Journal of Engineering and Applied Science - RUET

*Corresponding author: Department of Electrical & Electronic Engineering, Rajshahi University of Engineering & Technology, Rajshahi-6204, Bangladesh E-mail addresses: rosnisayed@gmail.com (R. Sayed)

66 Journal of Engineering and Applied Science Vol. 03, No. 01, pp. 66–79, June 2019

Performance Comparison of Linear Equalization Techniques in Secure Wireless Multicasting

R. Sayed*, M. Z. I. Sarkar, D. K. Sarker

Department of Electrical & Electronic Engineering, Rajshahi University of Engineering & Technology, Rajshahi-6204, Bangladesh

ARTICLE INFORMATION ABSTRACT

Received date: 15 Jan 2019 Revised date: 02 May 2019 Accepted date: 07June 2019

This paper deals with the investigation of the impact of minimum mean- square error (MMSE) and zero-forcing (ZF) equalizations on the performance of secure wireless multicasting. A secure multiple-input multiple-output (MIMO) multicasting scenario is considered in which a source S transmits a common stream of information to a group of M receivers in the presence of an eavesdropper. In order to analyze the secure outage performance of the proposed model considering MMSE and ZF equalizations, we derive the closed-form analytical expressions for the ergodic secrecy multicast capacity, the probability of non-zero secrecy multicast capacity and the secure outage probability for multicasting with MMSE and ZF equalizations. Finally, the performance comparison of secure wireless multicasting with MMSE and ZF equalization techniques are shown to understand which one plays significant role in enhancing the security of wireless multicasting.

Keywords

Zero-forcing

Secrecy multicast capacity Outage performance Confidential information

1. Introduction

Wireless network is very much susceptible to unauthorized access i.e., eavesdropping due to the openness of wireless medium. For this reason, it becomes a challenging task of transmitting personal and confidential information through wireless medium. That’s why the issues of security comprising multicast network have attracted a lot of interests to the researchers [1-4]. On the other hand, in the recent years enormous efforts had been done for enhancing the capacity and outage performances of wireless communication networks utilizing the various techniques of linear equalization [5-11]. A user cooperated multicast network with single eavesdropper was proposed in [1] deriving the expression of secure outage probability. A stochastic MIMO multicast network was studied in [2] and found the expressions of space-time capacity and secrecy rate. The security issues of artificial noise aided MIMO multicast networks were studied in [3] solving the problem of secrecy rate maximization. Large-scale MIMO networks were studied in [4] with ZF, MMSE and Maximal Ratio Combining (MRC) receivers and compared the outage performances of the proposed system with these three techniques. Classical multiuser detection algorithms were designed in [5] for ZF and MMSE receivers and solved the problem of complexity of the ZF and MMSE detectors in the case of multiuser MIMO systems. An efficiency improvement technique was incorporated into the ZF and MMSE algorithms for a MIMO system in [6] and the gain provided by the proposed approach was estimated via simulations. The bit error rate (BER)

Journal of Engineering and Applied Science

Contents are available at www.jeas-ruet.ac.bd

performance of a spatially multiplexed MIMO system were studied in [7] with ZF and MMSE detections considering quadrature phase shift keying (QPSK) and 16-ary phase shift keying (16-PSK) modulations and showed via simulation that QPSK modulation was more effective than 16-PSK both for ZF and MMSE detections. The capacity performances of ZF, MMSE and MRC detectors were studied and compared in [8] for co-located MIMO (C-MIMO) and distributed MIMO (D-MIMO) networks over Nakagami-m fading channels. The performances of MRC, MMSE and ZF detectors were also studied and compared in [9] for C-MIMO and D-MIMO networks considering the effect of interference. A MIMO AF cooperative relay network was studied in [10] with ZF equalization and derived the expression of symbol error probability.

Although, a lot of papers are available on the study of wireless communication network systems considering ZF and MMSE equalizers and recently, the security issues of multicast networks was addressed in [11] with ZF equalization, but to the best of author’s knowledge, there are no works in the literature which address the problems of security of MIMO multicast networks with ZF and MMSE equalizations and compare the secure outage and capacity performances of multicast networks considering these two techniques. Hence the difference between this work and the previous works can be summarized as follows;

• the previous works did not consider the security issues in multicasting using MMSE and ZF equalizations. But the key objective of this work is to investigate how the MMSE and ZF equalizations at the receivers enhances the secrecy capacity of multicast networks.

