# On the Level Crossing Rate
of Fluid Antenna Systems

Priyadarshi Mukherjee, Constantinos Psomas, and Ioannis Krikidis  Department
of Electrical and Computer Engineering, University of Cyprus
Email: {mukherjee.priyadarshi, psomas<EMAIL_ADDRESS>
###### Abstract

Multiple-input multiple-output (MIMO) technology has significantly impacted
wireless communication, by providing extraordinary performance gains. However,
a minimum inter-antenna space constraint in MIMO systems does not allow its
integration in devices with limited space. In this context, the concept of
fluid antenna systems (FASs) appears to be a potent solution, where there is
no such restriction. In this paper, we investigate the average level crossing
rate (LCR) of such FASs. Specifically, we derive closed-form analytical
expressions of the LCR of such systems and extensive Monte-Carlo simulations
validate the proposed analytical framework. Moreover, we also demonstrate that
under certain conditions, the LCR obtained coincides with that of a
conventional selection combining-based receiver. Finally, the numerical
results also provide insights regarding the selection of appropriate
parameters that enhance the system performance.

###### Index Terms:

Fluid antenna systems, spatial correlation, level crossing rate, selection
combining, diversity.

##  I Introduction

Multiple-input multiple-output (MIMO) can be considered as one of the most
popular wireless technologies in recent years. The concepts of diversity and
multiplexing gain form the basis of MIMO systems, which led to its
extraordinary performance for wireless communication links. However, there
must be a minimum distance of $\frac{\lambda}{2}$ between the antennas in MIMO
systems, where $\lambda$ is the transmission wavelength [1]. This limits the
integration of MIMO systems inside mobile devices such as tablets and mobile
phones, where the physical space is very limited.

To overcome the aforementioned limitation of MIMO, the novel concept of fluid
antenna system (FAS) was proposed in [2]. FAS is essentially a single antenna
system with $N$ fixed locations (referred as “port”) distributed over a given
space. The idea of FAS is originally motivated by the increasing trend of
using ionized solutions or liquid metals for antennas [3, 4, 5]. The most
interesting aspect of FAS is that an antenna element is no longer kept fixed
at a particular location, but it can switch to a relatively more favorable
location inside the boundaries, if required. Apparently, the objective of FAS
resembles that of traditional transmit antenna selection (TAS) systems [6],
where multiple antennas are deployed at different locations and the antenna
with the strongest signal is selected. However, unlike TAS systems, the single
antenna element in FAS can change position among the predetermined ports. In
this way, an FAS exploits the phenomenon of spatial diversity and the received
signal from the port with the strongest channel condition is selected.
Furthermore, there is no limitation of maintaining a minimum inter-port
distance and as a result, the space making up the FAS may be small with large
number of ports. The work in [2] evaluates the theoretical performance of such
systems in terms of outage probability. It is shown that even with a small
space and a practically feasible number of ports, FAS can significantly
outperform conventional maximum ratio combining-based systems. The work in [7]
investigates the second order statistics of a FAS-based receiver. In
particular, the work evaluates its performance in terms of the level crossing
rate (LCR), average fade duration (AFD), and also ergodic capacity.

As stated above, FAS does not have any constraint on the inter-port distance.
This, unlike in conventional MIMO systems, makes the aspect of spatial
correlation a crucial factor in characterizing its performance limits.
However, the work in [7] does not make this consideration. Motivated by this,
in this work, we investigate the FAS second order statistics by taking the
spatial correlation into account. Specifically, we derive closed-form
analytical expressions of the LCR as a function of both the number of ports
and the associated spatial correlation. We demonstrate that in certain
scenarios, the FAS LCR coincides with that of a conventional selection
combining (SC)-based system with independent and identically distributed
(i.i.d.) channels. To the best of our knowledge, this is the first work that
presents a complete analytical framework to characterize the LCR for a FAS, by
taking into account its practical constraints and limitations.

##  II System model

Figure 1: The considered topology consisting of a single antenna transmitter
and a FAS-based mobile device.

We consider a simple point-to-point topology, where we have a single antenna
transmitter and a $N$-port FAS-based mobile receiver, as depicted in Fig. 1.
As shown in the figure, a typical FAS-based receiver is essentially a single
antenna system with a single radio-frequency (RF) chain. The $N$ ports are
evenly distributed in a linear space of $W\lambda$, where $\lambda$ is the
transmission wavelength and the antenna can switch locations instantly among
the ports [2]. Fig. 1 illustrates that the first port is the reference and the
liquid antenna is always switched to the port with best channel conditions.
The channels at these ports are characterized as [2]

$\displaystyle\begin{cases}\\!h_{1}=\sigma x_{0}+j\sigma y_{0}\\\
\\!h_{k}=\sigma\left({\sqrt{1-\mu_{k}^{2}}x_{k}+\mu_{k}x_{0}}\right)\\\
\quad\>\>+j\sigma\left({\sqrt{1-\mu_{k}^{2}}y_{k}+\mu_{k}y_{0}}\right)\>\>\text{for
}k=2,{\cdots},N,\end{cases}$ (1)

where $x_{0},\cdots,x_{N},y_{0},\cdots,y_{N}$ are independent Gaussian random
variables with zero mean and variance $\frac{1}{2}$, and $\mu_{k}$ $\forall$
$k$ are the parameters that control the spatial correlation between the
channels. Accordingly, we have $\mathbb{E}[|h_{k}|^{2}]=\sigma^{2}$ $\forall$
$k$ and [2]

$\mu_{k}=J_{0}\left(\frac{2\pi(k-1)}{N-1}W\right),\qquad\text{for
}k=2,{\dots},N,$ (2)

where $J_{0}(\cdot)$ is the zero-order Bessel function of the first kind. Note
that due to the oscillatory nature of $J_{0}(\cdot)$, for a given $N$,
$\mu_{k}$ $\forall$ $k$ is not a monotonically decreasing function of $W$. It
may happen that, multiple values of $W$ result in identical $\mu_{k}$. Hence,
to obtain a reasonable range of $W,$ we consider $J_{0}(\cdot)$ till the point
where it reaches the first zero. Accordingly, irrespective of $N$, we obtain
$W\in[0,0.38]$. The FAS always selects the port with the strongest channel
condition, i.e.

$|h_{\mathrm{FAS}}|=\max\\{|h_{1}|,|h_{2}|,{\dots},|h_{N}|\\}.$ (3)

As the ports in an FAS are very close to each other, the aspect of spatial
correlation plays a crucial role in determining which of the $N$ ports is
selected. Accordingly, the joint probability distribution function (PDF) of
$|h_{1}|,|h_{2}|,\cdots,|h_{N}|$ is [2]

$\displaystyle p_{|h_{1}|,\cdots,|h_{N}|}(x_{1},\dots,x_{N})$
$\displaystyle\quad=\prod_{\begin{subarray}{c}k=1\\\ (\mu_{1}\triangleq
0)\end{subarray}}^{N}\\!\\!\frac{2x_{k}}{\sigma^{2}(1-\mu_{k}^{2})}e^{-\frac{x_{k}^{2}+\mu_{k}^{2}x_{1}^{2}}{\sigma^{2}(1-\mu_{k}^{2})}}I_{0}\left({\frac{2~{}\mu_{k}x_{1}x_{k}}{\sigma^{2}(1-\mu_{k}^{2})}}\right)\\!,$
(4)

for $x_{1},{\dots},x_{N}\geq 0$, where $I_{0}(\cdot)$ is the zero-order
modified Bessel function of the first kind. It is important to note that the
mutual coupling does not affect an FAS, as only one antenna element is
activated at each time. Hence, (II) is not a conventional $N$-variate random
variable, but a product of $N$ bi-variate random variables.

##  III Level Crossing Rate for FAS

In this section, we characterize the LCR of an FAS, which is an important
parameter in characterizing the dynamics of any random process. It facilitates
to evaluate the impact of the time-varying channel on the FAS performance. The
LCR enables to estimate the statistics of error occurrence in signal
detection. The LCR of a random process $r$ at threshold $r_{\rm th}$
essentially gives the number of times per unit duration that $r$ crosses
$r_{\rm th}$ in the negative (or positive) direction [8]. Mathematically it
defined as

$L(r_{\rm th})=\int_{0}^{\infty}\dot{r}p_{\dot{R}R}(\dot{r},r_{\rm
th})d\dot{r},$ (5)

where $\dot{r}$ is the time derivative of $r$ and $p_{\dot{R}R}(\dot{r},r)$ is
the joint PDF of $r(t)$ and $\dot{r}(t)$ in an arbitrary instant $t$. For an
isotropic scattering scenario, the time derivative of the signal envelope is
Gaussian distributed with zero mean, irrespective of the fading distribution
[9]. As we aim to analyze the LCR of a FAS, we propose the following theorem
in this direction.

######  Theorem 1.

The LCR for a $N$-port FAS is given by (III).

