# Performance Analysis for Covert Communications Under Faster-than-Nyquist
Signaling

Yuan Li, Yuchen Zhang, Wanyu Xiang, Jianquan Wang, Sa Xiao, Liang Chang and
Wanbin Tang Yuan Li, Yuchen Zhang, Wanyu Xiang, Jianquan Wang, Sa Xiao, and
Wanbin Tang are with the National Key Laboratory of Science and Technology on
Communications, University of Electronic Science and Technology of China,
Chengdu, 611731 China (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>wbtang@uestc.edu.cn). Liang Chang is with Beijing ATV
Technology Co.,Ltd., Beijing, China (e-mail: changliang@aotev.com).

###### Abstract

In this letter, we analyze the performance of covert communications under
_faster-than-Nyquist_ (FTN) signaling in the Rayleigh block fading channel.
Both Bayesian criterion- and _Kullback-Leibler_ (KL) divergence-based
covertness constraints are considered. Especially, for KL divergence-based
one, we prove that both the maximum transmit power and covert rate under FTN
signaling are higher than those under Nyquist signaling. Numerical results
coincide with our analysis and validate the advantages of FTN signaling to
realize covert data transmission.

###### Index Terms:

Covert communications, faster-than-Nyquist signaling, Bayesian criterion,
Kullback-Leibler divergence

##  I Introduction

THANKS to the higher symbol rate at the transmitter side, _faster-than-
Nyquist_ (FTN) signaling has been regarded as a promising technology to
improve the _spectrum efficiency_ (SE) in the future wireless communication
systems [1, 2, 3]. However, the higher symbol rate can also incur severe
_inter-symbol-interference_ (ISI), which may degrade the transmit rate if has
not been carefully dealt with. Therefore, the ISI cancellation schemes to
improve the SE of FTN signaling have been extensively studied [3, 4, 5, 6, 7,
8].

On the other hand, besides degrading the transmit rate of the legitimate data
transmission, the incurred ISI can also aggravate the detection or decoding of
illegal eavesdroppers. Therefore, some existing work has exploited the ISI
incurred by FTN signaling to enhance the security of the data transmission [6,
7, 8]. In [6], the eigenvalue decomposition based scheme had been investigated
to improve the secrecy rate of FTN signaling in quasi-static fading channel.
In [7] and [8], the time-varied coefficients of shaped filter and symbol
duration were introduced to incur more severe interference to ensure secure
transmission, respectively. However, the aforementioned work all focuses on
exploiting FTN signaling to avoid the information from being decoded by
illegal eavesdroppers, i.e., physical layer security. In contrast, covert
communications, which aim to prevent the communication between the transmitter
Alice and the receiver Bob, from being detected by the eavesdropper Willie,
can provide stronger security [9, 10, 11, 12, 13]. For instance, in military
wireless communications, preventing communications from being detected has a
higher priority. Also, Alice can transmit no more than $\mathcal{O}(\sqrt{n})$
bits to Bob covertly and reliably for $n$ channel uses, i.e., the square root
law [9], has been proved in covert communications under _additive white
Gaussian noise_ (AWGN) channel. To the best of our knowledge, the covert issue
has not been studied under FTN signaling. Therefore, the covert performance of
FTN signaling is worth further exploring.

In this letter, we analyze the performance of covert communications under FTN
signaling in the Rayleigh block fading channel. We consider both the Bayesian
criterion- and _Kullback-Leibler_ (KL) divergence-based covertness
constraints, and theoretically prove the transmit power and covert rate under
FTN signaling with the latter constraint are higher than those under Nyquist
signaling. Numerical results coincide with our analysis and show that the
maximum transmit power and covert rate can be increased via FTN signaling and
decrease with the value of the acceleration factors.

##  II System Model

Figure 1: The system model of covert communications under FTN siganling.

