# The Optimal Carrier-Secret Ratio for Wireless Covert Channels Based on Constellation Shaping Modulation

Authors: Sen Qiao; Guangjie Liu; Xiaopeng Ji; Weiwei Liu
Journal: Security and Communication Networks
 (2021)

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## Abstract

Wireless covert communication is an emerging communication technique that prevents eavesdropping. This paper considers the bit error ratio (BER) problem of covert communication based on constellation shaping modulation (CSM). The impact of carrier-secret ratio (CSR) on BER is studied and the approximate solution of optimal CSR is obtained. Then, we extended the conclusion to typical communication scenarios with one and more relays where the undetectability and reliability were analyzed and inspected. It is proved that there also exists the optimal CSR in scenarios with relays. Additionally, it is found that the undetectability under the constraints of constant total power depends on the eavesdropper’s position, and we found an undetectability deterioration area (UDA) in the scenario of relays. Simulation results show the existence of optimal CSR and its impact on transmission performance.


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## Body

## 1. Introduction

Due to the openness of wireless channels, wireless communication systems are extremely vulnerable to attacks, counterfeiting, and eavesdropping. With the advent of the Internet of Things (IoT) era, a large number of smart devices are connected and controlled to meet various requirements. Hence, it is very important to safeguard the information against security breaches and to ensure the privacy of communication.To ensure the security of personal information, some efficient anonymous authentication schemes have been proposed to adapt to different scenarios [1–3]. Multiple technologies are integrated to promote the realization of the Internet of Things (IoT), including wireless sensor networks (WSNs), radio frequency identification (RFID), machine to machine (M2M), and low-power personal area networks (PANs) [4]. To ensure the high efficiency of information channels from the clients to the cloud server, Ahmad proposed a new variant of the optimistic concurrency control protocol to avoid using the upstream communication channel all the time [5]. The most important precondition of secure and reliable group communication is an efficient group key distribution. Azees and Vijayakumar proposed a computationally efficient group key distribution scheme for secure group communication based on bilinear pairing [6]. As the number of devices connected to supporting platforms continues to increase, some proper means for access control are demanding, such as authentication and authorization method [7, 8], image watermarking [9], and cloud computing [10].However, the challenges of information security and privacy are not limited to the above. Count on the rapid growth of telecommunication field new challenges arises [11]. Eavesdroppers can intercept the wireless communication signals and try to get the communication contents, which poses a great threat to the security and privacy of the communication. In order to ameliorate the undetectability of private information, information hiding technology gradually becomes necessary. As a branch of modern information hiding technology in the field of wireless communication, wireless covert channels hide the transmission process of information that needs to be kept secret in the process of normal wireless communication. Even if an eavesdropper intercepts the communication signal, it cannot be distinguished from normal wireless communication.Based on the ubiquitous channel noise phenomenon, modulation-type wireless covert communication modulates the secret information into an artificial noise signal, which is superimposed on the normal communication signal. It is the most widely used physical-layer wireless covert communication at present. The basic theory and performance limit of the covert communication in AWGN channels are discussed in Reference [12]. It is indicated that at most Οn bits can be transmitted to the receiver reliably without being detected by the detector.


### 1.1. Motivations

In modulation-type wireless covert communication, the bit error ratio (BER) of covert information is usually much greater than that of the carrier signal. The problems of BER are always solved by means of coding or increasing the power of covert signals. Yet, the difficulty of decryption and the transmission rate of the covert messages will deteriorate with encoding. By means of increasing the transmission power, undetectability will deteriorate [13]. In Reference [14], relays are proposed to increase the power of covert signal received. But it has not been simulated with specific modulation methods, and no one has considered whether the optimal power ratio of covert signal exists. We plan to research the undetectability and reliability of wireless covert communication in the scenario of relays based on a specific modulation method. And consider whether there exists an optimal carrier-secret ratio (CSR), which can ameliorate the reliability of covert communication under the premise of meeting the requirements of undetectability.


### 1.2. Contributions

The contributions of our work are as follows:(1)
We investigated the relationship between BER, CSR, and SNR in wireless covert channels with constellation shaping modulation. We obtained the approximate solution of optimal CSR and extended it to several scenarios with relays. With the approximate solution of optimal CSR, the process of searching for an actual optimal CSR can be accelerated when some optimization algorithms are adopted such as gradient descent and conjugate gradient.(2)
We found an undetectability deterioration area (UDA) in the scenario of one relay and two relays, and the undetectability deteriorates when an eavesdropper is in it. The UDA can be used to avoid the deterioration of undetectability with an improper set of relays. Otherwise, eavesdroppers can detect in the UDA to improve detection efficiency.The remainder of this paper is organized as follows: in the next section, some background including wireless covert channel with dirty constellation and wireless covert channel with constellation shaping modulation is introduced; in Section3, we introduced the basis of our scheme including the classic system model and binary hypothesis testing; in Section 4, the relationship between BER, CSR, and SNR in wireless covert channels with constellation shaping modulation is investigated. The approximate solution of optimal CSR is obtained and extended to several scenarios of relays, in which undetectability deterioration areas (UDAs) were found and analyzed; Section 5 gives the experimental results on undetectability and reliability; and finally, Section 6 concludes the whole paper.


## 1.1. Motivations

In modulation-type wireless covert communication, the bit error ratio (BER) of covert information is usually much greater than that of the carrier signal. The problems of BER are always solved by means of coding or increasing the power of covert signals. Yet, the difficulty of decryption and the transmission rate of the covert messages will deteriorate with encoding. By means of increasing the transmission power, undetectability will deteriorate [13]. In Reference [14], relays are proposed to increase the power of covert signal received. But it has not been simulated with specific modulation methods, and no one has considered whether the optimal power ratio of covert signal exists. We plan to research the undetectability and reliability of wireless covert communication in the scenario of relays based on a specific modulation method. And consider whether there exists an optimal carrier-secret ratio (CSR), which can ameliorate the reliability of covert communication under the premise of meeting the requirements of undetectability.


## 1.2. Contributions

The contributions of our work are as follows:(1)
We investigated the relationship between BER, CSR, and SNR in wireless covert channels with constellation shaping modulation. We obtained the approximate solution of optimal CSR and extended it to several scenarios with relays. With the approximate solution of optimal CSR, the process of searching for an actual optimal CSR can be accelerated when some optimization algorithms are adopted such as gradient descent and conjugate gradient.(2)
We found an undetectability deterioration area (UDA) in the scenario of one relay and two relays, and the undetectability deteriorates when an eavesdropper is in it. The UDA can be used to avoid the deterioration of undetectability with an improper set of relays. Otherwise, eavesdroppers can detect in the UDA to improve detection efficiency.The remainder of this paper is organized as follows: in the next section, some background including wireless covert channel with dirty constellation and wireless covert channel with constellation shaping modulation is introduced; in Section3, we introduced the basis of our scheme including the classic system model and binary hypothesis testing; in Section 4, the relationship between BER, CSR, and SNR in wireless covert channels with constellation shaping modulation is investigated. The approximate solution of optimal CSR is obtained and extended to several scenarios of relays, in which undetectability deterioration areas (UDAs) were found and analyzed; Section 5 gives the experimental results on undetectability and reliability; and finally, Section 6 concludes the whole paper.


## 2. Related Works

### 2.1. Background

Wireless covert communication mainly involves three factors of inspection: undetectability, reliability, and communication rate. At present, there is no special detection work to measure undetectability for noisy wireless covert communication. References [13, 15, 16] take KL divergence between residual and ambient noise as parameters to inspect undetectability. Reference [17] inspects the undetectability with KS distance between residual and ambient noise.Reliability refers to the ability of wireless covert communication to resist channel interference. Channel interference may come from the natural fading of the channel, or from the jammer. To resist channel interference, multihop relaying is a frequently used method [18]. Reference [19] evaluated and optimized the covert communications by designing the parameters of the multihop network, including the coding rates, transmit power, and required number of hops.The researchers further analyzed the covert communication capacity of multiple scenarios with multiple unfavorable factors to the eavesdropper, including three aspects of the transmitter [20, 21], receiver [22–24], and additional nodes [25–28]. The methods of covert communication include artificial additional signal noise, artificial coding domain error, insertion of additional signal band, etc. The research results in this field have also been further extended to other communication scenarios such as relay communication [14, 29], multiantenna [30, 31], and broadcast communication [32–34].