$\displaystyle L(x_{\rm th})$ $\displaystyle=\frac{\sqrt{2\pi}x_{\rm
th}f_{D}}{\sigma}\Biggl{\\{}e^{-\frac{x_{\rm
th}^{2}}{\sigma^{2}}}\prod_{k=2}^{N}\left[1-Q_{1}\left(\sqrt{\frac{2\mu_{k}^{2}}{\sigma^{2}(1-\mu_{k}^{2})}}x_{\rm
th},\sqrt{\frac{2}{\sigma^{2}(1-\mu_{k}^{2})}}x_{\rm
th}\right)\right]+\sum_{i=2}^{N}\frac{1}{(1-\mu_{i}^{2})}e^{-\frac{x_{\rm
th}^{2}}{\sigma^{2}(1-\mu_{i}^{2})}}$
$\displaystyle\quad\times\int_{0}^{x_{\rm
th}}\frac{2x_{1}}{\sigma^{2}}e^{-\frac{x_{1}^{2}}{\sigma^{2}(1-\mu_{i}^{2})}}I_{0}\left({\frac{2~{}\mu_{i}x_{\rm
th}x_{1}}{\sigma^{2}(1-\mu_{i}^{2})}}\right)\prod_{\begin{subarray}{c}k=2\\\
k\neq
i\end{subarray}}^{N}\left[1-Q_{1}\left(\sqrt{\frac{2\mu_{k}^{2}}{\sigma^{2}(1-\mu_{k}^{2})}}x_{1},\sqrt{\frac{2}{\sigma^{2}(1-\mu_{k}^{2})}}x_{\rm
th}\right)\right]dx_{1}\Biggr{\\}}.$ (6)

###### Proof.

See Appendix A. ∎

We observe from (III) that the LCR is a function of the spatial correlation,
the number of ports, the decision threshold, and the maximum Doppler frequency
of the channel. This LCR of an FAS is different from that of a conventional
SC-based receiver, primarily because of the unique PDF of the channels at the
$N$ ports (as it can be seen from (II)) and also the aspect of the associated
spatial correlation. For the sake of completeness, we consider the following
two extreme cases: $\mu_{k}=0,1$ $\forall$ $k=1,\cdots,N$. The LCR $L(x_{\rm
th})$ corresponding to $\mu_{k}=0$ $\forall$ $k$, i.e. for a no spatial
correlation scenario, is given below.

######  Corollary 1.

For a scenario without spatial correlation, i.e. $\mu_{k}=0$ $\forall$ $k$,
$L(x_{\rm th})$ is

$L(x_{\rm th})=N\sqrt{2\pi}f_{D}\frac{x_{\rm th}}{\sigma}e^{-\frac{x_{\rm
th}^{2}}{\sigma^{2}}}\left(1-e^{-\frac{x_{\rm
th}^{2}}{\sigma^{2}}}\right)^{N-1}.$ (7)

The above corollary follows directly from Theorem 1, by replacing $\mu_{k}=0$
$\forall$ $k=1,\cdots,N$ and using

$Q_{1}(0,b)=\int_{b}^{\infty}xe^{-\frac{x^{2}}{2}}dx=e^{-\frac{b^{2}}{2}}\quad\text{for}\quad
b\geq 0.$ (8)

Without a spatial correlation, for a given set of system parameters, $L(x_{\rm
th})$ obtained in (7) coincides with the LCR of a conventional SC-based
receiver with i.i.d. channels [10, Eq. 18]. The case with $\mu_{k}\\!=\\!1$
$\forall$ $k$ essentially corresponds to the scenario where the $N$ ports are
identical, i.e., there is no need of any switching of the liquid antenna among
the ports.

######  Corollary 2.

For a scenario with $\mu_{k}\\!=\\!1$ $\forall$ $k$, we have

$L(x_{\rm th})=\frac{\sqrt{2\pi}}{\sigma}f_{D}x_{\rm th}e^{-\frac{x_{\rm
th}^{2}}{\sigma^{2}}}.$ (9)

###### Proof.

See Appendix B. ∎

It is interesting to note from the above corollary, that in case of identical
channels, the FAS LCR is independent of $N$. Nevertheless, the analytical
expression of $L(x_{\rm th})$ derived in Theorem 1, being too involved, does
not provide any insightful analysis. Hence, we consider a simple case of
$N=2$.

######  Corollary 3.

For a two-port FAS, the LCR is given by

$\displaystyle L(x_{\rm th})$ $\displaystyle=\frac{2\sqrt{2\pi}f_{D}x_{\rm
th}}{\sigma^{3}(1-\mu^{2})}e^{\left(\\!\\!-\frac{x_{\rm
th}^{2}}{\sigma^{2}(1-\mu^{2})}\right)}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\times\sum_{k=0}^{\infty}\frac{\left(\mu
x_{\rm
th}\right)^{2k}}{(k!)^{2}\left(\sigma^{2}(1-\mu^{2})\right)^{k-1}}\gamma\left(k+1,\frac{x_{\rm
th}^{2}}{\sigma^{2}(1-\mu^{2})}\right).$ (10)

###### Proof.

See Appendix C. ∎

The above corollary demonstrates the effect of parameters such as $x_{\rm
th},\mu,$ and $\sigma$ on the LCR. For example, we observe from (3) that
$L(x_{\rm th})$ is the product of an unimodal function and an increasing
function with respect to $x_{\rm th}$. This implies that $L(x_{\rm th})$ is
also unimodal, i.e. $L(x_{\rm th})$ initially increases with $x_{\rm th}$, but
it starts decreasing after a certain point.

##  IV Numerical Results

We now validate our theoretical analysis with extensive Monte-Carlo
simulations. Without any loss of generality, we consider unit power channels,
i.e. $\mathbb{E}[|h_{k}|^{2}]=\sigma^{2}=1$ $\forall$ $k=1,\cdots,N$, where
$N$ is the number of ports in the FAS and a carrier frequency of $900$ MHz.

Figure 2: Verification of proposed analysis via Monte-Carlo simulations.

Fig. 2 demonstrates the variation of the normalized LCR (NLCR) $L(x_{\rm
th})/f_{D}$ versus the decision threshold $x_{\rm th}$ for two scenarios with
$W=0.1$ and $0.3$, respectively. In this figure, we have considered a $N$-port
FAS, with $N=2,3,$ and $4$, respectively. Note that the values considered are
solely for illustration. We observe that the theoretical results (lines) match
very closely with the simulation results (markers); this verifies our proposed
analytical framework. The figure supports our claim that LCR depends on both
$x_{\rm th}$ and $\mu$. It can be further noted that the LCR increases with
increase of $x_{\rm th}$ until it reaches its maximum and then it decreases.
Moreover, this particular value of $x_{\rm th}$ depends on the value of $W$,
which corroborates the claims made in [11] regarding the effect of optimum
threshold selection. Furthermore, we observe that, the choice of $W$ also has
an impact on the NLCR. Finally, the closed-form expression for NLCR derived in
[7] significantly deviates from the simulation results.

Figure 3: Impact of spatial correlation on NLCR; $x_{\rm th}=28$ dBm.

Fig. 3 illustrates the effect of $N$ on the NLCR performance. For an identical
$N$, a marginal increase of $W$ leads to a significant improvement in
performance; for example, observe the performance gap at $N=16$ between
$W=0.1,0.2,$ and $0.3,$ respectively. It is worth to note, that the best
performance is observed when the channels are independent at the ports, i.e.
$\mu_{k}=0$ $\forall$ $k=1,\cdots,N$. Finally, the figure demonstrates that
irrespectively of the choice of $W$, an FAS asymptotically attains the optimal
performance in terms of NLCR, as $N\rightarrow\infty$; greater the value of
$W$, faster is the convergence. However, from (2), we know that $W$ cannot be
increased arbitrarily due to the oscillatory nature of $J_{0}(\cdot)$; in this
way, the spatial correlation affects the FAS performance. Furthermore, we also
observe that the change of $W$ or $N$ does not affect the NLCR, as in [7].

Fig. 4 depicts the variation of NLCR with $N$ for multiple values of $x_{\rm
th}$. We observe that for both the cases, i.e. (III) and [7], a lower
threshold results in a lower NLCR. Furthermore, we observe that, as also seen
in Fig. 3, [7] is invariant to $N$ (from the dashed lines). On the contrary,
NLCR as derived in (III) is significantly affected with increasing $N$; the
NLCR decreases with $N$. This demonstrates the key benefit of an FAS, where it
is advantageous to have higher $N$ without any inter-port distance constraint.

##  V Conclusion

In this paper, motivated by the practical constraints of a FAS, we proposed a
novel and general analytical framework for the exact evaluation of an
important second order statistical parameter of FAS, namely the LCR. In
particular, by considering the effect of the time-varying nature of fading
channels, we investigated the aspect of spatial correlation in characterizing
this performance metric. Closed form expressions for the LCR were analytically
derived and it was demonstrated that in certain scenarios, they coincide with
the LCR of an SC-based receiver with i.i.d. channels. Finally, we validate our
proposed framework by extensive Monte-Carlo simulations. Based on the
framework presented in this paper, an immediate extension of this work is to
investigate the second order statistics of a multiple FAS-based topology with
a heterogeneous geometry.

