Secrecy at the Physical Layer in TWC with Two Half-Duplex DF Relays in the Presence of an External Eavesdropper
System Model of Two-Way Communication with DF Relays
In this model, two sources share their information signal with each other through two half-duplex DF relays as in Sharma et al. (2019). The complete system model of this TWC network has been shown in Figure 12.7. In this communication, an external eavesdropper efforts to eavesdrop the message of both the sources, but not simultaneously at a time. Each of the relays has two directional antennas, one is used for receiving the information signal, while the other is for transmitting the information signal. The directional antennas at each relay assist the relay for two-way communication. The rest of the communication nodes has omnidirectional antennas. The EAV keep its position in such a way that it presents at almost equal distance from both the sources, and this position provides the proper receiving of the information signal from both the sources. We have found this particular position of EAV on a line that is perpendicular to another line that connects to both the sources. Due to directional antennas, the EAV is unable to receive the relayed signal. The communication completes in two phases; first is a broadcasting phase, and another is the relaying phase as shown in Figure 12.8. In the broadcast phase, both the sources transmit the signals. In broadcasting phase, the information signal of source one (SI) is received by the
FIGURE 12.7 System model of two-way communication with two half-duplex DF relay.
FIGURE 12.8 The time frame of two-way communication with two half-duplex DF relay.
relay one (R1), and the information signal of source two (S2) is received by the relay two (R2). In the relaying phase, the S2 receives the signal from Rl, and SI receives the signal from R2.
All channels are Rayleigh fading. The channel has Gaussian noise with zero mean and variance N0. The channel coefficients are denoted by h>r the channel gain is denoted by gtj and mean channel gains are denoted by Q1;, where / = (51. R1,52, R2, E), here subscript E indicates for EAV, and j = (51,R1,52.R2,£). All the channels are independent and nonidentically distributed random variables.
The signal strength at any receiving node is represented in signal-to-noise ratio (SNR) or signal-to-interference plus noise ratio (SINR) parameter. In the broadcasting phase, the information signal of the SI is received by the Rl, and the information signal of the S2 is received by the (R2). The received signal at the Rl is
where Psl is the 51 ’s transmit power, xsl message of 51, and n0 is the Gaussian noise. From Eq. (12.36), SNR at the Rl is
In the same way, the SNR at the R2 is
The SINR of information signal S1 at the EAV is
The SINR of S2 at the EAV is
In the relaying phase, the information signal strength of 51 at the 52 via R1 is
where a is the fraction of relay power, i.e., aPR has assigned to R1 and (1 - a)PR has assigned to R2. The information signal strength of 52 at the 51 via R2 is
The end-to-end information signal strength of 51 at the 52 is and end-to-end information signal strength of 52 at 51 is
Global secrecy capacity
Using the Shannon channel capacity formula, the channel capacity of information signal one is
and channel capacity of information signal two is
The channel capacity of the information signal one at the EAV is The channel capacity of the information signal two at the EAV is
Now, the secrecy capacity of information signal one is defined as
The secrecy capacity of information signal two is defined as (Pan et al. 2015; Kalamkar and Banerjee 2017)
The global secrecy capacity is expressed as (Zhang et al. 2017; Sharma et al. 2019)
SOP Formulation with DF Relays and Optimality of Source Power and Fraction of Relay Power
We have defined the SOP in the previous Section 12.2.2. Here, we express the SOP as
Equation (12.52) is further simplified as
The term P(c|ff2 < CthSEC) (let P(Cjf£ < CthSEC) = h ) in closed form is expressed as
where all constants are given as
The term I2 in closed form is expressed as
where all constants are given as
We put /, from Eq. (12.54) and U from Eq. (12.56) in Eq. (12.53); we get the SOP in final closed-form expression as
Optimality of source power and fraction of relay power
The optimal value of source power and fraction of relay power is obtained by differentiating the SOP with respect to source power and fraction of relay power, respectively Sharma et al. 2019). First, the SOP is differentiated with respect to PSI keeping the fixed value of ^SI and a. For finding the optimal value, we equate this differentiation to zero. The expression is given as
The expression of can be expressed as
and the expression of can be expressed as
We can find the final expression of Eq. (12.59) by putting —— from Eq. (12.60)
and —— from Eq. (12.61) in Eq. (12.59). The obtained expression is a transcenden- dPn
tal equation, which the bisection method of the numerical method is used to solve. The solution of the optimal value is shown in the next subsection.
Next, the optimality of « can be found following the same procedure as we have followed to find the optimality of PS1. The derivatives of the SOP with respect to «is expressed as
where — can be expressed as
and —can be expressed as 3a
From the expression in Eqs. (12.63) and (12.64), we obtain the final expression of Eq. (12.62) as a transcendental equation, which the bisection method of the numerical method is used to solve. The solution of the optimal value is shown in the next subsection.