### 2.2. Wireless Covert Channel with Dirty Constellation

In the wireless covert channel with dirty constellation (WCC-DC), the secret message bits can be transmitted as the constellation error of the normal signal in order to reduce the suspicion by all uninformed detectors.The framework of a wireless covert channel with dirty constellation is shown in Figure1(a). The wireless covert channel is implemented on the wireless communication physical layer with OFDM structure. The transmitter divides all OFDM subcarriers into secret subcarriers and normal subcarriers. On the secret subcarrier, the carrier information is modulated in QPSK to obtain the carrier signal, and then the covert signal modulated by QPSK is superposed on the carrier signal. The covert constellation points are rotated at a certain angle around normal constellation points, as shown in Figure 1(b). On the normal subcarrier, carrier information is modulated in QPSK to obtain the carrier signal, and then the random noise is superimposed. The purpose of the rotation of signal-loaded signals and the superimposition of random noise is to remove the regularity of secret signals in the constellation. The subcarrier partition results of the wireless covert channel should be shared between the transmitter and the receiver.Figure 1
The schematic diagram of WCC-DC: (a) the framework of WCC-DC and (b) rotation of covert constellation.
(a)(b)However, the wireless covert channel with dirty constellation has a high BER when the power of covert signal is low. When we increase the power of covert signal, the undetectability of covert communication deteriorates. Therefore, Cao et al. [35] proposed a covert communication method based on constellation shaping modulation (WCC-CSM).


### 2.3. Wireless Covert Channel with Constellation Shaping Modulation

The general framework for the wireless communication system with constellation shaping modulation is demonstrated in Figure2. We suppose that each subcarrier mc of the OFDM wireless communication is modulated by QPSK. In the proposed scheme, we can use all subcarriers to establish the wireless covert communication. With constellation shaping modulation, the secret information ms is modulated into an artificial noise signal Ss. Then, the artificial noise signal Ss is superimposed on the carrier signal Sc to generate the secret subcarrier Sct.Figure 2
The framework of WCC-CSM wireless covert channel.To generate the secret artificial noise signalSs, the cumulative distribution function (CDF) FCDF of noise is estimated with the reference channel noise data S0.The secret information is denoted byms = ms,1,ms,2,…ms,N, and the artificial noise signals Ss are divided into I/Q vectors, which are denoted by xsI+j⋅xsQ. Here, xsI is the I vector of artificial noise signals denoted by xsI=xs,1I,xs,2I,xs,3I,…xs,NI, and xsQ is the Q vector denoted byxsQ=xs,1Q,xs,2Q,xs,3Q,…xs,NQ. The constellation shaping modulation function is defined as(1)FSMFms=Ss=xsI+j⋅xsQ.For shaping modulation, the transmitter firstly transforms the secret informationms into continuous variables di, and then di are mapping to artificial noise signal Ss with CDF of the reference channel noise Snormal.The transform function ofdi is defined as follows:(2)di=ms,i+r2.We denoter as a random number distributed in the interval (0, 1). And the mapping function which transforms ms into Ss is defined as(3)Ss=FCDF−1di.The mapping functionFCDF−1 is the inverse function of cumulative distribution function of Snormal. Then, the artificial noise signal Ss is superimposed on carrier signal Sc to generate secret subcarrier Sct.The received secret subcarrier is denoted byS^ct. The I/Q vectors of secret subcarrier S^ct are denoted by x^ctI+j⋅x^ctQ, the subcarrier m^c can be demodulated with QPSK:(4)m^c=Fde−QPSKx^ctI+j⋅x^ctQ,and the subcarrier m^c will be modulated by QPSK again to acquire the ideal the subcarrier S^c:(5)S^c=FQPSKm^c=x^cI+j⋅x^cQ.We denotex^cI+j⋅x^cQ as the I/Q vectors of S^c. The receiver can obtain the ideal subcarrier x^cI+j⋅x^cQ, and then the residual signal (i.e., artificial noise S^s) can be extracted with(6)S^s=S^ct−S^c=x^ctI−x^cI+j⋅x^ctQ−x^cQ.We denote theI/Q vectors of artificial noise S^s as x^sI+j⋅x^sQ. The artificial noise S^s can be transformed into d^i by the cumulative distribution function FCDF with(7)d^i=FCDFS^s.The receiver can demodulate the secret informationm^s by the covert demodulation constellation (CDC), which is illustrated in Figure 3.Figure 3
Covert demodulation constellation of WCC-CSM.The four black points are the ideal constellation points; the red regions are the distribution areas of secret subcarrier with artificial noise. The function of covert demodulation constellation is denoted byFCDC⋅:(8)m^s=FCDCdi.


## 2.1. Background

Wireless covert communication mainly involves three factors of inspection: undetectability, reliability, and communication rate. At present, there is no special detection work to measure undetectability for noisy wireless covert communication. References [13, 15, 16] take KL divergence between residual and ambient noise as parameters to inspect undetectability. Reference [17] inspects the undetectability with KS distance between residual and ambient noise.Reliability refers to the ability of wireless covert communication to resist channel interference. Channel interference may come from the natural fading of the channel, or from the jammer. To resist channel interference, multihop relaying is a frequently used method [18]. Reference [19] evaluated and optimized the covert communications by designing the parameters of the multihop network, including the coding rates, transmit power, and required number of hops.The researchers further analyzed the covert communication capacity of multiple scenarios with multiple unfavorable factors to the eavesdropper, including three aspects of the transmitter [20, 21], receiver [22–24], and additional nodes [25–28]. The methods of covert communication include artificial additional signal noise, artificial coding domain error, insertion of additional signal band, etc. The research results in this field have also been further extended to other communication scenarios such as relay communication [14, 29], multiantenna [30, 31], and broadcast communication [32–34].


## 2.2. Wireless Covert Channel with Dirty Constellation

In the wireless covert channel with dirty constellation (WCC-DC), the secret message bits can be transmitted as the constellation error of the normal signal in order to reduce the suspicion by all uninformed detectors.The framework of a wireless covert channel with dirty constellation is shown in Figure1(a). The wireless covert channel is implemented on the wireless communication physical layer with OFDM structure. The transmitter divides all OFDM subcarriers into secret subcarriers and normal subcarriers. On the secret subcarrier, the carrier information is modulated in QPSK to obtain the carrier signal, and then the covert signal modulated by QPSK is superposed on the carrier signal. The covert constellation points are rotated at a certain angle around normal constellation points, as shown in Figure 1(b). On the normal subcarrier, carrier information is modulated in QPSK to obtain the carrier signal, and then the random noise is superimposed. The purpose of the rotation of signal-loaded signals and the superimposition of random noise is to remove the regularity of secret signals in the constellation. The subcarrier partition results of the wireless covert channel should be shared between the transmitter and the receiver.Figure 1
The schematic diagram of WCC-DC: (a) the framework of WCC-DC and (b) rotation of covert constellation.
(a)(b)However, the wireless covert channel with dirty constellation has a high BER when the power of covert signal is low. When we increase the power of covert signal, the undetectability of covert communication deteriorates. Therefore, Cao et al. [35] proposed a covert communication method based on constellation shaping modulation (WCC-CSM).


