ , A 2 = μ ( 1 + k ) γ ¯ , A 3 = μ 2 k ( 1 + k ) ( μ k + m ) γ ¯ and A 4 = Γ ( μ + 1 ) Γ ( m + 1 ) ∑ n = 0 ∞ Γ ( m + n ) Γ ( μ + n ) A 3 n n ! .
Let γ s , k denotes the SNR of source-to-kth relay. Then, the PDF of γ s , k denoted by f γ s , k ( γ s , k ) for the κ-μ shadowed fading channels is given by,
f γ s , k ( γ s , k ) = ∑ n = 0 ∞ s 1 s 4 γ s , k n + μ − 1 e − s 2 γ s , k , (3)
where s 1 = μ μ m m ( 1 + k ) μ Γ ( μ ) ( m + μ k ) m γ ¯ s μ , s 2 = μ ( 1 + k ) γ ¯ s , s 3 = μ 2 k ( 1 + k ) ( μ k + m ) γ ¯ s and s 4 = Γ ( μ + 1 ) Γ ( m + 1 ) ∑ n = 0 ∞ Γ ( m + n ) Γ ( μ + n ) s 3 n n ! .
Let γ k , i denotes the SNR of kth relay-to-ith destination receiver. Then, the PDF of the SNR of γ k , i denoted by f γ k , i ( γ k , i ) for the κ-μ shadowed fading channel is given by,
f γ k , i ( γ k , i ) = ∑ n = 0 ∞ a 1 a 4 γ k , i n + μ − 1 e − a 2 γ k , i , (4)
where a 1 = μ μ m m ( 1 + k ) μ Γ ( μ ) ( m + μ k ) m γ ¯ a μ , a 2 = μ ( 1 + k ) γ ¯ a , a 3 = μ 2 k ( 1 + k ) ( μ k + m ) γ ¯ a and a 4 = Γ ( μ + 1 ) Γ ( m + 1 ) ∑ n = 0 ∞ Γ ( m + n ) Γ ( μ + n ) a 3 n n ! .
Let γ k , j denotes the SNR of kth relay-to-jth eavesdropper. Then, the PDF of the SNR of γ k , j denoted by f γ k , j ( γ k , j ) for κ-μ shadowed fading channels is given by,
f γ k , j ( γ k , j ) = ∑ n = 0 ∞ b 1 b 4 γ k , j n + μ − 1 e − b 2 γ k , j , (5)
where b 1 = μ μ m m ( 1 + k ) μ Γ ( μ ) ( m + μ k ) m γ ¯ b μ , b 2 = μ ( 1 + k ) γ ¯ b , b 3 = μ 2 k ( 1 + k ) ( μ k + m ) γ ¯ b and b 4 = Γ ( μ + 1 ) Γ ( m + 1 ) ∑ n = 0 ∞ Γ ( m + n ) Γ ( μ + n ) b 3 n n ! .
Let γ i denotes the SNR of best relay for source-to-destination link. Then, the
CDF of γ i denoted by F γ ∑ d i ( γ i ) for the κ-μ shadowed fading channels is given by,
F γ ∑ d i ( γ i ) = { F γ e q k ( γ i ) } R = { 1 − P r ( γ s , k > γ i ) P r ( γ k , i > γ i ) } R , (6)
where P r ( γ s , k > γ i ) ≜ ∫ γ i ∞ f γ s , k ( γ s , k ) d γ s , k and P r ( γ k , i > γ i ) ≜ ∫ γ i ∞ f γ k , i ( γ k , i ) d γ k , i . Substituting the values of f γ s , k ( γ s , k ) and f γ k , i ( γ k , i ) , and using the following identity of [
∫ u ∞ x n e − μ x d x = e − μ u ∑ k = 0 n ( n ! u k k ! μ n − k + 1 ) (7)
the value of P r ( γ s , k > γ i ) and P r ( γ k , i > γ i ) can be calculated in closed-form. Then, substituting the values of P r ( γ s , k > γ i ) and P r ( γ k , i > γ i ) in Equation (6) and using the following identities of [
{ ∑ k = 0 ∞ ( a k x k ) } n = ∑ k = 0 ∞ ( c k x k ) (8)
( a + x ) n = ∑ i = 0 n ( n i ) x i a n − i , (9)
where c m = 1 m a 0 ∑ k = 1 m ( k n − m + k ) a k c m − k for m ≥ 1 and c 0 = a 0 n for m = 0 and n is an integer number, the closed-form expression for the F γ ∑ d i ( γ i ) can be derived as
F γ ∑ d i ( γ i ) = ∑ k 1 = 0 ∞ ∑ k 2 = 0 R ω 4 γ i 2 k 1 e − ω 5 k 2 γ i , (10)
where ω 4 = ∑ k 2 = 0 R R ! ( − 1 ) k 2 R ! ( R − k 2 ) ! [ ∑ n = 0 ∞ ω 1 ( n + μ − 1 ) ! q n + μ ] 2 k 2 , ω 5 = q γ ¯ s + q γ ¯ m and q = μ ( 1 + k ) .