In this paper, we consider a covert communication system under FTN signaling
as shown in Fig. 1, where a legitimate transmitter Alice transmits signals to
its receiver Bob, while an eavesdropper Willie wants to detect the existence
of the data transmission. In addition, there is a secret key shared between
Alice and Bob to ensure their communication [11, 12]. For instance, the secret
key can help Bob synchronize data or choose codebook. We assume that channel
A-B and A-W follow the Rayleigh block fading. The transmitted signals of Alice
can be represented by

$x(t)=\sum_{n=-\infty}^{\infty}{a_{n}h(t-n\tau T)},$ (1)

where $a_{n}\sim\mathcal{CN}(0,\sigma_{a}^{2})$ is the transmitted symbol.
This assumption is widely adopted in literature [2, 3, 4] to obtain the
capacity of FTN signaling. In addition, we use this assumption to obtain the
upper bound of the performance analysis of using FTN signaling in covert
communications, which provides a theoretical guidance for practical cases.
$\tau\in(0,1)$ is the acceleration factor, $h(t)$ is the corresponding _root
raised cosine_ (RRC) shaped filter with unit energy and roll-off factor
$\alpha$. $T$ is the orthogonal interval of $h(t)$. Accordingly, the transmit
power $P$ is

$P=\frac{\sigma_{a}^{2}}{\tau T}.$ (2)

In this letter, in order to provide more stringent covertness requirement for
the legitimate data transmission, $h(t)$ and $\tau$ are assumed to be
perfectly known by Willie. After the process of matched filter, the received
signal can be represented by

$r_{w}(t)=\sum_{n=-\infty}^{\infty}{h_{aw}a_{n}g(t-n\tau T)}+\eta_{w}(t),$ (3)

where $h_{aw}\sim\mathcal{CN}(0,1)$ is the fading coefficients of channel A-W,
$g(t)=h(t)*h^{*}(-t)$, $\eta_{w}(t)=z_{w}(t)*h^{*}(-t)$, and $z_{w}(t)$ is the
AWGN at Willie’s side.

The hypothesis test at Willie’s side can be represented by

$\left\\{\begin{aligned} \mathcal{H}_{0}:~{}&r_{w}[n]=\eta_{w}[n]\\\
\mathcal{H}_{1}:~{}&r_{w}[n]=\sum_{l=-\infty}^{\infty}h_{aw}a[n-l]g[l]+\eta_{w}[n]\end{aligned}\right.$
(4)

according to the Ungerboeck observation model [14]. Since $r_{w}[n]$ is
correlated to each other, we apply $\textbf{r}_{w}\in\mathbb{C}^{N\times 1}$
as the observation value, where $N$ is the received symbol block length. Thus,
(4) can be rewritten as

$\left\\{\begin{aligned}
\mathcal{H}_{0}:~{}&\textbf{r}_{w}=\boldsymbol{\eta}_{w}\\\
\mathcal{H}_{1}:~{}&\textbf{r}_{w}=h_{aw}\textbf{G}\textbf{a}+\boldsymbol{\eta}_{w},\end{aligned}\right.$
(5)

where
$\boldsymbol{\eta}_{w}\sim\mathcal{CN}(\textbf{0},\sigma_{w}^{2}\textbf{G}$),
$\textbf{G}\in\mathbb{R}^{N\times N}$ is the corresponding positive definite
ISI matrix [4, 5], and $E[\textbf{aa}^{H}]=\sigma_{a}^{2}\textbf{I}$.

##  III Covertness Constraints Analysis

In this section, we analyze the covertness constraints under Bayesian
criterion and KL divergence with FTN signaling.

###  III-A Covertness Constraint Analysis under Bayesian Criterion

The likelihood function of $\textbf{r}_{w}$ can be represented by

$L(\textbf{r}_{w})=\frac{p_{1}(\textbf{r}_{w})}{p_{0}(\textbf{r}_{w})},$ (6)

where $p_{0}(\textbf{r}_{w})=p\left(\textbf{r}_{w}|\mathcal{H}_{1}\right)$ and
$p_{1}(\textbf{r}_{w})=p\left(\textbf{r}_{w}|\mathcal{H}_{0}\right)$ are the
_probability density function_ (PDF) of $\textbf{r}_{w}$ under
$\mathcal{H}_{0}$ and $\mathcal{H}_{1}$, respectively.

