# Characterizing the Ambiguity Function of Constant-Envelope OFDM Waveforms
††thanks: David G. Felton’s efforts were supported by the Naval Research
Enterprise Internship Program (NREIP) and David A. Hague’s efforts were
supported by the Naval Undersea Warfare Center’s In-House Laboratory
Independent Research (ILIR) program.

David G. Felton1 David A. Hague2 1Radar Systems Lab (RSL), University of
Kansas, Lawrence, KS
2Sensors and Sonar Systems Deptartment, Naval Undersea Warfare Center,
Newport, RI

###### Abstract

This paper investigates the radar Ambiguity Function (AF) properties of
Constant Envelope Orthogonal Frequency Division Multiplexing (CE-OFDM)
waveforms employing Phase-Shift Keying (PSK). The CE-OFDM is in fact a special
case of the Multi-Tone Sinusoidal Frequency Modulated (MTSFM) waveform which
allows for applying many of the same mathematical techniques of the MTSFM
model to the CE-OFDM model. This results in novel compact closed-form
expressions for the spectrum, AF, and Auto-Correlation Function (ACF) of the
CE-OFDM waveform. The mainlobe structure of the CE-OFDM’s AF is characterized
by the Ellipse of Ambigiuity (EOA) model. This produces precise closed-form
expressions for the CE-OFDM’s Root-Mean Square (RMS) bandwidth and the degree
of range-Doppler coupling present in the waveform’s AF mainlobe. These
expressions show that a CE-OFDM waveform employing PSK as the symbol encoding
scheme will possess a fixed RMS bandwidth for fixed modulation index $h$ and
number of sub-carriers $L$. Additionally, we show that the EOA model predicts
that a CE-OFDM waveform employing PSK encoding will almost always possess a
“Thumbtack-Like” AF shape.

###### Index Terms:

CE-OFDM, Waveform Design, Generalized Bessel Functions, Dual-Function
Radar/Communications

##  I Introduction

The Constant-Envelope Orthogonal Frequency Division Multiplexing (CE-OFDM)
waveform is a constant envelope analogue of the standard OFDM waveform model
where OFDM modulation is performed in either the instantaneous phase or
frequency domain [1, 2]. The CE-OFDM waveform encoding is essentially a form
of Frequency Modulation (FM) which guarantees a constant envelope. This
constant envelope property makes these waveforms better suited for
transmission on real-world radar and communications transmitters than standard
OFDM waveforms [3] whose complex envelope can vary substantially [4]. CE-OFDM
waveforms have recently been proposed for use in Dual-Function
Radar/Communication (DFRC) applications [5, 6, 7] as the encoded communication
symbols can also be utilized as a discrete set of parameters that realize
waveforms with desirable Ambiguity Function (AF) properties [8].

The majority of the literature on CE-OFDM waveforms initially focused on its
properties and performance for use as a communications waveform [2, 3, 1, 9,
10]. More recent efforts in the literature have studied CE-OFDM as a potential
radar waveform and have begun to characterize its properties from the radar
waveform design perspective[11, 8]. These efforts have mainly focused on
analyzing the structure of the waveform’s spectrum, AF, and Auto/Cross-
Correlation Functions (ACF/CCF). These first-principle design metrics
fundamentally determine a radar waveform’s performance. However, to the best
of the authors’ knowledge, the properties of the CE-OFDM waveform’s AF has not
been fully characterized. Doing so might provide insight into the appropriate
selection of symbol encoding for DFRC applications.

This paper takes a first step in more fully characterizing the structure of
the CE-OFDM waveform’s AF. We utilize a CE-OFDM waveform model that employs
Phase-Shift Keying (PSK) to encode symbols. We first show that the CE-OFDM
waveform is a special case of the Multi-Tone Sinusoidal FM (MTSFM) waveform,
an adaptive transmit waveform model from the active sonar literature [12, 13].
Leveraging results from the efforts in [12], we derive exact closed-form
expressions for the CE-OFDM waveform’s spectrum, AF, and ACF which to the best
of the authors’ knowledge appear to be novel. We additionally fully
characterize the mainlobe structure of the CE-OFDM waveform’s AF using the
Ellipse of Ambiguity (EOA) model from the well established radar literature
[14, 15]. We show that a CE-OFDM waveform employing PSK essentially always
realizes a “Thumbtack-Like” AF shape. These results also provide an exact
closed-form expression for the Root Mean Square (RMS) bandwidth that unlike
previous approximations [3] is free from any restrictions on the choice of CE-
OFDM waveform parameters. Finally, we show that the metrics describing the CE-
OFDM’s ACF sidelobe structure are multi-modal functions of the waveform’s PSK
symbols. Thus, the search for “optimal” codes that produce desirably low
AF/ACF sidelobes will require either a completely different code construction
scheme from that of Phase-Coded waveforms [4] or direct numerical optimization
methods.

##  II The Structure of the Ambiguity Function

In general, a basebanded FM waveform is expressed in continuous time as

$s\left(t\right)=\frac{\operatorname{rect}\left(t/T\right)}{\sqrt{T}}e^{j\varphi\left(t\right)}$
(1)

where $T$ is the waveform’s duration and $\varphi\left(t\right)$ is its phase
modulation function. The time support is $|t|\leq T/2$, and $1/\sqrt{T}$
normalizes the signal energy to unity. The waveform’s frequency modulation
function $m\left(t\right)$ maps its instantaneous frequency as a function of
time and is expressed as

$m\left(t\right)=\dfrac{1}{2\pi}\dfrac{\partial\varphi(t)}{\partial t}.$ (2)

The AF measures the response of the waveform’s MF to its Doppler shifted
versions and is defined as [15, 14]

$\displaystyle\chi\left(\tau,\nu\right)=\int_{-\infty}^{\infty}s\left(t-\frac{\tau}{2}\right)s^{*}\left(t+\frac{\tau}{2}\right)e^{j2\pi\nu
t}dt$ (3)

where $\nu$ is the Doppler shift. Lastly, the ACF is the zero Doppler cut of
the AF

$\displaystyle
R\left(\tau\right)=\chi\left(\tau,0\right)=\int_{-\infty}^{\infty}s\left(t-\frac{\tau}{2}\right)s^{*}\left(t+\frac{\tau}{2}\right)dt.$
(4)

There are a series of design metrics that characterize a waveform’s AF shape
in terms of its mainlobe and sidelobe structure. These metrics are described
below.

####  II-1 Mainlobe Structure

The AF mainlobe structure determines a waveform’s ability to estimate the
range and Doppler of a target and to resolve multiple targets in range and
Doppler. The AF mainlobe can be approximated by a second order Taylor series
expansion [14, 15]. The EOA is the contour of this AF mainlobe approximation
at some height $\xi$ and is always a coupled ellipse [14]. The EOA for the AF
is expressed as [14, 15, 16]

$1-|\chi\left(\tau,\nu\right)|^{2}=\xi=\beta_{rms}^{2}\tau^{2}+2\rho\tau\nu+\tau_{rms}^{2}\nu^{2}$
(5)

where $\beta_{rms}^{2}$ is the waveform’s RMS bandwidth and determines time-
delay (range) sensitivity, $\tau_{rms}^{2}$ is the RMS pulse-length which
determines Doppler (range-rate) sensitivity, and $\rho$ is the Range-Doppler
Coupling Factor (RDCF) for the AF mainlobe.