Let γ j denotes the SNR of best relay for source-to-eavesdropper link. Then, the
CDF of γ j denoted by F γ ∑ e j ( γ j ) , for κ-μ shadowed fading channels is given by,
F γ ∑ e j ( γ j ) = { F γ q k ( γ j ) } R = { 1 − P r ( γ s , k > γ i ) P r ( γ k , j > γ j ) } R , (11)
where P r ( γ s , k > γ j ) ≜ ∫ γ j ∞ f γ s , k ( γ s , k ) d γ s , k and P r ( γ k , j > γ j ) ≜ ∫ γ j ∞ f γ k , j ( γ k , j ) d γ k , j . Substituting the values of f γ s , k ( γ s , k ) and f γ k , j ( γ k , j ) , and using the identities of Equations (7)-(9), the closed-form expression for the F γ ∑ e j ( γ j ) can be defined as
F γ ∑ e j ( γ j ) = ∑ k 1 = 0 ∞ ∑ k 2 = 0 R β 3 γ j 2 k 1 e − β 2 k 2 γ j , (12)
where β 2 = q γ ¯ s + q γ ¯ e , q = μ ( 1 + k ) and
β 3 = ∑ k 2 = 0 R R ! k 2 ! ( R − k 2 ) ! ( − 1 ) k 2 [ ∑ n = 0 ∞ ω 1 ( n + μ − 1 ) ! q n + μ ] 2 k 2 .
The PDF of γ i denoted by f γ ∑ d i ( γ i ) for κ-μ shadowed fading channel can be obtained as
f γ ∑ d i ( γ i ) = R f γ e q k ( γ i ) [ F γ e q k ( γ i ) ] R − 1 , (13)
where the value of f γ e q k ( γ i ) can be calculated from the following expression
f γ e q k ( γ i ) = d d γ i { F γ e q k ( γ i ) } . (14)
Substituting the value of F γ e q k ( γ i ) in Equation (14) and after differentiation, the closed-form expression of f γ e q k ( γ i ) is given by,
f γ e q k ( γ i ) = ( − 2 k 1 γ i 2 k 1 − 1 + ω 5 γ i 2 k 1 ) ω 3 e − ω 5 γ i , (15)
where ω 3 = ∑ k 1 = 0 ∞ ω 2 2 ( γ ¯ s γ ¯ m ) k 1 and ω 5 = q γ ¯ s + q γ ¯ m . Finally, substituting the values of f γ e q k ( γ i ) and F γ e q k ( γ i ) in Equation (13) we have
f γ ∑ d i ( γ i ) = ( ω 7 γ i v 1 + ω 8 γ i v 2 ) e − ω 9 γ i , (16)
where ω 7 = ∑ k 1 = 0 ∞ − 2 ω 3 ω 6 k 1 R , ω 8 = ω 3 ω 5 ω 6 R , ω 9 = ∑ k 3 = 0 R − 1 ω 5 ( 1 + k 3 ) , v 1 = v 2 − 1 and v 2 = 4 ∑ k 1 = 0 ∞ k 1 .
The PDF of γ j denoted by f γ ∑ e j ( γ j ) for κ-μ shadowed fading channel can be defined as
f γ ∑ e j ( γ j ) = R f γ q k ( γ j ) [ F γ q k ( γ j ) ] R − 1 . (17)
The value of f γ q k ( γ j ) is calculated as follows,
f γ q k ( γ j ) = d d γ j { F γ q k ( γ j ) } = ( − 2 k 1 γ j 2 k 1 − 1 + β 2 γ j 2 k 1 ) β 1 e − β 2 γ j , (18)
where β 1 = ∑ k 1 = 0 ∞ ω 2 2 ( γ ¯ s γ ¯ e ) k 1 and β 2 = q γ ¯ s + q γ ¯ e . Substituting the values of f γ q k ( γ j ) and F γ q k ( γ j ) in Equation (17), the closed-form expression for the f γ ∑ e j ( γ j ) can be derived as,
f γ ∑ e j ( γ j ) = ( β 5 γ j v 1 + β 6 γ j v 2 ) e − β 7 γ j , (19)
where β 5 = ∑ k 1 = 0 ∞ − 2 k 1 β 1 β 4 , β 6 = β 1 β 2 β 4 , β 7 = ∑ k 3 = 0 R − 1 β 2 ( 1 + k 3 ) , v 1 = v 2 − 1 and v 2 = 4 ∑ k 1 = 0 ∞ k 1 .
