# Apparent evidence for Hawking points in the CMB Sky

Daniel An1, Krzysztof A. Meissner2, Paweł Nurowski3 and Roger Penrose4
1Science Department SUNY Maritime College,
6 Pennyfield Av., New York 10465, USA
2Faculty of Physics, University of Warsaw
Pasteura 5, 02-093 Warsaw, Poland
3Center for Theoretical Physics of PAS
Al. Lotników 32/46, 02-688 Warsaw, Poland
4Mathematical Institute, Oxford University,
Radcliffe Observatory Quarter,
Woodstock Rd., Oxford OX2 6GG, UK

###### Abstract

Dedicated to the memory of Stephen Hawking

This paper presents strong observational evidence of numerous previously
unobserved anomalous circular spots, of significantly raised temperature, in
the CMB sky. The spots have angular radii between 0.03 and 0.04 radians (i.e.
angular diameters between about 3 and 4 degrees). There is a clear cut-off at
that size, indicating that each anomalous spot would have originated from a
highly energetic point-like source, located at the end of inflation – or else
point-like at the conformally expanded Big Bang, if it is considered that
there was no inflationary phase.

The significant presence of these anomalous spots, was initially noticed in
the Planck 70 GHz satellite data by comparison with 1000 standard simulations,
and then confirmed by extending the comparison to 10000 simulations. Such
anomalous points were then found at precisely the same locations in the WMAP
data, their significance confirmed by comparison with 1000 WMAP simulations.
Planck and WMAP have very different noise properties and it seems exceedingly
unlikely that the observed presence of anomalous points in the same directions
on both maps may come entirely from the noise. Subsequently, further
confirmation was found in the Planck data by comparison with 1000 FFP8.1 MC
simulations (with $l\leqslant 1500$).

The existence of such anomalous regions, resulting from point-like sources at
the conformally stretched-out big bang, is a predicted consequence of
conformal cyclic cosmology (CCC), these sources being the Hawking points of
the theory, resulting from the Hawking radiation from supermassive black holes
in a cosmic aeon prior to our own.

###### pacs:

04.20.Ha,04.70.Dy,98.80.Bp,98.80.Ft

1\. Conformal Cyclic Cosmology

The proposal of conformal cyclic cosmology (CCC) P1 ; P2 ; P3  provides an
alternative to the initial inflationary epoch of current cosmology, where CCC
replaces inflation by the $\Lambda$-driven exponential expansion of a pre-Big
Bang universe epoch, referred to as a previous aeon (compare Gasperini and
Veneziano GV ). $\Lambda$ is taken as an absolute positive (cosmological)
constant and the entire universe history is taken to be an unending sequence
of such aeons, each beginning with a big bang and ending with a
$\Lambda$-driven exponential expansion. The conformal infinity (see P64 ) of
each aeon joins conformally smoothly to the conformally expanded big bang
origin of the subsequent aeon.

The conformal infinity of each aeon is spacelike because $\Lambda>0$ P65  and
the conformally stretched big bang of each is also spacelike, so the matching
has some geometrical rationale. Moreover this identification is physically as
well as geometrically plausible. The conformal squashing of the exceedingly
cold and rarefied remote future results in an enormous increase in the
temperature and density. Correspondingly, the conformal stretching of each big
bang provides an enormous reduction in the temperature and density, so the
conformal matching is not physically implausible. Moreover, the matter content
of the remote future of each aeon is essentially dominated by photons, these
being governed by the conformally invariant Maxwell equations so that, with
regard to this dominant matter component of the remote future, this conformal
picture of space-time is physically appropriate. Correspondingly, as we
proceed back in time into the big bang of each aeon, we find that massive
particles attain kinetic energies so large that their masses become physically
irrelevant and act as conformally invariant massless entities in the limit as
that aeon’s big bang is approached. It is thus argued that the conformal
joining of each aeon’s conformal future infinity to the conformally stretched
big bang of a subsequent aeon may be considered as making both geometrical and
physical sense.

There are two additional issues concerning the remote future of each aeon. One
of these is the presence of a certain proportion of massive particles, atoms,
or molecules, that one could expect to survive to the indefinite future of
each aeon, such as hydrogen atoms, protons (and/or positrons), electrons, and
neutrinos (or neutrons, etc. in neutron stars), which in current theory retain
their mass indefinitely. This is dealt with in CCC by the proposal of a
$\Lambda$-driven ultimate mass fade-out, where it is argued that the masses of
all particles asymptotically fade to zero in the remote future of each aeon.
One possible rationale for this is, with $\Lambda$ being fundamental, that the
most basic group of particle physics should be the de Sitter rather than
Poincaré group, and that mass, not being a Casimir operator of the de Sitter
group, need not be an absolute constant, for a stable particle, in the
presence of $\Lambda$ (see P3 ). Charge remains constant in CCC, as does
$\hslash$, from which it follows that the ground state of hydrogen will
gradually dissociate as the electron mass and proton mass gradually fade away.
Protons themselves might decay, but since charge is conserved one may expect
that the ultimate positively charged decay product would be positrons,
subject, as with electrons, to ultimate asymptotic mass fade-out.

This proposal of ultimate mass fade-out is indeed an assumption of CCC. The
remaining remote-future issue is the ultimate fate of black holes. Accepted
theory H74 ; H75  asserts that these will eventually evaporate away by Hawking
radiation as the universe finally cools to near absolute zero. CCC is in
accordance with this view, whose remarkable consequences will be discussed
further in Section 3. For now, it may be remarked, first, that galactic
clusters remain essentially bound as the universe (exponentially) expands, and
it may be expected that the majority of the mass in a cluster should
eventually be swallowed by a hugely supermassive black hole of perhaps up to
some $10^{14}$ solar masses. The expectation is Page  that although it might
take up to some $10^{106}$ years, virtually the entire mass of that black
hole, and therefore of that galactic cluster, will ultimately be radiated away
in the form of photons, or perhaps other particles that will have become
effectively massless due to mass fade-out. Because of the extremely late
stage, in each aeon, of this energy release, and owing to the huge conformal
squashing involved, it all becomes effectively concentrated at a single point
H (as marked in Fig.1.) of the crossover 3-surface X (as marked in Fig.1.)
which joins that aeon’s conformal infinity to the conformally stretched big
bang of the subsequent aeon. We refer to such a point H as a Hawking point.

Figure 1: A conformal diagram representing the effect of a highly energetic
event occurring at the space-time point H. In CCC, H is taken to be a Hawking
point, where virtually the entire Hawking radiation of a previous-aeon
supermassive black hole is concentrated at H by the conformal compression of
the hole’s radiating future. The horizontal line at the bottom stands for the
crossover surface dividing the previous cosmic aeon from our own and describes
our conformally stretched Big Bang. In conventional inflationary cosmology, X
would represent the graceful exit turn-off of inflation. In each case, the
future light cone of H represents the outer causal boundary of physical
effects initiated at H, and such effects can reach D only within the roughly
0.08 radian spread indicated at the top of the diagram.

Before coming to the observational implications of this picture, it will be
helpful to explain certain other aspects of CCC. It is important, first, to
understand that CCC is not, in any major way, in conflict with conventional
$\Lambda$-CDM cosmology, from some $10^{-35}$ s (the presumed turn-off moment
of inflation, in conventional theory) after our Big Bang (where the
capitalized “Big Bang” refers to the specific moment that initiated our own
particular aeon). From this moment onwards, within our current aeon, CCC is in
basic agreement with the conventional $\Lambda$-CDM picture – though the
Hawking points would modify things in a way that we argue here has some direct
observational support. CCC’s ultimate mass fade-out does not affect this
agreement. The presence of Hawking points adds an intriguing new ingredient to
the conventional picture, though not grossly altering it. Most particularly,
CCC would not appear to alter, in any major way, the conventional description
of the $\sim$380000-year period following the turn-off of inflation (graceful
exit) and decoupling, being beautifully confirmed by agreement with the
Planck-data power spectrum for $l$-values greater than about 40.

The actual input from inflation to this $\sim$380000-year evolution, in
standard cosmology, is fairly minimal, but its role in providing a source for
the almost scale-invariant temperature variations in the CMB needs an
alternative explanation in CCC. This is provided by the $\Lambda$-driven
exponential expansion of the previous aeon, where inflation theory’s inflaton
quantum fluctuations are replaced by the effects, coming through from the
previous aeon, of gravitation wave impulses that are the products of erebon
decay in that previous aeon, erebons being the particles that constitute dark
matter in CCC. Such dark matter is a necessary consequence of the equations of
CCC (P2 ; GP1 ; P3 ). With a decay half-life of the order of $10^{11}$ years,
both the near scale invariance of the CMB temperature fluctuations and the
spectral index can find a CCC explanation. Details will be published
elsewhere, but for a preliminary discussion, see P3 .

It should also be pointed out that the arguments in support of inflation that
it supplies an explanation for the temperature fluctuations that would appear
to be acausal in a cosmology without inflation are easily explained within CCC
. The existence of an aeon prior to ours gives ample scope for such
correlations. For example, the gravitational wave signals considered in the
following section would certainly provide correlations of this apparently
acausal kind.

2\. Observable effects of previous-aeon black-hole encounters

There are two quite distinct types of event concerning supermassive black
holes in the previous aeon, that should be observable, according to CCC, and
it is important not to confuse them. One of these would be the effects of
gravitational waves coming from inspiralling pairs of supermassive black holes
(discussed in this section) and the other the effects of the evaporation of
supermassive black holes (discussed in the next section). The observable
signals for the first type would be narrow-width circular rings, within which
would be the observed effect of the final burst of this radiation, containing
almost the entire energy emitted. These rings would often occur in concentric
sets since several such events could often take place within the history of a
single galactic cluster, whose extended world-line would reach the crossover
surface at the central point of the rings. Such rings wold be distinguished by
having anomalously low temperature variance around them (GP1 , appendix B), or
of significantly differing average temperature around the ring from that of a
neighbouring ring concentric with it. Such rings could be of large angular
diameter across the sky, but not of greater than around $40^{\circ}$ Tod ;
Nelson . Evidence for the existence of such rings in the WMAP data was found
by Gurzadyan and Penrose GP1  and by Meissner, Nurowski and Ruszczycki MNR  in
the WMAP data, the latter providing a 99.7% statistical confidence level for
the reality of the effect. Subsequently, An, Meissner and Nurowski AMN
examined the Planck satellite data, again finding a significant effect, at a
99.4% confidence level.

