# Do the Outburst Properties of M31N 2008-12a Depend on the Time Since the
Previous Eruption?

William A. Burris Department of Astronomy, San Diego State University, 5500
Campanile Drive, San Diego, CA 92182, USA  Allen W. Shafter Department of
Astronomy, San Diego State University, 5500 Campanile Drive, San Diego, CA
92182, USA  Kamil Hornoch Astronomical Institute of the Czech Academy of
Sciences, Fričova 298, CZ-251 65 Ondřejov, Czech Republic

###### Abstract

Photometric observations spanning the UV to the near IR during the nine most
recent eruptions (2014–2022) of the extragalactic nova M31N 2008-12a are
presented and analyzed in order to explore whether the lightcurve properties
for a given eruption, specifically the peak magnitudes and fade rates, are
correlated with the time interval since the previous eruption. No significant
correlation between the pre-eruption interval and the rate of decline was
found, however it appears that the brightness at the peak of an outburst may
be positively correlated with the time interval since the previous eruption.

Andromeda Galaxy (39) – Novae (1127) – Recurrent Novae (1366)

††journal: (and accepted) RNAAS††facilities: Swift:
https://swift.gsfc.nasa.gov/archive/

##  1 Introduction

M31N 2008-12a was discovered by Koichi Nishiyama and Fujio Kabashima, on 2008
Dec. 26.48 UT in the outskirts of
M31111http://www.cbat.eps.harvard.edu/iau/CBAT_M31.html#2008-12a. After the
object was seen again in the fall of 2011 and 2012, Shafter et al. (2012)
speculated that the object was either a recurrent nova or a slow nova
undergoing multiple rebrightenings. When the object erupted again in the fall
of 2013, the recurrent nova nature of the object became clear Darnley et al.
(2014); Tang et al. (2014). Based on timings of the 15 eruptions observed
every year since 2008, the mean recurrence time is $364.18\pm 2.18$ d, or
$0.997\pm 0.006$ yr, which is the shortest of any known nova.

The $\sim 1$ yr recurrence time offers an exceptional opportunity to study how
the properties of a nova progenitor affects the outburst behavior. In the case
of 2008-12a, the short recurrence time requires that the white dwarf in the
system must be near the Chandrashekhar mass and accreting at a rate a few
times $10^{-7}~{}M_{\odot}$ yr-1 (e.g., Kato et al., 2014).

Starting in 2014 a worldwide campaign to monitor 2008-12a has produced a
wealth of photometric data (e.g., see Darnley & Henze, 2020, and references
therein). Here, we analyze lightcurve data from 2014 to the most recent
eruption in 2022. Our primary goal is to explore whether properties of the UV
and optical lightcurves are affected by the time since the previous eruption
($t_{\mathrm{pre}}$).

Figure 1: Top Left: The peak optical magnitudes reached by 2008-12a as a
function of $t_{\mathrm{pre}}$. For values of $t_{\mathrm{pre}}$ representing
multiple eruptions, the average peak magnitude is shown. A likely trend of
peak brightness with $t_{\mathrm{pre}}$ is evident. The $B$ peak magnitude for
the 2016 eruption, which was obtained $\sim$5 hr after the other filters, may
have faded somewhat during that interval. Bottom Left: Peak magnitude as a
function of $t_{\mathrm{pre}}$ for the Swift UVOT bandpasses. No clear trend
is observed, likely because the data were typically obtained after optical
identification. Top (Bottom) Right: The $t_{2}$ time as a function of
$t_{\mathrm{pre}}$ for the optical (UVOT) bandpasses. In both cases no
correlation between $t_{2}$ time and $t_{\mathrm{pre}}$ is observed. (See
Appendix for DBF)

##  2 Photometric Data

Optical (Johnson $B,V,R,I$ and Sloan $r$) and UVOT (UVW1 and UVW2) photometry
for eruptions from 2014 to 2022 have been analyzed to determine lightcurve
parameters. The optical data were gleaned from the literature, while the UVOT
photometry was obtained from the Swift archive. The UVOT data were reanalyzed
using the HEASoft tool uvotmaghist to achieve the largest temporal coverage
available, while using smaller apertures ($3^{\prime\prime}$ instead of the
$5^{\prime\prime}$ used in the default Swift calibration) and multiple
background regions to maximize the signal-to-noise. Unfortunately, the UVOT
data were not always obtained using the same filters making direct comparisons
between the various eruptions difficult. Both the UVW1
($\lambda_{\mathrm{eff}}\sim 2600$ Å) and UVW2 ($\lambda_{\mathrm{eff}}\sim
1928$ Å) filters were used in 2014, while in subsequent years only one filter
was employed: UVW1 in 2015, and UVW2 from 2016 onwards.

##  3 LightCurve Parameters

Linear least-square fits to $\sim$2 mag below peak were performed on the rise
(when observed) and fall from maximum light. When the rise was observed, the
intersection of the two linear fits was taken to be the time of maximum light,
and the extrapolated magnitude at that time was taken to be the peak
magnitude. When the rise was not captured, we took the brightest observed
point to represent maximum light222Formally, this represents a lower limit to
the actual peak magnitude, but due to the sustained monitoring around the
world, observations were typically made within hours of discovery, and thus
likely close to peak brightness.. In all cases, the rate of decline from
maximum light was used to determine the corresponding $t_{2}$ times (the time
to decline by 2 mag from maximum light) for each eruption. There were some
years where only a few data points in a particular filter were measured, not
enough to measure a meaningful $t_{2}$ time. In these cases, we also adopted
the brightest measured magnitude as maximum light.

The peak magnitudes and $t_{2}$ times are shown in Figure 1 where we have
plotted the parameters as a function of the time interval since previous
eruption, $t_{\mathrm{pre}}$. Given that $t_{\mathrm{pre}}$ was (remarkably!)
nearly identical for several epochs, we averaged the maximum magnitudes and
$t_{2}$ times for those pre-eruption intervals. This process resulted in five
mean intervals of $\langle t_{\mathrm{pre}}\rangle$ = 310.49 d (2014/2018),
329.83 d (2015), 362.34 d (2019/2020), 382.09 d (2017/2021/2022), and 471.34 d
(2016).

No correlation is apparent between the $t_{2}$ and $t_{\mathrm{pre}}$ times,
however we do find tentative evidence for a correlation between the peak
brightness and $t_{\mathrm{pre}}$. The most compelling evidence is provided by
the 2016 eruption, which had the longest pre-eruption interval of
$t_{\mathrm{pre}}=471.34$ d. In 4 out of the 5 optical filters, and in the
UVW2 filter, the peak brightness of the 2016 eruption is largest seen, even
though the peak was not captured in any filter that year. On the other hand,
the 2014 and 2018 eruptions, which had the shortest pre-eruption mean interval
($\langle t_{\mathrm{pre}}\rangle=310.49$ d), were the faintest of the nine
most recent eruptions (in all but the $I$ and $R$ filters), although the 2018
lightcurve morphology was somewhat complicated exhibiting a double-peaked
maximum: a dip after the original peak followed by a short resurgence. It is
unclear whether the double-peak structure of the outburst is related to the
short pre-eruption interval, but it will be interesting to see if 2008-12a
exhibits similar behavior during future eruptions with short
$t_{\mathrm{pre}}$. There is no obvious double-peaked structure in the 2014
eruption, but the data are limited.

##  4 Conclusions

Our analysis has failed to reveal any obvious relationship between the $t_{2}$
time and $t_{\mathrm{pre}}$. However, we did find tentative evidence that the
peak optical brightness reached by M31N 2008-12a increases with the time since
the previous eruption. The trend is not obvious in the UV likely because the
Swift observations were always triggered after the eruption was discovered in
the optical giving the nova more time to fade before the UV photometry could
be performed.

The accreted mass required to trigger an eruption depends on $M_{\mathrm{WD}}$
and the mean accretion rate onto the white dwarf’s surface (e.g., Townsley &
Bildsten, 2005). A longer $t_{\mathrm{pre}}$ implies a lower $\langle
dM/dt\rangle$ between eruptions. The observed trend of peak brightness with
$t_{\mathrm{pre}}$ may be the result of increased degeneracy in the accreted
layer that would be expected for this slightly lower accretion rate. Continued
monitoring of 2008-12a during future eruptions will be required in order to
confirm this finding, and if warranted, to explore its origin.

This work has been supported by NASA grant 80NSSC20K0547 (AWS) and by the
project RVO:67985815 (KH). We thank K. Page for her advice on the Swift
reductions.

## References

  * Darnley & Henze (2020) Darnley, M. J., & Henze, M. 2020, Advances in Space Research, 66, 1147, doi: 10.1016/j.asr.2019.09.044
  * Darnley et al. (2014) Darnley, M. J., Williams, S. C., Bode, M. F., et al. 2014, A&A, 563, L9, doi: 10.1051/0004-6361/201423411
  * Kato et al. (2014) Kato, M., Saio, H., Hachisu, I., & Nomoto, K. 2014, ApJ, 793, 136, doi: 10.1088/0004-637X/793/2/136
  * Shafter et al. (2012) Shafter, A. W., Hornoch, K., Ciardullo, J. V. R., Darnley, M. J., & Bode, M. F. 2012, The Astronomer’s Telegram, 4503, 1
  * Tang et al. (2014) Tang, S., Bildsten, L., Wolf, W. M., et al. 2014, ApJ, 786, 61, doi: 10.1088/0004-637X/786/1/61
  * Townsley & Bildsten (2005) Townsley, D. M., & Bildsten, L. 2005, ApJ, 628, 395, doi: 10.1086/430594

