Covert Millimeter-Wave Communication via a Dual-Beam Transmitter

08/20/2019
by   Mohammad Vahid Jamali, et al.
University of Michigan
0

In this paper, we investigate covert communication over millimeter-wave (mmWave) frequencies. In particular, a dual-beam mmWave transmitter, comprised of two independent antenna arrays, attempts to reliably communicate to a receiver Bob when hiding the existence of transmission from a warden Willie. In this regard, operating over mmWave bands not only increases the covertness thanks to directional beams, but also increases the transmission data rates given much more available bandwidths and enables ultra-low form factor transceivers due to the lower wavelengths used compared to the conventional radio frequency (RF) counterpart. We assume that the transmitter Alice employs one of its antenna arrays to form a directive beam for transmission to Bob. The other antenna array is used by Alice to generate another beam toward Willie as a jamming signal with its transmit power changing independently from a transmission block to another block. We characterize Willie's detection performance with the optimal detector and the closed-form of its expected value from Alice's perspective. We further derive the closed-form expression for the outage probability of the Alice-Bob link, which enables characterizing the optimal covert rate that can be achieved using the proposed setup. Our results demonstrate the superiority of mmWave covert communication, in terms of covertness and rate, compared to the RF counterpart.

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I Introduction

Rapid growth of wireless networks and emerging variety of applications, including Internet of Things (IoT) and critical controls, necessitate sophisticated solutions to secure the data transmission. Traditionally, the main objective of security schemes, using either cryptographic or information-theoretic approaches, is to secure data in the presence of adversary eavesdroppers. However, a stronger level of security can be obtained in wireless networks if the existence of communication is hidden from the adversaries. To this end, recently, there has been increasing attention to investigate covert communication, also referred to as communication with low probability of detection (LPD), in different setups with the goal of hiding the existence of communication [1, 2, 3, 4, 5, 6, 7, 8]. Generally speaking, covert communication refers to the problem of reliable communication between a transmitter Alice and a receiver Bob while maintaining a low probability of detecting the existence of communication from the perspective of a warden Willie [2].

The information-theoretic limits on the rate of covert communication have been presented in [1] for additive white Gaussian noise (AWGN) channels, where it is proved that bits of information can be transmitted to Bob, reliably and covertly, in uses of the channel. The same square root law have been developed for binary symmetric channels (BSCs) [3] and discrete memoryless channels (DMCs) [4]. Moreover, the principle of channel resolvability has been used in [5] to develop a coding scheme that can reduce the number of required shared key bits. Also, the first- and second-order asymptotics of covert communication over binary-input discrete memoryless channels are studied in [6]. The covert communications setup is also extended to a broadcast scenario [7] and mutiple-access scenarios [8] from an information-theoretic perspective.

The achievable covert rate in the aforementioned framework is zero in the limit of large when noting that . However, it is demonstrated that positive covert rates can be achieved by introducing additional uncertainty, from Willie’s perspective, into the system. For instance, it is shown in [9, 10] that Willie’s uncertainty about his noise power helps in achieving positive covert rates. Moreover, by considering slotted AWGN channels, it is proved in [11] that positive covert rates are achievable if the warden does not know when the transmission is taking place. The possibility of achieving positive-rate covert communication is further demonstrated for amplify-and-forward (AF) relaying networks with a greedy relay attempting to transmit its own information to the destination on top of forwarding the source’s information [12], dual-hop relaying systems with channel uncertainty [13], a downlink scenario under channel uncertainty and with a legitimate user as cover [14], and a single-hop setup with a full-duplex receiver acting as a jammer [15].