Let d min = min 1 ≤ i ≤ M γ i denotes the SNR of multicast channel. Then, the PDF of d min denoted by f d m i n ( γ i ) , for the κ-μ shadowed fading channel is given by,
f d min ( γ i ) = M f γ ∑ d i ( γ i ) { 1 − F γ ∑ d i ( γ i ) } M − 1 . (20)
Substituting the value of f γ ∑ d i ( γ i ) and F γ ∑ d i ( γ i ) in Equation (20), the closedform analytical expression for the f d min ( γ i ) can be derived as
f d min ( γ i ) = ( η 3 γ i k 6 + η 4 γ i k 7 ) e − η 5 γ i , (21)
where
η 3 = ∑ k 4 = 0 M − 1 M ( M − 1 ) ! k 4 ! ( ( M − 1 ) − k 4 ) ! ( ω 4 ) k 4 ω 7 ,
η 4 = ∑ k 4 = 0 M − 1 M ( M − 1 ) ! k 4 ! ( ( M − 1 ) − k 4 ) ! ( ω 4 ) k 4 ω 8 ,
η 5 = ∑ k 2 = 0 R ∑ k 4 = 0 M − 1 ( ω 9 + ω 5 k 2 k 4 ) ,
k 6 = ∑ k 1 = 0 ∞ ∑ k 4 = 0 M − 1 ( v 1 + 2 k 1 k 4 ) and
k 7 = ∑ k 1 = 0 ∞ ∑ k 4 = 0 M − 1 ( v 2 + 2 k 1 k 4 ) .
Let d max = max 1 ≤ j ≤ N γ j denotes the maximum SNR of eavesdropper’s channel. Then, the PDF of d max denoted by f d max ( γ j ) , for the κ-μ shadowed fading channel is given by,
f d max ( γ j ) = N f γ ∑ e j ( γ j ) { F γ ∑ e j ( γ j ) } N − 1 . (22)
Substituting the values of f γ ∑ e j ( γ j ) and F γ ∑ e j ( γ j ) in Equation (22), the closed-form analytical expression for the f d max ( γ j ) can be derived as
f d max ( γ j ) = η 1 γ j v 1 + k 5 + η 2 γ j v 2 + k 5 e β 8 γ j , (23)
where β 8 = ∑ k 2 = 0 R { β 7 + β 2 k 2 ( N − 1 ) } , η 1 = N β 3 N − 1 β 5 and η 2 = N β 3 N − 1 β 6 .
The PNSMC is defined as
P r ( C s m c a s t > 0 ) = ∫ 0 ∞ f d min ( γ i ) ∫ 0 γ i f d max ( γ j ) d γ j d γ i . (24)
Substituting the values of f d min ( γ i ) and f d max ( γ j ) in Equation (24) and using the following identities of [
∫ 0 u x n e − μ x d x = n ! μ n + 1 − e − μ u ∑ k = 0 n ( n ! u k k ! μ n − k + 1 ) (25)
∫ 0 ∞ x n e − μ x d x = n ! μ − n − 1 (26)
the derived analytical expression in the closed-form for the P r ( C s m c a s t > 0 ) is shown in Equation (27) at the top of the next page, where
δ 1 = η 1 ( v 1 + k 5 ) ! β 8 v 1 + k 5 + 1 + η 2 ( v 2 + k 5 ) ! β 8 v 2 + k 5 + 1 , δ 2 = ( − 1 ) ∑ k 9 = 0 v 1 + k 5 ( v 1 + k 5 ) ! k 9 ! η 1 β 8 v 1 − k 9 + 1 and δ 3 = ( − 1 ) ∑ k 8 = 0 v 2 + k 5 ( v 2 + k 5 ) ! k 8 ! η 2 β 8 v 2 − k 8 + 1 .
P r ( C s m c a s t > 0 ) = δ 1 k 6 ! η 5 k 7 { η 3 + η 4 k 7 η 5 } + ∑ k 9 = 0 v 1 + k 5 δ 2 ( k 6 + k 9 ) ! ( β 8 + η 5 ) k 7 + k 9 { η 3 + η 4 ( k 7 + k 9 ) β 8 + η 5 } + ∑ k 8 = 0 v 2 + k 5 δ 3 ( k 6 + k 8 ) ! ( β 8 + η 5 ) k 7 + k 8 { η 3 + η 4 ( k 7 + k 8 ) β 8 + η 5 } (27)
The SOPM is defined as
P o u t ( R s m c a s t ) = 1 − ∫ 0 ∞ f d max ( γ j ) ∫ x ∞ f d min ( γ i ) d γ i d γ j , (28)
where x = e 2 R s m c a s t ( 1 + γ j ) − 1 , x = H + e 2 R s m c a s t γ j , H = e 2 R s m c a s t − 1 and R s m c a s t denotes the target secrecy multicast rate which is indicated as RS in the figure.