For Willie, the detection problem is to distinguish $\mathcal{H}_{0}$ and
$\mathcal{H}_{1}$. Assuming
$\mathbb{P}(\mathcal{H}_{0})=\mathbb{P}(\mathcal{H}_{1})=0.5$, which is widely
adopted in detection theory. Then, in the sense of minimizing the error
probability, the optimal detector is the likelihood ratio test by the Bayesian
criterion [15], which can be represented as

$L\left(\textbf{r}_{w}\right)\mathop{\gtrless}\limits_{\mathcal{D}_{0}}^{\mathcal{D}_{1}}1,$
(7)

where $\mathcal{D}_{0}$ and $\mathcal{D}_{1}$ are the decision regions of
$\mathcal{H}_{0}$ and $\mathcal{H}_{1}$, respectively. Then, the false alarm
and miss detection probabilities of Willie are
$\mathbb{P}_{FA}=\text{Pr}(\mathcal{D}_{1}|\mathcal{H}_{0})$ and
$\mathbb{P}_{MD}=\text{Pr}(\mathcal{D}_{0}|\mathcal{H}_{1})$, respectively.

Since
$E[\boldsymbol{\eta}_{w}\boldsymbol{\eta}_{w}^{H}]=\sigma_{w}^{2}\textbf{G}$
and
$E[(h_{aw}\textbf{Ga}+\boldsymbol{\eta}_{w})(h_{aw}\textbf{Ga}+\boldsymbol{\eta}_{w})^{H}]=|h_{aw}|^{2}\sigma_{a}^{2}\textbf{G}\textbf{G}^{H}+\sigma_{w}^{2}\textbf{G}$,
we have

$p_{0}(\textbf{r}_{w})=\frac{1}{\pi^{N}\text{det}(\sigma_{w}^{2}\textbf{G})}\text{exp}\\{-\textbf{r}_{w}^{H}(\sigma_{w}^{2}\textbf{G})^{-1}\textbf{r}_{w}\\},$
(8)

and

$p_{1}(\textbf{r}_{w})=\frac{\text{exp}\\{-\textbf{r}_{w}^{H}(|h_{aw}|^{2}\sigma_{a}^{2}\textbf{G}\textbf{G}^{H}+\sigma_{w}^{2}\textbf{G})^{-1}\textbf{r}_{w}\\}}{\pi^{N}\text{det}(|h_{aw}|^{2}\sigma_{a}^{2}\textbf{G}\textbf{G}^{H}+\sigma_{w}^{2}\textbf{G})}.$
(9)

By substituting (8) and (9) into (6), the logarithm of $L(\textbf{r}_{w})$ can
be written as

$l(\textbf{r}_{w})=\text{log}\beta_{1}+\textbf{r}_{w}^{H}\left[\frac{\textbf{G}^{-1}}{\sigma_{w}^{2}}-(|h_{aw}|^{2}\sigma_{a}^{2}\textbf{G}\textbf{G}^{H}+\sigma_{w}^{2}\textbf{G})^{-1}\right]\textbf{r}_{w},$
(10)

where

$\beta_{1}=\frac{\text{det}(\sigma_{w}^{2}\textbf{G})}{\text{det}(|h_{aw}|^{2}\sigma_{a}^{2}\textbf{GG}^{H}+\sigma_{w}^{2}\textbf{G})}.$
(11)

To be noticed, when $\tau<1/(1+\alpha)$ and $N\to\infty$, G has some
asymptotically-zero eigenvalues [3]. However, asymptotically-zero is not zero.
Therefore, the derivations in this section hold for any feasible $\tau$
theoretically.

As mentioned in [9, 16, 13, 17], the covertness constraint under Bayesian
criterion is given by

$\xi^{min}=\mathbb{P}_{FA}+\mathbb{P}_{MD}\geq 1-\epsilon,$ (12)

where $\epsilon\in[0,1)$ is a given value.

Furthermore, we have the following theorem for FTN signaling.