The RMS bandwidth is expressed as [16]

$\displaystyle\beta_{rms}^{2}$
$\displaystyle=\int_{\infty}^{\infty}\left(f-f_{0}\right)^{2}|S\left(f\right)|^{2}df$
(6)
$\displaystyle=\int_{\Omega_{t}}|\dot{s}\left(t\right)|^{2}dt-\left|\int_{\Omega_{t}}s\left(t\right)\dot{s}^{*}\left(t\right)dt\right|^{2}$
(7)
$\displaystyle=\dfrac{1}{T}\int_{-T/2}^{T/2}\left[\dot{\varphi}\left(t\right)\right]^{2}dt-\left|\dfrac{1}{T}\int_{-T/2}^{T/2}j\dot{\varphi}\left(t\right)dt\right|^{2}$
(8)

where $f_{0}$ is the waveform’s spectral centroid $\langle f\rangle$,
$S\left(f\right)$ is the waveform’s Fourier transform, $\dot{s}\left(t\right)$
is the first time derivative of the waveform $s\left(t\right)$, $\Omega_{t}$
represents the region of support in time of the waveform, and
$\dot{\varphi}\left(t\right)=2\pi m\left(t\right)$ is the first time
derivative of the waveform’s instantaneous phase. Note that (8) results from
inserting (1) into (7). The RMS pulse-length term is expressed as

$\tau_{rms}^{2}=4\pi^{2}\int_{\Omega_{t}}\left(t-t_{0}\right)^{2}|s\left(t\right)|^{2}dt$
(9)

where $t_{0}$ is the first time moment $\langle t\rangle$ of the the waveform
$s\left(t\right)$ and is zero for waveforms such as (1) that are even-
symmetric in time. The RDCF is expressed as

$\rho=-2\pi\Im\Biggl{\\{}\int_{\Omega_{t}}ts\left(t\right)\dot{s}^{*}\left(t\right)dt\Biggr{\\}}=2\pi\int_{-T/2}^{T/2}t\dot{\varphi}\left(t\right)dt.$
(10)

where $\Im\\{\\}$ denotes the imaginary component of the integral. The second
expression in (10) results from inserting (1) into the first expression in
(10).

For the waveform model in (1), the RMS pulse-length $\tau_{rms}^{2}$ is solely
dependent upon the pulse-length $T$ of the waveform. The RMS bandwidth and
RDCF EOA parameters are solely dependent upon the modulation function (2). If
the waveform’s modulation function is known, $\beta_{rms}^{2}$ and $\rho$ can
be calculated in exact closed form which along with $\tau_{rms}^{2}$ provides
full characterization of the waveform’s AF mainlobe structure [14, 15, 16].
Furthermore, if the waveform’s modulation function is composed of a discrete
set of parameters, those parameters can be chosen to design waveforms with
specific EOA parameters thus directly shaping the AF mainlobe structure [17].

####  II-2 Sidelobe Structure

This paper focuses specifically on the sidelobe structure of the ACF. Two of
the most common metrics are the Peak-to-Sidelobe Level Ratio (PSLR) and the
Integrated Sidelobe Level (ISL). The PSLR is expressed as

$\displaystyle\text{PSLR}=\dfrac{\underset{\Delta\tau\leq|\tau|\leq
T}{\text{max}}\bigl{\\{}\left|R\left(\tau\right)\right|^{2}\bigr{\\}}}{\underset{0\leq|\tau|\leq\Delta\tau}{\text{max}}\bigl{\\{}\left|R\left(\tau\right)\right|^{2}\bigr{\\}}}=\underset{\Delta\tau\leq|\tau|\leq
T}{\text{max}}\bigl{\\{}\left|R\left(\tau\right)\right|^{2}\bigr{\\}}.$ (11)

where $\Delta\tau$ is the null of the ACF mainlobe. Thus, the ACF’s null-to-
null mainlobe width is $2\Delta\tau$. Note that the rightmost expression in
(11) results from the assumption that the waveform is unit energy and thus the
maximum value of $|R\left(\tau\right)|^{2}$ is unity which occurs at $\tau=0$.
The ISL is the ratio of the area under the sidelobe region $A_{\tau}$ of
$|R\left(\tau\right)|^{2}$ to the area under mainlobe region $A_{0}$ of
$|R\left(\tau\right)|^{2}$ expressed as

$\displaystyle\text{ISL}~{}=\dfrac{A_{\tau}}{A_{0}}=\dfrac{\int_{\Delta\tau}^{T}\left|R\left(\tau\right)\right|^{2}d\tau}{\int_{0}^{\Delta\tau}\left|R\left(\tau\right)\right|^{2}d\tau}.$
(12)

Note that the integration is performed only over positive time-delays since
the ACF is even-symmetric in $\tau$. A lower ISL corresponds to an ACF with
lower overall sidelobe levels but does not necessarily translate to a lower
PSLR.

##  III CE-OFDM Waveform Model

This section describes how the CE-OFDM waveform model is derived from the
standard OFDM waveform. Additionally, this section shows how the CE-OFDM
waveform model is closely related to the MTSFM waveform model [12] and shares
many of its mathematical properties.

###  III-A The CE-OFDM Waveform

The CE-OFDM waveform is realized as a phase-modulated waveform whose
instantaneous phase itself takes on the form of an OFDM waveform

$\varphi\left(t\right)=2\pi h\sum_{\ell=-L}^{L}c_{\ell}e^{j\frac{2\pi\ell
t}{T}}$ (13)

where $c_{\ell}$ are the complex valued PSK symbols, $L$ is the number of
unique complex sub-carriers, and $h$ is the waveform’s modulation index which
in conjunction with $L$ controls the bandwidth of the waveform. Converting the
complex Fourier series in (13) to the real-valued Fourier series results in
the modified expression for the CE-OFDM’s phase modulation function

$\varphi\left(t\right)=\dfrac{\tilde{\alpha}_{0}}{2}+2\pi
h\sum_{\ell=1}^{L}\tilde{\alpha}_{\ell}\cos\left(\dfrac{2\pi\ell
t}{T}\right)+\tilde{\beta}_{\ell}\sin\left(\dfrac{2\pi\ell t}{T}\right)$ (14)

where $\tilde{\alpha}_{\ell}$ and $\tilde{\beta}_{\ell}$ are the real Fourier
series coefficients expressed as

$\tilde{\alpha}_{0}=c_{0},~{}\tilde{\alpha}_{\ell}=\dfrac{c_{\ell}+c_{-\ell}}{2},~{}\tilde{\beta}_{\ell}=\dfrac{j\left(c_{\ell}-c_{-\ell}\right)}{2}.$
(15)

Assuming PSK symbols are used in the CE-OFDM model, the real Fourier series
coefficients lie on the unit circle such that
$\sqrt{\tilde{\alpha}_{\ell}^{2}+\tilde{\beta}_{\ell}^{2}}=1$. Lastly,
assuming $\tilde{\alpha}_{0}=0$ and expressing (14) in terms of the amplitude-
phase representation for real-valued Fourier series results in the final form
of the CE-OFDM’s phase modulation function

$\varphi\left(t\right)=2\pi
h\sum_{\ell=1}^{L}{|\Gamma_{\ell}|\cos\left(\dfrac{2\pi\ell
t}{T}+\phi_{\ell}\right)}$ (16)

where
$|\Gamma_{\ell}|=\sqrt{\tilde{\alpha}_{\ell}^{2}+\tilde{\beta}_{\ell}^{2}}=1$
and the symbol phases $\phi_{\ell}$ are expressed as

$\phi_{\ell}=\operatorname{atan}\left\\{\dfrac{\tilde{\beta}_{\ell}}{\tilde{\alpha}_{\ell}}\right\\}.$
(17)

Using (2), the CE-OFDM waveform’s frequency modulation function is expressed
as

$m(t)=-\dfrac{2\pi
h}{T}\sum_{{\ell}=1}^{L}\ell|\Gamma_{\ell}|\sin\left(\dfrac{2\pi\ell
t}{T}+\phi_{\ell}\right).$ (18)

We wish to note here that the CE-OFDM’s instantenous phase (16), which has the
equivalent representation given by (14), is in fact a special case of the
MTSFM waveform model described in [12]. The MTSFM’s instantaneous phase is a
more general version of (14) where $\tilde{\alpha}_{\ell}$ and
$\tilde{\beta}_{\ell}$ are allowed to vary in magnitude rather than strictly
lie on the unit circle. Therefore, many of the same mathematical methods used
to describe the MTSFM waveform are directly applicable to the CE-OFDM waveform
model.

