Joint Packet Length and Power Optimization for Covert Short-Packet D2D Communications

Abstract

This paper proposes a joint optimization mechanism for packet length and power in the scenario of covert short packet D2D communication, so as to effectively improve the communication performance between D2D pairs subject to the covert constraint. Specifically, we construct a short-packet D2D communication model assisted by covert communication, and further propose the notion of effective covert throughput (ECT) to quantitatively characterize the trade-off between the covertness and reliability of IoT state monitoring information transmission. Secondly, in the constructed communication scenario, we analyze the detection error probability of the warden and clarify that the existing equal power transmission of pilot and data signals can minimize the detection performance of the warden. However, this strategy is achieved by compromising the transmission performance of the system, which means that the ECT of D2D pair may not be optimal. Thirdly, we aim to maximize the ECT of D2D pair and construct a joint optimization problem for pilot transmission power, data transmission power, and packet length. Furthermore, a joint optimization algorithm based on the 2D search is adopted to obtain the optimal solution of the established optimization problem. Simulation results demonstrated that the transmission performance of the joint optimization algorithm is better than that of the scheme of the equal power scheme on the premise of ensuring the covertness.

1. Introduction

The vigorous development of IoT technology has brought great convenience to people’s life. At the same time, a large number of node devices that support the IoT application are connected to the mobile Internet [1]. Their large number, wide distribution and diverse demands bring enormous pressure to cellular networks, and make the contradiction between the scarce wireless resources and the accelerating resource consumption more prominent [2,3,4]. In view of this, short-packet D2D communication has been widely used. Its core idea is that in D2D communication, information exchange is conducted between adjacent devices through short packets [5]. Due to the characteristics of the short communication distance and direct communication, it has advantages such as high data rate [6], low delay [7] and low power consumption [8].

Compared with traditional cellular communication, the D2D communication based on short packets faces severe challenges in terms of security. (i) Because D2D communication is not verified by the management of the base station, its communication link is vulnerable to attack [9]. (ii) Due to the limited computing capacity and energy reserve of user terminal devices in D2D networks, it is difficult to enhance data security through high-complexity key encryption algorithms [10]. (iii) The data carried by short data packets are generally key core information, such as the control information of IoT nodes, the highly compressed basis of semantic communication, the privacy data of patients in telemedicine, etc. Once the data packets containing such information are intercepted or even deciphered, the blow to communication security will be devastating [11]. Therefore, in order to enhance the security of D2D communication, covert communication technology is added to the existing secure communication system to ensure that the warden can only detect the signals transmitted between D2D pairs with a very low probability, so as to achieve the purpose of hiding communication signals or communication behaviors [12,13].

Unfortunately, most existing studies focus on covert D2D communication and have not yet explored the impact of using short data packets on covert D2D communication [14]. In essence, the covert D2D communication hides the communication behavior by using the co-channel interference generated by CUEs (Cellular Users) and the inherent noise of the physical environment, so that the warden cannot accurately judge whether the change of the received signal is from the occurrence of D2D communication behavior. In a word, since the covertness performance in covert D2D communication depends on the warden’s ability to recognize the changes in its received signals [15,16], most of the existing works are to ensure the covert performance between D2D pairs by introducing uncertainty factors. According to the working mode of D2D pairs, the relevant studies are described as follows:

In the dedicated mode: Ref. [17] utilizes the co-channel interference generated by multiple D2D pairs sharing the same resource block, causing the uncertainty of the warden to the interference signal to realize covert D2D communication. The results demonstrate that the number of D2D pairs in the same channel and the transmission power are important factors affecting the communication covertness. In [18], the D2D link dedicates orthogonal spectrum resources, and delimits a security zone. The D2D transmitter acts as a dynamic repeater to forward covert information to cell users outside the security zone. The results demonstrate that as long as the transmission power of the transmitter does not exceed the maximum value, the probability of detection by the warden is small enough.

In the multiplexed mode: Ref. [10] uses the co-channel interference generated by spectrum multiplexing to achieve covertness. Considering the freshness of information and energy consumption, a high energy transmission probability-power control method under age of information (AoI) constraints is proposed. Ref. [19] makes use of the channel uncertainty of the warden and introduces the power domain NOMA and continuous interference cancellation techniques to study the covert performance of the D2D transmitter when it knows and does not know the detection decision threshold of the warden. In addition to using cochannel interference generated by CUEs, in order to further enhance the covertness of D2D communication, Ref. [20] deploys an antenna array at the base station to send artificial noise to disturb the warden. The research demonstrates that the covert throughput can be further improved by setting the appropriate number of base station antennas as well as using artificial noise transmission power.

It is worth noting that while the above methods implement covert transmission between the D2D pair, they do not take into account the effect of short packets on covert D2D communication, as mentioned earlier [21,22]. In fact, compared with the long packet, the decoding error is inevitable in short packet communication, and the influence of background noise on the decoding error is difficult to eliminate due to the limited observation samples at the D2D receiver [23]. Therefore, the existing covert D2D communication cannot be directly applied to covert short-packet D2D communication. Moreover, the D2D communication rate is not only affected by the received signal to the interference plus noise ratio (SINR), but also by the combination of the packet length and packet error rate [24]. Therefore, the optimization of the packet length as well as the transmission power should be considered in the implementation of covert short-packet D2D communication. In addition, the existing research still has the following deficiencies:

(1) Most existing studies assume that the receiver has mastered the channel state information (CSI) between the D2D pair, without considering the acquisition process. It is precisely the process of obtaining CSI that may lead to the exposure of the D2D user communication behavior [25]. Therefore, it is necessary to consider the acquisition process of CSI under covert conditions [26]. Especially, considering the special role of the pilot in the process of channel estimation, it is a feasible solution to achieve covert channel estimation by reasonably allocating resources such as the pilot packet length and transmission power [24].

(2) In covert D2D communication, there is mutual interference between D2D pairs and CUEs in the spectrum multiplexing mode. D2D users not only need to ensure their own communication quality, but also need ensure that the interferences to CUEs are within a reasonable range [10]. Therefore, it is necessary to comprehensively consider the communication link quality of CUEs and D2D pairs.

(3) In existing covert D2D communication, there is not much analysis on the tradeoff between communication covertness and reliability, which leads to the improvement of the communication covert performance at the expense of communication quality. Moreover, different scenarios have different requirements for communication covertness and reliability. Therefore, we need to properly control and allocate resources, such as transmission power and packet length based on the preferences of communication scenarios for performance indicators to ensure both the D2D communication covertness performance and transmission performance.

Inspired by the above analysis, we consider using a power allocation of pilot signals to realize the channel estimation under covert conditions in covert short-packet D2D communication scenarios, so as to obtain accurate CSI. Furthermore, we consider the quality of cellular user communication links as one of the constraints and propose a joint scheme of packet length and transmission power to achieve the optimal compromise between the covertness and reliability of the D2D pair. The main work is summarized as follows:

• Through the uncertainty of the background noise power of the warden and the interference of CUE in the detection process, the warden cannot accurately judge whether the changes in the received signal come from the occurrence of the D2D communication behavior. We construct a short-packet D2D communication scene assisted by covert communication. In this model, the ECT is proposed to quantitatively characterize the tradeoff between the covertness and reliability of the D2D pair under the short packet communication. We consider the process of obtaining CSI under covert conditions, comprehensively considering the average decoding error and channel estimation error of the short packet communication receiver, and analyze the transmission performance of the system.
• In the covert short-packet D2D communication scenario, we derive a strict covert constraint lower bound based on the KL divergence of the observed signal probability, and analyze the factors affecting the covert performance such as the allocation factor and pilot packet length. On this basis, the analysis of the existing equal power distribution scheme between the pilot and data signal can ensure the best covert performance of D2D communication, but it damages the reliability of D2D communication.
• To ensure both communication covertness and reliability, we aim to maximize the ECT of D2D pairs and construct a joint optimization problem for the pilot transmission power, data transmission power, and packet length. Furthermore, we propose a 2D numerical search algorithm to obtain the optimal solution of the established optimization problem. The simulation results demonstrate that compared with the scheme of the equal power transmission for the pilot and data, under the general model, the transmission performance of the system has been improved to a certain extent while ensuring covertness.