The WMAP analysis provided in GP1  was repeated by DeAbreu, Contreras, and
Scott DCS , and although a virtually identical picture of the circular
features was obtained, these authors argue that the presence of such features
is not statistically significant. They did not, however, repeat the further
analysis actually given by Gurzadyan and Penrose, which showed that when the
analysis was modified so as to search for somewhat elliptical rather than
circular rigs, the numbers drop dramatically, and continued to drop
consistently, with increasing ellipticity, showing that concentric circular
rings predicted by CCC have a preferential significance. The procedure adopted
here is to us the same concentric-circle search as before, but applied to a
CMB sky that is twisted about the galactic polar axis, this being equivalent
to a search for concentric elliptical shapes in the real sky. This procedure
is misunderstood by DeAbreu, et al, who argue that the temperature variance
profile of the twisted sky is different from that of the real sky, a point
which is actually irrelevant to the analysis of GP1 , which depends only on
that for the real sky. We should also point out that other earlier claims
(such as WE ; HA ) that they do not find statistically significant evidence
for the signals later argued for more forcefully by Gurzadyan and Penrose GP1
are not demonstrations that such signals do not actually exist, but indicate
that a more sophisticated analysis is needed, such as that of GP1 ; MNR ; AMN
.

More importantly, DeAbreu, et al. provide no explanation for the striking
departure from isotropy in the observed features, where they comment only on
the relatively minor feature that more centres are found in the galactic
southern than in the northern hemisphere. However, the departure from isotropy
is far more striking than this, which is made particularly obvious in the
search for centres of triples of low-variance rings in the Planck satellite
data as displayed in Fig. 2 of GP2 , which exhibits the centres of concentric
triples (at least) of low-variance circular rings. We notice that the vast
majority (in fact around 83%) of the 1134 centres found lie within three
roughly elliptical regions (two in the southern and one in the northern
galactic hemisphere), each of not more than about 3% of the entire region of
the sky being examined.

It should be pointed out that such anisotropy/inhomogeneity is not a
prediction of CCC, but is perfectly consistent with that theory. The trouble
lies more with the conventional inflationary picture, according to which, such
gross departures from uniformity are hard to explain. In the CCC picture,
there is no inflation, and therefore no strong requirement that the large-
scale spatial universe be particularly homogeneous or isotropic, and the
strong clumping depicted in Fig. 2 of GP2  could be explained by the presence,
in the previous aeon, of enormously large superclusters of galaxies in the
directions of these three concentrated regions. Such an interpretation is
enhanced by the fact that there is also a strong “clumping” in the colour
coding in Fig. 2 of GP2 , a feature whose strong significance for CCC, and
also its relevance to Hawking points, will be discussed elsewhere.

3\. Hawking points in CCC

We now come to the essential purpose of this paper: the observational
implications of the ultimate fate of supermassive black holes in the previous
aeon, and the observational implications for our current aeon. The picture
that we should bear in mind is that whereas superclusters gradually disperse,
owing to the exponential expansion of the later stages of each aeon,
individual galactic clusters remain bound. The supermassive black holes within
the galaxies of each cluster, will begin to feel each other out and then
spiral into one another to form a single hugely massive black hole. As time
evolves, that remaining supermassive black hole would be expected to swallow
perhaps the majority of the matter in that galactic cluster, though some fair
portion might escape into outer space, to join intergalactic hydrogen and dark
matter already there. Eventually, the mass content of the aeon would consist,
to large extent, of huge supermassive black holes of perhaps up to $10^{14}$
solar masses, which would last for periods of up to perhaps $10^{106}$ years
Page  after which it should have Hawking-evaporated away completely.

We must bear in mind that, despite Hawking radiation being of an absurdly tiny
temperature, over the aeon’s history the entire mass-energy of the hole will
be finally radiated away and, in the conformal picture, this will take place
within what would effectively be a single Hawking point H, only
infinitesimally beneath X. Being mainly in the form of photons (and some
neutrinos and other mass-faded particles) this radiation comes directly
through X to heat the initial material in the succeeding aeon enormously, just
to the future of the Hawking point H, depositing the hole’s entire mass-energy
there. This hugely heated region would then gradually spread out in our aeon
until reaching the decoupling 3-surface D, providing something like a Gaussian
distribution on D, centred at a point G on D, just to the future of H in the
conformal picture, where the spread, duly constrained by the speed of light,
should, from our vantage point, be no greater than the maximum causally
allowed diameter of around 4∘ (i.e. a radius $\leq 0.035$ radians). What we
see from our current vantage point would be the intersection of our past light
cone C with this small distribution on D, which appears to us as a small
Gaussian-like distribution centred at the Hawking point H, having spread out
to subtend an angular distance of no greater than $0.035$ radians on either
side of G (i.e. within H’s light cone), see Fig.1.

The actual temperature profile depends on the detailed particle physics
involved, in accordance with conventional theory, and, for the enormously
highly energetic particles involved, we would expect that it should appear to
us like a Gaussian with maximum temperature at at G, and cooling off as it
spreads out to something a bit less than the causally allowed maximum of 2∘
radius – less, because C would normally not quite pass through G (see Fig 1.)
We call such a (necessarily circular) region of raised temperature a Hawking
disc, the presence of such features in the actual CMB sky being a critical
issue for CCC. In Section 4, we describe the details of our search for Hawking
discs in the actual CMB where, according to the above discussion, we take note
of the fact that the temperature of a Hawking disc ought to have a maximum at
its centre, and which falls off in a Gaussian-like way towards its outer edge.
In order to identify such discs we look for annular regions in the CMB sky,
taken to be concentric with such a proposed Hawking disc, and see if we find a
significant drop in temperature from the inner to outer boundary of the
annulus. The test is to see whether these features are found in the actual CMB
sky with parameter values as are predicted by the theory described above.

Accordingly, we should indeed expect to find such annuli, but where the outer
edge has a radius no larger than about 0.035 radians, and where we anticipate
a sharp cut-off in numbers for such discs having a greater outer radius than
this, and not so many of them with outer radii of less than around 0.02
radians. These are expectations that we infer from Fig.1. As we shall see in
Section 4, our findings appear to be strongly in accordance with these
expectations.

4\. Anomalous small regions in the CMB map

In order not to prejudice our search in favour of the particular signals
anticipated by CCC as described above, we broadened our search, so as to be
for annuli with inner radii $r_{1}$ of between zero and 0.04 radians and outer
radii $r_{2}$ from 0.01 up to 0.08 radians. The annular width
$\varepsilon=r_{2}-r_{1}$ is taken to run from 0.01 to 0.04 radians, all taken
in steps of 0.01 radians. Moreover we allow for the temperature slope from
inner to outer radius to be positive, rather than the CCC-expected negative
slopes.

Remarkably, we find, at a 99.98% confidence level (of agreement between this
particular aspect of CCC theory and the Planck satellite data) the existence
of this previously unnoticed family of anomalously energetic small circular
regions in the CMB sky. A comparison with 1000 conventional Planck simulations
provided a powerful case for the possibility of there being actual signals of
the kind that would be consistent with the theoretical considerations that we
have described earlier in this paper, the signal appearing to be outstandingly
strong. Accordingly, we compared our CMB findings with a further independent
9000 simulations. Again we found the same strong signal as before, at the same
parameter values, and our confidence level of better than 99.98%, referred to
above, is based solely on these subsequent 9000 simulations.

To provide additional confirmation of this result we also used 1000 FFP8.1 MC
simulations (with $l\leqslant 1500$) and obtained the same result. As an
independent test, we compared the WMAP ‘deconvolved’ real map with 1000
simulations and again arrived at the same result, noting that the five
strongest anomalous regions of the Planck map were found to occupy precisely
the same locations as five of the strongest locations found in the WMAP data.
Planck and WMAP have very different noise properties and it seems unlikely
that the observed presence of anomalous points in the same directions in both
maps could come entirely from the noise.

The details and the particular motivations underlying our current search are
given below. In this section we simply provide a brief description of the
particular anomalous regions of the CMB sky that we appear to be seeing, and
raise some of the significant implications of this. What our search reveals is
a multitude of distinctive circular spots in the CMB, of increasing
temperature towards their centres, having an intensity somewhat more than an
order of magnitude greater than the standard $10^{-5}$ temperature
fluctuations.

A striking and noteworthy feature of these anomalous spots is that within the
range of diameters that we examine, there is a sharp cut-off at an angular
diameter of around $0.08$ radians. To understand the puzzle for conventional
inflationary theory that is raised by this finding, it is helpful to examine
the conformal diagram of Fig. 1. This depicts the 380000-year period between a
powerful source of energy at the space-time point H and the decoupling surface
D. We note that if the source had not been in the extremely early universe, as
depicted in the figure, but within the following 380000 years before reaching
D, then the spreading out of the signal at D could indeed be constrained to
within around $0.08$ radians in the observed CMB. But the physics within that
380000 year period is well understood, being superbly confirmed by the close
agreement with the CMB power spectrum at $l$ values larger than about 40, so
it is hard to see how the signals we see could originate in this way. On the
other hand, any hugely energetic disturbance that took place much earlier than
the turn-off of inflation (the so-called graceful exit moment) would have
spread to a far larger diameter when reaching D. In conventional inflationary
theory, unless H were indeed constrained to be very close to the turn-off
surface of inflation represented by the horizontal line X, the spread of this
energy to the future of H would be expected to be much larger than the
observed angular diameter of around $0.08$ radians. Accordingly, H would have
to be very close to the turn-off of inflation, as depicted in the figure,
which would seem to be problematic for inflationary theory. On the other hand,
as has been argued above, the expectations of CCC are well in accordance with
these observations and, indeed, such features are predictions of that theory.