_Theorem 1:_ Let G = $\textbf{V}^{T}\boldsymbol{\Lambda}\textbf{V}$, where
$\textbf{V}^{T}\textbf{V}$ = I and $\boldsymbol{\Lambda}$ =
diag$\\{$$\lambda_{0}$, $\lambda_{1}$, $\cdots$, $\lambda_{N-1}$$\\}$. Then,
the false alarm and miss detection probabilities of Willie can be represented
by

$\mathbb{P}_{FA}=\frac{1}{2\pi}\int_{\theta}^{\infty}\int_{-\infty}^{\infty}\prod_{n=0}^{N-1}{\frac{1}{1-2j\alpha_{n}\omega}\text{exp}(-j\omega
t)}d\omega dt$ (13)

and

$\mathbb{P}_{MD}=\frac{1}{2\pi}\int^{\theta}_{-\infty}\int_{-\infty}^{\infty}\prod_{n=0}^{N-1}{\frac{1}{1-2j\gamma_{n}\omega}\text{exp}(-j\omega
t)}d\omega dt,$ (14)

respectively, where $\theta=-\text{log}\beta_{1}$,

$\alpha_{n}=\frac{1}{2}\left(\frac{|h_{aw}|^{2}\sigma_{a}^{2}\lambda_{n}}{|h_{aw}|^{2}\sigma_{a}^{2}\lambda_{n}+\sigma_{w}^{2}}\right),$
(15)

and

$\gamma_{n}=\frac{1}{2}\left(\frac{|h_{aw}|^{2}\sigma_{a}^{2}}{\sigma_{w}^{2}}\lambda_{n}\right).$
(16)

_Proof:_ Replace G as $\textbf{V}^{T}\boldsymbol{\Lambda}\textbf{V}$, (10) can
be represented by

$l(\textbf{r}_{w})=\text{log}\beta_{1}+T(\textbf{r}_{w}),$ (17)

where

$\beta_{1}=\prod_{n=0}^{N-1}{\frac{\sigma_{w}^{2}}{|h_{aw}|^{2}\sigma_{a}^{2}\lambda_{n}+\sigma_{w}^{2}}}$
(18)

and

$T(\textbf{r}_{w})=\textbf{r}_{w}^{H}\textbf{V}\left[\frac{\boldsymbol{\Lambda}^{-1}}{\sigma_{w}^{2}}-(|h_{aw}|^{2}\sigma_{a}^{2}\boldsymbol{\Lambda}^{2}+\sigma_{w}^{2}\boldsymbol{\Lambda})^{-1}\right]\textbf{V}^{T}\textbf{r}_{w}$
(19)

is the test statistic.

Let $\textbf{y}=\textbf{V}^{T}\textbf{r}_{w}$, the PDF of y can be represented
by

$\textbf{y}\sim\left\\{\begin{aligned}
&\mathcal{H}_{0}:~{}\mathcal{CN}(0,\sigma_{w}^{2}\boldsymbol{\Lambda})\\\
&\mathcal{H}_{1}:~{}\mathcal{CN}(0,|h_{aw}|^{2}\sigma_{a}^{2}\boldsymbol{\Lambda}^{2}+\sigma_{w}^{2}\boldsymbol{\Lambda}).\end{aligned}\right.$
(20)

Then, the logarithm of (7) can be rewritten as

$T_{1}(\textbf{y})\mathop{\gtrless}\limits_{\mathcal{D}_{0}}^{\mathcal{D}_{1}}\theta,$
(21)

where

$\displaystyle T_{1}(\textbf{y})$
$\displaystyle=\textbf{y}^{H}\left[\frac{\boldsymbol{\Lambda}^{-1}}{\sigma_{w}^{2}}-(|h_{aw}|^{2}\sigma_{a}^{2}\boldsymbol{\Lambda}^{2}+\sigma_{w}^{2}\boldsymbol{\Lambda})^{-1}\right]\textbf{y}$
(22)
$\displaystyle=\sum_{n=0}^{N-1}\left(\frac{|h_{aw}|^{2}\sigma_{a}^{2}}{\sigma_{w}^{2}(|h_{aw}|^{2}\sigma_{a}^{2}\lambda_{n}+\sigma_{w}^{2})}\right)|y[n]|^{2}.$
(23)