The remaining parts of this paper are organized as follows. Section 2 introduces the system model and analyzes the channel estimation process for covert D2D communication. Section 3 analyzes the covertness performance to demonstrate the influencing factors of communication covertness. Section 4 puts forward the optimization scheme and analyzes the performance of the communication system. We conduct numerical simulations and analyze the results in Section 5. Finally, the conclusion is drawn in Section 6.

2. System Model

As shown in Figure 1, we consider a D2D-based IoT State monitoring communication model, which consists of the base station, multiple CUEs, multiple D2D pairs and a warden (Willie). The spectrum resources occupied by CUEs are uniformly allocated by the base station and are mutually orthogonal. The model assumes that the frequency spectrum resources occupied by each CUE can only be reused by one D2D pair and the CUE do not interfere with each other. In this system model, DT (D2D transmitter) periodically generates the node monitoring information and tries to send it to the DR (D2D receiver), while Willie keeps monitoring whether information is being transferred between D2D pairs. Once the warden detects the transmission of information between D2D pairs, it may take steps to block or eavesdrop on the transmission.

In this system model, the D2D pair transmits information with short packets, and their structure is shown in Figure 2. The D2D channel is modeled as the quasi-static Rayleigh fading, and the channel coefficient is constant in one slot, and changes independently between slots. Before decoding, the DR must perform the channel estimation to accurately obtain the CSI. DT transmits L symbols to DR within a time slot, where the first $L_p$ symbols are pilot symbols for channel estimation from DT to DR, and the remaining $L_d=L-L_p$ symbols are data symbols. We also assume that Willie does not know the pilot sequence and the information codebook, and the D2D pair shares the secret sequence to randomly arrange the transmitted pilot and data symbols [7,24].

The channel between any two users i and j is labeled $h_{ij}$, and the subscripts can be $cr$, $cw$, $tr$ and $tw$, corresponding to the CUE-DR, CUE-Willie, DT-DR and DT-Willie channels, respectively. The additive white Gaussian noise of the DR, Willie and base station are denoted as ${\mathbf{n}}_r\left[l\right]$, ${\mathbf{n}}_w\left[l\right]$ and ${\mathbf{n}}_b\left[l\right]$, respectively, and ${\mathbf{n}}_r\left[l\right]\sim \mathcal{C}\mathcal{N}\left(0,{\sigma }_r^2\right)$, ${\mathbf{n}}_w\left[l\right]\sim \mathcal{C}\mathcal{N}\left(0,{\sigma }_w^2\right)$ and ${\mathbf{n}}_b\left[l\right]\sim \mathcal{C}\mathcal{N}\left(0,{\sigma }_b^2\right)$. Supposing $h_{tr}\sim \mathcal{C}\mathcal{N}\left(0,{\beta }_{tr}\right)$ is a zero mean cyclic symmetric complex Gaussian random variable with variance $\mathbb{E}\left[{\left|h_{tr}\right|}^2\right]={\beta }_{tr}$.

2.1. Channel Estimation Modeling

It is assumed that the D2D receiver uses the Minimum Mean Square Error (MMSE) estimator when performing channel estimation. The actual fading coefficient of the channel can be expressed as the sum of the estimated channel and the estimated error [27], which are denoted by ${\widehat{h}}_{tr}$ and ${\tilde{h}}_{tr}$, respectively [26], i.e.,

$$h_{tr}={\widehat{h}}_{tr}+{\tilde{h}}_{tr}$$

(1)

and

$${\tilde{h}}_{tr}\sim \mathcal{C}\mathcal{N}\left(0,\frac{{\beta }_{tr}{\sigma }_r^2}{{\beta }_{tr}{L}_p{P}_p+{\sigma }_r^2}\right)$$

(2)

where $P_p$ represents the average power of the pilot signal. For convenience of expression, the variance of the estimated channel and error are denoted as ${\widehat{\beta }}_{tr}$ and ${\tilde{\beta }}_{tr}$, respectively, and the variance of the channel from CUE to DR is denoted as ${\beta }_{cr}$.

In the decoding stage, the signal vector at DR can be expressed as:

$${\mathbf{y}}_{rd}\left[l\right]=\sqrt{P_d}{\widehat{h}}_{tr}{\mathbf{x}}_d\left[l\right]+\sqrt{P_d}{\tilde{h}}_{tr}{\mathbf{x}}_d\left[l\right]+\sqrt{P_c}{h}_{cr}{\mathbf{x}}_c\left[l\right]+{\mathbf{n}}_r\left[l\right]$$

(3)

where ${\mathit{P}}_d$ represents the average power of data symbols, ${\mathit{P}}_c$ is the transmission power of CUEs, and ${\mathbf{x}}_d\in {\mathcal{C}}^{1\times {\mathit{L}}_d}$ and ${\mathbf{x}}_c\in {\mathcal{C}}^{1\times {\mathit{L}}_d}$ represent data symbols and CUE transmission signals, respectively. The SINR of DR can be expressed as:

$$\gamma \stackrel{\Delta }{=}\frac{{\mathit{P}}_d{\widehat{\beta }}_{tr}}{{\sigma }_r^2+{\mathit{P}}_c{\beta }_{cr}+{\mathit{P}}_d{\tilde{\beta }}_{tr}}=\frac{{\mathit{P}}_d{\beta }_{tr}^2{\mathit{L}}_p{\mathit{P}}_p}{{\zeta }_r^2+{\mathit{P}}_d{\beta }_{tr}{\zeta }_r+{\zeta }_r{\beta }_{tr}{\mathit{L}}_p{\mathit{P}}_p}$$

(4)

In order to simplify the expression, ${\zeta }_r={\sigma }_r^2+P_c{\beta }_{cr}$, $\mu$ denotes the power fraction allocated to data in the total power, and the power fraction allocated to pilot symbols is denoted as $1-\mu$, which is specifically expressed as [23]:

$${\mathit{P}}_d{\mathit{L}}_d=\mu \mathit{PL},{\mathit{P}}_p{\mathit{L}}_p=(1-\mu )\mathit{PL},0<\mu <1$$

(5)

where P represents the average power of all transmitted signals. Therefore, substituting (5) into (4), the SINR of DR can be further expressed as:

$$\gamma =\frac{{\beta }_{tr}^2\mu (1-\mu )\mathit{P}L}{\left(L-L_p\right){\zeta }_r\left[\frac{{\zeta }_r}{PL}+\frac{\mu {\beta }_{tr}}{L-L_p}+{\beta }_{tr}(1-\mu )\right]}$$

(6)

In the process of channel estimation, considering its cost, the effective SINR of a single channel is

$$\begin{array}{c}{\gamma }_d=\frac{L-L_p}{L}\times \gamma \\ =\frac{{\beta }_{tr}^2\mu (1-\mu )\mathit{P}}{({\sigma }_r^2+{\mathit{P}}_c{\beta }_{cr})\left[\frac{{\sigma }_r^2+{\mathit{P}}_c{\beta }_{cr}}{\mathit{P}L}+\frac{\mu {\beta }_{tr}}{L-L_p}+{\beta }_{tr}(1-\mu )\right]}\end{array}$$

(7)

2.2. Communication Link Quality Assessment

In the D2D communication underlying cellular networks, the CUEs and D2D users sharing the same channel interfere with each other. To ensure the basic communication requirements, the SINR of the D2D link and cellular link must meet certain constraints. It is assumed that each D2D pair multiplexes, at most, one orthogonal frequency Resource Block (RB). At the same time, each RB can be reused by a D2D pair. The SINR and its restrictions for cellular users and D2D users are shown, as follows [10]:

$$SIN{R}_j^c=\frac{P_j^c{\left|h_{cb}\right|}^2}{P_i^d{\left|h_{t_{i}b}\right|}^2+{\sigma }_b^2},SIN{R}_j^c⩾SIN{R}_{min}^c$$

(8)

$$SIN{R}_i^d=\frac{P_i^d{\left|h_{cr}\right|}^2}{P_j^c{\left|h_{t_{j}r}\right|}^2+{\sigma }_r^2},SIN{R}_i^d⩾SIN{R}_{min}^d$$

(9)

where $P_j^c$ represents the transmission power of the j-th cellular user, $P_i^d$ represents the transmission power of the i-th cellular user, $SIN{R}_{min}^c$ and $SIN{R}_{min}^d$ represent the minimum signal to the interference plus noise power ratio (SINR) of the cellular user’s link and D2D user, respectively. In this paper, the links’ quality constraints for D2D users are mainly reflected in the average bit error rate (ABER) of the receiver, which will be introduced below.