Table A1: M31N 2009-12a Lightcurve Parameters (Figure 1) $t_{\mathrm{pre}}$ (d) | Mag |  $t_{2}$ (d) | Filter | Sourceaa 1: ATel #12181, 2: ATel #12190, 3: ATel #12200, 4: ATel #12205, 5: ATel #12203, 6: ATel #6543, 7: ATel #6546, 8: ATel #6535, 9: ATel #7965, 10: ATel #7967, 11: ATel #7969, 12: ATel #7980, 13: ATel #7984, 14: ATel #8029, 15: ATel #8038, 16: ATel #7974, 17: ATel #7976, 18: ATel #7979, 19: ATel #14130, 20: ATel #14131, 21: ATel #13279, 22: ATel #13281, 23: ATel #13302, 24: ATel #15037, 25: ATel #15039, 26: ATel #15068, 27: ATel #15038, 28: ATel #15797, 29: ATel #15786, 30: ATel #15788, 31: ATel #15789, 32: ATel #15795, 33: ATel #15802, 34: ATel #15902, 35: ATel #11118, 36: ATel #11125, 37: ATel #11126, 38: ATel #11124, 39: ATel #9864, 40: ATel #9861, 41: ATel #9857, 42: ATel #9881.
---|---|---|---|---
310.35 | $17.20\pm 0.07$ | $3.96\pm 0.51$ | $UVW2$ | This work
310.35 | $18.83\pm 0.01$ | $\dots$ | $B$ | 1,2,3,4
310.35 | $18.79\pm 0.01$ | $3.18\pm 0.73$ | $V$ | 1,2,3,4
310.35 | $18.35\pm 0.08$ | $2.49\pm 0.34$ | $R$ | 1,2,3,4
310.35 | $18.66\pm 0.02$ | $2.60\pm 0.44$ | $r^{\prime}$ | 4,5
310.35 | $18.21\pm 0.05$ | $\dots$ | $I$ | 1,2,3,4
310.64 | $18.78\pm 0.11$ | $2.81\pm 0.62$ | $UVW1$ | This work
310.64 | $16.81\pm 0.06$ | $3.45\pm 0.19$ | $UVW2$ | This work
310.64 | $18.65\pm 0.15$ | $\dots$ | $B$ | 6,7
310.64 | $18.52\pm 0.10$ | $\dots$ | $V$ | 6,7
310.64 | $18.24\pm 0.10$ | $3.42\pm 0.40$ | $R$ | 6,7
310.64 | $18.47\pm 0.01$ | $\dots$ | $r^{\prime}$ | 8
310.64 | $18.24\pm 0.11$ | $\dots$ | $I$ | 6,7
329.83 | $16.91\pm 0.04$ | $2.44\pm 0.11$ | $UVW1$ | This work
329.83 | $18.70\pm 0.01$ | $2.20\pm 0.17$ | $B$ | 9,10,11,12,13,14,15
329.83 | $18.60\pm 0.03$ | $2.21\pm 0.22$ | $V$ | 9,10,11,12,13,14,15,16,17,18
329.83 | $18.32\pm 0.02$ | $1.96\pm 0.12$ | $R$ | 10,11,17
329.83 | $18.50\pm 0.02$ | $\dots$ | $r^{\prime}$ | 9,12,13
329.83 | $18.50\pm 0.08$ | $1.34\pm 0.24$ | $I$ | 10,11,17
359.37 | $17.46\pm 0.14$ | $3.71\pm 0.71$ | $UVW2$ | This work
359.37 | $18.38\pm 0.08$ | $\dots$ | $R$ | This work
359.37 | $18.29\pm 0.07$ | $\dots$ | $r^{\prime}$ | 19,20
359.37 | $18.15\pm 0.10$ | $\dots$ | $I$ | This work
365.31 | $17.05\pm 0.08$ | $3.32\pm 0.18$ | $UVW2$ | This work
365.31 | $18.34\pm 0.16$ | $2.47\pm 0.72$ | $B$ | 21,22,23
365.31 | $18.35\pm 0.05$ | $2.35\pm 0.33$ | $V$ | 21,22,23
365.31 | $18.02\pm 0.05$ | $2.33\pm 0.63$ | $R$ | 21,22,23
365.31 | $18.18\pm 0.03$ | $\dots$ | $I$ | 21,23
379.18 | $17.69\pm 0.09$ | $5.42\pm 0.60$ | $UVW2$ | This work
379.18 | $18.44\pm 0.05$ | $\dots$ | $B$ | 24
379.18 | $18.39\pm 0.06$ | $3.22\pm 0.34$ | $V$ | 24,25,26
379.18 | $18.25\pm 0.04$ | $2.40\pm 0.07$ | $R$ | This work
379.18 | $18.28\pm 0.06$ | $\dots$ | $r^{\prime}$ | 24,25,26,27
383.16 | $18.02\pm 0.09$ | $4.11\pm 0.46$ | $UVW2$ | This work
383.16 | $18.52\pm 0.06$ | $3.19\pm 0.17$ | $B$ | 28
383.16 | $18.35\pm 0.27$ | $3.12\pm 0.20$ | $V$ | 28
383.16 | $18.26\pm 0.01$ | $6.21\pm 0.33$ | $R$ | 28
383.16 | $18.23\pm 0.10$ | $3.02\pm 0.63$ | $r^{\prime}$ | 29,30,31,32,33,34
383.98 | $17.92\pm 0.09$ | $6.08\pm 0.50$ | $UVW2$ | This work
383.98 | $18.40\pm 0.25$ | $2.31\pm 0.26$ | $B$ | 35,36,37
383.98 | $18.06\pm 0.09$ | $1.83\pm 0.09$ | $V$ | 35,36,37,38
383.98 | $18.18\pm 0.09$ | $4.44\pm 0.38$ | $R$ | 35,36,37,38
383.98 | $18.21\pm 0.10$ | $\dots$ | $I$ | 36,37,38
471.37 | $16.51\pm 0.06$ | $3.56\pm 0.24$ | $UVW2$ | This work
471.37 | $18.50\pm 0.10$ | $2.38\pm 0.04$ | $B$ | 39
471.37 | $17.95\pm 0.09$ | $1.77\pm 0.61$ | $V$ | 39,40
471.37 | $17.76\pm 0.05$ | $2.46\pm 0.18$ | $R$ | 39,40,41
471.37 | $17.98\pm 0.04$ | $2.11\pm 0.07$ | $r^{\prime}$ | 40,41,42
471.37 | $17.68\pm 0.08$ | $2.50\pm 0.21$ | $I$ | 39,40


# Movable Frequency Diverse Array for Wireless Communication Security

Zihao Cheng, Jiangbo Si, _Member, IEEE_ , Zan Li, _Senior Member, IEEE_ ,
Pengpeng Liu, Yangchao Huang and Naofal Al-Dhahir, _Fellow, IEEE_

###### Abstract

Frequency diverse array (FDA) is a promising antenna technology to achieve
physical layer security by varying the frequency of each antenna at the
transmitter. However, when the channels of the legitimate user and
eavesdropper are highly correlated, FDA is limited by the frequency constraint
and cannot provide satisfactory security performance. In this paper, we
propose a novel movable FDA (MFDA) antenna technology where the positions of
antennas can be dynamically adjusted in a given finite region. Specifically,
we aim to maximize the secrecy capacity by jointly optimizing the antenna
beamforming vector, antenna frequency vector and antenna position vector. To
solve this non-convex optimization problem with coupled variables, we develop
a two-stage alternating optimization (AO) algorithm based on block successive
upper-bound minimization (BSUM) method. Moreover, to evaluate the security
performance provided by MFDA, we introduce two benchmark schemes, i.e., phased
array (PA) and FDA. Simulation results demonstrate that MFDA can significantly
enhance security performance compared to PA and FDA. In particular, when the
frequency constraint is strict, MFDA can further increase the secrecy capacity
by adjusting the positions of antennas instead of the frequencies.

###### Index Terms:

Frequency diverse array, Movable antenna, Phased array, Physical layer
security.

##  I Introduction

Frequency diverse array (FDA) is a promising antenna technology used at the
multi-antenna transmitter in wireless communization systems [1, 2, 3, 4, 5].
Compared with conventional phased array (PA), FDA introduces a small frequency
increment across array elements to yield distinct accumulating phase lags
among array antennas and forms a special beampattern related to distance and
angle. By tuning the antenna frequency vector (AFV) at the transmitter, FDA
can enable the transmitted signals to constructively superimpose at the
desired positions and destructively cancel at the undesired positions
simultaneously [6, 7, 8]. Especially in covert communication, FDA can achieve
perfect covertness by optimizing AFV with the instantaneous channel state
information (CSI) of the warden [9, 10, 11]. Moreover, FDA develops an angle-
range dependent beampattern which helps decouple the correlated channels to
enhance physical layer security [12, 13, 14]. Specifically, by optimizing AFV,
the received signal levels of the legitimate user and eavesdropper can be
distinguished clearly. When facing multi-eavesdropper scenario, a larger
frequency increment is needed to decouple the correlated channels [15, 16].
However, the frequency increment is usually assumed to be much smaller than
the carrier frequency due to receiver restriction which greatly limits FDA
performance. For instance, when the legitimate and illegitimate users are
proximal, such that their channels are highly correlated, the frequency
increment needs to be greater than the carrier frequency to completely
decouple their channels [12, 11].

It is worth mentioning that both FDA and PA are fixed-distance arrays which
means that the distance between each antenna is unchangeable and no smaller
than a half-wavelength [17, 18, 19]. Recently, movable antenna (MA) technology
was proposed to achieve better communication performance where the antennas at
the transmitter were movable within a confined region [20, 21, 22, 23, 24, 25,
26, 27]. By smartly designing the antenna positions vector (APV), MA can
achieve higher spatial diversity gains by employing much fewer antennas
compared with the conventional antenna selection (AS) technology [20, 21, 22].
Specially, with known global instantaneous CSI, MA can achieve full array gain
with null steering [23]. Moreover, MA is capable of mitigating the
interference from an adversary when MA is deployed at the receiver [28]. In a
multi-user and multi-interference scenario, MA can move in a three-dimensional
(3D) region to obtain higher rate gain from the users and reduce the impact
from different interference sources simultaneously. Besides, MA has great
flexibility which can be deployed in one-dimensional (1D) line, two-
dimensional (2D) surface and even six-dimensional (6D) region [24, 25, 26,
27]. In summary, MA outperforms fixed-distance antenna array and can be widely
utilized in physical layer security.

In light of the above considerations, we propose a novel antenna technology,
namely movable frequency diverse array (MFDA), which can further exploit the
spatial degrees of freedom (DoFs) in transmit antennas based on FDA. Our main
contributions are summarized as follows:

  * 1.

MFDA-aided multi-input single-output (MISO) wireless communication system is
firstly investigated in this paper. Different from FDA, MFDA can not only
adjust the frequency of each antenna but also adjust the position, such that
it achieves two-dimensional security. Specifically, we jointly optimize the
antenna beamforming vector (ABV), AFV and APV at the transmitter to maximize
the secrecy capacity subject to the given constraints. Moreover, we
investigate the influences of imperfect CSI at the eavesdropper on security
performance.

  * 2.

Since the formulated optimization problem is non-convex and has multiple
coupled variables, we first reformulate it into a more tractable form. Then,
we propose a low-complexity two-stage alternating optimization (AO) algorithm
to solve this problem. Specifically, in the first stage, we alternately
optimize AFV and APV by utilizing the block successive upper-bound
minimization (BSUM) method. In the second stage, with optimized AFV and APV,
the optimal ABV expression can be obtained in closed form.

  * 3.

In the case of imperfect CSI, we assume that the exact coordinate of the
eavesdropper is unknown, but the suspicious region of the eavesdropper is
known. We first divide the region into multiple discrete samples and consider
the worst sample. Then, the proposed two-stage AO algorithm is utilized to
solve the problem. Different from the case of perfect CSI, the block
coordinate descent (BCD) method replaces the BSUM to optimize AFV and APV in
the first stage, and the AO and CVX method is utilized to optimize ABV in the
second stage.

  * 4.

MFDA can achieve better security performance than FDA and PA. Specifically,
when the channels of the legitimate user and eavesdropper are highly
correlated, FDA is limited by the frequency constraint and cannot provide
satisfactory performance. By contrast, MFDA can adjust the positions of
antennas instead of the frequencies to further improve the performance.

The rest of this paper is organized as follows. Section II describes the MFDA-
aided MISO system model and formulates the optimization problem. Section III
develops a two-stage AO algorithm to solve the problem and investigates the
influences of beamformer refresh frequency on secrecy capacity. Section VI
considers the case of imperfect CSI at the eavesdropper. Numerical results are
presented and discussed in Section V, and conclusions are drawn in Section IV.

_Notations_ : Boldface uppercase and lowercase letters denote matrices and
vectors, respectively. Italic letters denote scalars; ${\bf{x}}^{\dagger}$
denotes the Hermitian of a vector $\bf{x}$. For a given matrix $\bf{A}$, we
denote its transpose, Hermitian and inverse by ${\bf{A}}^{T}$,
${\bf{A}}^{{\dagger}}$ and ${\bf{A}}^{-1}$, respectively. Denote
${\left\|.\right\|^{2}}$ as ${\ell_{2}}$-norm. $\mathbb{C}^{M\times N}$
denotes the complex $M\times N$ matrices.
$x\sim{\cal{CN}}\left({\bf{\Phi},\bf{\Omega}}\right)$ denotes the circular
symmetric complex Gaussian random vector with mean vector $\bf{\Phi}$ and
covariance matrix $\bf{\Omega}$. $\left\langle{\cdot,\cdot}\right\rangle$
denotes the inner product. ${\bf{tr}}\left(\cdot\right)$ is trace operator of
a matrix. $\left\lfloor\cdot\right\rfloor$ and $\left\lceil\cdot\right\rceil$
are the nearest integer towards $-\infty$ and $+\infty$, respectively.
$c=a{\kern 1.0pt}\bmod{\kern 1.0pt}b$ means $c$ is the remainder of integer
$a$ divided by integer $b$.

##  II System Model

Figure 1: MFDA-aided MISO wireless communication system model

###  II-A System Description and Channel Model

In this paper, we consider a MISO wireless communication system where 1D
linear MFDA is deployed to assist in the communication from the legitimate
user Alice to Bob, while preventing being eavesdropped by the illegitimate
user Eve. The number of antennas at Alice is denoted by $M$, and both Bob and
Eve are assumed to be equipped with a single antenna. As shown in Fig. 1(a), a
smart MFDA controller helps adjust APV
${\bf{x}}\buildrel\Delta\over{=}{\left[{{x_{1}},...,{x_{M}}}\right]}\in{{\mathbb{C}}^{1\times
M}}$ and AFV
${\bf{f}}\buildrel\Delta\over{=}{\left[{{f_{1}},...,{f_{M}}}\right]}\in{{\mathbb{C}}^{1\times
M}}$ at Alice via a separate wireless link. Compared with PA, MFDA introduces
small frequency increments across the array antennas, such that the frequency
of the $m$-th antenna is given by

$\displaystyle\ {f_{m}}={f_{C}}+\Delta{f_{m}},m=1,...,M,$ (1)

where $f_{C}$ is the carrier frequency and $\Delta{f_{m}}$ is the introduced
frequency increment. We assume that $0\leq\Delta{f_{m}}\leq\Delta F$ and the
frequency increment is much smaller than the carrier frequency $\Delta
F\ll{f_{C}}$.

