# Fundamental Trade-Offs in Monostatic ISAC: A Holistic Investigation Towards
6G

Musa Furkan Keskin, Member, IEEE, Mohammad Mahdi Mojahedian, Member, IEEE,
Carina Marcus, Olof Eriksson, Andrea Giorgetti, Senior Member, IEEE, Joerg
Widmer, Fellow, IEEE, Henk Wymeersch, Fellow, IEEE This work is supported by
the Vinnova RADCOM2 project under Grant 2021-02568.

###### Abstract

Integrated sensing and communication (ISAC) emerges as a cornerstone
technology for the upcoming 6G era, seamlessly incorporating sensing
functionality into wireless networks as an inherent capability. This paper
undertakes a holistic investigation of two fundamental trade-offs in
monostatic OFDM ISAC systems—namely, the time-frequency domain trade-off and
the spatial domain trade-off. To ensure robust sensing across diverse
modulation orders in the time-frequency domain, including high-order QAM, we
design a linear minimum mean-squared-error (LMMSE) estimator tailored for
sensing with known, randomly generated signals of varying amplitude. Moreover,
we explore spatial domain trade-offs through two ISAC transmission strategies:
concurrent, employing joint beams, and time-sharing, using separate, time-non-
overlapping beams for sensing and communications. Simulations demonstrate
superior performance of the LMMSE estimator in detecting weak targets in the
presence of strong ones under high-order QAM, consistently yielding more
favorable ISAC trade-offs than existing baselines. Key insights into these
trade-offs under various modulation schemes, SNR conditions, target radar
cross section (RCS) levels and transmission strategies highlight the merits of
the proposed LMMSE approach.

Index Terms– OFDM, ISAC, monostatic sensing, LMMSE estimator, time-frequency
trade-off, spatial trade-off, concurrent transmission, time-sharing
transmission.

##  I Introduction

###  I-A Background and Motivation

As research and standardization efforts for 6G intensify, integrated sensing
and communications (ISAC) stands out as a key enabler that can simultaneously
facilitate high-quality connectivity and endow networks with intrinsic sensing
capability [1, 2, 3, 4, 5, 6]. This convergence promises to revolutionize
network functionalities that extend beyond traditional communication-only
paradigms by making sensing a fundamentally built-in component rather than an
add-on feature [7, 8]. In the broad spectrum of ISAC configurations, the
monostatic approach is gaining momentum, especially given its ability to
exploit the entire communication data for sensing purposes and its potential
to mitigate the stringent synchronization challenges that often plague
bistatic and multistatic systems [9, 10, 11]. Among various waveform
candidates for ISAC implementation, the orthogonal frequency-division
multiplexing (OFDM) waveform emerges as a natural choice owing to its
widespread adoption across prevailing wireless standards such as 5G,
5G-Advanced, WiFi/WLAN and DVB-T [12]. The inherent characteristics of OFDM,
including its high spectral efficiency, immunity to multipath effects, high-
accuracy and low-complexity radar operation and design flexibility, make it an
ideal choice for ISAC applications [13, 14, 15, 16, 17, 18]. There also exist
studies on single-carrier ISAC that utilize the preamble [19, 20] or both the
preamble and payload [21, 22] of IEEE802.11ad waveforms for automotive
monostatic sensing.

In the context of monostatic ISAC, OFDM radar sensing with 5G waveforms has
recently received increasing attention. In [9], radar performance evaluation
of a BS acting as monostatic radar has been carried out, considering 5G NR
OFDM waveform compliant with 3GPP Release 16 and using various radar
performance metrics, including angle, range, Doppler and localization
accuracies and probability of detection. The authors have also assessed the
impact of self-interference due to imperfect isolation in full-duplex
operation on the sensing performance. In [23], the sensing performance (range
and velocity estimation) of 5G PRS signals is investigated using the Cramér-
Rao bound (CRB) and the root mean-squared error (RMSE). The authors have also
investigated the CRB of range estimation for positioning using PRS signals.
The feasibility and superiority of PRS over other pilot signals in radar
sensing is demonstrated via simulations. Inspired by the results for 5G PRS-
based sensing, the authors also provide recommendations on 5G-Advanced and 6G
frame structure design (i.e., where and how much to place sensing reference
signals). Moreover, in [24], a 5G-specification-compliant beam management
scheme based on sensing is proposed using SSB and RS signals and a time-
frequency resource allocation scheme is presented to minimize beam
misalignment probability. The study in [25] performs a link-level analysis of
5G NR downlink V2I sensing-assisted communication system by explicitly
considering 5G NR frame structure containing Physical Downlink Control Channel
(PDCCH) and Physical Downlink Shared Channel (PDSCH) (consisting of data,
DMRS, CSI-RS and PTRS). In [26], different 5G signals and channels (SSB,
PDCCH, PDSCH and CSI-RS) are investigated in terms of their delay-Doppler AF
characteristics in both monostatic and bistatic configurations. A passive
bistatic sensing demonstration with a transmitter BS and a receiver with
surveillance and reference antennas has been performed to experimentally
validate the sensing capability of 5G signals. Furthermore, the study in [27]
investigates the joint utilization of CSI-RS, PRS and DMRS signals for radar
sensing and showcases corresponding improvements in sensing accuracy.

