# On the Impact of Phase Noise on Monostatic Sensing in OFDM ISAC Systems

Musa Furkan Keskin1, Carina Marcus2, Olof Eriksson2, Henk Wymeersch1, and Visa
Koivunen3
1Chalmers University of Technology, Sweden, 2Veoneer Sweden AB, Sweden, 3Aalto
University, Finland

###### Abstract

Phase noise (PN) can become a major bottleneck for integrated sensing and
communications (ISAC) systems towards 6G wireless networks. In this paper, we
consider an OFDM ISAC system with oscillator imperfections and investigate the
impact of PN on monostatic sensing performance by performing a misspecified
Cramér-Rao bound (MCRB) analysis. Simulations are carried out under a wide
variety of operating conditions with regard to SNR, oscillator type (free-
running oscillators (FROs) and phase-locked loops (PLLs)), 3-dB bandwidth of
the oscillator spectrum, PLL loop bandwidth and target range. The results
provide valuable insights on when PN leads to a significant degradation in
range and/or velocity accuracy, establishing important guidelines for hardware
and algorithm design in 6G ISAC systems.

Index Terms– OFDM, integrated sensing and communications, phase noise,
misspecified Cramér-Rao bound.

##  I Introduction

Integrated sensing and communications (ISAC) is expected to be one of the
enablers for 6G communication systems, both to support existing communication
functions and to enable novel sensing applications [1]. Dual-functioning
monostatic systems are of particular interest, e.g., in automotive and IoT
applications, due to their tight integration, low cost, and efficient spectrum
usage [2]. On the other hand, since such systems operate at mmWave frequencies
(30–100 GHz), they are sensitive to hardware impairments, such as phase noise
(PN), carrier frequency offsets, power amplifier nonlinearity [3, 4], which
can severely degrade the sensing performance, both in terms of detection and
tracking.

Among the various hardware impairments, PN stands out, since it is time-
varying and, in monostatic sensing, has target-dependent statistics [5]. The
importance of PN is underlined by the intense research activities in radar-
centric systems, such as FMCW-based radars [6, 7]. For communication-centric
ISAC systems, PN is well understood only from the communication receiver
perspective, where the PN statistics are independent of the objects in the
propagation channel, and standard methods (either pilot-based or blind) can be
utilized to mitigate the PN effect [8, 9, 10, 11]. In sharp contrast, the
impact of PN on OFDM ISAC sensing remains relatively underexplored [5, 12,
13].

In this paper, we carry out a detailed investigation of monostatic sensing
performance in OFDM ISAC systems under the impact of PN by leveraging tools
from estimation theory under model misspecification [14, 15]. Our specific
contributions are as follows:

  * •

We derive the misspecified Cramér-Rao bound (MCRB) and the corresponding
lower-bound (LB) [15] on range and velocity estimation for OFDM radar with PN,
considering mismatch between a true model with PN and an assumed model without
PN.

  * •

Under correct specification of the observation model (i.e., when the receiver
is aware of the existence of PN), we derive the hybrid CRB [16, 17] on range,
velocity and PN estimation.

  * •

To quantify PN-induced accuracy degradations through a comparative analysis of
the LB and the hybrid CRB, we conduct extensive simulations under a broad
range of operating conditions pertaining to SNR, oscillator type/parameters
and target range, which offer valuable insights into scenarios with
significant impact of PN on radar performance.

##  II System Model

We consider an OFDM ISAC system consisting of an ISAC transceiver and a
communications receiver (RX), as shown in Fig. 1. The ISAC transceiver
includes (i) a conventional OFDM transmitter (TX) to send data symbols to the
communications RX, and (ii) a radar RX on the joint hardware platform to
process the backscattered signals for target detection and parameter
estimation [18]. The sensing configuration is monostatic since the TX and the
radar RX are co-located. In addition, the communications RX performs standard
OFDM receive operations, such as channel estimation, frequency synchronization
and data detection [19]. In the joint ISAC transceiver hardware, the TX and
radar RX shares the same oscillator, which is assumed to be imperfect and
impaired by phase noise (PN) [20, 21]. In this section, we derive OFDM
transmit and radar receive signal models in the presence of PN and provide a
statistical characterization of PN in the radar receiver.

Figure 1: OFDM ISAC system with an ISAC transceiver and a communications
receiver. Self-correlated (differential) PN process in the radar receiver due
to the use of the shared oscillator in the ISAC transceiver leads to the range
correlation effect (i.e., range-dependent PN statistics), unlike the range-
independent non-stationary PN process in the communications receiver.