Prior studies on covert communication in wireless setups often consider transmission over conventional radio frequency (RF) wireless links. However, a superior performance, in terms of covertness and achievable rates, can be attained when performing the covert communication over millimeter-wave (mmWave) bands. This makes mmWave bands a suitable option for covert communication to increase the security level of wireless applications involving critical data, motivating the investigation of mmWave covert communications. Given the significant differences between the channel models and system architectures of mmWave communication and that of RF communication, the existing results on covert communications can not be immediately extended to covert communication over mmWave band. In particular, communication over mmWave bands can exploit directive beams, thanks to the deployment of massive antenna arrays, to compensate for the path losses over these ranges of frequencies. In this paper, we study covert communication over mmWave channels from a communication theory perspective, i.e., by analyzing the performance of the system in the limit of large block-lengths , and show the superiority of mmWave covert communication, in terms of covertness and rate, compared to the RF counterpart.

Ii Channel and System Models

Ii-a Channel Model

Recent experimental studies have demonstrated that mmWave links are highly sensitive to blocking effects [16, 17]. In order to model this characteristic, a proper channel model should differentiate between the LOS and non-LOS (NLOS) channel models since the path loss in the NLOS links can be much higher than that of the LOS path due to the weak diffractions in the mmWave band. Therefore, two different sets of parameters are considered for the LOS and NLOS mmWave links, and a deterministic function , that is a non-increasing function of , the link distance (in meters) between nodes and , is the probability that an arbitrary link of length is LOS. We consider a generic throughout our analysis and use the model , suggested in [17], for our numerical analysis.

Similar to [17], we express the channel coefficient for an arbitrary mmWave link between the transmitter and receiver as , where , , and are respectively the channel fading coefficient, the total directivity gain (including both the transmitter and receiver beamforming gains), and the path loss of the - mmWave link.

To characterize the path loss of the link with the length , we consider different path loss exponents () and intercepts () for the LOS and NLOS links, respectively. Consequently, denoting the path losses associated with the LOS and NLOS links as and , respectively, the path loss is either with probability or with probability .

To ascertain the total directivity gain , we use the common sectored-pattern antenna model adopted in [18, 19] which approximates the actual array beam pattern by a step function, i.e., with a constant main lobe over the beamwidth and a constant side lobe otherwise, where and . Then, for a given link, if the spatial arrangement of the beams of the transmitter and receiver are known, the total directivity gain can be known from the product of the gains of the transmitter and receiver. In a particular case where the main lobe of a node

(either transmitter or receiver) is pointed to another node, we assume some additive beamsteering errors denoted by a symmetric random variable (RV)

around the transmitter-receiver direction. Similar to [19], it is assumed that node will have a gain of if , i.e., with probability and a gain of otherwise, where

is the cumulative distribution function (CDF) of the RV

.

Finally, it is common in the literature to model the fading amplitude of mmWave links as independent Nakagami-distributed RVs each with shape parameter and scale parameter , and consider different Nakagami parameters for the LOS and NLOS links as and , respectively [17, 18]. In the case of Nakagami- fading with parameters , , and ,

has a normalized gamma distribution with shape and scale parameters of

and , respectively, i.e., .

Note that, from an information-theoretic perspective, mmWave communications, and in general wideband communications, can be viewed as low-capacity scenarios [20] suggesting a natural framework for mmWave covert communications.

Ii-B System Model

We consider the common setup of covert communication comprised of three parties: a transmitter Alice is intending to covertly communicate to a receiver Bob over mmWave bands when a warden Willie is attempting to realize the existence of this communication. Alice employs a dual-beam mmWave transmitter consisting of two antenna arrays. The first antenna array is used for the transmission to Bob while the second array is exploited as a jammer to enable positive-rate mmWave covert communication. Therefore, when Bob is not in the main lobe of Alice-Willie link, he receives the jamming signal gained with the side lobe of the second array in addition to receiving the desired signal from Alice with the main lobe of the first array. Similarly, when Willie is not in the main lobe of Alice-Bob link, he receives the desired signal gained with the side lobe of the first array in addition to receiving the jamming signal from Alice with the main lobe of the second array. On the other hand, when Bob is in the main lobe of Willie’s link, or vice versa, both of their received signals are gained with main lobes. Throughout our analysis we assume that Alice, Bob, and Willie are in some fixed locations (hence having some given directivity gains) and evaluate the impact of changing Willie’s location in our numerical results.