In [28], zero-padded OFDM has been proposed to deal with the issue of full-
duplex ISAC by placing a guard interval between OFDM symbols, in which the
transmitter is off and the sensing receiver is collecting echoes in the
absence of self-interference. In [29], detection probability vs. rate trade-
off has been investigated via power allocation optimization for the
sensing/communication weight. In [30], a 3D joint range-angle-velocity
estimation algorithm has been proposed for MIMO-OFDM ISAC systems, using DFT
and symbol-division operations. The study in [18] adopts a communication-
centric ISAC philosophy, aiming to maximize the communication rate under radar
beampattern matching constraints. In particular, a subcarrier allocation
strategy has been developed to improve the ISAC trade-offs in a MU-MIMO ISAC
system, and per-subcarrier digital precoders, the analog precoder and
subcarrier selection are jointly designed using convex optimization techniques
and Riemannian manifold optimization.

Recent works have focused on inherent trade-offs arising in OFDM ISAC systems,
including the time-frequency trade-off and the spatial trade-off, also known
as the deterministic-random trade-off [31] and the subspace trade-off [6],
respectively. The time-frequency trade-off arises from selecting the
modulation order for random data: higher-order QAM enhances communication
rates but compromises sensing performance, as the random, non-constant-modulus
data leads to elevated side-lobe levels. On the other hand, the spatial trade-
off is linked to the selection of ISAC beamforming strategies that focus on
illuminating specific angular sectors for sensing and communication purposes.
In [31, 32], the time-frequency trade-off is investigated by formulating a
generic ISAC trade-off optimization problem to optimize the input data
distribution. The study in [33] considers sensing with random ISAC signals and
proposes a new sensing metric to characterize the accuracy of sensing channel
estimation. In [34], a monostatic ISAC system with sensing and communication
waveform superposition has been proposed. The idea is to superpose the sensing
signal onto a 5G NR OFDM communication signal via an optimized power
allocation between sensing and communication components. To illustrate the
benefits of superposition, the authors have also performed experimental
demonstrations at 60 GHz. Moreover, [35] studies the time-frequency trade-off
by considering the ISAC waveform as a random but known nuisance parameter in
the sensing signal model and characterizing the CRB-rate region. In [36],
time-sharing schemes have been considered for an ISAC system from an
information-theoretic perspective and the capacity-distortion region has been
proposed to characterize the ISAC trade-offs, resulting from the common choice
of input distribution for both functionalities. Finally, the study in [17]
proposes a probabilistic constellation shaping (PCS) approach to maximize the
achievable rate under constraints on the variance of the radar ambiguity
function with random OFDM data.

###  I-B Research Gaps

In light of the existing literature on monostatic ISAC systems, several
fundamental questions remain unanswered:

  * •

How can robust sensing algorithms be designed for monostatic OFDM ISAC
systems, accommodating known yet varying-amplitude random data symbols in the
time-frequency domain generated by high-order modulation schemes? To what
extent can we mitigate the masking effect in both single- and multi-target
environments, resulting from increased side-lobe levels induced by non-
constant-modulus data?

  * •

What spatial domain trade-offs arise from the choice of ISAC transmit
beamformers? How do different transmission strategies, namely, concurrent and
time-sharing transmission, perform under various settings concerning the
sensing algorithm, percentage of sensing-dedicated pilots and modulation order
of random data?

  * •

Given the inherent tension between sensing and communication (S&C) in time-
frequency and spatial domains, how can communication data rates and
accompanying ISAC trade-offs be realistically evaluated under diverse
modulation orders and power allocations between sensing and communication
beams?

Figure 1: Monostatic ISAC system featuring a monostatic ISAC transceiver,
which integrates an ISAC transmitter and a sensing receiver on the same
hardware platform, and a communications receiver on a separate device. In
concurrent transmission, sensing receiver utilizes all communication data,
with transmit power distributed between sensing and communication beams. In
time-sharing transmission, dedicated sensing and communications beams are
transmitted in a time-multiplexed fashion.

###  I-C Contributions

With the aim of addressing the identified research gaps for the forthcoming 6G
era, this study performs a comprehensive investigation of fundamental trade-
offs in monostatic OFDM ISAC systems by introducing novel sensing algorithms,
beamforming/transmission strategies and holistic performance evaluations
across a diverse spectrum of 5G/6G settings. The main contributions can be
summarized as follows:

  * •

Sensing with Non-Constant-Modulus Data in OFDM ISAC: Time-Frequency Domain
Trade-Offs: We propose a novel OFDM radar sensing algorithm based on an LMMSE
estimate of the time-frequency domain radar channel, specifically tailored for
detection and delay-Doppler-angle estimation with non-constant-modulus random
communication data in monostatic ISAC systems. The proposed algorithm can
effectively deal with diverse modulation orders and broad range of multi-
target sensing environments.

  * •

ISAC Transmission Strategies: Spatial Domain Trade-Offs: To evaluate the
spatial domain trade-offs, we investigate two ISAC transmission strategies:
concurrent transmission, where the same beam is used for S&C, and time-sharing
transmission, which uses dedicated beams for S&C non-overlapping in time. We
also introduce a mutual information (MI) approximation method that leverages
Monte Carlo sampling techniques to evaluate data rates and the resulting ISAC
trade-offs under various modulation schemes.