###  II-A Transmit Signal Model

Consider an OFDM frame with $M$ symbols, $N$ subcarriers, total symbol
duration $T_{\rm{sym}}=T_{\rm{cp}}+T$, elementary symbol duration $T$, cyclic
prefix (CP) duration $T_{\rm{cp}}$ and subcarrier spacing $\Delta f=1/T$. The
complex baseband OFDM transmit signal can be expressed as [18, 22]

$\displaystyle
s(t)=\frac{1}{\sqrt{N}}\sum_{m=0}^{M-1}\sum_{n=0}^{N-1}x_{n,m}\,e^{j2\pi
n\Delta
ft}\Pi\left(\frac{t-mT_{\rm{sym}}}{T_{\rm{sym}}}\right)\leavevmode\nobreak\ ,$
(1)

where $x_{n,m}\in\mathbb{C}$ is the data symbol on the ${n^{\rm{th}}}$
subcarrier of the ${m^{\rm{th}}}$ symbol and $\Pi(t)$ is a rectangular pulse
assuming the value $1$ for $t\in\left[0,1\right]$ and $0$ otherwise. Let
$\phi(t)$ denote the PN process in the oscillator and $f_{c}$ the carrier
frequency. Then, the upconverted transmit signal is given by [23]

$\widetilde{s}(t)=\Re\left\\{s(t)e^{j\left[2\pi
f_{c}t+\phi(t)\right]}\right\\}\leavevmode\nobreak\ .$ (2)

###  II-B Radar Receive Signal Model

We consider a single target with round-trip delay $\tau=2R/c$, normalized
Doppler shift $\nu=2v/c$ and complex channel gain $\alpha$, where $R$, $v$ and
$c$ represent the distance, radial velocity and speed of propagation,
respectively. For the transmit signal in (2), the passband signal at the radar
RX is given by

$\displaystyle\widetilde{y}(t)=\Re\left\\{\alpha\,s(t-\tau(t))e^{j\left[2\pi
f_{c}(t-\tau(t))+\phi(t-\tau(t))\right]}\right\\}\leavevmode\nobreak\ ,$ (3)

where the Doppler shift causes a time-varying delay $\tau(t)=\tau-\nu t$. The
downconversion of (3) occurs through the noisy oscillator, which is equivalent
to multiplication by $e^{-j(2\pi f_{c}t+\phi(t))}$ [24], leading to the
equivalent complex baseband signal [25]

$\displaystyle y(t)$ $\displaystyle=\alpha\,s(t-\tau(t))e^{-j2\pi
f_{c}\tau}e^{j2\pi f_{c}\nu
t}e^{j\left[\phi(t-\tau(t))-\phi(t)\right]}\leavevmode\nobreak\ .$ (4)

Assuming $\lvert\nu\rvert\ll 1/N$ [26, 27, 28] (which holds, e.g., for typical
vehicular ISAC scenarios [5]) and small time-bandwidth product [29], the
signal in (4) becomes

$\displaystyle y(t)$ $\displaystyle=\alpha\,s(t-\tau)e^{-j2\pi
f_{c}\tau}e^{j2\pi f_{c}\nu t}w(t,\tau)\leavevmode\nobreak\ ,$ (5)

where $w(t,\tau)$ is the multiplicative PN process, i.e.,

$\displaystyle w(t,\tau)\triangleq
e^{j\left[\phi(t-\tau)-\phi(t)\right]}\leavevmode\nobreak\ .$ (6)

###  II-C Fast-Time/Slow-Time Representation with Phase Noise

Following traditional OFDM radar processing for (5) (i.e., removing the CP in
the ${m^{\rm{th}}}$ symbol and sampling at $t=mT_{\rm{sym}}+T_{\rm{cp}}+\ell
T/N$ for $\ell=0,\ldots,N-1$), and adopting the standard assumptions
$T_{\rm{cp}}\geq\tau$ [26, 30, 31] and $f_{c}T\nu\ll 1$ [32, 33, 34], the
received signal for the ${m^{\rm{th}}}$ symbol can be written as [27, 28, 5]

$\displaystyle y_{\ell,m}$ $\displaystyle=\alpha\,e^{j2\pi
f_{c}mT_{\rm{sym}}\nu}w_{\ell,m}(\tau)$ (7) $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ \times\frac{1}{\sqrt{N}}\sum_{n=0}^{N-1}x_{n,m}\,e^{j2\pi
n\frac{\ell}{N}}e^{-j2\pi n\Delta f\tau}\leavevmode\nobreak\ ,$

where the sampled version of the PN term $w(t,\tau)$ in (5) for
$t=mT_{\rm{sym}}+T_{\rm{cp}}+\ell T/N$ is represented by $w_{\ell,m}(\tau)$.
Let us define the frequency-domain and temporal (slow-time) steering vectors
as

$\displaystyle\mathbf{b}(\tau)$
$\displaystyle\triangleq{\left[1,e^{-j2\pi\Delta
f\tau},\ldots,e^{-j2\pi(N-1)\Delta f\tau}\right]}^{T}\leavevmode\nobreak\ ,$
(8) $\displaystyle\mathbf{c}(\nu)$ $\displaystyle\triangleq{\left[1,e^{-j2\pi
f_{c}T_{\rm{sym}}\nu},\ldots,e^{-j2\pi
f_{c}(M-1)T_{\rm{sym}}\nu}\right]}^{T}\leavevmode\nobreak\ .$ (9)

Then, we can organize the observations in (7) into a fast-time/slow-time
observation matrix [28]