## Appendix

###  V-A Proof of Theorem 1

Figure 4: Impact of spatial correlation on NLCR; $W=0.2$.

In this context, the $N$-variate joint PDF $p_{|\dot{h}|,|h|}(\dot{x},x)$ is
given by [12, 8.42]

$\displaystyle p_{|\dot{h}|,|h|}(\dot{x},x)$
$\displaystyle=\sum_{i=1}^{N}p_{|\dot{h_{i}}|}(\dot{x})$
$\displaystyle\\!\\!\times\underbrace{\int_{0}^{x}\\!\\!\cdots\\!\\!\int_{0}^{x}}_{(N-1)-{\rm
fold}}\\!\\!\\!\\!p_{|h_{1}|,\cdots,|h_{N}|}(x_{1},\dots,x_{i}=x,\cdots,x_{N})$
$\displaystyle\times\underbrace{dx_{1}\cdots dx_{k}\cdots
dx_{N}}_{\begin{subarray}{c}(N-1)-{\rm fold}\\\ k\neq i\end{subarray}},$ (11)

where $|\dot{h}_{i}|$ is the time derivative of the signal envelope at the
$i$-th port. Hence, from (5), we obtain the LCR as

$\displaystyle L(x_{\rm th})$
$\displaystyle=\int_{0}^{\infty}\dot{x}p_{|\dot{h}|,|h|}(\dot{x},x)d\dot{x}$
$\displaystyle=\int_{0}^{\infty}\\!\\!\\!\\!\dot{x}\sum_{i=1}^{N}p_{|\dot{h_{i}}|}(\dot{x})\\!\\!$
$\displaystyle\times\underbrace{\int_{0}^{x_{\rm
th}}\\!\\!\\!\\!\cdots\\!\\!\int_{0}^{x_{\rm th}}}_{(N-1)-{\rm
fold}}\\!\\!\\!\\!\\!\\!p_{|h_{1}|,\cdots,|h_{N}|}(x_{1},\dots,x_{i}=x_{\rm
th},\cdots,x_{N})$ $\displaystyle\times\underbrace{dx_{1}\cdots dx_{k}\cdots
dx_{N}}_{\begin{subarray}{c}(N-1)-{\rm fold}\\\ k\neq
i\end{subarray}}d\dot{x}.$ (12)

Furthermore, in case of an identically distributed Rayleigh fading scenario,
$p_{|\dot{h_{i}}|}(\dot{x})$ $\forall$ $i=1,\cdots,N,$ follows a zero mean
Gaussian PDF [9] with variance
$\sigma_{\dot{X}}^{2}=\pi^{2}\sigma^{2}f_{D}^{2}$, where $f_{D}$ is the
maximum Doppler frequency. As a result, we obtain

$\int_{0}^{\infty}\\!\\!\dot{x}p_{|\dot{h_{i}}|}(\dot{x})d\dot{x}=\frac{\sigma_{\dot{X}}}{\sqrt{2\pi}}=\sqrt{\frac{\pi}{2}}\sigma
f_{D},\>\>\forall i=1,\cdots,N.$ (13)

By combining (V-A) and (13), we have

$\displaystyle L(x_{\rm th})=\sqrt{\frac{\pi}{2}}\sigma f_{D}$
$\displaystyle\times\sum_{i=1}^{N}\underbrace{\int_{0}^{x_{\rm
th}}\\!\\!\cdots\int_{0}^{x_{\rm th}}}_{(N-1)-{\rm
fold}}p_{|h_{1}|,\cdots,|h_{N}|}(x_{1},\dots,x_{i}=x_{\rm th},\cdots,x_{N})$
$\displaystyle\times\underbrace{dx_{1}\cdots dx_{k}\cdots
dx_{N}}_{\begin{subarray}{c}(N-1)-{\rm fold}\\\ k\neq i\end{subarray}}.$ (14)

The summation term in (V-A) is alternatively written as

$\displaystyle\sum_{i=1}^{N}\underbrace{\int_{0}^{x_{\rm
th}}\\!\\!\cdots\int_{0}^{x_{\rm th}}}_{(N-1)-{\rm
fold}}p_{|h_{1}|,\cdots,|h_{N}|}(x_{1},\dots,x_{i}=x_{\rm th},\cdots,x_{N})$
$\displaystyle\quad\times\underbrace{dx_{1}\cdots dx_{k}\cdots
dx_{N}}_{\begin{subarray}{c}(N-1)-{\rm fold}\\\ k\neq i\end{subarray}}$
$\displaystyle=\underbrace{\int_{0}^{x_{\rm th}}\\!\\!\cdots\int_{0}^{x_{\rm
th}}}_{(N-1)-{\rm fold}}p_{|h_{1}|,\cdots,|h_{N}|}(x_{1}=x_{\rm
th},\cdots,x_{N})\underbrace{dx_{2}\cdots dx_{N}}_{(N-1)-{\rm fold}}$
$\displaystyle\quad+\sum_{i=2}^{N}\underbrace{\int_{0}^{x_{\rm
th}}\\!\\!\cdots\\!\\!\int_{0}^{x_{\rm th}}}_{(N-1)-{\rm
fold}}\\!\\!p_{|h_{1}|,\cdots,|h_{N}|}(x_{1},\dots,x_{i}=x_{\rm
th},\cdots,x_{N})$ $\displaystyle\quad\times\underbrace{dx_{1}\cdots
dx_{k}\cdots dx_{N}}_{\begin{subarray}{c}(N-1)-{\rm fold}\\\ k\neq
i\end{subarray}}.$ (15)

The first term of (V-A) is evaluated as

$\displaystyle\underbrace{\int_{0}^{x_{\rm th}}\cdots\int_{0}^{x_{\rm
th}}}_{(N-1)-{\rm fold}}p_{|h_{1}|,\cdots,|h_{N}|}(x_{1}=x_{\rm
th},\cdots,x_{N})\underbrace{dx_{2}\cdots dx_{N}}_{(N-1)-{\rm fold}}$
$\displaystyle\overset{(a)}{=}\frac{2x_{\rm th}}{\sigma^{2}}e^{-\frac{x_{\rm
th}^{2}}{\sigma^{2}}}\int_{0}^{x_{\rm th}}\cdots\int_{0}^{x_{\rm
th}}\prod_{k=2}^{N}\frac{2x_{k}}{\sigma^{2}(1-\mu_{k}^{2})}e^{-\frac{x_{k}^{2}+\mu_{k}^{2}x_{\rm
th}^{2}}{\sigma^{2}(1-\mu_{k}^{2})}}$ $\displaystyle\quad\times
I_{0}\left({\frac{2~{}\mu_{k}x_{\rm
th}x_{k}}{\sigma^{2}(1-\mu_{k}^{2})}}\right)dx_{2}\cdots dx_{N}$
$\displaystyle=\frac{2x_{\rm th}}{\sigma^{2}}e^{-\frac{x_{\rm
th}^{2}}{\sigma^{2}}}\prod_{k=2}^{N}\int_{0}^{x_{\rm
th}}\frac{2x_{k}}{\sigma^{2}(1-\mu_{k}^{2})}e^{-\frac{x_{k}^{2}+\mu_{k}^{2}x_{\rm
th}^{2}}{\sigma^{2}(1-\mu_{k}^{2})}}$ $\displaystyle\quad\times
I_{0}\left({\frac{2~{}\mu_{k}x_{\rm
th}x_{k}}{\sigma^{2}(1-\mu_{k}^{2})}}\right)dx_{k}$
$\displaystyle\overset{(b)}{=}\frac{2x_{\rm th}}{\sigma^{2}}e^{-\frac{x_{\rm
th}^{2}}{\sigma^{2}}}\prod_{k=2}^{N}\left[1-Q_{1}\left(\sqrt{\frac{2\mu_{k}^{2}}{\sigma^{2}(1-\mu_{k}^{2})}}x_{\rm
th},\right.\right.$
$\displaystyle\quad\left.\left.\sqrt{\frac{2}{\sigma^{2}(1-\mu_{k}^{2})}}x_{\rm
th}\right)\right],$ (16)

where $(a)$ follows from (II), $Q_{1}(\cdot,\cdot)$ is the first-order Marcum
$Q$-function, and $(b)$ follows from [13, Eq. 10]. Hereafter, the second term
of (V-A) is expanded as (V-A), where $(c)$ is based on [13, Eq. 10] and the
fact that the joint PDF stated in (II) is not a regular multivariate Rayleigh
PDF. This distribution is a product of $N$ pairs of bi-variate Rayleigh PDFs,
with the first port of the FAS being the reference point for all the remaining
$N-1$ ports. By combining (V-A), (V-A), (V-A), and (V-A), we obtain (III),
which completes the proof.

