# Multibeam Sparse Tiled Planar Array for Joint Communication and Sensing

Hadi Alidoustaghdam∗, André Kokkeler, Yang Miao
Department of Electrical Engineering, University of Twente, the Netherlands
✉ _ ∗<EMAIL_ADDRESS>
###### Abstract

Multibeam analog arrays have been proposed for millimeter-wave joint
communication and sensing (JCAS). We study multibeam planar arrays for JCAS,
providing time division duplex communication and full-duplex sensing with
steerable beams. In order to have a large aperture with a narrow beamwidth in
the radiation pattern, we propose to design a sparse tiled planar array (STPA)
aperture with affordable number of phase shifters. The modular tiling and
sparse design of the array are non-convex optimization problems, however, we
exploit the fact that the more irregularity of the antenna array geometry, the
less the side lobe level. We propose to first solve the optimization by the
maximum entropy in the phase centers of tiles in the array; then we perform
sparse subarray selection leveraging the geometry of the sunflower array.
While maintaining the same spectral efficiency in the communication link as
conventional uniform planar array (CUPA), the STPA improves angle of arrival
estimation when the line-of-sight path is dominant, e.g., the STPA with 125
elements distinguishes two adjacent targets with 20∘ difference in the
proximity of boresight whereas CUPA cannot. Moreover, the STPA has a 40%
shorter blockage time compared to the CUPA when a blocker moves in the
elevation angles.

###### Index Terms:

6G, joint communication and sensing, tiled planar array, multibeam, phased
array.

##  I Introduction

In migration to 6G and connected smart networks, communication and sensing
devices have to be co-designed to maintain the reliability of high data-rate
connectivity and to have accurate information on the surrounding operation
environment [1]. There are many open challenges for joint communication and
sensing (JCAS), including the waveform design, the array aperture design, and
the resource optimization [1, 2, 3]. In this paper, our interest lies in the
base station (BS) array design. To situate both the communication and sensing
functionality, we focus on enabling time division duplex (TDD) communication
and a phased array radar in one array aperture.

Conventionally, the free-space radiation parameters, i.e., the directivity,
half-power beamwidth and side lobe level (SLL) are the figure-of-merits for
designing antenna arrays, which is valid for the channels with dominant line-
of-sight (LOS) path in communication and radar. However, in case of channels
with considerable non-line-of-sight (NLOS) paths or multi-user multiple-input-
multiple-output (MIMO) communication, the spectral efficiencies of
communication links are the reliable performance metrics [4, 5]. Therefore,
the radiation pattern in the BS is determined by the channel dynamics and is
configured for maximizing spectral efficiency of the communication link; it
might be narrow or wide beamwidth, single or multi beams. Additionally, a
phased array radar has a channel with a dominant LOS path, hence single beam
with low SLL and narrow beamwidth is always beneficial for improving signal-
to-noise ratio (SNR) and cross-range resolution [6]. In regard to JCAS, a
dynamic radiation pattern in BS might be desired as either communication or
sensing might dictate to react to the dynamic channel conditions. To enable
JCAS, the antenna selection, i.e., the synthetic change in aperture size [7]
could be necessary. But as far as the LOS path is dominant in JCAS, which is
considered in this paper, the exploitation of all antennas in the BS is
beneficial for increasing the SNR and performance of JCAS. It is stated in [8]
that increasing the size of aperture while preserving the same antenna numbers
does not improve the directivity, therefore directivity is not included in our
study, and we focus on designing a JCAS array with low SLL and narrow
beamwidth in the radiation pattern while maintaining a relatively appropriate
number of radiating elements to lower the costs. In order to manage the number
of array elements without deteriorating the performance of communication and
sensing, array tiling [9, 10, 11, 12, 5, 4] sparse arrays [13] and modular
design [8] are potential solutions.

###  I-A State-of-The-Art of Large Array Design for Communication or Sensing

####  I-A1 Array tiling

By grouping antennas into the so-called “tiles” and feeding each tile with one
phase shifter, it guarantees a large aperture with minimum phase shifters and
reduces the hardware cost. Until now, the investigations on array tiling and
the discussions on the resultant performance are mainly for communication by
phased array antennas [9, 10, 11, 12] or massive MIMO communication [5, 4], in
which genetic algorithm, iterative convex programming or information-theoretic
entropy concept are used for tiling the array. Among these solutions, authors
in [9] have proposed that two adjacent antennas can be excited by the same
phase shifter hence regarded as one domino tile, and if the phase centers of
these dominoes are distributed with maximum entropy in the aperture, the
radiation pattern has a low SLL. We choose this method due to its simplicity
and manageable time cost in the optimization.

####  I-A2 Sparse modular subarray

A narrow beamwidth in radiation pattern of a phased array requires a large
array aperture size. To decrease the complexity of large aperture, sparse
modular arrays are studied in [8] where the positions of modular uniform
subarrays are optimized for desired beamwidth and SLL. When sparse arrays are
considered, clustering antennas is another solution to avoid complexity of
transmission lines from the RF front-end to antennas, e.g., in [14] and [15],
the BS antenna array is designed based on irregular clustering and sequential
rotation where the optimization problem considers the inter-element spacing of
elements, aperture size and modular layout design. Sparse subarrays are also
proposed in non-terrestrial networks. For instance, due to a large
transmission distances in satellite communications, a larger aperture is
desirable to lower beamwidth; a sunflower sparse array of uniform hexagonal
subarrays is designed for satellite application in [16, 13].

Depending on the separation [17] or sharing of waveforms [18] between
communication and sensing, the requirements of apertures for JCAS are
different, e.g., number and locations of antennas or interference mitigation,
but in case of dominant LOS path, the sparse large array is beneficial for
better angle of arrival (AoA) estimation [6, 8], therefore we intend to
exploit the aforementioned solutions to design a large aperture with
affordable number of phase shifters for a BS. We propose a sparse tiled planar
array (STPA) with a shared waveform for JCAS. Firstly, a large aperture with
uniformly distributed antennas is tiled by dominoes based on the maximum
entropy of tile centers in the aperture. Then the sunflower array is exploited
to locate the center of subarrays and lower the number of tiles. When the
tiles in subarrays are chosen, the positions of the subarrays are optimized to
lower SLL.

###  I-B Contributions of This Paper

As a solution for JCAS, we propose a single STPA which is exploited for both
TDD communication and sensing, besides an auxiliary aperture for radar full-
duplex operation. The main contributions of this paper are:

  * •

In literature, JCAS is studied by employing conventional uniform planar arrays
(CUPA) for either downlink [19] or uplink communication [20], whereas in this
paper, STPA is proposed for both downlink and uplink communication at a BS
since a narrow beamwidth of the transmit beam (in downlink) or receive beam
(in uplink) is required for proper AoA estimation in sensing. The numerical
results are provided for downlink, however STPA can be adopted for uplink
sensing by the method in [21].

  * •

The angular resolution of the JCAS aperture is improved by suitable adaptation
of tiling and then sparse design. As it is stated in [22], the tiled array has
the benefits of modular design and low SLL in wide scanning angles compared to
solely sparse design. The narrow beamwidth of STPA compared to CUPA ensures
illuminating and receiving from only the desired angle during downlink and
uplink sensing, respectively. This narrow beamwidth is profitable for
increasing the number of beams for multi-user communication as well. The
narrow beamwidth is achieved with the cost of higher SLL compared to CUPA,
however this drawback is tolerable in the channels with dominant LOS path.

  * •

Considering the large bandwidth in JCAS for 6G, the irregularity introduced by
tiling and sparse subarray geometries, compared to the conventional uniform
arrays, results in a low SLL in the radiation pattern over a larger frequency
band [23]. The numerical results show that STPA has a proper SLL when the
frequency is scaled up to 2.5 times the designed frequency of operation, which
results in a 42% increase in the operational frequency of the array.

  * •

The narrow beamwidth of STPA is beneficial for decreasing the blockage time
since the spectral efficiency is dropped when the blocker reaches the narrower
communication beam, i.e., we show in a JCAS scenario that if a tracked target
blocks the communication link, the designed STPA can have a $40\%$ shorter
blockage time than CUPA in elevational movements of the blocker.

The following notations are used in this paper: $(.)^{T}$ and $(.)^{*}$ denote
the transpose and conjugate transpose operator, respectively.
$\left\lVert.\right\rVert_{F}$ is the Frobenius norm.

Figure 1: The configuration for JCAS in BS while operating in downlink
communication.

##  II Proposed Design Methodology

In this section, the structure of a BS with an analog beamforming array for
JCAS is studied. After that, a realization of STPA is proposed via a flowchart
and finally the metrics adopted for JCAS performance are introduced.

###  II-A Problem Formulation

Let us consider a TDD base station based on fully connected analog array as in
Fig. 1, where a JCAS aperture with $N$ antennas is supposed to communicate
with a single-antenna user and illuminate the target via two beams. This
aperture serves as the transmitter during downlink communication and as the
receiver during uplink communication. Without loss of generality, we study the
performance of only one RF chain of a fully connected analog array, since
other RF chains are also connected to the same antennas. Due to long
communication frames, full-duplex radar sensing functionality is mandatory for
perpetual sensing of environment, therefore auxiliary apertures with $N_{aux}$
antennas are needed to serve as radar receiver during downlink communication,
and as radar transmitter during uplink communication111The reason for transmit
antennas at the uplink is that sensing solely by the communication signal has
the following problems, i.e., the power resources of communication users are
limited so is the SNR for sensing, these users have also beamforming toward
BS, therefore the environment is not illuminated for sensing, besides the
sensing environment is dependent on the location of user which could be
unknown, therefore transmit antennas at BS during uplink communication are
beneficial for sensing [21].. The signal processing for downlink and uplink
JCAS are different, i.e., reflected signals from sensing target and uplink
communication signal from the user interfere at JCAS aperture which operates
as the receiver, hence they need to be separated, whereas downlink sensing
does not need such separation since the auxiliary Rx only receives the
reflection from the target. In this paper, we choose downlink communication
and sensing to demonstrate the performance of STPA. This performance is
representative to the uplink communication and sensing, since the narrow
beamwidth of the JCAS aperture as a receiver is beneficial for the sensing,
and the spectral efficiency of uplink communication with the STPA is similar
to the CUPA.

Radar has a two-way radiation pattern, and an ideal radar radiation pattern
should be narrow in both transmit and receive pattern for optimal power usage,
hence we study the challenging part where both communication and sensing are
considered in a single JCAS aperture, the auxiliary arrays can be designed
accordingly. One of the features of millimeter wave (mmWave) communication is
the large number of antennas, which provides larger gain towards the user.
Besides, the antennas in the JCAS aperture are used as transmitter for
downlink sensing, therefore the transmit beam for sensing is already narrow
and the auxiliary Rx can have less antenna numbers to decrease implementation
costs, hence we assume $N\gg N_{aux}$. Therefore, the tiling is targeted only
to the $N$ antennas at the JCAS aperture, and we simply choose $N_{aux}=1$ and
focus on the JCAS aperture for the performance study of STPA. We use
communication signals with orthogonal frequency division multiplexing (OFDM)
waveforms to communicate and sense.

Figure 2: The flowchart of designing STPA.