  * •

Holistic Performance Investigation: To offer comprehensive and insightful
guidelines for the design of 6G systems, we carry out extensive simulations to
investigate fundamental ISAC trade-offs within the time-frequency and spatial
domains, considering a wide array of transmission settings and channel
conditions. These settings include factors such as modulation order,
concurrent vs. time-sharing transmission, power and time allocation between
S&C and the SNR of the sensing environment. The proposed LMMSE estimator
demonstrates substantial improvements in the trade-offs between probability of
detection and achievable rate compared to existing baselines. It also
increases the dynamic range in sensing by up to $8\,\rm{dB}$, significantly
mitigating the masking effect (due to QAM signaling) and effectively
recovering the performance attainable with constant-modulus symbols.
Furthermore, we identify an intricate, scenario-dependent interplay between
concurrent and time-sharing strategies in ISAC trade-off dynamics, determined
by two opposing factors: full-data utilization for sensing and impact of QAM
data on sensing, which are beneficial and detrimental, respectively, for the
concurrent strategy.

##  II System Model and Problem Description

Consider a monostatic OFDM ISAC system consisting of two entities, as shown in
Fig. 1: a multiple-antenna dual-functional ISAC transceiver and a single-
antenna communications receiver (RX). The ISAC transceiver contains (i) an
ISAC transmitter (TX) with an $N_{\rm{T}}$-element uniform linear array (ULA)
that sends data/pilot symbols to the communications RX and (ii) a sensing RX
with an $N_{\rm{R}}$-element ULA that performs radar sensing by processing the
backscattered signals for target detection, estimation and tracking [7, 5,
13]. In the monostatic configuration under consideration, the sensing RX is
co-located on the same device as the ISAC TX, thereby sharing the same
oscillator and having access to the entire OFDM transmit signal [37, 9, 11].
To ensure that the sensing RX experiences no self-interference during full-
duplex operation, we assume that the TX/RX antennas at the ISAC transceiver
are sufficiently isolated [13, 38, 39, 19, 40, 7, 41]. In this section, we
introduce the OFDM transmit signal model, different ISAC
transmission/beamforming strategies and provide received signal models at both
the sensing and communication RXs. Additionally, we formulate the problems of
interest tackled throughout the paper.

###  II-A Transmit Signal Model

We consider an OFDM frame (e.g., in 5G NR [42, 43]) with $N$ subcarriers and
$M$ symbols. The complex baseband transmit signal for the ${m^{\rm{th}}}$
symbol can be expressed as [13, 14, 9]

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

where $x_{n,m}$ is the data/pilot on the ${n^{\rm{th}}}$ subcarrier and the
${m^{\rm{th}}}$ symbol, $\Delta f=1/T$ is the subcarrier spacing with $T$
representing the elementary symbol duration, $T_{\rm{sym}}=T+T_{\rm{cp}}$ is
the total symbol duration including the cyclic prefix (CP) $T_{\rm{cp}}$, and
$g(t)$ is a rectangular pulse shape that takes the value $1$ for $t\in[0,1]$
and $0$ otherwise. Employing a single-stream beamforming model [19, 44, 45,
46, 9], the passband transmit signal over the TX array for the entire OFDM
frame is given by

$\displaystyle\Re\left\\{\sum_{m=0}^{M-1}\mathbf{f}_{m}s_{m}(t)e^{j2\pi
f_{c}t}\right\\}\,,$ (2)

where $\mathbf{f}_{m}\in\mathbb{C}^{N_{\rm{T}}\times 1}$ is the TX beamforming
(BF) vector applied for the ${m^{\rm{th}}}$ symbol and $f_{c}$ is the carrier
frequency. Denoting by ${P_{\rm{T}}}$ the transmit power, we set
$\left\lVert\mathbf{f}_{m}\right\rVert^{2}={P_{\rm{T}}}\,\forall m$ and
$\mathbb{E}\\{\lvert x_{n,m}\rvert^{2}\\}=1$.

###  II-B ISAC Transmission Strategies

We investigate two ISAC transmission strategies concerning the choice of
$x_{n,m}$ and $\mathbf{f}_{m}$ in (2).

####  II-B1 Concurrent Transmission

In the concurrent transmission, a common beam is utilized simultaneously for
S&C. Using the multibeam approach, the TX BF vector employed at the
${m^{\rm{th}}}$ symbol is given by [47, 9]

$\displaystyle\mathbf{f}_{m}=\sqrt{\rho}\,\mathbf{f}_{m,{\rm{s}}}+\sqrt{1-\rho}\,\mathbf{f}_{m,{\rm{c}}}\,,$
(3)

where $\mathbf{f}_{m,{\rm{s}}}\in\mathbb{C}^{N_{\rm{T}}\times 1}$ and
$\mathbf{f}_{m,{\rm{c}}}\in\mathbb{C}^{N_{\rm{T}}\times 1}$ represent,
respectively, the sensing and communication BF vectors, and $0\leq\rho\leq 1$
denotes the ISAC weight that enables controlling the trade-off between S&C. In
this strategy, all $x_{n,m}$’s are assumed to be data symbols111For ease of
exposition and analysis, the communication channel is assumed to be perfectly
estimated a-priori, eliminating the need for pilot symbols. While the proposed
framework can theoretically be extended to account for the impact of
communication pilots on ISAC performance trade-offs, exploring this aspect is
beyond the scope of the current study and is reserved for future research.
intended for the communications RX, while the sensing RX exploits the entire
frame for radar sensing.