$\displaystyle\sum_{i=2}^{N}\underbrace{\int_{0}^{x_{\rm
th}}\\!\\!\cdots\int_{0}^{x_{\rm th}}}_{(N-1)-{\rm
fold}}p_{|h_{1}|,\cdots,|h_{N}|}(x_{1},\dots,x_{i}=x_{\rm
th},\cdots,x_{N})\underbrace{dx_{1}\cdots dx_{k}\cdots
dx_{N}}_{\begin{subarray}{c}(N-1)-{\rm fold}\\\ k\neq i\end{subarray}}$
$\displaystyle=\sum_{i=2}^{N}\int_{0}^{x_{\rm
th}}\\!\\!\cdots\\!\\!\int_{0}^{x_{\rm th}}\\!\\!\frac{2x_{\rm
th}}{\sigma^{2}(1-\mu_{i}^{2})}e^{-\frac{x_{\rm
th}^{2}+\mu_{i}^{2}x_{1}^{2}}{\sigma^{2}(1-\mu_{i}^{2})}}I_{0}\left({\frac{2~{}\mu_{i}x_{\rm
th}x_{1}}{\sigma^{2}(1-\mu_{i}^{2})}}\right)\\!\\!\prod_{\begin{subarray}{c}k=1\\\
k\neq
i\end{subarray}}^{N}\frac{2x_{k}}{\sigma^{2}(1-\mu_{k}^{2})}e^{-\frac{x_{k}^{2}+\mu_{k}^{2}x_{1}^{2}}{\sigma^{2}(1-\mu_{k}^{2})}}I_{0}\left({\frac{2~{}\mu_{k}x_{k}x_{1}}{\sigma^{2}(1-\mu_{k}^{2})}}\right)dx_{1}\cdots
dx_{k}\cdots dx_{N}$ $\displaystyle\overset{(c)}{=}\sum_{i=2}^{N}\frac{2x_{\rm
th}}{\sigma^{2}(1-\mu_{i}^{2})}e^{-\frac{x_{\rm
th}^{2}}{\sigma^{2}(1-\mu_{i}^{2})}}\\!\\!\int_{0}^{x_{\rm
th}}\frac{2x_{1}}{\sigma^{2}}e^{-\frac{x_{1}^{2}}{\sigma^{2}(1-\mu_{i}^{2})}}I_{0}\left(\\!{\frac{2~{}\mu_{i}x_{\rm
th}x_{1}}{\sigma^{2}(1-\mu_{i}^{2})}}\\!\right)\\!\\!\prod_{\begin{subarray}{c}k=2\\\
k\neq
i\end{subarray}}^{N}\left[\\!1-Q_{1}\left(\sqrt{\frac{2\mu_{k}^{2}}{\sigma^{2}(1-\mu_{k}^{2})}}x_{1},\sqrt{\frac{2}{\sigma^{2}(1-\mu_{k}^{2})}}x_{\rm
th}\right)\\!\right]dx_{1}.$ (17)

###  V-B Proof of Corollary 2

The case of $\mu_{k}=1$ $\forall$ $k$ implies identical channels at all the
ports. As a result, we obtain

$\displaystyle L(x_{\rm th})$
$\displaystyle=\int_{0}^{\infty}\\!\\!\dot{x}p_{|\dot{h}|,|h|}(\dot{x},x_{\rm
th})d\dot{x}\overset{(a)}{=}p_{|h|}(x_{\rm
th})\\!\\!\int_{0}^{\infty}\dot{x}p_{|\dot{h}|}(\dot{x})d\dot{x}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=\frac{2x_{\rm
th}}{\sigma^{2}}e^{-\frac{x_{\rm
th}^{2}}{\sigma^{2}}}\\!\\!\int_{0}^{\infty}\\!\\!\dot{x}p_{|\dot{h_{i}}|}(\dot{x})d\dot{x}\overset{(b)}{=}\frac{\sqrt{2\pi}}{\sigma}f_{D}x_{\rm
th}e^{-\frac{x_{\rm th}^{2}}{\sigma^{2}}},$ (18)

where $(a)$ follows from
$p_{|\dot{h}|,|h|}(\dot{x},x)=p_{|\dot{h}|}(\dot{x})p_{|h|}(x)$ [9, Eq. 2.97]
and $(b)$ follows from (13). Hence, the proof.

###  V-C Proof of Corollary 3

Since we are considering a two-port scenario, we take into account the joint
PDF of the channel at these two ports. Hence, by replacing $N=2$ in (II), the
joint PDF becomes

$\displaystyle p_{|h_{1}|,|h_{2}|}(x_{1},x_{2})$
$\displaystyle=\frac{4x_{1}x_{2}}{\sigma^{4}(1-\mu_{2}^{2})}e^{-\frac{x_{1}^{2}+x_{2}^{2}}{\sigma^{2}(1-\mu_{2}^{2})}}$
$\displaystyle\\!\\!\\!\\!\\!\\!\times
I_{0}\left({\frac{2\mu_{2}x_{1}x_{2}}{\sigma^{2}(1-\mu_{2}^{2})}}\right)\\!,\quad\text{for}\quad
x_{1},x_{2}\geq 0.$ (19)

Thus, by replacing $N=2$ in (III) and after some trivial algebraic
manipulations, we obtain

$\displaystyle L(x_{\rm th})$ $\displaystyle=\sqrt{\frac{\pi}{2}}\sigma
f_{D}\left(\int_{0}^{x_{\rm th}}p_{|h_{1}|,|h_{2}|}(x_{\rm
th},x_{2})dx_{2}\right.$ $\displaystyle\quad\left.+\int_{0}^{x_{\rm
th}}p_{|h_{1}|,|h_{2}|}(x_{1},x_{\rm th})dx_{1}\right)$
$\displaystyle\overset{(a)}{=}\sqrt{2\pi}\sigma f_{D}\\!\\!\int_{0}^{x_{\rm
th}}p_{|h_{1}|,|h_{2}|}(x_{\rm th},x_{2})dx_{2}$
$\displaystyle=\frac{4\sqrt{2\pi}\sigma f_{D}x_{\rm
th}}{\sigma^{4}(1-\mu^{2})}\exp\left(\\!\\!-\frac{x_{\rm
th}^{2}}{\sigma^{2}(1-\mu^{2})}\right)\int_{0}^{x_{\rm th}}\\!\\!x_{2}$
$\displaystyle\quad\times\\!\exp\left(\\!\\!-\frac{x_{2}^{2}}{\sigma^{2}(1-\mu^{2})}\\!\\!\right)\\!\\!I_{0}\left(\\!{\frac{2\mu
x_{\rm th}x_{2}}{\sigma^{2}(1-\mu^{2})}}\\!\right)\\!dx_{2},$ (20)

where $(a)$ follows from (V-C) and the integral $\int_{0}^{x_{\rm
th}}x_{2}\\!\exp\left(\\!\\!-\frac{x_{2}^{2}}{\sigma^{2}(1-\mu^{2})}\\!\\!\right)I_{0}\left(\\!{\frac{2\mu
x_{\rm th}x_{2}}{\sigma^{2}(1-\mu^{2})}}\\!\right)\\!dx_{2}$ is obtained as

$\displaystyle\int_{0}^{x_{\rm
th}}x_{2}\exp\left(-\frac{x_{2}^{2}}{\sigma^{2}(1-\mu^{2})}\right)I_{0}\left({\frac{2\mu
x_{\rm th}x_{2}}{\sigma^{2}(1-\mu^{2})}}\right)dx_{2}$
$\displaystyle\overset{(b)}{=}\\!\\!\int_{0}^{x_{\rm
th}}\\!\\!x_{2}\exp\left(\\!\\!-\frac{x_{2}^{2}}{\sigma^{2}(1-\mu^{2})}\right)\\!\\!\sum_{k=0}^{\infty}\frac{1}{(k!)^{2}}\left(\frac{\mu
x_{\rm th}x_{2}}{\sigma^{2}(1-\mu^{2})}\right)^{\\!\\!2k}\\!\\!dx_{2}$
$\displaystyle\overset{(c)}{=}\sum_{k=0}^{\infty}\frac{1}{(k!)^{2}}\left(\frac{\mu
x_{\rm th}}{\sigma^{2}(1-\mu^{2})}\right)^{2k}$
$\displaystyle\quad\times\int_{0}^{x_{\rm
th}}\exp\left(-\frac{x_{2}^{2}}{\sigma^{2}(1-\mu^{2})}\right)x_{2}^{2k+1}dx_{2}$
$\displaystyle=\frac{1}{2}\\!\sum_{k=0}^{\infty}\\!\frac{\left(\mu x_{\rm
th}\right)^{2k}}{(k!)^{2}\left(\sigma^{2}(1-\mu^{2})\right)^{k-1}}\gamma\left(\\!k+1,\frac{x_{\rm
th}^{2}}{\sigma^{2}(1-\mu^{2})}\\!\right).$ (21)

Here $(b)$ follows from [14, 8.445] and by assuming $0<\mu<1$, $(c)$ follows
from changing the order of summation and integration, and
$\gamma(\cdot,\cdot)$ denotes the lower incomplete Gamma function. As a
result, by substituting (V-C) in (V-C), we finally obtain (3).