###  II-B Realization of STPA

Note that the inherent problems of tiling of the JCAS aperture for the least
SLL and a sparse aperture for the best AoA estimation are non-convex
optimizations which are difficult to solve, therefore the optimization has to
be based on the foreknowledge, hence local optimization is employed, as it is
shown in the flowchart of Fig. 2. The main idea of this design is based on two
scales of irregularity, namely the geometry of subarrays (local scale), and
the phase centers of tiles in the whole aperture (global scale). Initially, a
large CUPA is tiled for the minimum SLLs in the radiation pattern. The tiling
problem is formulated based on maximum entropy in the distribution of the
phase centers of the tiles in the array. In the JCAS aperture shown in Fig. 3,
assume $N_{\text{sub}}$ antennas are grouped in tiles and fed by power
dividers. The center of mass of such a tile could be considered as the center
of phase for the radiation from the whole tile. The irregularity in the
distributions of these phase centers is measured with an information-theoretic
entropy-based objective function, i.e., if the entropy of phase centers at
each row and column of the aperture is maximized, then the resulting phased
array aperture has a desired SLL while scanning. To obtain more mathematical
insight into tiling based on the maximum entropy, refer to [24]. The resultant
tiled array can still be thinned by grouping some tiles in subarrays and
removing others. A subarray is determined by a circle circumventing its tiles,
the centers of these circles (subarrays) are obtained by the geometry of the
sunflower array since the positions of elements are controllable, although it
is an aperiodic sparse array (see appendix A). The radius of a circle is a
thinning parameter for the aperture, which can be tuned to reach the desired
SLL with minimum number of tiles. It is assumed that the phase centers of
tiles are still distributed with a maximum entropy after thinning of aperture,
but the inter-distances among subarrays increase the SLL, however the
resultant empty space among subarrays can be exploited to optimize the
location of subarrays efficiently to decrease the SLL (see appendix B). Once
the JCAS aperture with the minimum number of phase shifters, the desired SLL
and beamwidth is achieved, its performance is compared to CUPA.

Figure 3: The initial configuration of antennas at the BS.

####  II-B1 Expanded Beam Pattern

In order to assure that the designed STPA bears the desired SLL and beamwidth
in various scanning scenarios, the expanded beam pattern (EBP) is employed,
which facilitates studying SLLs for all scanning angles in a single radiation
pattern. The radiation field of an array with isotropic radiating elements in
$(u,v)$-space can be obtained by:

$f(u,v)=\frac{1}{N}\sum_{n=1}^{N}\alpha_{n}e^{j\Omega_{n}}e^{jk(ux_{n}+vy_{n})},u=\cos\theta\cos\phi,v=\cos\theta\sin\phi$
(1)

where $N$ is the number of radiating elements, $\alpha_{n}$ and $\Omega_{n}$
are the amplitude and phase of excitation, respectively. $k=2\pi/\lambda$ is
the wave-number at the wavelength $\lambda$, $x_{n}$ and $y_{n}$ are the
positions of antennas in Cartesian coordinates. $\theta$ is the elevation
angle measured from $xy$-plane and $\phi$ is the azimuth angle measured from
$x$-axis. The radiation field in (1) can scan toward the angle $(u_{s},v_{s})$
by the proper weighting of amplitudes and phases. In case of conventional
beamforming, which is based on only phase excitation in the elements, the
weights are:

$\alpha_{n}=1,\Omega_{n}=-jk(u_{s}x_{n}+v_{s}y_{n}),$

then the radiation pattern is:

$F(u,v;u_{s},v_{s})=\frac{1}{N^{2}}\Bigg{|}\sum_{n=1}^{N}e^{jk\big{(}(u-u_{s})x_{n}+(v-v_{s})y_{n}\big{)}}\Bigg{|}^{2}$
(2)

In order to simplify studying the above radiation pattern for all
$(u_{s},v_{s})$s, the EBP is defined as [8]:

$F_{\zeta}(\tilde{u},\tilde{v})=\frac{1}{N^{2}}\Bigg{|}\sum_{n=1}^{N}e^{jk\zeta(\tilde{u}x_{n}+\tilde{v}y_{n})}\Bigg{|}^{2},\zeta=1+\cos(\theta_{min})$

where $\tilde{u}=u-u_{s}$, $\tilde{v}=v-v_{s}$. The scanning of beam in the
whole $0^{\circ}\leq\theta\leq 90^{\circ}$ without grating lobes is not
practical, therefore the desired angle of view is $\theta_{min}\leq\theta\leq
90^{\circ}$ where $\theta_{min}$ is the minimum of angle $\theta$. The SLL for
$F_{\zeta}(\tilde{u},\tilde{v})$ is studied to ensure an appropriate SLL in
all scanning angles.

Figure 4: The flowchart of performance measurement. Figure 5: The scenarios
for JCAS in downlink communication, (a) without overlapping of communication
and sensing beams, (b) with overlapping of communication and sensing beams,
when the UE blocks the sensing link and (c) forthcoming overlapping of
communication and sensing beams, when the sensing target can block the
communication link.

###  II-C Performance Metrics

The performance metrics for communication and sensing are spectral efficiency
and angular resolution, respectively, as it is shown in the flowchart of Fig.
4. The communication performance (CP) includes cases without and with
overlapping of beams when the UE blocks the sensing link and also when the
sensing target blocks the communication link. The overlapping occurs when:

$(u_{c}-u_{s})^{2}+(v_{c}-v_{s})^{2}<r_{b}^{2}$ (3)

where $(u_{c},v_{c})$ and $(u_{s},v_{s})$ are the angles of communication and
sensing beams, respectively, and $r_{b}$ is the radius of the beamwidth in
$(u,v)$-space. In pursuit of simplicity, we suppose the user has no reflection
of electromagnetic waves and all the energy transmitted in the communication
beam of the BS is captured by the user.

####  II-C1 CP without overlapping communication and sensing beams

Let us consider the scenario in Fig. 5(a). A geometric model for both
communication and sensing including $L$ multipath signals with the angle of
departure $(\theta_{T,l},\phi_{T,l})$ and AoA $(\theta_{R,l},\phi_{R,l})$ is
assumed, then the time varying channel $\mathbf{H}\in\mathbb{C}^{N\times
N_{R}}$ would be:

$\begin{split}&\mathbf{H}=\sum_{l=1}^{L}b_{l}\delta(t-\tau_{l})e^{j2\pi
f_{D,l}t}\mathbf{a}_{T}(\theta_{T,l},\phi_{T,l})\mathbf{a}^{T}_{R}(\theta_{R,l},\phi_{R,l})\\\
\end{split}$ (4)

where $b_{l}$ is the complex value modelling signal attenuation, $\tau_{l}$ is
the propagation delay and $f_{D,l}$ is the Doppler frequency. $\mathbf{a}_{T}$
and $\mathbf{a}_{R}$ are the steering vectors for transmit and receive arrays.
The channel (4) might also include the LOS path for communication. The
multibeam weighting vector at the JCAS aperture is [18]:

$\mathbf{w}_{JCAS}=\sqrt{\rho}\mathbf{w}_{c}+\sqrt{(1-\rho)}\mathbf{w}_{s}$
(5)

where $0\leq\rho\leq 1$ is the power allocation factor between beams which
equals to 0.5 in this paper, $\mathbf{w}_{c}\in\mathbb{C}^{N\times 1}$ is the
weighting vector for communication and $\mathbf{w}_{s}\in\mathbb{C}^{N\times
1}$ denotes the weighting vector for sensing to a desired direction
$(\theta_{s},\phi_{s})$. For simplicity, $\mathbf{w}_{c}$ is considered as the
right singular vector of the channel corresponding to the maximum singular
value of $\mathbf{H}$ with $\left\lVert\mathbf{w}_{c}\right\rVert_{F}^{2}=1$
whereas
$\mathbf{w}_{s}=\mathbf{a}_{T}^{*}(\theta_{s},\phi_{s})/\left\lVert\mathbf{a}_{T}(\theta_{s},\phi_{s})\right\rVert_{F}$.
In this paper, the performance metric for communication is spectral
efficiency, and when Gaussian symbols are transmitted in a mmWave channel, the
spectral efficiency is [25]:

$R=\text{log}_{2}\text{det}\Big{(}1+\frac{\mathbf{H}^{*}\mathbf{w}_{JCAS}^{*}\mathbf{w}_{JCAS}\mathbf{H}}{\sigma_{n}^{2}}\Big{)}$
(6)

where $\sigma_{n}^{2}$ denotes the variance of a zero-mean Gaussian noise.

####  II-C2 CP with overlapping beams when the UE blocks the sensing link

When the UE blocks the sensing link, the two beams are united and the whole
energy is captured by the UE, then the spectral efficiency is increased. This
happens either in the scanning mode of sensing or the tracking mode with the
range of target being larger than the range of UE as in Fig. 5(b). The model
for this case, due to its simplicity, is not considered here and will only be
shown via simulation.

####  II-C3 CP with overlapping beams when the sensing target blocks the
communication link

Let us study the overlapping when the range of the target is smaller than the
range of a single-antenna UE, as in Fig. 5(c). It is supposed that the target
occupies only one angle step, is tracked, and the sensing beam is aligned with
the angle of the target. When this target moves toward the communication link,
the beams are overlapping, the LOS link is vanished and blockage occurs. In
order to model the blockage, the following channel model is considered with
and without blockage [26, 27]:

$\begin{split}&\mathbf{H}=\begin{cases}\mathbf{H}_{LOS}+\mathbf{H}_{NLOS}+\mathbf{H}_{target,1}&\text{
if no overlapping beams,}\\\ \mathbf{H}_{NLOS}+\mathbf{H}_{target,2}&\text{ if
beams overlapping,}\end{cases}\\\ \end{split}$ (7)

where:

$\begin{split}&\mathbf{H}_{LOS}=\mathbf{a}_{T}(\theta_{UE},\phi_{UE})\frac{\lambda}{4\pi
r_{UE}}e^{-j\frac{2\pi}{\lambda}r_{UE}},\\\
&\mathbf{H}_{target,s}=\mathbf{a}_{T}(\theta_{t},\phi_{t})\frac{\lambda}{4\pi
d_{t}}\Gamma_{t,s}e^{-j\frac{2\pi}{\lambda}d_{t}},s\in\\{1,2\\},d_{t}=r_{t}+r_{r},\end{split}$
(8)

where $\mathbf{H}_{NLOS}$ is the same channel (4) as described in section C.1,
$\mathbf{H}_{target,1}$ and $\mathbf{H}_{target,2}$ denote the channel of BS-
target-UE before or after the blockage and during the blockage, respectively.
$r_{t}$ and $r_{r}$ are the distance between the target and the BS, and the
target and the UE, respectively. Let us suppose a human as the target,
therefore $\Gamma_{t,1}$ which models the reflection and scattering is
calculated by [26]:

$\Gamma_{t,1}(\delta)=\frac{\epsilon_{b}\sin\delta-\sqrt{\epsilon_{b}-\cos^{2}\delta}}{\epsilon_{b}\sin\delta+\sqrt{\epsilon_{b}-\cos^{2}\delta}}+\mathcal{CN}(0,\sigma_{b}^{2})$
(9)

where $\delta$ is the incident angle on the body, $\epsilon_{b}$ is the
complex permittivity of the body and a zero-mean Gaussian random variable with
a standard deviation of $\sigma_{b}$ models the non-flat body and swinging
limbs. Besides, when the human blocks the LOS link, the result is $20$ dB
attenuation of power at the receiver UE [28], therefore $\Gamma_{t,2}$ which
denotes the diffraction coefficient, simply equals to $0.1$ at the operation
frequency of $28$ GHz. We assume the target moves in a line perpendicular to
the LOS path where $\phi_{t}=\phi_{UE}$ and only the angle $\theta_{t}$
varies, and its Doppler effects on $\mathbf{H}_{target,s}$ are negligible. The
channel modeled in $\eqref{BlockedChannel}$ is used for calculating the
spectral efficiency.