####  II-B2 Time-Sharing Transmission

In the time-sharing transmission, dedicated beams for sensing and
communications are used in non-overlapping time slots [36]. Thus, at each OFDM
symbol, the ISAC TX transmits either a sensing beam or a communication beam.
More formally,

$\displaystyle\mathbf{f}_{m}=\begin{cases}\mathbf{f}_{m,{\rm{s}}},&\leavevmode\nobreak\
m\in\mathcal{S}\\\ \mathbf{f}_{m,{\rm{c}}},&\leavevmode\nobreak\
m\in\mathcal{C}\end{cases}\,,\leavevmode\nobreak\
x_{n,m}=\begin{cases}{\rm{pilot}},&\leavevmode\nobreak\ m\in\mathcal{S}\\\
{\rm{data}},&\leavevmode\nobreak\ m\in\mathcal{C}\end{cases}\,,$ (4)

where $\mathcal{S}\cup\mathcal{C}=\\{0,\ldots,M-1\\}$ and
$\mathcal{S}\cap\mathcal{C}=\emptyset$. In (4), ‘pilot’ refers to dedicated
sensing pilots with unit amplitude, whereas ‘data’ could be either unit-
amplitude (e.g., QPSK) or varying-amplitude (e.g., QAM). The sensing RX
utilizes only the pilots for sensing, while the communications rate might be
compromised due to the replacement of a portion of data by pilots. The time-
sharing ratio $\lvert\mathcal{S}\rvert/M$ allows tuning the trade-off between
S&C.

###  II-C Sensing Signal Model

Given the transmit signal in (1) and (2), the backscattered signal at the
sensing RX array over subcarrier $n$ and symbol $m$ after the CP removal and
FFT operations can be written as [48, 9, 41, 16]

$\displaystyle\mathbf{y}_{n,m}=\mathbf{H}_{n,m}\mathbf{f}_{m}x_{n,m}+\mathbf{n}_{n,m}\in\mathbb{C}^{N_{\rm{R}}\times
1}\,,$ (5)

where $\mathbf{H}_{n,m}\in\mathbb{C}^{N_{\rm{R}}\times N_{\rm{T}}}$ denotes
the channel matrix at the ${n^{\rm{th}}}$ subcarrier and the ${m^{\rm{th}}}$
symbol, and $\mathbf{n}_{n,m}\in\mathbb{C}^{N_{\rm{R}}\times 1}$ is the
additive white Gaussian noise (AWGN) component with
$\mathbf{n}_{n,m}\sim{\mathcal{CN}}({{\bm{0}}},\sigma^{2}{\bm{\mathrm{I}}})$
and $\sigma^{2}=N_{0}N\Delta f$, with $N_{0}$ representing the noise power
spectral density (PSD).

Considering the presence of $K$ point targets in the environment, the sensing
channel can be expressed as

$\displaystyle\mathbf{H}_{n,m}=\sum_{k=0}^{K-1}\alpha_{k}e^{-j2\pi n\Delta
f\tau_{k}}e^{j2\pi
mT_{\rm{sym}}\nu_{k}}\mathbf{a}_{\rm{R}}(\theta_{k})\mathbf{a}_{\rm{T}}^{T}(\theta_{k})\,,$
(6)

where $\alpha_{k}$, $\tau_{k}$, $\nu_{k}$ and $\theta_{k}$ denote,
respectively, the complex channel gain (including the effects of path
attenuation and radar cross section (RCS)), round-trip delay, Doppler shift
and angle-of-arrival (AOA)/ angle-of-departure (AOD) of the ${k^{\rm{th}}}$
target. Here, $\tau_{k}=2d_{k}/c$ and $\nu_{k}=2v_{k}/\lambda$, with $c$,
$\lambda=c/f_{c}$, $d_{k}$ and $v_{k}$ denoting the speed of propagation, the
wavelength, the range and velocity of the ${k^{\rm{th}}}$ target,
respectively. In addition, the channel gain is given by the radar range
equation
$\lvert\alpha_{k}\rvert^{2}={\sigma_{{\rm{rcs}},k}}\lambda^{2}/[(4\pi)^{3}d_{k}^{4}]$
[49, Eq. (2.8)], where ${\sigma_{{\rm{rcs}},k}}$ denotes the RCS of the
${k^{\rm{th}}}$ target. Moreover, the ULA steering vectors are defined as

$\displaystyle\mathbf{a}_{\rm{T}}(\theta)$
$\displaystyle={\left[1\leavevmode\nobreak\
e^{j\frac{2\pi}{\lambda}d\sin(\theta)}\leavevmode\nobreak\
\ldots\leavevmode\nobreak\
e^{j\frac{2\pi}{\lambda}d(N_{\rm{T}}-1)\sin(\theta)}\right]}^{T}\,,$ (7)
$\displaystyle\mathbf{a}_{\rm{R}}(\theta)$
$\displaystyle={\left[1\leavevmode\nobreak\
e^{j\frac{2\pi}{\lambda}d\sin(\theta)}\leavevmode\nobreak\
\ldots\leavevmode\nobreak\
e^{j\frac{2\pi}{\lambda}d(N_{\rm{R}}-1)\sin(\theta)}\right]}^{T}\,,$ (8)

where $d=\lambda/2$ denotes the antenna element spacing.