$\displaystyle\mathbf{Y}=\alpha\,\mathbf{W}\odot\mathbf{F}_{N}^{H}\Big{(}\mathbf{X}\odot\mathbf{b}(\tau)\mathbf{c}^{H}(\nu)\Big{)}+\mathbf{Z}\leavevmode\nobreak\
,$ (10)

where $\mathbf{W}\in\mathbb{C}^{N\times M}$ with
$\left[\mathbf{W}\right]_{\ell,m}\triangleq w_{\ell,m}(\tau)$ is the
multiplicative PN matrix, $\mathbf{F}_{N}\in\mathbb{C}^{N\times N}$ is the
unitary DFT matrix with
$\left[\mathbf{F}_{N}\right]_{\ell,n}=\frac{1}{\sqrt{N}}e^{-j2\pi
n\frac{\ell}{N}}$, $\mathbf{X}\in\mathbb{C}^{N\times M}$ contains the data
symbols with $\left[\mathbf{X}\right]_{n,m}\triangleq x_{n,m}$,
$\mathbf{Y}\in\mathbb{C}^{N\times M}$ with
$\left[\mathbf{Y}\right]_{\ell,m}\triangleq y_{\ell,m}$, and
$\mathbf{Z}\in\mathbb{C}^{N\times M}$ is additive complex white Gaussian
sensor noise with
${\rm{vec}}\left(\mathbf{Z}\right)\sim{\mathcal{CN}}({{\bm{0}}},\allowbreak
2\sigma^{2}{\bm{\mathrm{I}}})$.

###  II-D Phase Noise Statistics

Let us define the differential/self-correlated PN process [35, 36, 23] in (6)
as

$\xi(t,\tau)\triangleq\phi(t)-\phi(t-\tau)\leavevmode\nobreak\ ,$ (11)

which is a stationary process with statistics that depend on the delay $\tau$
[35, Sec. IV], [23, 5]:

$\xi(t,\tau)\sim{\mathcal{N}}(0,\sigma^{2}_{\xi}(\tau))\leavevmode\nobreak\ .$
(12)

In (12), $\sigma^{2}_{\xi}(\tau)$ represents the delay-dependent variance of
$\xi(t,\tau)$ whose form depends on the oscillator type, i.e., free-running
oscillator (FRO) or phase-locked loop (PLL). In particular, we have [35, 23,
20]

$\displaystyle\sigma^{2}_{\xi}(\tau)=\begin{cases}4\pi
f_{\rm{3dB}}\lvert\tau\rvert\leavevmode\nobreak\ ,&\text{FRO}\\\
\frac{2f_{\rm{3dB}}}{f_{\rm{loop}}}\left(1-e^{-2\pi
f_{\rm{loop}}\lvert\tau\rvert}\right)\leavevmode\nobreak\
,&\text{PLL}\end{cases}\leavevmode\nobreak\ ,$ (13)

where $f_{\rm{3dB}}$ and $f_{\rm{loop}}$ denote the $3\,\rm{dB}$ bandwidth of
the Lorentzian oscillator spectrum and the loop bandwidth of PLL,
respectively.

We denote by $\bm{\xi}\in\mathbb{R}^{NM\times 1}$ the discrete-time version of
the PN process in (11) over $NM$ time-domain samples in the considered OFDM
frame. According to (6) and (10), we have the relation
${\rm{vec}}\left(\mathbf{W}\right)=e^{-j\bm{\xi}}$. Based on [5], the
statistics of $\bm{\xi}$ are given by

$\displaystyle\bm{\xi}\sim{\mathcal{N}}({{\bm{0}}},\mathbf{R}(\tau))\leavevmode\nobreak\
,$ (14)

where $\mathbf{R}(\tau)\in\mathbb{R}^{NM\times NM}$ is the delay-dependent
covariance matrix of $\bm{\xi}$, with the entries

$\displaystyle\left[\mathbf{R}(\tau)\right]_{i_{1},i_{2}}=R_{\xi\xi}(\Delta
t_{i_{1}i_{2}},\tau)\leavevmode\nobreak\ .$ (15)

In (15),

$\displaystyle R_{\xi\xi}(\Delta t,\tau)$
$\displaystyle=\frac{\sigma^{2}_{\xi}(\tau+\Delta
t)+\sigma^{2}_{\xi}(\tau-\Delta t)}{2}-\sigma^{2}_{\xi}(\Delta
t)\leavevmode\nobreak\ ,$ (16)

is the correlation function of $\xi(t,\tau)$ [5, Lemma 1] and $\Delta
t_{i_{1}i_{2}}\triangleq(i_{1}-i_{2})T_{\rm{s}}+(m_{1}-m_{2})T_{\rm{cp}}$ for
$(i_{1},i_{2})=(n_{1}+m_{1}N,n_{2}+m_{2}N)$, where $0\leq n_{1},n_{2}\leq N-1$
and $0\leq m_{1},m_{2}\leq M-1$ are the fast-time and slow-time sample
indices, respectively, and $T_{\rm{s}}=T/N$ is the sampling interval.

###  II-E Goals

Given the observation model in (10) and the PN statistics in (14), our goal is
to characterize the performance of range and velocity estimation in the
presence of PN under a wide variety of operating conditions with regard to
SNR, target range and oscillator quality (i.e., $f_{\rm{3dB}}$ and
$f_{\rm{loop}}$). To accomplish this goal, we resort to standard CRB and MCRB
as analytical tools.

##  III CRB Analysis under Phase Noise

In this section, we derive the theoretical bounds on range and velocity
estimation with and without PN, and under different levels of knowledge on the
presence of PN and its statistics.