###  II-D Sensing performance of JCAS aperture

The main sensing feature of the designed JCAS BS in Fig. 5(a) is the angular
resolution due to its narrow beamwidth, which will be demonstrated by two
angularly adjacent targets. The sensing here is based on the phased array
where at each angle, the range and velocity are estimated. Similar to [18],
the same OFDM waveform is used for JCAS and sensing should be performed during
multiple sets of symbols; $N_{d}$ symbols are required for the coherent
processing for Doppler estimation at each angle, where $N_{e}$ and $N_{a}$ are
the number of elevation and azimuth angles, respectively, resulting in
$N_{e}N_{a}$ angles for full sensing of space. Here, we suppose that self-
interference is already cancelled by a combination of passive [29] and active
methods [21, 30], therefore the received signal at the auxiliary Rx is
simplified by:

$\tilde{y}_{n,n_{a},k}=\sum_{l=1}^{L}b_{l}\mathbf{w}^{T}_{JCAS}\mathbf{a}_{T}(\theta_{T,l},\phi_{T,l})e^{-j2\pi
n\tau_{l}f_{0}}e^{j2\pi
f_{D,l}(kT_{s}+(n_{a}-1)T_{f})}+\tilde{z}_{n}/\tilde{s}_{n}\\\ $ (10)

where $\tilde{y}_{n,n_{a},k}$ is the received signal at $n$th subcarrier when
sensing the angle indexed by $(n_{a},k)$ where $k=(n_{e}-1)N_{d}+n_{d}$,
$1\leq n_{e}\leq N_{e}$ and $1\leq n_{d}\leq N_{d}$. While $T_{s}$ is the
symbol duration, $f_{0}=1/T_{s}$. $\tilde{z}_{n}$ is the noise sample and
$\tilde{s}_{n}$ is the baseband signal for subcarrier $n$. It is assumed that
the communication beam is constant during $T_{f}$, the time of one
communication packet. At each scanning angle, $\mathbf{w}_{s}$ and
consequently $\mathbf{w}_{JCAS}$ in (5) are calculated, then the inverse
discrete Fourier transform (DFT) and DFT are applied to the received signal
$\tilde{y}_{n,n_{a},k}$ to extract the range and Doppler of targets,
respectively.

Figure 6:  (a) The tiled array with dominoes (the colors are only for
differentiating the tiles), (b) its EBP with the blue circle with radius
$\tilde{r}_{b}=0.043$ denotes the 3 dB-beamwidth of the main beam in
$(\tilde{u},\tilde{v})$-space, then the beamwidth in $(u,v)$-space is
$r_{b}=\zeta\tilde{r}_{b}=0.065$.  Figure 7: The subarrays chosen based on the
sunflower array, where the crosses denote the phase centers of the dominoes,
red circles show the centers of the subarrays, and the blue squares
demonstrate the chosen dominoes in each subarray.

##  III Numerical Analysis

###  III-A JCAS Array Aperture – A Design Example

Let us consider a uniform array for the JCAS aperture with $N_{x}\times
N_{y}=30\times 30$, with $\lambda/2$ inter-element distance where $\lambda$ is
the wavelength at the center frequency of operation $f_{c}=28$ GHz. Firstly,
these antennas are tiled with dominoes based on the maximum entropy, as in
Fig. 6(a). In order to show the capability of this large tiled array when
scanning to $\theta_{min}=60^{\circ}$ according to 5G base station
requirements [12], its EBP which corresponds to the scanning by linear
beamforming is shown in Fig. 6(b), where the maximum SLL is $-13.24$ dB and
the beamwidth is $r_{b}=0.065$. Then, the sunflower array is used to thin
these dominoes. In order to cover the whole aperture, the distance between
elements of the sunflower array is $s=3.5\lambda$, while the angle steps of
these elements are golden ratio $\tau=1.618$. The domino tiles which are in
$r_{c}$ distance from the elements of sunflower are chosen for the subarrays
and the thinning rate depends on this $r_{c}$. The simulation showed that
$r_{c}=1.15\lambda_{c}$ provides the best performance for low SLL and minimum
required tiles, which results in overall 125 dominoes as shown in Fig. 7. This
elimination of dominoes increases maximum SLL, and reaches $-10.93$ dB,
however the locations of subarrays can also be optimized for improving the SLL
via iterative convex optimization (see appendix B). The parameters in the
position optimization of subarrays for an EBP with $\zeta=1.5$ are
$\tilde{r}_{b}=0.043$ for the main beamwidth and $\mu=\lambda/25$ for the
maximum of position steps. The angles $0^{\circ}\leq\theta\leq 90^{\circ}$ and
$-180^{\circ}\leq\phi<180^{\circ}$ are swept with $1^{\circ}$. The objective
function is to minimize the maximum SLL in the EBP. As is shown in Fig.
8(a)-(b), the optimization reaches a maximum SLL of $-13.95$ dB. For mutual
coupling and implementation considerations, the inter-distances of each tile
in a subarray with the adjacent tiles in other subarrays are checked to be
larger than $\lambda$ which is shown in Fig. 8(c). The EBP of the final
geometry, has an SLL of $-13.5$ dB (due to the linear approximation in the
optimization) while having a beamwidth of $\tilde{r}=0.043$ as it is shown in
Fig. 8(d). This final array geometry in Fig. 8(a) will be employed as the JCAS
aperture.

Figure 8: (a) The final geometry of subarrays where the blue squares are the
centers of domino tiles, (b) maximum SLL during iterative optimization on the
position of subarrays, (c) $d^{t_{i},t_{j}}_{s}$, the distance of tiles
$t_{i}$ and $t_{j}$ when they are not in the same subarray and $x$-axis is the
enumeration of $\\{t_{i},t_{j}\\}$ pairs and (d) the EBP of aperture with the
blue circle on main beam indicating the 3dB-beamwidth.

####  III-A1 Robustness of STPA to wideband operation

The array factor is the radiated field of the array when antennas are
isotropic, and is critical for wideband operation. Fig. 9 demonstrates how
this designed aperture performs with an increase in the frequency of
operation. Where the EBP of STPA at $f/f_{c}=2.5$ still shows a maximum SLL of
-11.44 dB, the CUPA with 144 antennas suffers from high SLL (grating lobes).
Considering 1.75$f_{c}$ and 2.5$f_{c}$ as maximum frequencies for CUPA and
STPA, respectively, STPA contributes to 42% improvement in maximum operational
frequency of the array.

Figure 9:  The level of maximum SLL in EBP with $\zeta=2$ for STPA and CUPA
with 144 antennas.

####  III-A2 Practical STPA design

An example of an STPA is simulated in CST Studio. The radiating elements are
patch antennas as in Fig. 10(a) and designed on the Rogers RO3003 substrate
with thickness of $0.25$ mm, relative permittivity of $\epsilon_{r}=3$, copper
cladding thickness of 35 $\mu$m and excited by a discrete port. At $f_{c}=28$
GHz, the 10dB-bandwidth is $863$ MHz and 3dB-beamwidth is
$89.3^{\circ}<\theta_{\text{3~{}dB}}<119.1^{\circ}$ for various $\phi$s. Upon
the knowledge of the positions of phase centers for tiles as shown in Fig.
8(a) and the orientation of dominoes (horizontal or vertical), the STPA has
obtained and shown in Fig. 10(b). The radiation from the STPA can be simulated
by simultaneous excitation of the two horizontally or vertically juxtaposed
patch antennas (corresponding to a tile) with the same phase. The number of
patch antennas is 250 whereas the number of phase shifters and tiles is 125.
As an example, the S11 parameters of 18 antennas (9 dominoes) at subarray 1
are shown in Fig. 10(c) when all antennas are excited simultaneously for
scanning $(\theta,\phi)=(90^{\circ},0^{\circ})$ and Fig. 10 (d) for scanning
$(\theta,\phi)=(60^{\circ},45^{\circ})$. The changes in the resonance
frequency and magnitude of S11 are noticeable due to the mutual coupling
effects. The radiation from the STPA at the frequency of $28$ GHz and
beamformed to boresight $(\theta,\phi)=(90^{\circ},0^{\circ})$ is shown in
Fig. 10(e) where the maximum directivity is $29.4$ dBi, the 3dB-beamwidth is
$4^{\circ}$ and the maximum SLL is $-13.6$ dB, whereas in case of beamformed
towards $(\theta,\phi)=(60^{\circ},45^{\circ})$, the maximum directivity is
$26.5$ dBi, the maximum SLL reaches $-11.9$ dB and beamwidth to $4.6^{\circ}$
as shown in Fig. 10(f). The decrease of directivity and SLL suppression in the
full-wave simulations is due to the non-isotropic radiation of patch antennas,
the mutual coupling effects and the asymmetric radiations of horizontal and
vertical tiles.

Figure 10: (a) The geometry of the patch antenna element in CST, with the red
dot denoting the feeding point (b) the arrays of antennas and the blue circle
denoting subarray 1, (c) S11-parameters of 18 antennas at subarray 1 when
scanning $(\theta=90^{\circ},\phi=0^{\circ})$, (d) S11-parameters of 18
antennas at subarray 1 when scanning $(\theta=60^{\circ},\phi=45^{\circ})$ (e)
the radiation pattern of the array when scanning
$(\theta=90^{\circ},\phi=0^{\circ})$ (f) the radiation pattern of array when
scanning $(\theta=60^{\circ},\phi=45^{\circ})$.

###  III-B Performance metrics

Once the practicality of STPA is demonstrated by full-wave simulation, we will
proceed with simple isotropic modelling of radiating elements for analyzing
the performance for the communication and sensing in comparison with CUPA.

####  III-B1 CP without overlapping communication and sensing beams

We consider the situation Fig. 5(a) and assume a user is at the range of
$r_{UE}=50$ m and the angle of
$(\theta_{UE},\phi_{UE})=(70^{\circ},50^{\circ})$. The channel in (4) from the
JCAS aperture to the user is modeled by a dominant LOS path, besides NLOS
paths which include $N_{cl}=8$ clusters each with $N_{ray}=10$ rays and having
10 dB power less than the LOS path. The overall power of the channel,
including LOS and NLOS path, is normalized for the channel as in [12]. The
scatters are distributed uniformly within all elevations
$0^{\circ}\leq\theta\leq 90^{\circ}$ and azimuth
$-180^{\circ}\leq\phi<180^{\circ}$ angles. The channel from BS to the user via
the target $\mathbf{H}_{target,1}$ is also considered in the NLOS path. Based
on the estimated channel and desired scanning angles, $\mathbf{w}_{JCAS}$ and
based on that, the spectral efficiency $R$ can be calculated. The average of
$R$ from 200 channel realizations is obtained for the designed STPA. We intend
to compare the performance of STPA with a symmetrical CUPA, so the closest
numbers of antennas of CUPA are $N=11\times 11$ and $N=12\times 12$. As it is
shown in Fig. 11, all three antenna configurations (STPA and 2 times CUPA)
have similar performances, where CUPA with $N=121$ and $N=144$ has slightly
better spectral efficiency due to their higher directivity.

Figure 11: The spectral efficiency for the designed STPA with $N=125$ and CUPA
with $N=121$ and $N=144$ when there is no overlapping of communication and
sensing beams. SNR denotes the SNR level in the LOS path.

Figure 12: The effect of the scanning beam on the spectral efficiency for the
STPA compared with CUPAs while scanning in (a) azimuth, and (b) elevation.

