# Towards rotating 2-2-holes

Bob Holdom<EMAIL_ADDRESS>Department of Physics, University of
Toronto, Toronto, Ontario, Canada M5S 1A7

###### Abstract

Static 2-2-hole solutions of quadratic gravity have been investigated to be a
possible horizonless replacement for black holes as the endpoint of
gravitational collapse. Realistically such objects will form with spin, but
rotating 2-2-hole solutions are currently not known. We take some steps here
to explore the existence and properties of such solutions. We employ an
expansion of the field equations where the expansion parameter is inversely
related to the size of the object. This expansion parameter appears explicitly
in the trial metrics, and we are able to find solutions of the leading order
field equations. These vacuum solutions are candidates to describe most of the
interior of a rotating 2-2-hole.

##  I Introduction

A 2-2-hole [1, 2, 3] is a horizonless solution of the classical field
equations of quadratic gravity, a $2+4$ derivative theory. The interior of a
2-2-hole involves high curvatures and both the two and four derivative terms
in the fields equations are needed to describe the interior solution. But
2-2-holes can be macroscopically large ($GM^{2}\gg 1$), and so one would have
expected an intrinsic suppression of the four derivative terms because of the
smallness of two extra derivatives $\sim(GM)^{-2}$ of the four derivative
terms as compared to the factor of $G^{-1}$ in the two derivative terms. This
suppression is overcome because the metric itself develops components that
become very small in the interior, by a factor of order
$\varepsilon\equiv(GM^{2})^{-1}$. This factor ends up enhancing the four
derivative terms by a relative factor of $\varepsilon^{-1}$, and this is what
causes the two sets of terms in the field equations to be similar in
magnitude. The interior metric of a static 2-2-hole can be expressed as

$\displaystyle ds^{2}=-\varepsilon B(r)dt^{2}+\varepsilon
A(r)dr^{2}+r^{2}d\theta^{2}+r^{2}\sin(\theta)^{2}d\phi^{2}.$ (1)

We have explicitly factored out the factor of $\varepsilon$, so that $A(r)$
and $B(r)$ are now of order one throughout most of the interior. They both
vanish at the origin like $r^{2}$, a feature that prompted the name
“2-2-hole”.

When $\varepsilon$ is minuscule, to a very good approximation the 2-2-hole
solution can be obtained by expanding the field equations to leading order in
$\varepsilon$ and solving these somewhat simplified equations for $A(r)$ and
$B(r)$. This will describe only the interior. Close to the would-be horizon
radius of $2GM$, $\varepsilon A(r)$ and $\varepsilon B(r)$ grow very rapidly,
and the curvature invariants drop very rapidly, such as to match onto the
Schwarzschild (Schd) solution at a radius just slightly larger than $2GM$. The
approximation of dropping the higher order terms breaks down in the transition
region and in particular the Schd solution of the exterior requires the full
Einstein equations and not just the terms leading in $\varepsilon$.

In this work we shall only consider rotating vacuum solutions. In particular
we restrict ourselves to the vacuum field equations of the Einstein-Weyl
theory,

$\displaystyle m_{\rm Pl}^{2}(R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R)-2\frac{m_{\rm
Pl}^{2}}{m_{G}^{2}}B_{\mu\nu}=0.$ (2)

$B_{\mu\nu}=0$ is the traceless Bach tensor, and we take the ghost mass to be
$m_{G}\approx m_{\rm Pl}$. Static 2-2-hole solutions were first obtained as
vacuum solutions [4] (see also [5]). More realistic and interesting static
solutions that are sourced by a thermal gas were found later [2] (see also
[3]). The associated entropy is easily understood and calculated, and it is
somewhat larger than the entropy of a similar size black hole when
$m_{G}\approx m_{\rm Pl}$. This then points to the stability of 2-2-holes.
Vacuum solutions, whether static or rotating, are probably not
stable,111Rotating solutions are still stationary and so like the static
solutions they are blind to the potential instabilities inherent in four
derivative field equations. For time dependent problems, with fluctuations of
Planck scale or smaller, input from quantum quadratic gravity, the associated
stable quantum field theory, would be needed. but in the static case the
vacuum and non-vacuum solutions are found to be at least qualitatively
similar.

Rotation brings in $\theta$ as well as $r$ dependence and an off-diagonal
component of the metric. Correspondingly there is an increased number of
components (six) of the field equations that need to be solved. In the static
case a hybrid approach based on a series expansion around the origin along
with numerical integration proved to be sufficient to study the 2-2-hole
solutions, both for the interior and transition regions and the matching to
the exterior Schd solution. A rather complete picture of the static thermal
2-2-holes is obtained in [6]. This approach does not seem practical in the
rotating case.

We shall exploit the $\varepsilon$ expansion by inserting explicit factors of
$\varepsilon$ into trial metrics, and then expanding the field equations in
powers of $\varepsilon$. This expansion proves to be quite helpful by reducing
the complexity of the leading order components of the field equations. We
shall focus on finding the leading order metric, that is the metric that
satisfies the different components of the field equations at leading order in
$\varepsilon$. Note that the different components may have leading order terms
that differ in the power of $\varepsilon$, but all these leading order terms
must vanish for a leading order solution. We shall also consider the effects
of higher orders in the $\varepsilon$ expansion, and for this we must modify
the metric with corrections that are relatively suppressed by $\varepsilon$.
But it is only when $r$ approaches the transition region from the inside will
the higher order effects be important. The leading order metric should be a
good approximation throughout most of the interior.

A neutral rotating 2-2-hole will be characterized by two dimensionful
constants, its size $R$ and its spin $a$ (having the same dimension as $R$).
Various functions define the various components of the metric and we shall
pull out explicit factors of $R$ to be left with dimensionless functions.
These are the $F_{i}(r,\theta)$ with $i=t,r,\theta,\phi,t\phi$ labelling one
of the four diagonal components or the one off-diagonal component of the
metric. These are actually functions of $r/R$, $a/R$ and $\theta$.
Alternatively we may use a dimensionless function $f_{i}(\theta)$, a function
only of $a/R$ and $\theta$.

We are able to make progress with the following two ansatzes for the leading
order metric. They differ depending on whether the $g_{tt}$ component is of
order $\varepsilon^{0}$ or $\varepsilon^{1}$.

Ansatz 1

$\displaystyle ds^{2}=-f_{t}(\theta)\,dt^{2}+\varepsilon
F_{r}(r,\theta)\,dr^{2}+R^{2}F_{\theta}(r,\theta)\,d\theta^{2}+R^{2}f_{\phi}(\theta)\,d\phi^{2}\pm
2Rf_{t\phi}(\theta)\,dt\,d\phi,$ (3)

Ansatz 2

$\displaystyle ds^{2}=-\varepsilon F_{t}(r,\theta)\,dt^{2}+\varepsilon
F_{r}(r,\theta)\,dr^{2}+R^{2}F_{\theta}(r,\theta)\,d\theta^{2}+R^{2}F_{\phi}(r,\theta)\,d\phi^{2}\pm
2RF_{t\phi}(r,\theta)\,dt\,d\phi.$ (4)

All functions are generally non-negative, except possibly $f_{t}(\theta)$ and
$F_{t}(r,\theta)$. The latter functions may describe ergoregions when
negative. Ansatz 2 involves more general functions of $r$ and $\theta$, and
with both $g_{tt}$ and $g_{rr}$ of order $\varepsilon$ it is more in line with
the static case. But it is still interesting to compare and contrast ansatz 1
with ansatz 2, and in the next two sections we consider these two ansatzes in
turn. In each case we will be able to solve the leading order equations and
thus obtain constraints on the leading order metric. The leading order metric
is not determined completely, and we expect that any freedom left in it will
be fixed in a full solution that matches the interior solution to the exterior
solution. This might only be possible for one of the two ansatzes.

We shall also consider corrections to these leading order metrics as defined
by the following corrected line element,

$\displaystyle ds^{2}_{\rm corrected}=ds^{2}$
$\displaystyle+\varepsilon^{p}G_{t}(r,\theta)\,dt^{2}+\varepsilon^{2}G_{r}(r,\theta)\,dr^{2}+\varepsilon
G_{\theta}(r,\theta)\,d\theta^{2}$ $\displaystyle+\varepsilon
G_{\phi}(r,\theta)\,d\phi^{2}+\varepsilon G_{t\phi}(r,\theta)\,dt\,d\phi,$ (5)

where $p=1,2$ for Ansatz 1 and 2 respectively. It will turn out to be useful
to know at what order the various components of the field equations depend on
the correction terms in the metric.

For both ansatzes we shall find a Ricci-flat metric at leading order in
$\varepsilon$. By this we mean that $R_{\mu\nu}$ is reduced in size by one
order of $\varepsilon$ after the leading order equations are solved. This
result implies that these solutions are in no way smoothly related to the
static 2-2-hole solutions, which are certainly not Ricci-flat at leading order
in $\varepsilon$. Our results for the rotating case are therefore pointing
towards a new class of solutions. We view the Ricci-flatness at leading order
as accidental, and we do not expect it to extend to next-to-leading order as
well. If it held to all orders then we should get back the unique rotating
vacuum solution of the Einstein equations, which is the Kerr metric. But our
leading-order solutions are certainly not Kerr-like.

One might wonder about other expectations for how the interior metric of a
2-2-hole should depend on $a$ and $\theta$. In this connection it is useful to
compare the Kerr metric (in Boyer-Lindquist coordinates) in the limit
$r\to\infty$ to the less familiar limit $r\to 0$,

$\displaystyle ds^{2}_{\rm Kerr}$
$\displaystyle\underset{r\to\infty}{=}-\,dt^{2}+dr^{2}+r^{2}\,d\theta^{2}+r^{2}\sin(\theta)^{2}\,d\phi^{2}\pm\frac{a}{r}4GM\sin\left(\theta\right)^{2}\,dt\,d\phi$
(6) $\displaystyle\underset{r\to
0}{=}-\,dt^{2}+\cos(\theta)^{2}\,dr^{2}+a^{2}\cos(\theta)^{2}\,d\theta^{2}+a^{2}\sin(\theta)^{2}\,d\phi^{2}\pm\frac{r}{a}\frac{4GM\sin\left(\theta\right)^{2}}{\cos\left(\theta\right)^{2}}\,dt\,d\phi.$

They are very different, and so we should be cautious about preconceived
notions for the $a$ and $\theta$ dependence inside a rotating 2-2-hole.

The main results for the two ansatzes are described in Sections II and III. We
also discuss the propagation of waves and particles on these spacetimes. In
Section IV we briefly describe an attempt to find a rotating 2-2-hole solution
that would be smoothly connected to the static metric solution when $a\to 0$.
Some conclusions are given in Section V.

##  II Anzatz 1

With the metric in (3), the Einstein and Bach tensors in the field equations
start at order $\varepsilon^{-1}$ and $\varepsilon^{-2}$ respectively. As we
have explained, this allows for the actual relative size of the Einstein and
Bach tensors to be comparable in the interior of macroscopically large
objects. The Einstein tensor has three non-vanishing components at this order,
the $tt$, $\phi\phi$ and $t\phi$ components. The $\theta\theta$ component
accidentally vanishes; accidental zeros arise for this ansatz because of the
lack of $r$ dependence in $g_{tt}$, $g_{\phi\phi}$ and $g_{t\phi}$. All three
nonzero components of the Einstein tensor turn out to be proportional to

$\displaystyle
F_{r}\\!\left(r,\theta\right)F_{\theta}^{\prime}\\!\left(r,\theta\right)^{2}+F_{\theta}\\!\left(r,\theta\right)F_{\theta}^{\prime}\\!\left(r,\theta\right)F_{r}^{\prime}\\!\left(r,\theta\right)-2F_{r}\\!\left(r,\theta\right)F_{\theta}\\!\left(r,\theta\right)F_{\theta}^{\prime\prime}\\!\left(r,\theta\right).$
(7)

(Our notation is such that a prime always denotes a derivative with respect to
the first argument of the function.) This vanishes when

$\displaystyle
F_{r}(r,\theta)=\frac{f_{r}(\theta)}{F_{\theta}(r,\theta)}R^{2}F_{\theta}^{\prime}(r,\theta)^{2}.$
(8)

$f_{r}(\theta)\geq 0$ is another dimensionless function of $a/R$ and $\theta$.