###  III-B Mathematical Properties of the CE-OFDM Waveform’s AF

This section describes a series of mathematical properties that characterize
the CE-OFDM waveform’s AF shape.

####  III-B1 Spectrum, AF, and ACF of the CE-OFDM Waveform

Much like the MTSFM waveform model [12], the CE-OFDM waveform time series can
be expressed in terms of a complex Fourier series. As shown in Appendix A,
exploiting the Jacobi-Anger expansion for Generalized Bessel Functions (GBFs)
[18, 19] results in the following expression for the CE-OFDM waveform’s time-
series

$s\left(t\right)=\frac{\operatorname{rect}\left(t/T\right)}{\sqrt{T}}\times\\\
\sum_{m=-\infty}^{\infty}j^{m}e^{jm\phi_{1}}\mathcal{J}_{m}\left(\left\\{z_{\ell}\right\\}_{\ell=1}^{L};\left\\{\gamma_{\ell}\right\\}_{\ell=2}^{L}\right)e^{j\frac{2\pi
mt}{T}}$ (19)

where
$\mathcal{J}_{m}\left(\left\\{z_{\ell}\right\\}_{\ell=1}^{L};\left\\{\gamma_{\ell}\right\\}_{\ell=2}^{L}\right)$
is the $L$-dimensional, $(L-1)$-parameter GBF with $L$-dimensional variables
$\left\\{z_{\ell}\right\\}_{\ell=1}^{L}=2\pi h|\Gamma_{\ell}|$ and $(L-1)$
parameters
$\left\\{\gamma_{\ell}\right\\}_{\ell=2}^{L}=j^{-(\ell-1)}e^{-j\ell\phi_{1}}e^{j\phi_{\ell}}$.
Throughout the rest of this paper, the GBF in (19) will be denoted as simply
$\mathcal{J}_{m}$ for brevity. Equation (19) is a more compact representation
of the CE-OFDM in time than previous efforts such as those in [8] which
utilize product-of-sums of 1-dimensional Bessel functions. More importantly,
(19) allows for deriving compact closed-form expressions of the CE-OFDM
waveform’s spectrum, AF, and ACF. These expressions, derived in Appendix B,
are shown below

$S\left(f\right)=\sqrt{T}\sum_{m=-\infty}^{\infty}j^{m}e^{jm\phi_{1}}\mathcal{J}_{m}\operatorname{sinc}\left[\pi
T\left(f-\frac{m}{T}\right)\right],$ (20)
$\chi\left(\tau,\nu\right)=\left(\frac{T-|\tau|}{T}\right)\sum_{m,n}j^{(m-n)}e^{j\phi_{1}(m-n)}\mathcal{J}_{m}\mathcal{J}_{n}^{*}\\\
\times
e^{-j\frac{\pi\left(m+n\right)\tau}{T}}\operatorname{sinc}\left[\pi\left(\frac{T-|\tau|}{T}\right)\left(\nu
T+\left(m-n\right)\right)\right].$ (21)

The ACF of the CE-OFDM follows from setting $\nu$ in (21) to zero. The
expression for the CE-OFDM’s spectrum (20) and AF (21) are closely related to
that of the MTSFM’s spectrum and AF derived in [12] with the only difference
being the type of GBF in the expression and the complex phase terms.

####  III-B2 Mainlobe Characterization

The EOA parameters of the CE-OFDM, derived in Appendix C, are expressed as

$\displaystyle\beta_{rms}^{2}$
$\displaystyle=\dfrac{4\pi^{4}h^{2}}{3T^{2}}\left(2L^{3}+3L^{2}+L\right),$
(22) $\displaystyle\tau_{rms}^{2}$ $\displaystyle=\dfrac{\pi^{2}T^{2}}{3},$
(23) $\displaystyle\rho$
$\displaystyle=4\pi^{2}h\sum_{\ell=1}^{L}|\Gamma_{\ell}|\left(-1\right)^{\ell}\cos\left(\phi_{\ell}\right).$
(24)

We’ll denote the RDCF $\rho$ in terms of the normalized RDCF $\tilde{\rho}$
which takes on the values $-1\leq\rho\leq 1$ [14, 15, 20] and is expressed as

$\tilde{\rho}=\dfrac{\rho}{\tau_{rms}\beta_{rms}}=\left(\dfrac{6}{\pi}\right)\dfrac{\sum_{\ell=1}^{L}|\Gamma_{\ell}|\left(-1\right)^{\ell}\cos\left(\phi_{\ell}\right)}{\sqrt{2L^{3}+3L^{2}+L}}.$
(25)

We note several observations regarding the results shown in (22)-(25). First,
as with all rectangularly tapered waveforms, the Doppler sensitivity is
proportional to the square of the waveform duration $T$ times a constant
($\pi^{2}/3$). Second, the expression for $\beta_{rms}^{2}$ in (22) is an
exact closed-form expression for all real values of $h$ and $L$. This result
strongly contrasts with the approximation to $\beta_{rms}^{2}$ commonly
encountered in the literature of $\beta_{rms}^{2}\approx
4\pi^{2}h^{2}L^{2}/T^{2}$ which holds for $2\pi h>1$ [3, 1]. Additionally,
equation (22) shows that the choice of PSK sequence $\phi_{\ell}$ does not
influence $\beta_{rms}^{2}$. Lastly, the normalized RDCF $\tilde{\rho}$ in
(25) is dependent only on the number of carriers $L$ and the PSK sequence
$\phi_{\ell}$ and not on modulation index $h$ or waveform duration $T$.

##  IV Implications for CE-OFDM Waveform Design

Figure 1 shows an example CE-OFDM waveform with $L=24$ sub-carriers with
$M_{\text{PSK}}=32$ symbols $\phi_{\ell}$ drawn randomly. The waveform’s
modulation index $h=0.1856$ and is chosen such that its RMS bandwidth (22) is
equivalent to that of an LFM waveform with a Time-Bandwidth Product (TBP) of
200. This is done by setting (22) equal to the LFM’s RMS bandwidth
$\pi^{2}\Delta f^{2}/3$ and solving for $h$

$\displaystyle\dfrac{4\pi^{4}h^{2}}{3T^{2}}\left(2L^{3}+3L^{2}+L\right)$
$\displaystyle=\dfrac{\pi^{2}\Delta f^{2}}{3}$ $\displaystyle h$
$\displaystyle=\frac{T\Delta f}{2\pi\sqrt{2L^{3}+3L^{2}+L}}$ (26)

where $\Delta f$ is the LFM waveform’s swept bandwidth. Note that while both
waveforms possess the same RMS bandwidth, the CE-OFDM sweeps through a band
that is slightly larger than $\pm\Delta f/2$. However, as can be seen in panel
(b) of Figure 1, both waveforms possess spectra whose magnitude falls off at
about the same rate outside the $\pm\Delta f/2$ band. The waveform possesses a
”Thumbtack-Like” AF shape with a largely uncoupled mainlobe
($\tilde{\rho}=0.0848$) and the ACF of the waveform attains a PSLR of -15.21
dB and an ISL of -0.17 dB.

Figure 1: Spectrogram (a), spectrum (b), AF (c), and ACF (d) of a CE-OFDM
waveform with a TBP of 200 ($h=0.1856$) and $L=24$ randomly generated PSK
symbols where $M_{\text{PSK}}=32$. The CE-OFDM waveform possesses a spectrum
that is densely concentrated in a compact band of frequencies and possesses a
“Thumbtack-Like” AF shape with $\tilde{\rho}=0.0848$.