## Acknowledgment

This work was co-funded by the European Regional Development Fund and the
Republic of Cyprus through the Research and Innovation Foundation, under the
project INFRASTRUCTURES/1216/0017 (IRIDA). It has also received funding from
the European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (Grant agreement No. 819819).

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# Reconfigurable Intelligent Surface-induced Randomness for mmWave Key
Generation

Shubo Yang, Han Han, Yihong Liu, , Weisi Guo, , Zhibo Pang, and Lei Zhang  S.
Yang is with the Glasgow College, University of Glasgow (e-mail:
2429400y@student.gla.ac.uk). H. Han is with Electrical and Computer
Engineering Department, University of Toronto, ON M5S 3G4, Canada (e-mail:
johnny.han@mail.utoronto.ca). Y. Liu and L. Zhang are with James Watt School
of Engineering, University of Glasgow, Glasgow G12 8QQ, U.K. (e-mail:
<EMAIL_ADDRESS>lei.zhang@glasgow.ac.uk). W. Guo is with School of
Aerospace, Transport and Manufacturing, Cranfield University, MK43 0AL
Cranfield, U.K. (e-mail: weisi.guo@cranfield.ac.uk). Z. Pang is with
Department of Automation Technology, ABB Corporate Research Sweden, Vasteras,
Sweden, and Department of Intelligent Systems, Royal Institute of Technology
(KTH), Stockholm, Sweden. (e-mail<EMAIL_ADDRESS>zhibo@kth.se).
Funding information: Zhibo Pang’s work is partly funded by the Swedish
Foundation for Strategic Research (SSF) through the project APR20-0023.

###### Abstract

Secret key generation in physical layer security exploits the unpredictable
random nature of wireless channels. The millimeter-wave (mmWave) channels have
limited multipath and channel randomness in static environments. In this
paper, for mmWave secret key generation of physical layer security, we use a
reconfigurable intelligent surface (RIS) to induce randomness directly in
wireless environments, without adding complexity to transceivers. We consider
RIS to have continuous individual phase shifts (CIPS) and derive the RIS-
assisted reflection channel distribution with its parameters. Then, we propose
continuous group phase shifts (CGPS) to increase the randomness specifically
at legal parties. Since the continuous phase shifts are expensive to
implement, we analyze discrete individual phase shifts (DIPS) and derive the
corresponding channel distribution, which is dependent on the quantization
bit. We then derive the secret key rate (SKR) to evaluate the randomness
performance. With the simulation results verifying the analytical results,
this work explains the mathematical principles and lays a foundation for
future mmWave evaluation and optimization of artificial channel randomness.

###### Index Terms:

Physical layer security, secret key generation, reconfigurable intelligent
surface, intelligent reflecting surface.

##  I Introduction

Wireless networks are becoming ubiquitous nowadays and in the future Internet
of Things (IoT) systems. However, their broadcast nature makes them vulnerable
to malicious attacks. Classic encryption schemes, such as advanced encryption
standard and public key cryptography, are dependent on cryptography
computation techniques [1]. The applications of classic schemes to IoT devices
and wireless sensor networks (WSNs) bring challenges, since the devices and
sensor nodes have small sizes and limited computational capability. Thus,
extensive research is carried out in secret key generation of physical layer
security, where the legitimate users extract keys from their correlated
observations of the reciprocal channel in a lightweight manner [2]. The
correlation of channels makes it possible to generate keys without key
exchange, and the dynamic uniqueness of the channel prevents eavesdroppers
from mimicking. While in dynamic environments the movements of users or
objects are sufficient to produce randomness, the randomness is limited in
static environments, such as in open terrain with no moving objects. Besides,
the millimeter wave (mmWave) communication is envisioned as a significant
technology for the fifth generation (5G) networks and beyonds [3]. Thus, the
security in mmWave static environments needs to be researched.

The newly developed reconfigurable intelligent surface (RIS), also known as
Intelligent Reflecting Surface (IRS), has the potential to produce artificial
randomness in mmWave static environments. RIS is a two-dimensional surface
consisting of a large number of passive low-cost reflecting elements [4]. Each
scattering element of RIS is independently capable of altering the amplitude
and/or phase of the incident signals. Additionally, RIS becomes more important
in high-frequency band communications, e.g., mmWave and THz communications
that have severe coverage issues. The existing RIS applications mainly target
indoor static scenarios [5]. Therefore, RIS can be easily incorporated in
mmWave static environments. By adding RIS, artificial randomness can be
produced. The randomness does not rely on dynamic environments and is produced
directly in the channel, without needing increased transceiver costs.

Research has been done on RIS-assisted key generation. Random shifting RIS is
applied to increase the secure transmission rate, and the time allocation for
key generation and transmission is designed [6]. RIS with discrete phase
shifts is adopted to generate secret keys, and the secret key rate (SKR) is
derived [7]. The practical implementation of using RIS in the OFDM system is
conducted [8]. However, most existing literature focuses on sub-6 GHz systems
and models on Gaussian channels. The mmWave channels have a poor scattering
nature and exhibit limited multipath, so they may not conform to Gaussian
channels. Moreover, the channel distribution resulted from RIS phase shifts
and element number is still unknown. Besides, most previous works assume each
RIS element phase shift is independent and identically distributed (i.i.d),
without changing phase shift distributions to improve randomness.

Therefore, in this paper, the RIS-assisted mmWave key generation is proposed.
We model RIS-induced randomness in mmWave key generation, and we focus on the
fundamental analytical derivations of channel distribution with random RIS
weights. For the random weights, we consider applying continuous individual
phase shifts (CIPS) on each element and continuous group phase shifts (CGPS)
on elements in groups to produce higher randomness. In addition, we consider
channels for both continuous111The continuous RIS weights are difficult to
implement. This is because more levels of weight phase shift result in more
costs, which is not scalable to a large number of elements [9]. The continuous
weights’ performance is the upper limit for discrete weights when weights’
quantization bits approach infinity. and discrete phase shifts and compare
their performance. To summarize, the main contributions of this paper are as
follows.

  * •

The RIS-induced channel distribution and its parameters are derived, given RIS
is a uniform rectangular array (URA) and the elements have CIPS. As the
result, an artificial Rayleigh/Rician fading is induced directly in the
environments.

  * •

To increase the amount of induced randomness, CGPS is proposed and the channel
is derived. The channel variance for legal parties increases, and there is a
tradeoff between group number and group size. The discrete individual phase
shifts (DIPS) is also discussed based on quantization bits.

  * •

The SKR is derived for CIPS, CGPS, and DIPS, to evaluate the performance of
artificial randomness.

_Notations:_ Bold-faced letters are used to denote matrix or column vectors,
while lightfaced letters are used to denote scalar quantities. Superscripts
$(\cdot)^{T}$, $(\cdot)^{*}$, and $(\cdot)^{H}$ represent the transpose,
conjugate and conjugate transpose operations, respectively. $\odot$ denotes
the point-wise multiplication. We use the notations shown in Table I in this
paper.

TABLE I: Notations

Symbol | Definition
---|---
$\lambda$ | Wavelength
$k$ | Wave number, where $k=\frac{2\pi}{\lambda}$
$d$ | Element spacing in RIS
$M$ | Element number of the RIS
$m$ | An integer in range $[1,M]$
$\phi_{m}$ | The phase shift of the $m$-th RIS element weight
$\psi$ | The incident or reflected azimuth angle
$\theta$ | The incident or reflected elevation angle
$B$ | RIS discrete weight quantization bit
$\bm{R}$ | The covariance between the real and imaginary parts of a complex Gaussian distribution
$R_{s}$ | Secret key rate
$U(a,b)$ | Indicate a uniform distribution on interval $(a,b)$
$N(\mu,\sigma^{2})$ | Indicate a Gaussian distribution with mean $\mu$ and variance $\sigma^{2}$
$h_{(\cdot)}$ | Indicate the channel
$(\cdot)_{x},(\cdot)_{y}$ | Indicate physical quantities on $x$-axis and $y$-axis, respectively
$(\cdot)_{real},(\cdot)_{imag}$ | Indicate quantities of the real and imaginary parts, respectively
$(\cdot)_{cos},(\cdot)_{sin}$ | Indicate quantities of cosine and sine functions, respectively
$(\cdot)_{i},(\cdot)_{o}$ | Indicate incident and reflected angles, respectively
$(\cdot)_{CI}$ | Indicate quantities when RIS weights have continuous individual phase shifts
$(\cdot)_{CG}$ | Indicate quantities when RIS weights have continuous group phase shifts
$(\cdot)_{DI}$ | Indicate quantities when RIS weights have discrete individual phase shifts

##  II System Model

Consider an RIS-assisted wireless communication between legitimate parties,
Alice and Bob. There is also an eavesdropper Eve, who passively listens to the
communication. Assume all the parties are operated under a mmWave static
environment. We consider the narrowband line-of-sight (LOS) transmission in
the far field between Alice/Bob and RIS.