## 2.3. Wireless Covert Channel with Constellation Shaping Modulation

The general framework for the wireless communication system with constellation shaping modulation is demonstrated in Figure2. We suppose that each subcarrier mc of the OFDM wireless communication is modulated by QPSK. In the proposed scheme, we can use all subcarriers to establish the wireless covert communication. With constellation shaping modulation, the secret information ms is modulated into an artificial noise signal Ss. Then, the artificial noise signal Ss is superimposed on the carrier signal Sc to generate the secret subcarrier Sct.Figure 2
The framework of WCC-CSM wireless covert channel.To generate the secret artificial noise signalSs, the cumulative distribution function (CDF) FCDF of noise is estimated with the reference channel noise data S0.The secret information is denoted byms = ms,1,ms,2,…ms,N, and the artificial noise signals Ss are divided into I/Q vectors, which are denoted by xsI+j⋅xsQ. Here, xsI is the I vector of artificial noise signals denoted by xsI=xs,1I,xs,2I,xs,3I,…xs,NI, and xsQ is the Q vector denoted byxsQ=xs,1Q,xs,2Q,xs,3Q,…xs,NQ. The constellation shaping modulation function is defined as(1)FSMFms=Ss=xsI+j⋅xsQ.For shaping modulation, the transmitter firstly transforms the secret informationms into continuous variables di, and then di are mapping to artificial noise signal Ss with CDF of the reference channel noise Snormal.The transform function ofdi is defined as follows:(2)di=ms,i+r2.We denoter as a random number distributed in the interval (0, 1). And the mapping function which transforms ms into Ss is defined as(3)Ss=FCDF−1di.The mapping functionFCDF−1 is the inverse function of cumulative distribution function of Snormal. Then, the artificial noise signal Ss is superimposed on carrier signal Sc to generate secret subcarrier Sct.The received secret subcarrier is denoted byS^ct. The I/Q vectors of secret subcarrier S^ct are denoted by x^ctI+j⋅x^ctQ, the subcarrier m^c can be demodulated with QPSK:(4)m^c=Fde−QPSKx^ctI+j⋅x^ctQ,and the subcarrier m^c will be modulated by QPSK again to acquire the ideal the subcarrier S^c:(5)S^c=FQPSKm^c=x^cI+j⋅x^cQ.We denotex^cI+j⋅x^cQ as the I/Q vectors of S^c. The receiver can obtain the ideal subcarrier x^cI+j⋅x^cQ, and then the residual signal (i.e., artificial noise S^s) can be extracted with(6)S^s=S^ct−S^c=x^ctI−x^cI+j⋅x^ctQ−x^cQ.We denote theI/Q vectors of artificial noise S^s as x^sI+j⋅x^sQ. The artificial noise S^s can be transformed into d^i by the cumulative distribution function FCDF with(7)d^i=FCDFS^s.The receiver can demodulate the secret informationm^s by the covert demodulation constellation (CDC), which is illustrated in Figure 3.Figure 3
Covert demodulation constellation of WCC-CSM.The four black points are the ideal constellation points; the red regions are the distribution areas of secret subcarrier with artificial noise. The function of covert demodulation constellation is denoted byFCDC⋅:(8)m^s=FCDCdi.


## 3. Basis of Our Scheme

### 3.1. System Model

Similar to the famous Alice–Bob model [36], the standard wireless covert channel system model includes the transmitter (i.e., Alice), the receiver (i.e., Bob), and the detector (i.e., Willie).Willie observes the channel to detect whether Alice transmits or not. Willie’s probability of detection error consists of two components: the probability of missed detection and the probability of false alarm.The literature as seen in the aforementioned works only mentioned the impact of finite samples (i.e., finitemi) on the detection performance at Willie. It is numerically shown that with noise uncertainty at Willie, there may exist an optimal number of samples that maximize the communication rate subject to ξ≥1−ε, where ξ is the sum of PF (i.e., false alarm rate) and PM(i.e., miss detection rate) at Willie and ε is an arbitrarily small number. We define 0<ε≤1 as the maximum acceptable detection rate of Willie.


### 3.2. Binary Hypothesis Testing at Willie

According to the system model shown in Figure4, the performance elements of the wireless covert channel mainly include two aspects: undetectability and reliability.Figure 4
The framework of wireless covert communication.In communication, Alice totally transmitsn symbols to Bob. We denote the finite block as mi(i∈1,n), which consists of normal information mci and secret information msi, while Willie is passively collecting mi observations on Alice’s transmission in order to detect the presence of her secret information (i.e., whether Alice is transmitting secret information). We denote the AWGN at Bob and Willie as ri∼CN0,σw2. The received signal at Willie for each signal symbol is given by(9)ywi=mi+ri.The main purpose of Willie is to confirm whether Alice transmits or not. We define two hypotheses,H0 and H1, to distinguish these two cases:(10)H0:mi=mci,H1:mi=mci+msi.H0 denotes the null hypothesis, where Alice is not transmitting secret information. H1 denotes the alternative hypothesis, where Alice is transmitting secret information. In the covert communication, the ultimate goal of Willie is to minimize the total error rate (i.e.ξ). We denote T and F as binary decisions that infer whether Alice is transmitting or not. The false alarm rate and miss detection rate are given by(11)PF=PrT|H0,PM=PrF|H1.Suppose Willie performs the optimal detect. Following Pinsker’s inequality [37, 38](12)PF+PM≥1−12DP0‖P1,where relative entropy DP0‖P1 (also called KL divergence) is defined as follows:(13)DP0‖P1=∫np0xlnp0xp1xdx,where n is the value range of x. We denote p1(x) as the distribution of a sequence m[i], which detected by Willie, and p0(x) denotes the distribution of a sequence mc[i], which detected when Alice is not transmitting.The KL divergence is always used to calculate the correlation between distributions. Except KL divergence, we can also use KS distance (also called Kolmogorov–Smirnov statistic) to calculate the distance between distributions. The KS distance is defined as follows:(14)DKS=maxF1x−F0x.m1[i] and m0[i] are both divided into K bins. The number of the elements in m1[i] and m0[i] are denoted by h1j and h0j, j∈1,2…K. The cumulative distribution functions of m1[i] and m0[i] are defined as follows:(15)F1x=∑j=1xh1jn,F0x=∑j=1xh0jn.Willie always set thresholdΓ of KL divergence and KS distance to judge whether there is communication.(16)PrT|DP0‖P1>Γ‖DKS>Γ,PrF|DP0‖P1<Γ‖DKS<Γ.To measure the reliability of the communication, we denote BER as follows:(17)Pe=nerrorn,where nerror is the number of error symbols.


## 3.1. System Model

Similar to the famous Alice–Bob model [36], the standard wireless covert channel system model includes the transmitter (i.e., Alice), the receiver (i.e., Bob), and the detector (i.e., Willie).Willie observes the channel to detect whether Alice transmits or not. Willie’s probability of detection error consists of two components: the probability of missed detection and the probability of false alarm.The literature as seen in the aforementioned works only mentioned the impact of finite samples (i.e., finitemi) on the detection performance at Willie. It is numerically shown that with noise uncertainty at Willie, there may exist an optimal number of samples that maximize the communication rate subject to ξ≥1−ε, where ξ is the sum of PF (i.e., false alarm rate) and PM(i.e., miss detection rate) at Willie and ε is an arbitrarily small number. We define 0<ε≤1 as the maximum acceptable detection rate of Willie.