####  III-B2 CP with overlapping communication and sensing beams when the UE
blocks the sensing link

As the same waveform is used for both beams, when overlapping occurs either in
the scanning mode, or in the tracking mode with the range of the tracking
target being larger than the range of the UE, which is the case in Fig. 5(b),
the signal power at the UE is increased, therefore spectral efficiency is
supposed to increase. Let us consider the user is at the angle
$(\theta_{UE},\phi_{UE})=(70^{\circ},50^{\circ})$, the sensing is in the
scanning mode and the sensing beam first scans over azimuth at the elevation
$\theta=70^{\circ}$ and then over elevation at azimuth $\phi=50^{\circ}$
angles. The average spectral efficiencies at an SNR of 10 dB is presented in
Fig. 12(a) and (b). The deviation from the spectral efficiency without
overlapping is noticeable for the STPA only when the difference of
communication angle and scanning angle is $10^{\circ}$, whereas this number
for the CUPA with $144$ and $121$ antennas is $25^{\circ}$. The Fig. 12(a) and
(b) and above discussion are also valid for the tracking mode of a target with
larger range than the UE, e.g., $r_{UE}=70$ m and $r_{t}=140$ m. Regarding
sensing performance in these cases, the received signal at the BS from the
scanning beam when it reaches the angle of the UE, is zero due to zero
reflection. In the overlapping when the tracked target is behind the UE, the
received reflected signal at the BS is again zero and the target is also lost.
The spectral efficiency of the communication link increased since the sensing
and communication beam have the same waveform, if one intends to use separate
waveforms in separate beams then beam overlapping is detrimental for the SNR
at the user.

####  III-B3 CP with overlapping communication and sensing beams when the
sensing target blocks the communication link

The range for UE is considered to be $r_{UE}=70$ m, the target body moves at
the constant azimuth of $\phi_{t}=50^{\circ}$, perpendicular to the LOS path
and crossing it at $r_{m}=35$ m as in Fig. 5(c). The parameters for the body
in (9) are assumed as $\epsilon_{b}=0.1-j2.33$, $\sigma_{b}=0.45$ dB [26] and
the incident angle on the body $\delta=\theta_{t}$. The variation of spectral
efficiency at an SNR of 10 dB for the STPA and CUPAs are shown in Fig. 13.
There is an increased oscillation in the spectral efficiency just before and
after blockage occurrence, which is due to constructive addition of
$\mathbf{H}_{target,1}$ and $\mathbf{H}_{LOS}$ [31]. The most noticeable
difference between the STPA and CUPAs is the time of blockage where the STPA,
due to its narrower beamwidth, has shorter time. In these scenarios, the
blockage time is proportional to the overlapping angles of communication and
sensing beam, hence it is observed in Fig. 13 that the STPA has a 40$\%$
shorter blockage time than CUPAs in the elevation sensing.

Figure 13: The spectral efficiency when the target blocks the LOS path at the angle $\theta_{t}=70^{\circ}$ . Figure 14: The scenario for sensing performance. TABLE I: OFDM simulation parameters  Parameter | Value | Description
---|---|---
$f_{c}$ | 28 GHz | Carrier frequency
$B$ | 40 MHz | Bandwidth
$N_{sc}$ | 2048 | Number of subcarriers
$T_{s}$ | 51.2 $\mu$s | Symbol duration
$\Delta r$ | 3.75 m | Range resolution
$\Delta v$ | 3.07 m/s | Velocity resolution
$v_{umax}$ | 52.32 m/s | Maximum unambiguous velocity
$r_{umax}$ | 7680 m | Maximum unambiguous range

####  III-B4 Sensing performance of JCAS aperture

One of the key performance metric for sensing is the angular resolution in the
scanning mode, which is studied for the STPA in comparison to CUPA with
$N=144$ antennas. Let us consider an outdoor scenario as in Fig. 14 where, in
the worst case, two targets are located at the same ranges of $r_{t}=70$ m,
velocities of $v_{t}=20$ m$/$s and the elevation of $\theta_{t}=70^{\circ}$
but in different azimuth angles. The OFDM simulation parameters are shown in
the Table I. The radiation gain of JCAS tiles and auxiliary antennas, the
radar cross-section of targets and the noise figure of the receiver are
assumed to be unit, then the two-way channel gain in (4) is simply
$b_{l}=\frac{\lambda^{2}}{(4\pi)^{3/2}r^{2}}$.

The coherent processing duration for Doppler estimation is $N_{d}T_{s}$ where
$N_{d}=34$. A communication packet includes $N_{d}N_{e}$ symbols, where
$N_{e}=19$ is the number of elevation angles and scanning in azimuth is
performed during $N_{a}=73$ communication packets, hence the elevation and
azimuth angles are swept by $5^{\circ}$. The variance of noise $\tilde{z}_{n}$
in (10) is considered to be 10 dB lower than the received target signal. As
the emphasis here is on the angular resolution of JCAS aperture, we assume the
angle differences of targets are $20^{\circ}$ only in azimuth and compare the
angle estimation of targets between the STPA and the CUPA with $N=144$. The
range, velocity and elevation angle of targets are shown in Fig. 15(a) and (c)
for the CUPA, and in Fig. 16(a) and (c) for the STPA. Both arrays provide a
true estimation of these features of targets. However, comparing the results
in Fig. 15(b) and (d) and Fig. 16(b) and (d) show that the designed STPA is
capable of distinguishing two adjacent targets in $\phi_{t}=0^{\circ}$ and
$20^{\circ}$ where the CUPA with more number of antennas cannot. Therefore,
the STPA provides a high angular resolution image during the scanning mode of
sensing.

Figure 15: The normalized sensing images for CUPA with 144 antennas, where two
targets and the communication user are at the same range, velocity, and
elevation $\theta_{t}=70^{\circ}$ but at different azimuths
$\phi_{t}=0^{\circ}$ and $20^{\circ}$. The two targets are _not_
distinguishable in $\phi$-axis. The red dots show the true feature of targets.
Figure 16: The normalized sensing images for the STPA, where two targets and
the communication user are at the same range, velocity, and elevation
$\theta_{t}=70^{\circ}$ but at different azimuths $\phi_{t}=0^{\circ}$ and
$20^{\circ}$. The two targets are distinguishable in $\phi$-axis. The red dots
show the true feature of targets.

###  III-C Discussion

In this paper, we assumed a scenario with a dominant LOS path, therefore the
increase in the average SLL due to tiling and sparse design is not
detrimental. The effect of interference on the JCAS performance of the STPA
for NLOS case is still an open question. As future work, the aperture
efficiency of the designed STPA should also be improved. This can be tackled
by assigning tiles with different sizes to outer subarrays. Besides, this
study can be extended to hybrid beamforming arrays combined with phased-MIMO
radar concepts.

##  IV Conclusion

Analog beamforming arrays can increase the gain of radiation and overcome
propagation attenuation in millimeter wave communication and sensing links,
therefore in alliance with the fully-connected analog beamforming arrays, a
tiled sparse planar array (STPA) is proposed with modular design, narrow
beamwidth and decreased number of phase shifters for joint communication and
sensing applications. The design is based on maximum entropy in the phase
centers of the tiles in the array and sparse subarrays. The results show that
an STPA with 125 domino tiles has stable radiation up to 2.5 times the center
frequency of operation, $f_{c}$, whereas the maximum frequency of operation is
$1.75f_{c}$ for CUPA with 144 antennas. A realistic radiation pattern of STPA,
including antenna imperfections, is shown by full-wave simulation. When the
sensing and communication beams are not overlapping, CUPA and STPA guarantee
the same spectral efficiency in the communication link, whereas a better
angular resolution in sensing is achieved by the STPA, e.g., two targets in
the proximity of boresight are distinguished while their azimuth is 20∘
different. The analysis also includes the overlapping of communication and
sensing beams and a blockage scenario by a tracked target. As the STPA has a
narrower beamwidth than the CUPA, the blockage time is 40$\%$ shorter while
the tracked target moves in the elevation angles.

##  V Acknowledgment

This work has received funding from the Dutch Sector Plan and the MSCA-IF-2020
V.I.P. project No. 101026885.

##  VI Appendix

###  VI-A Sunflower array

The sunflower array can demonstrate the same radiation pattern as a uniform
array, but with a lower number of antennas. In polar coordinates, the $m$th
element of the sunflower array $(\rho_{m},\psi_{m})$ is obtained by [13]:

$\rho_{m}=s\sqrt{\frac{m}{\pi}},\psi_{m}=2\pi m\tau\text{ where }m=1,2,...,M$
(11)

where $s$ is the inter-distance of elements, $\tau=1.618$ denotes the golden
ratio and $M$ is the number of elements.

###  VI-B Position optimization

The positions of subarrays are optimized based on an iterative convex
optimization. Let us consider the expanded radiation field of the array in
$(\tilde{u},\tilde{v})$-space as:

$f_{\zeta}(\tilde{u},\tilde{v})=\frac{1}{N}\sum_{n=1}^{N}e^{jk\zeta(\tilde{u}x_{n}+\tilde{v}y_{n})},$
(12)

As it is discussed in [32], when the positions of antennas are optimization
variables, the problem is nonlinear and non-convex, however it can be solved
as a convex problem iteratively by first-order Taylor expansion as in:

$\begin{split}&f_{\zeta}^{i}(\tilde{u},\tilde{v})=\frac{1}{N}\sum_{n=1}^{N}e^{jk\zeta\tilde{u}x_{n}^{i-1}}(1+jk\zeta\tilde{u}\epsilon_{n}^{i})\\\
&e^{jk\zeta\tilde{v}y_{n}^{i-1}}(1+jk\zeta\tilde{v}\beta_{n}^{i})\\\
&\approx\frac{1}{N}\sum_{n=1}^{N}e^{jk\zeta\tilde{u}x_{n}^{i-1}}e^{jk\zeta\tilde{v}y_{n}^{i-1}}\\\
&(1+jk\zeta\tilde{u}\epsilon_{n}^{i}+jk\zeta\tilde{v}\beta_{n}^{i}),\\\
&\text{ where }|\beta_{n}^{i}|\ll 1,|\epsilon_{n}^{i}|\ll 1\\\ \end{split}$
(13)

where $i$ denotes the iteration index, and $\epsilon^{i}_{n}$ and $\beta^{i}$
are the movement in $x$ and $y$ axes at $i$th iteration, respectively.
$x^{i-1}_{n}$ and $y^{i-1}$ are the position in $x$ and $y$ axes at $(i-1)$th
iteration, respectively. (13) is a linear convex function that can be solved
by Matlab CVX [33]. To enforce only subarray movements,
$\\{\epsilon^{i}_{n},\beta^{i}_{n}\\}$s are equal for all antennas in a
subarray. Finally, the optimization can be written as:

$\displaystyle\min_{E^{i},B^{i}}$ $\displaystyle\gamma^{i}$ (14) s.t.
$\displaystyle\max(|f_{\zeta}^{i}(\\{\mathbf{\tilde{u},\tilde{v}}\\}_{{\mathbf{SL}}})|)\leq\gamma^{i},$
$\displaystyle\text{real}(f_{\zeta}^{i}(0,0))=1,$
$\displaystyle|\epsilon^{i}_{n}|\leq\mu,|\beta^{i}_{n}|\leq\mu,\hskip
19.91684pt\forall n\in\\{1,2,...,N\\}$

where $E^{i}=\\{\epsilon^{i}_{1},\epsilon^{i}_{2},...,\epsilon^{i}_{N}\\}$ and
$B^{i}=\\{\beta^{i}_{1},\beta^{i}_{2},...,\beta^{i}_{N}\\}$.
$\\{\mathbf{\tilde{u},\tilde{v}}\\}_{{\mathbf{SL}}}$ is the set of all side
lobe angles $(\tilde{u},\tilde{v})_{\text{SL}}$ and $\mu$ is a predetermined
number. As the elements are domino tiles, there should be a distance of at
least $\lambda$ between the phase center of each domino at one subarray with
the phase centers of other dominoes at adjacent subarrays for fabrication
possibility. This could also be added as a constraint to (14) as discussed in
[32].