• the closed-form analytical expressions of the (i) ergodic secrecy multicast capacity (ii) the probability of non- zero secrecy multicast capacity and (iii) the secure outage probability for multicasting, considering MMSE and ZF equalizations are not available in the literature. On the other hand, this paper has derived the closed-form analytical expressions of the aforementioned parameters that are the positive additions in the literature.

However, the outline of this paper can be summarized as follows:

• At first, we propose a multicast network in the presence of an eavesdropper considering linear ZF and MMSE equalizers both at the receivers and eavesdropper and define the secrecy multicast capacity of the proposed system so that eavesdropper is unable to decode information from the main channel.

• Second, we derive the closed-form expressions of the probability density function (PDF) of signal-to-noise ratios (SNRs) and capacities for multicast and eavesdropper’s channels considering ZF and MMSE equalizers at the receivers.

• Third, using these PDFs we derive the closed-form expressions for the ergodic secrecy multicast capacity, the probability of non-zero secrecy multicast capacity and the secure outage probability for multicasting.

• Finally, we compare the capacity and outage performances of the proposed system with ZF and MMSE equal- izations.

The rest of the paper is organized as follows. Section 2 describes the system model and problem formulation. The analytical expressions for the ergodic secrecy multicast capacity, the probability of non-zero secrecy multicast capacity and the secure outage probability for multicasting are described in sections 3, 4 and 5, respectively. Section 6 provides the numerical results and section 7 draws the conclusions of this work.

2. System Model and Problem Formulation

We consider a MIMO multicast network shown in Fig.1, where a sourceStransmits a common stream of information to a group of M users in the presence of an eavesdropper. The source, each user and eavesdropper are equipped withn_T,n_R andn_E antennas, respectively. The channel matrix between the source andith user is denoted byH_i∈ C^nR×nT (i=1,2,3, . . . ,M)and that between source and eavesdropper is denoted byG_e∈C^nE×nT. All the channels are considered as Rayleigh fading. Let,xdenotes the transmitted signal vector. Hence, the received signals at theith user and eavesdropper denoted respectively byy_mi andy_ecan be expressed as

y_mi =H_ix+z_i, (1)

y_e=G_ex+w_e, (2)

1 õ õ õ

M Users

Eavesdropper H1

I I I

HM

S

E Ge

Source

Figure 1.The scenario of secure wireless multicasting in the presence of an eavesdropper.

wherez_i∼Nf(0,N_m0I_nR)andw_e∼Nf(0,N_e0I_nE)are the vectors of Gaussian noises imposed on the receivers ofith user and eavesdropper, respectively.I_nR∈C^nR×nR andI_nE ∈C^nE×nE are the identity matrices, andN_m0 andN_e0 denote the powers of the noises at the receivers of user and eavesdropper, respectively. In the following subsections, we describe the PDF of SNRs with MMSE and ZF equalizations at the receivers for users and eavesdropper channels.

2.1. MMSE equalization at the Receivers

With MMSE equalization at the receivers, the post-processing SNR at theith user is given by [12]

γm^mmse_i = γ¯m0

(

I_nR+P_uH^†_iH_i )₋₁

kk

−1, (3)

where ¯γm₀ is the average SNR of the main channel andk=1,2,3, . . . ,n_R. The PDF ofγm^mse_i is given by [12]

f_γ^mmse

mi (γmi) = γm_iαi−1e⁻

γ_mi θiγ¯m0

Γ(αi)(θiγ¯m₀)^αi, (4) whereαi= ⁽ⁿ_n^{R−nT+1+(nT−1)µ)2}

R−nT+1+(nT−1)κ , θi= ⁿ_n^{R−nT+1+(nT−1)κ}

R−nT+1+(nT−1)µP_uβi, P_u is the average transmit power and √

βi models the geometric attenuation and shadow fading. The parametersµ^andκcan be determined by the following equations;