5\. Details of the analysis

In the present paper the procedure we use to analyse Planck and WMAP data is
similar to the one used by us before in MNR  and AMN  (Galactic equatorial
belt excluded, imposed masks etc.) with one crucial difference. To explain
this difference we recall that in the previous searches (i.e. looking for the
ring-type structures) the assumed profile consisted of two contiguous
concentric annuli, the inner with negative weight and the outer with positive
weight. The convolution of the profile (with different angular radii and width
of the annuli) with the actual temperature $T$ was calculated for rings in
different directions in the sky. Such calculations were performed both for the
real maps as measured by WMAP and Planck (70 GHz, SMICA, SEVEM…) and,
initially, for 1000 artificial maps generated with the observed CMB power
spectrum. Then the CDF’s of the results were compared using the procedure
described in KAM . The results showed that in some cases none of the
artificial maps out of this 1000 performed better than the real map.

The difference between our previous approach and the present one is that now
we are looking for the slopes of T around a given direction for an annulus of
inner angular radius $r_{1}$ and width $\varepsilon$. The slope in a given
direction is calculated by the formula (minimizing $\sum(\delta
T_{i}-a\,x_{i}-b)^{2}$ with respect to $a$ and $b$)

$a=\frac{n\sum(x_{i}\,\delta T_{i})-(\sum x_{i})(\sum\delta T_{i})}{n\sum
x_{i}^{2}-(\sum x_{i})^{2}}$ (1)

where $x_{i}$ is the angular distance of point $i$ from the given direction,
$\delta T_{i}$ is the temperature at this point and $n$ is the number or
points in the annulus. The sums run over all points in an annulus around the
given direction. Having the slopes for all directions on the sky we create the
CDF for a given map. Then we create a “theoretical” CDF by averaging the CDFs
of all these 10000 artificial maps and use the formula from KAM  to calculate
2 numbers describing extremal, positive and negative, parts of CDF. Then we
check how many artificial maps “outperform” the real map, separately for
positive and negative slopes.

Table I, below, gives the results for smallest annuli. Values $r_{1}$ for the
inner radii are given in column 1, and for widths $\varepsilon$, in columns 2
and 7. The 3rd column shows the number $N_{-}^{1k}$ of artificial maps (out of
this 1000) outperforming the real map with large positive slopes and the 4th
and 9th columns, the number of artificial maps (out of this 1000)
outperforming the real map with large negative slopes. The zeroes in columns 4
and 9 show that for annuli of widths $\varepsilon=0.02$ or $\varepsilon=0.03$
(with inner radius $r_{1}=0.01$) there are no artificial maps outperforming
the real map with large negative slopes i.e. with the temperature decreasing
outwards. The appearance of these zeroes led us to perform an analysis with a
much larger set of maps (9 000 more, in addition to the original 1000) either
to confirm or reject the hypothesis of the physical validity of the strong
signal seen in the true data, for $r_{1}=0.01$ and widths either
$\varepsilon=0.02$ or $\varepsilon=0.03$ both absent in the first set of 1000
artificial maps. We see in the 11th and 6th ($N_{-}^{10k}$) columns that the
hypothesis is confirmed and we can estimate the probability of the purely
random appearance of annuli ($0.01,\ 0.02$) as approximately $0.01$% and
($0.01,\ 0.03$) as approximately $0.02$%, providing us with a confidence level
of better than $99.98$% that there is a genuine signal in the true CMB at
$r_{1}=0.01$ and widths at either $\varepsilon=0.02$ or $\varepsilon=0.03$.

Table 1: # of artificial maps outperforming the real Planck 70 GHz map

$r_{1}$ | $\varepsilon$ | $N_{+}^{1k}$ | $N_{-}^{1k}$ | $N_{+}^{10k}$ | $N_{-}^{10k}$
---|---|---|---|---|---
0.0 | 0.01 | 921 | 242 | 9358 | 2406
0.01 | 0.01 | 952 | 139 | 9592 | 1422
0.02 | 0.01 | 215 | 831 | 2199 | 8330
0.03 | 0.01 | 625 | 905 | 6398 | 9062
0.04 | 0.01 | 182 | 910 | 1926 | 9310

$\varepsilon$ | $N_{+}^{1k}$ | $N_{-}^{1k}$ | $N_{+}^{10k}$ | $N_{-}^{10k}$
---|---|---|---|---
0.02 | 710 | 186 | 7110 | 1700
0.02 | 734 | 0 | 7232 | 1
0.02 | 384 | 110 | 4056 | 1119
0.02 | 258 | 978 | 2627 | 9810
0.02 | 991 | 921 | 9929 | 9169

$r_{1}$ | $\varepsilon$ | $N_{+}^{1k}$ | $N_{-}^{1k}$ | $N_{+}^{10k}$ | $N_{-}^{10k}$
---|---|---|---|---|---
0.0 | 0.03 | 681 | 63 | 6649 | 595
0.01 | 0.03 | 875 | 0 | 8904 | 2
0.02 | 0.03 | 756 | 601 | 7567 | 5855
0.03 | 0.03 | 180 | 666 | 1979 | 6602
0.04 | 0.03 | 162 | 412 | 1664 | 4271

$\varepsilon$ | $N_{+}^{1k}$ | $N_{-}^{1k}$ | $N_{+}^{10k}$ | $N_{-}^{10k}$
---|---|---|---|---
0.04 | 608 | 6 | 5939 | 43
0.04 | 779 | 968 | 7737 | 9722
0.04 | 289 | 513 | 3131 | 5115
0.04 | 749 | 597 | 7531 | 6093
0.04 | 42 | 378 | 315 | 3745

In table II, the galactic coordinates (in radians, latitude from the North
Galactic Pole) of the annuli with the most significant negative slopes are
given for inner-radius 0.01 and widths 0.02 and 0.03. (Sometimes the same
point appears in both tables, as with $(\theta,\phi)=(2.219,0.012)$). The
slopes are strikingly large apparently pointing to some novel phenomenon.

Table 2: galactic coordinates of 10 annuli (Planck)
with most negative slopes ($T_{\rm CMB}$/rad)
left: $r_{1}=0.01,\ \varepsilon=0.02$, right: $r_{1}=0.01,\ \varepsilon=0.03$

$T_{\rm CMB}$/rad | $\theta$ | $\phi$
---|---|---
-0.01403 | 2.219 | 0.012
-0.01266 | 0.204 | 2.405
-0.01215 | 0.703 | 2.777
-0.01166 | 2.988 | 4.908
-0.01164 | 2.545 | 0.051
-0.01154 | 2.949 | 0.052
-0.01151 | 2.678 | 5.388
-0.01129 | 0.716 | 3.698
-0.01113 | 0.689 | 0.844
-0.01103 | 0.799 | 0.785

$T_{\rm CMB}$/rad | $\theta$ | $\phi$
---|---|---
-0.01033 | 0.204 | 2.405
-0.00891 | 2.962 | 0.056
-0.00804 | 2.219 | 0.012
-0.00735 | 0.140 | 4.783
-0.00721 | 2.383 | 0.744
-0.00701 | 2.795 | 2.705
-0.00677 | 0.799 | 0.785
-0.00662 | 2.949 | 0.052
-0.00660 | 2.756 | 5.209
-0.00658 | 2.545 | 0.051

The difference between the temperatures of the outer and inner boundaries for
the most significant annulus $(0.01,0.02)$ is $-2.8\cdot 10^{-4}$ K and for
$(0.01,0.03)$ it is $-3.1\cdot 10^{-4}$ K i.e. more than an order of magnitude
more than the average CMB fluctuation.

To confirm the result we also used 1000 maps from FFP8.1 MC simulations (with
$l\leqslant 1500$) produced and made available by the PLANCK team PMC . These
maps put many factors into consideration, such as beam, noise, scanning
strategy and gravitational lensing. The simulations have given the results
very similar to those given in Table I and are given below in Table III. In
particular for $(0.01,0.02)$ and $(0.01,0.03)$ there were no (or 1) simulated
maps that outperformed the real map. In addition, we can see that for
$(0.00,0.04)$ the number is also significantly low. Therefore, we can draw the
conclusion that our previous result was not influenced by gravitational
lensing or noise.

Table 3: # of artificial FFP8.1 MC maps outperforming the real Planck 70 GHz map $r_{1}$ | $\varepsilon$ | $N_{+}^{1k}$ | $N_{-}^{1k}$ | $\varepsilon$ | $N_{+}^{1k}$ | $N_{-}^{1k}$ | $\varepsilon$ | $N_{+}^{1k}$ | $N_{-}^{1k}$
---|---|---|---|---|---|---|---|---|---
0.00 | 0.02 | 733 | 187 | 0.03 | 665 | 55 | 0.04 | 581 | 4
0.01 | 0.02 | 705 | 1 | 0.03 | 878 | 0 | 0.04 | 780 | 974
0.02 | 0.02 | 387 | 117 | 0.03 | 772 | 594 | 0.04 | 315 | 540
0.03 | 0.02 | 260 | 985 | 0.03 | 163 | 672 | 0.04 | 736 | 635
0.04 | 0.02 | 983 | 911 | 0.03 | 143 | 408 | 0.04 | 31 | 373

We also compared our result with the WMAP ’deconvolved’ 1000 maps with the
theoretical Planck power spectrum. The results are given below in Table IV.
Again for $(0.01,0.02)$ and $(0.01,0.03)$ there were no simulated maps that
outperformed the real map. In table V we give the directions of the most
intense annuli from the WMAP data. Note that most intense 5 annuli listed in
Table II (from the Planck data) are also in Table V (from the WMAP data).