## 3.2. Binary Hypothesis Testing at Willie

According to the system model shown in Figure4, the performance elements of the wireless covert channel mainly include two aspects: undetectability and reliability.Figure 4
The framework of wireless covert communication.In communication, Alice totally transmitsn symbols to Bob. We denote the finite block as mi(i∈1,n), which consists of normal information mci and secret information msi, while Willie is passively collecting mi observations on Alice’s transmission in order to detect the presence of her secret information (i.e., whether Alice is transmitting secret information). We denote the AWGN at Bob and Willie as ri∼CN0,σw2. The received signal at Willie for each signal symbol is given by(9)ywi=mi+ri.The main purpose of Willie is to confirm whether Alice transmits or not. We define two hypotheses,H0 and H1, to distinguish these two cases:(10)H0:mi=mci,H1:mi=mci+msi.H0 denotes the null hypothesis, where Alice is not transmitting secret information. H1 denotes the alternative hypothesis, where Alice is transmitting secret information. In the covert communication, the ultimate goal of Willie is to minimize the total error rate (i.e.ξ). We denote T and F as binary decisions that infer whether Alice is transmitting or not. The false alarm rate and miss detection rate are given by(11)PF=PrT|H0,PM=PrF|H1.Suppose Willie performs the optimal detect. Following Pinsker’s inequality [37, 38](12)PF+PM≥1−12DP0‖P1,where relative entropy DP0‖P1 (also called KL divergence) is defined as follows:(13)DP0‖P1=∫np0xlnp0xp1xdx,where n is the value range of x. We denote p1(x) as the distribution of a sequence m[i], which detected by Willie, and p0(x) denotes the distribution of a sequence mc[i], which detected when Alice is not transmitting.The KL divergence is always used to calculate the correlation between distributions. Except KL divergence, we can also use KS distance (also called Kolmogorov–Smirnov statistic) to calculate the distance between distributions. The KS distance is defined as follows:(14)DKS=maxF1x−F0x.m1[i] and m0[i] are both divided into K bins. The number of the elements in m1[i] and m0[i] are denoted by h1j and h0j, j∈1,2…K. The cumulative distribution functions of m1[i] and m0[i] are defined as follows:(15)F1x=∑j=1xh1jn,F0x=∑j=1xh0jn.Willie always set thresholdΓ of KL divergence and KS distance to judge whether there is communication.(16)PrT|DP0‖P1>Γ‖DKS>Γ,PrF|DP0‖P1<Γ‖DKS<Γ.To measure the reliability of the communication, we denote BER as follows:(17)Pe=nerrorn,where nerror is the number of error symbols.


## 4. Optimal Carrier-Secret Ratio for WCC-CSM

### 4.1. Classic Scenario

After the secret subcarrierS^ct transmitting through the wireless channel, the receiver can get the output of the slow-fading channel as S^=S^ct1,S^ct2,…S^N with(18)S^cti=Pt⋅λ0dtrα⋅hA⋅Scti+nAi,where Pt denotes the transmission energy of transmitter; λ0 denotes the wavelength of the signal; hA denotes the complex baseband equivalent channel coefficient of the main channel between the transmitter (i.e., Alice) and receiver (i.e., Bob); and nA denotes zero-mean circularly symmetric complex Gaussian noise. And we also have two real variables dtr and α∈R that denote the distance and path-loss exponent of the channel between Alice and Bob, respectively. The path-loss exponent α takes a value between 2 and 4. In free space, microwave transmission has path-loss exponentα=2.The detector (i.e., Willie) can receive the output asy^willie=y^willie1,y^2,...y^willieN with(19)y^williei=Pt⋅λ0dWillie2⋅hW⋅Scti+nWi.hW denotes the complex baseband equivalent channel coefficient of the main channel between the transmitter (i.e., Alice) and detector(i.e., Willie); dWillie denotes the distance exponent between Alice and Willie. The noise nW also follows a zero-mean circularly symmetric complex Gaussian distribution.Theorem 1.
The undetectability of wireless covert communication deteriorates with the increase of CSR.Proof.
As is mentioned above, the probability of detection error must satisfy a lower boundary of(20)PF+PM≥1−12DP0‖P1.
Considering equations (13) and (20) jointly, we can obtain the lowest boundary is at the lowest “KL divergence,” We denote p1(x) as the distribution of residual signal S^s, which detected by Willie, and p0(x) denotes the distribution of the reference channel noise S0. The artificial noise Ss is the mapping of S0∼N0,σω2; thus, we can obtain the residual signal S^s∼N0,σω2∼0,P+σω2. P denotes the energy of the signal, which received by Bob or Willie. The “KL divergence” of S0 and S^s can be expressed as(21)DS0‖S^s=∫np0xlnp0xp1xdx,=∫n12πσωe−x2/σω2⋅lnσsσω⋅ex2/σω2−−x2/σsdx=∫n12πσωe−x2/σω2⋅lnP+σω2σω+x2⋅PP+σω2⋅σω2dx=12lnP+σω2σω2−PP+σω2
As is illustrated in equation (21), the “KL divergence” increases with the increase of P. Even the zero-mean circularly symmetric complex Gaussian noise with the same variance has different distributions. In order to calculate the “KL divergence”, we divide both S0 and S^s into K bins, the probability in j bins are denoted by PS0j and PSresj, j∈1,2…K. The expression (21) can be expressed as(22)DS0‖S^s=∫nPS0xlnPS0xPSresxdx.
With the increase ofK, PS0j and PSresj will approach p0x and p1x. However, if the K is too great, the PF will be great. If the K is too small, the PM will be great. We need to choose a suitable value of K. In this paper, we set K = 100.
PWillie denotes the signal energy detected by Willie. Pideal denotes the ideal signal energy. The “KL divergence” can be expressed as(23)DS0‖S^s=12lnP+σω2σω2−PP+σω2=12lnPWillie−Pidealσω2−PWillie−Pideal−σω2PWillie−Pideal.
ΔP denotes PWillie−Pideal. Taking the partial derivative with respect toΔP, equation (23) can be expressed as(24)∂DS0‖S^s∂ΔP=12ΔP−σω2ΔP2.
ΔP can be expressed as(25)ΔP≈1−1CSR⋅PWillie+σω2.
Analysis: Considering expression equations (24) and (25) jointly, we can get ∂DS0‖S^s/∂ΔP>0. The KL divergence increases with the increase of ΔP. With the PWillie unchanged, ΔP increases with the increase of CSR. Hence, the KL divergence increases with the increase of CSR.
The KL divergence increases with the increase ofPWillie. When the KL divergence is greater than Γ, Willie judges there is covert communication. When the KL divergence equals to Γ, the corresponding SNR0 can be expressed as(26)SNR0=P0⋅λ0⋅h02d02.
If SNR < SNR0, the covert communication will not be detected. When the transmission power is constant, the threshold detection distance d0 can be illustrated in Figure 5. The purple dotted circle is the equipower line in which the covert communication will not be detected with “KL divergence.”
The probability of undetectedPud can be expressed as(27)Pud=PrPt⋅λ0⋅hW2dW2<SNR0.Figure 5
Location diagram of Alice, Bob, and Willie.Theorem 2.
There exists an optimal ratio between the carrier signal and secret signal no matter what the value of SNR is. The BER minimizes at the optimal ratio.Proof.
The reliability of the system is inspected by the BER. The BER of QPSK is(28)PeQPSK=12erfcr.
erfc⋅ denotes the Gauss error function, r denotes the signal-noise ratio. The BER of covert communication is denoted as Pecov, which can be expressed as(29)Pecov=1−1−Pe,mc⋅1−Pe,sc=1−1−erfcSNR⋅CSRSNR+CSR+11−erfcSNRCSR+1.
SNR denotes the signal-noise ratio ofS^ct to S0, CSR denotes the carrier-secret ratio of Sc to Ss. Considering the range of SNR and CSR, the expression (29) can be approximated as(30)Pecov=1−1−Pe,mc⋅1−Pe,sc≈1−erf0.75×CSR⋅erfSNRCSR.
Taking the partial derivative with respect to CSR, the equation (30) can be expressed as(31)∂Pecov∂CSR=−1π⋅e−0.75CSR⋅0.75CSR⋅erfSNR0.75CSR−erfCSR⋅1π⋅e−SNR/CSR⋅SNR⋅CSR−3/2.
Analysis: Let the∂Pecov/∂CSR equals zeros. We can obtain(32)1π⋅e−0.75CSR⋅0.75CSR⋅erfSNR0.75CSR=erfCSR⋅1π⋅e−SNR/CSR⋅SNR⋅CSR−3/2erfSNR0.75CSR⋅e−0.75CSR⋅0.75CSR=erfCSR⋅e−SNR/CSR⋅SNRCSR.
The equality can be established whenCSR=4/3SNR, which is the optimal CSR. We can obtain the lowest BER at the optimal CSR. The expression (29) can be expressed as Figure 6.
As can be seen from Figure6, Pecov minimizes at the optimal CSR. And the approximate solution we obtained is consistent with the theoretical Pecov in Figure 6. With the approximate solution of optimal CSR, the process of searching for an actual optimal CSR can be accelerated when some optimization algorithms are adopted such as gradient descent and conjugate gradient.
As wireless communication is affected by channel fading, it is often necessary to set one or more relays to extend the communication distance. Therefore, the relay communication scenarios are described in details in the following subsections.Figure 6
BER curve in AWGN channel.