###  II-D Communications Signal Model

The signal received at the communications RX over subcarrier $n$ and symbol
$m$ is given by

$\displaystyle
y^{{\rm{com}}}_{n,m}=(\mathbf{h}^{{\rm{com}}}_{n,m})^{T}\mathbf{f}_{m}x_{n,m}+z_{n,m}\in\mathbb{C}\leavevmode\nobreak\
,$ (9)

where $\mathbf{h}^{{\rm{com}}}_{n,m}\in\mathbb{C}^{N_{\rm{T}}\times 1}$ is the
communication channel over subcarrier $n$ and symbol $m$. In addition,
$z_{n,m}$ is AWGN with $z_{n,m}\sim{\mathcal{CN}}(0,{\sigma^{2}_{\rm{c}}})$.
Assuming the existence of $\widetilde{K}$ paths between the ISAC TX and the
communications RX, $\mathbf{h}^{{\rm{com}}}_{n,m}$ can be modeled as

$\displaystyle\mathbf{h}^{{\rm{com}}}_{n,m}=\sum_{k=0}^{\widetilde{K}-1}\widetilde{\alpha}_{k}e^{-j2\pi
n\Delta f\widetilde{\tau}_{k}}e^{j2\pi
mT_{\rm{sym}}\widetilde{\nu}_{k}}\mathbf{a}_{\rm{T}}(\widetilde{\theta}_{k})\,,$
(10)

where $\widetilde{\alpha}_{k}$, $\widetilde{\tau}_{k}$, $\widetilde{\nu}_{k}$
and $\widetilde{\theta}_{k}$ denote, respectively, the complex channel gain,
delay, Doppler shift and AOD of the ${k^{\rm{th}}}$ path. Here, $k=0$
represents the line-of-sight (LOS) path. Accordinly, the channel gains are
given by
$\lvert\widetilde{\alpha}_{0}\rvert^{2}=\lambda^{2}/(4\pi\widetilde{d}_{0})^{2}$
and
$\lvert\widetilde{\alpha}_{k}\rvert^{2}={\widetilde{\sigma}_{{\rm{rcs}},k}}\lambda^{2}/[(4\pi)^{3}\widetilde{d}^{2}_{k,1}\widetilde{d}^{2}_{k,2}]$
for $k>0$ [50, Eq. (45)], where ${\widetilde{\sigma}_{{\rm{rcs}},k}}$ denotes
the RCS of the scatterer associated with the ${k^{\rm{th}}}$ path,
$\widetilde{d}_{k,1}$ and $\widetilde{d}_{k,2}$ are the distances between TX-
scatterer and scatterer-RX.

###  II-E Beamformers for Sensing and Communications

To search for potential targets in the environment, the TX sensing beam
$\mathbf{f}_{m,{\rm{s}}}$ in (3) and (4) sweeps an angular range
$[-\theta_{\rm{max}},\theta_{\rm{max}}]$. We assume the use of $B$ different
sensing beams over $M$ symbols. Let ${\mathcal{M}}_{b}$ denote the set of
indices of symbols for which the ${b^{\rm{th}}}$ beam is employed, i.e.,
${\mathcal{M}}_{1}=\\{1,\ldots,M/B\\},\,\ldots\,,{\mathcal{M}}_{B}=\\{M-M/B+1,\ldots,M\\}$.
Hence, the sensing beams are given by
$\mathbf{f}_{m,{\rm{s}}}=\sqrt{{P_{\rm{T}}}/N_{\rm{T}}}\mathbf{a}_{\rm{T}}^{\ast}(\theta_{b})$
for $m\in{\mathcal{M}}_{b}$, where
$\theta_{b}=-\theta_{\rm{max}}+2(b-1)/(B-1)\theta_{\rm{max}}$.

For communications, we assume a LOS beam tracking scenario [51] where the TX
communication beam $\mathbf{f}_{m,{\rm{c}}}$ in (3) and (4) is aligned with
$\widetilde{\theta}_{0}$ and stays constant during the entire frame, i.e.,
$\mathbf{f}_{m,{\rm{c}}}=\sqrt{{P_{\rm{T}}}/N_{\rm{T}}}\mathbf{a}_{\rm{T}}^{\ast}(\widetilde{\theta}_{0})\,\forall
m$.

###  II-F Impact of Non-Constant-Modulus Data on Sensing

To demonstrate the impact of different modulation orders on sensing
performance, we provide an illustrative example of range profiles in Fig. 2,
obtained with QPSK and $1024-$QAM modulations. The range profiles are
generated by applying reciprocal filtering and 2-D FFT based standard OFDM
radar processing (e.g., [13, 39, 52, 9]). From the figure, it is evident that
while high-order modulations provide high data rates, they also tend to result
in elevated side-lobe levels, which significantly impairs the detection of
weak targets. This example sheds light on one of the fundamental trade-offs in
ISAC systems, namely the time-frequency (or, deterministic-random) trade-off,
from the perspective of detection.