Table 4: # of artificial maps outperforming the real WMAP 70 GHz map $r_{1}$ | $\varepsilon$ | $N_{+}^{1k}$ | $N_{-}^{1k}$ | $\varepsilon$ | $N_{+}^{1k}$ | $N_{-}^{1k}$ | $\varepsilon$ | $N_{+}^{1k}$ | $N_{-}^{1k}$
---|---|---|---|---|---|---|---|---|---
0.00 | 0.02 | 636 | 187 | 0.03 | 690 | 47 | 0.04 | 485 | 4
0.01 | 0.02 | 776 | 0 | 0.03 | 942 | 0 | 0.04 | 731 | 973
0.02 | 0.02 | 535 | 102 | 0.03 | 726 | 461 | 0.04 | 442 | 552
0.03 | 0.02 | 186 | 941 | 0.03 | 341 | 685 | 0.04 | 642 | 733
0.04 | 0.02 | 995 | 728 | 0.03 | 240 | 500 | 0.04 | 14 | 326
Table 5: galactic coordinates of annuli (WMAP)
with most negative slopes ($T_{\rm CMB}$/rad), $r_{1}=0.01$

$\varepsilon$ | $\theta$ | $\phi$
---|---|---
0.02 | 0.204 | 2.405
0.02 | 0.582 | 5.323
0.02 | 2.219 | 0.012
0.02 | 0.703 | 2.778
0.02 | 2.118 | 1.853
0.02 | 2.988 | 4.909

$\varepsilon$ | $\theta$ | $\phi$
---|---|---
0.02 | 0.595 | 3.159
0.02 | 2.962 | 0.056
0.03 | 0.204 | 2.405
0.03 | 2.962 | 0.056
0.03 | 2.118 | 1.853
0.03 | 2.219 | 0.012

Outlook

It seems to us that our analysis of slopes of the temperature in the CMB maps
gives us a significant initial indication of the nature of the anomalous
regions and provides an important new input into cosmology, irrespective of
the validity of CCC. It is hard to see, however, that they find a natural
explanation in the currently conventional inflationary picture.

Acknowledgments: We thank Paweł Bielewicz, Arthur Kosowsky, Don Page and
Robert Wald for discussions. R.P. thanks J.P. Moussouris for financial
assistance through a personal endowment. K.A.M. was partially supported by the
Polish National Science Centre grant DEC-2017/25/B/ST2/00165. We acknowledge
the help from the Świerk Computing Centre in the National Centre for Nuclear
Research (NCBJ, Otwock, Poland).

## References

  * (1) R. Penrose, Before the big bang: an outrageous new perspective and its implications for particle physics. In EPAC 2006 Proc., Edinburgh, Scotland, pp.2759-62 ed. C.R.Prior., (2006)
  * (2) R. Penrose, Cycles of Time: An Extraordinary New View of the Universe, Bodley Head, London, 2010
  * (3) R. Penrose, Found. Phys. 6 (2018) 1
  * (4) M. Gasperini, and G. Veneziano, Astropart. Phys. 1 (1993) 317
  * (5) R. Penrose, Conformal approach to infinity, in Relativity, Groups and Topology: The 1963 Les Houches Lectures, eds. B.S. DeWitt and C.M. DeWitt (Gordon and Breach, New York) (1964)
  * (6) R. Penrose, Proc. Roy. Soc. London Ser.A 284 (1965) 159
  * (7) S.W. Hawking, Nature, 248 (1974) 30.
  * (8) S.W. Hawking, Comm. Math. Phys. 43 (1975) 199
  * (9) D.N. Page, Phys. Rev. D13 (1976) 198
  * (10) V.G. Gurzadyan and R. Penrose, Eur. Phys. J. Plus 128 (2013) 22.
  * (11) K.P. Tod, Gen. Relativ. Gravit. 44 (2012) 2933
  * (12) W. Nelson and E. Wilson-Ewing, Phys.Rev. D84 (2011) 043508
  * (13) K.A. Meissner, P. Nurowski and B. Ruszczycki, Proc. R. Soc. A469 (2013) 2155
  * (14) D. An, K.A. Meissner and P. Nurowski, Mon.Not.Roy.Astron.Soc. 473 (2018) no.3, 3251
  * (15) A. DeAbreu, C. Contreras and D. Scott, JCAP 12 (2015) 031
  * (16) I.K. Wehus and H.K. Eriksen, ApJ 733 (2010) L29
  * (17) A. Hajian, ApJ 740 (2011) 52
  * (18) V.G. Gurzadyan and R. Penrose, Eur.Phys.J.Plus 131 (2016) 11
  * (19) K.A. Meissner, A Tail Sensitive Test for Cumulative Distribution Functions, arXiv: 1206.4000 [math.ST].
  * (20) Planck Collaboration, Astron. Astrophys. 594 (2016) A12, arXiv:1509.06348


# RIS-Assisted Secure Transmission Exploiting Statistical CSI of Eavesdropper

Cen Liu, Chang Tian, and Peixi Liu  State Key Laboratory of Advanced Optical
Communication Systems and Networks,
Department of Electronics, Peking University, Beijing, China
Email: {cen, tianch<EMAIL_ADDRESS>
###### Abstract

We investigate the reconfigurable intelligent surface (RIS) assisted downlink
secure transmission where only the statistical channel of eavesdropper is
available. To handle the stochastic ergodic secrecy rate (ESR) maximization
problem, a deterministic lower bound of ESR (LESR) is derived. We aim to
maximize the LESR by jointly designing the transmit beamforming at the access
point (AP) and reflect beamforming by the phase shifts at the RIS. To solve
the non-convex LESR maximization problem, we develop a novel penalty dual
convex approximation (PDCA) algorithm based on the penalty dual decomposition
(PDD) optimization framework, where the exacting constraints are penalized and
dualized into the objective function as augmented Lagrangian components. The
proposed PDCA algorithm performs double-loop iterations, i.e., the inner loop
resorts to the block successive convex approximation (BSCA) to update the
optimization variables; while the outer loop adjusts the Lagrange multipliers
and penalty parameter of the augmented Lagrangian cost function. The
convergence to a Karush-Kuhn-Tucker (KKT) solution is theoretically guaranteed
with low computational complexity. Simulation results show that the proposed
PDCA scheme is better than the commonly adopted alternating optimization (AO)
scheme with the knowledge of statistical channel of eavesdropper.

###### Index Terms:

reconfigurable intelligent surface, physical layer security, statistical CSI,
penalty dual decomposition, block successive convex approximation.

##  I Introduction

Reconfigurable intelligent surface (RIS), which has given rise to the brand-
new concept of “smart radio environment”, is regarded as a promising
technology in the future 5G beyond and 6G cellular network[1, 2, 3, 4, 5, 6].
RIS is comprised of many uniformly distributed low-cost reflecting elements
that can adjust both amplitude and phase of the impinging signal, so as to
form a directional beam in a passive manner. It is compatible with existing
wireless communication systems and can bring many distinctive advantages, such
as low power consumption, noise-free and self-interference-free under full-
duplex mode. Therefore, RIS-assisted wireless transmission systems are getting
growing interest from both academia and industry.

Physical layer security, as a key technology aiming at providing confidential
message transmission and solving the privacy protection problem in the
physical layer, has received considerable attention in the past decade[7, 8].
For the enhancement of security performance, a basic approach is to improve
the spectral efficiency of the legitimate users or deteriorate that of the
eavesdroppers (Eve). In an RIS-assisted wireless system, the received signal
can be suppressed at the Eve while being boosted at the user. Thus, the
deployment of RIS can bring a new degree of freedom (DoF) in the space domain,
and the security performance can be further improved.

Several works related to the RIS-assisted physical layer security have emerged
recently [9, 10, 11, 12, 13, 14]. In [9, 10] and [11], an RIS-assisted single-
user single-Eve multi-input single-output (MISO) communication system was
investigated with or without taking the direct links of access point (AP)-user
and AP-Eve into account, respectively. Artificial noise was jointly designed
with transmit beamforming and RIS phase shifts both in the multi-user single-
Eve [12] and single-user multi-Eve MISO systems [13]. Furthermore, the
security performance of an RIS-assisted multi-input multi-output (MIMO) system
was analyzed in [14]. Existing literature usually assume the global
instantaneous channel state information (CSI) is perfectly known, which is
impractical in many cases. At first, the Eve-related instantaneous CSI is hard
to be fully available since it is impractical to implement the Eve-related
channel estimation in a real-time manner due to the uncertainty of Eve’s
position. In addition, the CSI between RIS and mobile user can not be
estimated directly by traditional channel estimation techniques due to the
RIS’s passive nature.

This paper is dedicated to the investigation of RIS-assisted secure
transmission where only the statistical CSI of Eve is available. The main
contributions are summarized as follows:

  1. 1.

The scenario we consider is more practical than the works mentioned above
since the statistical CSI is relatively easy to obtain through long-term
observation;

  2. 2.

A lower bound of the ergodic secrecy rate is derived to cope with the
formulated stochastic optimization problem by resorting to its more tractable
deterministic counterpart;

  3. 3.

The proposed penalty dual convex approximation (PDCA) algorithm can provably
solve the induced non-convex rate maximization problem to a Karush-Kuhn-Tucker
(KKT) point. Meanwhile, it possesses lower computational complexity and better
performance than the commonly adopted alternating optimization (AO) algorithm.

_Notations:_ $\left(\cdot\right)^{\mathsf{H}}$,
$\left(\cdot\right)^{\mathsf{T}}$, and $\left(\cdot\right)^{*}$ stand for the
Hermitian, the transpose and the conjugate operations, respectively. $|\cdot|$
and $\|\cdot\|$ represent the modulus of a complex number and the
$\mathcal{\ell}_{2}$ norm of a complex vector. ${\rm diag}\left(\cdot\right)$
denotes the diagonal matrix whose diagonals are the elements of the input
vector.

Figure 1: RIS-assisted single-user MISO wireless system in the presence of a
single-antenna eavesdropper.