Figure 1: RIS-assisted secret key generation model.

This assumption can be justified since the multipath effect is constrained and
the high path loss makes non-line-of-sight (NLOS) paths’ power small. The RIS
provides cascaded LOS paths. Additionally, the path loss is a constant related
to distance and can be easily predicted by eavesdropper [7]. Since the
transmission channel between Alice and Bob is reciprocal, we only consider the
situation where Alice is transmitting signals to Bob to avoid redundancy.

As shown in Fig. 1. Alice and Bob aim at establishing a random secret
cryptographic key based on their reciprocal communication channel and reducing
the key leakage to Eve. They measure the common channel through their
exchanged signals and generate keys using parameters of the received signals,
e.g. channel state information (CSI) and received signal strength (RSS). When
Alice transmits signal $s$, the received signal at Bob can be expressed as

$y=(h_{ab}+\bm{h_{rb}^{H}}\bm{W}\bm{h_{ar}})s+z\ ,$ (1)

where $h_{ab}\in\mathbb{C}^{1\times 1}$, $\bm{h_{ar}}\in\mathbb{C}^{M\times
1}$, $\bm{h_{rb}}\in\mathbb{C}^{M\times 1}$ represent the direct channel
between Alice and Bob, the channel between Alice and RIS, and the channel
between RIS and Bob, $z\sim\mathcal{CN}(0,\sigma_{z}^{2})$ is the additive
white Gaussian noise (AWGN), and $\bm{W}$ is the diagonal weight matrix with
each entity on the diagonal being the RIS weight of each surface element. The
weight of the $m$-th surface element is represented as $e^{j\phi_{m}}$.

To generate shared keys in static environments and reduce the key leakage to
Eve, it is crucial to increase the channel randomness through RIS. As in (1),
though $\bm{h_{ar}}$ and $\bm{h_{rb}}$ are static in mmWave static
environments, when RIS weights $\bm{W}$ vary,
$\bm{h_{rb}^{H}}\bm{W}\bm{h_{ar}}$ becomes dynamic. Thus, RIS becomes the
major source of channel randomness through varying the reflection channel
$\widetilde{H}=\bm{h_{rb}^{H}}\bm{W}\bm{h_{ar}}$. In this case, we focus on
$\widetilde{H}$, which can be expressed using steering vectors and weight
vector [10] as

$\widetilde{H}=\bm{w}^{H}\cdot[\bm{a}(\bm{\Omega}_{i})\odot\bm{a}(\bm{\Omega}_{o})]\
,$ (2)

where $\bm{w}$ is the weight vector with each entity being the diagonal entity
of $\bm{W}$, i.e., weight of each surface element. Besides, the steering
vector of the channels between Alice and RIS and between RIS and Bob are

$\bm{a}(\bm{\Omega}_{i})=[a(\bm{\Omega}_{i,1}),\cdots,a(\bm{\Omega}_{i,m}),\cdots,a(\bm{\Omega}_{i,M})]^{T}$
(3)
$\bm{a}(\bm{\Omega}_{o})=[a(\bm{\Omega}_{o,1}),\cdots,a(\bm{\Omega}_{o,m}),\cdots,a(\bm{\Omega}_{o,M})]^{T}\
,$ (4)

where $\bm{\Omega}_{i,m}$ and $\bm{\Omega}_{o,m}$ are the terms characterizing
the incident spatial information from Alice at RIS $m$-th element and the
reflected spatial information from $m$-th element to Bob, respectively. Since
the environment is static, $\bm{\Omega}_{i}$ and $\bm{\Omega}_{o}$ are fixed,
and $\widetilde{H}$ becomes a function of solely $\bm{w}$. Note that averaging
$\widetilde{H}$ derived from all possible pairs of $\bm{\Omega}_{i}$ and
$\bm{\Omega}_{o}$ provides an estimation of the overall channel distribution.

##  III RIS-Assisted Key Generation

In this section, we first derive the probability density function (p.d.f) of
reflection channel $\widetilde{H}$ with a given random RIS having CIPS and
CGPS. Individual shifting means that the RIS phase shifts
$\\{\bm{\phi}=[\phi_{1},\cdots,\phi_{m},\cdots,\phi_{M}],m\in[1,M]\\}$ are
i.i.d. Group shifting means the elements phase shifts in a group are first
implemented maximum ratio transmission (MRT) and then added a random phase
shift. Next, we consider DIPS. The channel for 1-bit is found different from
other quantization bits. Since group phase shifts need MRT to compensate for
the phase difference caused by element distance, the discrete group phase
shifts (DGPS) need multiple levels to realize the compensation. Implementing
multiple-level discrete phase shifts is expensive compared to low-level phase
shifts. Thus, we only discuss CGPS, and DGPS will be our future work.

Note that a uniform linear array (ULA) can be considered a special case of URA
while URA is a typical planar array structure. Thus, we consider RIS in URA
configuration, which can be further extended into other complex RIS shapes.

###  III-A Continuous Individual Phase Shifts

Consider a URA RIS with $M_{x}$ and $M_{y}$ elements equally spaced on x-axis
and y-axis, respectively, where the total element number is $M=M_{x}\cdot
M_{y}$. The incident and reflected angles are described by
$\bm{\Omega}_{i}=(\psi_{i},\theta_{i})$ and
$\bm{\Omega}_{o}=(\psi_{o},\theta_{o})$. Assume the i.i.d RIS weight phase
shifts $\bm{\phi}$ are continuous uniformly distribution $\phi_{m}\sim
U(0,2\pi)$. Then, the multiplication of steering vectors in (2) with elements
spacing along x-axis $d_{x}$ and y-axis $d_{y}$ can be expressed as

$\displaystyle\bm{a}(\bm{\Omega}_{i})\odot\bm{a}(\bm{\Omega}_{o})=$
$\displaystyle[1,\cdots,e^{j(\xi_{x}\cdot(m_{x}-1)+\xi_{y}\cdot(m_{y}-1))},$
(5)
$\displaystyle\cdots,e^{j(\xi_{x}\cdot(M_{x}-1)+\xi_{y}\cdot(M_{y}-1))}]^{T}\
,$

where $m_{x}\in[1,M_{x}]$ and $m_{y}\in[1,M_{y}]$ are integers [11].
Additionally, $k=\frac{2\pi}{\lambda}$ is the wave number and

$\xi_{x}=kd_{x}(\cos\psi_{i}\sin\theta_{i}+\cos\psi_{o}\sin\theta_{o})\ ,$ (6)
$\xi_{y}=kd_{y}(\sin\psi_{i}\sin\theta_{i}+\sin\psi_{o}\sin\theta_{o})\ .$ (7)

Consequently, the reflection channel following (2) can be constructed as

$\displaystyle\widetilde{H}\\!\\!=$ $\displaystyle
1+\cdots+\\!e^{j[\phi_{m}+(\xi_{x}\cdot(m_{x}-1)+\xi_{y}\cdot(m_{y}-1))]}$ (8)
$\displaystyle+\cdots+e^{j[\phi_{M}+(\xi_{x}\cdot(M_{x}-1)+\xi_{y}\cdot(M_{y}-1))]}$
$\displaystyle=$
$\displaystyle\sum_{m_{y}=1}^{M_{y}}\sum_{m_{x}=1}^{M_{x}}e^{j(\phi_{m}+\alpha)}$
$\displaystyle=$
$\displaystyle\sum_{m_{y}=1}^{M_{y}}\sum_{m_{x}=1}^{M_{x}}\cos(\phi_{m}+\alpha)+j\sum_{m_{y}=1}^{M_{y}}\sum_{m_{x}=1}^{M_{x}}\sin(\phi_{m}+\alpha)\
,$

where

$m=(m_{y}-1)\cdot M_{x}+m_{x}\ ,\
\alpha=(m_{x}-1)\cdot\xi_{x}+(m_{y}-1)\cdot\xi_{y}\ .$ (9)

According to the central limit theorem (CLT), $\widetilde{H}$ converges to a
complex Gaussian distribution when $M$ is large enough. This assumption could
be justified since a practical RIS usually has an extremely large number of
elements, e.g. more than tens of elements [5]. Therefore, the real and
imaginary parts both converge to Gaussian distributions that are fully
determined by means and variances. The mean and variance of the real part of
$\widetilde{H}_{CI}$ can be expressed as

$\mu_{real,CI}\\!=\\!\sum_{m_{y}=1}^{M_{y}}\\!\\!\sum_{m_{x}=1}^{M_{x}}\\!\\!\mu_{cos,CI}\
,\
\sigma^{2}_{real,CI}\\!=\\!\\!\sum_{m_{y}=1}^{M_{y}}\\!\\!\sum_{m_{x}=1}^{M_{x}}\\!\\!\sigma^{2}_{cos,CI}\
,$ (10)

where

$\mu_{cos,CI}=\int_{0}^{2\pi}\cos(\phi_{m}+\alpha)f(\phi_{m})d\phi_{m}=0\ .$
(11)
$\sigma^{2}_{cos,CI}\\!\\!=\\!\\!\int_{0}^{2\pi}\\!\\!\\!\\!\cos^{2}(\phi_{m}+\alpha)f(\phi_{m})d\phi_{m}-\mu_{cos,CI}^{2}\\!=\\!\frac{1}{2}.$
(12)