### 4.2. One-Relay Scenario

Covert information is always transmitted with low power; we can set relays to extend the transmission distance. Each relay employs the amplify-and-forward (AF) protocol and has two phases. Alice transmits signal in one phase; the relay amplifies the signal and forwards to Bob in another phase. We can set the positions of Alice, Bob, Willie, and relay as illustrated in Figure7(a). dtr denotes the distance between Alice and Bob; dtrr denotes the distance between Alice and relay; and drb denotes the distance between relay and Bob. If we keep the total transmission energy constant, the transmission energy of Alice and relay are both 0.5Pt, and we can obtain the output of the covert channel:(33)y^relayi=Pt2⋅λ0dtrr⋅hA⋅Scti+nAi,y^Bobi=Pt2⋅λ0drb⋅hB⋅y^relayi+nBi.Figure 7
Diagram of relay location: (a) relay at random position and (b) relay in middle.
(a)(b)As is illustrated in Figure7(a),(34)drb+dtrr=dtr.We can obtain the signal which is received by Bob:(35)y^Bobi=Pt2⋅λ0drb⋅hB⋅Pt2⋅λ0dtrr⋅hA⋅Scti+nAi+nBi,=Pt2⋅λ0drb⋅dtrr⋅hB⋅hA⋅Scti+nCiIn free space, the signal which is received by Bob is only about the energy and distance. So the optimization position of the relay is in the middle of Alice and Bob (i.e.,drb = dtrr), as can be seen in Figure 7(b).The distance between Willie, Alice, and relay are denoted bydW and dWR. The probability of undetectability Pud can be expressed as(36)Pud=PrPt2⋅λ0⋅hA2dW2<SNR0⋅PrPt2⋅λ0⋅hB2dWR2<SNR0.Construct a coordinate system with Alice as the origin of the coordinate axis. We can obtain the coordinates of Alice (0, 0), Bob (2dtrr, 0), relay (dtrr, 0), and Willie (XWillie, YWillie). In the AWGN channel, the power of signal received by Willie is just related to distance and transmit power. We can get “equipower lines” in the scenario of no relay and one relay.Then, the power detected by Willie can be expressed as(37)2XWillie−dtrr2+YWillie2=XWillie2+YWillie2.Equation (37) can be expressed as(38)XWillie−2dtrr2+YWillie2=2dtrr2.We denoted the circle expressed in equation (38) as the undetectability deterioration area (UDA). The undetectability deteriorates with setting relay when Willie is in the UDA. And the eavesdropper can detect in the UDA to improve detection efficiency.It is illustrated in Figure8, the green dotted line is UDA. If Willie is in the UDA, the Pud will decrease in the scenario of one relay and the consequent deterioration of undetectability. Correspondingly, the undetectability will ameliorate if Willie is outside the green dotted circle. When William is on the green dotted line, the Pud will be constant.Figure 8
Power comparison of classic scenario and one relay scenario.The BER of covert communicationPecov.r1 can be expressed as(39)Pecov.r1=1−1−Pe,mc1⋅1−Pe,sc11−Pe,mc2⋅1−Pe,sc2+oΔ≈1−1−erfc2SNR⋅CSR2SNR+CSR+11−erfc2SNRCSR+12.Referring expression (32), we can obtain the local minimum of Pecov.r1 at CSR=4/3SNR, which is the optimal CSR.


### 4.3. Two-Relay Scenario

Based on the above, we discuss the two-relay scenario. If we keep the total transmission energy constant, the transmission energy of Alice, relay1, and relay2 are allPt/3. We can obtain the output of the covert signal:(40)y^relay1i=Pt3⋅λ0dtrr⋅hA⋅Scti+nAi,y^relay2i=Pt3⋅λ0dtrr⋅hB⋅y^relay1i+nBi,y^Bobi=Pt3⋅λ0dtrr⋅hC⋅y^relay2i+nCi.As can be seen in Figure9, the distances between Willie and Alice, relay1, and relay2 are denoted by dWR, dWR1, and dWR2, respectively.Figure 9
Diagram of two relays location.The probability of undetectedPud can be expressed as(41)Pud=PrPt3⋅λ0⋅hA2dW2<SNR0⋅PrPt3⋅λ0⋅hB2dWR12<SNR0⋅PrPt3⋅λ0⋅hC2dWR22<SNR0.Further extension, the probability of undetectability forn hops can be expressed as(42)Pud=∏nPrPtn⋅λ0⋅hn2dWn2<SNR0.The UDA of relay1 can be expressed as(43)3XWillie−dtrr2+YWillie2=XWillie2+YWillie2.Equation (43) can be converted to(44)XWillie−32dtrr2+Y2=32dtrr2.The UDA of relay2 can be expressed as(45)3XWillie−2dtrr2+YWillie2=XWillie2+YWillie2.Equation (45) can be converted to(46)XWillie−3dtrr2+Y2=3dtrr2.As can be seen in Figure10, the circles expressed in equations (44) and (46) are UDAs. If Willie is in the blue or green dotted circle, the undetectability will deteriorate. Correspondingly, the undetectability will ameliorate if Willie is outside the blue and green dotted circles.Figure 10
Power comparison of the classic scenario and two-relay scenario.The BER of covert communicationPecov.r2 can be expressed as(47)Pecov.r2=1−1−Pe,mc1⋅1−Pe,sc11−Pe,mc2⋅1−Pe,sc21−Pe,mc3⋅1−Pe,sc3+oΔ≈1−1−erfc3SNR⋅CSR3SNR+CSR+11−erfc3SNRCSR+13.It has been proved that there exists an optimal CSR, and we can obtain the minimum ofPecov.r2 at CSR=4/3SNR.