Without loss of generality, we define the first antenna of Alice as the origin
of the coordinate. Thus, the coordinates of Bob and Eve are denoted by
$\left({{r_{b}},{\theta_{b}}}\right)$ and
$\left({{r_{e}},{\theta_{e}}}\right)$, respectively. To characterize the
theoretical performance gain brought by the MFDA, we assume that
$\left({{r_{b}},{\theta_{b}}}\right)$ and
$\left({{r_{e}},{\theta_{e}}}\right)$ are perfectly known by Alice and all
channels are line-of-sight (LoS) channels as in [12, 13, 14]. In addition, the
channels of Alice$\to$Bob and Alice$\to$Eve are defined as

$\displaystyle\
{{\bf{h}}_{ab}}\left({t,{r_{b}},{\theta_{b}}}\right)={\rm{Lfs}}\left({{r_{b}}}\right){\left[{{h_{1}}\left({t,{r_{b}},{\theta_{b}}}\right),...,{h_{M}}\left({t,{r_{b}},{\theta_{b}}}\right)}\right]^{T}}$
(2)

and

$\displaystyle\
{{\bf{h}}_{ae}}\left({t,{r_{e}},{\theta_{e}}}\right)={\rm{Lfs}}\left({{r_{e}}}\right){\left[{{h_{1}}\left({t,{r_{e}},{\theta_{e}}}\right),...,{h_{M}}\left({t,{r_{e}},{\theta_{e}}}\right)}\right]^{T}},$
(3)

where
${h_{m}}\left({t,{r_{b}},{\theta_{b}}}\right)={e^{-j2\pi{f_{m}}\left[{t-\frac{{{r_{b}}-{x_{m}}\sin{\theta_{b}}}}{c}}\right]}}$
and
${h_{m}}\left({t,{r_{e}},{\theta_{e}}}\right)={e^{-j2\pi{f_{m}}\left[{t-\frac{{{r_{e}}-{x_{m}}\sin{\theta_{e}}}}{c}}\right]}}$,
$\forall m$. $c$ is the speed of light and ${\rm{Lfs}}\left(\cdot\right)$ is
the path loss factor.

###  II-B Problem Formulation

We observe from (2) and (3) that the channels of MFDA are dependent on time
$t$. Thus, the ABV at Alice is assumed to be time-varying to adapt to the
variable channels which can be defined as

$\displaystyle\
{\bf{w}}\left(t\right)={\left[{{w_{1}}\left(t\right),...,{w_{M}}\left(t\right)}\right]^{T}}.$
(4)

Then, the received signals at Bob and Eve can be written as

$\displaystyle\
{{\bf{y}}_{b}}\left(t\right)={\bf{h}}_{ab}^{H}\left({t,{r_{b}},{\theta_{b}}}\right){\bf{w}}\left(t\right)+{{\bf{n}}_{b}}$
(5)

and

$\displaystyle\
{{\bf{y}}_{e}}\left(t\right)={\bf{h}}_{ae}^{H}\left({t,{r_{e}},{\theta_{e}}}\right){\bf{w}}\left(t\right)+{{\bf{n}}_{e}},$
(6)

where ${{\bf{n}}_{b}}$ and ${{\bf{n}}_{e}}$ are the additive white Gaussian
noise (AWGN) at Bob and Eve with mean zero and variances $\sigma_{b}^{2}$ and
$\sigma_{e}^{2}$, i.e.,
${{\bf{n}}_{b}}\sim{\cal{CN}}\left({0,\sigma_{b}^{2}}\right)$ and
${{\bf{n}}_{e}}\sim{\cal{CN}}\left({0,\sigma_{e}^{2}}\right)$. Thus, the
instantaneous achievable rates at Bob and Eve are given by

$\displaystyle\
{R_{b}}\left(t\right)=\log\left({1+\sigma_{b}^{-2}{{\left|{{\bf{h}}_{ab}^{H}\left({t+\frac{{{r_{b}}}}{c},{r_{b}},{\theta_{b}}}\right){\bf{w}}\left(t\right)}\right|}^{2}}}\right)$
(7)

and

$\displaystyle\
{R_{e}}\left(t\right)=\log\left({1+\sigma_{e}^{-2}{{\left|{{\bf{h}}_{ae}^{H}\left({t+\frac{{{r_{e}}}}{c},{r_{e}},{\theta_{e}}}\right){\bf{w}}\left(t\right)}\right|}^{2}}}\right).$
(8)

Next, the instantaneous secrecy capacity can be expressed as

$\displaystyle\ {C_{\sec}}\left(t\right)$
$\displaystyle={R_{b}}\left(t\right)-{R_{e}}\left(t\right)$
$\displaystyle=\log\left({\frac{{1+\sigma_{b}^{-2}{{\left|{{\bf{h}}_{ab}^{H}\left({t+\frac{{{r_{b}}}}{c},{r_{b}},{\theta_{b}}}\right){\bf{w}}\left(t\right)}\right|}^{2}}}}{{1+\sigma_{e}^{-2}{{\left|{{\bf{h}}_{ae}^{H}\left({t+\frac{{{r_{e}}}}{c},{r_{e}},{\theta_{e}}}\right){\bf{w}}\left(t\right)}\right|}^{2}}}}}\right).$
(9)

To improve the secrecy performance gain, we aim to maximize the instantaneous
secrecy capacity by jointly optimizing ABV, APV and AFV at Alice with the
perfect global CSI. The optimization problem can be formulated as follows:

$\displaystyle\mathop{\max}\limits_{\left\\{{{\bf{w}}\left(t\right),{\bf{x}},{\bf{f}}}\right\\}}{\rm{}}{C_{\sec}}\left(t\right)$
(10a)
$\displaystyle{s.t.}\quad{\left\|{\bf{w}}\left(t\right)\right\|^{2}}\leq{P_{max}},$
(10b) $\displaystyle\quad\quad{x_{1}}=0,{x_{M}}\leq{D_{\max}},$ (10c)
$\displaystyle\quad\quad{x_{m}}-{x_{m-1}}\geq{D_{0}},\forall m,$ (10d)
$\displaystyle\quad\quad{f_{C}}\leq{f_{m}}\leq{f_{C}}+\Delta F,\forall m,$
(10e)

where (10b) is the maximum power constraint and ${P_{max}}$ is the maximum
transmit power. (10c) is 1D linear range constraint which ensures that MFDA is
moved within the predetermined region $\left[{0,{D_{\max}}}\right]$ and
${{D_{\max}}}$ is the movable boundary of MFDA. (10d) aims to avoid antenna
coupling and ${D_{0}}$ is the minimum distance between any adjacent antennas
of MFDA. Besides, (10e) is the frequency constraint of MFDA.

##  III Proposed Algorithm for Optimization Problem

We observe that the optimization problem (10a) contains three optimized
variables which are are tightly coupled and make the problem challenging to
solve. Usually, AO algorithm is utilize to solve this problem [29, 30].
However, when there are multiple variables, the complexity of AO algorithm
greatly increases and the convergence accuracy decreases. To avoid these
problems, we propose a low-complexity two-stage AO algorithm to solve problem
(10).

###  III-A Optimizing ${\bf{w}}\left(t\right)$ with Given ${\bf{x}}$ and
${\bf{f}}$

According to (II-B), problem (10) can be reformulated as

$\displaystyle\mathop{\max}\limits_{{\bf{w}}\left(t\right)}{\rm{}}\frac{{1+{{\bf{w}}^{H}}\left(t\right){\bf{A}}{\bf{w}}\left(t\right)}}{{1+{{\bf{w}}^{H}}\left(t\right){\bf{B}}{\bf{w}}\left(t\right)}}$
(11a)
$\displaystyle{s.t.}\quad{{\bf{w}}^{H}}\left(t\right){\bf{w}}\left(t\right)\leq{P_{\max}},$
(11b)

where
${\bf{A}}=\frac{1}{{\sigma_{b}^{2}}}{{\bf{h}}_{ab}}\left({t+\frac{{{r_{b}}}}{c},{r_{b}},{\theta_{b}}}\right){\bf{h}}_{ab}^{H}\left({t+\frac{{{r_{b}}}}{c},{r_{b}},{\theta_{b}}}\right)$
and
${\bf{B}}=\frac{1}{{\sigma_{e}^{2}}}{{\bf{h}}_{ae}}\left({t+\frac{{{r_{e}}}}{c},{r_{e}},{\theta_{e}}}\right){\bf{h}}_{ae}^{H}\left({t+\frac{{{r_{e}}}}{c},{r_{e}},{\theta_{e}}}\right)$.
The optimal ABV is given by [31]

$\displaystyle\
{{\bf{w}}_{opt}}\left(t\right)=\sqrt{{P_{\max}}}{{\bf{u}}_{\max}},$ (12)

where ${{\bf{u}}_{\max}}$ is the normalized eigenvector corresponding to the
largest eigenvalue of the matrix
${\left({{\bf{B}}+P_{\max}^{-1}{{\bf{I}}_{M}}}\right)^{-1}}\left({{\bf{A}}+P_{\max}^{-1}{{\bf{I}}_{M}}}\right)$.

By substituting ${{\bf{w}}_{opt}}\left(t\right)$ into (II-B), we rewrite
problem (10) into a more tractable form as

$\displaystyle\mathop{\min}\limits_{\left\\{{{\bf{x}},{\bf{f}}}\right\\}}{\left|{\left\langle{{{\bf{h}}_{ab}}\left({t,{r_{b}},{\theta_{b}}}\right),{{\bf{h}}_{ae}}\left({t,{r_{e}},{\theta_{e}}}\right)}\right\rangle}\right|^{2}}$
(13a) $\displaystyle{s.t.}\quad{x_{1}}=0,{x_{M}}\leq{D_{\max}},$ (13b)
$\displaystyle\quad\quad{x_{m}}-{x_{m-1}}\geq{D_{0}},\forall m,$ (13c)
$\displaystyle\quad\quad{f_{C}}\leq{f_{m}}\leq{f_{C}}+\Delta F,\forall m.$
(13d)

The detailed reformulation proof is presented in Appendix A. According to (2)
and (3), we have

$\displaystyle{\left|{\left\langle{{{\bf{h}}_{ab}}\left({t,{r_{b}},{\theta_{b}}}\right),{{\bf{h}}_{ae}}\left({t,{r_{e}},{\theta_{e}}}\right)}\right\rangle}\right|^{2}}$
$\displaystyle={\left|{\sum\limits_{m=1}^{M}{{e^{j2\pi{f_{m}}\left({{x_{m}}\frac{{\sin{\theta_{e}}-\sin{\theta_{b}}}}{c}+\frac{{{r_{b}}-{r_{e}}}}{c}}\right)}}}}\right|^{2}}$
$\displaystyle=\sum\limits_{m=1}^{M}{\sum\limits_{n=1}^{M}{{e^{j2\pi\left({{f_{m}}{\tau_{m}}-{f_{n}}{\tau_{n}}}\right)}}}}$
$\displaystyle=M+{\sum\limits_{m=1}^{M}{{\sum\limits_{\scriptstyle{n=1}\hfill\atop\scriptstyle{n\neq
m}\hfill}^{M}{{\cos\left[{2\pi\left({{f_{m}}{\tau_{m}}-{f_{n}}{\tau_{n}}}\right)}\right]}}}}}.$
(14)

where
${\tau_{m}}={x_{m}}\frac{{\sin{\theta_{e}}-\sin{\theta_{b}}}}{c}+\frac{{{r_{b}}-{r_{e}}}}{c}$.
Next, problem (13) can be reformulated as

$\displaystyle\mathop{\min}\limits_{\left\\{{{\bf{x}},{\bf{f}}}\right\\}}{\sum\limits_{m=1}^{M}{{\sum\limits_{\scriptstyle{n=1}\hfill\atop\scriptstyle{n\neq
m}\hfill}^{M}{\cos\left[{2\pi\left({{f_{m}}{\tau_{m}}-{f_{n}}{\tau_{n}}}\right)}\right]}}}}$
(15a) $\displaystyle{s.t.}\quad{x_{1}}=0,{x_{M}}\leq{D_{\max}},$ (15b)
$\displaystyle\quad\quad{x_{m}}-{x_{m-1}}\geq{D_{0}},\forall m,$ (15c)
$\displaystyle\quad\quad{f_{C}}\leq{f_{m}}\leq{f_{C}}+\Delta F,\forall m.$
(15d)

_Remark 1:_ After utilizing the optimal ABV in (12), (13a) reduces to
minimizing the inner product between
${{\bf{h}}_{ab}}\left({t,{r_{b}},{\theta_{b}}}\right)$ and
${{\bf{h}}_{ae}}\left({t,{r_{e}},{\theta_{e}}}\right)$. This implies that the
adjusted APV and AFV aim to make the channels of Bob and Eve as orthogonal as
possible. In particular, the ideal case is
${{\bf{h}}_{ab}}\left({t,{r_{b}},{\theta_{b}}}\right)\perp{{\bf{h}}_{ae}}\left({t,{r_{e}},{\theta_{e}}}\right)$.
Unfortunately, due to the frequency constraint in (15d), the ideal case is
difficult to achieve [12]. By contrast, MFDA introduces a new dimension. When
the frequency constraint is strict, MFDA can optimize APV to further improve
the security performance instead of optimizing AFV.