The Bach tensor has all four of the $tt$, $\phi\phi$, $t\phi$ and
$\theta\theta$ components non-vanishing at this order. Each is a complicated
function involving up to four derivatives of $F_{\theta}(r,\theta)$ and three
derivatives of $F_{r}(r,\theta)$, but we have verified that these four
components of the Bach tensor also vanish when (8) is satisfied. Thus at this
leading order the field equations can be satisfied by having the Einstein and
Bach terms separately vanishing, with the simple relationship between
$F_{r}(r,\theta)$ and $F_{\theta}(r,\theta)$ in (8).

With relation (8) in place, we can now inspect the field equations at next
order, that is at $\varepsilon^{0}$ and $\varepsilon^{-1}$ for the Einstein
and Bach tensors respectively. At this order the field equations will have
dependence on the next order corrections in the metric, as in (5). The $tt$,
$\phi\phi$, $t\phi$ and $\theta\theta$ components of the Einstein and Bach
tensors are nonzero and we have verified that each of these components are
sensitive to at least some or all of the various $G_{i}(r,\theta)$’s and their
derivatives. The expressions are lengthy and complicated, especially for the
Bach tensor. But the many undetermined functions leaves open the possibility
that the field equations at next order can be satisfied. In this case we
expect that this will occur through cancellations between the Einstein and
Bach terms that are separately non-vanishing.

There is one additional component that becomes nonzero at this order, and this
is the $r\theta$ component of the Einstein tensor. (This component of the Bach
tensor is accidentally zero.) This nonzero Einstein component is receiving its
leading contribution at this order, and so it is not sensitive to the
$G_{i}(r,\theta)$’s. It therefore implies another constraint on the leading
order metric. This component is proportional to

$\displaystyle
f_{\phi}(\theta)f_{t}^{\prime}(\theta)+f_{t}(\theta)f_{\phi}^{\prime}(\theta)+2f_{t\phi}(\theta)f_{t\phi}^{\prime}(\theta),$
(9)

which vanishes when

$\displaystyle f_{t}(\theta)f_{\phi}(\theta)+f_{t\phi}(\theta)^{2}=C,$ (10)

for some dimensionless constant $C$. Note that $C$ can also depend on $a/R$.

If we expand to one further order (orders $\varepsilon^{1}$ and
$\varepsilon^{0}$ respectively) then the $rr$ component finally shows up. But
this component at this order is sensitive to the corrected metric, meaning
that it was accidentally zero at the previous order. It therefore provides no
further constraint on the leading order metric.

###  II.1 Invariants

With the results in (8) and (10) we find that the volume element is

$\displaystyle\sqrt{-g}$ $\displaystyle=\left(\varepsilon
CR^{6}f_{r}(\theta)F_{\theta}^{\prime}(r,\theta)^{2}\right)^{\frac{1}{2}}.$
(11)

We see a $\sqrt{\varepsilon}$ suppression of the volume element, to be
compared with a $\varepsilon$ suppression for the static 2-2-hole. We have
already seen that $f_{r}(\theta)\geq 0$ and thus we must have $C>0$.
$f_{t}(\theta)$ is the only function that could be negative, and we see that
(10) with $C>0$ still permits $f_{t}(\theta)$ to have either sign. When
negative then $g_{tt}\geq 0$ would correspond to an ergoregion that extends to
the origin of the 2-2-hole.

Let us now consider the quadratic curvature invariants. We find that two are
enhanced by $\varepsilon^{-1}$,

$\displaystyle
R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}=C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}=$
$\displaystyle\frac{1}{\varepsilon}\frac{\left((C-f_{t\phi}(\theta)^{2})f_{t}^{\prime}(\theta)^{2}+2f_{t\phi}^{\prime}(\theta)f_{t\phi}(\theta)f_{t}^{\prime}(\theta)f_{t}(\theta)-f_{t\phi}^{\prime}(\theta)^{2}f_{t}(\theta)^{2}\right)}{R^{4}Cf_{t}(\theta)^{2}f_{r}(\theta)F_{\theta}\\!\left(r,\theta\right)^{2}}$
(12)

We have used (10) to eliminate $f_{\phi}(\theta)$. This result is stable under
corrections to the leading order metric, that is it is independent of the
$G_{i}(r,\theta)$’s in (5). $R_{\mu\nu}R^{\mu\nu}$, like $R$, turns out to be
of order $\varepsilon^{0}$, and it does depend on the $G_{i}(r,\theta)$’s.
From (12) we notice that $f_{t}(\theta)$ and $f_{r}(\theta)$ must not vanish
to avoid a singularity at all $r$.

###  II.2 Propagation

We first consider the wave equation for a massless scalar field
$\varphi(r,\theta,\phi,t)$ on this 2-2-hole background, $\Box\varphi=0$. At
leading order in $\varepsilon$ only the $t$\- and $r$-derivative terms remain.
The neglect of the angular derivative terms at this stage assumes that the
angular momenta are not so large as to compensate for the $\varepsilon$
suppression. The single $r$-derivative terms can be eliminated if we define

$\displaystyle\varphi(r,\theta,\phi,t)=e^{-i\omega
t}\sqrt{{\frac{F_{\theta}^{\prime}\left(r,\theta\right)}{F_{\theta}\left(r,\theta\right)}}}\Phi(r,\theta,\phi).$
(13)

Then the leading order wave equation takes the form

$\displaystyle\left({\frac{\partial^{2}}{\partial{r}^{2}}}+\omega^{2}-V(r,\theta)\right)\Phi\left(r,\theta,\phi\right)=0,$
(14)

where the potential is

$\displaystyle
V(r,\theta)={\frac{2F_{\theta}\left(r,\theta\right)^{2}F_{\theta}^{\prime}\left(r,\theta\right)F_{\theta}^{\prime\prime\prime}\left(r,\theta\right)-3F_{\theta}\left(r,\theta\right)^{2}F_{\theta}^{\prime\prime}\left(r,\theta\right)^{2}+F_{\theta}^{\prime}\left(r,\theta\right)^{4}}{4F_{\theta}\left(r,\theta\right)^{2}F_{\theta}^{\prime}\left(r,\theta\right)^{2}}}.$
(15)

We see that both the wave equation and the curvatures can be singular at the
origin if $F(r,\theta)$ vanishes there. It is also possible for the potential
to be singular while the curvatures are not, when $F^{\prime}(r,\theta)$ but
not $F(r,\theta)$ vanishes at the origin; for example if
$F^{\prime}(r,\theta)\sim r$ then $V(r,\theta)\sim r^{-2}$ with a negative
residue. The static 2-2-hole is quite different, there the curvatures are
singular while the wave equation, including the angular part, is regular at
the origin.

Particle-like motion is described by the geodesic equations, corresponding to
a small wave-length (eikonal) limit of the wave equations. This limit is
essentially the opposite to that used to obtain the form of the wave equation
above. The geodesic equations for a massless particle describes a path
$(t(\zeta),r(\zeta),\theta,\phi(\zeta))$ as a function of the affine parameter
$\zeta$, where for simplicity we have set $\theta$ to a constant. We then have

$\displaystyle\frac{d^{2}t(\zeta)}{d\zeta^{2}}=0\quad\frac{d^{2}\phi(\zeta)}{d\zeta^{2}}=0,$
(16)

and a radial equation that is solved by

$\displaystyle\sqrt{F_{\theta}(r(\zeta),\theta)}=c_{0}(\theta)+c_{1}(\theta)\zeta.$
(17)

The existence and stability of possible light ring orbits can be found via the
formalism discussed in [7]. Two potentials are defined by

$\displaystyle
H_{\pm}(r,\theta)=\frac{-g_{t\phi}\pm\sqrt{g_{t\phi}^{2}-g_{tt}g_{\phi\phi}}}{g_{\phi\phi}}.$
(18)

The existence of a light ring corresponds to a stationary point occurring for
either potential, and a stable light ring corresponds to this being a local
minimum of that potential. Stable light rings are thought to imply a spacetime
instability that may destroy the object.222Realistic 2-2-holes are sourced by
a hot thermal gas, and the effect of light rings occurring within such a
medium may need to be considered further. The metric components in (18)
precisely those that are $r$-independent in anzatz 1, and so these potentials
are independent of $r$,

$\displaystyle H_{\pm}(r,\theta)=\frac{f_{t}(\theta)}{(\sqrt{C}\mp
f_{\phi}(\theta))R}.$ (19)

This degeneracy is expected to be broken at next order in $\varepsilon$, and
so at this stage we can say little about interior light rings. Ansatz 2 will
give $r$-dependent potentials at leading order.

###  II.3 A particular choice

$F_{\theta}(r,\theta)$ is the remaining undetermined $r$-dependent function in
the metric of ansatz 1. Of all the metric components, $g_{\theta\theta}$ is
the only one that could possibly be the same as the Kerr metric. The static
2-2-hole solution already strongly modifies $g_{tt}$ and $g_{rr}$ from the
Schd metric, and then rotation implies a non-vanishing $g_{t\phi}$ that points
to a modification of $g_{\phi\phi}$ as well. This leaves $g_{\theta\theta}$,
and so for ansatz 1 we consider the Kerr choice
$g_{\theta\theta}=R^{2}F_{\theta}(r,\theta)=r^{2}+a^{2}\cos(\theta)^{2}$. Then
the curvature invariants in (12) are singular only when both $r=0$ and
$\theta=\pi/2$, as with the Kerr metric. They also show that the leading order
metric is asymptotically flat, although this is not relevant for its role as
an approximate interior metric. The metric in this case is

$\displaystyle ds^{2}=$ $\displaystyle-
f_{t}(\theta)\,dt^{2}+\varepsilon\frac{4f_{r}(\theta)r^{2}}{r^{2}+a^{2}\cos(\theta)^{2}}\,dr^{2}+(r^{2}+a^{2}\cos(\theta)^{2})\,d\theta^{2}$
$\displaystyle+R^{2}\frac{C-f_{t\phi}(\theta)^{2}}{f_{t}(\theta)}\,d\phi^{2}\pm
2Rf_{t\phi}(\theta)\,dt\,d\phi.$ (20)

The various $f_{i}(\theta)$ and $C$ can depend on $a/R$. We see $g_{rr}\sim
r^{2}$, at least for small $r/a$, which is similar to the $g_{rr}\sim r^{2}$
behaviour of a static 2-2-hole. In addition

$\displaystyle\sqrt{-g}$ $\displaystyle=2Rr\left(\varepsilon
Cf_{r}(\theta)\right)^{\frac{1}{2}}.$ (21)

##  III Ansatz 2

This ansatz for the leading order metric is given in (4). Once again the
leading order terms of the Einstein and Bach tensors are of order
$\varepsilon^{-1}$ and $\varepsilon^{-2}$ respectively, making them of similar
size. At this order, and for each of the two tensors, the three non-vanishing
components are $\theta\theta$, $\phi\phi$ and $\phi t$. Looking first at the
Einstein tensor, the $\theta\theta$ component is proportional to

$\displaystyle
4F_{t\phi}^{\prime\prime}\\!\left(r,\theta\right)F_{r}\\!\left(r,\theta\right)F_{t\phi}\\!\left(r,\theta\right)-F_{t\phi}^{\prime}\\!\left(r,\theta\right)^{2}F_{r}\\!\left(r,\theta\right)-2F_{t\phi}^{\prime}\\!\left(r,\theta\right)F_{r}^{\prime}\\!\left(r,\theta\right)F_{t\phi}\\!\left(r,\theta\right).$
(22)

This vanishes when

$\displaystyle
F_{r}(r,\theta)=\frac{f_{r}(\theta)}{\sqrt{F_{t\phi}(r,\theta)}}R^{2}F_{t\phi}^{\prime}(r,\theta)^{2}.$
(23)

When this result is inserted into the $\phi t$ component, the latter becomes
proportional to

$\displaystyle
F_{r}\\!\left(r,\theta\right)^{2}F_{t\phi}^{\prime}\\!\left(r,\theta\right)^{3}-4F_{r}\\!\left(r,\theta\right)F_{t\phi}\\!\left(r,\theta\right)^{2}F_{t\phi}^{\prime}\\!\left(r,\theta\right)F_{r}^{\prime\prime}\\!\left(r,\theta\right)$
$\displaystyle+4F_{r}\\!\left(r,\theta\right)F_{t\phi}\\!\left(r,\theta\right)^{2}F_{t\phi}^{\prime\prime}\\!\left(r,\theta\right)F_{r}^{\prime}\\!\left(r,\theta\right)$
(24)
$\displaystyle-3F_{r}\\!\left(r,\theta\right)F_{t\phi}\\!\left(r,\theta\right)F_{t\phi}^{\prime}\\!\left(r,\theta\right)^{2}F_{r}^{\prime}\\!\left(r,\theta\right)+2F_{t\phi}\\!\left(r,\theta\right)^{2}F_{t\phi}^{\prime}\\!\left(r,\theta\right)F_{r}^{\prime}\\!\left(r,\theta\right)^{2}.$