µ^{= 1} n_T−1

nT

l=1,l̸=i

∑

1 n_RP_uβl

(

1−^nT_n⁻¹_R +^nT_n⁻¹

R µ^{)+1, (5)}

κ



1+

nT

l=1,l̸=i

∑

P_uβl

( nRPuβl

(

1−^nT_n⁻¹_R +^nT_n⁻¹

R µ⁾⁺¹⁾²



=

nT

l=1,l̸=i

∑

P_uβlµ⁺¹ (

nRPuβl

(

1−^nT_n⁻¹_R +^nT_n⁻¹

R µ⁾⁺¹⁾²

. (6)

Similar to equations (3) and (4), the post-processing SNR at the eavesdropper and it’s corresponding PDF are given, respectively by

γe^mmse= γ¯e₀

(

In_E+PuG^†_eG_e )₋₁

ll

−1, (7)

f_γ^mmse_e (γe) = γeαe−1e⁻

γe θeγ¯e0

Γ(αe)(θeγ¯e₀)^αe, (8) Journal of Engineering and Applied Science Vol. 03, No. 01, pp. 66–79, June 2019

where ¯γe₀is the average SNR of the eavesdropper channel,l=1,2,3, . . . ,n_E,αe=⁽ⁿ_n^{E−nT+1+(nT−1)δ)2}

E−nT+1+(n_T−1)λ ,θe=ⁿ_n^{E−nT+1+(nT−1)λ}

E−nT+1+(n_T−1)δP_uηi

and√

ηimodels the geometric attenuation and shadow fading. The parametersδ ^andλ can be determined by the fol- lowing equations;

δ ^{= 1} n_T−1

nT

l=1,l̸=i

∑

1 n_EP_uηl

(

1−^nT_n⁻¹_E +^nT_n⁻¹

E δ^{)+1, (9)}

λ



1+

nT

l=1,l̸=i

∑

P_uηl

( nEPuηl

(

1−^nT_n⁻¹_E +^nT_n⁻¹

E δ⁾⁺¹⁾²



 =

nT

l=1,l̸=i

∑

P_uηlδ⁺¹ (

nEPuηl

(

1−^nT_n⁻¹_E +^nT_n⁻¹

E δ⁾⁺¹⁾²

. (10)

2.2. ZF equalization at the Receivers

With ZF equalization at the receivers, the post-processing SNR at theith user is given by [13]

γm^{z f}_i = γ¯m₀

( H^†_iH_i

)₋₁

kk

, (11)

where ¯γm₀ is the average SNR and the PDF ofγm^{z f}i is given by [13]

f_γ^{z f}

mi(γmi) = γmi

n_R−nTe⁻

γ_mi γ¯m0

(n_R−n_T)! ¯γm^nR0^−nT+1

. (12)

Similar to equations (11) and (12), the post-processing SNR at the eavesdropper and it’s corresponding PDF are given by

γe^{z f} = γ¯e₀

( G^†_eG_e

)₋₁

ll

, (13)

f_γ^{z f}_e(γe) = γen_E−nTe⁻

γe γ¯e0

(n_E−n_T)! ¯γeⁿ0^E−nT+1

. (14)

3. Ergodic Secrecy Multicast Capacity

In this section, we derive the closed-form expressions for the ergodic secrecy multicast capacity considering MMSE and ZF equalizers at the receivers of multicast users and eavesdropper.

3.1. With MMSE equalization at the Receivers

From equation (1), the capacity at theith user can be defined as C_m^mmse

i =log₂(

1+γm^mmse_i

). (15)

Therefore, the multicast capacity forMusers is given by C_m^mmse= min

1≤i≤MC_m^mmse_i =log₂(1+ min

1≤i≤Mγm^mmsei ). (16)

From equation (2), the capacity at the eavesdropper can be defined as

C_e^mmse=log₂(1+γe^mmse). (17)

Hence, the secrecy multicast capacity is given by C^mmse−s_mcast =max

f(x_k)

[C_m^mmse−C_e^mmse] =log₂

[1+min_1≤i≤Mγm_i

1+γe

]

. (18)

Definingd_min^mmse=min_1≤i≤Mγmi,Cm^mmse=log₂(1+d_min^mmse)andCe^mmse=log₂(1+γe), we have