_Remark 2:_ According to problem (15), it is clear that ABV cannot affect the
optimization of APV or AFV. It means that ABV is not coupled with APV and AFV,
but APV and AFV are coupled with each other. Thus, problem (10) can be recast
to a two-stage algorithm. Specifically, the first stage involves obtaining the
optimal APV and AFV in problem (15) using the AO algorithm. The second stage
is obtaining the optimal ABV by (12). Based on this fact, we propose a two-
stage AO algorithm in this paper. Moreover, it is worth mentioning that the
optimal APV and AFV are independent on time, but the optimal ABV is dependent
on time.

###  III-B Optimizing ${\bf{x}}$ with Given ${\bf{w}}\left(t\right)$ and
${\bf{f}}$

With given ${\bf{f}}$, problem (15) can be rewritten as

$\displaystyle\mathop{\min}\limits_{\bf{\tau}}{\sum\limits_{m=1}^{M}{{\sum\limits_{\scriptstyle{n=1}\hfill\atop\scriptstyle{n\neq
m}\hfill}^{M}\cos\left[{2\pi\left({{\tau_{m}}{f_{m}}-{\tau_{n}}{f_{n}}}\right)}\right]}}}$
(16a) $\displaystyle{s.t.}\quad{x_{1}}=0,{x_{M}}\leq{D_{\max}},$ (16b)
$\displaystyle\quad\quad{x_{m}}-{x_{m-1}}\geq{D_{0}},\forall m,$ (16c)

where ${\tau}=\left\\{{{\tau_{1,}}{\tau_{2,...,}}{\tau_{M}}}\right\\}$.
Problem (16a) is clearly non-convex, thus we resort to the BSUM method to
solve it. BSUM solves the problem by formulating a new convex function to
approach the optimization target. More specifically, in the $s$-th iteration
of BSUM, we first optimize ${{{\tau_{m}}}}$ where $m=s{\kern 1.0pt}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\rm{mod}}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}M$, and obtain the optimal position $\tau_{m}^{opt}$. Then, we update
${{\bf{\tau}}^{s-1}}$ with $\tau_{m}^{opt}$, i.e.,
${{\bf{\tau}}^{s}}={{\bf{\tau}}^{s-1}}$. The problem can be written as

$\displaystyle\mathop{\min}\limits_{{\tau_{m}}}\quad{{\sum\limits_{\scriptstyle{n=0}\hfill\atop\scriptstyle{n\neq
m}\hfill}^{M-1}{{\cos\left[{2\pi\left({{\tau_{m}}{f_{m}}-{\tau_{n}^{s-1}}{f_{n}}}\right)}\right]}}}}$
(17a) $\displaystyle{s.t.}\quad{x_{1}}=0,{x_{M}}\leq{D_{\max}},$ (17b)
$\displaystyle\quad\quad{x_{m}}-{x_{m-1}}\geq{D_{0}},\forall m.$ (17c)

Next, we define the non-convex function in (17a) as

$\displaystyle\
{y_{m,n}}\left({{x}}\right)=\cos\left[{2\pi\left({{x}{f_{m}}-{\tau_{n}^{s-1}}{f_{n}}}\right)}\right].$
(18)

Following the BSUM method, the objective function (18) is approximated by the
upper-bound quadratic function ${u_{m,n}}\left({{x}}\right)$, which is defined
by

$\displaystyle\
{u_{m,n}}\left({{x}}\right)={k_{m,n}}{\left({{x}-{\zeta_{m,n}}}\right)^{2}}+{\delta_{m,n}},$
(19)

where ${k_{m,n}}\in\mathbb{R}$, ${k_{m,n}}>0$, ${{\zeta_{m,n}}}\in\mathbb{R}$
and ${\delta_{m,n}}\in\mathbb{R}$ are the parameters of the new quadratic
function. For a given point ${\tau_{m}^{s-1}}$ in (18), the approximate
function (19) should satisfy the following constraints:

$\displaystyle\
\left\\{{\begin{array}[]{*{20}{l}}{{u_{m,n}}\left({\tau_{m}^{s-1};\tau_{n}^{s-1}}\right)={y_{m,n}}\left({\tau_{m}^{s-1};\tau_{n}^{s-1}}\right)}\\\
{u_{m,n}^{{}^{\prime}}\left({\tau_{m}^{s-1};\tau_{n}^{s-1}}\right)=y_{m,n}^{{}^{\prime}}\left({\tau_{m}^{s-1};x_{n}^{s-1}}\right)}\\\
{{u_{m,n}}\left({{\zeta_{m,n}};\tau_{n}^{s-1}}\right)\in\left\\{{1,-1}\right\\}},\end{array}}\right.$
(23)

Next, we can obtain the parameters of the new quadratic function in (19) by
utilizing Lemma 1.

_Lemma 1:_ If
$y_{m,n}^{{}^{\prime}}\left({\tau_{m}^{s-1};\tau_{n}^{s-1}}\right)\neq 0$, we
have

$\displaystyle\
\left\\{{\begin{array}[]{*{20}{l}}{{k_{m,n}}=\frac{{-\pi{f_{m}}\sin\left[{2\pi\left({\tau_{m}^{s-1}{f_{m}}-\tau_{n}^{s-1}{f_{n}}}\right)}\right]}}{{\tau_{m}^{s-1}-{\zeta_{m,n}}}}}\\\
{{\zeta_{m,n}}=\left\\{{\begin{array}[]{*{20}{l}}{\frac{{\left\lfloor{2\left({\tau_{m}^{s-1}{f_{m}}-\tau_{n}^{s-1}{f_{n}}}\right)}\right\rfloor}}{{2{f_{m}}}}+\frac{{\tau_{n}^{s-1}{f_{n}}}}{{{f_{m}}}}}\\\
{if\quad
y_{m,n}^{{}^{\prime}}\left({\tau_{m}^{s-1};\tau_{n}^{s-1}}\right)>0}\\\
{\frac{{\left\lceil{2\left({\tau_{m}^{s-1}{f_{m}}-\tau_{n}^{s-1}{f_{n}}}\right)}\right\rceil}}{{2{f_{m}}}}+\frac{{\tau_{n}^{s-1}{f_{n}}}}{{{f_{m}}}}}\\\
{if\quad
y_{m,n}^{{}^{\prime}}\left({\tau_{m}^{s-1};\tau_{n}^{s-1}}\right)<0}\end{array}}\right.}\\\
{\delta_{m,n}}=\cos\left[{2\pi\left({\tau_{m}^{s-1}{f_{m}}-\tau_{n}^{s-1}{f_{n}}}\right)}\right]\\\
\quad\quad-{k_{m,n}}{\left({\tau_{m}^{s-1}-{\zeta_{m,n}}}\right)^{2}}.\end{array}}\right.$
(32)

If $y_{m,n}^{{}^{\prime}}\left({\tau_{m}^{s-1};\tau_{n}^{s-1}}\right)=0$, we
have

$\displaystyle\
\left\\{{\begin{array}[]{*{20}{l}}{{k_{m,n}}=-2{\pi^{2}}f_{m}^{2}{y_{m,n}}\left({\tau_{m}^{s-1};\tau_{n}^{s-1}}\right)}\\\
{{\zeta_{m,n}}=\tau_{m}^{s-1}}\\\
{{\delta_{m,n}}={y_{m,n}}\left({\tau_{m}^{s-1};\tau_{n}^{s-1}}\right)\in\left\\{{-1,1}\right\\}}.\end{array}}\right.$
(36)

The proof of Lemma 1 is presented in Appendix B. Therefore problem (17a) can
be replaced by

$\displaystyle\mathop{\min}\limits_{{\tau_{m}}}\quad{{\sum\limits_{\scriptstyle{n=0}\hfill\atop\scriptstyle{n\neq
m}\hfill}^{M-1}{k_{m,n}}{\left({\tau_{m}-{\zeta_{m,n}}}\right)^{2}}+{\delta_{m,n}}}}$
(37a)

It is clear that the optimal solution of problem (37a) is given by

$\displaystyle\
\tau_{m}^{opt}=\frac{{{\sum\limits_{\scriptstyle{n=0}\hfill\atop\scriptstyle{n\neq
m}\hfill}^{M-1}{{{k_{m,n}}{\zeta_{m,n}}}}}}}{{{\sum\limits_{\scriptstyle{n=0}\hfill\atop\scriptstyle{n\neq
m}\hfill}^{M-1}{{{k_{m,n}}}}}}}.$ (38)

Thus, the optimal position of the $m$-th antenna is given by

$\displaystyle\
x_{m}^{opt}=\left({\tau_{m}^{opt}-\frac{{{r_{b}}-{r_{e}}}}{c}}\right){\left({\sin{\theta_{e}}-\sin{\theta_{b}}}\right)^{-1}}c.$
(39)

According to the constraints (58b) and (58c), the optimal position of the
$m$-th antenna is given by

$\displaystyle\ x_{m}^{opt}=\left\\{{\begin{array}[]{*{20}{l}}0&{m=1}\\\
{x_{m-1}^{s}+{D_{0}}}&{{x^{opt}}\leq x_{m-1}^{s}+{D_{0}}}\\\
{{x^{opt}}}&{{D_{1}}\geq{x^{opt}}>x_{m-1}^{s}+{D_{0}}}\\\
{{D_{1}}}&{{D_{1}}<{x^{opt}}.}\end{array}}\right.$ (44)

where ${D_{1}}={D_{\max}}-\left({M-m}\right){D_{0}}$. The convergence of the
BSUM is verified in Fig. 2.

###  III-C Optimizing ${\bf{f}}$ with Given ${\bf{w}}\left(t\right)$ and
${\bf{x}}$

With given ${\bf{w}}\left(t\right)$ and ${\bf{x}}$, the optimization problem
can be formulated as

$\displaystyle\mathop{\min}\limits_{\bf{f}}{\sum\limits_{m=1}^{M}{{\sum\limits_{\scriptstyle{n=1}\hfill\atop\scriptstyle{n\neq
m}\hfill}^{M}\cos\left[{2\pi\left({{\tau_{m}}{f_{m}}-{\tau_{n}}{f_{n}}}\right)}\right]}}}$
(45a) $\displaystyle{s.t.}\quad{f_{C}}\leq{f_{m}}\leq{f_{C}}+\Delta F,\forall
m.$ (45b)

Problem (26) is non-convex which can be solved by the BSUM method developed in
the previous subsection. Due to the similar mathematical derivation, we omit
the proof here for brevity. The optimal frequency of the $m$-th antenna is
given by

$\displaystyle\
f_{m}^{opt}=\left\\{{\begin{array}[]{*{20}{l}}{{f_{C}}}&{{f_{opt}}<{f_{C}}}\\\
{{f_{opt}}}&{{f_{C}}\leq{f_{opt}}<{f_{C}}+\Delta F}\\\ {{f_{C}}+\Delta
F}&{{f_{C}}+\Delta F\leq{f_{opt}}.}\end{array}}\right.$ (49)

where ${{f_{opt}}}$ can be obtained by Lemma 1.

###  III-D Overall Two-Stage AO Algorithm

Inspired by _Remark 1_ , we propose a two-stage AO algorithm with low
complexity. Specifically, the first stage aims to alternately optimize the APV
and AFV by utilizing the BSUM method which is summarized in Algorithm 1. The
second stage is to obtain the optimal ABV with the optimized APV and AFV. The
overall algorithm is summarized in Algorithm 2.