Furthermore, the false alarm probability of Willie can be represented by

$\mathbb{P}_{FA}=\text{Pr}\\{T_{1}(\textbf{y})\geq\theta|\mathcal{H}_{0}\\}.$
(24)

Let $v[n]=y[n]/\sqrt{\frac{\sigma_{w}^{2}\lambda_{n}}{2}}$, we have
$v[n]\sim\mathcal{CN}(0,2)$. Then (24) can be rewritten as

$\mathbb{P}_{FA}=\text{Pr}\\{\sum_{n=0}^{N-1}\alpha_{n}|v[n]|^{2}\geq\theta|\mathcal{H}_{0}\\}.$
(25)

Furthermore, according to Appendix 5A of [15] and [18], the characteristic
function of $T_{1}(\textbf{y})$ can be represented by

$\displaystyle\phi_{T_{1}}(\omega)$ $\displaystyle=E[\text{exp}(j\omega
T_{1})]$ (26)
$\displaystyle=\prod_{n=0}^{N-1}{\frac{1}{1-2j\alpha_{n}\omega}}.$ (27)

Then, the PDF of $T_{1}(\textbf{y})$ can be represented by

$p_{T}(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\phi_{T}(\omega)\text{exp}(-j\omega
t)d\omega,t>0.$ (28)

Therefore, (25) can be rewritten as (13).

Similarly, let
$v[n]=y[n]/\sqrt{\frac{|h_{aw}|^{2}\sigma_{a}^{2}\lambda_{n}^{2}+\sigma_{w}^{2}\lambda_{n}}{2}}$,
and $\mathbb{P}_{MD}$ can be represented by (14).$\hfill\blacksquare$

It should be noted that $\mathbb{P}_{FA}$ and $\mathbb{P}_{MD}$ cannot be
directly derived from (13) and (14), respectively. Therefore, we carry out
Monte Carlo simulations to calculate them according to (24) and
$\mathbb{P}_{MD}=\text{Pr}\\{T_{1}(\textbf{y})<\theta|\mathcal{H}_{1}\\}$,
respectively.

###  III-B Covertness Constraint Analysis under KL divergence

In this subsection, we first derive the KL divergence-based covertness
constraint under FTN signaling, and then compare the maximum transmit powers
under FTN and Nyquist signaling.

For the optimal test at Willie,

$\xi^{min}=1-\mathcal{V}_{T}(p_{0},p_{1}),$ (29)

where $\mathcal{V}_{T}(p_{0},p_{1})$ is the total variation between
$p_{0}(\textbf{r}_{w})$ and $p_{1}(\textbf{r}_{w})$. To avoid the intractable
expressions for $\mathcal{V}_{T}(p_{0},p_{1})$, we use KL divergence that is
widely adopted in the literature to limit the detection performance at Willie
[9, 13]. Specially, according to Pinsker’s inequality, we have

$\mathcal{V}_{T}(p_{0},p_{1})\leq\sqrt{\mathcal{D}(p_{1}||p_{0})/2},$ (30)

where

$\mathcal{D}(p_{1}||p_{0})=\int_{\textbf{r}_{w}}p_{1}(\textbf{r}_{w})\text{log}\frac{p_{1}(\textbf{r}_{w})}{p_{0}(\textbf{r}_{w})}d\textbf{r}_{w}$
(31)

is the KL divergence from $p_{1}(\textbf{r}_{w})$ to $p_{0}(\textbf{r}_{w})$.
Then, (12) can be obtained by (29) and (30) as

$\mathcal{D}(p_{1}||p_{0})\leq 2\epsilon^{2}.$ (32)

To be noticed, the KL divergence-based covertness constraint is stricter than
(12), which means it is fully operational in practice [13]. Then we have the
following theorem.

_Theorem 2:_ The KL divergence-based covertness constraint under FTN signaling
can be represented by

$\displaystyle\mathcal{D}(p_{1}||p_{0})=$
$\displaystyle\sum_{n=0}^{N-1}{\frac{|h_{aw}|^{2}\sigma_{a}^{2}}{\sigma_{w}^{2}}\lambda_{n}-\text{log}\left(\frac{|h_{aw}|^{2}\sigma_{a}^{2}}{\sigma_{w}^{2}}\lambda_{n}+1\right)}\leq
2\epsilon^{2},$ (33)

where $\lambda_{n}$ is the eigenvalue mentioned in _Theorem 1_.