The “Thumbtack-like” AF shape is essentially the only AF shape attainable for
a CE-OFDM waveform utilizing PSK. The max value for the RDCF as a function of
$L$, derived in Appendix C, is expressed as

$\tilde{\rho}_{\text{max}}=\left(\dfrac{6}{\pi}\right)\dfrac{L}{\sqrt{2L^{3}+3L^{2}+L}}$
(27)

and is achieved when the PSK sequence is

$\phi_{\ell}=\left\\{\begin{array}[]{ll}\pi,&\ell~{}\text{odd}\\\ \\\
0,&\ell~{}\text{even.}\\\ \end{array}\right.$ (28)

Figure 2 plots equation (27) and several mainlobe ellipses of CE-OFDM
waveforms with a TBP of 200 and varying $\tilde{\rho}$ for $\xi=0.9$. Note
that for increasing $L$, $\tilde{\rho}_{\text{max}}$ falls off at a rate of
roughly $L^{-1/2}$ which is for the specific set of PSK symbols $\phi_{\ell}$
shown in (28). As a design example, consider the case where $L=24$ like the
waveform in Figure 1 resulting in $\tilde{\rho}_{\text{max}}=0.2673$. As can
be seen in Figure 2, this AF mainlobe shape does not substantially differ from
a mainlobe with exactly zero coupling. Even the maximum
$\tilde{\rho}_{\text{max}}$ of 0.7797, achieved when $L=1$, produces only a
moderately coupled AF mainlobe. In general, CE-OFDM waveforms utilizing
pseudo-random PSK codes tend to produce smaller $\tilde{\rho}$ values than
$\tilde{\rho}_{\text{max}}$. For most CE-OFDM waveform design examples in the
literature [8, 10, 11], $L>24$. Therefore, CE-OFDM waveforms using PSK
encoding and practical values of $L$ will essentially always possess a
”Thumbtack-Like” AF shape.

Figure 2: Maximum normalized RDCF $\tilde{\rho}_{\text{max}}$ (a) for CE-OFDM
waveforms as a function of number of carriers $L$ and several example mainlobe
ellipses (b) for varying values of $\tilde{\rho}$. For practical values of
$L$, the CE-OFDM almost always possesses a ”Thumbtack-Like” AF shape.

Figure 3 shows the ISL and PSLR of an example CE-OFDM waveform with an
equivalent TBP of 200 like the example shown in Figure 1 but with only $L=2$
sub-carriers and $h=5.81$. Both sidelobe metrics exhibit multiple local
extrema and odd-symmetry in $\phi_{1}$ and $\phi_{2}$. This shows that
appropriate choice of $\phi_{\ell}$ can indeed influence the CE-OFDM’s AF/ACF
sidelobe structure. The symmetry in these metrics suggest that there may exist
certain PSK codes that produce more favorably low sidelobe levels. An
alternative to deriving specific codes is to develop waveform optimization
methods that produce CE-OFDM waveforms with desirably low ACF sidelobes. This
is the topic of another paper [21].

Figure 3: Plot of ISL and PSLR vs. $\phi_{\ell}$ for a CE-OFDM waveform with
equivalent TBP of 200 and $L=2$ sub-carriers. Both sidelobe metrics exhibit
odd-symmetry in $\phi_{1}$ and $\phi_{2}$ as well as multiple local extrema
showing that the choice of PSK symbols $\phi_{\ell}$ directly impacts the CE-
OFDM waveform’s ACF sidelobe structure.

##  V Conclusion

This paper derived several exact closed form expressions that more fully
characterize the AF shape of CE-OFDM waveforms employing PSK encoding. These
results show that such CE-OFDM waveforms almost always possess a ”Thumbtack-
Like” AF. Additionally, we show that the choice of PSK code indeed influences
the waveform’s ACF sidelobe structure. Designing CE-OFDM waveforms with
desirably low AF/ACF sidelobes will likely involve novel code construction
techniques or waveform optimization methods. Future efforts will explore these
code construction and waveform optimization methods as well as extend the
analysis performed in this paper to other forms of digital modulation such as
Quadrature Amplitude Modulation (QAM).

##  Appendix A Generalized Bessel Functions and the CE-OFDM Waveform Model

The $L$-dimensional, $(L-1)$-parameter GBF possesses the following generating
function [18]

$\exp\left[\frac{z_{1}}{2}\left(q-\frac{1}{q}\right)+\sum_{\ell=2}^{L}\frac{z_{\ell}}{2}\left(q^{\ell}\gamma_{\ell-1}-\frac{1}{q^{\ell}\gamma_{\ell-1}}\right)\right]\\\
=\sum_{m=-\infty}^{\infty}\mathcal{J}_{m}\left(\left\\{z_{\ell}\right\\}_{\ell=1}^{L};\left\\{\gamma_{\ell}\right\\}_{\ell=2}^{L}\right)q^{m}$
(29)

where $\\{z_{\ell}\\}_{\ell=1}^{L}$ are the $L$ variables and
$\\{\gamma_{\ell}\\}_{\ell=2}^{L}$ are the $(L-1)$ parameters for the GBF as
shown in (19). Letting $q=je^{j\theta}e^{j\phi_{1}}$ and
$\gamma_{\ell}=-j^{-\left(\ell-1\right)}e^{-j\ell\phi_{1}}e^{j\phi_{\ell}}$
results in

$\left(q-\frac{1}{q}\right)=je^{j\theta}e^{j\phi_{1}}+je^{-j\theta}e^{-j\phi_{1}}=2j\cos\left(\theta+\phi_{1}\right)\\\
$ (30)

and

$\displaystyle\left(q^{\ell}\gamma_{\ell-1}-\frac{1}{q^{\ell}\gamma_{\ell-1}}\right)$
$\displaystyle=je^{j\ell\theta}e^{j\phi_{\ell}}+je^{-j\ell\theta}e^{-j\phi_{\ell}}$
$\displaystyle=2j\cos\left(\ell\theta+\phi_{\ell}\right).$ (31)

Inserting the results of (30) and (31) into (29) results in the Jacobi-Anger
expansion for the $L$-dimensional $(L-1)$-parameter GBF [18]

$\exp\left[j\sum_{\ell=1}^{L}z_{\ell}\cos\left(\ell\theta+\phi_{\ell}\right)\right]=\\\
\sum_{m=-\infty}^{\infty}j^{m}e^{jm\phi_{1}}\mathcal{J}_{m}\left(\left\\{z_{\ell}\right\\}_{\ell=1}^{L};\left\\{\gamma_{\ell}\right\\}_{\ell=2}^{L}\right)e^{jm\theta}.$
(32)

##  Appendix B Derivation of the CE-OFDM Waveform’s Spectrum and AF

###  B-A CE-OFDM Spectrum

Using (19), the Fourier transform of the CE-OFDM waveform is expressed as

$S\left(f\right)=\dfrac{1}{\sqrt{T}}\sum_{m=-\infty}^{\infty}j^{m}e^{jm\phi_{1}}\mathcal{J}_{m}\int_{-T/2}^{T/2}e^{-j2\pi\left(f-\frac{m}{T}\right)t}dt.$
(33)

The integral in (33) evaluates to $T\operatorname{sinc}\left[\pi
T\left(f-m/T\right)\right]$ resulting in the final expression

$S\left(f\right)=\sqrt{T}\sum_{m=-\infty}^{\infty}j^{m}e^{jm\phi_{1}}\mathcal{J}_{m}\operatorname{sinc}\left[\pi
T\left(f-\frac{m}{T}\right)\right].$ (34)

###  B-B CE-OFDM AF

Inserting (19) into (3) results in the expression

$\chi\left(\tau,\nu\right)=\dfrac{1}{T}\sum_{m,n}j^{(m-n)}e^{j\phi_{1}(m-n)}\mathcal{J}_{m}\mathcal{J}_{n}^{*}e^{j\frac{\pi\left(m+n\right)\tau}{T}}\\\
\times\int_{-\infty}^{\infty}\operatorname{rect}\left(\dfrac{t-\tau/2}{T}\right)\operatorname{rect}\left(\dfrac{t+\tau/2}{T}\right)e^{j2\pi\left[\nu+\left(m-n\right)\right]t}dt.$
(35)

The integral in (35) simplifies to

$\int_{-\left(\frac{T-|\tau|}{2}\right)}^{\left(\frac{T-|\tau|}{2}\right)}e^{j2\pi\left[\nu+\left(m-n\right)\right]t}dt.$
(36)

Evaluating (36) results in the final expression for the CE-OFDM waveform’s AF

$\chi\left(\tau,\nu\right)=\left(\frac{T-|\tau|}{T}\right)\sum_{m,n}j^{(m-n)}e^{j\phi_{1}(m-n)}\mathcal{J}_{m}\mathcal{J}_{n}^{*}\\\
\times
e^{j\frac{\pi\left(m+n\right)\tau}{T}}\operatorname{sinc}\left[\pi\left(\frac{T-|\tau|}{T}\right)\left(\nu
T+\left(m-n\right)\right)\right].$ (37)

The ACF is then found by setting $\nu=0$.