###  III-A Deterministic CRB without Phase Noise

In this part, we assume that the generative model in (10) does not contain PN,
i.e.,

$\displaystyle\mathbf{Y}=\alpha\,\mathbf{F}_{N}^{H}\Big{(}\mathbf{X}\odot\mathbf{b}(\tau)\mathbf{c}^{H}(\nu)\Big{)}+\mathbf{Z}\leavevmode\nobreak\
.$ (17)

In vector form, (17) can be written as

$\displaystyle\mathbf{y}=\alpha\,\mathbf{q}(\tau,\nu)+\mathbf{z}\leavevmode\nobreak\
,$ (18)

where
$\mathbf{y}\triangleq{\rm{vec}}\left(\mathbf{Y}\right)\in\mathbb{C}^{NM\times
1}$,
$\mathbf{z}\triangleq{\rm{vec}}\left(\mathbf{Z}\right)\in\mathbb{C}^{NM\times
1}$, and

$\displaystyle\mathbf{q}(\tau,\nu)$
$\displaystyle\triangleq{\rm{vec}}\left(\mathbf{F}_{N}^{H}\Big{[}\mathbf{X}\odot\mathbf{b}(\tau)\mathbf{c}^{H}(\nu)\Big{]}\right)\in\mathbb{C}^{NM\times
1}\leavevmode\nobreak\ .$ (19)

The unknown parameter vector in (18), which involves only deterministic
parameters, is given by

$\displaystyle{\bm{\eta}}=\left[\tau,\nu,\alpha_{\rm{R}},\alpha_{\rm{I}}\right]^{T}\in\mathbb{R}^{4\times
1}\leavevmode\nobreak\ ,$ (20)

where $\alpha_{\rm{R}}\triangleq\Re\left\\{\alpha\right\\}$ and
$\alpha_{\rm{I}}\triangleq\Im\left\\{\alpha\right\\}$. Then, the deterministic
CRB on the variance of an unbiased estimator $\widehat{\bm{\eta}}(\mathbf{y})$
of ${\bm{\eta}}$ in (20) can be expressed as [37]

$\displaystyle\mathbb{E}_{\mathbf{y}}\left[\left(\widehat{\bm{\eta}}(\mathbf{y})-{\bm{\eta}}\right)\left(\widehat{\bm{\eta}}(\mathbf{y})-{\bm{\eta}}\right)^{T}\right]\succeq{\mathbf{J}}^{-1}\leavevmode\nobreak\
,$ (21)

where ${\mathbf{J}}\in\mathbb{R}^{4\times 4}$ is the Fisher information matrix
(FIM), given by (using the Slepian-Bangs formula [37, Eq. (15.52)] through the
Gaussianity of the measurement in (18))

$\displaystyle\left[{\mathbf{J}}\right]_{i,j}$
$\displaystyle=\frac{1}{\sigma^{2}}\Re\left\\{\frac{\partial\bm{\mu}^{H}({\bm{\eta}})}{\partial\eta_{i}}\frac{\partial\bm{\mu}({\bm{\eta}})}{\partial\eta_{j}}\right\\}\leavevmode\nobreak\
,$ (22)

where $\bm{\mu}({\bm{\eta}})\triangleq\alpha\,\mathbf{q}(\tau,\nu)$ is the
mean of $\mathbf{y}$ in (18) and $\eta_{i}\triangleq[{\bm{\eta}}]_{i}$. Using
the FIM in (22), the CRBs on delay and Doppler estimation accuracy can be
computed as

$\displaystyle{\rm{CRB}}_{\tau}=[{\mathbf{J}}^{-1}]_{1,1}\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\
{\rm{CRB}}_{\nu}=[{\mathbf{J}}^{-1}]_{2,2}\leavevmode\nobreak\ .$ (23)

###  III-B Hybrid CRB under Phase Noise

In this part, we consider the generative model in (10) with PN, which can be
expressed in vector form as

$\displaystyle\mathbf{y}=\alpha\,\mathbf{\Xi}\mathbf{q}(\tau,\nu)+\mathbf{z}\leavevmode\nobreak\
,$ (24)

where
$\mathbf{\Xi}\triangleq{\rm{diag}}\left(e^{-j\bm{\xi}}\right)\in\mathbb{C}^{NM\times
NM}$. For the observation in (24), the unknown parameter vector is given by

$\displaystyle{\bm{\eta}}=\left[\tau,\nu,\alpha_{\rm{R}},\alpha_{\rm{I}},\bm{\xi}\right]^{T}\in\mathbb{R}^{(NM+4)\times
1}\leavevmode\nobreak\ ,$ (25)

which consists of both deterministic delay-Doppler-gain parameters and random
PN. Then, the hybrid CRB on the variance of an unbiased estimator
$\widehat{\bm{\eta}}(\mathbf{y})$ of ${\bm{\eta}}$ in (25) can be written as
[16, 17]

$\displaystyle\mathbb{E}_{\mathbf{y},\bm{\xi}}\left[\left(\widehat{\bm{\eta}}(\mathbf{y})-{\bm{\eta}}\right)\left(\widehat{\bm{\eta}}(\mathbf{y})-{\bm{\eta}}\right)^{T}\right]\succeq{\mathbf{J}}_{\rm{hyb}}^{-1}\leavevmode\nobreak\
,$ (26)

where ${\mathbf{J}}_{\rm{hyb}}\in\mathbb{R}^{(NM+4)\times(NM+4)}$ is the
hybrid FIM (HFIM), given by