Algorithm 1 Optimizing ${\bf{x}}$ by BSUM

1: Initialization: Initializing ${\bf{f}}$ via Algorithm 1 .

2: Set: $s=0$.

3: Repeat:a) $s=s+1$, and $m=s{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\rm{mod}}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\rm{M}}$.b) The new
quadratic function ${u_{m,n}}\left({{x}}\right)$ can be obtained by Lemma 1.
Then, the optimal position of $m$-th antenna can be obtained by (44).c) We
update ${{\bf{x}}^{s-1}}$ with $x_{m}^{opt}$ and obtain ${{\bf{x}}^{s}}$.

4: Until
$\left|{{C_{\sec}}\left({{{\bf{w}}_{opt}},{{\bf{x}}^{s}},{\bf{f}}}\right)-{C_{\sec}}\left({{{\bf{w}}_{opt}},{{\bf{x}}^{s-1}},{\bf{f}}}\right)}\right|\leq\varsigma$
is satisfied where $\varsigma$ is a small convergence value.

Algorithm 2 Overall two-stage AO algorithm

1: Initialization: Initializing ${{{\bf{w}}_{opt}}}$ via (12).

2: Set: $i=0$.

3: Repeat:a) $i=i+1$.b) Optimizing ${\bf{x}}$ and ${\bf{f}}$ by BSUM.c)
Obtaining ${{{\bf{w}}_{opt}}}$ with the optimized ${{\bf{x}}^{s}}$ and
${{\bf{f}}^{s}}$.

4: Until
$\left|{{C_{\sec}}\left({{{\bf{w}}_{opt}},{{\bf{x}}^{s}},{{\bf{f}}^{s}}}\right)-{C_{\sec}}\left({{{\bf{w}}_{opt}},{{\bf{x}}^{s-1}},{{\bf{f}}^{s-1}}}\right)}\right|\leq\varsigma$
is satisfied.

The complexity of Algorithm 1 is ${\cal{O}}\left({{I_{BSUM}}}\right)$ where
${{I_{BSUM}}}$ denotes the number of iterations required for BSUM convergence.
The overall complexity of Algorithm 2 is
${\cal{O}}\left({{I_{AO}}\left({{I_{BSUM}}+{I_{BSUM}}}\right)}\right)$ where
${{I_{AO}}}$ denotes the number of iterations required for AO algorithm
convergence.

###  III-E Beamformer Refresh Frequency

According to (10a), the formulated optimization problem is an instantaneous
problem and the optimal solution is dependent on time. Besides, from _Remark
2_ , we know that the optimized AFV and APV are independent on time, and only
the optimized ABV is dependent on time. Based on these facts, the optimal
solution is related to the speed of ABV optimization or the so-called
beamformer refresh frequency (BRF) [12]. In particular, if BRF can keep up
with channel refresh frequency (CRF), the instantaneous secrecy capacity can
be maximized. However, due to the device restriction and implementation cost,
high beamformer refresh is difficult to achieve. Hence, it is necessary to
investigate the influences of BRF and CRF on security performance.

More specifically, we define BRF and CRF as $\frac{1}{{\Delta{T_{BRF}}}}$ and
$\frac{1}{{\Delta{T_{CRF}}}}$, respectively, where $T$ is defined as the whole
transmission time. Then, the channel refresh time points are given by
$t_{l}^{c}=\left({l-1}\right)\frac{1}{L},l=1,2,...,L$ where
$L=\frac{T}{{\Delta{T_{CRF}}}}$, and the beamformer refresh time points are
given by $t_{k}^{w}=\left({k-1}\right)\frac{1}{K},k=1,2,...,K$ where
$K=\frac{T}{{\Delta{T_{BRF}}}}$. Thus, the secrecy capacity is given by

$\displaystyle\ {C_{\sec}}$
$\displaystyle=\frac{1}{T}\int_{0}^{T-1}{{C_{\sec}}\left({{{\bf{w}}_{opt}}\left(t\right),{{\bf{x}}_{opt}},{{\bf{f}}_{opt}}}\right)dt}$
$\displaystyle=\frac{1}{{LK}}\sum\limits_{l=1}^{L}{\sum\limits_{k=1}^{K}{{{\log}_{2}}}}\left({\frac{{1+{{\bf{w}}^{H}}\left({t_{k}^{w}}\right){\bf{A}}\left({t_{l}^{c}}\right){\bf{w}}\left({t_{k}^{w}}\right)}}{{1+{{\bf{w}}^{H}}\left({t_{k}^{w}}\right){\bf{B}}\left({t_{l}^{c}}\right){\bf{w}}\left({t_{k}^{w}}\right)}}}\right).$
(50)

##  IV Imperfect CSI at Eve

In this section, we consider a more practical scenario where Alice has
imperfect CSI knowledge about Eve. According to (3), we conclude that the
imperfect CSI at Eve is mainly caused by an estimation error in Eve’s range
and angle. Without loss of generality, we assume that the exact coordinate of
Eve is unknown, but the suspicious region for Eve is known. The uncertainty
regions in Eve’s range and angle is denoted as
${\cal{R}}\buildrel\Delta\over{=}\left[{{r_{e}}-\Delta r,{r_{e}}+\Delta
r}\right]$ and
${\cal{A}}\buildrel\Delta\over{=}\left[{{\theta_{e}}-\Delta\theta,{\theta_{e}}+\Delta\theta}\right]$,
where ${\Delta r}$ and ${\Delta\theta}$ is the estimation error of the range
and angle, respectively. Since ${\cal{R}}$ and ${\cal{A}}$ are continuous
sets, we propose to uniformly generate some discrete samples over the sets
which are given as ${r_{y}}={r_{e}}-\Delta r+\left({y-1}\right)\frac{{2\Delta
r}}{{Y-1}},y\in\left\\{{1,2,...,Y}\right\\}$ and
${\theta_{z}}={\theta_{e}}-\Delta\theta+\left({z-1}\right)\frac{{2\Delta\theta}}{{Z-1}},z\in\left\\{{1,2,...,Z}\right\\}$.
The optimization problem can be formulated as

$\displaystyle\mathop{\max}\limits_{\left\\{{{\bf{w}}\left(t\right),{\bf{x}},{\bf{f}}}\right\\}}{\rm{}}\left\\{{{R_{b}}\left(t\right)-\mathop{\max}\limits_{Y,Z}{R_{{e_{y,z}}}}\left(t\right)}\right\\}$
(51a)
$\displaystyle{s.t.}\quad{\left\|{\bf{w}}\left(t\right)\right\|^{2}}\leq{P_{max}},$
(51b) $\displaystyle\quad\quad{x_{1}}=0,{x_{M}}\leq{D_{\max}},$ (51c)
$\displaystyle\quad\quad{x_{m}}-{x_{m-1}}\geq{D_{0}},\forall m,$ (51d)
$\displaystyle\quad\quad{f_{C}}\leq{f_{m}}\leq{f_{C}}+\Delta F,\forall m.$
(51e)

Problem (30) is non-convex function with the coupled variables which is
challenging to solve. Inspired by _Remark 2_ , we conclude that the
optimization for ABV cannot affect that for APV and AFV. Thus, the proposed
two-stage AO algorithm can effectively solve it.

###  IV-A Optimizing ${\bf{w}}\left(t\right)$ with Given ${\bf{x}}$ and
${\bf{f}}$

For given APV and AFV, problem (29) can be converted to

$\displaystyle\mathop{\max}\limits_{\bf{w}}{\rm{}}\left\\{{{{\log}_{2}}\left({1+{{\bf{w}}^{H}}{\bf{Aw}}}\right)-\mathop{\max}\limits_{Y,Z}{{\log}_{2}}\left({1+{{\bf{w}}^{H}}{{\bf{B}}_{y,z}}{\bf{w}}}\right)}\right\\}$
(52a) $\displaystyle{s.t.}\quad{\left\|{\bf{w}}\right\|^{2}}\leq{P_{max}}.$
(52b)

We define a matrix as ${\bf{W}}={\bf{w}}{{\bf{w}}^{H}}$ which follows that
${\rm{rank}}\left({\bf{W}}\right)=1$, and the rank-1 constraint is non-convex
which can be relaxed by the semi-definite relaxation (SDR) method. Then,
problem (31) can be converted to

$\displaystyle\mathop{\max}\limits_{\bf{W}}\left\\{{{{\log}_{2}}\left({1+{\bf{Tr}}\left({{\bf{A}}{\bf{W}}}\right)}\right)-\mathop{\max}\limits_{Y,Z}{{\log}_{2}}\left({1+{\bf{Tr}}\left({{{\bf{B}}_{y,z}}{\bf{W}}}\right)}\right)}\right\\}$
(53a) $\displaystyle{s.t.}\quad{\bf{Tr}}\left({\bf{W}}\right)\leq{P_{\max}}.$
(53b)

Problem (32) is still non-convex and difficult to solve, we next resort to
Lemma 2 via Fenchel conjugate arguments [32].

_Lemma 2:_ Considering the function $\varphi\left(u\right)=-uv+\ln u+1$ for
any $v>0$.

$\displaystyle\ \mathop{\max}\limits_{u>0}\varphi\left(u\right)=-\ln v$ (54)

with the optimum value $u^{opt}=1/v$. By utilizing Lemma 2, we have

$\displaystyle\ \ln
2{\log_{2}}\left({1+Tr\left({{{\bf{B}}_{y,z}}{\bf{W}}}\right)}\right)$
$\displaystyle=\ln\left({1+Tr\left({{{\bf{B}}_{y,z}}{\bf{W}}}\right)}\right)$
$\displaystyle={u_{e}}\left({Tr\left({{{\bf{B}}_{y,z}}{\bf{W}}}\right)+1}\right)$
$\displaystyle-\ln{u_{e}}-1.$ (55)

where
${u_{e}}={\left({Tr\left({{{\bf{B}}_{y,z}}{\bf{W}}}\right)+1}\right)^{-1}}$.
It is worth mentioning that ${u_{e}}$ and $\bf{W}$ are coupled optimization
variables, and we apply the AO method to solve it. Specifically, with fixed
$\bf{W}$, the expression of the optimal $u_{e}^{opt}$ can be given in closed
form as

$\displaystyle\
u_{e}^{opt}={\left({Tr\left({{{\bf{B}}_{y,z}}{\bf{W}}}\right)+1}\right)^{-1}}.$
(56)

Then, with given ${u_{e}}$, the optimization problem can be formulated as

$\displaystyle\mathop{\max}\limits_{{\bf{W}},g}\quad\ln\left({1+{\bf{Tr}}\left({{\bf{AW}}}\right)}\right)-g$
(57a)
$\displaystyle{s.t.}\quad\quad{u_{e}}\left({Tr\left({{{\bf{B}}_{y,z}}W}\right)+1}\right)-\ln{u_{e}}-1\leq
g,\forall y,z,$ (57b)
$\displaystyle\quad\quad\quad{\bf{Tr}}\left({\bf{W}}\right)\leq{P_{\max}}.$
(57c)

Problem (36) is a semi-definite program (SDP) problem which can be efficiently
solved by convex solvers such as CVX [33]. We observe that the optimal
solution ${{\bf{W}}^{opt}}$ may not satisfy the rank-1 constraint. Thus, we
apply the Gussian randomization method to recover ${{\bf{w}}^{opt}}$
approximately [34].

###  IV-B Optimizing ${\bf{x}}$ with Given ${\bf{w}}\left(t\right)$ and
${\bf{f}}$.

Different from (12), we cannot get the expression of the optimal ABV in closed
form from problem (36). Hence, there is no simplified optimization object when
optimizing APV and AFV. Inspired by _Remark 1_ , we conclude that the
optimization problems for APV and AFV are actually to minimize the inner
product between the channels of Bob and Eve. Therefore, the optimization
problem can be formulated as [9]

$\displaystyle\mathop{\min}\limits_{{{{\bf{x}}}}}\mathop{\max}\limits_{Y,Z}{\left|{\left\langle{{{\bf{h}}_{ab}},{{\bf{h}}_{ae}}\left({{r_{y}},{\theta_{z}}}\right)}\right\rangle}\right|^{2}}$
(58a) $\displaystyle{s.t.}\quad{x_{1}}=0,{x_{M}}\leq{D_{\max}},$ (58b)
$\displaystyle\quad\quad{x_{m}}-{x_{m-1}}\geq{D_{0}},\forall m,$ (58c)

where
$\mathop{\max}\limits_{Y,Z}{\left|{\left\langle{{{\bf{h}}_{ab}},{{\bf{h}}_{ae}}\left({{r_{y}},{\theta_{z}}}\right)}\right\rangle}\right|^{2}}$
aims to consider the worst sample in the suspicious region of Eve. According
to (45a),
${\left|{\left\langle{{{\bf{h}}_{ab}},{{\bf{h}}_{ae}}\left({{r_{y}},{\theta_{z}}}\right)}\right\rangle}\right|^{2}}$
is not monotonic for $\bf{x}$ which means that we cannot find the worst sample
directly. Thus, we propose to utilize the BCD method to search the worst
sample and solve problem (37).