_Proof:_ By substituting (8) and (9) into (31), we have

$\displaystyle\mathcal{D}(p_{1}||p_{0})=\text{log}\beta_{1}$
$\displaystyle+E\left[\textbf{r}_{w}^{H}\left[\frac{\textbf{G}^{-1}}{\sigma_{w}^{2}}-(|h_{aw}|^{2}\sigma_{a}^{2}\textbf{G}\textbf{G}^{H}+\sigma_{w}^{2}\textbf{G})^{-1}\right]\textbf{r}_{w}\right].$
(34)

According to [19],
$E[\textbf{x}^{T}\textbf{Q}\textbf{x}]=\text{Tr}(\textbf{Q}\boldsymbol{\Sigma})+\boldsymbol{\mu}^{T}\textbf{Q}\boldsymbol{\mu}$,
where $\boldsymbol{\mu}$ and $\boldsymbol{\Sigma}$ are the mean and covariance
of x, respectively. Then (34) can be written as

$\mathcal{D}(p_{1}||p_{0})=\text{log}\beta_{1}+\text{Tr}\left(\frac{|h_{aw}|^{2}\sigma_{a}^{2}}{\sigma_{w}^{2}}\textbf{G}^{H}\right),$
(35)

and thus we have (33). $\hfill\blacksquare$

##  IV Covert Rate Analysis

According to [2, 3, 4], the instantaneous maximum achievable rate of the
communication link between Alice and Bob under FTN signaling can be
represented by

$R_{FTN}=\int_{-1/2\tau T}^{1/2\tau
T}\text{log}_{2}\left(1+\frac{2|h_{ab}|^{2}P}{N_{0}}H_{fo}(f)\right)df,$ (36)

where $h_{ab}\sim\mathcal{CN}(0,1)$ is the fading coefficient of channel A-B,
$H_{fo}(f)=\sum_{k=-\infty}^{\infty}{|H(f-k/(\tau T))|^{2}}$ and $N_{0}$ is
one-side spectral density of noise, and $H(f)$ is the Fourier transform of
$h(t)$.

Following Section III, the instantaneous covert rate under Bayesian criterion-
or KL divergence-based covertness constraint can be represented by

$\displaystyle\max_{P}~{}R_{FTN}$ (37)
$\displaystyle\text{s.t.}~{}(\ref{Bayesian_rule})~{}\text{or}~{}(\ref{D2}).$

Since $R_{FTN}$ in (36) is monotonically increasing with $P$, the optimal $P$
to Problem (37) should be the maximum transmit power satisfying (12) or (33).
For (12), the maximum transmit power can be obtained by solving (13) and (14)
via MATLAB. For (33), $P_{F}^{max}$ can be obtained from (38). By substituting
the maximum transmit power into (36), we can obtain the instantaneous maximum
covert rate under FTN signaling. Specially, for the KL divergence based case,
we have the following theorem.

_Theorem 3:_ Let $R_{N}$ be the instantaneous covert rate of Nyquist signaling
under the KL divergence-based covertness constraint. For a given $\epsilon$
satisfying (32), we have $R_{FTN}>R_{N}$.

_Proof:_ Assume $P_{F}$ and $P_{N}$ to be the transmit powers of Alice under
FTN and Nyquist signaling with the KL divergence-based covertness constraint,
respectively. For the same detection time duration at Willie, (33) can be
rewritten as

$\displaystyle\mathcal{D}_{1}=$
$\displaystyle\sum_{n=0}^{N-1}{\frac{|h_{aw}|^{2}\tau
TP_{F}\lambda_{n}}{\sigma_{w}^{2}}-\text{log}\left(\frac{|h_{aw}|^{2}\tau
TP_{F}\lambda_{n}}{\sigma_{w}^{2}}+1\right)}\leq 2\epsilon_{1}^{2}$ (38)

and

$\mathcal{D}_{2}=\sum_{n=0}^{N^{\prime}-1}{\frac{|h_{aw}|^{2}P_{N}T}{\sigma_{w}^{2}}-\text{log}\left(\frac{|h_{aw}|^{2}P_{N}T}{\sigma_{w}^{2}}+1\right)}\leq
2\epsilon_{2}^{2}$ (39)

for FTN and Nyquist signaling, respectively, where $N^{\prime}=N\tau$.