Substituting the values of f d min ( γ i ) and f d max ( γ j ) in Equation (28) and using the identities of Equations (25) and (26), the derived analytical expression in the closed-form for the P o u t ( R s m c a s t ) is shown below in Equation (29) as
P o u t ( R s m c a s t ) = 1 − [ ∑ k 11 = 0 k 6 ∑ k 13 = 0 k 11 { η 1 η 6 ( v 1 + k 5 + k 13 ) ! ( β 8 + η 8 ) v 1 + k 5 + k 13 + 1 } + ∑ k 11 = 0 k 6 ∑ k 13 = 0 k 11 { η 2 η 6 ( v 2 + k 5 + k 13 ) ! ( β 8 + η 8 ) v 2 + k 5 + k 13 + 1 } + ∑ k 12 = 0 k 7 ∑ k 14 = 0 k 12 { η 1 η 7 ( v 1 + k 5 + k 14 ) ! ( β 8 + η 8 ) v 1 + k 5 + k 14 + 1 } + ∑ k 12 = 0 k 7 ∑ k 14 = 0 k 12 { η 2 η 7 ( v 2 + k 5 + k 14 ) ! ( β 8 + η 8 ) v 2 + k 5 + k 14 + 1 } ] , (29)
where, η 6 = ∑ k 11 = 0 k 6 ∑ k 13 = 0 k 11 η 3 ( H ) k 11 − k 13 ( e η 5 H ) ( e − 2 R s ) k 13 k 6 ! k 11 ! k 11 ! k 13 ! ( k 11 − k 13 ) ! η 5 k 6 − k 11 + 1 , η 7 = ∑ k 12 = 0 k 7 ∑ k 14 = 0 k 12 η 4 ( H ) k 12 − k 14 ( e η 5 H ) ( e − 2 R s ) k 14 k 7 ! k 12 ! k 12 ! k 14 ! ( k 12 − k 14 ) ! η 5 k 7 − k 12 + 1 and η 8 = η 5 e 2 R s .
The PNSMC represented by P r ( C s m c a s t > 0 ) is depicted in
The P r ( C s m c a s t > 0 ) is depicted in
(dot line) to 1 (solid line) and M decrease from 2 (dash dot line) to 1 (short dash line) when M = 1 and N = 1 , respectively. This is due to the fact that when the number of multicast users increases, each user’s bandwidth decreases, which in turn lowers the multicast user’s capacity.
The P r ( C s m c a s t > 0 ) is depicted in
In
The impact of μ illustrates with R that is described in
and decreases if the value of R increase from 1 to 2 when R = 1 and μ = 1.5 , respectively.
In
In
The P o u t ( R s m c a s t ) is plotted against γ s which shows in
The P o u t ( R s m c a s t ) is depicted in
Therefore, based on the closed-form analytical expressions for the
P r ( C s m c a s t > 0 ) and the P o u t ( R s m c a s t ) and from the observations of numerical
results, the main findings of this paper can be summarized as follows: The P r ( C s m c a s t > 0 ) decreases with κ, μ, m, M and N. Therefore, increase in κ, μ, m, M and N degrades the security of the system. On the other hand, if R increases P r ( C s m c a s t > 0 ) increases and P o u t ( R s m c a s t ) decreases. Therefore, increasing in R enhances the security of the system.
The creation of an analytical mathematical model to guarantee security in wireless multicasting using κ-μ shadowing fading channels with opportunistic relaying is the main topic of this article. The parameters such as κ, μ, m, the number of multicast users, eavesdroppers, and relays that assist in determining the values of N and R are used to derive the analytical formulas in the closed-form for the PNSMC and the SOPM. Our results conclude that the security in wireless multicasting through κ-μ fading channel degrades with the κ, μ, m, M and N. But increasing the number of relays helps to mitigate this loss of security which takes part in competition of opportunistic relaying technique to be the best relay. The effects of interferences on the security in multicasting through κ-μ shadowing fading channel are not considered in this paper. The zero-forcing precoding can be used to eliminate the effects of interferences and the selection and phase alignment precoding techniques can be used to enhance the security by using interference power. Finally, the validity of analytical expressions is verified via Monte-Carlo simulation.
The authors declare no conflicts of interest regarding the publication of this paper.
Alam, A. and Sarkar, M.Z.I. (2022) Enhancement of Multicast Security with Opportunistic Relaying Technique over κ-μ Shadowed Fading Channels. Journal of Computer and Communications, 10, 121-137. https://doi.org/10.4236/jcc.2022.1011009