Specifically, in the $s$-th iteration of BCD, we first optimize ${x_{m}}$
where $m=s{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\rm{mod}}{\kern
1.0pt}{\kern 1.0pt}{\kern 1.0pt}M$, and the optimization problem is rewritten
as follows:

$\displaystyle\mathop{\min}\limits_{{x_{m}}}\mathop{\max}\limits_{Y,Z}{{\sum\limits_{\scriptstyle{n=0}\hfill\atop\scriptstyle{n\neq
m}\hfill}^{M-1}{{\cos\left[{2\pi\left({{{\tau_{m,y,z}}}{f_{m}}-{\tau_{n,y,z}}{f_{n}}}\right)}\right]}}}}$
(59a) $\displaystyle{s.t.}\quad{x_{1}}=0,{x_{M}}\leq{D_{\max}},$ (59b)
$\displaystyle\quad\quad{x_{m}}-{x_{m-1}}\geq{D_{0}},\forall m,$ (59c)

where
${\tau_{m,y,z}}={x_{m}}\frac{{\sin{\theta_{z}}-\sin{\theta_{b}}}}{c}+\frac{{{r_{y}}-{r_{e}}}}{c}$.
Here, we utilize the linear search method to solve this problem, and the
search range for $x_{m}$ is $\left[{x_{m-1}^{s}+{D_{0}},{D_{1}}}\right]$.
Then, we update ${{\bf{x}}^{s-1}}$ with $x_{m}^{opt}$ where $x_{m}^{opt}$ is
the optimal solution, i.e., ${{\bf{x}}^{s}}={{\bf{x}}^{s-1}}$.

###  IV-C Optimizing ${\bf{f}}$ with Given ${\bf{w}}\left(t\right)$ and
${\bf{x}}$.

Similar to problem (37), the optimization problem in this case is formulated
as

$\displaystyle\mathop{\min}\limits_{{{{\bf{f}}}}}\mathop{\max}\limits_{Y,Z}{\left|{\left\langle{{{\bf{h}}_{ab}},{{\bf{h}}_{ae}}\left({{r_{y}},{\theta_{z}}}\right)}\right\rangle}\right|^{2}}$
(60a) $\displaystyle{s.t.}\quad{f_{C}}\leq{f_{m}}\leq{f_{C}}+\Delta F,\forall
m.$ (60b)

We propose to utilize the BCD and linear search method to solve this problem,
and the search range for $f_{m}$ is $\left[{{f_{C}},{f_{C}}+\Delta F}\right]$.

_Remark 3:_ According to (III-A), it is clear that
${{{\left|{\left\langle{{{\bf{h}}_{ab}},{{\bf{h}}_{ae}}\left({{r_{y}},{\theta_{z}}}\right)}\right\rangle}\right|}^{2}}}$
is not monotonic for ${r_{y}}$ and ${\theta_{z}}$, which means that we cannot
directly find the worst-case coordination for Eve in the estimated region
$\left({\cal{R}},{\cal{A}}\right)$. To solve it, we utilize the discretization
method to simplify the region to some samples and search the worst case when
optimizing APV and AFV. This method can effectively to deal with the problems
of the position uncertainty in FDA [9]. It is worth mentioning that multiple
samples can make the results accurate, but it also leads to the complexity of
the optimization problem.

###  IV-D Overall Two-Stage AO Algorithm

We propose to utilize the two-stage AO algorithm to solve the problem (30).
Specifically, the APV and AFV at the transmitter are optimized in the first
stage by utilizing the BCD method which is summarized in Algorithm 3. We
consider the worst case among all cases and aim to minimize the inner product
between the channels of Bob and Eve. In the second stage, we utilize the CVX
method to obtain the optimal ABV which is summarized in Algorithm 4. The
overall algorithm is the same as Algorithm 2.

Algorithm 3 Optimizing ${\bf{x}}$ by BCD

1: Initialization: Initializing ${\bf{f}}$ via Algorithm 3 and
${{\bf{w}}_{opt}}$ via Algorithm 4.

2: Set: $s=0$.

3: Repeat:a) $s=s+1$, and $m=s{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern
1.0pt}{\rm{mod}}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\rm{M}}$.b) Searching
$x_{m}$ in the given region $\left[{x_{m-1}^{s}+{D_{0}},{D_{1}}}\right]$ to
minimize
$\mathop{\max}\limits_{Y,Z}{\left|{\left\langle{{{\bf{h}}_{ab}},{{\bf{h}}_{ae}}\left({{r_{y}},{\theta_{z}}}\right)}\right\rangle}\right|^{2}}$.
c) We update ${{\bf{x}}^{s-1}}$ with $x_{m}^{opt}$ and obtain
${{\bf{x}}^{s}}$.

4: Until
$\left|{{C_{\sec}}\left({{{\bf{w}}_{opt}},{{\bf{x}}^{s}},{\bf{f}}}\right)-{C_{\sec}}\left({{{\bf{w}}_{opt}},{{\bf{x}}^{s-1}},{\bf{f}}}\right)}\right|\leq\varsigma$
is satisfied.

The complexity of Algorithm 3 mainly depends on the number of iterations for
the BCD method and samples for the coordinate uncertainty at Eve. We denote
the increment of the position search as $\Delta x$, thus the complexity of the
position search is
${\cal{O}}\left({\frac{{{D_{1}}-x_{m-1}^{s}-{D_{0}}}}{{\Delta x}}}\right)$.
Moreover, $Y\times Z$ and ${\rm{I}}_{BCD}^{\bf{x}}$ denote the number of the
samples and iterations required for BCD convergence, respectively. The
complexity of Algorithm 3 is
${\cal{O}}\left({\frac{{\left({{D_{1}}-x_{m-1}^{s}-{D_{0}}}\right)YZ}}{{\Delta
x}}{{\rm{I}}_{BCD}^{\bf{x}}}}\right)$. When optimizing AFV, the complexity of
Algorithm 3 is ${\cal{O}}\left({\frac{{\Delta F}}{{\Delta
f}}YZ{{\rm{I}}_{BCD}^{\bf{f}}}}\right)$.

Algorithm 4 Optimizing ${\bf{w}}$ by CVX and AO algorithm

1: Initialization: Initializing ${{\bf{w}}_{0}}$ according to the maximum
transmit power constraint ${\left\|{{\bf{w}}_{0}}\right\|^{2}}\leq{P_{max}}$.

2: Set: $s=1$ and ${{\bf{W}}}={\bf{w}}_{0}^{H}{{\bf{w}}_{0}}$.

3: Repeat:a) We obtain
$u_{e,y,z}^{opt}={\left({Tr\left({{{\bf{B}}_{y,z}}{\bf{W}}}\right)+1}\right)^{-1}}$
and ${{\bf{u}}^{s}}=\left[{u_{e,1,1}^{opt},...,u_{e,Y,Z}^{opt}}\right]$b) With
given ${{\bf{u}}^{s}}$, we solve problem (36) by CVX and obtain the optimized
${\bf{W}}^{s}$. Then, we have $s=s+1$.

4: Until
$\left|{{C_{\sec}}\left({{{\bf{W}}^{s}},{{\bf{u}}^{s}}}\right)-{C_{\sec}}\left({{{\bf{W}}^{s-1}},{{\bf{u}}^{s-1}}}\right)}\right|\leq\varsigma$
is satisfied.

5: If rank$\left({{{\bf{W}}^{s}}}\right)=1$, the optimal ABV is given by
${{\bf{w}}^{opt}}={\rm{eig}}\left({{{\bf{W}}^{s}}}\right)$ where
eig$\left(\cdot\right)$ is eigenvector decomposition. If
rank$\left({{{\bf{W}}^{s}}}\right)\neq 1$, we apply the Gussian randomization
method.

The complexity of Algorithm 4 is
${\cal{O}}\left({{M^{3.5}}\log\left({\frac{1}{\varsigma}}\right)I_{AO}^{\bf{W}}}\right)$
where ${I_{AO}^{\bf{W}}}$ denotes the number of the iterations required for AO
convergence.

##  V Numerical Results

In this section, numerical results for MFDA-aided MISO wireless communication
are presented. Unless otherwise specified, the maximum transmission power is
${P_{max}}=10$dBm and the carrier frequency is ${f_{C}}=10$GHz. The noise
variances at Bob and Willie are $\sigma^{2}_{b}=-100$dBm and
$\sigma^{2}_{e}=-100$dBm, respectively. The position coordinates of Bob and
Eve are
$\left({{r_{b}},{\theta_{b}}}\right)=\left({1000{\rm{m}},{{30}^{\circ}}}\right)$
and
$\left({{r_{e}},{\theta_{e}}}\right)=\left({1000{\rm{m}},{{35}^{\circ}}}\right)$.
The minimum antenna distance is ${D_{0}}=\frac{\lambda}{2}$ where
$\lambda=\frac{c}{{{f_{C}}}}$ is the wavelength of MFDA. The movable antenna
boundary is ${D_{\max}}=30\lambda$ and the frequency increment range is
$\Delta F=1$ GHz. Moreover, the path loss factor is given as [34]

$\displaystyle\
{\rm{Lfs}}\left(r\right)=C{\left({\frac{r}{R}}\right)^{-\alpha}},$ (61)

where $C=-30$dBm is the path loss at the reference distance $R=1{\rm{m}}$ and
$r$ is the individual link distance. $\alpha$ is the path loss exponent where
${\alpha_{ab}}=2$ and ${\alpha_{ae}}=3$. Besides, the whole time is $T=1s$ and
CRF is $\Delta{T_{CRF}}={10^{-6}}s$.

For comparison, the security performance of three benchmark strategies are
also investigated as follows: 1) PA, i.e., only optimizing the ABV at Alice.
2) FDA, i.e., optimizing the ABV and AFV at Alice. 3) Upper bound, i.e.,
${C_{\sec}}={\log}\left({1+\sigma_{b}^{-2}{\rm{Lf}}{{\rm{s}}^{2}}\left({{r_{b}}}\right){P_{\max}}M}\right)$.

###  V-A MFDA with Perfect Eve CSI

Figure 2: Typical converging traces of the BSUM for optimizing APV with
$M=10$.

Figure 2 depicts the convergence performance of the BSUM algorithm for APV
where each trace starting from random initial APV. As expected, the secrecy
capacity increases with the iteration index and converges at about 60
iterations which verifies the convergence of BSUM. Moreover, it is worth
mentioning that the BUSM utilized for AFV has the same convergence curve as
that for APV, thus we omit the figure here for brevity.

Figure 3: Secrecy capacity ${C_{\sec}}$ versus number of antennas $M$.

Figure 3 depicts the secrecy capacity ${C_{\sec}}$ versus the number of
antennas $M$ for different strategies. We observe that ${C_{\sec}}$ increases
with $M$, because that brings more spatial DoFs. In particular, the secrecy
capacity of MFDA is larger than that of PA and FDA, and can reach the upper
bound with much fewer antennas. This is due to the fact that MFDA can provide
larger secrecy capacity via adjusting the position and frequency of each
antenna. However, FDA can only adjust the frequency, and is easily limited by
the frequency constraint in a highly correlated channel scenario. This
indicates that two-dimensional security achieved by MFDA can effectively
handle different channel scenarios to enhance the security performance.
Moreover, PA only optimizes the ABV at Alice which results in the worst
security performance than other strategies as expected. Hence, PA requires
more antennas to reach the upper bound.

Figure 4: Secrecy capacity ${C_{\sec}}$ versus different Eve’s angel
${\theta_{e}}$.

