Security and privacy play a major role in contemporary wireless communication systems. Much security research focuses on employing encryption [1] or information-theoretic [2] security approaches to protect the content of messages from interception, but there are applications where hiding even the existence of a signal can be crucial, for example in hiding the presence of an implanted medical device or in military communication. In covert communication, the transmitter Alice tries to transmit a message reliably to the receiver Bob in the presence of an adversary Willie, who tries to detect the communication.
Bash et. al. [3] established the fundamental limits of covert communication for the discrete-time additive white Gaussian noise (AWGN) channel. According to the square root law (SRL) of [3], Alice can transmit at most O( √ n) bits to Bob in n channel uses with low probability of detection by adversary Willie. This work motivated [4] and [5], which established that if Willie has uncertainty in the statistics of the noise impacting his receiver, O(n) bits can be transmitted covertly in n channel uses. Since such uncertainty might not arise naturally in a given system [6], Sobers et. al. [7] employed jamming to achieve O(n) covert throughput in n channel uses.
The results discussed above assume a standard discretetime communication channel. Researchers generally consider a discrete-time channel for analytical convenience, with the understanding that it is functionally equivalent to the true continuous-time channel. However, these techniques that exploit noise uncertainty or cooperative jamming are vulnerable to detection (i.e. not covert) when the constructions developed for the discrete-time channel are mapped to a true continuoustime channel. Li et. al. [8] showed that the technique of [7] cannot send O(n) bits covertly in n channel uses on a continuous-time channel if Willie employs interference cancellation, even in the presence of a jamming signal as proposed in [7], emphasizing the more realistic continuous-time channel.
Here, we consider the performance of the constructions of [4] and [5] when they are mapped to a continuous-time channel in an obvious manner. By having Willie employ a cyclostationary detector, we show that the performance results established in [4] and [5] are indeed optimistic. By conducting an asymptotic characterization of the cyclostationary detector, we demonstrate that schemes employing standard bauded digital communication approaches (i.e., randomly drawn symbols placed on a standard pulse shape with a fixed symbol interval) are limited to sending at most O( √ n) covert bits to Bob in n channel uses. Furthermore, we show that Alice would indeed be able to send O(n) bits in n channel uses covertly to Bob on the continuous-time channel if Willie were restricted to a power detector, hence demonstrating that it is the availability of other detectors on the continuous-time channel that leads to the limitation of the throughput of schemes suggested by the design for the discrete-time channel.
Section II presents the natural extension of the constructions of [4] and [5] to the continuous-time channel. Section III presents a sketch of the proofs for the two main results of the paper. Section IV presents the conclusions.
Consider the scenario presented in Figure 1, where Alice wants to transmit a message to Bob over (continuoustime) interval [-T /2, T /2] while keeping the transmission hidden from adversary Willie, with both channels experiencing additive white Gaussian noise (AWGN). Suppose Alice sends an independent and identically distributed (i.i.d.) sequence of symbols f -m/2 , f -m/2+1 ,. . . , f m/2 , such that f i ∈ CN (0, σ 2 a ), using a digital communication signal in the standard format with pulse shape q(t):
where T s is the symbol period and m = T /T s , hence extending the Lee et. al. [4] discrete-time channel in the natural way to continuous-time channel. We take the pulse shape q(t) to be a square-root raised cosine (SRRC) pulse with bandwidth (1 + β)/2T s , where β > 0 is the excess bandwidth or roll-off factor, but our results are valid for any pulse shape that decays sufficiently fast to zero for t → ∞. The pulse shape q(t) is normalized such that Consider H 0 the hypothesis when Alice is not transmitting and H 1 the hypothesis when Alice is transmitting. Then, the received signal y w (t) at Willie is given under each hypothesis as:
where n w (t) ∞ t=-∞ , the noise at Willie, is a zero-mean stationary Gaussian random process with power spectral density S w (f ) = N 0 /2. Per Section I, we consider the case where N 0 is not perfectly known to Willie [4], [5].