The imaginary part can be derived similar to the real part. By substituting
(11) (12) into (10), the channel real and imaginary parts are

$\Re{(\tilde{H}_{CI})}\sim N(0,\frac{M}{2})\ ,\ \Im{(\tilde{H}_{CI})}\sim
N(0,\frac{M}{2})\ .$ (13)

Since the real and imaginary parts of $\widetilde{H}_{CI}$ constitute a joint
Gaussian distribution and the covariance of them equals 0, the real and
imaginary parts are independent. The distribution of $\widetilde{H}_{CI}$ is
also independent of $\bm{\Omega}_{i}$ and $\bm{\Omega}_{o}$. With any pair of
incident and reflected angles, the mean of $\widetilde{H}_{CI}$ is
$\mu_{CI}=0$ while the variance of $\widetilde{H}_{CI}$ can be calculated as

$\sigma^{2}_{CI}=\mathbb{E}[(\widetilde{H}_{CI}-\mu_{CI})(\widetilde{H}_{CI}-\mu_{CI})^{*}]=M\
.$ (14)

Due to the channel parameters CSI and RSS are normally used to generate keys,
the magnitude and phase distributions are also important. According to the
joint Gaussian distribution, the result magnitude p.d.f and phase p.d.f of
$\widetilde{H}_{CI}$ are Rayleigh and uniform distributions. When the direct
path and reflection path are both considered, the direct path adds a non-zero
mean, and the induced Rayleigh fading will turn into Rician fading. Hence, the
CIPS RIS produces randomness by inducing an artificial Rayleigh/Rician fading
in the environments.

###  III-B Continuous Group Phase Shifts

Different from the i.i.d elements in CIPS, group shifting means first using
MRT to achieve maximum transmission rate at Alice and Bob, then shifting the
phase randomly. This induces more randomness at Alice and Bob.

The total elements $M$ is divided into groups of $q$ elements. There are
$N=\lfloor\frac{M}{q}\rfloor$ groups in total. The $n$-th group has the added
random phase $\phi_{n,rand}\sim U(0,2\pi)$. When the $m$-th element is in
$n$-th group, the phase shift for $m$-th element is

$\phi_{m,CG}=\phi_{m,MRT}+\phi_{n,rand}\ ,$ (15)

where

$\phi_{m,MRT}=\frac{[\bm{a}(\bm{\Omega}_{i,m})\cdot\bm{a}(\bm{\Omega}_{o,m})]^{H}}{||\bm{a}(\bm{\Omega}_{i,m})\cdot\bm{a}(\bm{\Omega}_{o,m})||}\
.$ (16)

The $\phi_{m,MRT}$ compensates the steering vector phase difference of
elements in the same group $n$ to achieve larger signal strength. The random
phase shift $\phi_{n,rand}$ produces the randomness. Then, according to the
channel in (8), the channel for CGPS can be expressed as

$\widetilde{H}_{CG}=\sum_{n=1}^{N}q\cos(\phi_{n,rand})+\
j\sum_{n=1}^{N}q\sin(\phi_{n,rand})\ .$ (17)

When RIS has a large number of elements, the number of groups $N$ can be large
when given a proper value of $q$. Then, $\widetilde{H}_{CG}$ converges to a
complex Gaussian distribution. Similarly to Section III-A, the mean
$\mu_{real,CG}=0$ and the variance of the real part can be expressed as

$\displaystyle\sigma^{2}_{real,CG}=\sum_{n=1}^{N}\sigma^{2}[q\cos(\phi_{n,rand})]=\frac{Nq^{2}}{2}\
.$ (18)

Thus, the channel distribution for CGPS RIS is

$\Re{(\tilde{H}_{CG})}\sim N(0,\frac{Nq^{2}}{2})\ ,\Im{(\tilde{H}_{CG})}\sim
N(0,\frac{Nq^{2}}{2})\ .$ (19)

The channel variance can be then calculated as

$\sigma^{2}_{CG}=Nq^{2}\ .$ (20)

There is a tradeoff between $N=\lfloor\frac{M}{q}\rfloor$ and $q$. When $q$ is
large the variance increases quadratically, but $N$ may become too small to
use CLT. The Gaussian distribution is the distribution that maximizes the
entropy at a given variance. The CGPS variance $\sigma^{2}_{CG}=Nq^{2}$ can
achieve larger randomness than the CIPS at Alice and Bob.

###  III-C Discrete Individual Phase Shifts

Since continuous phase shifts are expensive to implement, discrete phase
shifts RIS need be researched. Assume RIS has discrete weight phase shifts
$\bm{\phi}$ with quantization bit $B$. $\bm{\phi}$ is i.i.d uniformly on
discrete values
$\\{0,\frac{2\pi}{2^{B}},\cdots,\frac{2\pi(2^{B}-1)}{2^{B}}\\}$. When $B$
approaches to infinity, the channel distribution approaches to the
distribution assisted by RIS with continuous phase shifts. Similar to the
CIPS, the DIPS reflection channel can be expanded as in (8) and (10). The mean
and variance of the real part $\mu_{cos,DI}$ and $\sigma^{2}_{cos,DI}$ are
expressed as

$\mu_{cos,DI}=\mathbb{E}\\{\cos(\phi_{m}+\alpha)\\}=0\ ,$ (21)
$\displaystyle\\!\\!\sigma^{2}_{cos,DI}$
$\displaystyle=\\!\mathbb{E}\\{\cos^{2}(\phi_{m}+\alpha)\\}-\mu_{cos,DI}^{2}$
(22)
$\displaystyle=\\!\frac{1}{2}+\frac{1}{2}\mathbb{E}\\{\cos(2\phi_{m}+2\alpha)\\}$
$\displaystyle=\\!\\!\frac{1}{2}\\!\\!+\\!\\!\frac{1}{2}\cos(2\alpha)\mathbb{E}\\{\\!\cos(2\phi_{m})\\!\\}\\!-\\!\frac{1}{2}\sin(2\alpha)\mathbb{E}\\{\\!\sin(2\phi_{m})\\!\\}\\!\
\\!,$

where the means of $\cos(2\phi_{m})$ and $\sin(2\phi_{m})$ determine
$\sigma^{2}_{cos,DI}$. According to quantization bits $B$, there are two cases
of $\sigma^{2}_{cos,DI}$.

####  III-C1 $B\geq 2$

$\mathbb{E}\\{\cos(2\phi_{m})\\}=0$ and $\mathbb{E}\\{\sin(2\phi_{m})\\}=0$,
which leads to $\sigma^{2}_{cos,DI}=\frac{1}{2}$. Therefore, same as CIPS, the
channel is a complex Gaussian random variable with real and imaginary parts
being same as in (13), that are, $\Re{(\tilde{H}_{DI})}\sim N(0,\frac{M}{2})$
and $\Im{(\tilde{H}_{DI})}\sim N(0,\frac{M}{2})$. The variance is the same as
in (14) $\sigma^{2}_{DI}=M$.

####  III-C2 $B=1$

$\mu_{real,DI}=\mu_{imag,DI}=0$. Different from $B\geq 2$ case, since
$\phi_{m}$ only takes values $\\{0,\pi\\}$,
$\mathbb{E}\\{\cos(2\phi_{m})\\}=1$ and $\mathbb{E}\\{\sin(2\phi_{m})\\}=0$.
Thus, according to (22), the real and imaginary parts of $\widetilde{H}_{DI}$
can be expressed as

$\displaystyle\Re{(\tilde{H}_{DI})}\sim
N(0,\frac{M}{2}+\frac{1}{2}\sum_{m_{y}=1}^{M_{y}}\sum_{m_{x}=1}^{M_{x}}\cos(2\alpha))\
,$ (23) $\displaystyle\Im{(\tilde{H}_{DI})}\sim
N(0,\frac{M}{2}-\frac{1}{2}\sum_{m_{y}=1}^{M_{y}}\sum_{m_{x}=1}^{M_{x}}\cos(2\alpha))\
.$

The real and imaginary parts of $\widetilde{H}_{DI}$ are correlated, with
covariance $\mathbf{R}_{DI}$ expressed as

$\displaystyle\mathbf{R}_{DI}$
$\displaystyle=\mathbb{E}\\{\sum_{m_{y}=1}^{M_{y}}\sum_{m_{x}=1}^{M_{x}}\cos(\phi_{m}+\alpha)\sin(\phi_{m}+\alpha)\\}$
(24)
$\displaystyle=\sum_{m_{y}=1}^{M_{y}}\sum_{m_{x}=1}^{M_{x}}\frac{1}{2}\sin(2\alpha)\
.$

The real and imaginary parts are independent when $\mathbf{R}_{DI}=0$. Assume
$m_{x}=m_{y}$ and $d_{x}=d_{y}$, which is a typical configuration of URA RIS.
Then, $\mathbf{R}_{DI}=0$ when the following condition is satisfied

$\gamma=\frac{b\pi}{2kd_{x}}\ ,b\in\mathbb{Z}\ ,$ (25)
$\gamma=\cos\psi_{i}\sin\theta_{i}+\cos\psi_{o}\sin\theta_{o}+\sin\psi_{i}\sin\theta_{i}+\sin\psi_{o}\sin\theta_{o}\
.$ (26)

Additionally, the reflection channel variance is $\sigma^{2}_{DI}\leq M$. The
less-equal sign is due to the covariance between the real and imaginary parts.
When $\mathbf{R}_{DI}=0$, 1-bit RIS provides the same variance. The magnitude
and phase distributions for 1-bit DIPS can be turned into deriving the
envelope and phase of correlated Gaussian quadratures according to [12].