Figure 4 illustrates how different angles ${\theta_{e}}$ affect the secrecy
capacity ${C_{\sec}}$. We first observe that ${C_{\sec}}$ increases with
${\theta_{e}}$ and gradually approaches the upper bound. It is because that,
when Eve approaches Bob, the channels of Bob and Eve are highly correlated
which increases the difficulty of channel decoupling, thus resulting in a
corresponding decrement in secrecy capacity. In particular, when
${\theta_{e}}={30^{\circ}}$, neither MFDA, FDA nor PA can achieve any secrecy
capacity. Fortunately, our proposed MFDA can fully utilize the frequency and
position dimensions of antennas to reduce the performance loss, thus the
secrecy capacity of MFDA drops sharply only when Eve and Bob are very close.
This also implies that MFDA has stronger decoupling ability for highly
correlated channel compared with FDA and PA.

Figure 5: Secrecy capacity ${C_{\sec}}$ versus frequency increment range
$\Delta F/{f_{C}}$ with $M=20$.

Figure 5 analyzes the secrecy capacity ${C_{\sec}}$ for different frequency
increment range $\Delta F/{f_{C}}$ and movable antenna boundary ${D_{\max}}$.
It is clear that larger ${D_{\max}}$ and $\Delta F$ can provide more
selectivity to optimize APV and AFV which can improve the secrecy capacity
correspondingly. In practice, due to the device limitation or complex channel
state, both APV and AFV will have constraints, such that relying on separately
optimizing either of them cannot achieve the best security performance. Our
proposed MFDA can jointly optimize APV and AFV to realize two-dimensional
security which leads to the larger secrecy capacity than FDA and PA. In
particular, when the AFV constraint is strict, MFDA can optimize APV to
further improve secrecy performance instead of optimizing AFV.

Figure 6: Secrecy capacity ${C_{\sec}}$ versus BRF with $M=20$, $CRF=1$MHz and
$T=1$s.

Figure 6 investigates the effects of BRF and CRF on secrecy capacity where the
ideal MFDA means that the refresh speed of beamformer is larger than or equal
to that of the channel. As expected, larger BRF can increase the secrecy
capacity, and BRF $=1$KHz is sufficient to achieve the ideal upper bound.
Moreover, when BRF $<{10^{1}}$Hz , PA outperforms FDA and MFDA. This is
because a small BRF results in the current optimized ABV not being able to
adapt to all the varying CSIs before the next optimization, thus cannot
exploit the advantages provided by the frequency and position dimensions.

###  V-B MFDA with Imperfect Eve CSI

In this subsection, we consider the case of imperfect CSI at Eve and
investigate the influences of the uncertainties in angle and range on secrecy
capacity.

(a) BCD algorithm for APV (b) AO algorithm for ABV

Figure 7: Typical converging traces of the BCD algorithm and the AO algorithm
based on CVX where $\Delta\theta={3^{\circ}}$ and $Z=13$.

Figure 7(a) verifies the convergence of the BCD algorithm for optimizing APV
where each trace starting from random initial APV and channel inner product is
denoted as
${\left|{\left\langle{{{\bf{h}}_{ab}},{{\bf{h}}_{ae}}\left({{r_{y}},{\theta_{z}}}\right)}\right\rangle}\right|^{2}}$.
The converging behaviour for optimizing AFV is similar to that for optimizing
APV. Thus, we omit the figure here for brevity. Figure 7(b) verifies the
convergence of the AO algorithm based on CVX for optimizing ABV where each
trace starting from random initial ABV subject to the maximum transmit power
constraint.

(a) Small uncertainty (b) Larger uncertainty

Figure 8: Secrecy capacity ${C_{\sec}}$ versus angle uncertainty
$\Delta\theta$ at Eve.

Figure 8 depicts the influences of angle uncertainty $\Delta\theta$ at Eve on
secrecy capacity ${C_{\sec}}$. We consider two cases of angle uncertainty,
i.e., small uncertainty and larger uncertainty. Specifically, small
uncertainty implies that the estimated CSI at Eve has a small error and larger
uncertainty implies that the estimated CSI has a larger error. It is clear
that ${C_{\sec}}$ decreases with $\Delta\theta$ which means that the
estimation error for Eve’s CSI can deteriorate security performance, and
larger estimation error leads to a more severe performance degradation. As
expected, MFDA can significantly decrease performance loss caused by imperfect
Eve CSI compared to FDA and PA. In particular, when the angle uncertainty is
small, almost no loss in security performance by utilizing MFDA. Moreover, the
influences of the range uncertainty $\Delta r$ at Eve on ${C_{\sec}}$ are the
same as that of $\Delta\theta$, thus we omit the figures here for brevity.

##  VI Conclusion

In this paper, we proposed a novel antenna technology MFDA to enhance security
performance in MISO wireless communication. Specifically, we aimed to maximize
the secrecy capacity by jointly optimizing the ABV, APV and AFV at the
transmitter, subject to the given constraints. Then, a two-stage AO algorithm
with lower complexity was developed to solve the optimization problem.
Moreover, we consider the case of imperfect CSI at Eve. Simulation results
demonstrated that MFDA introduces a new dimension which can further improve
security performance based on conventional FDA. Moreover, MFDA has strong
channel identification ability to decouple the highly correlated channel, and
can effectively reduce the security performance loss caused by CSI estimation
error.

## Appendix A

By substituting the optimal beamforming vector
${{\bf{w}}_{opt}}\left(t\right)$ into (11a), we have
$\frac{{1+{\bf{w}}_{opt}^{H}\left(t\right){\bf{A}}{{\bf{w}}_{opt}}\left(t\right)}}{{1+{\bf{w}}_{opt}^{H}\left(t\right){\bf{B}}{{\bf{w}}_{opt}}\left(t\right)}}={\lambda_{\max}}$
where ${\lambda_{\max}}$ is the maximum eigenvalue of
${\left({{\bf{B}}+P_{\max}^{-1}{{\bf{I}}_{M}}}\right)^{-1}}\left({{\bf{A}}+P_{\max}^{-1}{{\bf{I}}_{M}}}\right)$.
By utilizing the matrix inverse lemma [35], we have

$\displaystyle{\left({{\bf{B}}+P_{\max}^{-1}{{\bf{I}}_{M}}}\right)^{-1}}\left({{\bf{A}}+P_{\max}^{-1}{{\bf{I}}_{M}}}\right)$
$\displaystyle={P_{\max}}\left({{{\bf{I}}_{M}}-\frac{{{P_{\max}}{\bf{h}}_{ae}^{H}{{\bf{h}}_{ae}}\sigma_{e}^{-2}}}{{\left({1+{P_{\max}}\left\|{{{\bf{h}}_{ae}}}\right\|_{2}^{2}}\right)}}}\right)\left({\frac{{{\bf{h}}_{ab}^{H}{{\bf{h}}_{ab}}}}{{\sigma_{b}^{2}}}+{\frac{{{{\bf{I}}_{M}}}}{{{P_{\max}}}}}}\right)$
$\displaystyle={P_{\max}}\left({\begin{array}[]{*{20}{l}}{\frac{{{\bf{h}}_{ab}^{H}{{\bf{h}}_{ab}}}}{{\sigma_{b}^{2}}}-\frac{{{P_{\max}}{\bf{h}}_{ae}^{H}{{\bf{h}}_{ae}}{\bf{h}}_{ab}^{H}{{\bf{h}}_{ab}}}}{{\left({1+{P_{\max}}\left\|{{{\bf{h}}_{ae}}}\right\|_{2}^{2}}\right)\sigma_{b}^{2}\sigma_{e}^{2}}}}\\\
{-\frac{{{\bf{h}}_{ae}^{H}{{\bf{h}}_{ae}}}}{{\left({1+{P_{\max}}\left\|{{{\bf{h}}_{ae}}}\right\|_{2}^{2}}\right)\sigma_{e}^{2}}}}\end{array}}\right)+{{\bf{I}}_{M}}.$
(64)

Next, we decompose ${{\bf{h}}_{ab}}$ and ${{\bf{h}}_{ab}}$ into two orthogonal
vectors

$\displaystyle\
\frac{{{{\bf{h}}_{ab}}}}{{{\sigma_{b}}}}=\alpha{{\bf{h}}_{e}}+\beta{{\bf{h}}_{e\bot}},\frac{{{{\bf{h}}_{ae}}}}{{{\sigma_{e}}}}={\left\|{{{\bf{h}}_{ae}}}\right\|_{2}}{{\bf{h}}_{e}}+0\cdot{{\bf{h}}_{e\bot}},$
(65)

where ${{\bf{h}}_{e}}$ and ${{\bf{h}}_{e\bot}}$ are parallel and orthogonal
unit vectors of ${{\bf{h}}_{ae}}$, respectively. In addition, $\alpha$ and
$\beta$ are scalar values of ${{\bf{h}}_{ab}}$ subject to
${\alpha^{2}}+{\beta^{2}}=\frac{{\left\|{{{\bf{h}}_{ab}}}\right\|_{2}^{2}}}{{\sigma_{b}^{2}}}$
where
${\alpha^{2}}=\frac{{{{\left|{\left\langle{{{\bf{h}}_{ab}},{{\bf{h}}_{ae}}}\right\rangle}\right|}^{2}}}}{{\sigma_{b}^{2}\left\|{{{\bf{h}}_{ae}}}\right\|_{2}^{2}}}$,
${\beta^{2}}=\frac{{\left\|{{{\bf{h}}_{ab}}}\right\|_{2}^{2}}}{{\sigma_{b}^{2}}}-\frac{{{{\left|{\left\langle{{{\bf{h}}_{ab}},{{\bf{h}}_{ae}}}\right\rangle}\right|}^{2}}}}{{\sigma_{b}^{2}\left\|{{{\bf{h}}_{ae}}}\right\|_{2}^{2}}}$,
$\left\|{{{\bf{h}}_{ab}}}\right\|_{2}^{2}={\rm{Lf}}{{\rm{s}}^{2}}\left({{r_{b}}}\right)M$
and
$\left\|{{{\bf{h}}_{ae}}}\right\|_{2}^{2}={\rm{Lf}}{{\rm{s}}^{2}}\left({{r_{e}}}\right)M$.

By substituting (65) into (A), we have

$\displaystyle\frac{{1+{P_{\max}}\left\|{{{\bf{h}}_{ae}}}\right\|_{2}^{2}}}{{{P_{\max}}}}{\left({B+P_{\max}^{-1}{{\bf{I}}_{M}}}\right)^{-1}}\left({A+P_{\max}^{-1}{{\bf{I}}_{M}}}\right)$
$\displaystyle=\left({1+{P_{\max}}\left\|{{{\bf{h}}_{ae}}}\right\|_{2}^{2}}\right)\frac{{{\bf{h}}_{ab}^{H}{{\bf{h}}_{ab}}}}{{\sigma_{b}^{2}}}-\frac{{{P_{\max}}{\bf{h}}_{ae}^{H}{{\bf{h}}_{ae}}{\bf{h}}_{ab}^{H}{{\bf{h}}_{ab}}}}{{\sigma_{b}^{2}\sigma_{e}^{2}}}$
$\displaystyle-\frac{{{\bf{h}}_{ae}^{H}{{\bf{h}}_{ae}}}}{{\sigma_{e}^{2}}}$
$\displaystyle=\left({{\alpha^{2}}-\left\|{{{\bf{h}}_{ae}}}\right\|_{2}^{2}}\right){\bf{h}}_{e}^{H}{{\bf{h}}_{e}}+\alpha\beta{\bf{h}}_{e\bot}^{H}{{\bf{h}}_{e}}+\alpha\beta{\bf{h}}_{e}^{H}{{\bf{h}}_{e\bot}}$
$\displaystyle+{\beta^{2}}{\bf{h}}_{e\bot}^{H}{{\bf{h}}_{e\bot}}+\left({{P_{\max}^{-1}}+\left\|{{{\bf{h}}_{ae}}}\right\|_{2}^{2}}\right){{\bf{I}}_{M}}$
$\displaystyle=\left[{{\bf{h}}_{e\bot}^{H},{\bf{h}}_{e}^{H}}\right]{\bf{C}}\left[{\begin{array}[]{*{20}{c}}{{{\bf{h}}_{e\bot}}}\\\
{{{\bf{h}}_{e}}}\end{array}}\right],$ (68)

where ${\bf{C}}$ is

$\displaystyle\
{\bf{C}}=\left({P_{\max}^{-1}+\left\|{{{\bf{h}}_{ae}}}\right\|_{2}^{2}}\right){{\bf{I}}_{2}}+\left[{\begin{array}[]{*{20}{c}}{{\beta^{2}}}&{\alpha\beta}\\\
{\alpha\beta}&{{\alpha^{2}}-\left\|{{{\bf{h}}_{ae}}}\right\|_{2}^{2}}\end{array}}\right].$
(71)