3. Covert Performance Analysis

3.1. Willie’s Detection Strategy

The signal received by Willie can be expressed as:

$${\mathbf{y}}_w\left[l\right]=\left\{\begin{array}{c}\sqrt{{\mathit{P}}_c}{h}_{cw}{\mathbf{x}}_c\left[l\right]+{\mathbf{n}}_w\left[l\right],\begin{array}{ccc}& & {\mathcal{H}}_0\end{array}\hfill \\ \begin{array}{c}\sqrt{{\mathit{P}}_c}{h}_{cw}{\mathbf{x}}_c\left[l\right]+\sqrt{{\mathit{P}}_d}{h}_{tw}{\mathbf{x}}_d\left[l\right]+{\mathbf{n}}_w\left[l\right],{\mathcal{H}}_1\end{array}\hfill \end{array}\right.$$

(10)

where the channel parameter $h_{ab}$ is a zero-mean cyclic symmetric complex Gaussian random variable, and $h_{ab}\sim \mathcal{C}\mathcal{N}\left(0,{\beta }_{ab}\right)$, where $ab\in \left\{tr,tb,tw,cr,cb,cw\right\}$. ${\mathbf{x}}_c\left[l\right]$ and ${\mathbf{x}}_d\left[l\right]$, respectively, represent normalized signals transmitted by cellular users and D2D transmitters and satisfy $E\left[\mathbf{x}\left[l\right]{\mathbf{x}}^H\left[l\right]\right]=1$, where $l=1,2,3,\cdots$ represents the channel use index, ${\mathbf{n}}_w\left[l\right]$ represents AWGN at the node w, subject to distribution ${\mathbf{n}}_w\left[l\right]\sim \mathcal{C}\mathcal{N}\left(0,{\sigma }_w^2\right)$, and ${\sigma }_w^2$ is Willie’s noise power.

Willie’s detection error probability can be expressed as

$$\xi \triangleq {\mathbb{P}}_{FA}+{\mathbb{P}}_{MD}$$

(11)

where

$${\mathbb{P}}_{MD}\stackrel{\Delta }{=}Pr\left({\mathcal{D}}_0\left|{\mathcal{H}}_1\right.\right),{\mathbb{P}}_{FA}\stackrel{\Delta }{=}Pr\left({\mathcal{D}}_1\left|{\mathcal{H}}_0\right.\right)$$

(12)

${\mathbb{P}}_{MD}$ is the missed detection probability and ${\mathbb{P}}_{FA}$ is the false alarm probability. To ensure covert communication, Willie’s detection error probability must be constrained by certain conditions, namely, ${\xi }^{*}\ge 1-\epsilon$, where $\epsilon \in \left[0,1\right]$ represents the detection probability tolerated by covert constraints.

Willie makes a binary decision based on the received signal. According to the Neyman–Pearson criterion, Willie’s optimal decision rule is the likelihood ratio detection (LRT), and the mathematical expression is as follows [27,28]:

$$T\left(y_w\right)=\frac{{\mathbb{P}}_1\triangleq f\left(y_w\left[l\right]\left|{\mathcal{H}}_1\right.\right)}{{\mathbb{P}}_0\triangleq f\left(y_w\left[l\right]\left|{\mathcal{H}}_0\right.\right)}\underset{{\mathcal{D}}_0}{\overset{{\mathcal{D}}_1}{\gtrless }}1$$

(13)

Under the condition of ${\mathcal{H}}_0$, Willie’s received signal vector only depends on AWGN. Let ${\zeta }_w={\sigma }_w^2+P_c{\beta }_{cw}$ and the likelihood function of the received signal ${\mathbf{y}}_w$ be:

$$f\left({\mathbf{y}}_w\left[l\right]\left|{\mathcal{H}}_0\right.\right)=\frac{1}{{\left(\pi {\zeta }_w\right)}^L}exp\left(-\sum _{l=1}^L{\left|{\mathbf{y}}_w^l\right|}^2/\phantom{\sum _{l=1}^L{\left|{\mathbf{y}}_w^l\right|}^2{\zeta }_w}{\zeta }_w\right)$$

(14)

${\mathbf{y}}_w^l$ represents the l-th channel use of ${\mathbf{y}}_w$. Under the condition of ${\mathcal{H}}_1$, Willie’s received signal vector can be written as:

$${\mathbf{y}}_w=\left\{y_w^1,\dots ,y_w^{L_p},y_w^{L_p+1},\dots ,y_w^L\right\}$$

(15)

The first $L_p$ symbols are composed of DT transmitting pilot symbols and AWGN, and the last $L_d$ symbols are composed of DT transmitting data symbols and AWGN.

Therefore, the likelihood function of the received signal ${\mathbf{y}}_w$ under the ${\mathcal{H}}_1$ condition is:

$$\begin{array}{c}f\left({\mathbf{y}}_w\left[l\right]\left|{\mathcal{H}}_1\right.\right)={\displaystyle \prod _{l=1}^{L_p}}f\left({\mathbf{y}}_w\left[l\right]\right){\displaystyle \prod _{l=L_p+1}^L}f\left({\mathbf{y}}_w\left[l\right]\right)\\ =\frac{1}{{\left(\pi (P_p{\beta }_{tw}+{\zeta }_w)\right)}^{L_p}}{e}^{\frac{-{\displaystyle \sum _{l=1}^{L_p}}{\left|{\mathbf{y}}_w^l\right|}^2}{P_p{\beta }_{tw}+{\zeta }_w}}\times \frac{1}{{\left(\pi (P_d{\beta }_{tw}+{\zeta }_w)\right)}^{L-L_p}}{e}^{\frac{-{\displaystyle \sum _{l=L_p+1}^L}{\left|{\mathbf{y}}_w^l\right|}^2}{P_d{\beta }_{tw}+{\zeta }_w}}\end{array}$$

(16)

3.2. Covert Constraints

By observing (15) and (16), we note that it is complicated to calculate the detection error probability directly through the likelihood function. We use the total variation of Willie’s observed signal probability under different conditions to calculate the lower bound of the error detection, and thus approach the optimal detection error probability [21,23]:

$${\xi }^{*}⩾1-{\mathcal{V}}_T\left({\mathbb{P}}_1∥{\mathbb{P}}_0\right)$$

(17)

It is not practical to use total variables when the observations are vectors and the metric parameters are products of probabilities. From the Pinsker inequality ${\mathcal{V}}_T\left({\mathbb{P}}_1∥{\mathbb{P}}_0\right)⩽\sqrt{\frac{1}{2}\mathcal{D}\left({\mathbb{P}}_1∥{\mathbb{P}}_0\right)}$, a lower bound of the detection error probability is obtained [13,21], i.e.,

$${\xi }^{*}⩾1-\sqrt{\mathcal{D}\left({\mathbb{P}}_0∥{\mathbb{P}}_1\right)/2}$$

(18)

where $\mathcal{D}\left({\mathbb{P}}_0∥{\mathbb{P}}_1\right)$ is the Kullback–Leibler (KL) divergence from ${\mathbb{P}}_0$ to ${\mathbb{P}}_1$. Thus, the covert constraint can be rewritten as $\mathcal{D}\left({\mathbb{P}}_0∥{\mathbb{P}}_1\right)⩽2{\epsilon }^2$. The KL divergence of this system can be calculated as [24]:

$$\begin{array}{c}\mathcal{D}\left({\mathbb{P}}_0∥{\mathbb{P}}_1\right)=L_p\left[ln(1+\frac{P_p}{{\zeta }_w})-\frac{P_p/\phantom{P_p{\zeta }_w}{\zeta }_w}{1+P_p{\left|h_{tw}\right|}^2/\phantom{P_p{\left|h_{tw}\right|}^2{\zeta }_w}{\zeta }_w}\right]+L_d\left[ln(1+\frac{P_d}{{\zeta }_w})-\frac{P_d/\phantom{P_d{\zeta }_w}{\zeta }_w}{1+P_d/\phantom{P_d{\zeta }_w}{\zeta }_w}\right]\\ \stackrel{a}{=}\frac{\left(1-\mu \right)LP{L}_p}{\left(\mu -1\right)LP-L_p{\zeta }_w}+L_d\left[ln(1+\frac{\mu LP}{L_d{\zeta }_w})-\frac{\mu LP}{\mu LP+L_d{\zeta }_w}\right]+L_{p}ln\left[1+\frac{\left(1-\mu \right)LP}{L_p{\zeta }_w}\right]\end{array}$$

(19)

$\stackrel{a}{=}$ indicates that relevant variables are substituted into the formula and obtained through a series of simple algebraic operations.