This vanishes when

$\displaystyle
F_{\theta}(r,\theta)=\frac{f_{\theta}(\theta)}{\sqrt{F_{t\phi}(r,\theta)}}.$
(25)

When the last two relations are inserted into the $\phi\phi$ component, the
latter becomes proportional to

$\displaystyle
F_{\phi}\\!\left(r,\theta\right)F_{t\phi}^{\prime}\\!\left(r,\theta\right)^{3}+F_{t\phi}\\!\left(r,\theta\right)^{2}F_{t\phi}^{\prime}\\!\left(r,\theta\right)F_{\phi}^{\prime\prime}\\!\left(r,\theta\right)$
$\displaystyle-
F_{t\phi}\\!\left(r,\theta\right)^{2}F_{\phi}^{\prime}\\!\left(r,\theta\right)F_{t\phi}^{\prime\prime}\\!\left(r,\theta\right)-F_{t\phi}\\!\left(r,\theta\right)F_{t\phi}^{\prime}\\!\left(r,\theta\right)^{2}F_{\phi}^{\prime}\\!\left(r,\theta\right).$
(26)

This vanishes when

$\displaystyle
F_{\phi}(r,\theta)=F_{t\phi}(r,\theta)\left(f_{\phi}(\theta)+\hat{f}_{\phi}(\theta)\ln(F_{t\phi}(r,\theta))\right).$
(27)

Now when we turn to the Bach tensor, we find that these three relations also
cause the vanishing of the order $\varepsilon^{-2}$ contributions to the
$\theta\theta$, $\phi\phi$ and $\phi t$ components.333Note that (24) has a
more general solution than that given in (25), but then the three Bach tensor
components do not all vanish. So once again the Einstein and Bach tensors
vanish independently, something that is not expected to hold at next order in
the $\varepsilon$ expansion.

With these relations in place then components start at order $\varepsilon^{0}$
and $\varepsilon^{-1}$ for the Einstein and Bach tensors respectively. We have
verified that the metric correction terms in (5) contribute to the
$\theta\theta$, $\phi\phi$ and $\phi t$ components at this order. In principle
these field equations at this order would help to determine the
$G_{i}(r,\theta)$ functions. We are more interested in the $tt$, $r\theta$ and
$rr$ components of both tensors, since these components first appear at this
order and so these do not receive corrections. The $r\theta$ component of the
Bach tensor accidentally vanishes and $rr$ component of both tensors happen to
vanish due to the relations we have already applied. The required vanishing of
the $tt$ and $r\theta$ components must be due to further constraints on the
leading order metric. We find that the $tt$ component of the Einstein tensor
is proportional to the same expression as in (26), but with
$F_{\phi}(r,\theta)$ replaced by $F_{t}(r,\theta)$. Thus this vanishes when

$\displaystyle
F_{t}(r,\theta)=F_{t\phi}(r,\theta)\left(f_{t}(\theta)+\hat{f}_{t}(\theta)\ln(F_{t\phi}(r,\theta))\right).$
(28)

This also causes the $tt$ component of the Bach tensor to vanish. Lastly, the
$r\theta$ component of the Einstein tensor is proportional to
$f_{r}^{\prime}(\theta)$, and thus we must replace $f_{r}(\theta)$ by a
constant $C$ that can still depend on $a/R$.

We can expand the Einstein and Bach tensors to one further order, that is to
order $\varepsilon^{1}$ and $\varepsilon^{0}$ respectively. The only nonzero
components for either tensor are $tt$, $rr$ and $r\theta$, but these
components have already appeared at lower order and so they will receive
corrections from the corrected metric. Thus we do not obtain any further
constraints on the leading order metric. From the constraints that we do have,
all the $r$-dependent functions have been expressed in terms of
$F_{t\phi}(r,\theta)$.

###  III.1 Invariants

The volume element at leading order in $\varepsilon$ is

$\displaystyle\sqrt{-g}$ $\displaystyle=\left(\varepsilon
R^{4}F_{r}(r,\theta)F_{\theta}(r,\theta)F_{t\phi}(r,\theta)^{2}\right)^{\frac{1}{2}}$
$\displaystyle=\left(\varepsilon
CR^{6}f_{\theta}(\theta)F_{t\phi}(r,\theta)F_{t\phi}^{\prime}(r,\theta)^{2}\right)^{\frac{1}{2}}.$
(29)

Again we have a $\sqrt{\varepsilon}$ dependence. The dependence on
$F_{t}(r,\theta)$ only occurs at higher order, and this is interesting because
it allows $F_{t}(r,\theta)$ to be negative. In fact from (28), appropriate
$f_{t}(\theta)$ and $\hat{f}_{t}(\theta)$ can give rise to an interior surface
of some shape where $F_{t}(r,\theta)$ changes sign. An ergoregion could lie
inside or outside this surface.

The curvature invariants take a simple form

$\displaystyle
R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}=C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}=\frac{1}{\varepsilon^{2}}\frac{3}{4C^{2}R^{4}F_{t\phi}(r,\theta)^{3}}.$
(30)

Again this is not sensitive to higher order corrections to the metric.
($R_{\mu\nu}R^{\mu\nu}$ on the other hand, like $R$, turns out to be of order
$\varepsilon^{0}$ and is sensitive to corrections.) Notice also that these
invariants are proportional to $\varepsilon^{-2}\sim R^{4}/\ell_{\rm Pl}^{4}$,
which means that they are now of order one in Planck units. As compared to
ansatz 1, where the curvature invariants are proportional to
$\varepsilon^{-1}$, ansatz 2 is thus more similar to the situation for a
static 2-2-hole.

###  III.2 Propagation

The scalar wave equation $\Box\varphi=0$ at leading order in $\varepsilon$
yields a very similar result to ansatz 1. If we define

$\displaystyle\varphi(r,\theta,\phi,t)=e^{-i\omega
t}\sqrt{{\frac{F_{t\phi}^{\prime}\left(r,\theta\right)}{F_{t\phi}\left(r,\theta\right)}}}\Phi(r,\theta,\phi),$
(31)

then we get the same wave equation as in (14) and (15), only with
$F_{\theta}(r,\theta)$ replaced by $F_{t\phi}(r,\theta)$. With this change the
previous discussion applies, and in particular the wave equation potential can
become singular when either $F_{t\phi}(r,\theta)$ or
$F^{\prime}_{t\phi}(r,\theta)$ vanishes.

The geodesic equations are complicated and so to permit a solution we set
$\hat{f}_{t}(\theta)=\hat{f}_{\phi}(\theta)=0$. In this case, for the same
path $(t(\zeta),r(\zeta),\theta,\phi(\zeta))$ as before, we find

$\displaystyle\frac{dt(\zeta)}{d\zeta}=-\frac{c_{1}f_{\phi}(\theta)}{2F_{t\phi}(r(\zeta),\theta)}\quad\frac{d\phi(\zeta)}{d\zeta}=-\frac{c_{1}}{RF_{t\phi}(r(\zeta),\theta)}.$
(32)

This result has also been constrained by the leading order part of the radial
equation. The next-to-leading part of the radial equation is then solved by

$\displaystyle\frac{d}{d\zeta}r\\!\left(\zeta\right)=\frac{\sqrt{CF_{t\phi}\\!\left(r\\!\left(\zeta\right),\theta\right)^{\frac{3}{2}}\left(c_{2}F_{t\phi}\\!\left(r\\!\left(\zeta\right),\theta\right)+f_{\phi}\\!\left(\theta\right)^{2}f_{t}\\!\left(\theta\right)\right)}}{2CF_{t\phi}\\!\left(r\\!\left(\zeta\right),\theta\right)F_{t\phi}^{\prime}\\!\left(r\\!\left(\zeta\right),\theta\right)R}.$
(33)

The potentials defined in (18) for the determination of light rings are

$\displaystyle H_{+}(r,\theta)$
$\displaystyle=\frac{\varepsilon}{2R}\left(f_{t}(\theta)+\hat{f}_{t}(\theta)\ln(F_{t\phi}(r,\theta))\right),$
(34) $\displaystyle H_{-}(r,\theta)$
$\displaystyle=-\frac{2}{R}\frac{1}{f_{\phi}(\theta)+\hat{f}_{\phi}(\theta)\ln(F_{t\phi}(r,\theta))}-\frac{\varepsilon}{2R}\left(f_{t}(\theta)+\hat{f}_{t}(\theta)\ln(F_{t\phi}(r,\theta))\right).$
(35)

These potentials are of a form that can easily avoid any stationary points.
Notice that the denominator of the first term of $H_{-}(r,\theta)$ is
proportional to $g_{\phi\phi}\geq 0$, and that one way to meet this constraint
is to set $\hat{f}_{\phi}(\theta)=0$. This is why the next order term is also
shown.

###  III.3 A particular choice

The results for the volume element and the curvature invariants in (29) and
(30), as well as the wave equation, are showing that $F_{t\phi}(r,\theta)$ for
ansatz 2 is playing a similar role to $F_{\theta}(r,\theta)$ for ansatz 1.
Thus it helps to compare the two ansatzes if the particular choice we made for
$F_{\theta}(r,\theta)$ is carried over to $F_{t\phi}(r,\theta)$ for ansatz 2,
that is $R^{2}F_{t\phi}(r,\theta)=r^{2}+a^{2}\cos(\theta)^{2}\equiv\Sigma$.
The metric becomes

$\displaystyle ds^{2}=$
$\displaystyle-\varepsilon\frac{\Sigma}{R^{2}}\left(f_{t}(\theta)+\hat{f}_{t}(\theta)\ln(\frac{\Sigma}{R^{2}})\right)\,dt^{2}+\varepsilon\frac{4Cr^{2}}{R\sqrt{\Sigma}}\,dr^{2}$
$\displaystyle+\frac{R^{3}f_{\theta}(\theta)}{\sqrt{\Sigma}}\,d\theta^{2}+\Sigma
f_{\phi}(\theta)\,d\phi^{2}\pm\frac{2\Sigma}{R}\,dt\,d\phi.$ (36)

The various $f_{i}(\theta)$ and $C$ can still depend on $a/R$. We have set
$\hat{f}_{\phi}(\theta)=0$ to ensure that $g_{\phi\phi}\geq 0$, since $\Sigma$
ranges from 0 to $\infty$. We again see $g_{rr}\sim r^{2}$ at small $r/a$. In
addition

$\displaystyle\sqrt{-g}$ $\displaystyle=2\sqrt{C\varepsilon
f_{\theta}(\theta)\Sigma r^{2}},$ (37)

which, other than the $\sqrt{\varepsilon}$, is rather similar to the Kerr
metric where $\sqrt{-g}=\sin(\theta)\Sigma$. The curvature invariants are
simply

$\displaystyle
R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}=C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}=\frac{1}{\varepsilon^{2}}\frac{3}{4}\frac{R^{2}}{C^{2}\Sigma^{3}}.$
(38)

These invariants for the Kerr metric also have a singularity at $\Sigma=0$ and
the same $1/r^{6}$ behaviour at large $r$. The main difference is the
$\varepsilon^{-2}$ enhancement for the 2-2-hole interior.