$\displaystyle{\mathbf{J}}_{\rm{hyb}}={{\mathbf{J}}^{(\rm{o})}}+{{\mathbf{J}}^{(\rm{p})}}\leavevmode\nobreak\
,$ (27)

with ${{\mathbf{J}}^{(\rm{o})}}$ and ${{\mathbf{J}}^{(\rm{p})}}$ representing
the observation-related and the prior information-related HFIMs, respectively.
Using [16, Eq. (7)] and the Slepian-Bangs formula [37, Eq. (15.52)], we obtain

$\displaystyle\left[{{\mathbf{J}}^{(\rm{o})}}\right]_{i,j}$
$\displaystyle=\mathbb{E}_{\bm{\xi}}\left[\frac{1}{\sigma^{2}}\Re\left\\{\frac{\partial\bm{\mu}^{H}({\bm{\eta}})}{\partial\eta_{i}}\frac{\partial\bm{\mu}({\bm{\eta}})}{\partial\eta_{j}}\right\\}\right]\leavevmode\nobreak\
,$ (28) $\displaystyle\left[{{\mathbf{J}}^{(\rm{p})}}\right]_{i,j}$
$\displaystyle=\mathbb{E}_{\bm{\xi}}\left[\frac{\partial\log
p_{\bm{\xi}}(\bm{\xi};\tau)}{\partial\eta_{i}}\frac{\partial\log
p_{\bm{\xi}}(\bm{\xi};\tau)}{\partial\eta_{j}}\right]\leavevmode\nobreak\ ,$
(29)

where
$\bm{\mu}({\bm{\eta}})\triangleq\alpha\,\mathbf{\Xi}\mathbf{q}(\tau,\nu)$ is
the mean of $\mathbf{y}$ given $\bm{\xi}$ in (24) and $p_{\bm{\xi}}$ is the
pdf of PN as a function of delay, obtained from (14) as

$\displaystyle p_{\bm{\xi}}(\bm{\xi};\tau)$
$\displaystyle=\frac{1}{\sqrt{(2\pi)^{NM}{{{\rm{det}}\left(\mathbf{R}(\tau)\right)}}}}\exp\left\\{-\frac{\bm{\xi}^{T}\mathbf{R}(\tau)^{-1}\bm{\xi}}{2}\right\\}\leavevmode\nobreak\
.$ (30)

Consequently, the hybrid CRBs on delay and Doppler estimation accuracy are
given by

$\displaystyle{\rm{CRB}}_{\tau}=[{\mathbf{J}}_{\rm{hyb}}^{-1}]_{1,1}\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\
{\rm{CRB}}_{\nu}=[{\mathbf{J}}_{\rm{hyb}}^{-1}]_{2,2}\leavevmode\nobreak\ .$
(31)

###  III-C Misspecified CRB under Phase Noise

The previous part considers a scenario in which the receiver is aware of the
existence of PN in the observations and has the knowledge of its statistics in
(14). In this part, considering the same generative model as in Sec. III-B, we
investigate a scenario where the receiver is unaware of the existence of PN
(or, the probability model for PN is unknown) and applies standard processing
suited to the ideal, PN-free model in (18). By leveraging the MCRB tool [15,
38, 39], we aim to quantify degradation in range/velocity accuracy under PN
due to model misspecification between the true model with PN and the assumed
model without PN.

####  III-C1 True and Assumed Models for Radar Observation

In the MCRB nomenclature [15], the true model corresponds to the one in (24),
which involves the effect of PN, i.e.,

$\displaystyle\mathbf{y}=\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\alpha}\,\mathbf{\Xi}\mathbf{q}(\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\tau},\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\nu})+\mathbf{z}\leavevmode\nobreak\ $

where $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\alpha}$, $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\tau}$ and $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\nu}$ represent the true values of the unknown parameters $\alpha$, $\tau$
and $\nu$, respectively. The pdf of the true observation model in (III-C1) for
a given PN realization $\bm{\xi}$ is given by

$\displaystyle
p(\mathbf{y})=\frac{1}{(2\pi\sigma^{2})^{NM}}\exp\left\\{-\frac{\left\lVert\mathbf{y}-\bm{\mu}\right\rVert^{2}}{2\sigma^{2}}\right\\}\leavevmode\nobreak\
,$ (32)

where $\bm{\mu}\triangleq\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\alpha}\,\mathbf{\Xi}\mathbf{q}(\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\tau},\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\nu})\in\mathbb{C}^{NM\times 1}$.

In practice, PN is usually ignored in radar processing, leading to the assumed
model without the impact of PN:

$\displaystyle\mathbf{y}=\alpha\,\mathbf{q}(\tau,\nu)+\mathbf{z}\leavevmode\nobreak\
.$ (33)

For the assumed model in (33), the misspecified parametric pdf is given by
[15]

$\displaystyle\widetilde{p}(\mathbf{y}\lvert{\bm{\eta}})=\frac{1}{(2\pi\sigma^{2})^{NM}}\exp\left\\{-\frac{\left\lVert\mathbf{y}-\widetilde{\bm{\mu}}({\bm{\eta}})\right\rVert^{2}}{2\sigma^{2}}\right\\}\leavevmode\nobreak\
,$ (34)

where
$\widetilde{\bm{\mu}}({\bm{\eta}})\triangleq\alpha\,\mathbf{q}(\tau,\nu)\in\mathbb{C}^{NM\times
1}$ and
${\bm{\eta}}=\left[\tau,\nu,\alpha_{\rm{R}},\alpha_{\rm{I}}\right]^{T}$ is the
unknown parameter vector.