## 4.1. Classic Scenario

After the secret subcarrierS^ct transmitting through the wireless channel, the receiver can get the output of the slow-fading channel as S^=S^ct1,S^ct2,…S^N with(18)S^cti=Pt⋅λ0dtrα⋅hA⋅Scti+nAi,where Pt denotes the transmission energy of transmitter; λ0 denotes the wavelength of the signal; hA denotes the complex baseband equivalent channel coefficient of the main channel between the transmitter (i.e., Alice) and receiver (i.e., Bob); and nA denotes zero-mean circularly symmetric complex Gaussian noise. And we also have two real variables dtr and α∈R that denote the distance and path-loss exponent of the channel between Alice and Bob, respectively. The path-loss exponent α takes a value between 2 and 4. In free space, microwave transmission has path-loss exponentα=2.The detector (i.e., Willie) can receive the output asy^willie=y^willie1,y^2,...y^willieN with(19)y^williei=Pt⋅λ0dWillie2⋅hW⋅Scti+nWi.hW denotes the complex baseband equivalent channel coefficient of the main channel between the transmitter (i.e., Alice) and detector(i.e., Willie); dWillie denotes the distance exponent between Alice and Willie. The noise nW also follows a zero-mean circularly symmetric complex Gaussian distribution.Theorem 1.
The undetectability of wireless covert communication deteriorates with the increase of CSR.Proof.
As is mentioned above, the probability of detection error must satisfy a lower boundary of(20)PF+PM≥1−12DP0‖P1.
Considering equations (13) and (20) jointly, we can obtain the lowest boundary is at the lowest “KL divergence,” We denote p1(x) as the distribution of residual signal S^s, which detected by Willie, and p0(x) denotes the distribution of the reference channel noise S0. The artificial noise Ss is the mapping of S0∼N0,σω2; thus, we can obtain the residual signal S^s∼N0,σω2∼0,P+σω2. P denotes the energy of the signal, which received by Bob or Willie. The “KL divergence” of S0 and S^s can be expressed as(21)DS0‖S^s=∫np0xlnp0xp1xdx,=∫n12πσωe−x2/σω2⋅lnσsσω⋅ex2/σω2−−x2/σsdx=∫n12πσωe−x2/σω2⋅lnP+σω2σω+x2⋅PP+σω2⋅σω2dx=12lnP+σω2σω2−PP+σω2
As is illustrated in equation (21), the “KL divergence” increases with the increase of P. Even the zero-mean circularly symmetric complex Gaussian noise with the same variance has different distributions. In order to calculate the “KL divergence”, we divide both S0 and S^s into K bins, the probability in j bins are denoted by PS0j and PSresj, j∈1,2…K. The expression (21) can be expressed as(22)DS0‖S^s=∫nPS0xlnPS0xPSresxdx.
With the increase ofK, PS0j and PSresj will approach p0x and p1x. However, if the K is too great, the PF will be great. If the K is too small, the PM will be great. We need to choose a suitable value of K. In this paper, we set K = 100.
PWillie denotes the signal energy detected by Willie. Pideal denotes the ideal signal energy. The “KL divergence” can be expressed as(23)DS0‖S^s=12lnP+σω2σω2−PP+σω2=12lnPWillie−Pidealσω2−PWillie−Pideal−σω2PWillie−Pideal.
ΔP denotes PWillie−Pideal. Taking the partial derivative with respect toΔP, equation (23) can be expressed as(24)∂DS0‖S^s∂ΔP=12ΔP−σω2ΔP2.
ΔP can be expressed as(25)ΔP≈1−1CSR⋅PWillie+σω2.
Analysis: Considering expression equations (24) and (25) jointly, we can get ∂DS0‖S^s/∂ΔP>0. The KL divergence increases with the increase of ΔP. With the PWillie unchanged, ΔP increases with the increase of CSR. Hence, the KL divergence increases with the increase of CSR.
The KL divergence increases with the increase ofPWillie. When the KL divergence is greater than Γ, Willie judges there is covert communication. When the KL divergence equals to Γ, the corresponding SNR0 can be expressed as(26)SNR0=P0⋅λ0⋅h02d02.
If SNR < SNR0, the covert communication will not be detected. When the transmission power is constant, the threshold detection distance d0 can be illustrated in Figure 5. The purple dotted circle is the equipower line in which the covert communication will not be detected with “KL divergence.”
The probability of undetectedPud can be expressed as(27)Pud=PrPt⋅λ0⋅hW2dW2<SNR0.Figure 5
Location diagram of Alice, Bob, and Willie.Theorem 2.
There exists an optimal ratio between the carrier signal and secret signal no matter what the value of SNR is. The BER minimizes at the optimal ratio.Proof.
The reliability of the system is inspected by the BER. The BER of QPSK is(28)PeQPSK=12erfcr.
erfc⋅ denotes the Gauss error function, r denotes the signal-noise ratio. The BER of covert communication is denoted as Pecov, which can be expressed as(29)Pecov=1−1−Pe,mc⋅1−Pe,sc=1−1−erfcSNR⋅CSRSNR+CSR+11−erfcSNRCSR+1.
SNR denotes the signal-noise ratio ofS^ct to S0, CSR denotes the carrier-secret ratio of Sc to Ss. Considering the range of SNR and CSR, the expression (29) can be approximated as(30)Pecov=1−1−Pe,mc⋅1−Pe,sc≈1−erf0.75×CSR⋅erfSNRCSR.
Taking the partial derivative with respect to CSR, the equation (30) can be expressed as(31)∂Pecov∂CSR=−1π⋅e−0.75CSR⋅0.75CSR⋅erfSNR0.75CSR−erfCSR⋅1π⋅e−SNR/CSR⋅SNR⋅CSR−3/2.
Analysis: Let the∂Pecov/∂CSR equals zeros. We can obtain(32)1π⋅e−0.75CSR⋅0.75CSR⋅erfSNR0.75CSR=erfCSR⋅1π⋅e−SNR/CSR⋅SNR⋅CSR−3/2erfSNR0.75CSR⋅e−0.75CSR⋅0.75CSR=erfCSR⋅e−SNR/CSR⋅SNRCSR.
The equality can be established whenCSR=4/3SNR, which is the optimal CSR. We can obtain the lowest BER at the optimal CSR. The expression (29) can be expressed as Figure 6.
As can be seen from Figure6, Pecov minimizes at the optimal CSR. And the approximate solution we obtained is consistent with the theoretical Pecov in Figure 6. With the approximate solution of optimal CSR, the process of searching for an actual optimal CSR can be accelerated when some optimization algorithms are adopted such as gradient descent and conjugate gradient.
As wireless communication is affected by channel fading, it is often necessary to set one or more relays to extend the communication distance. Therefore, the relay communication scenarios are described in details in the following subsections.Figure 6
BER curve in AWGN channel.


## 4.2. One-Relay Scenario

Covert information is always transmitted with low power; we can set relays to extend the transmission distance. Each relay employs the amplify-and-forward (AF) protocol and has two phases. Alice transmits signal in one phase; the relay amplifies the signal and forwards to Bob in another phase. We can set the positions of Alice, Bob, Willie, and relay as illustrated in Figure7(a). dtr denotes the distance between Alice and Bob; dtrr denotes the distance between Alice and relay; and drb denotes the distance between relay and Bob. If we keep the total transmission energy constant, the transmission energy of Alice and relay are both 0.5Pt, and we can obtain the output of the covert channel:(33)y^relayi=Pt2⋅λ0dtrr⋅hA⋅Scti+nAi,y^Bobi=Pt2⋅λ0drb⋅hB⋅y^relayi+nBi.Figure 7
Diagram of relay location: (a) relay at random position and (b) relay in middle.
(a)(b)As is illustrated in Figure7(a),(34)drb+dtrr=dtr.We can obtain the signal which is received by Bob:(35)y^Bobi=Pt2⋅λ0drb⋅hB⋅Pt2⋅λ0dtrr⋅hA⋅Scti+nAi+nBi,=Pt2⋅λ0drb⋅dtrr⋅hB⋅hA⋅Scti+nCiIn free space, the signal which is received by Bob is only about the energy and distance. So the optimization position of the relay is in the middle of Alice and Bob (i.e.,drb = dtrr), as can be seen in Figure 7(b).The distance between Willie, Alice, and relay are denoted bydW and dWR. The probability of undetectability Pud can be expressed as(36)Pud=PrPt2⋅λ0⋅hA2dW2<SNR0⋅PrPt2⋅λ0⋅hB2dWR2<SNR0.Construct a coordinate system with Alice as the origin of the coordinate axis. We can obtain the coordinates of Alice (0, 0), Bob (2dtrr, 0), relay (dtrr, 0), and Willie (XWillie, YWillie). In the AWGN channel, the power of signal received by Willie is just related to distance and transmit power. We can get “equipower lines” in the scenario of no relay and one relay.Then, the power detected by Willie can be expressed as(37)2XWillie−dtrr2+YWillie2=XWillie2+YWillie2.Equation (37) can be expressed as(38)XWillie−2dtrr2+YWillie2=2dtrr2.We denoted the circle expressed in equation (38) as the undetectability deterioration area (UDA). The undetectability deteriorates with setting relay when Willie is in the UDA. And the eavesdropper can detect in the UDA to improve detection efficiency.It is illustrated in Figure8, the green dotted line is UDA. If Willie is in the UDA, the Pud will decrease in the scenario of one relay and the consequent deterioration of undetectability. Correspondingly, the undetectability will ameliorate if Willie is outside the green dotted circle. When William is on the green dotted line, the Pud will be constant.Figure 8
Power comparison of classic scenario and one relay scenario.The BER of covert communicationPecov.r1 can be expressed as(39)Pecov.r1=1−1−Pe,mc1⋅1−Pe,sc11−Pe,mc2⋅1−Pe,sc2+oΔ≈1−1−erfc2SNR⋅CSR2SNR+CSR+11−erfc2SNRCSR+12.Referring expression (32), we can obtain the local minimum of Pecov.r1 at CSR=4/3SNR, which is the optimal CSR.