Finally we give an example of how the explicit $a$ dependence in the metric
can change without changing the results in (37) and (38). We could choose
$F_{t\phi}(r,\theta)=\Sigma/(aR)$ and then redefine
$(f_{\theta}(\theta),C)\to(\frac{a}{R})^{\frac{3}{2}}(f_{\theta}(\theta),C)$
and
$(f_{\phi}(\theta),f_{t}(\theta),\hat{f}_{t}(\theta))\to\frac{a}{R}(f_{\phi}(\theta),f_{t}(\theta),\hat{f}_{t}(\theta))$.
In this case the metric is

$\displaystyle ds^{2}=$
$\displaystyle-\varepsilon\frac{\Sigma}{R^{2}}\left(f_{t}(\theta)+\hat{f}_{t}(\theta)\ln(\frac{\Sigma}{aR})\right)\,dt^{2}+\varepsilon\frac{4Cr^{2}}{R\sqrt{\Sigma}}\,dr^{2}$
$\displaystyle+\frac{a^{2}Rf_{\theta}(\theta)}{\sqrt{\Sigma}}\,d\theta^{2}+\Sigma
f_{\phi}(\theta)\,d\phi^{2}\pm\frac{2\Sigma}{a}\,dt\,d\phi.$ (39)

##  IV An attempt with a more direct approach

A recent new representation of the Kerr metric has appeared [8], and it leads
to an obvious starting point in the search for a rotating 2-2-hole solution.
Using their notation, these authors find that

$\displaystyle(g_{\rm Kerr})_{ab}=g_{AB}{e^{A}}_{a}{e^{B}}_{b},$ (40)

where the “co-tetrad” is

$\displaystyle{e^{A}}_{a}=\left[\begin{array}[]{cccc}\sqrt{\frac{a^{2}+x^{2}}{x^{2}+a^{2}\cos\left(\theta\right)^{2}}}&0&0&-\frac{\sqrt{a^{2}+x^{2}}\,a\sin\left(\theta\right)^{2}}{\sqrt{x^{2}+a^{2}\cos\left(\theta\right)^{2}}}\\\
0&\sqrt{\frac{x^{2}+a^{2}\cos\left(\theta\right)^{2}}{a^{2}+x^{2}}}&0&0\\\
0&0&\sqrt{x^{2}+a^{2}\cos\\!\left(\theta\right)^{2}}&0\\\
-\frac{a\sin\left(\theta\right)}{\sqrt{x^{2}+a^{2}\cos\left(\theta\right)^{2}}}&0&0&\frac{\left(a^{2}+x^{2}\right)\sin\left(\theta\right)}{\sqrt{x^{2}+a^{2}\cos\left(\theta\right)^{2}}}\end{array}\right],$
(45)

and the “tetrad-component metric” is

$\displaystyle g_{AB}=\left[\begin{array}[]{cccc}-f(r)&0&0&0\\\
0&\displaystyle\frac{1}{f(r)}&0&0\\\ 0&0&1&0\\\
0&0&0&1\end{array}\right],\quad f(r)=1-\frac{2mr}{r^{2}+a^{2}}.$ (50)

The essence of the Kerr metric is contained in $g_{AB}$, since replacing
$g_{AB}$ by $\eta_{AB}$ would yield the metric for flat space in oblate
spheroidal coordinates.

This suggests that in the same way a static 2-2-hole is obtained from the Schd
metric with the replacements $-g_{tt}=1-2m/r\to\varepsilon B(r)$ and
$g_{rr}=1/(1-2m/r)\to\varepsilon A(r)$, we should make the same replacements
in the tetrad-component metric $g_{AB}$, that is $-g_{tt}=f(r)\to\varepsilon
B(r)$ and $g_{rr}=1/f(r)\to\varepsilon A(r)$. With this ansatz the
dimensionless functions $A(r)$ and $B(r)$ are functions of $r/R$ and $a/R$,
but not $\theta$. This construction is guaranteed to produce a metric that
goes smoothly to the form of the static 2-2-hole metric in (1) when $a\to 0$.
A point of interest for our discussion in section II is that the metric from
this ansatz has $g_{\theta\theta}=r^{2}+a^{2}\cos(\theta)^{2}$.

With this metric we have expanded the field equations in (2) to leading order
in $\varepsilon$. For $a\neq 0$ the leading order terms of the Einstein and
Bach tensors are now of order $\varepsilon^{-2}$ and $\varepsilon^{-4}$
respectively. Derivatives of $A(r)$ and $B(r)$ happen not to appear at leading
order, and this makes it quite easy to see that no solution of the vacuum
equations for $A(r)$ and $B(r)$ is possible. Some attempt to use a more
general form of $g_{AB}$ was also not successful. So thus far we have found no
evidence for a rotating solution that smoothly connects to the static
solution.

##  V Conclusion

We have utilized what is effectively a large size or large mass expansion to
approach the problem of describing the interior of a rotating 2-2-hole. The
expansion parameter is a very small quantity $\varepsilon\equiv(GM^{2})^{-1}$
that appears explicitly in the metric. This causes the four derivative terms
in the field equations to be of the same order as the two derivative terms.

We have solved the leading order field equations for two ansatzes. Ansatz 2
produces a result that is qualitatively more similar to the static 2-2-hole,
since like the later, two quadratic curvature invariants end up enhanced by a
factor of $\varepsilon^{-2}$. This is what produces Planckian curvatures over
a macroscopically large region. On the other hand the spacetime for both
anzatzes is Ricci-flat up to corrections that are next order in $\varepsilon$.
This is not the case for the static 2-2-hole solution.

For each of the two rotating ansatzes we displayed a particular choice of the
remaining $r$-dependent function that in turn gives a curvature singularity at
$r^{2}+a^{2}\cos(\theta)^{2}=0$. We note that the static solution exhibits a
curvature singularity at $r=0$ that is benign with respect to the wave
equation. In fact this singularity is associated with the fact that there are
no stable light rings in the static 2-2-hole spacetime. For the rotating case
it also appears that stable light rings can be avoided, but we cannot fully
answer this question without having the full matching to the exterior metric.

We have not proven that one of our solutions provides the true leading order
description of the rotating 2-2-hole interior. A proof would entail
demonstrating a successful matching to the external Kerr solution. On the
other hand there should be some leading order description, and at this stage
it is difficult to see how other analytical solutions could be found. If we
are on the way to the true description, then this would imply that the known
static 2-2-hole solution may be an isolated solution that only applies when
$a$ is exactly zero. This may not be a physically realizable situation, and so
it could be that one should rather take the $a\to 0$ limit of the rotating
solutions to approach the static case.

If it turns out that we are not yet on the right track to find the rotating
solutions, then it would seem to require a more clever approach to generalize
the static case, and in particular an approach more clever than the one
presented in Section IV.

## References

  * [1] B. Holdom and J. Ren, “Not quite a black hole”, Phys. Rev. D 95, no. 8, 084034 (2017). [arXiv:1612.04889 [gr-qc]].
  * [2] B. Holdom, “A ghost and a naked singularity; facing our demons,” arXiv:1905.08849 [gr-qc].
  * [3] J. Ren, “Anatomy of a thermal black hole mimicker,” Phys. Rev. D 100, no. 12, 124012 (2019) [arXiv:1905.09973 [gr-qc]].
  * [4] B. Holdom, “On the fate of singularities and horizons in higher derivative gravity,” Phys. Rev. D 66, 084010 (2002) [arXiv:hep-th/0206219 [hep-th]].
  * [5] H. Lü, A. Perkins, C. N. Pope and K. S. Stelle, “Spherically Symmetric Solutions in Higher-Derivative Gravity,” Phys. Rev. D 92, no.12, 124019 (2015) [arXiv:1508.00010 [hep-th]].
  * [6] B. Holdom, “2-2-holes simplified,” Phys. Lett. B 830, 137142 (2022) [arXiv:2202.08442 [gr-qc]].
  * [7] P. V. P. Cunha, E. Berti and C. A. R. Herdeiro, “Light-Ring Stability for Ultracompact Objects,” Phys. Rev. Lett. 119, no.25, 251102 (2017) [arXiv:1708.04211 [gr-qc]].
  * [8] J. Baines and M. Visser, “Physically motivated ansatz for the Kerr spacetime,” [arXiv:2207.09034 [gr-qc]].


# Pitfalls of Conditional Batch Normalization for Contextual Multi-Modal
Learning

Ivaxi Sheth
Mila, ÉTS Montréal
Aamer Abdul Rahman
Mila, ÉTS Montréal
Mohammad Havaei
ICH
Samira Ebrahimi Kahou
Mila, ÉTS Montréal, CIFAR AI Chair
Corresponding author<EMAIL_ADDRESS>
###### Abstract

Humans have perfected the art of learning from multiple modalities through
sensory organs. Despite their impressive predictive performance on a single
modality, neural networks cannot reach human level accuracy with respect to
multiple modalities. This is a particularly challenging task due to variations
in the structure of respective modalities. Conditional Batch Normalization
(CBN) is a popular method that was proposed to learn contextual features to
aid deep learning tasks. This technique uses auxiliary data to improve
representational power by learning affine transformations for convolutional
neural networks. Despite the boost in performance observed by using CBN
layers, our work reveals that the visual features learned by introducing
auxiliary data via CBN deteriorates. We perform comprehensive experiments to
evaluate the brittleness of CBN networks to various datasets, suggesting that
learning from visual features alone could often be superior for
generalization. We evaluate CBN models on natural images for bird
classification and histology images for cancer type classification. We observe
that the CBN network learns close to no visual features on the bird
classification dataset and partial visual features on the histology dataset.
Our extensive experiments reveal that CBN may encourage shortcut learning
between the auxiliary data and labels.

##  1 Introduction

Learning about complex relationships between real-world variables has been the
key to progression of human kind goertzel2012architecture . While some of
these relationships are now intuitive to us, some are more complex
mathematical relationships. With advancement of deep learning in the field of
computer vision he2016deep , natural language processing
vaswani2017transformer , and speech recognition bahdanau2016end , learning
complex relationships across modalities is still an open research question.
Most of the current state-of-the-art algorithms, learn from single modality
dosovitskiy2020image ; raffel2020exploring  which might limit the
representation learnt by the model.

Processing data from multiple modalities can provide richer context to the
respective modalitywang2020learning . For examples, video action recognition
carreira2017quo ; feichtenhofer2019slowfast ;
DBLP:journals/corr/abs-2104-13051  can be benefited by learning from the audio
along with the visual featuresshi2021multi . Such factors motivate context-
based learning, where auxiliary features are used to enhance the primary
feature representation. To learn such contextual information, designing of
methods that can learn from each individual modality along with their fusion
is critical but a challenging task. In computer vision, Conditional Batch
Normalization (CBN) based models that perform feature-wise affine
transformations have shown to achieve strong visual reasoningperez2018film .

CBNde2017modulating  relies on modulating the visual feature stream with
auxiliary features in the intermediary layers of a CNN. The normalized
activations produced by auxiliary network has been considered as a popular
method to compute contextual information. Recent works have shown
unreliability of batch normalization during distributional shifts, adversarial
attacks and dependency on the datasethuang2022delving ; galloway2019batch .
Lack of robustness of BN poses an immediate challenge for its conditional
version, CBN. With adoption of CBN in various applicationsabdelnour2018clear ;
brock2018large ; perez2018film , it is important to study the quality of
visual features learnt in presence of auxiliary data.

In this work, we empirically and visually show that while CBN network which
learns auxiliary data may surpass the performance of standard Batch
Normalization(BN) network, they often fail to learn visual features altogether
or partially. We devise a set of experiments to perform surgery into the CBN
network. We observe that the CBN network is brittle to the type and quality of
the contextual information and fails to generalize realistic scenarios by
learning shortcuts.

##  2 Background

Since the introduction of Batch Normalizationioffe2015batch , they have become
an integral part of CNNs for faster convergence. The intermediary
convolutional layers stabilise the variation of activations by normalising the
input, $x_{i}$ characterised by the following equation:

$x_{i}^{N}=\frac{x_{i}-\mu_{i}}{\sigma_{i}}$ (1)

Batch normalization introduces learnable parameters, $\beta_{i}$ and
$\gamma_{i}$ for each normalised layer $L_{i}^{N}$ :

$x_{i}^{N}=(\frac{x_{i}-\mu_{i}}{\sigma_{i}+\epsilon})\gamma_{i}+\beta_{i}$
(2)

Conditional batch normalizationde2017modulating , aims to learn the learnable
parameters from meta-data to "ground" visual features. It directly predicts
affine scaling parameter of the affine transformation from meta-data. A
shallow multi-layer perceptron, we call it auxiliary network, allows the meta-
data to predict the $\beta^{\prime}$ and $\gamma^{\prime}$ that shifts and
scales the normalised batch features. Formally, for meta-data $a_{i}$ and
auxiliary network for $\mathcal{H}$ and $\mathcal{O}$ respectively for affine
parameter prediction, $\mathcal{H}(a_{i})=\beta^{\prime}$ and
$\mathcal{O}(a_{i})=\gamma^{\prime}$.

$\hat{\beta_{i}}=\beta_{i}+\mathcal{H}(a_{i})$ (3)

$\hat{\gamma_{i}}=\gamma_{i}+\mathcal{O}(a_{i})$ (4)

These updated $\hat{\beta_{i}}$ and $\hat{\gamma_{i}}$ are used as parameters
for batch normalization. Modulation of a neural network activations using
conditional information through CBN has been proven successful in multi-modal
setting perez2018film . While CBN is a computationally efficient method to
fuse different modalities, the robustness of this affine layer is unexplored.
The addition of $\beta^{\prime}$ and $\gamma^{\prime}$, might seemingly be
small yet showing a boost in performance of the task, we observe that it
degrades in the quality of visual features learnt.