####  III-C2 Pseudo-True Parameter

To derive the expression for MCRB, we first define the pseudo-true parameter
[15]

${\bm{\eta}}_{0}=\arg\min_{{\bm{\eta}}}\leavevmode\nobreak\
\mathcal{D}\left(p(\mathbf{y})\leavevmode\nobreak\ \|\leavevmode\nobreak\
\widetilde{p}(\mathbf{y}\lvert{\bm{\eta}})\right),$ (35)

where $\mathcal{D}\left(p(\mathbf{y})\leavevmode\nobreak\
\|\leavevmode\nobreak\ \widetilde{p}(\mathbf{y}\lvert{\bm{\eta}})\right)$ is
the Kullback-Leibler (KL) divergence between the true pdf in (32) and the
assumed pdf in (34). Based on [40, Lemma 1], (35) is equivalent to

$\displaystyle{\bm{\eta}}_{0}$
$\displaystyle=\arg\min_{{\bm{\eta}}}\leavevmode\nobreak\
\left\lVert\bm{\mu}-\widetilde{\bm{\mu}}({\bm{\eta}})\right\rVert^{2}\leavevmode\nobreak\
,$ (36) $\displaystyle(\alpha_{0},\tau_{0},\nu_{0})$
$\displaystyle=\arg\min_{\alpha,\tau,\nu}\leavevmode\nobreak\
\left\lVert\bm{\mu}-\alpha\,\mathbf{q}(\tau,\nu)\right\rVert^{2}\leavevmode\nobreak\
.$ (37)

In (37), $\alpha_{0}$ is readily obtained as
$\alpha_{0}=\mathbf{q}(\tau,\nu)^{\dagger}\bm{\mu}$ as a function of $\tau$
and $\nu$, where
$\mathbf{A}^{\dagger}=(\mathbf{A}^{H}\mathbf{A})^{-1}\mathbf{A}^{H}$ denotes
the pseudo-inverse of $\mathbf{A}$. Plugging $\alpha_{0}$ back into (37), we
have

$\displaystyle(\tau_{0},\nu_{0})$
$\displaystyle=\arg\max_{\tau,\nu}\leavevmode\nobreak\
\frac{\lvert\bm{\mu}^{H}\mathbf{q}(\tau,\nu)\rvert^{2}}{\left\lVert\mathbf{q}(\tau,\nu)\right\rVert^{2}}=\arg\max_{\tau,\nu}\leavevmode\nobreak\
\lvert\bm{\mu}^{H}\mathbf{q}(\tau,\nu)\rvert^{2}\leavevmode\nobreak\ ,$

where
$\left\lVert\mathbf{q}(\tau,\nu)\right\rVert^{2}=\left\lVert\mathbf{X}\right\rVert_{F}^{2}$
is constant.

####  III-C3 MCRB and LB Derivation

Using the pseudo-true parameter in (35), the MCRB matrix is given by [15, 38]

$\displaystyle{\rm{MCRB}}({\bm{\eta}}_{0})=\mathbf{A}_{{\bm{\eta}}_{0}}^{-1}\mathbf{B}_{{\bm{\eta}}_{0}}\mathbf{A}_{{\bm{\eta}}_{0}}^{-1}\in\mathbb{R}^{4\times
4}\leavevmode\nobreak\ ,$ (38)

where the entries of $\mathbf{A}_{{\bm{\eta}}_{0}}\in\mathbb{R}^{4\times 4}$
and $\mathbf{B}_{{\bm{\eta}}_{0}}\in\mathbb{R}^{4\times 4}$ are given by

$\displaystyle[\mathbf{A}_{{\bm{\eta}}_{0}}]_{ij}$
$\displaystyle=\mathbb{E}_{p}\left\\{\frac{\partial^{2}}{\partial\eta_{i}\partial\eta_{j}}\log\widetilde{p}(\mathbf{y}\lvert{\bm{\eta}})\Big{|}_{{\bm{\eta}}={\bm{\eta}}_{0}}\right\\},$
(39) $\displaystyle[\mathbf{B}_{{\bm{\eta}}_{0}}]_{ij}$
$\displaystyle=\mathbb{E}_{p}\left\\{\frac{\partial}{\partial\eta_{i}}\log\widetilde{p}(\mathbf{y}\lvert{\bm{\eta}})\frac{\partial}{\partial\eta_{j}}\log\widetilde{p}(\mathbf{y}\lvert{\bm{\eta}})\Big{|}_{{\bm{\eta}}={\bm{\eta}}_{0}}\right\\},$
(40)

with $\mathbb{E}_{p}\left\\{\cdot\right\\}$ denoting the expectation over the
true pdf in (32) (see [40] for derivation of $\mathbf{A}_{{\bm{\eta}}_{0}}$
and $\mathbf{B}_{{\bm{\eta}}_{0}}$). The MCRB provides a lower bound on the
covariance matrix of any misspecified-unbiased (MS-unbiased) estimator of
${\bm{\eta}}_{0}$, i.e., [15]