Next, the influencing factors of the KL divergence are analyzed and discussed in Equation (19). Firstly, the first derivative of the $\mathcal{D}\left({\mathbb{P}}_0∥{\mathbb{P}}_1\right)$, with respect to the $\mu$, is calculated as:

$$\frac{\partial \mathcal{D}\left({\mathbb{P}}_0∥{\mathbb{P}}_1\right)}{\partial \mu }=\frac{L^2\left(L\mu -L_d\right)\left(L^2{\left(P{\beta }_{tw}\right)}^2{\mu }^2-L^2{\left(P{\beta }_{tw}\right)}^2\mu +L_p{L}_d{\zeta }_w^2\right)}{{\left(\mu LP{\beta }_{tw}+L_d{\zeta }_w^2\right)}^2{\left[\left(\mu -1\right)LP{\beta }_{tw}-L_p{\zeta }_w\right]}^2}$$

(20)

Let $\partial \mathcal{D}\left({\mathbb{P}}_0∥{\mathbb{P}}_1\right)/\phantom{\partial \mathcal{D}\left({\mathbb{P}}_0∥{\mathbb{P}}_1\right)\partial \mu }\partial \mu =0$ obtain the three possible values:

$${\mu }_1=L_d/\phantom{L_{d}L}L$$

(21)

$${\mu }_2=\frac{1}{2}-\frac{\sqrt{{\left(P{\beta }_{tw}\right)}^2{L}^2-4L_p{L}_d{\zeta }_w^2}}{2P{\beta }_{tw}L}$$

(22)

$${\mu }_3=\frac{1}{2}+\frac{\sqrt{{\left(P{\beta }_{tw}\right)}^2{L}^2-4L_p{L}_d{\zeta }_w^2}}{2P{\beta }_{tw}L}$$

(23)

Since $L_p{L}_d=L_p(L-L_p)$, and $L_d$, $L_p$ are positive integers, $L_d{L}_p$ obtains the minimum $L-1$ when $L_p=1$ or $L_d=1$, and when $L_p=L_d$ or $\left|L_p-L_d\right|=1$, $L_d{L}_p$ takes the maximum $L^2/\phantom{L^{2}4}4$ or $(L^2-1)/\phantom{(L^2-1)4}4$. Therefore, when ${\left(P{\beta }_{tw}\right)}^2{L}^2-4\left(L-1\right){\zeta }_w^2<0$, That is $PL{\beta }_{tw}<2\sqrt{L-1}{\zeta }_w$; at this time, ${\mu }_2$, ${\mu }_3$ are both complex numbers, and it is not advisable. ${\mu }^{*}={\mu }_1=L_d/\phantom{L_{d}L}L$. When ${\left(P{\beta }_{tw}\right)}^2{L}^2-L^2{\zeta }_w^2>0$, that is, $P{\beta }_{tw}>{\zeta }_w$, the transmission power must be higher than the equivalent noise power, which is contrary to the original intention of covert communication, and the energy of the application scenario is limited. Therefore, based on the above analysis, the best value of $\mu$ is ${\mu }^{*}=L_d/\phantom{L_{d}L}L$. Moreover, because $L_p{P}_p=(1-\mu )LP$, $L_d{P}_d=\mu LP$, we can obtain $P_p=P_d=P$ [23].

When the condition $PL{\beta }_{tw}<2\sqrt{L-1}{\zeta }_w$ is satisfied, $L^2{\left(P{\beta }_{tw}\right)}^2{\mu }^2-L^2{\left(P{\beta }_{tw}\right)}^2\mu$ + $L_p{L}_d{\zeta }_w^2$ in (20) is greater than 0. Therefore, $\partial \mathcal{D}\left({\mathbb{P}}_0∥{\mathbb{P}}_1\right)/\phantom{\partial \mathcal{D}\left({\mathbb{P}}_0∥{\mathbb{P}}_1\right)\partial \mu }\partial \mu <0$ when $\mu <{\mu }_1$ and $\partial \mathcal{D}\left({\mathbb{P}}_0∥{\mathbb{P}}_1\right)/\phantom{\partial \mathcal{D}\left({\mathbb{P}}_0∥{\mathbb{P}}_1\right)\partial \mu }\partial \mu >0$ when $\mu >{\mu }_1$. Thus, we obtain that ${\mu }_1$ is the optimal solution that minimizes $\mathcal{D}\left({\mathbb{P}}_0∥{\mathbb{P}}_1\right)$.

Therefore, when $PL{\beta }_{tw}<2\sqrt{L-1}{\zeta }_w$, the minimum value of the KL divergence $\mathcal{D}\left({\mathbb{P}}_0∥{\mathbb{P}}_1\right)$ can be obtained when pilot and data symbols are transmitted with equal power; that is, the maximum lower bound of the Willie detection error probability can be realized. Simultaneously, $PL{\beta }_{tw}<2\sqrt{L-1}{\zeta }_w$ indicates that this condition limits the total transmission power of the D2D communication system, and the value is related to the product of the packet length square root and interference noise, which is the key to ensure the optimal communication covertness of the D2D pair.

Therefore, when analyzing the D2D covert communication system based on channel estimation, we first consider optimizing the packet length to maximize the system performance under the condition of equal power transmission.

4. Communication Performance Analysis

4.1. D2D Receiver Error Packet Rate

Because the packet length of DT is limited, the decoding error is inevitable on the receiver. For the short packet communication with an encoding rate of R (nits /channel use), the packet error rate of DR can be expressed as [21,23]:

$$\delta =Q\left(\frac{\sqrt{L_d}\left(ln\left(1+{\gamma }_d\right)-R\right)}{\sqrt{1-\frac{1}{{\left({\gamma }_d+1\right)}^2}}}\right)=Q\left(\frac{\sqrt{L_d}\left(1+{\gamma }_d\right)\left(ln\left(1+{\gamma }_d\right)-R\right)}{\sqrt{{\gamma }_d\left({\gamma }_d+2\right)}}\right)$$

(24)

where ${\gamma }_d$ represents the effective received SINR of DR, the encoding rate $R=D/\phantom{D{L}_d}{L}_d$, and D is the information content of a single packet. For fading channels, the packet error rate varies with the channel coefficient. Therefore, the average packet error rate of DR can be used to characterize its receiving performance [23]:

$${\mathbb{E}}_{{\gamma }_d}\left(\delta \right)={\int }_0^{\infty }Q\left(\frac{\sqrt{L_d}\left(1+{\gamma }_d\right)\left(ln\left(1+{\gamma }_d\right)-R\right)}{\sqrt{{\gamma }_d\left({\gamma }_d+2\right)}}\right)f_{{\gamma }_d}\left(x\right)dx$$

(25)

where $Q\left(x\right)={\int }_x^{\infty }\frac{1}{\sqrt{2\pi }}exp\left(\frac{-t^2}{2}\right)dt$ and $f_{{\gamma }_d}\left(x\right)$ is the probability density function of SINR at the receiver of D2D. The expression of the packet error rate based on the Gaussian Q function can be approximately rewritten as [25,27]:

$$\begin{array}{c}\delta =Q\left(\frac{\sqrt{L_d}\left(ln\left(1+{\gamma }_d\right)-R\right)}{\sqrt{1-\frac{1}{{\left({\gamma }_d+1\right)}^2}}}\right)=Q\left(\frac{\sqrt{L_d}\left(1+{\gamma }_d\right)\left(ln\left(1+{\gamma }_d\right)-R\right)}{\sqrt{{\gamma }_d\left({\gamma }_d+2\right)}}\right)\\ \approx \left\{\begin{array}{cc}1,\hfill & {\gamma }_d<\vartheta +\frac{1}{2\phi }\hfill \\ \phi ({\gamma }_d-\vartheta )+\frac{1}{2},\vartheta +\frac{1}{2\phi }\le \hfill & {\gamma }_d\le \vartheta -\frac{1}{2\phi }\hfill \\ 0,\hfill & {\gamma }_d>\vartheta -\frac{1}{2\phi }\hfill \end{array}\right.\end{array}$$

(26)

Among them, $\phi =-\sqrt{L_d/\phantom{L_d\left(2\pi \left(exp\left(2R\right)-1\right)\right)}\left(2\pi \left(exp\left(2R\right)-1\right)\right)}$, and $\vartheta =exp\left(R\right)-1$. The average error packet rate of the D2D receiver can be obtained by the following theorem.