## 4.3. Two-Relay Scenario

Based on the above, we discuss the two-relay scenario. If we keep the total transmission energy constant, the transmission energy of Alice, relay1, and relay2 are allPt/3. We can obtain the output of the covert signal:(40)y^relay1i=Pt3⋅λ0dtrr⋅hA⋅Scti+nAi,y^relay2i=Pt3⋅λ0dtrr⋅hB⋅y^relay1i+nBi,y^Bobi=Pt3⋅λ0dtrr⋅hC⋅y^relay2i+nCi.As can be seen in Figure9, the distances between Willie and Alice, relay1, and relay2 are denoted by dWR, dWR1, and dWR2, respectively.Figure 9
Diagram of two relays location.The probability of undetectedPud can be expressed as(41)Pud=PrPt3⋅λ0⋅hA2dW2<SNR0⋅PrPt3⋅λ0⋅hB2dWR12<SNR0⋅PrPt3⋅λ0⋅hC2dWR22<SNR0.Further extension, the probability of undetectability forn hops can be expressed as(42)Pud=∏nPrPtn⋅λ0⋅hn2dWn2<SNR0.The UDA of relay1 can be expressed as(43)3XWillie−dtrr2+YWillie2=XWillie2+YWillie2.Equation (43) can be converted to(44)XWillie−32dtrr2+Y2=32dtrr2.The UDA of relay2 can be expressed as(45)3XWillie−2dtrr2+YWillie2=XWillie2+YWillie2.Equation (45) can be converted to(46)XWillie−3dtrr2+Y2=3dtrr2.As can be seen in Figure10, the circles expressed in equations (44) and (46) are UDAs. If Willie is in the blue or green dotted circle, the undetectability will deteriorate. Correspondingly, the undetectability will ameliorate if Willie is outside the blue and green dotted circles.Figure 10
Power comparison of the classic scenario and two-relay scenario.The BER of covert communicationPecov.r2 can be expressed as(47)Pecov.r2=1−1−Pe,mc1⋅1−Pe,sc11−Pe,mc2⋅1−Pe,sc21−Pe,mc3⋅1−Pe,sc3+oΔ≈1−1−erfc3SNR⋅CSR3SNR+CSR+11−erfc3SNRCSR+13.It has been proved that there exists an optimal CSR, and we can obtain the minimum ofPecov.r2 at CSR=4/3SNR.


## 5. Experimental Result

### 5.1. Experimental Setup

In this section, we inspect the undetectability and reliability to benchmark the proposed scheme. We set the wireless communication on an 802.11a/g PHY layer. The wireless covert channel is performed on all 100000 symbols. In transmissions, there are 48 subcarriers in a symbol. Simulation experiments are carried out in wireless channel models of AWGN channel models [39]. In some simulations, the wireless covert channel with dirty constellation (WCC-DC) is chosen for comparison. The undetectability is inspected by “KL divergence” and “KS distance”. The undetectability measures of I vectors, Q vectors, magnitudes, and phases of constellation errors are presented in the range of transmission power SNR = 10, …, 40 dB. The reliability is measured by BERs.


### 5.2. Undetectability

Willie (i.e., detector) observes the channel to judge whether Alice (i.e., transmitter) is transmitting in the covert channel or not. There must be a thresholdΓ to compare the value of “KL divergence” and “KS distance”. If the Γ is great, the PF will be too great. If the Γ is small, the PM will be too great. In this paper, we set the Γ of four measures of “KL divergence” as [0.04, 0.04, 0.055, 0.055], and the Γ of “KS distance” as [0.025, 0.025, 0.025, 0.025].In this section, we set the number of binsK = 100. Four samples were chosen for comparison, and the samples are as follows: WCC-DC with CSR = 5 dB and CSR = 10 dB, WCC-CSM with CSR = 5 dB, and CSR = 10 dB.As can be seen in Figure11, the “KL divergence” of WCC-CSM with CSR = 5 dB and CSR = 10 dB meet the threshold Γ [0.04, 0.04, 0.055, 0.055] in the range of SNR = 10, …, 40 dB. The resulting KL divergence is lower than the KL divergence achieved with WCC-DC.Figure 11
KL divergence of constellation errors with different CSRs: (a) KL divergence of I vectors, (b) KL divergence of Q vectors, (c) KL divergence of magnitudes, and (d) KL divergence of phase.
(a)(b)(c)(d)In Figure11(a), WCC-DC with CSR = 10 dB meets the threshold Γ of I vectors in the range of SNR = 10, …, 30 dB. The “KL divergence” of WCC-DC with CSR = 10 dB exceeds the threshold Γ when SNR = 40 dB, and WCC-DC with CSR = 5 dB exceeds the threshold in the range of SNR = 10,…40 dB. In Figure 11(b), the “KL divergence” of WCC-DC with CSR = 5 dB exceeds the threshold Γ at SNR = 23 dB. The “KL divergence” of WCC-DC with CSR = 10 dB exceeds the threshold Γ at SNR = 33 dB. In Figures 11(c) and 11(d), WCC-DC with CSR = 5 dB exceeds the threshold Γ at SNR = 23 dB and SNR = 38 dB, and WCC-DC with CSR = 10 dB exceeds the threshold Γ at SNR = 28 dB and SNR = 37 dB.As can be seen in Figure12, WCC-CSM meets the threshold Γand WCC-CSM has a smaller “KS distance” than WCC-DC in the range of SNR = 10, …, 40 dB. The “KS distance” of WCC-CSM changes little with different CSRs. And we can regulate the CSR without exceeding the threshold Γ of “KS distance.” In Figures 12(a)–12(c), the “KS distance” of WCC-DC with CSR = 5 dB exceeds the threshold at SNR = 10 dB. In Figure 12(c), the “KS distance” of WCC-DC with CSR = 10 dB exceeds the threshold at SNR = 13 dB.Figure 12
KS distances of constellation errors with different CSRs: (a) KS distances of I vectors, (b) KS distances of Q vectors, (c) KS distances of magnitudes, and (d) KS distances of phase.
(a)(b)(c)(d)We can come to a stage conclusion, WCC-CSM meets the threshold of “KL divergence” and “KS distance” in the range of SNR = 10, …, 40 dB. Hence, we can regulate the CSR to reduce the BER without exceeding the threshold of “KL divergence” or “KS distance.”


### 5.3. Reliability

The reliability of the system is measured by the BER. As is illustrated in expression (28), the BER decreases with the increase of SNR. In Section4.1, the optimal CSR was proposed. And the approximate solution of optimal CSR is obtained and extended to several scenarios (one relay, two relays). The BER curves of wireless covert channels in several scenarios are shown in Figure 13. The position of Alice, Bob, relay1, and relay 2 was shown in Section 4.Figure 13
BER of wireless covert channel in typical scenarios. (a) BER of classic scenario with different SNR. (b) BER of one relay with different SNR. (c) BER of two relays with different SNR. (d) Comparison of BER of several scenarios.
(a)(b)(c)(d)In Figure13(a), the BER of the classic scenario with different SNRs was presented in the range of CSR = 1, 2, …, 20 dB. The BER of WCC-CSM with SNR = 15 dB minimizes at CSR = 7 dB, which is 10% lower than the BER at CSR = 15 dB. The approximate solution of optimal CSR is 5 dB. The minimizing BER of WCC-CSM with SNR = 30 dB is 4% at CSR = 8 dB, and the approximate solution of optimal CSR is 7 dB. The minimizing BER of WCC-CSM with SNR = 40 dB is 0.05% at CSR = 9 dB. The theoretical approximation of the optimal CSR·with SNR = 40 dB is 7.5 dB.It is proved that an optimal CSR exists and the BER minimizes at the optimal CSR. With the increase of SNR, the optimal CSR gradually increases. But the theoretical approximate value of optimal CSR is slightly lower than the simulation results. As can be seen in Figures13(b) and 13(c), the optimal CSR exists and is slightly higher than the theoretical approximation in scenarios of relays. The BER minimizes at the optimal CSR.Comparing the BER under the constraints of constant total power, suppose that the SNR in the classic scenario is 20 dB. The SNR in the scenario of one relay is 23 dB, and the SNR in the scenario of two relays is 25 dB. As can be seen in Figure13(d), the reliability of classic scenario is optimal and the BER minimizes at the optimal CSR. No error correction coding is used in the paper, and the reliability deteriorates in the scenarios of relays. It is proved that simply setting relays without increasing the total power, the BER of covert communication will deteriorate.Simulation experiments are carried out in wireless channel models of AWGN channel. As can be seen in Figure13, we can obtain minimum Pecov, Pecov.r1, and Pecov.r2 at the optimal CSR in the AWGN channel. And the optimal CSR achieved in simulation is slightly higher than the theoretical approximation.