##  3 Related Works

Batch normalizationioffe2015batch  layers were introduced to alleviate the
problem of internal covariate shift, also leading to faster model convergence.
However, santurkar2018does  have argued that BN might not alleviate the
internal covariate shift. Recent studies chang2015batch ;
yazdanpanah2022revisiting  have shown standard BN training is detrimental to
performance in domain shift task. CBN layer was introduced to provide
contextual information through affine transformation. perez2018film  uses CBN
layer to modulate visual features to answer compositional questions on CLEVR
datasetjohnson2017clevr . Conditional Instance Normalization
dumoulin2016learned  was introduced in the context of style transfer. In
reinforcement learning, DBLP:journals/corr/abs-1806-01946  used CBN layer to
condition NMN and LSTM to follow compositionality in a 2D grid world. In
generative modeling, brock2018large ; zhang2019self  uses condition biasing
through image description. oreshkin2018tadam  used CBN layer to make input
distribution features robust to variations in few shot setting. While CBN has
proved it efficacy through its acceptance in different applications but a
thorough analysis of its potency is yet to be learntmichalski2019empirical .
We make the first attempt to uncover the potential risk of using CBN via a
series of experiments.

##  4 Experiments

CBN is widely used for various applications such as multi-modal language
acoustic learningabdelnour2018clear , meta-learningrequeima2019fast ;
zintgraf2019fast  and image generationbrock2018large . Due to its varied
applications, we design our methodology for a supervised image classification
task with visual and auxiliary features allowing for an easier surgery of the
neural network. The aim of these experiments is to expose the potential
pitfalls and impacts of fusing meta-data with images using CBN.

###  4.1 Experimental Setup

We design our experiments to answer some fundamental questions posed by fusing
of differently structured auxiliary data via CBN. We design a small experiment
for each question posed although the major question, we aim to ask is - Does
the addition of auxiliary data improve visual feature representation?

#### CBN training - Baseline

In this setting, auxiliary data is passed as the input to the shallow
auxiliary network that predicts the affine CBN parameters. This baseline
experiment is mainly performed to establish effectiveness of CBN for
classification.

#### Effect on CBN in absence of test-time auxiliary data

The additional features passing through auxiliary network and its fusion with
spatial featuresperez2018film  has been reported to improve accuracy across
tasks. We aim to look at the neural network beyond final predictive accuracy
with this experiment. In medical applications, the meta-data collection
differs across hospitals, and often among different medical practitioners. We
simulate this scenario by masking the attributes at test-time. Due to masking,
the auxiliary network is switched off and the network only relies on its own
affine transformation parameters. By reducing the dependence of network on
auxiliary data, it is easier to establish its independence from visual
features during test-time. We eventually completely switch off the auxiliary
data stream and notice the difference in reliance of this auxiliary stream
versus the spatial stream.

#### Effect on CBN with masked train-time auxiliary data

While in previous experiment, we attempted to establish the independence of
the visual stream, in this experiment, we aim to understand the quality of
visual features learnt in absence of meta-data. We mask the attributes
randomly to aid the network to learn and utilize visual features more. During
masking, we essentially pass a flag that prevents the addition of
$\mathcal{H}(a_{i})$ and $\mathcal{O}(a_{i})$ to $\hat{\beta_{i}}$ and
$\hat{\gamma_{i}}$ respectively.

#### Introduction of Supplementary network

We hypothesize that the presence of auxiliary data challenges the learning of
visual features. Therefore we introduce a supplementary network to ground and
assist with the optimization of CBN network to learn visual features. The
supplementary network has same network architecture as the CBN network but
uses standard batch normalization. We optimize a loss consists of: cross-
entropy loss of the CBN network, and distance between CBN network and
supplementary network distribution. More formally, let $\mathcal{F}$ represent
the CBN network that maps input $x_{i}$ to its label $y_{i}$ using subsidiary
features, $a_{i}$ and $\mathcal{G}$ represent the BN network that maps input
$x_{i}$ to its label $y_{i}$. Let the cross entropy loss of CBN network be
represented by $\mathcal{L}^{CBN}$. We define the total sum as follows:

$Loss=m*\mathcal{L}^{CBN}(x_{i},y_{i})+(1-m)*D_{kl}(\mathcal{F}||\mathcal{G})$
(5)

where $m$ is a hyperparameter. It is important to note that the aim of
supplementary network is to simply assist in optimization and grounding the
CBN network, therefore the weighting over the distance between supplementary
and CBN network is very small.

#### Removing data from the CBN network

Datasets such as CUB-200-211wah2011caltech  and Animals with
Attributesxian2018zero  2, which are popular in few-shot classification
problem, have attributes which are highly correlated with the images and
label. We use these datasets to better study the learning capability of visual
features in isolation from auxiliary data.

###  4.2 Dataset

We use natural images and medical images to test the effectiveness of CBN
using our proposed training loss. We use wah2011caltech  dataset for the task
of bird identification. It contains 11,788 images taken from 200 different
species of bird. Every image of the dataset contains 312 binary and continuous
concept labels. This dataset has been widely used in Few-Shot Classification
and Conditional Image Generation, making it suitable for our analysis. The
auxiliary data comes from the attributes. In our experiments we use binary
attributes.

For medical images, Whole Slide Pathology images were used. The tumor-
infiltrating lymphocytes (TILs) saltz2018spatial  contains histological
mappings based on each type of cancer. We use all 13 subsets of TCGA dataset,
therefore constituting 13 cancer types. Although the popular task for such a
dataset is necrosis classification doi:10.1200/PO.18.00043  but since it is a
binary classification task and is prone to overfitting without truly
emphasising the effectiveness of our analysis, we modify the task to be a
classification of different (here, thirteen) tumor types. The advantage of
medical images is that their meta-data is readily available from diagnosis. A
doctor uses whole slide images along with the meta-data for diagnostics
indicating importance of fusion of different modalities. The meta-data
includes information such as origin of tumor, age, gender, and size of tumor
cells.

###  4.3 Results and Analysis

In this section we evaluate the performance of networks that perform affine
transformations using CBN and standard Batch Normalization. Due to BN’s
sensitivity to the batch size lian2019revisit , we perform all our experiments
on batch size of 64. We perform our experiments on ResNet-18 he2015deep
backbone. For CBN network, we modify the ResNet-18 architecture to replace
batch normalization layers with CBN layers. The auxiliary network that uses
auxiliary data is a two layer MLP network to predict the affine parameters for
CBN networks. The experiments in this paper were carried out using the
V100SXM2 GPU. The results reported are averaged over 10 runs. Table 1
summarises the results on test set and we breakdown the analysis according to
the set of different experiments below.

#### CBN Baseline

We train both CBN network and BN network to perform classification on the CUB
and TIL dataset. In principle we expect the CBN network to learn from the
spatial feature along with the features learnt from the other auxiliary
modality. This suggests that, ideally, the representative power of this
baseline model is greater than the representation from the standard vision
only model. One may suspect that this should in turn lead to better
generalization. From, Figure1 we observe that the validation accuracy of the
CBN model is greater than the BN model by a great margin of $6.9\%$ in CUB and
over $10\%$ in TIL dataset. This proves the effectiveness of modulating visual
features by the auxiliary data. We interestingly also notice that the CBN
network achieves convergence much faster than the BN network. Although
convergence is a desired property, it is worrisome if the network converges in
just 4 epochs achieving $100\%$ accuracy in the case of CUB dataset! Although,
the same is not observed in TIL dataset, suggesting the difference in dataset
and/or auxiliary data type. This particularly motivates the upcoming
experiments to make an attempt at uncovering the black-box of conditional
learning from auxiliary information stream.

Figure 1: Validation accuracy versus epoch plotted for CUB dataset and TIL
dataset respectively. Models with CBN layer perform better

#### Dependence of CBN on test-time auxiliary data

In a realistic scenario, due to data privacy and differences in data
acquisition, meta-data collected for each patient maybe varied. If we were to
leverage the CBN model in such a setting, it is likely that some of the
auxiliary information is missing. To test this we limit the auxiliary
information during validation and test-time in increments from 0 to 1 (full
CBN network) of $10\%$ supplementary data. From Figure2 we observe that in the
case of CUB, with less than $20\%$ of auxiliary data, the network performs as
well as in the presence of all of the auxiliary data. While in the case of TIL
dataset, there is a sharp increase in the accuracy with greater fraction of
meta-data. We hypothesize that this is due to the difference in structure and
diversity of meta-data. Naturally histopathology data is very diverse
suggestive of its difficulty in learning. In addition it must be noted that in
complete absence of auxiliary information during testing time, the CBN network
has very low accuracy, $0.5\%$ in the case of CUB and $22.3\%$ for TIL. This
is suggestive of the network _not learning visual_ feature in the case of CUB
dataset. While, simultaneously the CBN network does achieve greater than
random accuracy in TIL dataset, its performance deteriorates highly in absence
of the meta-data.

Figure 2: Accuracy versus the fraction of auxiliary data masked during
inference.

#### Masking of train-time auxiliary data

From Figure 3, the dependence of accuracy on the auxiliary features is
apparent. We therefore attempt to establish the relationship between model’s
performance and the auxiliary data during training phase. In this experiment,
we incrementally increase the fraction of auxiliary data fed in to the CBN
model. With no auxiliary data at train-time, the network behaves similar to a
BN model since auxiliary affine predictions are turned off, where as with all
of the meta-data at train-time, the network behaves and performs similar to
the baseline CBN model. In Figure 3, we observe the general trend of achieving
higher accuracy in presence of higher auxiliary data, although in the
experiment with CUB dataset, the model converges much earlier with TIL data.
Intuitively Figure 3 also signifies the inverse change in dependency of visual
feature representation with increase in auxiliary data. It is observed that,
with affine parameter prediction switches on, the neural network attempts to
use the auxiliary data for label prediction.

Figure 3: Accuracy versus the fraction of auxiliary data masked during
training.

#### Introduction of supplementary network

We hypothesized that supplementary network may help to ground and bring the
CBN network closer to BN network, implying learning of visual features.
Hyperparamaters $m$ equals $0.9$. We observe that the hypothesis is proven
incorrect and supplementary network is not able to reduce the dependence of
auxiliary data on CBN network. In Figure6 we observe that the accuracy curves
for both CUB and TIL datasets are very similar to CBN curve, although more
stable. Due to ineffectiveness of the supplementary network on baseline
experiment, we perform all our experiments with CBN network.

#### Removing spatial data

Table 1 present that even in the absence of image data, the network is able to
perform as well as the baseline CBN network for CUB dataset. This is highly
suggestive of presence of shortcuts in data and the visual features not
contributing to the CBN model’s predictive performance. On the contrary, CBN
network on the TIL dataset learns some feature in absence of visual features,
therefore indicating partial learning of vision features.

#### Discussion

From an exhaustive set of experiments, we observe that although the
attenuating the affine batch normalization parameter via a second modality
leads is higher predictive performance as compared to standard batch
normalization training of vision modality, it leads to deterioration of visual
features. In the case of both TIL and CUB, there is an increased dependency on
the auxiliary data. This suggests that the affine parameters of CBN network
allow for a shortcut learning of auxiliary data leading to poor
generalization. Visually we compare the features learnt by CBN network with
all attributes, CBN network with no attributes and BN network for both CUB and
TIL dataset. We observe that the features learnt in each of the cases are very
different signifying the potential difference in their respective minima. It
is particularly interesting to note that in the absence of visual features in
CUB, the network has random predictive performance signified by 4(b). We
observe that the visual features presented in 4(a) is less explanatory of the
bird features as compared to 4(b).

(a) CBN model

(b) CBN with 0 attributes at test-time

(c) BN model

Figure 4: A visualization of GradCAMselvaraju2017grad  features for Horned
Grebe class. Although the CBN model achieve $100\%$ accuracy, the feature of
BN model are visually more descriptive.

(a) CBN model

(b) CBN with 0 attributes at test-time

(c) BN model

Figure 5: A visualization of GradCAMselvaraju2017grad  features for BLCA
cancer type class.

#### Ablation study - Continuous vs Discrete attributes in CUB

The CUB dataset contains continuous attribute values between 0 and 1. We
threshold the values greater than $50$ to $1$ and rest to $0$. Although in the
case of continuous attributes, we observe from Figure 7 that the neural
network converges within 2 epochs which suggests that the continuous
attributes leads to memorization, highly undesirable for generalization.
Therefore we use binary attributes in all of our experiments.