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# A Simple Proof of Bohr’s Inequality

Vern I. Paulsen and Dinesh Singh

######  ABSTRACT.

The classical inequality of Bohr concerning Taylor coeficients of bounded
holomorphic functions on the unit disk, has proved to be of significance in
answering in the negative the conjecture that if the non-unital von Neumann
inequality held for a Banach algebra then it was necessarily an operator
algebra. Here we provide a rather short and easy proof of the inequality.

##  1 Introduction

11footnotetext: 2000 _Mathematics Subject Classification_ Primary 46J10,
46J15; Secondary 30B1022footnotetext: _Keywords and Phrases_ Bohr’s inequality

Harald Bohr in 1914 obtained the following inequality in [4]:

######  Theorem.

Let $f$ be in the disk algebra $A$ with Taylor expansion
$f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$. Then
$\sum_{n=0}^{\infty}|a_{n}|r^{n}\leq\|f\|_{\infty}$ for $0\leq
r\leq\frac{1}{3}$ and $\frac{1}{3}$ is the best possible constant.

Here $z=re^{i\theta}$, and $A$ is the Banach algebra of holomorphic functions
on the open unit disk that have continuous boundary values and
$\|f\|_{\infty}=sup\\{|f(z)|:~{}|z|\leq 1\\}$. Actually, Bohr obtained the
inequality for $0\leq r\leq\frac{1}{6}$. Wiener, M. Riesz and Schur gave
independent proofs and each established that $\frac{1}{3}$ is the best
possible constant. See [1], [2], [3] and [5]. Other proofs were also given by
Sidon [10] and by Tomic [11].

Operator algebraists began to take an interest in Bohr’s inequality when Dixon
[5] used it to settle in the negative the conjecture that if the non-unital
von Neumann inequality held for a Banach algebra then it was necessarily an
operator algebra.

In [7], we (along with Popescu) also established, using Bohr’s inequality,
that every Banach algebra has an equivalent norm that satisfies the non-unital
von Neumann inequality thus providing large classes of Banach algebras that
are not operator algebras and yet they satisfy the non-unital von Neumann
inequality. In the same paper, Paulsen-Popescu-Singh also established versions
of Bohr’s inequality in the context of higher dimensional domains, for the non
commutative disk algebra as well as for the reduced and full group
$C^{*}-$algebras of the free group on $n$ generators. In [8] we provide a very
general version of Bohr’s inequality in the context of uniform algebras from
which the classical inequality follows easily. In [9] we have extended Bohr’s
inequality to operator valued holomorphic functions as well as to the case of
holomorphic functions on an annulus. For further connections see Aizenberg
[1], Boas and Khavinson [2] and Boas [3].

##  2 Preliminary observations

In this note we provide a rather short and easy proof of Bohr’s inequality
using the well known property that the unit ball of the disk algebra $A$ is
the closed convex hull of the set of finite Blaschke products, see [6, cor
2.4, page 196].

A finite Blaschke product is a function of the form

$\prod_{k=1}^{n}\dfrac{z-z_{k}}{1-\bar{z}_{k}z}$

where $|z_{k}|<1$ for each $k$. We also need the following two easy and
elementary observations: Setting
$M(f,r)=\sum\limits_{n=0}^{\infty}|a_{n}|r^{n}$ where
$f(z)=\sum\limits_{n=0}^{\infty}a_{n}z^{n}$, we see easily that

  1. 1.

$M(\alpha f+\beta g,r)\leq|\alpha|M(f,r)+|\beta|M(g,r)$

  2. 2.

$M(fg,r)\leq M(f,r)M(g,r)$ $(0\leq r<1)$

##  3 Proof of the Theorem

We assume with no loss of generality that $\|f\|_{\infty}=1$. Next, it is
elementary to check that the inequality is true for a single Blaschke factor,
$\varphi_{a}(z)=(z-a)/(1-\bar{a}z)=-a+\sum_{n=1}^{\infty}\bar{a}^{n-1}(1-a\bar{a})z^{n}.$
Indeed,

$M(\varphi_{a},r)=\frac{|a|+(1-2|a|^{2})r}{1-|a|r}\leq 1,$

for $0\leq r\leq\frac{1}{3}$ and $|a|<1$, as a little calculus exercise shows.
Next we note that if $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$ and
$g(z)=\sum_{n=0}^{\infty}b_{n}z^{n}$ are in $A$ then
$||a_{n}|-|b_{n}||\leq|a_{n}-b_{n}|=\left|\frac{1}{2\pi}\int_{0}^{2\pi}(f-g)e^{-in\theta}d\theta\right|\leq\|f-g\|_{\infty}$.
Thus for $r\leq\frac{1}{3}$, $|M(f,r)-M(g,r)|\leq\frac{3}{2}\|f-g\|_{\infty}$.
So in view of this inequality and by the norm density of the convex
combinations of finite Blaschke factors in the unit ball of $A$ it is enough
to establish Bohr’s inequality for any convex combination of finite Blaschke
products. However, by observation (1) above, it is enough to establish the
inequality for a finite Blaschke product. But then again by observation (2),
it suffices to establish Bohr’s inequality for a single Blaschke product,
which we have already done. Hence the proof is complete.

To see that $\frac{1}{3}$ is the best possible constant, verify that for
$r>\frac{1}{3},$ the maximum of $M(\varphi_{a},r)$ is attained for some
$|a|<1$ and is strictly greater than $1.$ This is most easily seen by noting
that the above formula is equal to $1$ at $|a|=1$ and is a decreasing function
of $|a|$ at $1.$

The second author wishes to thank the Department of Mathematics, University of
Houston and the Mathematical Sciences Foundation, Delhi, for their support.

## References

  * [1] L. Aizenberg, _Multidimensional analouges of Bohr’s theorem on power series_ , Proc. Amer. Math. Soc. 128(2000), 1147-1155.
  * [2] H. P. Boas and D. Khavinson, _Bohr’s power series theorem in several variables_ , Proc. Amer. Math. Soc. 125(1997), 2975-2979.
  * [3] H. P. Boas, _Majorant Series_ , J. Korean Math. Soc. 2(37)2000, 321-337.
  * [4] H. Bohr, _A theorem concerning power series_ , Proc. London Math. Soc. 2(13)1914, 1-5.
  * [5] P. G. Dixon, _Banach algebra satisfying the non-unital von Neumann inequality_ , Bull. London Math. Soc. (27)1995, 359-362.
  * [6] J. B. Garnett, _Bounded Analytic Functions_ , Academic Press, 1981.
  * [7] V. I. Paulsen, G. Popascu and D. Singh, _On Bohr’s inequality_ , Proc. London Math. Soc. 3(85)2002, 493-512.
  * [8] V. I. Paulsen and D. Singh, _Bohr’s inequality for uniform algebras_ , Proc. Amer. Math. Soc. 132(2004), 3577-3579.
  * [9] V. I. Paulsen and D. Singh, _Extensions of Bohr’s inequality_ , Bull. London Math. Soc. (to appear).
  * [10] S. Sidon, _Uber einen satz von Herrn Bohr_ , Math. Zeit. (26)1927, 731-732.
  * [11] M. Tomic, _Sur un theoreme de H. Bohr_ , Math. Scand. (11)1962, 103-106.

Department of Pure Mathematics and Institute for Quantum Computing, University
of Waterloo, Waterloo N2L 3G1
_E-mail address_<EMAIL_ADDRESS>
Center for Lateral Innovation, Creativity and Knowledge, SGT University,
Gurugram, Haryana 122001
_E-mail address_<EMAIL_ADDRESS>


# Information theory and Electron Spin Resonance dating

C. Tannous1∗ and J. Gieraltowski2
1 Université de Brest, Lab-STICC, CNRS-UMR 6285, F-29200 Brest, FRANCE
2Laboratoire Géosciences-Océan (LGO), CNRS-UMR 6538, 29280 Plouzané, FRANCE

∗To whom correspondence should be addressed; E-mail<EMAIL_ADDRESS>
> Chronometric dating is becoming increasingly important in areas such as the
> Origin and evolution of Life on Earth and other planets, Origin and
> evolution of the Earth and the Solar System… Electron Spin Resonance (ESR)
> dating is based on exploiting effects of contamination by chemicals or
> ionizing radiation, on ancient matter through its absorption spectrum and
> lineshape. Interpreting absorption spectra as probability density functions
> (pdf), we use the notion of Information Theory (IT) distance allowing us to
> position the measured lineshape with respect to standard limiting pdf’s
> (Lorentzian and Gaussian). This paves the way to perform dating when several
> interaction patterns between unpaired spins are present in geologic,
> planetary, meteorite or asteroid matter namely classical-dipolar (for
> ancient times) and quantum-exchange-coupled (for recent times). In addition,
> accurate bounds to age are provided by IT from the evaluation of distances
> with respect to the Lorentz and Gauss distributions. Dating arbitrary
> periods of times (?) and exploiting IT to introduce rigorous and accurate
> date values might have interesting far reaching implications not only in
> Geophysics, Geochronology (?), Planetary Science but also in Mineralogy,
> Archaeology, Biology, Anthropology (?), Paleoanthropology (?, ?)…

## Introduction

ESR (or EPR for Electron Paramagnetic Resonance) absorption spectroscopy is a
non-interfering versatile technique that allows to explore interaction between
unpaired spins and an applied magnetic field in condensed matter (?). These
unpaired spins pertain to electrons in general or electrons as well as holes
in semiconducting materials. CW-ESR (Continuous wave) method measures
concentrations of paramagnetic centers and free radicals by shining a sample
with microwaves at a fixed frequency while simultaneously sweeping the
magnetic field. Pulsed-ESR, CW-ESR successor allowed to reduce measurement
time, increase sensitivity and resolution, better separate different
interactions and detect different types of spin relaxation mechanisms (?).

ESR provides precious information about local structures and dynamic processes
of the paramagnetic centers within the sample under study.

Different frequencies are used such as S band (3.5 GHz), X band (9.25 GHz), K
band (20 GHz), Q band (35 GHz) and W band (95 GHz). Each frequency has its own
advantages and drawbacks and increasing it might increase its sensitivity. The
latter is the minimal concentration of unpaired spins ESR can detect. For
instance, going from the X to W band may result into a 30,000 enhancement in
sensitivity. Since X band static sensitivity is around 1012 spins/cm3 this
means the detection limit is decreased to 3.33 $\times$ 107 spins/cm3 (?).
Other means for improving static and dynamic sensitivity (in
spins/Gauss/$\sqrt{\rm Hz}$) involve reducing temperature or miniaturizing
experimental parts such as the resonant cavity or even employing cQED (circuit
Quantum Electrodynamics) devices such as Josephson junctions, and SQUIDS,
potentially reaching single-spin sensitivity by using high quality factor
superconducting micro-resonators along with Josephson Parametric Amplifiers
(?).

Importance of interest in ESR absolute dating capabilities stemmed from the
fact ionizing radiation ($\alpha,\beta$ and $\gamma$) creates unpaired spins
that might have extremely long lifetimes in certain materials (?). Ionizing
radiation occurs since rocks, sediments 111Earth surface processes are best
studied with optically stimulated luminescence (OSL) of sediments (?). With
OSL one can date deposits aged from one year to several hundred thousand
years., minerals and deposits contain radioactive elements, such as potassium
and isotopes produced by the 238U (Uranium), 235U (Actinium) and 232Th
(Thorium) decay series.