Willie conducts a binary hypothesis test to determine whether Alice is transmitting or not. Let P F A , the probability of false alarm, denote the probability that Willie chooses H 1 when H 0 is true, and P M D , the probability of missed detection, denote the probability that he chooses H 0 when H 1 is true. Using the now standard definition of covertness introduced in [3], a system is said to be covert if
III. MAIN RESULT Theorem 1: Given the continuous-time channel of Section II, if Alice sends ω( √ T ) bits over (continuous-time) interval [-T /2, T /2], then Willie can detect her with a low probability of error, P M D + P F A → 0, as T → ∞.
Proof: The receiver at Willie is shown in Figure 2. It employs a cyclostationary detector to determine whether Alice is transmitting. After front-end matched filtering to form y(t) = y w (t) * q(-t), here y w (t) is the signal received by Willie, the receiver forms
Ts dt Fig. 2. Receiver construction: cyclostationary detection at Willie Willie then employs a threshold test on |z| with threshold γ to detect whether Alice is transmitting or not, where γ is a constant that will be determined below. If |z| > γ, Willie decides that Alice is transmitting; if |z| ≤ γ, Willie decides that Alice is not transmitting. Statistical Analysis |z| under H 0 We first consider the statistics of |z| when Alice is not transmitting. To do such, consider E[|z| 2 |H 0 ].
Let the receiver noise be n(t) = n w (t) * q(-t), the signal obtained after passing the noise n w (t) on the received signal through the front-end filter at the receiver. Then:
e -j2πt/Ts e j2πs/Ts dtds.
For jointly Gaussian W, X, Y and Z,
Hence:
Consider the behavior of the first term of (2). With
where S n (f ) is the power spectral density of n(t) = n w (t) * q(-t). Letting F{q(t)} = Q(f ) be the Fourier transform of q(t),
and one can readily show that:
e -j2πt/Ts e j2πs/Ts dt ds
The random phase of the second term of (2) causes the expectation to be 0 for all s and t, so all that remains is to consider the third term, which we will see is the dominant term:
e -j2πt/Ts e j2πs/Ts dtds
/Ts e j2πs/Ts dtds where R n (t -s) is the autocorrelation function of n(t). Letting v = s + t and u = s -t and applying the change of variables to the above equation results in:
where we have used |e -j2πu/Ts | ≤ 1 and noted R 2 n (u) ≥ 0 to get the second line above. Employing Parseval's theorem
where we have defined the constant
Hence, the random variable |z| converges to 0 in the mean squared sense, and the rate of convergence of the mean squared value given in (10) will be important in establishing a tight upper bound on Alice's throughput. We note that the (probabilistic) convergence of |z| to 0 captures a key feature of the cyclostationary detector relative to a power detector when Willie does not know the noise variance N 0 . For a power detector, the mean is not zero when Alice is not present; rather it depends on N 0 , thus making it difficult for Willie to select a threshold when N 0 is unknown, as shown below in the proof of Theorem 2.
Under H 1 , the signal at the output of Willie's front-end matched filter is given by:
where p(t) = q(t) * q(-t). We desire to lower bound E[|z||H 1 ] by a constant so that Willie can set a threshold γ between 0 and that constant; combined with the rates at which the variance under H 0 and H 1 go to zero, this will establish the main result, as shown below.
Given that
we consider the more tractable E[z|H 1 ], which we will show is sufficient. Noting that the noise is zero mean and independent of the signal yields:
Using analogous arguments from our work above under H 0 , we know that the second term containing
For the first term, using
p 2 (t -kT s )e -j2πt/Ts dt (11)
The second and third terms of (11) will go to zero as T → ∞, as the tails of the raised cosine pulse are absolutely integrable.