##  5 Conclusion

In this work, we explored the efficacy of Conditional Batch Normalization to
fuse an auxiliary network with a vision only model. We observe that the CBN
network motivates learning of highly undesirable shortcuts learning between
the auxiliary data and the label. This leads to a boost in performance but a
drop in visual features. We observed that the effectiveness of CBN is
dependent on the dataset. We hope that this work inspires research in the
direction of learning generalizable visual features while conditioning in
presence of differently structured modality.

##  6 Acknowledgements

Authors would like to thank Vincent Michalski for fruitful discussions. We
would like to thank the Digital Research Alliance of Canada for compute
resources, Google and CIFAR for research funding. IS acknowledges funding from
Mitacs and Imagia Canexia Health. AAR acknowledges funding from FRQS.

## References

  * [1] Jerome Abdelnour, Giampiero Salvi, and Jean Rouat.  Clear: A dataset for compositional language and elementary acoustic reasoning.  arXiv preprint arXiv:1811.10561, 2018.
  * [2] Dzmitry Bahdanau, Jan Chorowski, Dmitriy Serdyuk, Philemon Brakel, and Yoshua Bengio.  End-to-end attention-based large vocabulary speech recognition.  In 2016 IEEE international conference on acoustics, speech and signal processing (ICASSP), pages 4945–4949. IEEE, 2016.
  * [3] Dzmitry Bahdanau, Felix Hill, Jan Leike, Edward Hughes, Pushmeet Kohli, and Edward Grefenstette.  Learning to follow language instructions with adversarial reward induction.  CoRR, abs/1806.01946, 2018.
  * [4] Andrew Brock, Jeff Donahue, and Karen Simonyan.  Large scale gan training for high fidelity natural image synthesis.  arXiv preprint arXiv:1809.11096, 2018.
  * [5] Joao Carreira and Andrew Zisserman.  Quo vadis, action recognition? a new model and the kinetics dataset.  In proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 6299–6308, 2017.
  * [6] Jia-Ren Chang and Yong-Sheng Chen.  Batch-normalized maxout network in network.  arXiv preprint arXiv:1511.02583, 2015.
  * [7] Harm De Vries, Florian Strub, Jérémie Mary, Hugo Larochelle, Olivier Pietquin, and Aaron C Courville.  Modulating early visual processing by language.  Advances in Neural Information Processing Systems, 30, 2017.
  * [8] Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, et al.  An image is worth 16x16 words: Transformers for image recognition at scale.  arXiv preprint arXiv:2010.11929, 2020.
  * [9] Vincent Dumoulin, Jonathon Shlens, and Manjunath Kudlur.  A learned representation for artistic style.  arXiv preprint arXiv:1610.07629, 2016.
  * [10] Christoph Feichtenhofer, Haoqi Fan, Jitendra Malik, and Kaiming He.  Slowfast networks for video recognition.  In Proceedings of the IEEE/CVF international conference on computer vision, pages 6202–6211, 2019.
  * [11] Angus Galloway, Anna Golubeva, Thomas Tanay, Medhat Moussa, and Graham W Taylor.  Batch normalization is a cause of adversarial vulnerability.  arXiv preprint arXiv:1905.02161, 2019.
  * [12] Ben Goertzel, Matt Iklé, and Jared Wigmore.  The architecture of human-like general intelligence.  In Theoretical foundations of artificial general intelligence, pages 123–144. Springer, 2012.
  * [13] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.  Deep residual learning.  Image Recognition, 7, 2015.
  * [14] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.  Deep residual learning for image recognition.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 770–778, 2016.
  * [15] Lei Huang, Yi Zhou, Tian Wang, Jie Luo, and Xianglong Liu.  Delving into the estimation shift of batch normalization in a network.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 763–772, 2022.
  * [16] Sergey Ioffe and Christian Szegedy.  Batch normalization: Accelerating deep network training by reducing internal covariate shift.  In International conference on machine learning, pages 448–456. PMLR, 2015.
  * [17] Justin Johnson, Bharath Hariharan, Laurens Van Der Maaten, Li Fei-Fei, C Lawrence Zitnick, and Ross Girshick.  Clevr: A diagnostic dataset for compositional language and elementary visual reasoning.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 2901–2910, 2017.
  * [18] Xiangru Lian and Ji Liu.  Revisit batch normalization: New understanding and refinement via composition optimization.  In The 22nd International Conference on Artificial Intelligence and Statistics, pages 3254–3263. PMLR, 2019.
  * [19] Emily Pei-Ying Lin, Tzu-Hung Hsiao, Jo-yang Lu, Siao-Han Wong, Tzu-Pin Lu, Konan Peck, Takashi Takahashi, and Pan-Chyr Yang.  Translating gene signatures into a pathologic feature: Tumor necrosis predicts disease relapse in operable and stage i lung adenocarcinoma.  JCO Precision Oncology, (2):1–13, 2018.
  * [20] Vincent Michalski, Vikram Voleti, Samira Ebrahimi Kahou, Anthony Ortiz, Pascal Vincent, Chris Pal, and Doina Precup.  An empirical study of batch normalization and group normalization in conditional computation.  arXiv preprint arXiv:1908.00061, 2019.
  * [21] Boris Oreshkin, Pau Rodríguez López, and Alexandre Lacoste.  Tadam: Task dependent adaptive metric for improved few-shot learning.  Advances in neural information processing systems, 31, 2018.
  * [22] Ethan Perez, Florian Strub, Harm De Vries, Vincent Dumoulin, and Aaron Courville.  Film: Visual reasoning with a general conditioning layer.  In Proceedings of the AAAI Conference on Artificial Intelligence, volume 32, 2018.
  * [23] Colin Raffel, Noam Shazeer, Adam Roberts, Katherine Lee, Sharan Narang, Michael Matena, Yanqi Zhou, Wei Li, Peter J Liu, et al.  Exploring the limits of transfer learning with a unified text-to-text transformer.  J. Mach. Learn. Res., 21(140):1–67, 2020.
  * [24] James Requeima, Jonathan Gordon, John Bronskill, Sebastian Nowozin, and Richard E Turner.  Fast and flexible multi-task classification using conditional neural adaptive processes.  Advances in Neural Information Processing Systems, 32, 2019.
  * [25] Joel Saltz, Rajarsi Gupta, Le Hou, Tahsin Kurc, Pankaj Singh, Vu Nguyen, Dimitris Samaras, Kenneth R Shroyer, Tianhao Zhao, Rebecca Batiste, et al.  Spatial organization and molecular correlation of tumor-infiltrating lymphocytes using deep learning on pathology images.  Cell reports, 23(1):181–193, 2018.
  * [26] Shibani Santurkar, Dimitris Tsipras, Andrew Ilyas, and Aleksander Madry.  How does batch normalization help optimization?  Advances in neural information processing systems, 31, 2018.
  * [27] Ramprasaath R Selvaraju, Michael Cogswell, Abhishek Das, Ramakrishna Vedantam, Devi Parikh, and Dhruv Batra.  Grad-cam: Visual explanations from deep networks via gradient-based localization.  In Proceedings of the IEEE international conference on computer vision, pages 618–626, 2017.
  * [28] Ivaxi Sheth.  Three-stream network for enriched action recognition.  CoRR, abs/2104.13051, 2021.
  * [29] Zhensheng Shi, Ju Liang, Qianqian Li, Haiyong Zheng, Zhaorui Gu, Junyu Dong, and Bing Zheng.  Multi-modal multi-action video recognition.  In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 13678–13687, 2021.
  * [30] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin.  Attention is all you need.  Advances in neural information processing systems, 30, 2017.
  * [31] Catherine Wah, Steve Branson, Peter Welinder, Pietro Perona, and Serge Belongie.  The caltech-ucsd birds-200-2011 dataset.  2011\.
  * [32] Yikai Wang, Fuchun Sun, Ming Lu, and Anbang Yao.  Learning deep multimodal feature representation with asymmetric multi-layer fusion.  In Proceedings of the 28th ACM International Conference on Multimedia, pages 3902–3910, 2020.
  * [33] Yongqin Xian, Christoph H Lampert, Bernt Schiele, and Zeynep Akata.  Zero-shot learning—a comprehensive evaluation of the good, the bad and the ugly.  IEEE transactions on pattern analysis and machine intelligence, 41(9):2251–2265, 2018.
  * [34] Moslem Yazdanpanah, Aamer Abdul Rahman, Muawiz Chaudhary, Christian Desrosiers, Mohammad Havaei, Eugene Belilovsky, and Samira Ebrahimi Kahou.  Revisiting learnable affines for batch norm in few-shot transfer learning.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 9109–9118, 2022.
  * [35] Han Zhang, Ian Goodfellow, Dimitris Metaxas, and Augustus Odena.  Self-attention generative adversarial networks.  In International conference on machine learning, pages 7354–7363. PMLR, 2019.
  * [36] Luisa Zintgraf, Kyriacos Shiarli, Vitaly Kurin, Katja Hofmann, and Shimon Whiteson.  Fast context adaptation via meta-learning.  In International Conference on Machine Learning, pages 7693–7702. PMLR, 2019.

## Appendix

Training Model | Testing Model | CUB | TIL
---|---|---|---
BN | BN | $\underset{\pm 1.1}{79.9}$ | $\underset{\pm 2.3}{51.1}$
CBN Baseline | CBN Baseline | $\underset{\pm 0.6}{99.2}$ | $\underset{\pm 3.4}{62.9}$
CBN Baseline | CBN-No attributes | $\underset{\pm 0.1}{0.5}$ | $\underset{\pm 3.8}{19.3}$
CBN + Supplementary Baseline | CBN + Supplementary Baseline | $\underset{\pm 0.5}{99.0}$ | $\underset{\pm 2.9}{63.1}$
CBN-No visual data | CBN-No visual data | $\underset{\pm 0.6}{99.1}$ | $\underset{\pm 10.7}{20.8}$
Table 1: Summary of results Figure 6: Accuracy versus epoch for CBN hybrid in
presence of supplementary network. Figure 7: Accuracy versus epoch for CBN
network in presence of continuous and binary attributes.


# Measurement Based Non-Line-Of-Sight
Vehicular Visible Light Communication Channel Characterization

Bugra Turan, Omer Narmanlioglu, Osman Nuri Koc, Emrah Kar, , Sinem Coleri, ,
and Murat Uysal B.Turan, O. N. Koc and E. Kar are with the Koc University Ford
Otosan Automotive Technologies Laboratory, Sariyer, Istanbul, 34450, Turkey.
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Coleri is with
the Department of Electrical and Electronics Engineering, Koc University,
Sariyer, Istanbul 34450, Turkey.
E-mail<EMAIL_ADDRESS>Narmanlioglu and M. Uysal are with the Department
of Electrical and Electronics Engineering, Ozyegin University, Istanbul 34794,
Turkey.
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>work was
supported by CHIST-ERA grant CHIST-ERA-18-SDCDN-001, the Scientific and
Technological Council of Turkey 119E350 and Ford Otosan.

###### Abstract

Vehicular visible light communication (V-VLC) aims to provide secure
complementary vehicle to everything communications (V2X) to increase road
safety and traffic efficiency. V-VLC provides directional transmissions,
mainly enabling line-of-sight (LoS) communications. However, reflections due
to nearby objects enable non-line-of-sight (NLoS) transmissions, extending the
usage scenarios beyond LoS. In this paper, we propose wide-band measurement
based NLoS channel characterization, and evaluate the performance of direct
current biased optical orthogonal frequency division multiplexing (DCO-OFDM)
V-VLC scheme for NLoS channel. We propose a distance based NLoS V-VLC channel
path loss model considering reflection surface characteristics and NLoS V-VLC
channel impulse response (CIR) incorporating the temporal broadening effect
due to vehicle reflections through weighted double gamma function. The
proposed path loss model yields higher accuracy up to 14 dB when compared to
single order reflection model whereas CIR model estimates the full width at
half maximum up to 2 ns accuracy. We further demonstrate that the target bit-
error-rate of $10^{-3}$ can be achieved up to 7.86 m, 9.79 m and 17.62 m
distances for black, orange and white vehicle reflection induced measured NLoS
V-VLC channels for DCO-OFDM transmissions.

###### Index Terms:

Visible light communication channel, DCO-OFDM, NLoS channel model.