Accurate dosimetry estimation is needed for radiometric dating in order to
estimate how much any sample was exposed to ionizing radiation (or by
extension to other processes) tying the dose absorbed by the sample to its
age. More precisely, the ESR signal intensity is proportional to the paleodose
$D$ (total radiation dose) given by $D=\int_{0}^{T}D_{R}(t)\,dt$ where
$D_{R}(t)$ is the natural dose rate (in Grays/year) and $T$ the exposure time
or estimated age (?, ?).

Dating is important in geochronology and in Planetary Science since asteroids
and meteorites contain organic materials that might provide important clues
for the sources of Life on Earth (?) and the formation of the Solar System.

It is to important to realize that ESR spectroscopy can handle radiometric
dating as well as chemical dating that delivers information about various
chemical processes, a mineral has been subjected to as in reference (?).

While several other dating methodologies exist (see for instance Geyh and
Schleicher (?) or Ikeya’s book (?)), the interest in ESR dating grew
considerably after realizing its wide time range since it spans from a few
thousand years to several million and even billions years which is far beyond
capabilities of 14C radio-carbon dating (?) (limited to about 50,000 years)
for instance and allows to study geochronology since the formation of the
Earth (about 4.5 billion years) until present time.

Historically Zeller (?) suggested for the first time in 1968 the use of ESR
for dating geological materials. In 1975 Ikeya (?) was able to successfully
date stalactite in Japanese caves with ESR and in 1978 Robins (?) was even
capable of identifying ancient heat treatment on flint artefacts with ESR.

More recently, Bourbin et al. (?) and Gourier et al. (?) introduced a
statistical approach based on estimating an average area separating any given
lineshape and the limiting Lorentz (or Breit-Wigner). This measure is a
statistical correlation factor called $R_{10}$ that we show has several
drawbacks and limitations. In sharp contrast, Information Theory (IT) provides
distances called divergence measures that are able of tackling most of the
cases a simple $R_{10}$ statistical correlation factor is unable to approach.
Moreover IT provides accurate bounds to age from the evaluation of distances
with respect to the limiting Lorentz and Gauss distributions.

This work is organized as follows: After describing ESR spectroscopy and spin
interactions classified as dipolar or exchange, we discuss the ESR lineshapes
arising in both situations then move on to the dating procedure based on the
evaluation of a statistical correlation factor $R_{10}$. The latter is unable
to describe several important cases (quantum exchange spin interaction or
Gaussian pdf..). Thus we move on to introducing powerful IT tools based on
evaluating distances (also called divergence measures or relative information
entropy) between different existing pdf’s. The latter originate from the ESR
lineshape by integrating it with respect to the magnetic field. Afterwards we
compare these IT results to $R_{10}$ values when available. We do not address
dosimetry procedures since this is beyond the goal of this work assuming that
dosimetry has been tackled properly by other works.

## Free Induction Decay function as a Stretched exponential

Free Induction Decay function $S(t)$ is a spin-spin correlation function
yielding time-dependent interactions between magnetic moments carried by
electronic spins in a material.

Since the ESR lineshape is the magnetic field derivative of $S(t)$ Fourier
Transform (FT) (as explained in Supplementary Material) it is capable of of
revealing these correlations.

In general, a spin system is expected to interact in two distinct ways:

  1. 1.

When electrons are far apart i.e. $r\gg$ Angström (i.e. several hundred
Angströms, microns and beyond) they can be considered as magnetic moments
interacting in a magnetostatic fashion as $\frac{1}{r^{3}}$. This dipolar
interaction (?) is considered as classical.

  2. 2.

Quantum exchange (?) if spins are very close (typical nearest neighbor
distance about a few Angströms). When electrons are close, overlap of their
wavefunctions leads to short-range Slater interaction of the form
$\exp(-r/r_{0})$ with $r$ the average inter-spin distance and $r_{0}$ on the
order of an Angström.

Actually there is a third mixed regime called DE (Dipolar-Exchange) when
inter-spin distances are intermediary between the Angström and large distances
as in the classical regime. This regime (?) which is beyond our scope is
complex since it is a mixture of classical and quantum types.

Figure 1: (Color on-line) ESR lineshape key parameters: $B_{res}$ resonance
field, $A_{pp}$ peak-to-peak amplitude and corresponding linewidth $\Delta
B_{pp}$.

In ESR or other resonance methods such as ferromagnetic (FMR) or nuclear
magnetic resonance (NMR), there are, in general, two spin relaxation times:
longitudinal $T_{1}$ (along applied external magnetic field) and transverse
$T_{2}$ (perpendicular to applied external magnetic field).

In the dipolar (?) case, Fel’dman and Lacelle (?) have shown that spin
correlation function is given by:

$S(t)=S(0)\exp[(-t/T_{2})^{\beta(D)}]$ (1)

with $T_{2}$ the transverse relaxation time, a measure of spin density in the
sample and $\beta(D)=\frac{D}{3}$ with $D$ the dimension of geometrical spin
arrangement (3 for full spatial, 2 for layers or thin films and 1 for spin
chains).

In the $D=3$ dipolar case, $\beta=1$ and $S(t)=S(0)\exp[(-t/T_{2})]$ whose FT
is a Lorentzian
$\hat{S}(\Delta\omega)=\frac{1}{\pi}\frac{T_{2}}{[1+(\Delta\omega
T_{2})^{2})]}$ where $\Delta\omega$ is frequency difference with respect to
resonance frequency.

In ESR spectroscopy, the Lorentzian and the Gaussian absorption curves are
considered as limiting absorption curves whose derivatives with respect to the
applied field yields the ESR lineshape.

A Lorentzian appears when we have homogeneous broadening whereas a Gaussian
results (in a solid) from thermal fluctuations of atomic/ionic constituents
causing changes in the local magnetic field (cf Jonas review (?)).

At lower dimensionality $D=1,2$ the rational exponent $\frac{D}{3}$ leads to a
”stretched exponential”, ”stretched Gaussian” or even ”stretched Lorentzian”
(cf fig 2) dependence 222One might be tempted to believe that since a
Lorentzian is given by $\frac{1}{\pi}\frac{1}{[1+x^{2}]}$, then a stretched
Lorentzian would be $\frac{1}{[1+x^{\alpha}]}$ with $\alpha~{}\neq~{}2$. In
fact, a stretched Lorentzian curve has several meanings: one might have a
superposition of Lorentzians or more complicated functions (possibly
containing terms such as $\frac{1}{[1+x^{\alpha}]}$). Generally, any function
whose width is larger than the Lorentzian is considered stretched..

Thus ”stretched” refers to a spatial arrangement of spins in planes or layers
($D=2$) or along linear chains ($D=1$).

To summarize, $\beta=1$ yields a pure exponential with Lorentzian FT, whereas
when $\beta=2$ we obtain a Gaussian with a Gaussian FT and for intermediate
values $1<\beta<2$, we recover the stretched curve varieties.

Performing the FT (as detailed in Supplementary Material) using
$S(t)=S(0)\exp[(-t/T_{2})^{\frac{D}{3}}]$ and $D=1,2,3,6$ to recover the
stretched Lorentzian ($D=1,2$), the Lorentzian ($D=3$) and finally the
Gaussian ($D=6$). In addition, one has to determine from the lineshape
$A_{pp}$ and $\Delta B_{pp}$ that depend on $D,T_{2},a$ (cf. fig. 1).

ESR lineshape results displayed in fig. 2 show that we evolve from Gaussian to
Lorentzian ($D=3$), to stretched Lorentzian ($D=2$) and finally stretched
Lorentzian ($D=1$). Comparing to the quantum or ”spin diffusion” case (fig. 3)
based on spin correlation function $S(t)=S(0)\exp[(-t/T_{2})^{\beta^{*}(D)}]$
with $\beta^{*}(D)$ function of spin arrangement dimension $D$.
$\beta^{*}(1)=3/2$ in 1D (see Dietz et al. (?)) and $\beta^{*}(2)=1$ in 2D (a
simplified version of the very complex 2D case (see Richards et al. (?)).

Classical dipolar interactions lead to spectrum broadening (?) i.e. damping.
In sharp contrast, quantum exchange among spins tends to reduce the ESR
linewidth (”Exchange Narrowing” effect) and consequently, damping.

Figure 2: (Color on-line) Illustration of dipolar broadening with ESR
lineshapes corresponding to spin correlation function
$S(t)=S(0)\exp[(-t/T_{2})^{\frac{D}{3}}]$ for a fixed value of parameter
$A_{pp}$. The Gaussian profile is the narrowest then we progress toward the
broader Lorentzian ($D=3$), stretched Lorentzian ($D=2$) and finally the
($D=1$) stretched Lorentzian. Note: Str means Stretched. Figure 3: (Color on-
line) Illustration of exchange narrowing with ESR lineshapes derived from
$S(t)=S(0)\exp[(-t/T_{2})^{\beta^{*}(D)}]$ with $\beta^{*}(D)$ function of
dimension $D$. Lineshapes evolve to broader (Stretched) from narrowest
(Gaussian) as dimensionality increases ($D=1,2,3$) in sharp contrast with
respect to the dipolar case (fig. 2). When $D=3$ the Lorentzian is recovered
as in the dipolar case.

## Dating with $R_{10}$ a statistical correlation measure

Characterizing the ESR lineshape around resonance, Bourbin et al. (?) and
Gourier et al. (?) introduced a statistical method based on an average area
estimation between any given lineshape and the 3D Lorentzian.

They introduced after ref. (?, ?) functional transformations on $B$ and
function $F(B)$, the derivative with respect to magnetic field of the $S(t)$)
FT in order to determine age from ESR lineshape from some universal behavior.

The double scaling transformation (cf. Supplementary Material) results in a
scaled magnetic field $B\rightarrow x$ and a scaled function $F(B)\rightarrow
f(x)$ such that ”Lorentzian derivative” $F(B)$ corresponds to linear function
$f_{L}(x)=x+3/4$ whereas ”Gaussian derivative” corresponds to
$f_{G}(x)=\exp(x-1/4)$. Then a correlation factor $R_{10}$ is extracted from:

$R_{10}=\frac{1}{10}\int_{0}^{10}[f(x)-f_{L}(x)]dx$ (2)

meaning that the area estimation is some kind of distance measure separating
the lineshape from the Lorentzian. The value 10 is the largest statistically
estimated $x$ value.

When $R_{10}>0$ we have close-neighbor spins regime (quantum exchange
interaction) with a mixed Gaussian-Lorentzian profile.

Finally, when $R_{10}<0$ we have distant spins (dipolar regime) in lower
dimension ($D=1,2$) with a stretched Lorentzian lineshape.

Figure 4: (Color on-line) $f(x)$ functions corresponding to the different
$F(B)$. ”Lorentzian derivative” corresponds to $f_{L}(x)=x+3/4$ whereas ”
Gaussian derivative” corresponds to $f_{G}(x)=\exp(x-1/4)$. Our results agree
with Bourbin et al. (?) numerical calculations. Figure 5: (Color on-line) In
the quantum case $D=1,2$ curves lie between the Lorentzian and the Gaussian
and our results agree with experimental data of Dietz et al. (?) in 1D for
(CH3)4NMnCl3 as well as with Richards et al. (?) in 2D for K2MnF4. Note: Num
stands for numerical and Exp for experimental.

In the dipolar case, $x,f(x)$ diagrams (cf fig. 4) serve to identify
$R_{10}>0$ region between Gaussian and Lorentzian whereas $R_{10}<0$ is below
the Lorentzian.