##  Appendix C Derivation of the CE-OFDM Waveform’s EOA Parameters

This section calculates the EOA parameters of the rectangularly windowed CE-
OFDM waveform model in (16)

###  C-A RMS Pulse Length

Inserting (1) into the RMS pulse-length expression (9) yields the expression

$\displaystyle\tau_{rms}^{2}=\dfrac{4\pi^{2}}{T}\int_{-T/2}^{T/2}t^{2}dt=\dfrac{\pi^{2}T^{2}}{3}$
(38)

which is the standard result for any rectangularly windowed FM waveform [14].

###  C-B RMS Bandwidth

The RMS bandwidth $\beta_{rms}^{2}$ is calculated via (8). Due to the odd-
symmetry of the frequency modulation function (18), the second integral in (8)
simplifies to 0. Inserting (14) into (8) results in the expression

$\beta_{rms}^{2}=\dfrac{16\pi^{4}h^{2}}{T^{3}}\int_{-T/2}^{T/2}\left[\sum_{\ell=1}^{L}\left(-\ell\tilde{\alpha}_{\ell}\right)\sin\left(\dfrac{2\pi\ell
t}{T}\right)+\right.\\\
\left.\left(-\ell\tilde{\beta}_{\ell}\right)\cos\left(\dfrac{2\pi\ell
t}{T}\right)\right]^{2}dt.$ (39)

Equation (39) can be rewritten as

$\beta_{rms}^{2}=\dfrac{8\pi^{4}h^{2}}{T^{2}}\left(\dfrac{2}{T}\right)\int_{-T/2}^{T/2}\left[\sum_{\ell=1}^{L}\left(-\ell\tilde{\alpha}_{\ell}\right)\sin\left(\dfrac{2\pi\ell
t}{T}\right)+\right.\\\
\left.\left(-\ell\tilde{\beta}_{\ell}\right)\cos\left(\dfrac{2\pi\ell
t}{T}\right)\right]^{2}dt.$ (40)

The scaling factor $2/T$ times the integral in (40) can be simplified by
utilizing the Parseval’s theorem for Fourier series resulting in the
expression

$\displaystyle\sum_{\ell=1}^{L}{\ell}^{2}\left(\tilde{\alpha}_{\ell}^{2}+\tilde{\beta}_{\ell}^{2}\right)=\sum_{\ell}{\ell}^{2}.$
(41)

Using the formula for the sum of the first $L$ integers squared yields the
final expression

$\beta_{rms}^{2}=\dfrac{4\pi^{4}h^{2}}{3T^{2}}\left(2L^{3}+3L^{2}+L\right).$
(42)

###  C-C Range-Doppler Coupling Factor

From (18),
$\dot{\varphi}\left(t\right)=-\dfrac{4\pi^{2}h}{T}\sum_{{\ell}=1}^{L}{|\Gamma_{\ell}|\sin\left(\dfrac{2\pi\ell
t}{T}+\phi_{\ell}\right)}$ and the RDCF is therefore expressed as

$\displaystyle\rho=-\dfrac{8\pi^{3}h}{T^{2}}\sum_{\ell=1}^{L}\ell|\Gamma_{\ell}|\int_{-T/2}^{T/2}t\sin\left(\frac{2\pi\ell
t}{T}+\phi_{\ell}\right)dt.$ (43)

Using the trigonometric identity $\sin\left(a\pm b\right)=\sin a\cos b\pm\cos
a\sin b$, (43) is expressed as

$\rho=-\dfrac{8\pi^{3}h}{T^{2}}\sum_{\ell=1}^{L}\ell|\Gamma_{\ell}|\left[\cos\left(\phi_{\ell}\right)\int_{-T/2}^{T/2}t\sin\left(\dfrac{2\pi\ell
t}{T}\right)+\right.\\\
\left.\sin\left(\phi_{\ell}\right)\int_{-T/2}^{T/2}t\cos\left(\dfrac{2\pi\ell
t}{T}\right)\right].$ (44)

The second integral evaluates to zero since the integrand is an odd-symmetric
function evaluated over an even-symmetric interval leaving only the first
integral to evaluate. Using integration by parts, the first integral in (44)
evaluates to

$\left(\dfrac{-T}{2\pi\ell}\right)\left[T\cos\left(\pi\ell\right)+T\cos(\left(-\pi\ell\right)\right]=\dfrac{-T^{2}}{2\pi\ell}\left(-1\right)^{\ell}.$
(45)

Inserting the result in (45) back into (44) results in the RDCF $\rho$ for the
CE-OFDM

$\rho=4\pi^{2}h\sum_{\ell}^{L}|\Gamma_{\ell}|\left(-1\right)^{\ell}\cos\left(\phi_{\ell}\right).$
(46)

For a CE-OFDM waveform employing PSK (i.e, $|\Gamma_{\ell}|=1$) on $L$ carrier
modulation frequencies, $\rho$ is maximized when the
$\left(-1\right)^{\ell}\cos\left(\phi_{\ell}\right)$ terms are $+1$ for all
$\ell$. This is achieved when $\phi_{\ell}$ takes the form

$\phi_{\ell}=\left\\{\begin{array}[]{ll}\pi,&\ell~{}\text{odd}\\\ \\\
0,&\ell~{}\text{even}\\\ \end{array}\right.$ (47)

resulting in the expression for $\rho_{\text{max}}$

$\rho_{\text{max}}=4\pi^{2}hL.$ (48)

Using (25), $\tilde{\rho}_{\text{max}}$ is expressed as

$\displaystyle\tilde{\rho}_{\text{max}}$
$\displaystyle=\dfrac{\rho_{\text{max}}}{\beta_{rms}\tau_{rms}}=\dfrac{\sqrt{3}\sqrt{3}T4\pi^{2}hL}{\left(\pi
T\right)\left(2\pi^{2}h\right)\sqrt{2L^{3}+3L^{2}+L}}$
$\displaystyle=\left(\dfrac{6}{\pi}\right)\dfrac{L}{\sqrt{2L^{3}+3L^{2}+L}}.$
(49)

## References

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# Mitigating the source-side channel vulnerability by characterization of
photon statistics

Tanya Sharma<EMAIL_ADDRESS>Quantum Science and Technology Laboratory,
Physical Research Laboratory, Ahmedabad, India 380009.  Indian Institute of
Technology, Gandhinagar, India 382355.  Ayan Biswas  School of Physics,
Engineering $\&$ Technology and York Centre for Quantum Technologies,
Institute for Safe Autonomy, University of York, YO10 5FT York, United Kingdom
Jayanth Ramakrishnan  Quantum Science and Technology Laboratory, Physical
Research Laboratory, Ahmedabad, India 380009.  Pooja Chandravanshi  Quantum
Science and Technology Laboratory, Physical Research Laboratory, Ahmedabad,
India 380009.  Ravindra P. Singh<EMAIL_ADDRESS>Quantum Science and
Technology Laboratory, Physical Research Laboratory, Ahmedabad, India 380009.