$\displaystyle\mathbb{E}_{p}\\{(\widehat{\bm{\eta}}(\mathbf{y})-{\bm{\eta}}_{0})(\widehat{\bm{\eta}}(\mathbf{y})-{\bm{\eta}}_{0})^{T}\\}\succeq{\rm{MCRB}}({\bm{\eta}}_{0}),$
(41)

where $\widehat{\bm{\eta}}(\mathbf{y})$ is an MS-unbiased estimator of
${\bm{\eta}}_{0}$, derived under the misspecified model, i.e.,
$\mathbb{E}_{p}\left\\{\widehat{\bm{\eta}}(\mathbf{y})\right\\}={\bm{\eta}}_{0}$.
Based on the MCRB, the covariance matrix of any MS-unbiased estimator
$\widehat{\bm{\eta}}(\mathbf{y})$ with respect to the true parameter
$\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{{\bm{\eta}}}\triangleq[\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\alpha}_{\rm{R}},\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\alpha}_{\rm{I}},\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\tau},\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\nu}]^{T}$ can be lower-bounded as [15]

$\displaystyle\mathbb{E}_{p}\\{(\widehat{\bm{\eta}}(\mathbf{y})-\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{{\bm{\eta}}})(\widehat{\bm{\eta}}(\mathbf{y})-\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{{\bm{\eta}}})^{T}\\}\succeq{\rm{LB}}({\bm{\eta}}_{0}),$ (42)

where

$\displaystyle{\rm{LB}}({\bm{\eta}}_{0})={\rm{MCRB}}({\bm{\eta}}_{0})+\overbrace{(\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{{\bm{\eta}}}-{\bm{\eta}}_{0})(\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{{\bm{\eta}}}-{\bm{\eta}}_{0})^{T}}^{\rm{Bias}}\leavevmode\nobreak\ .$ (43)

####  III-C4 Expected Performance under PN

The LB in (43) is a deterministic bound that can be used to characterize the
performance of delay-Doppler estimation in the presence of PN for a given
realization $\bm{\xi}$, where the observations are generated using the true
model in (32) and the estimator design is based on the assumed model in (34).
We can obtain the expected performance by averaging the LB over the
distribution of PN in (14) [14], i.e.,

$\displaystyle{\rm{LB}}^{{\rm{avg}}}({\bm{\eta}}_{0})=\mathbb{E}_{p_{\bm{\xi}}}\\{{\rm{LB}}({\bm{\eta}}_{0})\\}\leavevmode\nobreak\
.$ (44)

Using (44), the expected LBs on delay and Doppler estimation accuracy are
given by

$\displaystyle{\rm{LB}}^{{\rm{avg}}}_{\tau}=[{\rm{LB}}^{{\rm{avg}}}({\bm{\eta}}_{0})]_{1,1}\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\
{\rm{LB}}^{{\rm{avg}}}_{\nu}=[{\rm{LB}}^{{\rm{avg}}}({\bm{\eta}}_{0})]_{2,2}\leavevmode\nobreak\
.$ (45)

##  IV Simulation Results

To evaluate the impact of PN on the radar performance, we consider the
simulation setup in Table I in compliance with 5G NR FR2 parameters [41]. For
the observations in (10), we define the SNR as
${\rm{SNR}}=\lvert\alpha\rvert^{2}/(2\sigma^{2})$ and the data symbols
$\mathbf{X}$ are drawn randomly from the QPSK alphabet. In addition, the
oscillator parameters are set to $f_{\rm{3dB}}=100\,\rm{kHz}$ and
$f_{\rm{loop}}=1\,\rm{MHz}$ (in the case of PLL) [35]. Moreover, a target with
range $R=50\,\rm{m}$ and velocity $v=20\,\rm{m/s}$ is considered. For
performance evaluation, we investigate the following theoretical bounds in our
simulations:

  * •

CRB (PN-free): The deterministic CRBs on delay and Doppler estimation in (23).

  * •

CRB: The hybrid CRBs on delay and Doppler estimation in (31).

  * •

LB: The LB on delay and Doppler estimation in (45), computed using numerical
averaging over $100$ PN realizations in (44).

The CRB (PN-free) serves as an ideal, hypothetical baseline to show
performance losses due to PN with respect to the case where the receiver is
aware/unaware of the existence of PN, corresponding to CRB/LB, respectively.