In sharp contrast, $x,f(x)$ diagrams in the exchange case for $D=1,2$ (region
$R_{10}>0$) are between the Lorentzian and the Gaussian (cf fig. 5). Thus one
has to distinguish the classical dipolar from the quantum exchange case when
encountering $R_{10}>0$.

Nevertheless, obtaining $R_{10}$ from ESR lineshape is interesting since it
allows determination of the nature of spin interactions as quantum or
classical (from $S(t)$) and their geometrical arrangement from the $D$ value
as done by Bourbin et al. (?) and Gourier et al. (?) who considered only the
classical dipolar case $D=1,2$, thus the need to examine the quantum picture
(cf fig. 3).

In the intermediate dipolar case (lower dimensionality) $D=1,2$ lines are
under the Lorentzian, whereas in the intermediate exchange (?, ?) case $D=1,2$
lines are between the Gaussian and the Lorentzian.

This is important in order to obtain a reliable and precise dating assessment
of rocks and sediments along the lines of Bourbin et al. (?) and Gourier et
al. (?) who derived two different sets of age formulae from the $R_{10}$
factor.

## Information theory distance measures

Introducing a measure based on Information Theory (?) helps to discriminate
between the classical and quantum pictures.

There are several divergence or distance measures between probability
densities such as the Kullback-Leibler (?, ?), squared Hellinger,
$\alpha$-divergence, Jensen-Shannon, total variation (?, ?)…(see Supplementary
Material).

We choose the Cauchy-Schwarz divergence (CSD) measure (?, ?) defined between
two probability distributions $P,Q$ as:

$\displaystyle D(P||Q)=-\ln\left(\int_{x\in{\mathcal{X}}}dx\;p(x)q(x)\right)+$
$\displaystyle\frac{1}{2}\ln\left(\int_{x\in{\mathcal{X}}}dx\;p^{2}(x)\right)+\frac{1}{2}\ln\left(\int_{x\in{\mathcal{X}}}dx\;q^{2}(x)\right)$
(3)

where ${\mathcal{X}}$ is the set of values taken by continuous probability
densities (pdf) $p(x),q(x),~{}x~{}\in~{}{\mathcal{X}}$.

CSD obeys several axioms of distance (see Supplementary Material) such as
positivity $D(P||Q)\geq 0$, and triangle inequality: $D(P||R)\leq
D(P||Q)+D(Q||R)$. It obeys also symmetry $D(P||Q)=D(Q||P)$ unlike the
Kullback-Leibler (KL) measure (?, ?) defined by:

$D_{KL}(P||Q)=\int_{x\in{\mathcal{X}}}dx\;p(x)\ln\frac{p(x)}{q(x)}$ (4)

Moreover CSD does not suffer from singularities whereas KL does.

The pdf are obtained from the ESR lineshape by integrating it with respect to
the magnetic field (Detailed procedure is described in Supplementary
Material).

In Table 1 we display CSD distances between a unit Lorentzian pdf and a set of
Dipolar and Exchange pdf along with the $R_{10}$ factor and corresponding
ages.

In Table 2 we display CSD distances between a unit Gaussian and a set of
Dipolar and Exchange pdf along with the $R^{*}_{10}$ factor and corresponding
ages that would correspond to formula:

$R^{*}_{10}=\frac{1}{10}\int_{0}^{10}[f(x)-f_{G}(x)]dx,\,f_{G}(x)=\exp(x-1/4)$
(5)

Actually, we did not use the above formula and rather extrapolated the
Lorentzian $R_{10}$ values to estimate the age by modifying Bourbin et al.
coefficient $\alpha_{B}$. In addition we provide an experimental method (in
Supplementary Material) to determine age and evaluate its bounds based on
(CSD) distances $d_{L}$ and $d_{G}$ with respect to unit Lorentzian and
Gaussian distributions labeled as $p(x)$ as well as from $\sigma_{q}$ the
standard deviation of $q(x)$ corresponding to the measured ESR spectrum.
Finally, age is expressed with the formula $10^{A_{L,G}}$ where
$A_{L,G}=d_{L,G}^{\xi}\sigma_{q}^{\eta}$ with exponents $(\xi,\eta)$
determined by optimization (procedure detailed in Supplementary Material).

Lorentzian $P$ | $D(P||Q)$ | $R_{10}$ | Age (Gyr)
---|---|---|---
Dipolar pdf $Q$ |  |  |
$D=1$ Str Lorentz | 0.21 | -2.94 | 3.54
$D=2$ Str Lorentz | 2.50 $\times 10^{-2}$ | -1.94 | 2.74
$D=3$ Lorentz | 1.19 $\times 10^{-6}$ | 0.0 | 1.67
Gauss | 2.55 $\times 10^{-2}$ | 1.71 $\times 10^{3}$ | Undefined
Exchange pdf $Q$ |  |  |
$D=1$ Str Gauss | 1.14 $\times 10^{-2}$ | 6.78 | 0.29
$D=2$ Str Gauss | 4.12 $\times 10^{-3}$ | 2.30 | 0.93
$D=3$ Lorentz | 1.19 $\times 10^{-6}$ | 0.0 | 1.67
Gauss | 2.55 $\times 10^{-2}$ | 1.71 $\times 10^{3}$ | Undefined
Table 1:  Cauchy-Schwarz distances separating a unit width $P$ Lorentzian and $Q$ distributions (Dipolar and Exchange) with corresponding $R_{10}$ factor yielding age in Gyr (Giga or billion years). Age in years is deduced from Bourbin et al. (?) formula $10^{\left(\frac{R_{10}-\beta_{B}}{\alpha_{B}}\right)}$ with $\alpha_{B}=-9.$ and $\beta_{B}=83.$. Note that Skrzypczak et al. (?) define another set of coefficients: $\alpha_{S}=-5.3$ and $\beta_{S}=48.9$ (Other examples are detailed in Supplementary Material). Gaussian $P$ | $D(P||Q)$ | $R^{*}_{10}$ | Age (Myr)
---|---|---|---
Dipolar pdf $Q$ |  |  |
$D=1$ Str Lorentz | 0.25 | -2.94 | 90.59
$D=2$ Str Lorentz | 5.11 $\times 10^{-2}$ | -2.89 | 89.61
$D=3$ Lorentz | 1.51 $\times 10^{-2}$ | 2.73 | 27.09
Gauss | 0.0 | 0.0 | 48.44
Exchange pdf $Q$ |  |  |
$D=1$ Str Gauss | 1.86 $\times 10^{-2}$ | -5.02 $\times 10^{-2}$ | 48.96
$D=2$ Str Gauss | 1.44 $\times 10^{-2}$ | 3.51 | 22.91
$D=3$ Lorentz | 1.51 $\times 10^{-2}$ | 2.73 | 27.09
Gauss | 0.0 | 0.0 | 48.44
Table 2:  Cauchy-Schwarz distances separating a unit standard deviation $P$
Gaussian and $Q$ distributions (Dipolar and Exchange). The corresponding
$R^{*}_{10}$ factor is obtained from extrapolating the above Lorentzian
$R_{10}$ value and the age obtained in Myr (Mega or million years) is deduced
from altering the Bourbin et al. coefficient $\alpha_{B}$ (Other examples are
detailed in Supplementary Material).

In this work we developed an ESR lineshape based chronometric dating
methodology handling both Dipolar (classical) and Exchange (quantum)
interactions between unpaired spins.

It is based on evaluating IT distances (divergence measures) between a
Lorentzian or a Gaussian (analytical) pdf’s and an experimental pdf obtained
from the measured ESR lineshape integrated over the applied magnetic field.

From the ESR detection point of view, ancient material age is extracted with
FT of a spin-correlation function $S(t)$ pertaining to unpaired interacting
spins that might have extremely long lifetimes and originate from
($\alpha,\beta$ and $\gamma$) ionizing radiation.

$S(t)$ is of a stretched exponential form with a characteristic exponent
$\beta(D)$ that depends on unpaired spin arrangement dimension $D$.

In the dipolar interaction case, $\beta(D)=\frac{D}{3}$ according to Fel’dman
et al. (?) whereas in the exchange interaction case, Dietz et al. (?) and
Richards et al. (?) found $\beta^{*}(D=1)=3/2$ and $\beta^{*}(D=2)=1$
respectively (in the ”spin diffusion” case).

Considering aging as an evolutionary IT distance (or relative information
entropy) between pdf’s stemming from ESR lineshapes integrated with respect to
the applied magnetic field brings a major paradigm shift to chronometric
dating.

IT provides an alternate look at dating based on using distances (with respect
to Gaussian, Lorentzian…) as well as arbitrary coupling between spins
(dipolar, exchange…) paving the way to cover any period of time with the
proviso of having performed previously an appropriate dosimetry analysis. IT
approach might be further extended to the intermediary Dipolar-Exchange regime
where classical and quantum interactions are intertwined.

## Acknowledgments

We would like to thank Prof. Hervé Bellon and Dr. Stéfan Lalonde for
suggesting this problem and many enlightening discussions.

Materials and Methods
Information theory effective distance measure
Experimental protocols for dating with IT distance
Figs. S1 to S4
Tables S1 to S2
References (19-26) and (29-35)

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## Supplementary Materials

## Materials and Methods

ESR absorption spectroscopy reveals time-dependent interactions between
electronic spins carrying magnetic moments in a material as reflected in the
correlation function $S(t)$ also called Free Induction Decay function.

The ESR lineshape is the field derivative of $S(t)$ Fourier Transform (FT). An
example is shown in the case of the Lorentzian in fig. S1.

Figure S1: (Color on-line) absorption line (in red) and its field derivative
the ESR lineshape (in green) in the Lorentzian case.

Magnetic resonance at a field $B_{res}$ is such that frequency $\omega=\gamma
B$ where $\gamma$ is gyromagnetic ratio given by
$\gamma=\mu_{0}g_{L}e\hbar/2m_{e}$ ($\mu_{0}$= vacuum permeability, $g_{L}=$
Landé factor, $e,m_{e}=$ charge and electron mass). This implies
$\Delta\omega\propto B-B_{res}$ with $B$ (applied field) and resonance field
$B_{res}$.

When $S(t)=S(0)\exp[(-t/T_{2})]$ its FT is a Lorentzian in $B$ such that:
$\hat{S}(B)=\frac{1}{\pi}\frac{1}{\left[1+\left(\frac{B-B_{res}}{\Delta
B_{pp}}\right)^{2}\right]}$.

In the general case, ESR lineshape is obtained by taking the derivative with
respect to applied field $B$) of $\hat{S}(B)$ such that
$F(B)=\frac{d\hat{S}(B)}{dB}$. Note that $\hat{S}(B)$ is often denoted as the
ESR power absorption, the imaginary part of the generalized susceptibility
$\chi"(B)$.

$S(t)$ FT is written as $\hat{S}(\omega)=\int_{0}^{+\infty}e^{-i\omega
t}S(t)dt$.

In ESR, $\omega$ is proportional to the applied magnetic field allowing us to
replace $\omega$ by $B-B_{res}$ resulting in
$\hat{S}(B-B_{res})=\int_{0}^{+\infty}e^{-ia(B-B_{res})t}S(t)dt$ with
$a=g_{L}\mu_{B}/\hbar$, $g_{L}$ the Landé factor, $\mu_{B}$ Bohr magneton and
$\hbar$ Planck constant.