If we move the integral through the summation in the first term, we note that the summand is the Fourier transform
= X(f ) of the (shifted) raised cosine pulse shape squared evaluated at the frequency f = 1
Ts . Thus,
Ts and noting that as T → ∞, m → ∞, yields:
Hence, for the cyclostationary detector, as T → ∞, E[z|H 1 ] → C, where C ̸ = 0, and the constant C is not dependent on N 0 . In contrast, we will see below in the proof of Theorem 2 that, for the power detector, E[z|H 1 ] depends upon N 0 , which is unknown. The variance of |z| under H 1 is given by:
+|n(s)| 2 e -j2πs/Ts ds * As the product of two expressions, each of which involves four terms, the above expression expands into sixteen terms. After detailed technical evaluation (omitted) the resulting variance has the following bound as T → ∞:
where K 1 is defined above, and K 3 is a (positive) constant that depends on the noise variance and the pulse shaping filter as
2Ts . The probability of false alarm P F A and the probability of missed detection P M D can then be readily bounded using Chebyshev's inequality. For T large,
2 and
√ T , then P F A → 0 and P M D → 0 as T → ∞. Therefore, following arguments in [3], given that Alice is using a signal power of ω 1/
T , indicating that Alice transmits ω( √ T ) bits in a time interval of [0, T ], then P M D + P F A → 0 as T → ∞, even with noise uncertainty at Willie. □ The above result demonstrates that a straightforward extension of constructions employed for covert communications on discrete-time channels might not be effective on the continuous-time channel. From [7], it is apparent that a power detector is an optimal receiver for Willie to attempt to detect Alice's signal for some noise uncertainty models on a discretetime channel. Next, we demonstrate that this is not true on the continuous-time channel by showing that Alice would be able to transmit at a constant rate to Bob while remaining covert if Willie employs a power detector. Hence, it is the existence of better detectors on the continuous-time channel that exploit the structure of Alice's signal that restricts her ability to communicate covertly on such.
Theorem 2: Consider the model of Section II, with the uncertainty in Willie's noise variance characterized by a uniform random variable: N 0 ∼ U [l, u], where l and u are lower and upper bounds on N 0 , respectively. Then, Alice can send O(T ) bits covertly to intended receiver Bob over (continuous-time) interval [-T /2, T /2], if Willie employs a power detector.
Proof: A power detector at Willie forms the variable:
and compares it to a threshold τ . The expectation of z p is given by
Similarly for H 1 :
p 2 (t -kT s ) be the average signal power, which is a constant; then,
Since we are deriving an achievability result, it is sufficient to demonstrate performance against a detector that upper bounds the performance of Willie. To do such, assume that a genie averages out the variation due to measurement noise on Willie's received signal. Hence, under H 0 , Willie always measures N0 2 , and under H 1 he always measures σ 2 a P x + N0 2 . But recall that Willie does not know the value of N 0 and thus struggles to choose the threshold τ . Consider any choice of that threshold τ , and define the event A to be the event that a "good" threshold is selected such that A = {τ : N0 2 ≤ τ ≤ N0 2 + σ 2 a P x }. Now, consider the probability that A occurs:
Now note that if A is not true, then P F A + P M D = 1. Hence,
and Alice can select a constant power σ 2 a small enough such that P M D + P F A ≥ 1 -ϵ, hence establishing that Alice can employ constant power and transmit O(n) bits covertly in n channel uses when Willie uses a power detector. □
IV. CONCLUSION If Alice attempts to employ bauded modulation to transmit ω( √ T ) bits covertly in a time interval of [0, T ] to receiver Bob over a continuous-time channel, an attentive adversary Willie can detect that communication as T → ∞, even in the face of noise uncertainty at Willie, as the bauded signal allows detection by a cyclostationary detector for which the parameters do not depend on the noise variance. In contrast, a power detector, which is the optimal detector in a discrete-time channel under common uncertainty models for the noise variance, has difficulty setting a detection threshold. Hence, the existence of better detectors in continuous-time significantly impacts the covert throughput versus what might be suggested by design for a discrete-time model. This work also helps to demonstrate the challenges of covert communications as we incorporate more faithful representations of the physical layer.