Figure 2: Range profiles obtained via conventional OFDM radar processing based
on reciprocal filtering (i.e., element-wise division of transmit symbols) and
2-D FFT [13, 39, 52, 9] with QPSK and $1024-$QAM modulations. The parameters
employed are $B=100\,\rm{MHz}$, $M=140$, with additional details as specified
in Table II. The scenario involves three targets, each moving at
$10\,\rm{m/s}$, with RCSs of $(0,-10,-5)\,\rm{dBsm}$ and located at ranges of
$(20,60,80)\,\rm{m}$ and angles of $(10^{\circ},5^{\circ},15^{\circ})$. It is
observed that the use of communication data symbols from high-order
modulations increases side-lobe levels, thereby masking the presence of weaker
targets.

###  II-G Problem Description

Given the OFDM frame $\mathbf{X}\in\mathbb{C}^{N\times M}$ with
$[\mathbf{X}]_{n,m}=x_{n,m}$ consisting of random data (and, potentially,
pilot symbols), the ISAC TX beamformers $\\{\mathbf{f}_{m}\\}_{m=0}^{M-1}$,
and the sensing and communications models in (5) and (9), the problems of
interest in this work are: (i) to develop a radar sensing algorithm to detect
the presence of multiple targets and estimate their delay-Doppler-angle
parameters from 3-D tensor observations $\\{\mathbf{y}_{n,m}\\}$ in (5) over
$N_{\rm{R}}$ receive antennas, $N$ subcarriers and $M$ symbols, (ii) to
evaluate the data rate for the signal model in (9) across diverse modulation
orders (e.g., QAM, QPSK), and (iii) to investigate ISAC trade-offs for
different configurations of $\mathbf{X}$ and different choices of
$\mathbf{f}_{m}$ (determined by the ISAC weight in (3), the data/pilot ratio
in (4) and the modulation order in both strategies). We begin by tackling the
first problem in Sec. III. The method to approximate the communication MI for
data rate evaluation will be introduced in Sec. IV. Finally, we present
comprehensive simulation results to explore sensing and ISAC trade-off
performances in Sec. V.

##  III Radar Sensing Algorithm

In this section, we introduce a novel sensing algorithm for multiple target
detection and accompanying parameter estimation using the observation in (5),
that can account for arbitrary $\mathbf{X}$ and TX beam sweeping.

###  III-A Beam-Specific Channel Estimation

Target detection from the observation (5) via coherent processing in the
spatial-frequency-time domains involves a computationally demanding 3-D search
over the delay-Doppler-angle tuples of potential targets [53, 54].
Furthermore, the beam sweeping procedure used for sensing, as discussed in
Sec. II-E, results in varying gains for targets across different beams. This
is because the overall gain of the ${k^{\rm{th}}}$ target in (6) is
represented by $\alpha_{k}\mathbf{a}_{\rm{T}}^{T}(\theta_{k})\mathbf{f}_{m}$,
which varies over symbols $m$ when $\mathbf{f}_{m}$ changes [9].

Given these two challenges in detection/estimation from (5), we propose to
formulate the problem of sensing as a series of beam-specific channel
estimation problems where for each beam the time-frequency domain radar
channel is estimated per RX element by treating it as an unstructured channel
rather than a delay-Doppler parameterized one as in (6), followed by
noncoherent integration [53, 54] of the resulting delay-Doppler images over
the RX array. More formally, for the ${b^{\rm{th}}}$ beam, the sensing
observations in (5) at the ${i^{\rm{th}}}$ RX element can be expressed using
(6) as

$\displaystyle\mathbf{Y}_{i,b}=\mathbf{H}_{i,b}\odot\mathbf{X}_{b}+\mathbf{N}_{i,b}\in\mathbb{C}^{N\times
M_{b}}\leavevmode\nobreak\ ,$ (11)

where $\odot$ denotes the Hadamard (element-wise) product,

$\displaystyle\mathbf{H}_{i,b}$
$\displaystyle\triangleq\sum_{k=0}^{K-1}\alpha_{b,k}\mathbf{b}(\tau_{k})\mathbf{c}_{b}^{H}(\nu_{k})[\mathbf{a}_{\rm{R}}(\theta_{k})]_{i}\in\mathbb{C}^{N\times
M_{b}}$ (12)

represents the time-frequency radar channel at the ${i^{\rm{th}}}$ RX element
for the ${b^{\rm{th}}}$ beam,
$\mathbf{X}_{b}=[\mathbf{X}]_{:,{\mathcal{M}}_{b}}\in\mathbb{C}^{N\times
M_{b}}$ is the transmit symbols for the ${b^{\rm{th}}}$ beam,
${\rm{vec}}\left(\mathbf{N}_{i,b}\right)\sim{\mathcal{CN}}({{\bm{0}}},\sigma^{2}{\bm{\mathrm{I}}})$
and $M_{b}=\lvert{\mathcal{M}}_{b}\rvert$. In (12),
$\alpha_{b,k}=\alpha_{k}\mathbf{a}_{\rm{T}}^{T}(\theta_{k})\widetilde{\mathbf{f}}_{b}$
denotes the overall gain of the ${k^{\rm{th}}}$ target for the ${b^{\rm{th}}}$
beam,