C^mmse−s_mcast = (C_m^mmse−C_e^mmse). (19)

The PDFs ofCm^mmseandCe^mmseare derived as follows:

3.1.1 . Probability Density Function ofCm^mmse

The PDF ofd^mmse_min can be defined as f_d^mmse

min (γmi) =M f_γ^mmse

mi (γmi)[1−F_γ^mmse

mi (γmi)]^M−1, (20)

whereF_γ^mmse

mi (γm_i)is the cumulative distribution function (CDF) of SNR and it can be obtained as F_γ^mmse

mi (γmi) =

∫ _γ

mi

0

f_γ^mmse

mi (γmi)dγmi. (21)

Substituting equation (4) into equation (21) and performing integration we have F_γ^mmse

mi (γmi) =1−^α

∑

ⁱ⁻¹

k₁=0

γm^k1ie

−γ_mi γ¯m0θ_i

(γ¯m₀θi)^k1k₁!. (22) Substituting equations (4) and (22) into equation (20) we obtain

f_d^mmse

min (γmi) =Mγmiαi−1e⁻

γ_mi θ_iγ^¯m0

Γ(αi)(θiγ¯m₀)^αi



^α

∑

ⁱ⁻¹

k₁=0

γ^mk1ie

−γ_mi γ¯m0θ_i

(γ¯m₀θi)^k1k₁!





M−1

. (23)

The PDF ofCm^mmsecan be obtained using the following Proposition;

Proposition 1:[14]Let, the probability density function of x is denoted by f(x).Then, the probability density function of C =log_e(1+Θx)is given by

q(C) =e^C Θ f

(e^C−1 Θ

)

. (24)

Using Proposition 1, the PDF ofC_m^mmseis obtained from equation (23) as

q(Cm^mmse) =e^Cmmmsef (

e^Cmmmse−1 )

=Me^Cmmmse(e^Cmmmse−1)^αi−1e⁻

(eCmmse

m −1)

γ¯m0θi

Γ(αi)(γ¯m0θi)^αi



^αi

∑

−1 k1=0

(e^Cmmmse−1)^k1e

−(eCmmse

m −1)

γ¯m0θi

(γ¯m0θi)^k1k₁!





M−1

=

(αi−1)ϑ k

∑

1=0

Mβk₁(αi,ϑ^)(eCmmmse−1)_a−1 e^Cmmmse Γ[αi](γ¯m₀θi)^a e⁻

M(^eCmmmse−1)

γ¯m0θi , (25)

whereΘ=1,ϑ ^=M−1,αi+k₁=aandβk1(αi,ϑ⁾are the coefficients of(

e^Cmmmse−1)k₁

in the expansion of







αi−1 k

∑

₁=0

(e^Cmmmse−1)k1

e

−(^eCmmmse−1)

γ¯m0θi

(γ¯m₀θi)^k1k₁!







M−1

,

and these can be calculated recursively [15].

3.1.2 . Probability Density Function ofC_e^mmse

Using Proposition 1, the pdf ofCe^mmseis obtained from equation (8) as

q(Ce^mmse) =e^Cemmsef (

e^Cemmse−1 )

=e^Cemmse(

e^Cemmse−1)_αe−1

e

−(^eCemmse−1)

γ¯e0θe

Γ(αe)(γ¯e₀θe)^αe . (26) Journal of Engineering and Applied Science Vol. 03, No. 01, pp. 66-79, June 2019

3.1.3 . Ergodic Secrecy Multicast Capacity

From equation (19), the ergodic secrecy multicast capacity can be defined as

⟨C_mcast^mmse−s⟩=E[C_m^mmse]−E[C_e^mmse]. (27) We have

E[Cm^mmse] =

∫ _∞

0 Cm^mmseq(Cm^mmse)dCm^mmse. Substituting equation (25), we obtain

E[Cm^mmse] =

(αi−1)ϑ k

∑

₁=0

∫ _∞ 0

[ Cm^mmse

Mβk1(αi,ϑ⁾ e^−Cmmmsee

M(^eCmmmse−1)

γ¯m0θi

(e^Cmmmse−1)_a−1 Γ[αi](γ¯m₀θi)^a

]

dCm^mmse. (28)

Defininge^Cmmmse−1=z, we have E[Cm^mmse] =

(αi−1)ϑ k

∑

₁=0

Mβk₁(αi,ϑ⁾ Γ[αi](γ¯m₀θi)^a

∫ _∞ 0

ln(1+z)z^a−1e⁻

M γ¯m0θ_iz

dz. (29)

Performing integration using the following identity of [16, eq. 4.222.8],

∫ _∞ 0

log_e(1+ax)x^be^−xdx=

∑

b m=0

b!