Theorem 1.

For the D2D pair short packet communication with an encoding rate of R, the average decoding error rate of the D2D receiver (DR) is:

$${\mathbb{E}}_{{\gamma }_d}\left(\delta \right)=1-\phi {\overline{\gamma }}_d\left(exp(-\frac{2\phi \vartheta -1}{2\phi {\overline{\gamma }}_d})-exp(-\frac{2\phi \vartheta +1}{2\phi {\overline{\gamma }}_d})\right)$$

(27)

Proof. 

The approximate expression of Q function is substituted into Formula (25) to obtain

$$\begin{array}{c}{\mathbb{E}}_{{\gamma }_d}\left(\delta \right)={\int }_0^{\vartheta +\frac{1}{2\phi }}{f}_{{\gamma }_d}\left(x\right)dx+{\int }_{\vartheta +\frac{1}{2\phi }}^{\vartheta -\frac{1}{2\phi }}\left(\phi (x-\vartheta )+\frac{1}{2}\right)f_{{\gamma }_d}\left(x\right)dx\\ =\Xi +F_{{\gamma }_d}\left(\vartheta +\frac{1}{2\phi }\right)-F_{{\gamma }_d}\left(0\right)+\left(\frac{1}{2}-\phi \vartheta \right)\left[F_{{\gamma }_d}\left(\vartheta -\frac{1}{2\phi }\right)-F_{{\gamma }_d}\left(\vartheta +\frac{1}{2\phi }\right)\right]\end{array}$$

(28)

where $\Xi ={\int }_{\vartheta +\frac{1}{2\phi }}^{\vartheta -\frac{1}{2\phi }}\phi x{f}_{{\gamma }_d}\left(x\right)dx$, $F_{{\gamma }_d}\left(x\right)=1-exp\left(-x/\phantom{-x{\gamma }_d}{\gamma }_d\right)$ represents the Cumulative Distribution Function (CDF) of SINR at the D2D receiver. According to the antiderivative $\int x{e}^{px}dx=e^{px}\left(x/\phantom{xp-1/\phantom{1p^2}{p}^2}p-1/\phantom{1p^2}{p}^2\right)$, we can obtain

$$\begin{array}{c}\Xi ={\int }_{\vartheta +\frac{1}{2\phi }}^{\vartheta -\frac{1}{2\phi }}\phi x{f}_{{\gamma }_d}\left(x\right)dx={\int }_{\vartheta +\frac{1}{2\phi }}^{\vartheta -\frac{1}{2\phi }}\frac{x}{{\overline{\gamma }}_d}xexp\left(-\frac{x}{{\overline{\gamma }}_d}\right)dx\\ =\phi exp\left(-\frac{2\phi {\overline{\gamma }}_d-1}{2\phi {\overline{\gamma }}_d}\right)\left({\overline{\gamma }}_d-\vartheta +\frac{1}{2\phi }\right)-\phi exp\left(-\frac{2\phi {\overline{\gamma }}_d+1}{2\phi {\overline{\gamma }}_d}\right)\left({\overline{\gamma }}_d-\vartheta -\frac{1}{2\phi }\right)\end{array}$$

(29)

Substitute the expression (29) into the expression (28) to obtain the proof.    □

4.2. Effective Covert Throughput

Considering covert constraints and decoding errors comprehensively, the D2D’s effective covert throughput (ECT) of the D2D receiver (DR) is defined as [28,29]:

$${\overline{\eta }}_c=(L-L_p)R\left(1-{\mathbb{E}}_{{\gamma }_d}\left(\delta \right)\right)$$

(30)

Its physical meaning is the average information transmitted reliably by the D2D pair in a single time slot under the condition of the average covert constraint, and ${\gamma }_d$ represents the average SINR of the D2D receiver. The optimization problem to maximize the effective covert throughput of the system can be modeled as:

$$\begin{array}{c}\mathcal{P}:max{\overline{\eta }}_c\hfill \\ s.t.{\xi }^{*}\ge 1-\epsilon \hfill \end{array}$$

(31)

Since the covert communication rate is a small value, $\phi =-\sqrt{L_d/\phantom{L_d\left(2\pi \left(exp\left(2R\right)-1\right)\right)}\left(2\pi \left(exp\left(2R\right)-1\right)\right)}\approx -L_d/\phantom{-L_d\left(2\sqrt{\pi D}\right)}\left(2\sqrt{\pi D}\right)$ and $\vartheta =exp\left(R\right)-1\approx R={D/\phantom{DL}L}_d$. When the independent variable x is small enough, $exp(-x)\approx 1-x+x^2/\phantom{x^{2}2}2$, the correct receiving probability of DR can be simplified, as follows [30].

$$\begin{array}{c}1-E_{{\gamma }_d}\left(\delta \right)=\phi {\overline{\gamma }}_d\left(exp(-\frac{2\phi \vartheta -1}{2\phi {\overline{\gamma }}_d})-exp(-\frac{2\phi \vartheta +1}{2\phi {\overline{\gamma }}_d})\right)\\ =\left(1-\frac{D}{L_d{\overline{\gamma }}_d}\right)\left(1-\frac{\sqrt{\pi D}}{L_d{\overline{\gamma }}_d}\right)\end{array}$$

(32)

Therefore, the optimization problem to maximize the effective covert throughput of the system can be simplified as:

$${\overline{\eta }}_c=D\left(1-\frac{D}{L_d{\overline{\gamma }}_d}\right)\left(1-\frac{\sqrt{\pi D}}{L_d{\overline{\gamma }}_d}\right)$$

(33)

In order to solve the above optimization problems, we first consider the problem of covert constraints.

(1) When $P_p=P_d=P$, according to the above analysis, Willie’s KL divergence $\mathcal{D}\left({\mathbb{P}}_0∥{\mathbb{P}}_1\right)$ can be minimized when pilot and data symbols are transmitted with the same power; that is, the maximum lower bound of Willie’s detection error rate can be obtained. In this case, the optimization problem of the D2D receiver to maximize the effective covert throughput can be written as:

$$\begin{array}{c}{\mathcal{P}}_1:\underset{L_p,P}{max}{\overline{\eta }}_c\hfill \\ s.t.C_1:{\xi }^{*}\ge 1-\epsilon \hfill \\ C_2:{\mathit{SINR}}_j^c\ge {\mathit{SINR}}_{min}^c\hfill \\ C_3:L_p+L_d=L,L_p=1,2,\dots L-1\hfill \end{array}$$

(34)

In the objective function, the transmission power P of the D2D pair and pilot symbol number $L_p$ are optimization variables. Constraint $C_1$ represents the constraint condition of the D2D communication covertness, constraint $C_2$ represents the SINR of the CUE communication link meeting the minimum requirement, constraint $C_3$ represents the constraint the total number of channels uses and is assigned to the pilot signal as an integer.

Firstly, we analyze Willie’s minimum detection error probability. When $P_p=P_d=P$, Willie’s optimal detection threshold, false alarm probability, missed detection probability and minimum detection error probability can be obtained by the following theorem, according to the LRT judgment result.

Theorem 2.