$\displaystyle\mathbf{b}(\tau)$
$\displaystyle\triangleq{\left[1\leavevmode\nobreak\ e^{-j2\pi\Delta
f\tau}\leavevmode\nobreak\ \ldots\leavevmode\nobreak\ e^{-j2\pi(N-1)\Delta
f\tau}\right]}^{T}\leavevmode\nobreak\ ,$ (13) $\displaystyle\mathbf{c}(\nu)$
$\displaystyle\triangleq{\left[1\leavevmode\nobreak\ e^{-j2\pi
f_{c}T_{\rm{sym}}\nu}\leavevmode\nobreak\ \ldots\leavevmode\nobreak\ e^{-j2\pi
f_{c}(M-1)T_{\rm{sym}}\nu}\right]}^{T}\leavevmode\nobreak\ ,$ (14)

represent the frequency-domain and temporal (slow-time) steering vectors,
respectively, and $\mathbf{c}_{b}(\nu)=[\mathbf{c}(\nu)]_{{\mathcal{M}}_{b}}$.
Moreover, $\widetilde{\mathbf{f}}_{b}\in\mathbb{C}^{N_{\rm{T}}\times 1}$
represents the ${b^{\rm{th}}}$ beam, i.e.,
$[\mathbf{F}]_{:,{\mathcal{M}}_{b}}=[\widetilde{\mathbf{f}}_{b}\,\ldots\,\widetilde{\mathbf{f}}_{b}]\in\mathbb{C}^{N_{\rm{T}}\times
M_{b}}$, with
$\mathbf{F}\triangleq[\mathbf{f}_{1}\,\ldots\,\mathbf{f}_{m}]\in\mathbb{C}^{N_{\rm{T}}\times
M}$ denoting the entire TX beamforming matrix.

We revisit the two commonly employed methods to estimate $\mathbf{H}_{i,b}$
from $\mathbf{Y}_{i,b}$ in (11) and present the proposed method.

####  III-A1 Reciprocal Filtering (RF)

The reciprocal filtering (RF) performs channel estimation by element-wise
division of received symbols by transmit symbols [13, 39, 52, 9, 23], i.e.,

$\displaystyle\widehat{\mathbf{H}}_{i,b}=\mathbf{Y}_{i,b}\oslash\mathbf{X}_{b}\,,$
(15)

where $\oslash$ denotes element-wise division. The RF estimator in (15) can be
derived as a result of the least-squares (LS) solution in (11) and thus
corresponds to the zero-forcing estimator [55].

####  III-A2 Matched Filtering (MF)

The matched filtering (MF) approach aims to maximize the SNR at the output of
the filter and applies conjugate multiplication of received symbols by
transmit symbols [52, 56], i.e.,

$\displaystyle\widehat{\mathbf{H}}_{i,b}=\mathbf{Y}_{i,b}\odot\mathbf{X}_{b}^{\ast}\,.$
(16)

####  III-A3 Proposed LMMSE Estimator

We propose to employ an LMMSE estimator to estimate $\mathbf{H}_{i,b}$ in (11)
by treating it as a random unknown parameter as opposed to deterministic
modeling in RF and MF strategies. To this end, let us assume that the
vectorized version of the channel in (12), given by

$\displaystyle\mathbf{h}_{i,b}\triangleq{\rm{vec}}\left(\mathbf{H}_{i,b}\right)=\sum_{k=0}^{K-1}\alpha_{b,k}\mathbf{c}_{b}^{\ast}(\nu_{k})\otimes\mathbf{b}(\tau_{k})[\mathbf{a}_{\rm{R}}(\theta_{k})]_{i}\,,$
(17)

has the following first and second moments:

$\displaystyle\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mathbf{h}}_{i,b}\triangleq\mathbb{E}\\{\mathbf{h}_{i,b}\\},\leavevmode\nobreak\
\mathbf{C}_{i,b}\triangleq\mathbb{E}\\{(\mathbf{h}_{i,b}-\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mathbf{h}}_{i,b})(\mathbf{h}_{i,b}-\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mathbf{h}}_{i,b})^{H}\\}\leavevmode\nobreak\ $

Based on (17), the vectorized observations in (11) can be written as

$\displaystyle\mathbf{y}_{i,b}=\mathbf{D}_{i,b}\mathbf{h}_{i,b}+\mathbf{n}_{i,b}\,,$
(18)

where $\mathbf{y}_{i,b}\triangleq{\rm{vec}}\left(\mathbf{Y}_{i,b}\right)$,
$\mathbf{D}_{i,b}\triangleq{\rm{diag}}\left({\rm{vec}}\left(\mathbf{X}_{b}\right)\right)$
and $\mathbf{n}_{i,b}\triangleq{\rm{vec}}\left(\mathbf{N}_{i,b}\right)$. Given
the statistics in (III-A3), the LMMSE estimate of the radar channel
$\mathbf{h}_{i,b}$ in (18) is given by [57, p. 389]

$\displaystyle\widehat{\mathbf{h}}_{i,b}$
$\displaystyle=\mathbf{C}_{i,b}\mathbf{D}_{i,b}^{H}(\mathbf{D}_{i,b}\mathbf{C}_{i,b}\mathbf{D}_{i,b}^{H}+\sigma^{2}{\bm{\mathrm{I}}})^{-1}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\times(\mathbf{y}_{i,b}-\mathbf{D}_{i,b}\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mathbf{h}}_{i,b})+\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mathbf{h}}_{i,b}\,.$ (19)

The following lemma helps simplification of (19).