##  II System Model

We consider an RIS-assisted single-user MISO system in the presence of a
single-antenna eavesdropper, as illustrated in Fig. 1. The AP and RIS are
equipped with $M$ antennas and $N$ reflecting elements, respectively. The AP
transmits confidential message to the legitimate single-antenna user through
the direct link $\mathbf{h}_{\rm AU}\in\mathbb{C}^{M}$ and the reflecting link
with the help of RIS. The reflecting link can be viewed as a cascaded channel
of AP-RIS and RIS-user channel which are denoted by $\mathbf{H}_{\rm
AI}\in\mathbb{C}^{N\times M}$ and $\mathbf{h}_{\rm IU}\in\mathbb{C}^{N}$,
respectively. The received signal at the user is

$\displaystyle y_{\rm U}$ $\displaystyle=\left(\mathbf{h}_{\rm
IU}^{\mathsf{H}}\mathbf{\Phi}\mathbf{H}_{\rm AI}+\mathbf{h}_{\rm
AU}^{\mathsf{H}}\right)\mathbf{w}s+n_{\rm U}$ (1)

where $\mathbf{\Phi}={\rm diag}\left(\bm{\phi}\right)$ ,
$\bm{\phi}=\left[e^{j\theta_{1}},\ldots,e^{j\theta_{N}}\right]^{\mathsf{T}}$
denotes the phase shifts induced by each reflecting element of RIS.
$\mathbf{w}\in\mathbb{C}^{M}$ denotes the transmit beamforming vector at the
AP. $s$ and $n_{\rm U}$ denote the transmitted information and additive white
Gaussian noise (AWGN) at the user, respectively, with
$\mathbb{E}\left[|s|^{2}\right]=1$ and $n_{\rm
U}\sim\mathcal{CN}\left(0,\sigma_{U}^{2}\right)$. It can be easily verified
that $\mathbf{h}_{\rm IU}^{\mathsf{H}}\mathbf{\Phi}\mathbf{H}_{\rm
AI}=\bm{\phi}^{\mathsf{T}}\mathbf{H}_{\rm U}$ where $\mathbf{H}_{\rm U}={\rm
diag}\left(\mathbf{h}_{\rm IU}^{*}\right)\mathbf{H}_{\rm AI}$ stands for the
cascaded channel from the AP to RIS to the user. Due to the passive nature of
RIS, the RIS-user channel $\mathbf{h}_{\rm IU}$ can hardly be obtained.
Nevertheless, the cascaded channel $\mathbf{H}_{\rm U}$ can be estimated more
easily[15, 16]. We assume the user-related channels $\mathbf{h}_{\rm AU}$,
$\mathbf{H}_{\rm U}$ and $\mathbf{H}_{\rm AI}$ are perfectly known at the AP
and experience quasi-static flat fading.

Similar to the user, the signal received by Eve is

$\displaystyle y_{\rm E}$ $\displaystyle=\left(\mathbf{h}_{\rm
IE}^{\mathsf{H}}\mathbf{\Phi}\mathbf{H}_{\rm AI}+\mathbf{h}_{\rm
AE}^{\mathsf{H}}\right)\mathbf{w}s+n_{\rm E}$ (2)

where $\mathbf{h}_{\rm IE}$ is the RIS-Eve channel and $\mathbf{h}_{\rm AE}$
is the AP-Eve channel. $n_{\rm E}\sim\mathcal{CN}\left(0,\sigma_{\rm
E}^{2}\right)$ is AWGN at the Eve. We can also replace $\mathbf{h}_{\rm
IE}^{\mathsf{H}}\mathbf{\Phi}\mathbf{H}_{\rm AI}$ in (2) by
$\bm{\phi}^{\mathsf{T}}\mathbf{H}_{\rm E}$, where $\mathbf{H}_{\rm E}={\rm
diag}\left(\mathbf{h}_{\rm IE}^{*}\right)\mathbf{H}_{\rm AI}$ stands for the
cascaded channel from the AP to RIS to the Eve. We denote the AP-Eve channel
$\mathbf{h}_{\rm AE}$ and the RIS-Eve channel $\mathbf{h}_{\rm IE}$ as follows
as in[17]

$\displaystyle\mathbf{h}_{i{\rm E}}$
$\displaystyle=\sqrt{\zeta_{0}\left(\frac{d_{i{\rm
E}}}{d_{0}}\right)^{-\alpha_{i{\rm E}}}}\left(\sqrt{\frac{K_{i{\rm
E}}}{K_{i{\rm E}}+1}}\bar{\mathbf{h}}_{i{\rm E}}+\sqrt{\frac{1}{K_{i{\rm
E}}+1}}\tilde{\mathbf{h}}_{i{\rm E}}\right),$ $\displaystyle i$
$\displaystyle\in\\{\rm A,\rm I\\}.$ (3)

where $\zeta_{0}$ denotes the path loss at reference distance $d_{0}$,
$d_{i{\rm E}}$ is the distance between AP/RIS and Eve. $\alpha_{i{\rm E}}$ and
$K_{i{\rm E}}$ denote path loss exponent and Rician $K$-factor of
corresponding channels. $\bar{\mathbf{h}}_{i{\rm E}}$ represents the line-of-
sight (LoS) component which is determined by the angle of departure (AoD) at
the AP/RIS and angle of arrival (AoA) at the Eve, while the non-line-of-sight
(NLoS) component is denoted by $\tilde{\mathbf{h}}_{i{\rm
E}}\sim\mathcal{CN}(0,\mathbf{I})$. The statistical CSI of Eve that can be
taken advantage of are the means and covariance matrices of the AP-Eve channel
$\mathbf{h}_{\rm AE}$ and the RIS-Eve channel $\mathbf{h}_{\rm IE}$ since they
are relatively easy to obtain[18].

##  III Problem Formulation

###  III-A Ergodic Secrecy Rate Maximization

The secrecy rate from AP to user has the following expression

$\displaystyle R_{\rm S}=\left(R_{\rm U}-R_{\rm E}\right)^{+}$ (4)

where $\left(\cdot\right)^{+}\triangleq\max\\{\cdot,0\\}$ is the non-negative
operator, and

$\displaystyle R_{\rm
U}=\log_{2}\left(1+\frac{\left|\left(\bm{\phi}^{\mathsf{T}}\mathbf{H}_{\rm
U}+\mathbf{h}_{\rm AU}^{\mathsf{H}}\right)\mathbf{w}\right|^{2}}{\sigma_{\rm
U}^{2}}\right)$ (5) $\displaystyle R_{\rm
E}=\log_{2}\left(1+\frac{\left|\left(\bm{\phi}^{\mathsf{T}}\mathbf{H}_{\rm
E}+\mathbf{h}_{\rm AE}^{\mathsf{H}}\right)\mathbf{w}\right|^{2}}{\sigma_{\rm
E}^{2}}\right)$ (6)

are the achievable rates at the user and Eve, respectively. Taking expectation
over Eve’s statistical channels, we get the ergodic secrecy rate (ESR) as
follows

$\displaystyle\bar{R}_{\rm S}=\mathbb{E}_{\mathcal{H}_{\rm
E}}\left[\left(R_{\rm U}-R_{\rm E}\right)^{+}\right]$ (7)

where $\mathcal{H}_{\rm E}\triangleq\\{\mathbf{h}_{\rm AE},\mathbf{h}_{\rm
IE}\\}$. Then the ESR maximization problem can be formulated as

$\displaystyle\max_{\bm{\phi},\mathbf{w}}\ $ $\displaystyle\bar{R}_{\rm S}$
(8a) $\displaystyle{\rm s.t.}\ $ $\displaystyle|\phi_{i}|=1,\ i=1,2,\ldots,N,$
(8b) $\displaystyle\|\mathbf{w}\|^{2}\leq P_{\rm max}.$ (8c)

where (8b) and (8c) are unit-modulus constraint of RIS and maximum transmit
power constraint at the AP, respectively.

###  III-B Problem Reformulation

With the expectation operator in the objective (8a), this stochastic
optimization problem can not be efficiently handled. Therefore, we convert it
into a more tractable form by firstly deriving a deterministic lower bound of
the ergodic secrecy rate (LESR), which is given by the following proposition.

_Proposition 1:_ The ESR of the RIS-assisted MISO system $\bar{R}_{\rm S}$ is
lower bounded by $\bar{R}_{\rm S}^{\rm lb}$ given in (9) which is placed on
the top of this page, where $\mathbf{G}_{\rm A}$, $\mathbf{G}_{\rm I}$ and
$\mathbf{G}_{\rm AI}$ are some parameters given by

$\displaystyle\mathbf{G}_{i}$ $\displaystyle=\zeta_{0}\left(\frac{d_{i{\rm
E}}}{d_{0}}\right)^{-\alpha_{i{\rm E}}}\left(\frac{K_{i{\rm E}}}{K_{i{\rm
E}}+1}\bar{\mathbf{h}}_{i{\rm E}}\bar{\mathbf{h}}_{i{\rm
E}}^{\mathsf{H}}+\frac{1}{K_{i{\rm E}}+1}\mathbf{I}\right),$ $\displaystyle i$
$\displaystyle\in\\{\rm A,\rm I\\}.$ (10) $\displaystyle\mathbf{G}_{\rm AI}=$
$\displaystyle\zeta_{0}\left(\frac{d_{\rm
AE}}{d_{0}}\right)^{-\frac{\alpha_{\rm AE}}{2}}\left(\frac{d_{\rm
IE}}{d_{0}}\right)^{-\frac{\alpha_{\rm IE}}{2}}$
$\displaystyle\times\sqrt{\frac{K_{\rm AE}K_{\rm IE}}{(K_{\rm AE}+1)(K_{\rm
IE}+1)}}\bar{\mathbf{h}}_{\rm AE}\bar{\mathbf{h}}_{\rm IE}^{\mathsf{H}}$ (11)

_Proof:_ See Appendix A.$\hfill\blacksquare$

$\displaystyle\bar{R}_{\rm S}^{\rm
lb}=\left[\log_{2}\left(\frac{\frac{1}{\sigma_{\rm
U}^{2}}\left|\left(\bm{\phi}^{\mathsf{T}}\mathbf{H}_{\rm U}+\mathbf{h}_{\rm
AU}^{\mathsf{H}}\right)\mathbf{w}\right|^{2}+1}{\frac{1}{\sigma_{\rm
E}^{2}}\left(\|\mathbf{G}_{\rm
A}^{\frac{1}{2}}\mathbf{w}\|^{2}+\|\mathbf{G}_{\rm
I}^{\frac{1}{2}}\mathbf{\Phi}\mathbf{H}_{\rm
AI}\mathbf{w}\|^{2}+2\Re\left[\mathbf{w}^{\mathsf{H}}\mathbf{G}_{\rm
AI}\mathbf{\Phi}\mathbf{H}_{\rm
AI}\mathbf{w}\right]\right)+1}\right)\right]^{+}$ (9)

By _Proposition 1_ , the stochastic ESR maximization problem can be addressed
by solving the following deterministic LESR maximization problem

$\displaystyle\max_{\mathbf{w},\bm{\phi}}\ $ $\displaystyle\bar{R}_{\rm
S}^{\rm lb}$ (12) $\displaystyle{\rm s.t.}\ $ (8b), (8c).