(b−m)!

[

(−1)^b−m−1 a^b−m e^1aEi

(

−1 a

) +

b−m

∑

k=1

(k−1)!

(−a)^b−m−k ]

, (30)

we have

E[C_m^mmse] =

(αi−1)ϑ k

∑

1=0

a−1

∑

k2=0

M^1−aβk₁(αi,ϑ⁾ Γ(αi)

(a−1)!

(a−1−k2)!

[

(−1)^a−k2−2e

M γ¯m0θi

(_γ¯

m0θi

M

)_a−1−k2

×Ei (

− M γ¯m₀θi

) +

a−1−k2

k

∑

₃=1

(k₃−1)!

(_γ¯

m0θi

M

)_{a−1−k2−k3} ]

, (31)

where Ei(·)denotes the exponential integral function. Again, we have E[Ce^mmse] =

∫ _∞

0 Ce^mmseq(Ce^mmse)dCe^mmse. Substituting equation (26), we obtain

E[Ce^mmse] =

∫ _∞ 0





 Ce^mmsee^Cemmsee

−(^eCemmse−1)

γ¯e0θe

(e^Cemmse−1)_1−αe

Γ(αe)(γ¯e0θe)^αe





dCe^mmse. (32)

Defininge^Cemmse−1=y, we have

E[Ce^mmse] =(γ¯e₀θe)^−αe Γ(αe)

∫ _∞ 0

ln(1+y)y^αe−1e⁻

y

γ¯e0dy. (33)

Performing integration using equation (30), we obtain E[C_e^mmse] =

αe−1 k

∑

₄=0

(αe−1)!

(b−1)!Γ(αe) [

(−1)^b−2 (γ¯e₀θe)^b−1

Ei (−_γ¯e¹

0θe

) e⁻

1 γ¯e0θe

+

b−1

∑

k₅=1

(k₅−1)!

(−γ¯e₀θi)^b−k5−1 ]

, (34)

whereαe−k₄=b. Substituting equations (31) and (34) into equation (27), the closed-form expression of ergodic secrecy multicast capacity at theith user with MMSE equalization is obtained as

⟨C^mmse−s_mcast ⟩=

(αi−1)ϑ k

∑

₁=0

∑

τ k₂=0

βk₁(αi,ϑ⁾ M^τΓ(αi)

τ^! (τ−k₂)!

[

(−1)^τ−k2−1e

M γ¯m0θi

(_γ¯

m0θi

M

)_τ−k2 Ei (

− M γ¯m0θi

)

+

τ−k2

k

∑

₃=1

(k₃−1)!

(_γ¯

m0θi

M

)_d ]

−^α

∑

^e−1

k₄=0

(αe−1)!

ς^!Γ(αe) [

(−1)^ς−1 (γ¯e₀θe)^ς

Ei (−_γ¯e¹

0θe

) e⁻

1 γ¯e0θe

+

∑

ς k₅=1

(k₅−1)!

(γ¯e₀θe)^ς−k5 ]

, (35)

whereτ^=a−1,d=τ−k₂−k₃andς^=b−1.

3.2. With ZF equalization at the Receivers

From equation (1), the capacity atith user can be defined as C_m^{z f}

i=log₂( 1+γm^{z f}_i

). (36)

Therefore, the multicast capacity forMusers is given by C_m^{z f}= min

1≤i≤MCm_i =log₂(1+ min

1≤i≤Mγm^{z f}i). (37)

From equation (2), the capacity at the eavesdropper can be defined as

C^{z f}_e =log₂(1+γe^{z f}). (38)

Hence, the secrecy multicast capacity is given by C_mcast^{z f−s} =max

f(x_k)