Willie’s LRT in a single time slot can be equivalent to energy detection, and the optimal detection threshold that minimizes Willie’s error detection probability and Willie’s false alarm probability and missed detection probability under the condition of an optimal detection threshold are expressed as:

$${\tau }^{*}=\frac{{\zeta }_w}{P}({\zeta }_w+P{\beta }_{tw})ln\left(1+\frac{P{\beta }_{tw}}{{\zeta }_w}\right)$$

(35)

$${\mathbb{P}}_{FA}=1-\frac{\gamma (L,L{\tau }^{*}/\phantom{L{\tau }^{*}{\zeta }_w}{\zeta }_w)}{\Gamma \left(L\right)}$$

(36)

$${\mathbb{P}}_{MD}=\frac{\gamma (L,L{\tau }^{*}/\phantom{L{\tau }^{*}(P{\beta }_{tw}+{\zeta }_w)}(P{\beta }_{tw}+{\zeta }_w))}{\Gamma \left(L\right)}$$

(37)

In the formula, $\Gamma \left(L\right)=(L-1)!$ is the complete gamma function, and $\gamma (L,x)={\int }_0^x{e}^{-t}{t}^{L-1}dt$ is the lower incomplete gamma function.

Proof. 

When $P_p=P_d=P$, there is the Likelihood Ratio Test (LRT) test development, as follows:

$$\begin{array}{c}T\left(y_w\right)=\frac{f\left(y_w\left[l\right]\left|{\mathcal{H}}_1\right.\right)}{f\left(y_w\left[l\right]\left|{\mathcal{H}}_0\right.\right)}={\left(\frac{{\zeta }_w}{P_p{\beta }_{tw}+{\zeta }_w}\right)}^{L}exp\left({\displaystyle \sum _{l=1}^L}\frac{{\left|y_w^l\right|}^2}{{\zeta }_w}-{\displaystyle \sum _{l=1}^L}\frac{{\left|y_w^l\right|}^2}{P{\beta }_{tw}+{\zeta }_w}\right)\\ ={\left(\frac{{\zeta }_w}{P_p{\beta }_{tw}+{\zeta }_w}\right)}^{L}exp\left(\frac{P{\beta }_{tw}}{{\zeta }_w\left(P{\beta }_{tw}+{\zeta }_w\right)}{\displaystyle \sum _{l=1}^L}{\left|y_w^l\right|}^2\right)\underset{{\mathcal{D}}_0}{\overset{{\mathcal{D}}_1}{\gtrless }}1\end{array}$$

(38)

A logarithm of both sides of the test formula is taken at the same time to obtain:

$$lnT\left(y_w\right)=Lln\left(\frac{{\zeta }_w}{P_p{\beta }_{tw}+{\zeta }_w}\right)+\frac{P{\beta }_{tw}}{{\zeta }_w\left(P{\beta }_{tw}+{\zeta }_w\right)}\sum _{l=1}^L{\left|y_w^l\right|}^2\underset{{\mathcal{D}}_0}{\overset{{\mathcal{D}}_1}{\gtrless }}0$$

(39)

Therefore, Willie’s likelihood ratio detection is equivalent to the following energy detection:

$$\Lambda \left(y_w\right)=\frac{1}{L}\sum _{l=1}^L{\left|y_w^l\right|}^2\underset{{\mathcal{D}}_0}{\overset{{\mathcal{D}}_1}{\gtrless }}\frac{{\zeta }_w}{P{\beta }_{tw}}\left(P{\beta }_{tw}+{\zeta }_w\right)ln\left(\frac{P_p{\beta }_{tw}+{\zeta }_w}{{\zeta }_w}\right)$$

(40)

$\Lambda \left(y_w\right)$ is the average symbol power received by Willie on a single channel use, and the right side of the decision is Willie’s optimal detection threshold, represented by symbol ${\tau }^{*}$.

Since $f\left(y_w\left[l\right]\left|{\mathcal{H}}_0\right.\right)\sim \mathcal{C}\mathcal{N}\left(0,{\zeta }_w\right)$, $f\left(y_w\left[l\right]\left|{\mathcal{H}}_1\right.\right)\sim \mathcal{C}\mathcal{N}\left(0,{\zeta }_w+P{\beta }_{tw}\right)$, and $\Lambda \left(y_w\right)$ is a chi-square distribution with degrees of freedom as $2L$, according to relevant literature, definitions of the false alarm, and missed detection probabilities, Willie’s false alarm and missed detection probabilities under ${\tau }^{*}$ are:

$${\mathbb{P}}_{FA}=1-\frac{\gamma (L,L{\tau }^{*}/\phantom{L{\tau }^{*}{\zeta }_w}{\zeta }_w)}{\Gamma \left(L\right)},{\mathbb{P}}_{MD}=\frac{\gamma (L,L{\tau }^{*}/\phantom{L{\tau }^{*}(P{\beta }_{tw}+{\zeta }_w)}(P{\beta }_{tw}+{\zeta }_w))}{\Gamma \left(L\right)}$$

(41)

   □

The minimum detection error rate of Willie is:

$${\xi }^{*}=1-\frac{\gamma (L,L{\tau }^{*}/\phantom{L{\tau }^{*}{\zeta }_w}{\zeta }_w)}{\Gamma \left(L\right)}+\frac{\gamma (L,L{\tau }^{*}/\phantom{L{\tau }^{*}(P{\beta }_{tw}+{\zeta }_w)}(P{\beta }_{tw}+{\zeta }_w))}{\Gamma \left(L\right)}$$

(42)

It can be observed from the expression that under $P_p=P_d=P$, the detection error probability of Willie is determined by the total number of packet lengths L, and is independent of $L_p$ and $L_d$.

Then, consider the maximum covert throughput ${\overline{\eta }}_c$ of the D2D communication system,

$${\overline{\eta }}_c=\frac{D(L-L_P)}{L_d}\phi {\gamma }_d\left(exp(-\frac{2\phi \vartheta -1}{2\phi {\gamma }_d})-exp(-\frac{2\phi \vartheta +1}{2\phi {\gamma }_d})\right)$$

(43)

When pilot and data symbols are sent at an equal power, SINR at the receiver of D2D is

$$\begin{array}{c}{\overline{\gamma }}_d=\frac{{\beta }_{tr}^2\mu (1-\mu )PL}{\left(L-L_p\right)({\sigma }_r^2+P_c{\beta }_{cr})\left[\frac{{\sigma }_r^2+P_c{\beta }_{cr}}{PL}+\frac{\mu {\beta }_{tr}}{L-L_p}+{\beta }_{tr}(1-\mu )\right]}\\ =\frac{{\beta }_{tr}^2{P}^2{L}_p}{({\sigma }_r^2+P_c{\beta }_{cr})\left[{\sigma }_r^2+P_c{\beta }_{cr}+P{\beta }_{tr}(L_p+1)\right]}\end{array}$$

(44)

$$1-\frac{D}{L_d{\overline{\gamma }}_d}=1-\frac{D({\sigma }_r^2+P_c{\beta }_{cr})\left[{\sigma }_r^2+P_c{\beta }_{cr}+P{\beta }_{tr}(L_p+1)\right]}{L_d{\beta }_{tr}^2{P}^2{L}_p}$$

(45)

$$\frac{\partial {\overline{\eta }}_c}{\partial L_p}=\frac{DP{\beta }_{tr}\left(D{\zeta }_r^2-LP{\beta }_{tr}\right)L_p^2+DL{\zeta }_r^2({\zeta }_r^2+P{\beta }_{tr})}{L^2{\beta }_{tr}^{2}P{L}_p^2}$$

(46)

In the formula, ${\zeta }_r^2={\sigma }_r^2+P_c{\beta }_{cr}$, it can be observed that the molecular formula is a quadratic term of one unary; $\partial {\overline{\eta }}_c/\phantom{\partial {\overline{\eta }}_c\partial L_p}\partial L_p=0$ has two real roots, one positive and one negative. The channel use of the pilot symbols is a positive integer; thus, the solution of $L_p$ can be obtained as follows:

$$L_p=\sqrt{\frac{{\zeta }_r^2({\zeta }_r^2+P^{*}{\beta }_{tr})}{P^{*}{\beta }_{tr}\left(L{P}^{*}{\beta }_{tr}-D{\zeta }_r^2\right)}}$$

(47)

$L_p$ is a positive integer. Therefore, when $P_p=P_d=P$, the optimal pilot symbol number $L_p$ of the D2D receiver to maximize ECT can be expressed as:

$$L_p^{*}=\left\{\begin{array}{c}{L}_p^{ceil},if{\overline{\eta }}_c^{ceil}\ge {\overline{\eta }}_c^{floor}\hfill \\ L_p^{floor},if{\overline{\eta }}_c^{floor}\ge {\overline{\eta }}_c^{ceil}\hfill \end{array}\right.$$

(48)

among them

$$L_p^{ceil}=⌈\sqrt{\frac{{\zeta }_r^2({\zeta }_r^2+P^{*}{\beta }_{tr})}{P^{*}{\beta }_{tr}\left(L{P}^{*}{\beta }_{tr}-D{\zeta }_r^2\right)}}⌉$$

(49)

$$L_p^{floor}=⌊\sqrt{\frac{{\zeta }_r^2({\zeta }_r^2+P^{*}{\beta }_{tr})}{P^{*}{\beta }_{tr}\left(L{P}^{*}{\beta }_{tr}-D{\zeta }_r^2\right)}}⌋$$

(50)

In this formula, the symbols ⌈⌉ and ⌊⌋ represent the function of rounding up and down, respectively.