Recall $\lambda_{n}\approx(\tau T)^{-1}H_{fo}(f_{n})$ in [3], where
$f_{n}=n/N\tau T$. For $N\to\infty$, (38) can be rewritten as

$\displaystyle\lim_{N\to\infty}\mathcal{D}_{1}=\lim_{N\to\infty}\sum_{n=0}^{N-1}{\rho
P_{F}H_{fo}(f_{n})-\text{log}\left(\rho P_{F}H_{fo}(f_{n})+1\right)}$ (40)
$\displaystyle=\lim_{N\to\infty}N\tau T\int_{0}^{\frac{1}{\tau T}}{\rho
P_{F}H_{fo}(f)-\text{log}\left(\rho P_{F}H_{fo}(f)+1\right)df}$ (41)
$\displaystyle=\lim_{N\to\infty}N\tau T\Delta_{1}\leq 2\epsilon_{1}^{2},$ (42)

where $\Delta_{1}=\rho P_{F}-\int_{0}^{\frac{1}{\tau T}}{\text{log}\left(\rho
P_{F}H_{fo}(f)+1\right)df}$ and $\rho=\frac{|h_{aw}|^{2}}{\sigma_{w}^{2}}$. To
be noticed, (41) to (42) is according to equation (29) in [3].

Similarly, (39) with $N\to\infty$ can be rewritten as

$\lim_{N\to\infty}\mathcal{D}_{2}=N\tau T\Delta_{2},$ (43)

where $\Delta_{2}=\int_{0}^{\frac{1}{T}}{\left(\rho
P_{N}T-\text{log}\left(\rho P_{N}T+1\right)\right)df}$. Since
$\mathcal{D}_{1}$ and $\mathcal{D}_{2}$ are both monotonically increasing with
respect to the transmit power, $P_{F}\in(0,P_{F}^{max}]$ and
$P_{N}\in(0,P_{N}^{max}]$, respectively, where
$P_{F}^{max}=\mathcal{D}_{1}^{-1}(\epsilon_{1})$ and
$P_{N}^{max}=\mathcal{D}_{2}^{-1}(\epsilon_{2})$. Then, we have

$\lim_{N\to\infty}\frac{\text{max}\\{\mathcal{D}_{1}\\}}{\text{max}\\{\mathcal{D}_{2}\\}}=\frac{\Delta_{1}(P_{F}^{max},\tau)}{\Delta_{2}(P_{N}^{max})}=\frac{\epsilon_{1}^{2}}{\epsilon_{2}^{2}}.$
(44)

It is easy to know that $\Delta_{1}=\Delta_{2}$ when $P_{F}=P_{N}$ and
$\tau=1$. Meanwhile, it is worth noting that $\int_{0}^{\frac{1}{\tau
T}}{\text{log}\left(\rho P_{F}H_{fo}(f)+1\right)df}$ in $\Delta_{1}$ is
closely related to (36), which means $\Delta_{1}$ increases as $\tau$
increases for $\tau\geq 1/(1+\alpha)$ and stays the same for
$\tau<1/(1+\alpha)$.

Therefore, when $\epsilon$ is given, i.e., $\epsilon_{1}=\epsilon_{2}$ and
$\tau<1$, we have $P_{F}^{max}>P_{N}^{max}$.

In addition, for the same transmit power, $R_{FTN}>R_{N}$ has been proved in
[2]. Since $R_{FTN}$ and $R_{N}$ are both monotonically increasing with
respect to the transmit power, we have $R_{FTN}>R_{N}$ for a given $\epsilon$.
$\hfill\blacksquare$

Although the superiority of FTN signaling over Nyquist signaling in covert
communications is proved for $N\to\infty$, the conclusion still holds for a
practical value of $N$, which will be shown in section V.