###### Abstract

Quantum key distribution (QKD) theoretically offers unconditional security.
Unfortunately, the gap between theory and practice threatens side-channel
attacks on practical QKD systems. Many well-known QKD protocols use weak
coherent laser pulses to encode the quantum information. These sources differ
from ideal single photon sources and follow Poisson statistics. Many
protocols, such as decoy state and coincidence detection protocols, rely on
monitoring the photon statistics to detect any information leakage. The
accurate measurement and characterization of photon statistics enable the
detection of adversarial attacks and the estimation of secure key rates,
strengthening the overall security of the QKD system. We have rigorously
characterized our source to estimate the mean photon number employing multiple
detectors for comparison against measurements made with a single detector.
Furthermore, we have also studied intensity fluctuations to help identify and
mitigate any potential information leakage due to state preparation flaws. We
aim to bridge the gap between theory and practice to achieve information-
theoretic security.

Suggested keywords

##  I Introduction

The advent of quantum computers significantly threatens existing classical
cryptographic algorithms, and we can no longer guarantee the security of
sensitive information with classical communication. QKD protocols [1, 2, 3, 4,
5] have emerged as a promising approach to circumvent potential security
concerns.

However, practical implementations involve using imperfect devices that may
assist an adversary in gaining partial information. The device imperfections
may lead to further security threats by an adversary who may exploit such
loopholes [6, 7, 8, 9, 10, 11, 12]. Characterising these devices is a crucial
step towards ensuring the effectiveness of QKD in real-world scenarios.

One crucial aspect of optical quantum cryptography is using single-photon Fock
states. While the experimental realisation of true single-photon sources
remains challenging, many practical options, such as weak coherent pulses
(WCPs), heralded single-photon sources, and the entangled single-photon
source, are employed [13].

Many implementations involve the use of WCPs as an approximation of single-
photon Fock states. These WCPs are realised by highly attenuating a pulsed
laser source using calibrated attenuators. It is a principle belief that laser
sources operating well above the threshold emit a coherent state.
Consequently, the faint laser pulses derived from such sources exhibit Poisson
statistics regarding the number of photons per pulse. The mean photon number
is the exclusive defining feature of Poissonian statistics. Therefore,
ensuring the accurate measurement of the mean photon number is crucial for
achieving secure QKD using a weak coherent source.

The Poisson statistics of the WCP source render a non-zero probability,
however low, to attain more than a single photon per pulse. Such a loophole
exposes our QKD system to adversarial attacks like the photon number splitting
attack. The decoy state protocol [14, 15, 16, 17], involving the use of decoy
pulses with slightly different mean photon numbers, has been proposed to
mitigate this vulnerability. These decoy pulses are also characterised by
Poisson statistics but with a slightly different mean photon number. Accurate
estimation and characterisation of photon statistics, including the mean
photon number, are crucial for implementing secure QKD protocols and detecting
potential infiltration.

The practical implementation of QKD protocols heavily depends on the extensive
utilisation of single-photon avalanche photodiodes (SPADs) [18]. When
evaluating the capabilities of a single-photon detector, it is essential to
consider its spectral range, dead time, dark count rate, detection efficiency,
timing jitters, and capacity to discern photon numbers [19]. The SPADs are
threshold detectors which cannot provide information about the exact number of
photons per pulse. They are also known as on-off detectors since they can only
detect the presence or absence of a pulse containing photons. Hence using a
single SPAD for characterising the source will give inaccurate estimates. Many
studies [20, 21, 22, 23] discuss various approaches to photon characterisation
using multiple on-off detectors.

In this study, we focus on characterising the photon statistics of WCPs for
QKD applications. Initially, we estimate the average photon number using a
single detector. Subsequently, we employ a setup with four detectors to
achieve a more accurate estimation of the average photon number. In a previous
study [23], the authors have introduced the utilisation of four detectors to
accurately characterise the photon statistics of pulses by providing accurate
probabilities for lower photon numbers $(n\leq 3)$. We adopt this
characterisation approach to determine the Poisson statistics of the WCPs and
obtain highly accurate estimations of the mean photon numbers. By comparing
these results with previous estimates, we investigate the deviation of the
mean photon number $(\mu)$ and analyse the resulting information leakage
caused by this miscalculation.

Several investigations have delved into diverse state preparation
deficiencies, offering security proofs that establish safeguarding measures
even in the presence of these flaws [24, 25, 26, 27, 28, 29, 30]. Lasers
inevitably exhibit statistical fluctuations, resulting in inherent variability
in the mean photon number they emit. However, it’s crucial to quantify the
magnitude of these fluctuations and comprehend their potential to facilitate
information leakage. It is essential to recognise how the distinguishability
of these fluctuations among different sources impacts the security of our
Quantum Key Distribution (QKD) process. Furthermore, we have examined the
fluctuations in intensity as a function of the source’s intensity.

The organisation of this paper is as follows: We discuss the theoretical
background to estimate the mean photon number per pulse and the fluctuations
of a WCP source in Sec. II. We then discuss the experimental setup and
procedures in Sec. III. In Sec. IV, we present the results for our source
characterisation of intensity and fluctuations. Finally, we conclude and
summarise our paper in Sec. V.

##  II Theoretical Background

###  II.1 Coherent State

The coherent state is specified in the Dirac notation as $\ket{\alpha}$, where
$\alpha$ is the complex field amplitude. The coherent state is represented in
terms of photon number states:

$\ket{\alpha}=\mathrm{e}^{-|\alpha|^{2}/2}\sum^{\infty}_{n=0}\frac{\alpha^{n}}{\sqrt{n!}}\ket{n},$
(1)

‘

To find the probability $\mathcal{P}(n)$ that there are $n$ photons in the
coherent state we evaluate $\innerproduct{n}{\alpha}$ :

$\displaystyle\innerproduct{n}{\alpha}$
$\displaystyle=\mathrm{e}^{-|\alpha|^{2}/2}\sum_{m=0}^{\infty}\frac{\alpha^{m}}{\sqrt{m!}}\innerproduct{n}{m}$
(2)
$\displaystyle=\mathrm{e}^{-|\alpha|^{2}/2}\sum_{m=0}^{\infty}\frac{\alpha^{m}}{\sqrt{m!}}\delta_{nm}$
$\displaystyle=\mathrm{e}^{-|\alpha|^{2}/2}\frac{\alpha^{n}}{\sqrt{n!}},$

where, $\innerproduct{n}{m}=\delta_{n,m}$, since number states are
orthonormal. On equating $\mathcal{P}(n)$ with
$|\innerproduct{n}{\alpha}|^{2}$, we find:

$\mathcal{P}(n)\equiv|\innerproduct{n}{\alpha}|^{2}=\mathrm{e}^{-|\alpha|^{2}}\frac{\left|\alpha^{2}\right|^{n}}{n!}.$
(3)

Taking, $|\alpha|^{2}=\mu$, we obtain a Poisson distribution
$\mathcal{P}(n,\mu)$:

$\mathcal{P}(n,\mu)=\frac{\mu^{n}}{n!}\mathrm{e}^{-\mu}.$ (4)

This establishes that coherent states have Poissonian photon statistics with
mean photon number $\mu$.

###  II.2 Weak Coherent Pulses (WCPs)

A pulsed laser of repetition rate $\nu_{rep}$ and wavelength $\lambda$ is
attenuated to generate WCPs. Since a laser is a coherent source, a faint laser
source also emits coherent states. The number of photons per pulse is not
deterministic but follows the Poisson distribution. Here, we discuss how we
generate the WCPs of a desired distribution.

It is necessary for QKD sources to have a mean photon number of less than one
to meet the security requirements of QKD protocols. We start by measuring the
average power $P_{avg}$ of the source, from which we evaluate the energy per
pulse $E_{pulse}$ as,

$E_{pulse}=\frac{P_{avg}}{\nu_{rep}}.$ (5)

The average number of photons per pulse $n_{avg}$ is given by,

$n_{avg}=\frac{E_{pulse}\lambda}{hc}.$ (6)

We utilize a neutral density filter with a defined optical density (OD) to
attain the target mean photon number. The OD governs the degree of attenuation
applied to the laser beam. Through the choice of an appropriate filter and
varying attenuation, we can create WCPs exhibiting the desired Poisson
statistics, characterized by a mean photon number denoted as $\mu_{desired}$,
suitable for Quantum Key Distribution (QKD) applications.