##  I Introduction

Vehicular communications enable exchange of traffic, road and vehicle
information between vehicles, infrastructure and other road users through
vehicle to vehicle (V2V), vehicle to infrastructure (V2I) and vehicle to
pedestrian (V2P) communications. Currently, radio frequency (RF) based
cellular vehicle to everything (C-V2X) [1] and IEEE 802.11p [2] technologies
are standardized to support V2X applications. On the other hand, omnipresence
of light emitting diodes (LEDs) in the vehicle lighting paves the way for
vehicular visible light communications (V-VLC) as a complementary technology
to the existing schemes. Since V-VLC provides RF interference, jam and spoof
free directional optical communications, it is foreseen to be a strong
candidate to enable secure vehicular communications [3].

V-VLC is mainly considered to be a line-of-sight (LoS) technology due to its
directional propagation characteristics. However, non-line-of-sight (NLoS)
V-VLC transmissions are also demonstrated to increase received signal strength
(RSS) of LoS V-VLC link through object reflections [4]. Therefore, NLoS V-VLC
can be considered to support close proximity V2V safety applications such as
blind spot warning, lane change warning and intention sharing by taking
advantage of nearby vehicle reflected optical signals.

V-VLC system performance mainly depends on the V-VLC propagation channel
characteristics, which distort the optical signals through attenuation,
scattering and reflection. Thus, V-VLC channel modeling is extensively studied
in the literature, based on the LoS transmission assumptions [5, 6, 7]. [5]
proposes a ray tracing simulation based LoS path loss model incorporating
headlight radiation pattern, [6] presents taillight measurement based path
loss model and [7] provides a statistical path loss distribution model with
the consideration of asymmetric headlight radiation pattern. On the other
hand, only a few studies investigate the NLoS V-VLC channel [4, 8], where [4]
demonstrates the incremental effect of highly reflective road surface on V-VLC
RSS and [8] shows the bit-error-rate (BER) performance degradation of V-VLC
due to road surface specular reflections. However, the current literature
lacks the characterization of NLoS V-VLC channel due to vehicle reflections,
where the vehicle surface mainly determines the path loss and signal
distortions. On the other hand, channel impulse response (CIR) temporal
broadening due to NLoS scattering, which determines the achievable data rate
and bandwidth of the V-VLC systems has not yet been investigated for V-VLC
channels. Moreover, empirical channel models based on the low bandwidth
optical power measurements by optical power meters [6] and RSS measurements at
single tone frequencies [4] fail to address the channel behavior for a wide
range of LED modulation frequencies.

In this paper, we propose wide-band frequency sweep measurement based CIR and
path loss characterization for NLoS V-VLC channels due to vehicle body
reflections. Moreover, we investigate the BER performance of direct current
biased optical orthogonal frequency division multiplexing (DCO-OFDM) based
V-VLC system for NLoS channels for different modulation order and background
noise levels. We demonstrate that the proposed NLoS V-VLC channel path loss
model provides higher accuracy than the widely considered single order
reflection based NLoS channel model [9] for indoor visible light
communications (VLC), since NLoS channel model for V-VLC has not yet been
proposed in the literature. The original contributions of this paper are
threefold:

  * •

We provide a distance based NLoS V-VLC channel path loss model with the
consideration of pure vehicle surface reflections, for the first time in the
literature.

  * •

We characterize CIR of the NLoS V-VLC through weighted double gamma function
(WDGF) by using measurement data from different vehicle reflection induced
channels. Thus, for the first time in the literature, we investigate the
temporal broadening effect of V-VLC CIR due to vehicle reflections.

  * •

We provide attainable NLoS V-VLC distance for the target BER performance of
DCO-OFDM scheme under the consideration of measured channel gains for a wide
range of LED modulation frequencies.

The rest of the paper is organized as follows. Section II describes the
considered DCO-OFDM transmission system model. Section III provides the
methodology for measurement based NLoS channel modeling. Section IV presents
the proposed path model and WDGF based NLoS CIR characterization. Section V
presents simulation results for DCO-OFDM V-VLC for the empirical NLoS V-VLC
channel. Finally, Section VI concludes the paper.

##  II System Model

In this section, DCO-OFDM based transmission model is provided, where the
performance of DCO-OFDM based V-VLC system is evaluated for the proposed NLoS
channels.

In DCO-OFDM, binary information is first mapped to complex symbols
$s_{1}\textrm{ }s_{2}\textrm{ }...\textrm{ }s_{N/2-1}$ based on the deployed
constellation scheme, such as $M-$order quadrature amplitude modulation (QAM),
where $N$ is the number of subcarriers. To ensure that the output of inverse
fast Fourier transform (IFFT) is real-valued, Hermitian symmetry is imposed on
the transmitted signal as,

${{\bf{X}}_{\bf{S}}}={\left[{0\textrm{ }{s_{1}}\textrm{ }{s_{2}}\textrm{
}...\textrm{ }{s_{N/2-1}}\textrm{ }0\textrm{ }s_{N/2-1}^{*}\textrm{
}...\textrm{ }s_{2}^{*}\textrm{ }s_{1}^{*}}\right]^{\rm{T}}},$ (1)

where $(.)^{*}$ and $[.]^{\rm T}$ denote complex conjugate and transpose
operations, respectively. After $N-$point IFFT operation where the output is
denoted by $x_{S}[n]$, a cyclic prefix with length $N_{\mathrm{CP}}$, which is
greater than or equal to the delay spread of the channel, is appended to the
beginning of each DCO-OFDM frame and parallel streams are converted to serial.
The resulting DCO-OFDM signal is

$x_{S}(t)=\sum_{n=0}^{N+N_{\mathrm{CP}}-1}x_{S}[n]\delta(t-nT_{s}),$ (2)

where $T_{s}$ is the sampling interval and $x_{S}(t)$ has the average
electrical transmit signal power of $P_{\rm T}$. The modulated signal is
direct current (DC) biased and transmitted via LED. The time domain received
signal at the photodetector (PD) is

$y(t)=Rx_{S}(t)\otimes{h}(t)+v_{n}(t),$ (3)

where $R$ is the responsivity of PD (A/W), $h(t)$ is the optical channel
response obtained by the measurements, and $v_{n}(t)$ is additive white
Gaussian noise (AWGN) with zero mean and $\sigma_{N}^{2}=N_{0}W$ variance,
where $N_{0}$ is noise power spectral density (PSD) and $W$ denotes the system
bandwidth. In (3), $\otimes$ denotes the convolution operation.

At the receiver, the captured optical signals are converted to parallel
streams and cyclic prefix is removed. Then, the $N-$point fast Fourier
transform (FFT) is applied on the output.

##  III Measurement Based NLoS Channel Modeling

Vehicle surfaces are made up of different materials and follow various
geometries, where V-VLC signal reflection characteristics vary between
vehicles. To incorporate the vehicle surface dependent reflection
characteristics, a measurement based approach is employed with the following
steps:  (i) characterize the LED modulation bandwidth and ambient noise to
determine measurement parameters of maximum sweep frequency and intermediate
frequency bandwidth $({\rm IF\;BW})$,  (ii) measure the LoS channel frequency
response to obtain channel gain values as a benchmark to evaluate NLoS
channel,  (iii) measure NLoS channel gains for path loss and CIR
characterization,  (iv) post-process measurement data to obtain NLoS channel
path loss and CIR.

The LED modulation bandwidth determines the maximum measurable frequency of
the channel, whereas vector network analyzer (VNA) intermediate frequency
bandwidth selection determines the noise floor of the measurements. The LoS
channel gain ($S_{21}$) values constitute reflection free, non-distorted
channel frequency response measurements, which are used to evaluate reflection
induced NLoS measurements. The NLoS $S_{21}$ measurements are post processed
to extract distance and reflection surface dependent path loss with CIR
temporal characteristics.

To capture NLoS components from vehicle reflections, a vehicle LED day time
running light (DRL) in the nearby lane is directed towards the rear of the
reflector vehicle. The receiver vehicle is positioned behind the reflector
vehicle in the same lane, where LoS link does not exist between the
transmitter and receiver vehicle. The transmitter-reflector vehicle distance
($D_{1}$) is fixed, whereas the distance between reflector and receiver
vehicle ($D_{2}$) is varied. Three different reflection surfaces, orange light
commercial vehicle (Ford Courier 2019), white (BMW 320i 2014) and black
passenger vehicle (Fiat Tipo Station Wagon 2018) bodies are considered as
reflection surfaces to model the NLoS channel. The measurement scenario is
depicted in Fig. 1.

Figure 1: NLoS Scenario

The LoS channel path loss of the measurement setup is obtained with the
limitation of PD’s field of view (FoV) with a tube similar to the work in [4],
at the same transmitter-receiver height. Then, NLoS channel measurements are
conducted to obtain LED modulation frequency dependent channel gains. The path
loss and CIR are calculated at the post-processing stage. To minimize random
variations of the received signal due to measurement equipment, all measured
frequency responses are averaged over 10 realizations per measurement
location.

At the transmitter side, the sounding signals are fed into the driver block
from Port 1 of the VNA. The driver block amplifies the sounding signals and
adds DC bias before feeding to the LED. At the receiver side, PD captures the
signals and feed into the Port 2 of the VNA, yielding frequency dependent
$S_{21}$ values.

#### Path Loss Characterization

The path loss is derived as the ratio of the received electrical power to the
transmitted electrical power over the measurement bandwidth. The path loss
from the measurement data is obtained by [10]

${\rm PL\;[dB]}=-10{\rm log}_{10}{\left(\frac{1}{N_{\rm f}}\sum_{n=0}^{N_{\rm
f}-1}{|S_{\rm 21}(f_{\rm min}+n\Delta f)|^{2}}\right)},$ (4)

where $N_{\rm f}$ is the total number of $S_{\rm 21}$ values from minimum
measurement frequency, $f_{\rm min}$, to maximum measurement frequency,
$f_{\rm max}$, due to LED modulation bandwidth limitations, and $\Delta f$ is
the step frequency.

The reflection coefficient ($\rho$) of the considered surfaces is calculated
by the ratio of the measured NLoS reflected surface channel gain to pure LoS
channel gain at the reference distance of 2 m. The reflection coefficient
demonstrates the reflectivity of the surface, where higher values indicate
more reflective surfaces (i.e., brighter color).

Table I: NLoS path loss model and reflection coefficients Surface | Parameters
---|---
$\alpha$ | $\beta$ | n |  | Reflection
---
Coefficient
White Car | 0.9185 | 4.703 | 0.7189 | 0.0774
Orange Car | 0.7871 | 5.477 | 0.9998 | 0.0243
Black Car | 0.7516 | 5.384 | 0.9238 | 0.0156

####  NLoS V-VLC CIR Characterization

The NLoS CIR is characterized to evaluate the temporal distortion effects of
reflections on V-VLC signals. To obtain CIRs of the channels under
consideration, non-measured frequency interval of 0-200 kHz due to VNA
limitations, are zero padded and Hermitian symmetry is imposed on the $N$
measured $S_{21}$ parameters.

The IFFT of the zero padded and Hermitian symmetry imposed samples yields CIR
of the NLoS channel. To increase the time resolution of the CIR, frequency
domain measurement data are zero padded for the frequencies between $f_{\rm
max}$, and the target frequency ($f_{\rm target}$), the bandwidth is given by
${\rm BW}=f_{\rm target}-f_{\rm min}$ and time resolution is given by $t_{\rm
res}=1/{\rm BW}$.

The measurement parameters are selected as $f_{\rm min}=200$ kHz, $f_{\rm
max}=3$ MHz, $\Delta f=700$ Hz, ${\rm IF\;BW}=$ $10$ Hz, noise floor $=-110$
dBm, whereas $t_{\rm res}=1$ ns is selected with the padding of measurement
samples.

##  IV Empirical CIR and Path Loss Model

In this section, we provide CIR tail characterization together with the least-
squares fitting based channel path loss model for NLoS V-VLC channel.

###  IV-A Channel Impulse Response

The increasing transmitter-receiver for LoS and reflector-receiver distances
for NLoS channels result in longer CIR tails due to increased photons
scattering (See Fig. 2). We characterize CIR temporal broadening as a function
of transmitter-receiver distance for LoS and reflector-receiver distance for
NLoS channels. We adopt full width at half maximum (FWHM) metric to quantify
CIR broadening [11] and NLoS V-VLC CIR is approximated by WDGF as [12]

$h(t)=C_{1}\Delta t^{\alpha}{\rm exp}(-C_{2}\Delta t)+C_{3}\Delta
t^{\beta}{\rm exp}(-C_{4}\Delta t),$ (5)

where $C_{1}$, $C_{2}$, $C_{3}$, $C_{4}$, $\alpha$, and $\beta$ are model
coefficients to be found by least-squares fitting to the measurement data.
$\Delta t$ denotes the time scale starting from the CIR peak, since the tail
determines the CIR broadening.