[C_m^{z f}−C_e^{z f}] =log₂ [

1+min_1≤i≤Mγm^{z f}_i

1+γ^{ez f} ]

. (39)

Definingd_min^{z f} =min_1≤i≤Mγm^{z f}_i,Cm^{z f} =log₂(1+d_min^{z f} )andCe^{z f} =log₂(1+γe^{z f}), we have C_mcast^{z f−s} =(

C_m^{z f}−C_e^{z f})

. (40)

The PDFs ofCm^{z f} andCe^{z f} are derived as follows:

3.2.1 . Probability Density Function ofCm^{z f}

The PDF ofd^{z f}_mincan be defined as f_d^{z f}

min(γmi) =M f_γ^{z f}

mi(γmi)[1−F_γ^{z f}

mi(γmi)]^M−1, (41)

whereF_γ^{z f}_mi(γmi)is the CDF of SNR and it can be defined as F_γ^{z f}

mi(γm_i) =

∫ _γ

mi

0

f_γ^{z f}

mi(γm_i)dγm_i. Substituting equation (12) and performing integration, we have

F_γ^{z f}

mi(γm_i) =1−^nR

∑

^−nT

l1=0

γm^l1ie⁻

γ_mi γ¯m0

γ¯m^l1₀l₁!

. (42)

Journal of Engineering and Applied Science Vol. 03, No. 01, pp. 66–79, June 2019

Substituting equations (12) and (42) into equation (41), we have

f_d^{z f}

min(γmi) =Mγmiζe⁻

γ_mi γ¯m0

ζ^{! ¯}γm^ζ+1₀





∑

^ζ

l₁=0

γm^l1ie

−γ_mi γ¯m0

γ¯m^l1₀l₁!





ϑ

, (43)

whereζ ⁼ⁿR−n_T. Using equation (24), the PDF ofCm^{z f} is obtained from equation (43) as

q(Cm^{z f}) =M e^{Cmz f}

( e^{Cmz f} −1

)_ζ e⁻

( eCz f

m −1 ) γ¯m0

ζ^{! ¯}γm^ζ+10









∑

ζ l₁=0

( e^{Cmz f}−1

)l1

e

− (

eCz f m−1

) γ¯m0

γ¯m^l10l₁!









ϑ

=

ζϑ

∑

l₁=0

Me^{Cmz f}Bl1(ζ^,ϑ^{)(eCmz f}−1 )_ζ+l₁

e

M (

eCz f m−1

)

γ¯m0 ζ^!l1! ¯γm^ζ+l₀ ¹⁺¹

, (44)

whereBl1(ζ^,ϑ⁾are the coefficients of (

e^{Cmz f} −1 )l₁

in the expansion of









∑

ζ l1=0

( e^{Cmz f}−1

)l₁

e

− (

eCz f m −1

) γ¯m0

γ¯m^l1₀l1!









ϑ

,

and these can be calculated recursively.

3.2.2 . Probability Density Function ofCe^{z f}

Using equation (24), the PDF ofCe^{z f} is obtained from equation (14) as

q(Ce^{z f}) =e^{Cez f} f (

e^{Cez f} −1 )

= e^{Cez f}

( e^{Cez f}−1

)_ξ e

− (

eCz f e −1

) γ¯e0

ξ^{! ¯}γe^ξ₀⁺¹

, (45)

whereξ ⁼ⁿE−n_T.

3.2.3 . Closed-Form Expression of Ergodic Secrecy Multicast Capacity From equation (40), the ergodic secrecy multicast capacity can be defined as

⟨C_mcast^{z f−s}⟩=E[ Cm^{z f}

]−E[ Ce^{z f}

]. (46)

We have

E[ Cm^{z f}

]=

∫ _∞

0 Cm^{z f}q(Cm^{z f})dCm^{z f}. Substituting equation (44), we obtain

E[ Cm^{z f}

]=M

∑

ζϑ l₁=0

∫ _∞ 0

Cm^{z f}Bl1(ζ^,ϑ^{)(eCmz f}−1 )_ζ+l₁

e

M (

eCz f m −1

) γ¯m0 −Cm^{z f}

ζ^!l1! ¯γm^ζ+l₀ ¹⁺¹

dCm^{z f}. (47)