At the same time, the optimal value of the average symbolic power $P^{*}$ can be obtained by the following equation.

$$1-\frac{\gamma (L,L{\tau }^{*}/\phantom{L{\tau }^{*}{\zeta }_w}{\zeta }_w)}{\Gamma \left(L\right)}+\frac{\gamma (L,L{\tau }^{*}/\phantom{L{\tau }^{*}(P+{\zeta }_w)}(P+{\zeta }_w))}{\Gamma \left(L\right)}=\epsilon$$

(51)

In the formula, ${\tau }^{*}=\frac{{\zeta }_w}{P}({\zeta }_w+P)ln\left(1+\frac{P}{{\zeta }_w}\right)$, $\epsilon$ is the covert constraint and given value under general conditions. According to the analysis of the literature, when the covertness requirement is higher, the number of data symbols sent should be reduced, and more pilots are needed to obtain accurate channel information.

(2) When $P_p\ne P_d$, the optimization problem to maximize the effective covert throughput of the system can be modeled as:

$$\begin{array}{c}{\mathcal{P}}_2:\underset{L_p,\mu ,P}{max}{\overline{\eta }}_c\hfill \\ s.t.C_1:{\xi }^{*}\ge 1-\epsilon \hfill \\ C_2:{\mathit{SINR}}_j^c\ge {\mathit{SINR}}_{min}^c\hfill \\ C_3:P_d{L}_d=\mu PL,P_p{L}_p=(1-\mu )PL\hfill \\ C_4:L_p+L_d=L,L_p=1,2,\dots L-1\hfill \end{array}$$

(52)

${\overline{\gamma }}_d$ represents the average SINR of a D2D pair at the receiving end. In the objective function, the transmission power P of DT and the number of pilot symbols $L_p$ are optimization variables. Constraint $C_1$ represents the constraint condition of communication covertness, constraint $C_2$ represents the communication link of cellular users meeting the minimum demand, constraint $C_3$ represents the distribution factor constraint of the total signal power, and constraint $C_4$ represents the total number of channels and the integer allocated to the pilot signal.

When $P_p\ne P_d$, the expression of ${\xi }^{*}$ in the optimization problem is different from $P_p=P_d$. In order to solve the above problems, the latest expression of ${\xi }^{*}$ should be first derived, and the statistic under ${\mathcal{H}}_1$ is

$$T=X_1+X_2$$

(53)

and

$$X_1=\frac{1}{L}\left(\sum _{l=1}^{L_p}\left({\left|{\mathbf{y}}_w\left[l\right]\right|}^2\right)\right)=\frac{P_p+{\zeta }_w}{2L}{\chi }_{2L_p}^2$$

(54)

$$X_2=\frac{1}{L}\left(\sum _{l=L_p+1}^L\left({\left|{\mathbf{y}}_w\left[l\right]\right|}^2\right)\right)=\frac{P_d+{\zeta }_w}{2L}{\chi }_{2L_d}^2$$

(55)

${\chi }_{2L_p}^2$ and ${\chi }_{2L_d}^2$ are chi-square random variables with degrees of freedom $2L_p$ and $2L_d$, respectively, and their probability density functions are as follows:

$$f_{X_1}\left(x\right)=\frac{x^{L_p-1}}{\Gamma \left(L_p\right)}{\left(\frac{L}{P_p+{\zeta }_w}\right)}^{L_p}{e}^{\frac{Lx}{P_p+{\zeta }_w}}$$

(56)

$$f_{X_1}\left(x\right)=\frac{x^{L_p-1}}{\Gamma \left(L_p\right)}{\left(\frac{L}{P_p+{\zeta }_w}\right)}^{L_p}{e}^{\frac{Lx}{P_p+{\zeta }_w}}$$

(57)

The likelihood function of T under ${\mathcal{H}}_1$ is [23]:

$$\begin{array}{c}{f}_{\left.T\right|{\mathcal{H}}_1}\left(t\right)=\frac{L^L{t}^{L-1}}{{\left(P_p+{\zeta }_w\right)}^{L_p}{\left(P_d+{\zeta }_w\right)}^{L_d}\Gamma \left(L\right)}{e}^{\frac{Lt}{P_p+{\zeta }_w}}{\times }_1{F}_1\left(L_p;L;\frac{Lt(P_p-P_d)}{\left(P_p+{\zeta }_w\right)\left(P_d+{\zeta }_w\right)}\right)\hfill \end{array}$$

(58)

where ${}_1{F}_1\left(a;b;c\right)$ represents the first kind of confluent hypergeometric function and is the degenerate form of the hypergeometric function ${}_2{F}_1\left(a;b;c\right)$; ${}_2{F}_1\left(a;b;c\right)$ is the solution of the confluent hypergeometric differential equation. The confluence hypergeometric function can be expressed as the following hypergeometric number.

$${}_1{F}_1\left(a;b;c\right)=1+\frac{a}{b}z+\frac{a(a+1)}{b(b+1)}\frac{z^2}{2!}+\dots +\sum _{k=0}^{\infty }\frac{{\left(a\right)}_k}{{\left(b\right)}_k}\frac{z^k}{k!}$$

(59)

${\left(a\right)}_k$ and ${\left(b\right)}_k$ are Pochhammer symbols. It can be observed that due to the complexity of the expression, it is difficult to directly derive the closed expression of Willie’s detection error probability. In order to achieve the goal of maximizing the effective covert throughput under covert conditions effectively, according to the analysis of [24] and combined with the actual situation of D2D communication scenarios, for $P_p\ne P_d$, the optimal detection threshold that minimizes the Willie detection error probability can be approximately expressed as:

$${\tau }^{*}=-\frac{{\zeta }_w(P_p+{\zeta }_w)}{P_p}\times W\left\{-1,-\frac{P_p}{L\left(P_p+{\zeta }_w\right)}{\left(\frac{\Gamma \left(L\right)}{\Gamma \left(L_p\right)}\right)}^{1/\phantom{1L}L}{e}^{\frac{P_d+{\zeta }_w}{P_p+{\zeta }_w}}\right\}$$

(60)

The minimum error probability ${\xi }^{*}$ is

$${\xi }^{*}=1-\frac{\gamma (L,L{\tau }^{*}/\phantom{L{\tau }^{*}{\zeta }_w}{\zeta }_w)}{\Gamma \left(L\right)}+\frac{\gamma (L_p,L\left({\tau }^{*}-P-{\zeta }_w\right)/\phantom{L\left({\tau }^{*}-P-{\zeta }_w\right)(P+{\zeta }_w)}(P+{\zeta }_w))}{\Gamma \left(L\right)}$$

(61)

where $W\left\{z\right\}$ is the Lambert W function and represents the solution t of the equation $x=t{e}^t$.

The above process approximates the packet length to a large enough number. However, in reality, the packet length may not be large under the condition of short packet communication, but the minimum detection error probability obtained by this approximation can be the lower bound of the minimum detection error probability. Therefore, the optimization problem can be solved directly by 2D (2-Dimension) numerical search algorithm (Algorithm 1) under the $P_p\ne P_d$ condition. In the numerical search, send power P is placed in the outermost layer of the cycle. For any P, according to the constraints (52), the relationship between $L_p$ and $\mu$ is found, represented by $\mu$ as a function of $L_p$, and $L_p$ is searched within the range of $1\le L_p\le L-1$ to maximize the objective function. The specific implementation of this search algorithm is shown in the table.