$\mu_{desired}=\frac{E_{pulse}\lambda}{hc}*10^{-OD}.$ (7)

###  II.3 Estimations

We discuss methods to estimate the mean photon number ($\mu$) of the source
used in QKD implementation.

####  II.3.1 Method-I : Using single detection

Single photon detectors based on avalanche photodiodes are used to detect the
number of detection per second.

$N=\mu*\nu_{rep}*\eta.$ (8)

Where $N$ is the number of detections per second, $\nu_{rep}$ is the
repetition rate of the laser, and $\eta$ is detection efficiency. We can
estimate the value of $\mu$ as,

$\mu=\frac{N}{\nu_{rep}*\eta}.$ (9)

Avalanche photodiodes are the prevailing choice for quantum signal detection,
capable of indicating the presence or absence of photons within a pulse.
However, relying on a single on-off detector could lead to underestimating
photon statistics. Therefore, using photon-resolving detectors or a more
comprehensive methodology becomes imperative in characterizing the QKD source.
Accurate estimation of $\mu$ assumes paramount significance for accurate key
rate calculations.

####  II.3.2 Method-II Rigorous Characterisation

Figure 1: The branching efficiencies are defined as the probability for a
photon to reach a particular detector. In the setup shown above the four
branching efficiencies can be given as
$\\{T_{1}T_{2},T_{1}R_{2},R_{1}T_{3},R_{1}R_{3}\\}$.

For source characterisation, splitting the pulse using an infinite number of
beam splitters and threshold detectors is essential. The mean photon number in
QKD is quite low; hence, we assume that multi-photon probability is minimal.
This assumption enables us to rigorously characterise the source using four
detectors ($D=4$). Figure 1 illustrates an example of the setup necessary for
conducting such comprehensive characterisation. The branching efficiencies
define the probability of a photon reaching a specific detector. Let the
branching efficiencies be $\eta_{b,i}$, the coupling efficiency be $\eta_{c}$,
and the quantum efficiency of the detectors be $\eta_{d}$.

Overall efficiency is given as,

$\eta_{i}=\eta_{b,i}*\eta_{c}*\eta_{d},$ (10)

where $i=1,2,3,4$ represents all four arms and respective detectors. The
average efficiency is given as,

$\eta:=\frac{1}{4}\sum_{i=1}^{D}\eta_{i}.$ (11)

We define $Z_{D}:={1,2,3,4}$ as a set of all the detectors. We want to record
the r fold coincidences $(r=2,3,4)$, where $r=1$ refers to the counts in a
single detector. Let us denote the observed r fold coincidence probability as
$c_{obs,r}$ [23].

$c_{obs,r}={\binom{D}{r}}^{-1}\sum_{W\subset I_{r}}c_{obs,W},$ (12)

where, $I_{r}:=\\{W\subset Z_{D}\ \big{|}\ |W|=r\\}$, $W$ is a subset of
$Z_{D}$ with cardinality $|W|$, $c_{obs,W}$ denote the total coincidence
probability of set $W$, where all $|W|$ detectors detect irrespective of the
detection events in remaining detectors. The averaged r-fold coincidences,
given the pulse, has $n$ photons, is,

$c_{obs,r}=c_{n,r}\coloneqq\sum^{r}_{j=0}(-1)^{j}\omega_{r,j}\sum_{W\in
I_{j}}(1-\sum_{i\in W}\eta_{i})^{n},$ (13)

where,

$\omega_{r,j}:=\frac{\binom{D-j}{r-j}}{\binom{D}{r}}.$ (14)

For a Poissonian distribution with mean photon number $\mu$, the averaged
$r(1,2,3,4)$ fold coincidences should satisfy,

$c_{obs,r}=\sum_{n=0}^{\infty}p_{n}c_{n,r},$ (15)

where,

$p_{n}=\frac{e^{-\mu}\mu^{n}}{n!}.$ (16)

To evaluate the coincidences from experimental data, we will employ equation
(12) as referenced in our work. Additionally, we will utilize the bounds
specified in the article [23] to experimentally validate the Poissonian
statistics of the WCPs employed in QKD implementations and accurately estimate
the mean photon number for the distribution.

###  II.4 Information Leakage

In principle, an eavesdropper may gain complete information on the bit value
encoded on a multi-photon pulse. Privacy amplification enables us to reduce
the information an eavesdropper might earn on the shared key between the
authenticated parties. It is essential to determine the optimum subtraction
bits for privacy amplification and avoid underestimating them. Since we cannot
know Eve’s attack strategy in advance, we consider an attack where Eve can
access all the information in the multi-photon pulses. We could ensure
absolute security by applying sufficient privacy amplification discarding all
the bit values associated with the multi-photon pulses. If a WCP source has
Poisson distribution with $\mu$ as the mean photon number per pulse, then the
probability of multi-photons per pulse is given as,

$p_{multi}=\sum_{p_{n}\geq 2}p_{n}$ (17)

While estimating the secure key rate, we consider this worst-case scenario,
discard the coincidences, and consider only the single detection with a
compatible basis. We estimate the subtraction terms as the multi-photon pulses
that contribute to the single detections to consider the information leakage.
Hence, we consider,

$I(A:E)=\sum_{p_{n}\geq 2}\Bigl{(}p_{n}*\frac{1}{2^{n}}\Bigr{)}$ (18)

Where $\frac{1}{2^{n}}$ is the probability that a pulse containing n photons
gives a single detection in the correct basis.

We encode using one degree of freedom, specifically polarization in our case.
Any other degree of freedom associated with our states that could be used to
discriminate the states could lead to a potential side-channel attack. Our
objective is to investigate the differences in the fluctuations exhibited by
the sources. When $N_{s}$ signal states with an intensity of $\mu$ are
emitted, we posit that precisely $N_{s}\mu e^{-\mu}$ out of the $N_{s}$ signal
states correspond to single photons. In practical situations, the quantity
$\mu e^{-\mu}$ represents the probability of a single photon in the pulse. The
observed count of single-photon signals will exhibit statistical fluctuations.
We employ the methodology proposed in [31] to quantitatively assess the extent
of information leakage to Eve. The correlation function $R$ characterizes the
similarity between two sources. We compare $R$ with Eve’s guessing probability
$p(e|b)$, allowing the calculation of mutual information in terms of cross-
correlation:

$I^{\prime}(A:E)=1+\sum_{\begin{subarray}{c}i,j\\\ i\neq
j\end{subarray}}\frac{R_{ij}}{4}\log_{2}\left(\frac{R_{ij}}{4}\right).$ (19)

Here, $R_{ij}$ signifies the cross-correlation between sources $i$ and $j$.
For perfectly identical functions, $I^{\prime}(A:E)$ becomes zero, as
indicated by Equation 19. The measurement of cross-correlation offers an
estimation of the extent of information leakage. Please note that to
differentiate the information leakages due to multi-photon pulses and the
information leakage due to side channels, we represent them by $I(A:E)$ and
$I^{\prime}(A:E)$, respectively.

##  III Experimental Method

Figure 2: Experimental setup for characterising photon statistics of the
source: V-NDF: Variable Neutral Density Filter; NDF: Neutral Density Filter;
HWP: Half Wave Plate; BS: Beam Splitter; PBS: Polarizing Beam Splitter; M:
Mirror; C: Coupler; D: Single Photon Counting Module; TDC: ID-900 Time
Controller.

The experimental setup comprises four diode lasers of 808 nm (L808P010
Thorlabs). The source setup is the standard BB84 transmitter, using four laser
diodes to contribute to the four polarisation states for QKD. The source
comprises the laser diodes along with the attenuating optics.