Nonetheless, problem (12) is very challenging to solve since it is highly non-
convex caused by the coupling optimization variables in the objective function
and the non-convex constraint (8b). By utilizing the novel penalty dual
decomposition (PDD) framework[19, 20], we propose an efficient PDCA algorithm
which can provably solve problem (12) to KKT points.

##  IV The Proposed Penalty Dual Convex Approximation Algorithm

Intuitively, AO can be applied to solve problem (12) as in [9, 10, 11, 12, 13,
14]. But to ensure the convergence of the block coordinate descent (BCD)
method (AO is a special case of BCD since there are only a pair of
optimization variables), the subproblem of each block variable need to be
constrained in a convex set and also to be solved to its unique global
optimal[21]. As to primal problem (12), the former requirement can not be
satisfied due to the non-convex constraint (8b) while the latter is hard to
meet since the objective of each subproblem is non-convex, so the global
optimality of the solutions can not be guaranteed. Specifically, semidefinite
relaxation (SDR) as adopted in[9, 12] and[13] can only provide an approximate
local optimal solution.

To address above issues, in this section we propose a penalty dual convex
approximation (PDCA) algorithm by employing the penalty dual decomposition
(PDD) optimization framework in which the sequence of solution are guaranteed
to converge to a KKT point[19, 20]. By applying the PDD technique, we first
obtain an alternative formulation to (12) where an augmented Lagrangian term
is incorporated into the objective in order to handle the unit-modulus
constraint. We then develop an block successive convex approximation (BSCA)
algorithm to solve the augmented Lagrangian problem in the inner loop of the
PDD framework. Finally, we summarize the PDCA joint beamforming algorithm and
analyze its computational complexity.

At the beginning, we drop the non-negative operator in the objective of (12)
since zero rate can be achieved at least by adopting $\mathbf{w}=\mathbf{0}$.
Also, the outer logarithmic function is omitted because of its monotonicity.

###  IV-A Augmented Lagrangian Problem

To start with, an augmented Lagrangian (AL) function with penalty parameter
$\varrho$ is denoted by
$\mathcal{L}_{\varrho}(\bm{\phi},\mathbf{w};\bm{\lambda})$ while the dual
variable $\bm{\lambda}$ is introduced to deal with the constraint (8b).
Further, let us define an AL problem ($P_{\varrho,\bm{\lambda}}$) as follows

$\displaystyle(P_{\varrho,\bm{\lambda}})\
\max_{\bm{\phi},\mathbf{w}\in\mathcal{W}}\ \bigg{\\{}$
$\displaystyle\mathcal{L}_{\varrho}(\bm{\phi},\mathbf{w};\bm{\lambda})\triangleq\bar{R}_{\rm
S}^{\rm lb}(\bm{\phi},\mathbf{w})$ $\displaystyle-\frac{1}{2\varrho}$
$\displaystyle\sum_{i=1}^{N}\left[(|\phi_{i}|-1-\varrho\lambda_{i})^{2}-(\varrho\lambda_{i})^{2}\right]\bigg{\\}}$
(13)

where $\mathcal{W}=\left\\{\mathbf{w}\ |\ \|\mathbf{w}\|^{2}\leq P_{\rm
max}\right\\}$. The proposed PDCA algorithm exhibits a double-loop structure,
where the outer loop updates the dual variable and the penalty parameter while
the inner loop seeks to optimize the primal variables by solving problem
($P_{\varrho,\bm{\lambda}}$). The key to the PDD method is the inner
iterations for solving augmented Lagrangian problems.

###  IV-B Proposed BSCA Algorithm for Solving AL Problem

In the following, we develop an BSCA algorithm to solve
($P_{\varrho,\bm{\lambda}}$) in the inner loop of the PDD optimization
framework. With only the convex constraint (8c) left, BCD-type algorithms can
be used to solve problem ($P_{\varrho,\bm{\lambda}}$) to ensure the
accessibility of a KKT solution of the primal problem (12) as suggested
in[19]. Here, an inexact variant of BCD named block successive convex
approximation (BSCA)[21] is adopted to solve above AL problem. Following a
simple cyclic rule, convex approximation of the objective of each block
variable is updated successively and in each iteration a sufficient decrease
of the objective value can be yielded. By fixing the transmit beamformer in
($P_{\varrho,\bm{\lambda}}$), the RIS phase shifts design problem is stated as

$\displaystyle\min_{\bm{\phi}}\ \bigg{\\{}h(\bm{\phi})\triangleq$
$\displaystyle-\frac{\bm{\phi}^{\mathsf{H}}\mathbf{C}\bm{\phi}+2\Re\left[\bm{\phi}^{\mathsf{H}}\mathbf{c}_{1}\right]+c_{2}}{\bm{\phi}^{\mathsf{H}}\mathbf{D}\bm{\phi}+2\Re\left[\bm{\phi}^{\mathsf{H}}\mathbf{d}_{1}\right]+d_{2}}$
$\displaystyle+\frac{1}{2\varrho}\sum_{i=1}^{N}\left(|\phi_{i}|-1-\varrho\lambda_{i}\right)^{2}\bigg{\\}}$
(14)

where

$\displaystyle\mathbf{C}=\frac{1}{\sigma_{\rm U}^{2}}\mathbf{H}_{\rm
U}^{*}\mathbf{w}^{*}\mathbf{w}^{\mathsf{T}}\mathbf{H}_{\rm U}^{\mathsf{T}},$
$\displaystyle\mathbf{D}=\frac{1}{\sigma_{\rm E}^{2}}{\rm
diag}(\mathbf{H}_{\rm AI}^{*}\mathbf{w}^{*})\mathbf{G}_{\rm I}{\rm
diag}(\mathbf{H}_{\rm AI}\mathbf{w}),$
$\displaystyle\mathbf{c}_{1}=\frac{1}{\sigma_{\rm U}^{2}}\mathbf{H}_{\rm
U}^{*}\mathbf{w}^{*}\mathbf{w}^{\mathsf{T}}\mathbf{h}_{\rm AU}^{*},\
\mathbf{d}_{1}=\frac{1}{\sigma_{\rm E}^{2}}{\rm diag}(\mathbf{H}_{\rm
AI}^{*}\mathbf{w}^{*})\mathbf{G}_{\rm AI}^{\mathsf{H}}\mathbf{w},$
$\displaystyle c_{2}=\frac{1}{\sigma_{\rm U}^{2}}\left|\mathbf{h}_{\rm
AU}^{\mathsf{H}}\mathbf{w}\right|^{2}+1,\ d_{2}=\frac{1}{\sigma_{\rm
E}^{2}}\|\mathbf{G}_{\rm A}^{\frac{1}{2}}\mathbf{w}\|^{2}+1.$ (15)

A strongly convex quadratic function $\hat{h}(\bm{\phi},\bm{\phi}^{k})$ is
selected as a surrogate function in the neighborhood of the $k$th iteration
point $\bm{\phi}^{k}$

$\hat{h}(\bm{\phi},\bm{\phi}^{k})\triangleq
h(\bm{\phi}^{k})+\left<h^{\prime}(\bm{\phi}^{k}),\bm{\phi}-\bm{\phi}^{k}\right>+\frac{1}{2\alpha^{k+1}}\|\bm{\phi}-\bm{\phi}^{k}\|^{2}$

who satisfies
$\hat{h}^{\prime}(\bm{\phi}^{k},\bm{\phi}^{k})=\frac{\partial\mathcal{L}_{\varrho}(\bm{\phi}^{k},\mathbf{w};\bm{\lambda})}{\partial\bm{\phi}}$
(Assumption (6.1) in [21]). The next iteration is generated by

$\displaystyle\hat{\bm{\phi}}^{k+1}=\arg\min_{\bm{\phi}}\hat{h}(\bm{\phi},\bm{\phi}^{k})=\bm{\phi}^{k}-\alpha^{k+1}\nabla
h(\bm{\phi}^{k})$ (16)

which is exactly the gradient descent step. Nevertheless,
$\hat{\bm{\phi}}^{k+1}$ may not be a point of sufficient decrease, which
differs from that in block successive upper-bound minimization (BSUM)[22]
since $\hat{h}$ is only a local approximation of $\mathcal{L}_{\varrho}$ in
BSCA. For this reason, backtracking line search approach with the Armijo step
size selection rule[23] is exploited to search for a point of sufficient
decrease, denoted by $\bm{\phi}^{k+1}$.

Fixing the RIS phase shifts in ($P_{\varrho,\bm{\lambda}}$) results in
following transmit beamforming design problem at the AP

$\displaystyle\min_{\mathbf{w}\in\mathcal{W}}\
\bigg{\\{}g(\mathbf{w})\triangleq-\frac{\mathbf{w}^{\mathsf{H}}\mathbf{A}\mathbf{w}+1}{\mathbf{w}^{\mathsf{H}}\mathbf{B}\mathbf{w}+1}\bigg{\\}}$
(17)

where $\mathbf{A}=\frac{1}{\sigma_{\rm
U}^{2}}\mathbf{a}\mathbf{a}^{\mathsf{H}}$ with
$\mathbf{a}=(\bm{\phi}^{\mathsf{T}}\mathbf{H}_{\rm U}+\mathbf{h}_{\rm
AU}^{\mathsf{H}})^{\mathsf{H}}$ and $\mathbf{B}=\frac{1}{\sigma_{\rm
E}^{2}}(\mathbf{G}_{\rm A}+\mathbf{B}_{1}^{\mathsf{H}}\mathbf{G}_{\rm
I}\mathbf{B}_{1}+\mathbf{B}_{2}+\mathbf{B}_{2}^{\mathsf{H}})$ with
$\mathbf{B}_{1}=\mathbf{\Phi}\mathbf{H}_{\rm AI}$,
$\mathbf{B}_{2}=\mathbf{G}_{\rm AI}\mathbf{B}_{1}$. The problem (17) can be
addressed by the same approach as stated above, with the only difference is
that gradient projection, rather than gradient descent as (16), is performed
as follows

$\displaystyle\hat{\mathbf{w}}^{k+1}=\mathbb{P}_{\mathcal{W}}\left[\mathbf{w}^{k}-\alpha^{k+1}\nabla
g(\mathbf{w}^{k})\right]$ (18)

where $\mathbb{P}_{\mathcal{W}}\left[\cdot\right]$ denotes the projection onto
the convex set $\mathcal{W}$.