ESR lineshape $F(B)$ is obtained from the derivative with respect to $B$ of
$\hat{S}(B-B_{res})$ yielding:

$\displaystyle F(B)$ $\displaystyle=$ $\displaystyle
Re\left[\int_{0}^{+\infty}\frac{d}{dB}e^{-ia(B-B_{res})t}S(t)dt\right]$ (S1)
$\displaystyle=$ $\displaystyle-a\int_{0}^{+\infty}t\sin[a(B-B_{res})t]S(t)dt$

Hence it suffices to evaluate the above integral using
$S(t)=S(0)\exp[(-t/T_{2})^{\frac{D}{3}}]$ and $D=1,2,3,6$ to recover the
stretched Lorentzian ($D=1,2$), the Lorentzian ($D=3$) and finally the
Gaussian ($D=6$).

Integrating requires special methods to reduce errors resulting from large
number of oscillations emanating from the term $\sin[a(B-B_{res})t]$. We have
used a number of numerical algorithms (?) in order to obtain accurate and
reliable results (?, ?, ?).

In order to characterize the ESR lineshape around resonance, Bourbin et al.
(?) and Gourier et al. (?) introduced a statistical method based on an average
area estimation between any given lineshape and the 3D Lorentzian.

They introduced after ref. (?, ?) a double scaling transformation acting on
the magnetic field $B$ and the ESR lineshape $F(B)$ such that:

$\displaystyle x$ $\displaystyle=$ $\displaystyle\left(\frac{B-B_{res}}{\Delta
B_{pp}}\right)^{2},$ $\displaystyle y$ $\displaystyle=$ $\displaystyle
f(x)=\sqrt{\frac{A_{pp}}{|F(B-B_{res})|}\frac{|B-B_{res}|}{\Delta B_{pp}}}$
(S2)

For example, ”Lorentzian derivative” $F(B)$ corresponds to linear function
$f_{L}(x)=x+3/4$ whereas ”Gaussian derivative” corresponds to
$f_{G}(x)=\exp(x-1/4)$.

Dating is based on a statistical correlation factor $R_{x_{m}}$ extracted from
the integral:

$R_{x_{m}}=\frac{1}{x_{m}}\int_{0}^{x_{m}}[f(x)-f_{L}(x)]dx$ (S3)

where $x_{m}$ is some assumed large $x$ value such that the area estimated
lying between $f(x)$ and $f_{L}(x)]$ is a distance measure separating the ESR
lineshape from the Lorentzian.

Note that it is possible to determine another statistical correlation factor
$R^{*}_{x_{m}}$ extracted from the integral:

$R^{*}_{x_{m}}=\frac{1}{x_{m}}\int_{0}^{x_{m}}[f(x)-f_{G}(x)]dx$ (S4)

where the area value indicates how much the ESR lineshape differs from the
Gaussian.

### Information theory effective distance measure

Information Theory (IT) provides an effective distance (?) or a divergence
measure separating two functions (in this case probability density functions
or pdf) to enable their comparison not only from the geometrical shape point
of view but also from the information content as well.

Going from a distance connecting two points to one linking two functions helps
determine how much the functions are dissimilar in several ways (analytical,
geometrical and informational). That is why $D(P||Q)$ is also called relative
information entropy that does not behave as a metric but rather as the square
of the Euclidian one (?).

Several examples of $D(P||Q)$ are displayed in Table S1.

Distance $D(P||Q)$ | Definition
---|---
Squared Hellinger | $\int_{x\in{\mathcal{X}}}dx\;(\sqrt{p(x)}-\sqrt{q(x)})^{2}$
Squared triangular | $\int_{x\in{\mathcal{X}}}dx\;\frac{(q(x)-p(x))^{2}}{p(x)+q(x)}$
Squared perimeter | $\int_{x\in{\mathcal{X}}}dx\;\sqrt{p^{2}(x)+q^{2}(x)}-\sqrt{2}$
$\alpha$-divergence | $\frac{4}{1-\alpha^{2}}\left(1-\int_{x\in{\mathcal{X}}}dx\;p^{\frac{1-\alpha}{2}}(x)q^{1+\alpha}(x)\right)$
Jensen-Shannon | $\frac{1}{2}\int_{x\in{\mathcal{X}}}dx\;\left(p(x)\ln\frac{2p(x)}{p(x)+q(x)}+q(x)\ln\frac{2q(x)}{p(x)+q(x)}\right)$
Pearson $\chi^{2}_{P}$ | $\int_{x\in{\mathcal{X}}}dx\;\frac{(q(x)-p(x))^{2}}{p(x)}$
Neyman $\chi^{2}_{N}$ | $\int_{x\in{\mathcal{X}}}dx\;\frac{(q(x)-p(x))^{2}}{q(x)}$
Total variation (metric) | $\frac{1}{2}\int_{x\in{\mathcal{X}}}dx\;|p(x)-q(x)|$
Table S1:  Examples of Information Theory distances between two pdf
$p(x),q(x)$. Note that $\alpha$-divergence, Pearson $\chi^{2}_{P}$ and Neyman
$\chi^{2}_{N}$ do not satisfy the axiomatic symmetry requirement of a distance
$D(P||Q)=D(Q||P)$.

The Cauchy-Schwarz distance (CSD) measure (?, ?) is convenient since it obeys
the axioms of distance and is free of singularities (see main article).

ESR absorption spectra when interpreted as a pdf (see fig. S1 for the
Lorentzian case), is considered as modified by aging due to time-dependent
interactions between electronic spins in an archaeological material.

The time correlation function $S(t)$ (Free Induction Decay function) yields
the ESR lineshape $F(B)$ since it is the field derivative of its Fourier
Transform (FT).

Thus an IT distance could be used as a measure of aging induced by ionizing
radiation and this supplement details and illustrates the procedure for
carrying the dating method.

### Experimental protocols for dating with IT distance

Experimentally, two procedures are possible:

  1. 1.

Direct treatment of the absorption spectra considered as a pdf and evaluating
the IT distance with respect to a given reference (Lorentzian, Gaussian…).

  2. 2.

Extraction of the absorption spectra from the measured ESR lineshape $F(B)$ by
integrating it with respect to the applied field $B$ and afterwards evaluating
the IT distance.

The extraction of the absorption spectrum from ESR lineshape $F(B)$ entails
several steps:

  1. 1.

Spline interpolation of the ESR lineshape $F(B)$.

  2. 2.

Symmetrization of the ESR lineshape using shifting by $\delta$ such that $\int
F(B)dB=\delta\int dB$.

  3. 3.

Field integration of the symmetrized ESR lineshape to get the absorption
spectrum.

  4. 4.

Filtering (smoothing) of the absorption spectrum.

  5. 5.

Normalization of the absorption spectrum and transformation into a pdf $p(x)$
with $x=\left(\frac{B-B_{res}}{\Delta B_{pp}}\right)$.

After transformation of the ESR lineshape into a pdf, the evaluation of the
CSD measure (?, ?) can be undertaken to determine the age from the evaluation
of the CSD measure taken between the different probability distributions (?,
?, ?) corresponding to the ESR spectrum and the Lorentzian or Gaussian
considered as limiting distributions.

Age is determined from (CSD) distances $d_{L}$ and $d_{G}$ with respect to
unit Lorentzian and Gaussian distributions labeled as $p(x)$ as well as from
$\sigma_{q}$ the standard deviation of $q(x)$ corresponding to the measured
ESR spectrum. Age is evaluated with the formula $10^{A_{L,G}}$ where
$A_{L,G}=d_{L,G}^{\xi}\sigma_{q}^{\eta}$ for the Gyr (Giga or billion year),
Myr (Mega or million year) cases whereas it is given by
$A_{L,G}=d_{L,G}^{\zeta}\sigma_{q}^{\eta}$ for the Kyr (thousand year) cases
with $d=d_{L}$ in the Lorentz or $d=d_{G}$ in the Gaussian case. Exponents
$\xi$=-1.28, $\eta=2.74\times 10^{-2}$, $\zeta=$-0.32 are determined by
optimization (see Table S2).

Sample | $d_{L}$ | $d_{G}$ | $\sigma_{q}$ | Lorentz Age | Gauss Age
---|---|---|---|---|---
Gunflint (?) | 0.18 | 0.23 | 8.17 | 2.34 Gyr | 8.91 $\times 10^{-3}$ Gyr
B4 (?) | 0.21 | 0.21 | 0.77 | 8.51 Myr | 11.78 Myr
Enamel (?) | 1.49 $\times 10^{-2}$ | 1.17 $\times 10^{-2}$ | 1.60 | 7.29 Kyr | 14.47 Kyr
Table S2:  Age is evaluated with the formula $10^{A_{L,G}}$ where
$A_{L,G}=d_{L,G}^{\xi}\sigma_{q}^{\eta}$ for the Gyr, Myr cases whereas it is
given by $A_{L,G}=d_{L,G}^{\zeta}\sigma_{q}^{\eta}$ for the Kyr cases with
$d=d_{L}$ for the Lorentz case or $d=d_{G}$ for the Gaussian case.

We illustrate the Information Theory dating procedure with a first example
pertaining to Gunflint from Schreiber beach locality (Port Arthur Homocline)
in Ontario Canada.

Figure S2: (Color on-line) Absorption line obtained from the ESR lineshape
Gunflint sample taken from Skrzypczak PhD thesis (?) fig.II.25

After extracting the absorption spectrum from the ESR lineshape, we find the
CSD between Skrzypczak Gunflint (?) $q(x)$ and a unit Lorentzian $p(x)$ as
0.18 whereas it is 0.23 between $q(x)$ and a unit Gaussian $p(x)$.

In order to extract the age based on the $R_{10}$ evaluation as explained in
the main work, we find: $R_{10}$= -1.765 with age=2.62 Gyr. Lorentz and Gauss
CSD ages are 2.34 Gyr and 8.91 $\times 10^{-3}$ Gyr respectively (cf. Table
S2) whereas Skrzypczak-Bonduelle estimated it to be about 1.88 Gyr (Gyr) in
her PhD thesis (?).

Another example is a latosol sediment called B4 drawn from Balan et al. (?)
fig.2 and extracted from 100 cm depth at a site 60 km North of the city of
Manaus capital of Amazonas state in North-Western Brazil.

Figure S3: (Color on-line) Absorption line obtained from the ESR lineshape B4
latosol sediment of Balan et al. (?) fig.2. The negative wing of the pdf is
taken care of by upward shifting of the curve.

The age is determined from the $R_{10}$ evaluation obtained as $R_{10}$=
-0.575 with age=1.932 Gyr Using the extrapolated procedure outlined in the
main text, we get an age= 54.75 Mega-years (Myr). Lorentz and Gauss CSD ages
are 8.51 Myr and 11.78 Myr respectively (cf. Table S2) whereas Balan et al.
(?) estimated it around 23.9 Myr.

The third example is drawn from Jonas (?) who worked extensively on fossil
tooth enamel dating (spanning several hundred thousand years up to 2 Myr).

From the ESR lineshape (calibrated spectrum of fig.7), we extract the
absorption line according to the above protocol with the result displayed in
fig. S4.

Figure S4: (Color on-line) absorption line obtained from the fossil tooth
enamel ESR lineshape of Jonas (?) fig.7. The negative part of the pdf is
treated as in the B4 sediment case.

We find the CSD between a Lorentzian (resp. Gaussian) $p(x)$ and Jonas (?)
based $q(x)$ as 1.49 $\times 10^{-2}$ and 1.17 $\times 10^{-2}$ yielding
respective ages 7.29 Kyr and 14.47 Kyr.

For the sake of comparison, if we use the $R_{10}$= -3.26 value (not to be
used for the tooth enamel era but for ancient carbonaceous matter as in
Bourbin et al. (?)), we get an age of 3.84 Gyr, a value larger than 3.5 Gyr
the maximum accepted age of organic matter. On the other hand, if we use the
interpolation method we get 97.08 Myr which is beyond the 2 Myr limiting age
of fossil tooth enamel.