Voltage pulses trigger each laser diode with a repetition rate of 1.25 MHz. A
half-wave plate and a polarising beam splitter (PBS) combination direct the
emitted beam from laser diodes, functioning as both a polariser and an
attenuator. The polarising beam splitter (PBS) transmits the horizontally
polarised light and reflects vertically polarised light. The rotation of the
half-wave plate determines the intensity of each beam. So far, the setup is
identical for both bases. To convert the polarisation to the diagonal basis,
we insert a half-wave plate and rotate it by $22.5^{\circ}$. This adjustment
ensures that the beam’s polarisation aligns with the diagonal basis. The two
beams combine at a beam splitter (BS1), and we only consider the output from
one arm, discarding the other. The beam intensity reduces to half due to the
combination of the four beams at this beam splitter.

Next, all four beams pass through a variable neutral density filter (NDF) with
a maximum optical density (OD) of 4 and a fixed NDF with an optical density of
6. These filters reduce the intensity of the beams accordingly. We will
analyse the photon statistics of the resulting beam from this final NDF
containing the signal. To characterise the photon statistics of each source
individually, we block the other sources and analyse them one by one.

The characterisation setup consists of three beam splitters and four
detectors. The detectors utilised in the setup are single photon counting
modules(Excelitas-SPCM-AQRH-14). The incoming beam is coupled to the SPADs
using fiber couplers (Thorlabs-CFC5-B) and multi-mode fibers
(Thorlabs-M42L02). Upon reaching beam splitter BS3, the signal divides into
two arms. These arms are further split into two more arms by the beam
splitters BS4 and BS5, resulting in four arms coupled to the four detectors.

For each source, we recorded timestamps and recorded two, three, and four-fold
coincidences among all four detectors. We rotated the variable NDF to acquire
various values of $\mu$. The estimation of $\mu$ was approximately determined
using equation (20).

$N=\mu\nu_{rep}\eta$ (20)

Where $N$ is the number of counts in the detector per second, $\mu$ is the
mean photon number, $\nu_{rep}$ is the repetition rate, i.e. 1.25 MHz, and
$\eta$ is the overall efficiency. As discussed, the overall efficiency given
by (10) consists of the quantum efficiency of detector $65\%$, coupling
efficiency and branching efficiency. The beam splitters are not 50-50; we also
included those experimental efficiencies in our calculations. We characterized
the beam splitters, and Table-1 contains the mentioned transmittance and
reflectance values.

Beam Splitter | $T^{2}$ | $R^{2}$
---|---|---
BS3 | $\ \ 0.494\ \ $ | $\ \ 0.453\ \ $
BS4 | $0.474$ | $0.446$
BS5 | $0.461$ | $0.456$
Table 1: Transmittance and reflectance of beam-splitters used in
characterisation setup

Thus using (20), we can roughly estimate if we have achieved the desired value
of $\mu$ by looking at the number of counts in a single detector as discussed
in II.3.1 since we know that these counts only signify the presence and
absence of a pulse containing photons. We now characterise the source using
the r-fold coincidences recorded to estimate $\mu$ using II.3.2. We repeat
this process for all four laser diodes emitting different polarisations.

We consider a single detector for each source to study the intensity
fluctuations of all four sources. Furthermore, we adjust the variable
attenuator by rotating it to attain the desired count rate, and the detectors
capture individual signals. This information is documented over multiple
cycles to examine the fluctuations in the sources. We repeat this experiment
for various source intensities to examine fluctuations as a function of
intensity.

##  IV Results and Discussion

We recorded the single and coincidence detections. The photon statistics were
verified as Poissonian, as discussed in II.3.2. We compared the calculated
values from Method-I (II.3.1) and Method-II (II.3.2), and we plotted the
difference as a function of mean photon number ($\mu$) in Figure-3. The
difference in the estimated values increases with an increase in the mean
photon number. This outcome is predictable, as a single detector lacks the
accuracy to measure accurately in the presence of multi-photon pulses.
Coincidences using multiple on-off detectors enable us to resolve the photon
numbers with higher accuracy.

Figure 3: Difference between the mean photon number ($\mu$) calculated using
Method-I and Method-II of mean photon number ($\mu$).

We calculate the mutual information for each case using (18). It’s important
to note that we are considering the worst-case scenario, where Eve can extract
information from all multi-photon pulses in the BB84 protocol. Researchers can
extend this study to encompass other prepare-and-measure protocols that
utilize WCPs as the source. Accurately estimating potential information
leakage is crucial, achievable only through a well-characterized understanding
of photon statistics. The errors in mean photon estimation lead to a wrong
estimation of I(A:E). An adversary can gain information from the part of the
information that goes un-estimated. The corresponding difference in the
information leakage I(A:E) as a function of mean photon number($\mu$) can be
seen in Figure-4

Figure 4: Information leakage due to miscalculated mean photon number ($\mu$)
as a function of mean photon number ($\mu$)

The variation in intensity fluctuations for all four sources relative to the
average photon count is illustrated in Figure-5. Using a consistent detector,
we derive these fluctuations from numerous iterations of single-count data
collected from each source. The error bars indicate the disparity between data
points and the fitted linear curve. As anticipated, augmenting the mean photon
count leads to a rise in intensity fluctuations.

Figure 5: The variation in intensity fluctuations for all four sources vs the
average photon count ($\mu$).

Figure-6 illustrates the distribution of all four sources at an average value
of $0.5$ photons per pulse. Using the fitted data, we create these plots to
compare all four sources by analyzing their fluctuations at a specific value
of $\mu$. The area of the shaded region signifies the probability of no
detection, as the number of counts is never negative.

Figure 6: The distribution of all four sources at an average value of $0.5$
photons per pulse

We report the correlations and potential information leakage in Table-2.
Ensuring the absence of disparities among the sources and their consistent
uniformity is crucial. Varied intensity fluctuations among the sources could
empower Eve to extract information, possibly resulting in a side-channel
attack. Eve’s attempt to gather information between the two bits or across the
two bases is the reason we have examined the one-on-one correlations among all
the sources. It is thus essential to estimate the amount of information
leakage to ensure secure quantum communication.

Table 2: The correlations $R$ and the potential information leakage $I(A:E)$ among different sources. Sources |  R |  I(A:E)
---|---|---
S1 $\&$ S2 |  0.9904 |  0.0027
S1 $\&$ S3 |  0.9715 |  0.0082
S1 $\&$ S4 |  0.9993 |  0.0002
S2 $\&$ S3 |  0.9949 |  0.0014
S2 $\&$ S4 |  0.9948 |  0.0014
S3 $\&$ S4 |  0.9796 |  0.0058

##  V  Conclusion

The weak coherent source used in implementing QKD follows Poisson
distribution. The characterisation of this source is essential for accurately
estimating any information leakage due to multi-photon pulses. The SPADs used
are not photon resolving; hence using multiple on-off detectors offers a
better resolution. Four SPADs suffice for the QKD applications as we are well
below one photon per pulse on average. The inaccurate measurement of the mean
photon number may lead to unaccounted information leakage, leading to an
undetected adversarial attack. It is essential to rigorously characterise mean
photon numbers in practical QKD systems based on WCPs. The difference in the
mean photon number and the corresponding information leakage is estimated. The
values increase with an increase in the mean photon number. We can use one
detector method for smaller $\mu$ values, i.e. $\mu\leq 0.3$, as, beyond that,
the approximation deviates from rigorous characterisation drastically.

The fluctuations among the four sources exhibit minor variations, resulting in
a maximum information leakage on the order of $10^{-2}$ bits per pulse. These
source fluctuations escalate as intensity increases, prompting a
reconsideration of the intensity discrepancy between the decoy and signal
states. Should the difference be substantial, it could expose vulnerabilities
to side-channel attacks by Eve. It is essential to analyze to determine the
potential for future attacks stemming from these variations in fluctuations.

## Acknowledgement

The authors acknowledge the partial funding support from DST through the QuST
program. The authors are also thankful to group members of the QST lab for
their valuable inputs.

## Disclosure

The authors declare no conflicts of interest.

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