###  IV-C Proposed PDCA Algorithm and Computational Complexity

The proposed PDCA algorithm performs double-loop iterations: the inner loop
resorts to the BSCA to update the optimization variables; while the outer loop
adjusts the Lagrange multipliers and penalty parameter of the AL function. The
PDCA algorithm is summarized in Algorithm 1. In this algorithm, the notation
‘optimize($P_{\varrho^{r},\bm{\lambda}^{r}},\bm{\phi}^{r},\mathbf{w}^{r}$)’
represents the optimization oracle which using the BSCA, as depicted in
Algorithm 2, to iteratively solve the AL problem
($P_{\varrho^{r},\bm{\lambda}^{r}}$) based on the current iteration point
{$\bm{\phi}^{r},\mathbf{w}^{r}$}. Compared with the SDR-based AO algorithm
proposed in[9] whose worst-case computational complexity is
$\mathcal{O}(I_{\text{AO}}I_{\text{SDR}}N^{4.5})$[24], that of our double-loop
PDCA algorithm is only $\mathcal{O}(I_{\text{PDD}}I_{\text{BSCA}}N^{2})$,
where $I_{\left[\cdot\right]}$ denotes the number of iterations of the
corresponding algorithm.

Algorithm 1 PDCA Algorithm for Problem (12)

1:initialization: Set $r=0$, constraint violation tolerance $\eta>0$, error
tolerance $\epsilon>0$, initialize feasible {$\bm{\phi}^{0}$,
$\mathbf{w}^{0}$}, $\bm{\lambda}^{0}$, $\varrho^{0}>0$ and assign
$\bar{R}_{\rm S}^{{\rm lb}(0)}=\infty$.

2:repeat

3:
$\\{\bm{\phi}^{r+1},\mathbf{w}^{r+1}\\}=optimize(P_{\varrho^{r},\bm{\lambda}^{r}},\bm{\phi}^{r},\mathbf{w}^{r})$

4: if $\||\bm{\phi}^{r+1}|-\mathbf{1}\|_{\infty}\leq\eta$

5:
$\bm{\lambda}^{r+1}=\bm{\lambda}^{r}+\frac{1}{\varrho^{r}}\left(|\bm{\phi}^{r+1}|-\mathbf{1}\right)$

6: $\varrho^{r+1}=\varrho^{r}$

7: else

8: $\bm{\lambda}^{r+1}=\bm{\lambda}^{r}$

9: update $\varrho^{r+1}$ by decreasing $\varrho^{r}$

10: end

11: $r=r+1$

12: Calculate $\bar{R}_{\rm S}^{{\rm lb}(r)}=\bar{R}_{\rm S}^{\rm
lb}\left(\bm{\phi}^{r},\mathbf{w}^{r}\right)$

13:until $\left|\bar{R}_{\rm S}^{{\rm lb}(r)}-\bar{R}_{\rm S}^{{\rm
lb}(r-1)}\right|\leq\epsilon$.

Algorithm 2 BSCA Algorithm for the AL Problem $(P_{\varrho,\bm{\lambda}})$

1:input: $\varrho^{r}$, $\bm{\lambda}^{r}$, $\bm{\phi}^{r}$, $\mathbf{w}^{r}$
given by Algorithm 1.

2:initialization: Set $k=0$, error tolerance $\epsilon^{\prime}>0$,
{$\bm{\phi}^{0},\mathbf{w}^{0}$} = {$\bm{\phi}^{r},\mathbf{w}^{r}$},
$\bm{\lambda}=\bm{\lambda}^{r}$, $\varrho=\varrho^{r}$,
$\alpha_{1}^{\text{ini}},\alpha_{2}^{\text{ini}}>0$,
$\rho_{1},\rho_{2}\in(0,1)$, $c_{1},c_{2}\in(0,1)$.

3:repeat

4: $\alpha_{1}=\alpha_{1}^{\text{ini}}$, $\alpha_{2}=\alpha_{2}^{\text{ini}}$

5: repeat // backtracking line search

6: $\alpha_{1}=\rho_{1}\alpha_{1}$

7: until $h(\hat{\bm{\phi}}^{k+1})\leq
h(\bm{\phi}^{k})-c_{1}\alpha_{1}\|\nabla h(\bm{\phi}^{k})\|^{2}$

8: $\bm{\phi}^{k+1}=\hat{\bm{\phi}}^{k+1}$

9: repeat // backtracking line search

10: $\alpha_{2}=\rho_{2}\alpha_{2}$

11: until $g(\hat{\mathbf{w}}^{k+1})\leq
g(\mathbf{w}^{k})-c_{2}\alpha_{2}\|\nabla g(\mathbf{w}^{k})\|^{2}$

12: $\mathbf{w}^{k+1}=\hat{\mathbf{w}}^{k+1}$

13: $k=k+1$

14: Calculate
$\mathcal{L}_{\varrho}^{k}=\mathcal{L}_{\varrho}(\bm{\phi}^{k},\mathbf{w}^{k};\bm{\lambda})$

15:until
$\left|\mathcal{L}_{\varrho}^{k}-\mathcal{L}_{\varrho}^{k-1}\right|\leq\epsilon^{\prime}$

16:output:
$\\{\bm{\phi}^{r+1},\mathbf{w}^{r+1}\\}=\\{\bm{\phi}^{k},\mathbf{w}^{k}\\}$

##  V Simulation Results

Figure 2: Simulation setup. TABLE I: Simulation Parameters Parameter | Value
---|---
Number of antennas |  $M=8$.
Normalized antenna or element separation |  $\Delta_{\rm A}=\Delta_{\rm I}=\frac{1}{2}$.
Path loss at 1m |  $\zeta_{0}=-30$dB.
Path loss exponent |  $\alpha_{\rm AU}=\alpha_{\rm AE}=3.67$, $\alpha_{\rm IU}=\alpha_{\rm IE}=2.2$, $\alpha_{\rm AI}=2$.
Rician $K$-factor |  $K_{\rm AU}=K_{\rm AE}=0$, $K_{\rm IU}=K_{\rm IE}=10^{0.9}$, $K_{\rm AI}=\infty$.
Noise power |  $\sigma_{\rm U}^{2}=\sigma_{\rm E}^{2}=-90$dBm.
Coordinate/m |  $\left({\rm A}_{x},{\rm A}_{y}\right)=\left(5,0\right)$, $\left({\rm I}_{x},{\rm I}_{y}\right)=\left(0,50\right)$, $\left({\rm U}_{x},{\rm U}_{y}\right)=\left(5,60\right)$, $\left({\rm E}_{x},{\rm E}_{y}\right)=\left(10,55\right)$.

(a) $N=160$

(b) $P_{\rm max}=5$dBm

Figure 3: ESR and LESR versus $P_{\rm max}$ and $N$.

Figure 4: ESR and LESR versus the $Y$-Coordinates of Eve, user and RIS when
$P_{\rm max}=5$dBm and $N=160$.

(a) Same setting as TABLE I.

(b) Only NLoS path between RIS and user/Eve.

Figure 5: ESR and SR versus $P_{\rm max}$ when $N=32$

As depicted in Fig. 2, we consider an uniform linear array (ULA) at the AP and
an uniform planar array (UPA) equipped with $N=N_{y}N_{z}$ reflecting elements
at the RIS. For ease of exposition, $N_{y}=16$ is fixed and $N_{z}$ is
increased linearly according to $N$. The $z$-coordinate of AP, RIS, user and
Eve are set to $0$. We assume the AP-user/Eve direct channel follows Rayleigh
fading while the RIS-assisted channels follow Rician fading as in[5]. The
simulation parameters are summarized in Table I. In particular, the path loss
exponent and Rician $K$-factor of AP/RIS-user/Eve channel are set according to
the 3GPP Urban Micro-cell(UMi) propagation environment[25] and those of AP-RIS
channel are the same as those in[4].

We consider following schemes for comparison: 1) PDCA: the proposed PDCA
joint-design scheme; 2) AO: the alternating optimization (AO) scheme proposed
in[9]. To be more specific, on the one hand, the RIS phase shifts design
problem is solved by semidefinite relaxation (SDR) with Gauss randomization
technique. On the other hand, the transmit beamforming design problem is
derived in closed-form by leveraging the solution of the generalized Rayleigh
quotient problem[8]; 3) Opt w/o RIS: the beamformer is designed to maximize
the LESR in the absence of RIS. By substituting the solution of the LESR
maximization problem into formula (7), ESR is computed by Monte Carlo
simulation through taking average of $10^{5}$ i.i.d. random $\mathcal{H}_{\rm
E}=\\{\mathbf{h}_{\rm AE},\mathbf{h}_{\rm IE}\\}$ realizations.

In Fig. 3, we compare the ESR and LESR of all schemes versus $P_{\rm max}$ and
$N$, respectively. It can be seen that the proposed PDCA scheme achieves the
highest ESR among all schemes. In Fig. 3(a), more than $1$bps/Hz performance
gain is achieved by adopting PDCA compared with AO. It is observed from Fig.
3(b) that the performance gap between ESR and LESR of same scheme is quite
small (less than $0.3$bps/Hz), which demonstrates that the LESR serves as a
relatively tight lower bound. The ESR of both RIS-assisted schemes are
increasing with $N$, owing to the _beamforming gain_ and _aperture gain_[4]
provided by RIS.