Denoting the channel coefficients between Alice’s first and second arrays and the node (representing Bob and Willie) as and , respectively, one can observe that the path loss gains are the same, i.e., , while the fading gains and are independent normalized gamma RVs. We assume quasi-static fading channels meaning that fading coefficients remain constant over a block of channel uses. We further assume that Alice transmits the desired signal with a publicly-known power while the jamming transmit power of its second array is not known and changes independently from a block to another block. In this paper, we assume that

varies according to a uniform distribution in the interval

while the results can be extended to other distributions using a similar approach. Let and denote the total directivity gains of the links between Alice’s first and second arrays and the node , respectively. Then, when Alice is transmitting to Bob, the received signals by Bob and Willie at each channel use can be expressed as

(1)
(2)

respectively, where and are Alice’s desired and jamming signals, respectively, satisfying . Moreover, and

are zero-mean Gaussian noises at Bob and Willie’s receiver with variances

and , respectively.

Iii Willie’s Detection Error Rate

As discussed earlier, Willie attempts to detect whether Alice is transmitting to Bob or not. We assume that Willie has a perfect knowledge about his channel from Alice and applies binary hypothesis testing when he is unaware of the value of

. The null hypothesis

states that Alice did not transmit to Bob while the alternative hypothesis specifies that Alice did transmit to Bob. There are two types of errors that can occur. Willie’s decision of hypothesis when is true is referred to as a false alarm and its probability is denoted by . On the other hand, Willie’s decision in favor of when

is true classifies a

missed detection, whose probability is denoted by . Then Willie’s overall detection error is denoted by . We say a positive-rate covert communication is possible if for any there exists a positive-rate communication between Alice and Bob satisfying as the number of channel uses .

(11)

Iii-a with the Optimal Detector at Willie

As it is proved in [21, Lemma 2] for AWGN channels and also pointed out in [14, Lemma 1], the optimal decision rule that minimizes Willie’s detection error is given by

(4)

where is Willie’s detection threshold for which we obtain its optimal value/range later in this subsection. Also, using (II-B), the

strong law of large numbers

, and noting that (see, e.g., the proof of [14, Theorem 3]), under hypotheses and is given by

(5)
(6)

respectively. The optimal threshold of Willie’s detector and his corresponding detection error are characterized in the following theorem.

Theorem 1.

The optimal threshold for Willie’s detector is in the interval

(7)

and the corresponding minimum detection error rate is

(8)

where and .

Proof: Using (5) the false alarm probability is given by

(9)

Moreover, using (6) the missed detection probability is given by

(10)

where . Using (III-A) and (III-A), and by following a similar argument as in the proof of [15, Theoem 1], the range of optimal threshold and the corresponding error rate are obtained as (7) and (8), respectively.   

Remark 1. Eq. (8) shows that for small values of such that Willie can attain a zero error rate negating the possibility of achieving a positive-rate covert communication in the limit of . Although increasing beyond can increase and enable a positive-rate covert communication ( as ), it also degrades the performance of the desired Alice-Bob link as we will see in Section IV. The superiority of mmWave covert communication to that of omni-directional RF communication can be observed by noticing the beneficial impact of beamforming. In fact, in the received signal by Willie, is gained with which is much larger than the gain of ; this significantly degrades the performance of Willie’s detector. The opposite situation happens for the Alice-Bob link where the desired signal is gained with which is much larger than the gain of the jamming signal.

Iii-B From Alice’s Perspective

Since Alice and Bob are unaware of the instantaneous realization of the channel between Alice and Willie, they should rely on the expected value of . Note also that the minimum error rate in (8) is independent of the beamforming gain of Willie’s receiver as it cancels out in the ratio of and also in the comparison between and . Furthermore, Alice perfectly knows the gain of the side lobe of her first array to Willie. However, she has uncertainty about the gain of the main lobe of the second array toward Willie due to the misalignment error; it is either with probability or with probability . Moreover, Alice and Bob do not know whether the Alice-Willie link is LOS or NLOS; hence, they should take into account two possibilities given the LOS probability . Next, the expected value of form Alice’s perspective is characterized in the following theorem.