Figure 2: LoS and NLoS CIR distance dependent temporal broadening.

Fig. 2 demonstrates the CIR temporal broadening of LoS and NLoS channels with
the increasing distance. For LoS transmissions at 2 m and 20 m transmitter-
receiver distances, 671 ns and 690 ns FWHM is measured, respectively. On the
other hand, white vehicle reflected NLoS transmissions yield 696 ns and 778 ns
FWHM at 2 m and 20 m distances, respecively. Thus, CIR broadening due to NLoS
transmissions can be considered substantial when compared to LoS transmissions
since delay spread at 2 m NLoS channel is almost equivalent to 20 m LoS
channel.

Figure 3: CIR temporal broadening with increasing distances.

Fig. 3 shows the distance dependent measured and WDGF modeled FWHM of the LoS
and NLoS channels under consideration. For orange and white vehicle reflected
NLoS channels, WDGF model well approximates the CIR FWHM, whereas for black
vehicle reflected NLoS channels, the model accuracy decreases for diffuse
reflections, where model overestimates the measured CIR FWHM by 160 ns. At 16
m distance, black vehicle body diffuse reflections help to preserve CIR FWHM
with 29 ns elongation, whereas the model estimates 261 ns broadening. However,
providing 2 ns and 5 ns FWHM estimation accuracy at 12 m and 18 m distances,
respectively, for specular reflections of black vehicle induced NLoS channel,
the WDGF model can be considered to approximate the NLoS V-VLC CIR for
different distances and reflection surfaces.

###  IV-B Path Loss Model

The least-squares fitting based empirical NLoS channel path loss model is
obtained as

$\displaystyle{\rm PL\;[dB]}=$ $\displaystyle 10{\rm
log}_{10}\left(\left(\alpha{{\rm
exp}(-n}\frac{d_{0}}{d})\right)^{d-d_{0}}\left(\frac{d}{d_{0}}\right)^{\beta}\right)$
$\displaystyle+{\rm PL_{REF}},$ (6)

where ${\rm PL_{REF}}$ is the path loss at the reference distance $d_{0}$, $d$
is the reflection surface to receiver distance, $\alpha$, $\beta$ and $n$ are
the reflection surface dependent model coefficients. The proposed equation
incorporates reflection surface dependent parameters into the path loss at
reference distance, since each reflection surface inherits varying distance
dependent reflection characteristics.

Figure 4: V-VLC NLoS path loss for measured, Lambertian single order
reflection and proposed model fits

Fig. 4 shows measured path loss along with the single order reflection model
detailed in Appendix, and proposed empirical NLoS channel path loss models for
three different vehicle reflection surfaces. The proposed path loss expression
provides 0.18 dB, 0.19 dB, and 0.22 dB root mean square error (RMSE) to the
measurement data. On the other hand, single surface reflection model
underestimates the path loss by 14 dB, 8 dB, and 4 dB at 20 m reflector-
receiver distance for white, orange and black cars, respectively. The proposed
model, incorporating wide range of modulation frequencies of the LED, yields
accurate path loss estimations to evaluate the performance of multi-carrier
based optical communication schemes such as DCO-OFDM.

Figure 5: Ratio of NLoS to LoS channel gains for different reflective surfaces
and distances.

Fig. 5 shows the normalized NLoS channel gain for varying transmitter-receiver
distances, to evaluate the effects of various surface and colors on NLoS
channel path loss. The ratio mainly depends on the reflection surface
properties, where the receiver captures reflections from various parts of the
vehicle surface, such as bumper, trunk lid and vehicle body, depending on the
type of the vehicle. Considering the PD’s FoV, mainly specular reflections
from vehicle body is captured up to 8 m. Then, reflections from bumpers,
partially made up of rough plastic, yield diffuse reflections between $8-12$
m. Moreover, depending on the trunk lid design, flat surface of the orange
vehicle yields specular reflections whereas the edges and curvatures of the
black and white vehicle trunk lids lead to diffuse reflections between 12 m to
20 m.

##  V DCO-OFDM V-VLC Performance Analysis

The aim of this section is to investigate the BER performance of the DCO-OFDM
V-VLC scheme for NLoS channel at varying distances with Monte Carlo
simulations. Since V-VLC performance highly depends on the ambient noise
levels, we investigate day and night performance of the DCO-OFDM scheme with
two different noise levels and modulation orders. The NLoS V-VLC measured
channel gain values are considered with the simulation parameters set
including $P_{\rm T}=-12$ dBm, $N=64$, $R=0.3$, $M=4$-QAM for day, 16-QAM for
night, noise power of $-100$ dBm for day and $-110$ dBm for night scenarios.

Figure 6: NLoS V-VLC DCO-OFDM BER performance for three different vehicle
surface reflections, day (4-QAM) and night (16-QAM) scenarios

Fig. 6 depicts the Monte Carlo simulation results for distance based BER
performance of the DCO-OFDM scheme with the consideration of measured channel
gain values. For black vehicle induced NLoS channel, achievable reflector-
receiver distances for target BER of $10^{-3}$ are 6.67 m and 7.86 m for day
and night conditions, respectively. Considering orange vehicle reflected NLoS
channel, the achievable distances for target BER of $10^{-3}$ are 8.22 m and
9.79 m, whereas for white vehicle NLoS channel, target BER is attained at
14.26 m and 17.62 m distances at day and night conditions, respectively. The
achievable distance performance of the white vehicle reflected NLoS channel is
substantially better than black and orange vehicle reflected channels, due to
stronger reflections. Moreover, considering day and night performance of the
NLoS channels under consideration, the achievable distance difference between
two scenarios (i.e., day 4-QAM and night 16-QAM) increases with stronger
reflections, where the difference is 1.19 m, 1.57 m and 3.36 m for black,
orange and white vehicle reflected NLoS channels, respectively.

##  VI Conclusion

In this paper, we investigate the NLoS V-VLC channel path loss, CIR and BER
performance for DCO-OFDM based transmissions. NLoS channel path loss
expression is proposed by using the channel measurement data obtained through
three different reflection surfaces, where it is demonstrated to well
approximate the measurement data. Moreover, CIR of the NLoS V-VLC channel is
demonstrated to broaden with the increasing distance due to scattering of
photons. Therefore, WDGF model is adopted to approximate the CIR of NLoS
channel, where it provides higher accuracy for specular reflections induced
NLoS channels. The diffuse reflections due to vehicle tailgate design is
demonstrated to yield channel gain and CIR FWHM fluctuations, where the
approximation accuracy of the WDGF CIR model decreases. The BER performance of
DCO-OFDM V-VLC scheme demonstrates that, white vehicle reflections yield
substantially better achievable distance for target BER, whereas NLoS channel
through the black and orange vehicle reflections are able to provide 6.64 m
and 8.20 m link distance under day conditions, which can be practical for
close proximity V2V safety applications. The proposed path loss model along
with the WDGF CIR models can be employed for system simulations.

[Lambertian Single Order Reflection Model]

In this appendix, widely used single order reflection model is provided to
evaluate the path loss and time domain parameters of NLoS VLC channels. The
single order reflection model is considered as a benchmark to evaluate the
NLoS V-VLC channel path loss. For single order reflection model, the $i^{\rm
th}$ NLoS component is obtained through the summation of the single order
reflected components from each reflection surface.

The channel gain of the single order reflection NLoS VLC channel is given by
[13]

$\displaystyle H_{i}(d_{1},d_{2})=$ $\displaystyle\frac{(m+1)A_{\rm
r}}{2\pi^{2}d_{\rm 1}^{2}d_{\rm 2}^{2}}\rho dA_{\rm r}f(\alpha,\beta,\Psi),$
(7) $\displaystyle f(\alpha,\beta,\Psi)=$ $\displaystyle{\rm cos}^{m}\phi{\rm
cos}(\alpha){\rm cos}(\beta)T_{\rm s}(\Psi)g(\Psi){\rm cos}(\Psi),$ (8)

where $A_{\rm r}$ is the active area of the PD; $d_{\rm 1}$ is the distance
between the transmitter and reflector vehicle; $d_{\rm 2}$ is the distance
between the receiver and reflector vehicle; $\Psi$ is the angle of incidence
at the receiver; $\Psi_{c}$ is the field of view (FOV); $\alpha$ and $\beta$
are the angle of irradiance to the reflector vehicle and incidence to the
receiver, respectively; $\rho$ and $dA_{\rm r}$ are the reflection coefficient
and small differential area of the reflector surface, respectively; $T_{\rm
s}(\Psi)$ is the gain of an optical filter; $g(\Psi)$ is the gain of a non
imaging optical concentrator with refractive index $n$ given by

$g(\Psi)=\frac{n^{2}}{sin^{2}(\Psi_{\rm c})}.$ (9)

Single order reflection model generally assumes uniform distribution of the
reflection points throughout the reflection area, which is applicable for
large surface areas such as walls. Thus, single order reflection model is
based on the assumption of perfect diffuse surface, which equally scatters
incident illumination in all directions.

## References

  * [1] R. Molina-Masegosa and J. Gozalvez, “Lte-v for sidelink 5g v2x vehicular communications: A new 5g technology for short-range vehicle-to-everything communications,” IEEE Vehicular Technology Magazine, vol. 12, no. 4, pp. 30–39, 2017.
  * [2] F. Arena, G. Pau, and A. Severino, “A review on ieee 802.11 p for intelligent transportation systems,” Journal of Sensor and Actuator Networks, vol. 9, no. 2, p. 22, 2020.
  * [3] A. Memedi and F. Dressler, “Vehicular visible light communications: A survey,” IEEE Communications Surveys & Tutorials, 2020.
  * [4] C. Tebruegge, A. Memedi, and F. Dressler, “Empirical characterization of the nlos component for vehicular visible light communication,” in 2019 IEEE Vehicular Networking Conference (VNC), pp. 1–4, IEEE, 2019.
  * [5] M. Karbalayghareh, F. Miramirkhani, H. B. Eldeeb, R. C. Kizilirmak, S. M. Sait, and M. Uysal, “Channel modelling and performance limits of vehicular visible light communication systems,” IEEE Transactions on Vehicular Technology, vol. 69, no. 7, pp. 6891–6901, 2020.
  * [6] H. B. Eldeeb, E. Eso, E. A. Jarchlo, S. Zvanovec, M. Uysal, Z. Ghassemlooy, and J. Sathian, “Vehicular vlc: A ray tracing study based on measured radiation patterns of commercial taillights,” IEEE Photonics Technology Letters, 2021\.
  * [7] F. Al-Sallami, N. Aigoro, A. A. Mahmoud, Z. U. Ahmad, P. Haigh, O. Haas, and S. Rajbhandari, “Impact of vehicle headlights radiation pattern on dynamic vehicular vlc channel,” Journal of Lightwave Technology, 2021.
  * [8] P. Luo, Z. Ghassemlooy, H. Le Minh, E. Bentley, A. Burton, and X. Tang, “Performance analysis of a car-to-car visible light communication system,” Applied Optics, vol. 54, no. 7, pp. 1696–1706, 2015.
  * [9] T. Komine and M. Nakagawa, “Fundamental analysis for visible-light communication system using led lights,” IEEE transactions on Consumer Electronics, vol. 50, no. 1, pp. 100–107, 2004.
  * [10] M.-T. Martinez-Ingles, D. P. Gaillot, J. Pascual-Garcia, J.-M. Molina-Garcia-Pardo, J.-V. Rodríguez, L. Rubio, and L. Juan-Llácer, “Channel sounding and indoor radio channel characteristics in the w-band,” EURASIP Journal on Wireless Communications and Networking, vol. 2016, no. 1, pp. 1–8, 2016.
  * [11] G. Chen, Z. Xu, and B. M. Sadler, “Experimental demonstration of ultraviolet pulse broadening in short-range non-line-of-sight communication channels,” Optics express, vol. 18, no. 10, pp. 10500–10509, 2010.
  * [12] Y. Dong, H. Zhang, and X. Zhang, “On impulse response modeling for underwater wireless optical mimo links,” in 2014 IEEE/CIC International Conference on Communications in China (ICCC), pp. 151–155, IEEE, 2014.
  * [13] V. Dixit and A. Kumar, “Performance analysis of non-line of sight visible light communication systems,” Optics Communications, vol. 459, p. 125008, 2020.