Algorithm 1 The 2D search algorithm for maximizing throughput

1: Input: the sets of D2D pairs average transmit power and channel allocation factors $\mathcal{P}$ and $\mathcal{U}$. 2: Output: the matching result, $\mathcal{M}$. 3: Initialization: packet length L, $L_p$, receiver noise ${\sigma }_w^2$, ${\sigma }_b^2$, ${\sigma }_r^2$, and channel coefficient ${\beta }_{tr}$.   4:    For any given $\mathit{P}$, according to the Formula (52), let $L_p$ be expressed as the function of $\mu$, obtain the ${\mu }^{\prime }$ and $L_p^{\prime }$ which maximize ${\overline{\eta }}_c$; 5:    Put P, ${\mu }^{\prime }$ and $L_p^{\prime }$ into ${\overline{\eta }}_c$ and obtain ${{\overline{\eta }}_c}^{\prime }$. 6: End for 7:    Find the maximum value of all ${{\overline{\eta }}_c}^{\prime }$, and obtain the corresponding ${\mu }^{*}$ and $P^{*}$ as the global optimal solution. 8: Return:

5. Simulation Results and Analysis

Next, we will verify the above results by numerical simulation, and observe the influence of different parameters on channel estimation in the D2D covert communication system. In the model considered, the coverage radius of the cellular user cell is 1000 meters without a special description. The specific parameter settings are shown in the

Table 1. Simulation Parameter.

Parameter	Value
The minimum distance between CUE and BS	30 m
The maximum distance of D2D users	50 m
The distance between DT and CUE	80 m
The distance between DT and Willie	80 m
The distance between CUEs and DR	150 m
Cell radius of CUEs	1000 m
D2D user’s maximum transmit power	24 dBm
The noise power at the receiver of Willie and DR	−80 dBm
The user’s minimum SINR	0∼10 dB
The transmitted power of CUEs	17∼23 dBm

. The communication distance of the D2D pair is 30 meters, and its transmission power range is 17∼23 dBm; the distance between CUEs and BS is 50 meters, and the distance between the D2D transmitter and the CUE and the Willie are 80 meters, respectively. The distance between the D2D transmitter and BS is 100 meters, and the distance between the CUE and the D2D receiver is 150 meters; the user’s minimum SINR requirement is 0∼10 dB; the noise power of Willie and D2D receivers are set as ${\sigma }_w^2={\sigma }_r^2=-80$ dBm [10,24].

In order to verify that the equal power transmission strategy of the pilot and data can achieve the best covertness of D2D communication, Willie’s detection performance must be the worst. Figure 3 shows the trend diagram of the variation of the Willie detection error probability with distribution parameters under different $L_p$. As can be observed from the figure, with the increase in allocation parameter, the detection error probability of Willie first increases and then decreases, reaching the maximum value at $\mu =L_d/L$. It can be observed that the equal power transmission strategy can maximize the detection error probability of Willie. Therefore, equal-power transmission is the best scheme if only the communication covert performance is considered.

Figure 3 shows the minimum detection error probability ${\xi }^{*}$ achieved by the two methods, namely the lower bound on ${\xi }^{*}$ determined by KL divergence and the likelihood ratio test. As can be observed from the figure, the variation trends obtained by the two methods are consistent, and despite some differences, the maximum value of ${\xi }^{*}$ is obtained at the same $\mu$. This indicates that $P_d=P_p$ can maximize the actual detection error probability; namely, the equal-power transmission strategy can indeed minimize the detection performance of Willie. In addition, we can also observe from the figure that when the allocation parameter and packet length are fixed, the detection error probability also increases with the increase in the pilot packet length. It can be concluded that when the covert requirement is high, it can be achieved by increasing the pilot packet length.

Figure 4 describes the change of the D2D communication covert throughput with the number of pilot symbols under the condition of the equal power transmission of the pilot and data. As can be observed from the figure, there exists an optimal value $L_p^{*}$ on the curve that maximizes the covert throughput ${\eta }_c$. At the same time, it can be observed that with the increase in the transmission power of the D2D transmitter, the covert throughput of the D2D pair also increases, but the covertness of the communication decreases correspondingly. It can be observed that in order to achieve high communication covertness, a certain communication performance needs to be sacrificed.

In Figure 5, we compare the variation curves of the average error packet rate of the D2D receiver versus the SINR of the D2D receiver at different coding rates R. Firstly, it is observed that the average error packet rate decreases with the increase in the SINR of the D2D receiver when the encoding rate R is fixed. In the case of the fixed SINR, the larger the encoding rate of the short packet communication, the larger the average packet error rate of the D2D receiver. It is not difficult to explain. When the SINR of the D2D receiver is large, it means that it is less affected by noise and has a strong anti-interference ability; thus, the average error packet rate is relatively low. At the same time, with the same SINR, the higher the encoding rate is, the more packets are received, and the higher the error packet rate is, naturally. Therefore, in order to ensure that the packet error rate of the D2D receiver is controlled within a certain range, the SINR of the receiver must be guaranteed to reach a certain threshold, which is consistent with the conclusion of the theoretical analysis.

Figure 6 mainly shows the influence of the change of transmission power P on the covert throughput ${\eta }_c$ of the D2D communication system when the package length of the pilot is $L_p=10,20,30$, $L=100$. It can be observed that with the increase in the power of the transmitter, the ECT of the system increases and remains unchanged after reaching a certain value. This is because with the continuous increase in the D2D transmission power, on the one hand, it will cause serious interference to the communication link of the cellular network and reduce the communication quality of cellular users. On the other hand, the increase in transmission power also leads to an increase in the risk of being detected by Willie, which reduces ECT at the D2D receiver. In addition, the pilot symbols $L_p$ also have a significant impact on ${\eta }_c$. Under the same transmission power condition, the larger the pilot packet length, the smaller the throughput. This is because pilots take up more packet length, which will inevitably lead to less data transfer in the case of limited packet length. The above analysis indirectly verifies the conclusion of Figure 4.

Figure 7 and Figure 8 mainly show the comparison graphs of the SINR and ECT of the D2D receiver obtained by an equal-power transmission strategy and joint optimization method. As can be observed from the figure, compared with the equal-power transmission scheme, the transmission performance of the non-equal-power transmission scheme has been significantly improved, because the non-equal-power transmission scheme can not only optimize the power allocation of pilot and data symbols, but also optimize the packet length allocation scheme. In the equal-power transmission scheme, in order to ensure a better system covertness, some effective SINRs and communication throughputs of the system are sacrificed. That also verifies the trade-off between the communication covertness and reliability proposed in this paper.

By observing Figure 7 and Figure 8, it can be found that with the increase in transmission power, the SINR and ECT of the system first demonstrate an upward trend, and then remain unchanged or decline after reaching a certain value, which is caused by the restriction of the communication quality of the cellular link.

6. Conclusions

In this work, the traditional MMSE channel estimation is applied to D2D covert communication scenario, and the resource allocation scheme of the pilot and packet length is studied. Specifically, it is assumed that the D2D pair communication channel and the Willie detection channel are Rayleigh fading channels. Firstly, it is proved that the equal power transmission of the pilot symbol and data symbol can minimize Willie’s detection performance, and, on this basis, the maximum ECT of the D2D communication system is analyzed. However, the transmission performance of the communication system may not be optimal under the condition of the optimal covertness. Then, the joint optimization of the transmission power and channel allocation factor are analyzed to maximize the ECT of the communication system. The simulation results demonstrate that the requirement of covertness is proportional to the number of optimal pilot symbols, which means that when the number of symbols is fixed and the requirement of covertness is high, the number of the pilot needs to be increased to obtain the correct CSI, while the proportion of the data information will inevitably decrease. Under the condition of the jointly optimized power and packet length allocation strategy, the transmission performance of the communication system is obviously better than that of equal power transmission schemes, but the complexity is higher. Therefore, the equal-power transmission strategy achieves a good compromise between the complexity and communication performance, while the joint optimization scheme is the good compromise of the communication covertness and reliability.