In Fig. 4, we compare the ESR and LESR versus the $Y$-coordinates of Eve, user
and RIS, respectively. In the optimal beamforming scheme without RIS, it is
natural to see that the ESR increases with the coordinate of Eve and decreases
with the coordinate of user. For the RIS-assisted PDCA scheme, Fig. 4 (a)
shows that the ESR increases with the coordinate of Eve at first and then goes
down to a local minimum at 70m (a location near both RIS and user). On the
contrary, Fig. 4 (b) and Fig. 4 (c) demonstrate that the ESR decreases at
first and then goes up to a local maximum at 50m (RIS’s location) and 60m
(user’s location), respectively. It inspires us that it is a wise choice to
place the RIS near the user out of the consideration for the security
performance.

In Fig. 5, two perfect CSI schemes with and without RIS-assisted are simulated
to investigate their relationship with corresponding statistical CSI schemes.
The RIS-assisted scheme with perfect CSI is also solved by the proposed PDCA
method. It can be seen that the performance with perfect CSI serve as upper
bounds for the corresponding statistical CSI cases. In Fig. 5(b), we set
$\alpha_{\rm IU}=\alpha_{\rm IE}=3.67$ and $K_{\rm IU}=K_{\rm IE}=0$ the same
as those of the AP-user/Eve channel, which means there is only NLoS path
between RIS and user/Eve. Due to the blockage of the LoS path between RIS and
user/Eve, it is clear from the Fig. 5(b) that there is nearly no performance
gain for the RIS-assisted perfect CSI scheme. Nevertheless, there still exists
a prominent ESR improvement between the proposed PDCA scheme and the optimal
beamforming without RIS scheme, owing to the exploitation of the Eve-related
statistical CSI.

From the simulation results above, it is satisfying to see that the security
performance of the RIS-assisted PDCA scheme is better than the optimal
beamforming scheme without RIS, which proves the power of RIS in physical
layer security.

##  VI Conclusion

We studied the RIS-assisted secure transmission with the knowledge of
statistical CSI of eavesdropper. A lower bound of ESR is derived. Therefore,
the stochastic ESR maximization problem is converted into a deterministic LESR
maximization problem. By leveraging the proposed PDCA algorithm, the solution
of this non-convex optimization problem is guaranteed to converge to a KKT
point. Simulation results demonstrate that the security performance achieved
by the proposed PDCA scheme is better than the commonly adopted AO scheme. The
performance under perfect CSI settings are also given for comparison to
acquire more useful insights.

##  Appendix A Proof of Proposition 1

By the definition of ESR in (7), we have

$\displaystyle\bar{R}_{\rm S}$ $\displaystyle\geq\left[R_{\rm
U}-\mathbb{E}_{\mathcal{H}_{\rm E}}\left[R_{\rm E}\right]\right]^{+}$
$\displaystyle\geq\left[R_{\rm U}-\log_{2}\left(1+\frac{1}{\sigma_{\rm
E}^{2}}\mathbb{E}_{\mathcal{H}_{\rm E}}\left[\left|\left(\mathbf{h}_{\rm
IE}^{\mathsf{H}}\mathbf{\Phi}\mathbf{H}_{\rm AI}+\mathbf{h}_{\rm
AE}^{\mathsf{H}}\right)\mathbf{w}\right|^{2}\right]\right)\right]^{+}$
$\displaystyle=\bar{R}_{\rm S}^{\rm lb}$ (19)

where the first inequality is due to the convexity of
$\left[\cdot\right]^{+}$, and the second inequality is due to Jensen’s
inequality. The equality holds because of the independence between
$\mathbf{h}_{\rm AE}$ and $\mathbf{h}_{\rm IE}$ together with
$\mathbb{E}[|\tilde{h}_{{\rm AE},i}|^{2}]=\mathbb{E}[|\tilde{h}_{{\rm
IE},j}|^{2}]=1,\ i=1,\ldots,M,\ j=1,\ldots,N$.

## References

  * [1] T. J. Cui, M. Q. Qi, X. Wan, J. Zhao, and Q. Cheng, “Coding metamaterials, digital metamaterials and programmable metamaterials,” _Light: Science & Applications_, vol. 3, no. 10, pp. e218–e218, 2014.
  * [2] M. Di Renzo, A. Zappone, M. Debbah, M. S. Alouini, C. Yuen, J. de Rosny, and S. Tretyakov, “Smart radio environments empowered by reconfigurable intelligent surfaces: How it works, state of research, and the road ahead,” _IEEE Journal on Selected Areas in Communications_ , vol. 38, no. 11, pp. 2450–2525, 2020.
  * [3] Q.-U.-A. Nadeem, A. Kammoun, A. Chaaban, M. Debbah, and M.-S. Alouini, “Asymptotic max-min SINR analysis of reconfigurable intelligent surface assisted MISO systems,” _IEEE Transactions on Wireless Communications_ , vol. 19, no. 12, pp. 7748–7764, 2020.
  * [4] Q. Wu and R. Zhang, “Intelligent reflecting surface enhanced wireless network via joint active and passive beamforming,” _IEEE Transactions on Wireless Communications_ , vol. 18, no. 11, pp. 5394–5409, 2019.
  * [5] H. Guo, Y. Liang, J. Chen, and E. G. Larsson, “Weighted sum-rate maximization for reconfigurable intelligent surface aided wireless networks,” _IEEE Transactions on Wireless Communications_ , vol. 19, no. 5, pp. 3064–3076, 2020.
  * [6] S. Zeng, H. Zhang, B. Di, Y. Tan, Z. Han, H. Vincent Poor, and L. Song, “Reconfigurable intelligent surfaces in 6G: Reflective, transmissive, or both?” _IEEE Communications Letters_ , pp. 1–1, 2021.
  * [7] A. Mukherjee, S. A. A. Fakoorian, J. Huang, and A. L. Swindlehurst, “Principles of physical layer security in multiuser wireless networks: A survey,” _IEEE Communications Surveys Tutorials_ , vol. 16, no. 3, pp. 1550–1573, 2014.
  * [8] A. Khisti and G. W. Wornell, “Secure transmission with multiple antennas I: The MISOME wiretap channel,” _IEEE Transactions on Information Theory_ , vol. 56, no. 7, pp. 3088–3104, 2010.
  * [9] M. Cui, G. Zhang, and R. Zhang, “Secure wireless communication via intelligent reflecting surface,” _IEEE Wireless Communications Letters_ , vol. 8, no. 5, pp. 1410–1414, 2019.
  * [10] H. Shen, W. Xu, S. Gong, Z. He, and C. Zhao, “Secrecy rate maximization for intelligent reflecting surface assisted multi-antenna communications,” _IEEE Communications Letters_ , vol. 23, no. 9, pp. 1488–1492, 2019.
  * [11] X. Yu, D. Xu, and R. Schober, “Enabling secure wireless communications via intelligent reflecting surfaces,” in _2019 IEEE Global Communications Conference (GLOBECOM)_ , 2019, pp. 1–6.
  * [12] D. Xu, X. Yu, Y. Sun, D. W. K. Ng, and R. Schober, “Resource allocation for secure IRS-assisted multiuser MISO systems,” in _2019 IEEE Globecom Workshops (GC Wkshps)_ , 2019, pp. 1–6.
  * [13] X. Guan, Q. Wu, and R. Zhang, “Intelligent reflecting surface assisted secrecy communication: Is artificial noise helpful or not?” _IEEE Wireless Communications Letters_ , vol. 9, no. 6, pp. 778–782, 2020.
  * [14] W. Jiang, Y. Zhang, J. Wu, W. Feng, and Y. Jin, “Intelligent reflecting surface assisted secure wireless communications with multiple- transmit and multiple-receive antennas,” _IEEE Access_ , vol. 8, pp. 86 659–86 673, 2020.
  * [15] Z. He and X. Yuan, “Cascaded channel estimation for large intelligent metasurface assisted massive MIMO,” _IEEE Wireless Communications Letters_ , vol. 9, no. 2, pp. 210–214, 2020.
  * [16] W. Zhang, J. Xu, W. Xu, D. W. K. Ng, and H. Sun, “Cascaded channel estimation for IRS-assisted mmWave multi-antenna with quantized beamforming,” _IEEE Communications Letters_ , vol. 25, no. 2, pp. 593–597, 2021.
  * [17] Y. Han, W. Tang, S. Jin, C. Wen, and X. Ma, “Large intelligent surface-assisted wireless communication exploiting statistical CSI,” _IEEE Transactions on Vehicular Technology_ , vol. 68, no. 8, pp. 8238–8242, 2019.
  * [18] J. Zhang, J. Liu, S. Ma, C. Wen, and S. Jin, “Transmitter design for large intelligent surface-assisted MIMO wireless communication with statistical CSI,” in _2020 IEEE International Conference on Communications Workshops (ICC Workshops)_ , 2020, pp. 1–5.
  * [19] Q. Shi and M. Hong, “Penalty dual decomposition method for nonsmooth nonconvex optimization—Part I: Algorithms and convergence analysis,” _IEEE Transactions on Signal Processing_ , vol. 68, pp. 4108–4122, 2020.
  * [20] Y. Cai, C. Zhao, Q. Shi, G. Y. Li, and B. Champagne, “Joint beamforming and jamming design for mmWave information surveillance systems,” _IEEE Journal on Selected Areas in Communications_ , vol. 36, no. 7, pp. 1410–1425, 2018\.
  * [21] M. Razaviyayn, M. Hong, and Z.-Q. Luo, “A unified convergence analysis of block successive minimization methods for nonsmooth optimization,” _SIAM Journal on Optimization_ , vol. 23, no. 2, pp. 1126–1153, 2013.
  * [22] M. Hong, M. Razaviyayn, Z. Luo, and J. Pang, “A unified algorithmic framework for block-structured optimization involving big data: With applications in machine learning and signal processing,” _IEEE Signal Processing Magazine_ , vol. 33, no. 1, pp. 57–77, 2016.
  * [23] J. Nocedal and S. Wright, _Numerical Optimization_. Springer Science & Business Media, 2006.
  * [24] Z. Luo, W. Ma, A. M. So, Y. Ye, and S. Zhang, “Semidefinite relaxation of quadratic optimization problems,” _IEEE Signal Processing Magazine_ , vol. 27, no. 3, pp. 20–34, 2010.
  * [25] _Further Advancements for E-UTRA Physical Layer Aspects (Release 9)_ , 3GPP TS 36.814, Mar. 2010.