Theorem 2.

The expected value of form Alice’s perspective can be characterized as (11) shown at the top of the next page where , , is the gamma function [22, Eq. (8.310.1)], and and are defined above for . Moreover, the function is defined as

(12)

and for and is defined as

(13)
(14)

Proof: Let , , and denote the values of , , and , respectively, conditioned on the blockage instance and the gain of Alice’s second array to Willie. Then using (8) we have

(15)

The closed form of can be derived as

(16)

where step follows from Alzer’s lemma [23], [18, Lemma 6] for a normalized gamma RV , which states that can tightly be approximated with where , and then applying the binomial theorem assuming is an integer [18]. Moreover, step

is derived using the moment generating function (MGF) of a normalized gamma RV

, i.e., for any .

Moreover, for the expectation term in (III-B) we have

(17)

where , and and

are the probability density functions of the fading coefficients

and , respectively. Applying the part-by-part integration rule to and then using Alzer’s lemma together with the binomial theorem yields

(18)

By plugging (III-B) into (III-B), using the MGF of the normalized gamma RV , and then noting that we have

(19)

Now given that the parameter of Nakagami- fading is always larger than or equal to and it is assumed an integer here, we have where stands for the set of natural numbers. For using [22, Eq. (3.351.3)] we have for any real . On the other hand, for using [22, Eq. (2.325.1)] we have , where is the exponential integral function defined as [22, Eq. (8.211.1)] for negative arguments. Therefore, following a similar approach to the proof of [24, Corollary 2] we can calculate the difference of the two integrals in (III-B) as which is equal to for (note that for , and ). This completes the proof of the theorem given the definition of in Theorem 2.   

Remark 2. In the derivation of Theorem 2 it is assumed that Willie is not in the main lobe of Alice’s first antenna array and hence receives the covert signal by a side lobe gain . However, if Willie is within the main lobe of the first array, we should include another averaging over the gain of the first array given the beamsteering error, i.e., that gain is either with probability or with probability .

Iv Performance of the Alice-Bob Link

Iv-a Outage Probability

We assume that Alice targets a rate requiring the Alice-Bob link to meet a threshold signal-to-interference-plus-noise ratio (SINR) . Then the outage probability in achieving can be characterized as Theorem 3, where the SINR of the Alice-Bob link is given using (II-B) as

(20)

Note that in addition to , and , the blockage instance and the antenna gains can also randomly change from a transmission block to another block. In particular, while we assume that the jamming signal arrives with the deterministic side lobe gain , there are still uncertainties in the gains of Alice’s first array and Bob’s receiver (they are pointing their main lobes) due to the beamsteering error. Therefore, the gain of the main lobe of Alice’s first array pointed to Bob is either with probability or with probability . Similarly, the gain of Bob’s receiver is either with probability or with probability . Furthermore, in the derivation of Theorem 3 we assume that Willie is not in the main lobe of Alice’s first array. However, if Willie is in the Alice-Bob direction, we should include another averaging of the gain of the main lobe of Alice’s second array carrying the jammer signal, i.e., instead of a deterministic we should consider two possibilities with probabilities , , defined in Section III-B.

Theorem 3.

The outage probability of the Alice-Bob link in achieving the target rate can be characterized as

(21)

where and . Moreover, for and is defined as

(22)
(23)

Proof: Given the SINR of the Alice-Bob link in (20), we have for the outage probability conditioned on the blockage instance , and the antenna gains and

(24)

where in step we have defined and . Moreover, step follows by Alzer’s lemma together with the binomial theorem, and step is derived using the MGF of the normalized gamma RV . Finally, taking the integral in step and recalling the definition of the function