##  IV Secret Key Rate

SKR is the upper bound of the information bits per channel sample generated
between Alice and Bob. Since when Eve is several wavelengths away from Alice
and Bob, its channel is considered uncorrelated with the channel between Alice
and Bob, which is tested in practical experiments [13]. The SKR can be
expressed as

$R_{s}=I(y_{A};y_{B}|y_{E})=I(y_{A};y_{B})\ ,$ (27)

where $y_{A}$, $y_{B}$, and $y_{E}$ are the channels estimated by Alice, Bob,
and Eve, respectively.

###  IV-A CIPS, CGPS, and $B\geq 2$ DIPS

The mutual information between $y_{A}$ and $y_{B}$ is determined by their
variances. The direct channel adds a mean to the reflection channel, without
influencing the variance. Therefore, based on the channel expressions in (1)
and Section III-A, the SKR $R_{s}$ for CIPS and $B\geq 2$ DIPS can be derived
as

$R_{s,CI}=\log_{2}(1+\frac{M/2}{2\sigma_{z}^{2}+\frac{2\sigma_{z}^{4}}{M}})\
,\vspace{-3mm}$ (28)

where $\sigma_{z}^{2}$ is the noise power.

The SKR for CGPS can be calculated as (28) by substituting $\sigma^{2}_{CI}=M$
by $\sigma^{2}_{CG}=Nq^{2}$.

(a) Variance when RIS has different element number.

(b) Magnitude distribution.

(c) Phase distribution.

Figure 2: The channel variance of CIPS, $q=2,4$ CGPS, and $B=3$ DIPS. The
magnitude distribution and phase distribution of the CIPS and $B=3$ DIPS.

###  IV-B $B=1$ DIPS

The estimated SKR under $B=1$ DIPS can be expressed as

$\displaystyle R_{s,DI}$
$\displaystyle\leq\\{I(\Re{(y_{A})};\Re{(y_{B})})+I(\Im{(y_{A})};\Im{(y_{B})})\\}$
(29)
$\displaystyle=\log_{2}\\{(1\\!+\\!\frac{\sigma^{2}_{real,DI}}{2\sigma_{z}^{2}+\frac{\sigma_{z}^{4}}{\sigma^{2}_{real,DI}}})(1\\!+\\!\frac{\sigma^{2}_{imag,DI}}{2\sigma_{z}^{2}+\frac{\sigma_{z}^{4}}{\sigma^{2}_{imag,DI}}})\\},$

where $\sigma^{2}_{real,D,I}$ and $\sigma^{2}_{imag,D,I}$ are the variances of
the real and imaginary parts in (23). The less-than sign is because in (24),
the 1-bit DIPS RIS channel real and imaginary parts are correlated. The sign
takes eqivalence when the real and imaginary parts are independent taking the
condition in (25).

(a) Variance when RIS has different element number.

(b) Real part distribution.

(c) Imaginary part distribution.

Figure 3: The variance, real part distribution and imaginary part distribution
of the channel assisted by $B=1$ DIPS RIS.

##  V Simulation Results

In this section, the analytical channel distributions, channel parameters, and
SKR for CIPS, CGPS, and DIPS are verified by simulations.

###  V-A Verification for CIPS and CGPS Channel

Element spacing is set to $d=\frac{\lambda}{2}$. Incident and reflection
angles are set as $\bm{\Omega_{i}}=(30^{\circ},30^{\circ})$, and
$\bm{\Omega_{o}}=(150^{\circ},60^{\circ})$. For CGPS, set group size $q=2,4$.

####  V-A1 Channel Variance

The simulation for the reflection channel variance with respect to $M$ is
plotted in Fig. 2(a). The simulation results match analytical results in (14)
(20). The variance means the RIS has a great potential to induce channel
variation since it is usually implemented with a huge number of elements. The
variance for the CGPS is larger than the CIPS, and $q=4$ CGPS variance is
larger than $q=2$ CGPS because the variance increases quadratically with $q$.

####  V-A2 Magnitude and Phase Distributions

Set $M=64$, with $M_{x}=M_{y}=8$. The magnitude and phase distributions for
CIPS are presented in Fig. 2(b) and Fig. 2(c), respectively. The magnitude and
phase distribution simulation results match the analytical results. The
magnitude and phase are a Rayleigh distribution and a uniform distribution
being independent of $\bm{\Omega}_{i}$ and $\bm{\Omega}_{o}$, respectively.
The independence of $\bm{\Omega}_{i}$ and $\bm{\Omega}_{o}$ means that
whatever angles Alice and Bob locate in, the channel randomness remains the
same. Note that the sharp drops around 0 and $2\pi$ in Fig. 2(c) result from
several adjacent probability density values outside the bound $[0,2\pi)$,
being equal to 0, are averaged to plot a smoother p.d.f. Additionally, the
channel distribution Eve receives stays unchanged, and no more information
leakage dependent on angles. Note that if Eve locates at the same angle as
Bob222The situation Eve locates at the same angle as Bob do not happen in most
cases., the information is fully leaked. This is because, in the far-field
environment, Eve and Bob are differentiated by their locating angles. The
uniform phase distribution means that the values in $[0,2\pi)$ are taken with
equal probability. Thus, when CSI is utilized to generate keys, it greatly
prevents Eve from guessing the particular phase of the channel between Alice
and Bob.

###  V-B Verification for DIPS

####  V-B1 RIS Weight $B\geq 2$

Set $B=3$. Other parameters remain the same as the continuous case. The
simulations for the variance, magnitude, and phase of the reflection channel
are plotted in Fig. 2(a), Fig. 2(b), and Fig. 2(c). The simulation results are
the same as the ideal CIPS. This means that the ideal channel distribution and
uniform phase distribution can be achieved in a more practical situation.

####  V-B2 RIS Weight $B=1$

Set $B=1$, and two pairs of input and output angles, with case 1:
$\bm{\Omega_{i}}=(30^{\circ},30^{\circ})$,
$\bm{\Omega_{o}}=(150^{\circ},60^{\circ})$, and case 2:
$\bm{\Omega_{i}}=(110^{\circ},50^{\circ})$,
$\bm{\Omega_{o}}=(310^{\circ},20^{\circ})$. Other parameters are the same as
continuous case. The simulation for the reflection channel variance is plotted
in Fig. 3(a), where the variances increase linearly with respect to $M$.

The real and imaginary part of the reflection channel for two cases when
$M_{x}=M_{y}=8$ are plotted in Figure. 3(b) and Figure. 3(c), respectively.
The simulation results match the analytical result in (23) well. the real and
imaginary part distributions are dependent on values of $\bm{\Omega}_{i}$ and
$\bm{\Omega}_{o}$.

###  V-C Secret Key Rate

The RIS element number is set to $M=64$. The _Information Theoretical
Estimators (ITE) Toolbox_ is used to simulate the mutual information [14]. The
SKR for CIPS, $q=2$ CGPS, $B=2$ DIPS, and $B=1$ DIPS against signal-to-noise
ratio (SNR) are plotted in Fig. 4. The simulation results match the analytical
results in (28) and (29). The CGPS SKR is larger than the individual shifting.

Figure 4: Secret key rate when RIS is CIPS, $q=2$ CGPS, $B=2$ DIPS, and $B=1$
DIPS.

##  VI Conclusions

In this paper, we utilize RIS for mmWave physical layer security secret key
generation, and completes the underlying mathematical principles by deriving
the channel distribution. Though mmWave static channels lack randomness and
multipath, the RIS can induce an artificial Rayleigh/Rician fading randomness,
without adding transceiver costs. Specifically, based on the RIS-assisted
secret key generation model, we consider RIS weights have CIPS, CGPS, and DIPS
to produce artificial channel randomness. We derive the channel distribution
and its parameters. The CGPS is able to produce more randomness at legal
parties Alice and Bob, compared to individual shifting. The DIPS channel
distribution is dependent on the quantization bit. The SKR for the above RIS
settings is derived to evaluate the performance. Our theoretical conclusions
are verified by simulations. While this work focuses more on the channel
between Alice and Bob, the detailed analysis of leakage to eavesdroppers and
DGPS will be our future work.

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