According to the properties of matrix determinant [35], we have
$\det\left({{\lambda_{\max}}{{\bf{I}}_{2}}-{\bf{C}}}\right)=0$, i.e.,

$\displaystyle\left({{\lambda_{\max}}-\left({P_{\max}^{-1}+\left\|{{{\bf{h}}_{ae}}}\right\|_{2}^{2}+{\beta^{2}}}\right)}\right)$
$\displaystyle\times\left({{\lambda_{\max}}-\left({P_{\max}^{-1}+{\alpha^{2}}}\right)}\right)-{\alpha^{2}}{\beta^{2}}=0.$
(72)

Then, the closed form expression of ${{\lambda_{\max}}}$ is given by

$\displaystyle\ {\lambda_{max}}$
$\displaystyle\buildrel\Delta\over{=}\frac{{{P_{\max}}\left({2P_{\max}^{-1}+\left\|{{{\bf{h}}_{ae}}}\right\|_{2}^{2}+\left\|{{{\bf{h}}_{ab}}}\right\|_{2}^{2}}\right)}}{{2\left({1+{P_{\max}}\left\|{{{\bf{h}}_{ae}}}\right\|_{2}^{2}}\right)}}$
$\displaystyle+\frac{{{P_{\max}}\sqrt{{{\left({\left\|{{{\bf{h}}_{ab}}}\right\|_{2}^{2}+\left\|{{{\bf{h}}_{ae}}}\right\|_{2}^{2}}\right)}^{2}}-4\left\|{{{\bf{h}}_{ae}}}\right\|_{2}^{2}{\alpha^{2}}}}}{{2\left({1+{P_{\max}}\left\|{{{\bf{h}}_{ae}}}\right\|_{2}^{2}}\right)}}.$
(73)

It is clear that ${\lambda_{max}}$ decreases with ${{\alpha^{2}}}$. Problem
(11) aims to maximize ${\lambda_{max}}$ which is equivalent to minimizing
${{\alpha^{2}}}$. Moreover, we note that
${\left|{\left\langle{{{\bf{h}}_{ab}},{{\bf{h}}_{ae}}}\right\rangle}\right|^{2}}={\alpha^{2}}\left\|{{{\bf{h}}_{ae}}}\right\|_{2}^{2}={\alpha^{2}}M$
and
${\left|{\left\langle{{{\bf{h}}_{ab}},{{\bf{h}}_{ae}}}\right\rangle}\right|^{2}}$
monotonically increases with ${\alpha^{2}}$, thus maximizing ${\lambda_{max}}$
is equivalent to minimizing
${\left|{\left\langle{{{\bf{h}}_{ab}},{{\bf{h}}_{ae}}}\right\rangle}\right|^{2}}$.
Overall, problem (11) can be converted to problem (13).

##  Appendix B Proof of Lemma 1

###  B-A Case 1

In this case, we have
$y_{m,n}^{{}^{\prime}}{\left({\tau_{m}^{s-1};\tau_{n}^{s-1}}\right)}>0$,
${y_{m,n}}\left({{\zeta_{m,n}};\tau_{n}^{s-1}}\right)=-1$ and
${\zeta_{m,n}}<\tau_{m}^{s-1}$. The relation
$-1<{y_{m,n}}{\left({\tau_{m}^{s-1};\tau_{n}^{s-1}}\right)}<1$ holds if and
only if there exists an integer ${\kappa_{1}}\geq 0$ such that

$\displaystyle\
\begin{array}[]{l}-1<\cos\left[{2\pi\left({\tau_{m}^{s-1}{f_{m}}-\tau_{n}^{s-1}{f_{n}}}\right)}\right]<1\\\
\begin{array}[]{*{20}{l}}\mathop{\Rightarrow}\limits^{\left(a\right)}\pi+2\pi{\kappa_{1}}<2\pi\left({\tau_{m}^{s-1}{f_{m}}-\tau_{n}^{s-1}{f_{n}}}\right)<2\pi+2\pi{\kappa_{1}}\\\
\Rightarrow\frac{{2\left({\tau_{m}^{s-1}{f_{m}}-\tau_{n}^{s-1}{f_{n}}}\right)-2}}{2}<{\kappa_{1}}<\frac{{2\left({\tau_{m}^{s-1}{f_{m}}-\tau_{n}^{s-1}{f_{n}}}\right)-1}}{2}\\\
{\mathop{\Rightarrow}\limits^{\left(b\right)}{\kappa_{1}}=\frac{{\left\lfloor{2\left({\tau_{m}^{s-1}{f_{m}}-\tau_{n}^{s-1}{f_{n}}}\right)}\right\rfloor-1}}{2}},\end{array}\end{array}$
(79)

where $\left(a\right)$ is due to ${\zeta_{m,n}}<\tau_{m}^{s-1}$ and
$\left(b\right)$ is because that ${\kappa_{1}}$ is an integer solution. Owing
to ${y_{m,n}}\left({{\zeta_{m,n}};\tau_{n}^{s-1}}\right)=-1$, we have

$\displaystyle\
2\pi\left({{\zeta_{m,n}}{\tau_{m}}-\tau_{n}^{s-1}{f_{n}}}\right)=\pi+2{\kappa_{1}}\pi$
(80)

Thus, ${\zeta_{m,n}}$ is given by

$\displaystyle\
{\zeta_{m,n}}=\frac{{\left\lfloor{2\left({\tau_{m}^{s-1}{f_{m}}-\tau_{n}^{s-1}{f_{n}}}\right)}\right\rfloor}}{{2{f_{m}}}}+\frac{{\tau_{n}^{s-1}{f_{n}}}}{{{f_{m}}}}.$
(81)

According to (23), we can obtain ${k_{m,n}}$ and ${\delta_{m,n}}$ through
simple mathematical manipulations which are given by

$\displaystyle\
{{k_{m,n}}=\frac{{-\pi{f_{m}}\sin\left[{2\pi\left({{\tau_{m}^{s-1}}{f_{m}}-\tau_{n}^{s-1}{f_{n}}}\right)}\right]}}{{{\tau_{m}^{s-1}}-{\zeta_{m,n}}}}}$
(82)

and

$\displaystyle\
{{\delta_{m,n}}=\cos\left[{2\pi\left({{\tau_{m}^{s-1}}{f_{m}}-\tau_{n}^{s-1}{f_{n}}}\right)}\right]-{k_{m,n}}{{\left({{\tau_{m}^{s-1}}-{\zeta_{m,n}}}\right)}^{2}}}.$
(83)

###  B-B Case 2

In this case, we have
$y_{m,n}^{{}^{\prime}}{\left({\tau_{m}^{s-1};\tau_{n}^{s-1}}\right)}>0$,
${y_{m,n}}\left({{\zeta_{m,n}};\tau_{n}^{s-1}}\right)=-1$ and
$\tau_{m}^{s-1}<{\zeta_{m,n}}$. The relation
$-1<{y_{m,n}}{\left({\tau_{m}^{s-1};\tau_{n}^{s-1}}\right)}<1$ holds if and
only if there exists an integer ${\kappa_{2}}\geq 0$ such that

$\displaystyle\
\begin{array}[]{l}-1<\cos\left[{2\pi\left({\tau_{m}^{s-1}{f_{m}}-\tau_{n}^{s-1}{f_{n}}}\right)}\right]<1\\\
\begin{array}[]{*{20}{l}}{\Rightarrow
2\pi{\kappa_{2}}<2\pi\left({\tau_{m}^{s-1}{f_{m}}-\tau_{n}^{s-1}{f_{n}}}\right)<\pi+2\pi{\kappa_{2}}}\\\
\Rightarrow{\kappa_{2}}=\frac{{\left\lceil{2\left({\tau_{m}^{s-1}{f_{m}}-\tau_{n}^{s-1}{f_{n}}}\right)}\right\rceil-1}}{2}.\end{array}\end{array}$
(88)

According to ${y_{m,n}}\left({{\zeta_{m,n}};\tau_{n}^{s-1}}\right)=-1$, we
have

$\displaystyle\
2\pi\left({{\zeta_{m,n}}{f_{m}}-\tau_{n}^{s-1}{f_{n}}}\right)=\pi+2{\kappa_{2}}\pi.$
(89)

Thus, ${\zeta_{m,n}}$ is given by

$\displaystyle\
{\zeta_{m,n}}=\frac{{\left\lceil{2\left({\tau_{m}^{s-1}{f_{m}}-\tau_{n}^{s-1}{f_{n}}}\right)}\right\rceil}}{{2{f_{m}}}}+\frac{{\tau_{n}^{s-1}{f_{n}}}}{{{f_{m}}}}.$
(90)

Similarly, based on (23), we can obtain ${k_{m,n}}$ and ${\delta_{m,n}}$ which
are the same as (82) and (83), respectively.

###  B-C Case 3

In this case, we have
$y_{m,n}^{{}^{\prime}}{\left({\tau_{m}^{s-1};\tau_{n}^{s-1}}\right)}=0$ and
${y_{m,n}}{\left({\tau_{m}^{s-1};\tau_{n}^{s-1}}\right)}=-1$ which means that
${\zeta_{m,n}}=\tau_{m}^{s-1}$ and ${\delta_{m,n}}=-1$. To find ${k_{m,n}}$,
we first introduce a small parameter $\kappa_{3}$ whose value approaches zero,
i.e., ${\kappa_{3}}\to 0$. Then, at the point ${\tau_{m}^{s-1}+{\kappa_{3}}}$,
we have
$y_{m,n}^{{}^{\prime}}\left({\tau_{m}^{s-1}+{\kappa_{3}};\tau_{n}^{s-1}}\right)=u_{m,n}^{{}^{\prime}}\left({\tau_{m}^{s-1}+{\kappa_{3}};\tau_{n}^{s-1}}\right)$.
Since ${\zeta_{m,n}}=\tau_{m}^{s-1}$, we can obtain

$\displaystyle\ {k_{m,n}}$
$\displaystyle=\frac{{-\pi{f_{m}}\sin\left[{2\pi\left({\tau_{m}^{s-1}{f_{m}}-\tau_{n}^{s-1}{f_{n}}}\right)+2\pi{\kappa_{3}}{f_{m}}}\right]}}{{{\kappa_{3}}}}$
$\displaystyle\mathop{=}\limits^{\left(c\right)}\frac{{\pi{f_{m}}\sin\left({2\pi{\kappa_{3}}{f_{m}}}\right)}}{{{\kappa_{3}}}}.$
(91)

where $\left(c\right)$ is due to the sum difference of product method [36]. It
is worth noting that $\mathop{\lim}\limits_{{\kappa_{3}}\to
0}\frac{{\pi{f_{m}}\sin\left({2\pi{\kappa_{3}}{f_{m}}}\right)}}{{{\kappa_{3}}}}=2{\pi^{2}}f_{m}^{2}$,
thus ${k_{m,n}}=2{\pi^{2}}f_{m}^{2}$.

###  B-D Case 4

In this case, we have
$y_{m,n}^{{}^{\prime}}{\left({\tau_{m}^{s-1};\tau_{n}^{s-1}}\right)}=0$ and
${y_{m,n}}\left({\tau_{m}^{s-1};\tau_{n}^{s-1}}\right)=1$ which means that
${\zeta_{m,n}}=\tau_{m}^{s-1}$ and ${\delta_{m,n}}=1$. Similar to case 3, we
have
$y_{m,n}^{{}^{\prime}}\left({\tau_{m}^{s-1}-{\kappa_{4}};\tau_{n}^{s-1}}\right)=u_{m,n}^{{}^{\prime}}\left({\tau_{m}^{s-1}-{\kappa_{4}};\tau_{n}^{s-1}}\right)$,
such that we have

$\displaystyle\ {k_{m,n}}$
$\displaystyle=\frac{{\pi{f_{m}}\sin\left[{2\pi\left({\tau_{m}^{s-1}{f_{m}}-\tau_{n}^{s-1}{f_{n}}}\right)-2\pi{\kappa_{4}}{f_{m}}}\right]}}{{{\kappa_{4}}}}$
$\displaystyle=-\frac{{\pi{f_{m}}\sin\left({2\pi{\kappa_{4}}{f_{m}}}\right)}}{{{\kappa_{4}}}}=-2{\pi^{2}}f_{m}^{2}.$
(92)

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