Correspondingly, from a long-term perspective, the ergodic covert rate can be
represented as

$\displaystyle\max_{P}~{}E_{|h_{ab}|^{2}}[R_{FTN}]$ (45)
$\displaystyle\text{s.t.}~{}E_{|h_{aw}|^{2}}[\xi^{min}]\geq
1-\epsilon~{}\text{or}~{}E_{|h_{aw}|^{2}}[\mathcal{D}(p_{1}||p_{0})\leq
2\epsilon^{2}].$

Recall _Theorem 3_ , FTN singaling has a higher instantaneous covert rate than
the Nyquist signaling for specific $h_{aw}$ and $h_{ab}$. It is
straightforward that the ergodic covert rate of FTN signaling is higher than
that of the Nyquist signaling. However, it is troublesome to analysis the
advantage of using FTN signaling in covert communications over the Nyquist
signaling under the Bayesian criterion-based covertness constraint directly.
Therefore, the covert performance of using FTN signaling in the block fading
channel will also be evaluated in the next section.

##  V Numerical Results

In this section, we evaluate the performance of the covert communication link
under FTN signaling via the simulation. The parameter settings are listed in
Table I.

TABLE I: The parameter settings for the simulation. Parameter | Value
---|---
Shaped filter type | RRC
$\alpha$ | $0.3$
$T$ |  $1$ second
$\tau$ |  $0.5$ to $1$ with step of $0.05$
$\epsilon$ | $\\{0.01,0.05\\}$
$N^{\prime}$ | $\\{1000,2000,\cdots,9000,10000\\}$
$h_{ab}$, $h_{aw}$ |  $h_{ab}\sim\mathcal{CN}(0,1)$, $h_{aw}\sim\mathcal{CN}(0,1)$
$\sigma_{w}^{2}$ |  $1$ W
$N_{0}$ | 2 W/Hz
Figure 2: The maximum $P$ versus Nyquist intervals for $\epsilon=0.01$.

Figure 2 shows $P$ versus $N^{\prime}$ for $\epsilon=0.01$ under different
covertness constraints in the AWGN channel with the same detection time of
both schemes, where $N^{\prime}$ is the Nyquist intervals, i.e.,
$N=N^{\prime}$ and $N=\lceil N^{\prime}/\tau\rceil$ for Nyquist and FTN
signaling, respectively. Although _Theorem 3_ is proved for $N\to\infty$, the
advantages of FTN signaling over Nyquist signaling still exist for large
enough $N$. However, the gain of maximum $P$ of FTN increases slowly as the
decrease of $\tau$. Therefore, we also plot the maximum $P$ versus $\tau$ for
$N^{\prime}=5000$ and $\epsilon=0.01$ in Fig. 3, where $\tau=1$ corresponds to
the Nyquist signaling. It can be observed that the maximum $P$ increases as
$\tau$ decreases for $\tau\geq 1/(1+\alpha)=1/1.3$ and stays the same for
$\tau<1/1.3$, which is coincide with the proof of _Theorem 3_.

Figure 3: The maximum $P$ versus $\tau$ for $N^{\prime}=5000$ and
$\epsilon=0.01$.

Figure 4 shows the ergodic covert rate under FTN and Nyquist signaling versus
$N^{\prime}$ for $\epsilon=0.05$ in the Rayleigh block fading channel. As we
can see, the ergodic covert rate under FTN signaling is higher than that under
the Nyquist signaling for both covertness constraints.

Figure 4: The ergodic covert rate versus $N^{\prime}$ for $\epsilon=0.05$ in
the Rayleigh block fading channel.

##  VI Conclusion

In this letter, we have analyzed the performance of covert communications
under FTN signaling in the Rayleigh block fading channel. Both Bayesian
criterion- and KL divergence-based covertness constraints have been
considered. Especially, for KL divergence-based one, we have proved that the
maximum transmit power and covert rate were higher using FTN signaling than
those under Nyquist signaling. Numerical results have coincided with our
analysis and validated the advantages of FTN signaling to realize covert data
transmission.

## References

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