Table I: OFDM Parameters Parameter | Value
---|---
Carrier Frequency, $f_{c}$ | $28\,\rm{GHz}$
Total Bandwidth, $B$ | $30.72\,\rm{MHz}$
Number of Subcarriers, $N$ | $256$
Number of Symbols, $M$ | $10$
Subcarrier Spacing, $\Delta f$ | $120\,\rm{kHz}$
Symbol Duration, $T$ | $8.33\,\mu\rm{s}$
Cyclic Prefix Duration, $T_{\rm{cp}}$ | $0.58\,\mu\rm{s}$
Total Symbol Duration, $T_{\rm{sym}}$ | $8.91\,\rm{\mu s}$

###  IV-A Performance with respect to SNR

In Fig. 2 and Fig. 3, we show the range and velocity estimation
performances111The y-axes are labeled as root-mean-squared error (RMSE) to
have a common name for all the bounds. with respect to SNR for FRO and PLL
architectures, respectively. Comparing the CRB (PN-free) and the CRB curves,
it is observed that when the receiver is aware of the existence of PN, range
accuracy is only slightly degraded, whereas velocity accuracy reaches a
plateau at a certain SNR value, leading to considerable losses in performance
at high SNRs. This results from high (low) correlation of PN in fast-time
(slow-time) [5]. Therefore, the impact of PN on ranging accuracy can be
mitigated by utilizing the PN statistics in the estimation process.

Looking at the LB curves, we see that being unaware of the existence of PN
causes performance saturation in both range and velocity estimation at medium
and high SNR regimes. This is due to the model mismatch between the true and
assumed models in Sec. III-C1, leading to the SNR-independent bias term in
(43), which dominates the LB expression as the SNR increases [40]. Hence, PN
can be safely ignored at low SNRs, while it should be compensated for in radar
processing in medium-to-high SNR regimes, especially for emerging 6G sensing
applications with stringent accuracy requirements [42]. Finally, a comparative
analysis of the CRB and LB curves indicates an important distinction between
FRO and PLL. Namely, in velocity estimation, a certain degree of performance
loss due to PN can be recovered in the case of PLL, whereas with FRO almost no
improvement over the PN-unaware case (i.e., LB) is possible by exploiting the
PN statistics. The reason is that PN in FRO has no slow-time correlation to
exploit, unlike in PLL (see (13) and [5, Lemma 2] for details).

Figure 2: Bounds on 2 range and 2 velocity accuracy with respect to SNR for
FRO with $f_{\rm{3dB}}=100\,\rm{kHz}$.

Figure 3: Bounds on 2 range and 2 velocity accuracy with respect to SNR for
PLL with $f_{\rm{3dB}}=100\,\rm{kHz}$ and $f_{\rm{loop}}=1\,\rm{MHz}$.

###  IV-B Performance with respect to Target Range

In this part, considering the delay-dependent PN statistics given in (14), we
evaluate the radar performance under PN with respect to target range for fixed
SNR, as illustrated in Fig. 4 for FRO with $f_{\rm{3dB}}=100\,\rm{kHz}$. For
the LB, we observe deteriorating performance in both range and velocity
estimation as the target moves away from the radar due to increasing PN
variance with range, as seen from (13). For the CRB, ranging accuracy stays
constant with target range because PN can be compensated effectively in fast-
time through its high level of correlation. On the other hand, the CRB on
velocity increases with target range due to the absence of slow-time PN
correlation for FRO.

Figure 4: Bounds on 4 range and 4 velocity accuracy with respect to target
range at ${\rm{SNR}}=20\,\rm{dB}$ for FRO with $f_{\rm{3dB}}=100\,\rm{kHz}$.

###  IV-C Performance with respect to Oscillator Quality

In this part, we assess the impact of PN for different levels of oscillator
quality. Fig. 5 and Fig. 6 show the theoretical limits on range and velocity
estimation against $f_{\rm{3dB}}$ and $f_{\rm{loop}}$ for FRO and PLL,
respectively. It is seen from the CRB curves that ranging accuracy remains
almost constant with respect to worsening oscillator quality (i.e., increasing
$f_{\rm{3dB}}$ and decreasing $f_{\rm{loop}}$), indicating that PN can be
mitigated in fast-time by using the PN statistics. However, due to similar
reasons as in the previous subsections, the CRB on velocity increases with
decreasing oscillator quality. Moreover, in the case of the LB, the range and
velocity accuracy exhibit similar trends due to model misspecification, as
expected.

Figure 5: Bounds on 6 range and 6 velocity accuracy with respect to
$f_{\rm{3dB}}$ of FRO at ${\rm{SNR}}=20\,\rm{dB}$.

Figure 6: Bounds on 6 range and 6 velocity accuracy with respect to
$f_{\rm{loop}}$ of PLL with $f_{\rm{3dB}}=100\,\rm{kHz}$ at
${\rm{SNR}}=20\,\rm{dB}$.

##  V Concluding Remarks

We have investigated the impact of PN on range and velocity accuracy in an
OFDM ISAC system by leveraging the CRB and MCRB tools. It has been observed
that when PN is ignored in radar processing (i.e., model mismatch), it leads
to an error floor in both range and velocity estimation, which can create a
bottleneck for high-accuracy ($\rm{cm}$-level or $\rm{dm/s}$-level) sensing
applications towards dual-functional 6G networks. When PN statistics are taken
into account in the estimation process, the radar receiver experiences only
marginal degradation in ranging accuracy due to high correlation of PN in
fast-time, as opposed to substantial loss in velocity accuracy, especially at
high SNRs. Moreover, the range correlation effect in monostatic sensing has
been observed through delay-dependent PN statistics, which leads to lower
accuracy for farther targets.

## Acknowledgments

This work is supported, in part, by Vinnova Grant 2021-02568 and MSCA-IF grant
888913 (OTFS-RADCOM).

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