## 5.1. Experimental Setup

In this section, we inspect the undetectability and reliability to benchmark the proposed scheme. We set the wireless communication on an 802.11a/g PHY layer. The wireless covert channel is performed on all 100000 symbols. In transmissions, there are 48 subcarriers in a symbol. Simulation experiments are carried out in wireless channel models of AWGN channel models [39]. In some simulations, the wireless covert channel with dirty constellation (WCC-DC) is chosen for comparison. The undetectability is inspected by “KL divergence” and “KS distance”. The undetectability measures of I vectors, Q vectors, magnitudes, and phases of constellation errors are presented in the range of transmission power SNR = 10, …, 40 dB. The reliability is measured by BERs.


## 5.2. Undetectability

Willie (i.e., detector) observes the channel to judge whether Alice (i.e., transmitter) is transmitting in the covert channel or not. There must be a thresholdΓ to compare the value of “KL divergence” and “KS distance”. If the Γ is great, the PF will be too great. If the Γ is small, the PM will be too great. In this paper, we set the Γ of four measures of “KL divergence” as [0.04, 0.04, 0.055, 0.055], and the Γ of “KS distance” as [0.025, 0.025, 0.025, 0.025].In this section, we set the number of binsK = 100. Four samples were chosen for comparison, and the samples are as follows: WCC-DC with CSR = 5 dB and CSR = 10 dB, WCC-CSM with CSR = 5 dB, and CSR = 10 dB.As can be seen in Figure11, the “KL divergence” of WCC-CSM with CSR = 5 dB and CSR = 10 dB meet the threshold Γ [0.04, 0.04, 0.055, 0.055] in the range of SNR = 10, …, 40 dB. The resulting KL divergence is lower than the KL divergence achieved with WCC-DC.Figure 11
KL divergence of constellation errors with different CSRs: (a) KL divergence of I vectors, (b) KL divergence of Q vectors, (c) KL divergence of magnitudes, and (d) KL divergence of phase.
(a)(b)(c)(d)In Figure11(a), WCC-DC with CSR = 10 dB meets the threshold Γ of I vectors in the range of SNR = 10, …, 30 dB. The “KL divergence” of WCC-DC with CSR = 10 dB exceeds the threshold Γ when SNR = 40 dB, and WCC-DC with CSR = 5 dB exceeds the threshold in the range of SNR = 10,…40 dB. In Figure 11(b), the “KL divergence” of WCC-DC with CSR = 5 dB exceeds the threshold Γ at SNR = 23 dB. The “KL divergence” of WCC-DC with CSR = 10 dB exceeds the threshold Γ at SNR = 33 dB. In Figures 11(c) and 11(d), WCC-DC with CSR = 5 dB exceeds the threshold Γ at SNR = 23 dB and SNR = 38 dB, and WCC-DC with CSR = 10 dB exceeds the threshold Γ at SNR = 28 dB and SNR = 37 dB.As can be seen in Figure12, WCC-CSM meets the threshold Γand WCC-CSM has a smaller “KS distance” than WCC-DC in the range of SNR = 10, …, 40 dB. The “KS distance” of WCC-CSM changes little with different CSRs. And we can regulate the CSR without exceeding the threshold Γ of “KS distance.” In Figures 12(a)–12(c), the “KS distance” of WCC-DC with CSR = 5 dB exceeds the threshold at SNR = 10 dB. In Figure 12(c), the “KS distance” of WCC-DC with CSR = 10 dB exceeds the threshold at SNR = 13 dB.Figure 12
KS distances of constellation errors with different CSRs: (a) KS distances of I vectors, (b) KS distances of Q vectors, (c) KS distances of magnitudes, and (d) KS distances of phase.
(a)(b)(c)(d)We can come to a stage conclusion, WCC-CSM meets the threshold of “KL divergence” and “KS distance” in the range of SNR = 10, …, 40 dB. Hence, we can regulate the CSR to reduce the BER without exceeding the threshold of “KL divergence” or “KS distance.”


## 5.3. Reliability

The reliability of the system is measured by the BER. As is illustrated in expression (28), the BER decreases with the increase of SNR. In Section4.1, the optimal CSR was proposed. And the approximate solution of optimal CSR is obtained and extended to several scenarios (one relay, two relays). The BER curves of wireless covert channels in several scenarios are shown in Figure 13. The position of Alice, Bob, relay1, and relay 2 was shown in Section 4.Figure 13
BER of wireless covert channel in typical scenarios. (a) BER of classic scenario with different SNR. (b) BER of one relay with different SNR. (c) BER of two relays with different SNR. (d) Comparison of BER of several scenarios.
(a)(b)(c)(d)In Figure13(a), the BER of the classic scenario with different SNRs was presented in the range of CSR = 1, 2, …, 20 dB. The BER of WCC-CSM with SNR = 15 dB minimizes at CSR = 7 dB, which is 10% lower than the BER at CSR = 15 dB. The approximate solution of optimal CSR is 5 dB. The minimizing BER of WCC-CSM with SNR = 30 dB is 4% at CSR = 8 dB, and the approximate solution of optimal CSR is 7 dB. The minimizing BER of WCC-CSM with SNR = 40 dB is 0.05% at CSR = 9 dB. The theoretical approximation of the optimal CSR·with SNR = 40 dB is 7.5 dB.It is proved that an optimal CSR exists and the BER minimizes at the optimal CSR. With the increase of SNR, the optimal CSR gradually increases. But the theoretical approximate value of optimal CSR is slightly lower than the simulation results. As can be seen in Figures13(b) and 13(c), the optimal CSR exists and is slightly higher than the theoretical approximation in scenarios of relays. The BER minimizes at the optimal CSR.Comparing the BER under the constraints of constant total power, suppose that the SNR in the classic scenario is 20 dB. The SNR in the scenario of one relay is 23 dB, and the SNR in the scenario of two relays is 25 dB. As can be seen in Figure13(d), the reliability of classic scenario is optimal and the BER minimizes at the optimal CSR. No error correction coding is used in the paper, and the reliability deteriorates in the scenarios of relays. It is proved that simply setting relays without increasing the total power, the BER of covert communication will deteriorate.Simulation experiments are carried out in wireless channel models of AWGN channel. As can be seen in Figure13, we can obtain minimum Pecov, Pecov.r1, and Pecov.r2 at the optimal CSR in the AWGN channel. And the optimal CSR achieved in simulation is slightly higher than the theoretical approximation.


## 6. Conclusions

Reliability and undetectability are the main aspects of wireless covert communication. We considered the BER problem of covert communication based on WCC-CSM. We studied the impact of carrier-secret ratio (CSR) on the BER and investigated the relationship between SNR, CSR, and BER. We obtained the approximate solution of optimal CSR and extended it to the scenario of relays. With the approximate solution of optimal CSR, the process of searching for an actual optimal CSR can be accelerated. Furthermore, we found that the undetectability under the constraints of constant total power depends on the eavesdropper’s position. And we found an undetectability deterioration area (UDA) in the scenario of relays, and undetectability deteriorates with setting relays when an eavesdropper is in the UDA.The simulation proved that there exists an optimal CSR in the AWGN channel. The transmitter can obtain greater reliability with great undetectability at the optimal CSR. Additionally, the reliability deteriorates with setting relays under the constraints of constant total power. Some error correction coding or other methods must be adopted, avoiding the deterioration of BER.To improve the detection capability of Willie, it is necessary to find a better way to detect the covert communication except “KL divergence” or “KS distance” in our future work.



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*Source: 6919530-2021-12-03.xml*