######  Lemma 1 (OFDM Radar Channel Statistics).

Suppose that the target parameters in the radar sensing channel (17) are
distributed independently as
$\alpha_{b,k}\sim{\mathcal{CN}}(0,\sigma_{\alpha_{b,k}}^{2}),\tau_{k}\sim\mathcal{U}[0,1/\Delta
f],\nu_{k}\sim\mathcal{U}[0,1/(f_{c}T_{\rm{sym}})],\theta_{k}\sim\mathcal{U}[-\pi/2,\pi/2]$
(i.e., delays, Dopplers and angles are drawn uniformly from their respective
unambiguous detection intervals). In addition, different targets are assumed
to have independent distributions222In each OFDM frame, we sample a
realization from these distributions, which we assume to stay constant during
the frame.. Then, the statistics of $\mathbf{h}_{i,b}$ in (III-A3) are given
by

$\displaystyle\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mathbf{h}}_{i,b}={{\bm{0}}}\,,\leavevmode\nobreak\ \leavevmode\nobreak\
\mathbf{C}_{i,b}=\sigma_{\alpha_{b}}^{2}{\bm{\mathrm{I}}}\,,$ (20)

where
$\sigma_{\alpha_{b}}^{2}\triangleq\sum_{k=0}^{K-1}\sigma_{\alpha_{b,k}}^{2}$.

######  Proof.

Please see Appendix A.

Based on Lemma 1, (19) becomes

$\displaystyle\widehat{\mathbf{h}}_{i,b}$
$\displaystyle=\sigma_{\alpha_{b}}^{2}\mathbf{D}_{i,b}^{H}(\sigma_{\alpha_{b}}^{2}\mathbf{D}_{i,b}\mathbf{D}_{i,b}^{H}+\sigma^{2}{\bm{\mathrm{I}}})^{-1}\mathbf{y}_{i,b}\,.$
(21)

Folding (21) back into matrix and plugging the definition of
$\mathbf{D}_{i,b}$, the LMMSE channel estimate is given by

$\displaystyle\widehat{\mathbf{H}}_{i,b}=\frac{\mathbf{Y}_{i,b}\odot\mathbf{X}_{b}^{\ast}}{\big{|}\mathbf{X}_{b}\big{|}^{2}+{\rm{SNR}}_{b}^{-1}}\,,$
(22)

where ${\rm{SNR}}_{b}\triangleq\sigma_{\alpha_{b}}^{2}/\sigma^{2}$, and the
$\lvert\cdot\rvert^{2}$ operation is performed element-wise.

####  III-A4 Interpretation of LMMSE Estimator

A comparative analysis of (22) with (15) and (16) yields the following
insights:

  * •

At high SNRs, i.e., as ${\rm{SNR}}_{b}\to\infty$, we have
$\widehat{\mathbf{H}}_{i,b}\approx\mathbf{Y}_{i,b}\oslash\mathbf{X}_{b}$.
Hence, the LMMSE estimator converges to the RF estimator in (15).

  * •

At low SNRs, i.e., as ${\rm{SNR}}_{b}\to 0$, we obtain
$\widehat{\mathbf{H}}_{i,b}\approx\mathbf{Y}_{i,b}\odot\mathbf{X}_{b}^{\ast}$.
This suggests that the LMMSE estimator converges to the MF estimator in (16).

  * •

The LMMSE estimator represents a generalization of the RF and MF receivers to
the entire SNR range.

  * •

When using unit-amplitude modulations, where $\lvert\mathbf{X}_{b}\rvert$ is
an all-ones matrix, all estimators become equivalent, regardless of the SNR
level.

####  III-A5 Illustrative Example

Figure 3: Comparative analysis of range profiles obtained by different channel
estimation strategies with $1024-$QAM modulation. 3 Low-SNR scenario with the
OFDM parameters in Table II except $B=100\,\rm{MHz}$ and $M=140$. The scenario
involves two targets, each moving at $15\,\rm{m/s}$, with RCSs of
$(-7,20)\,\rm{dBsm}$ and located at ranges of $(75,85)\,\rm{m}$ and angles of
$(10^{\circ},10^{\circ})$. 3 High-SNR scenario with the OFDM parameters in
Table II except $B=200\,\rm{MHz}$ and $M=140$. The scenario involves two
targets, each moving at $15\,\rm{m/s}$, with RCSs of $(15,20)\,\rm{dBsm}$ and
located at ranges of $(75,5)\,\rm{m}$ and angles of $(10^{\circ},10^{\circ})$.
As opposed to the RF and MF estimators, which fail, respectively, at low and
high SNRs, the proposed LMMSE approach provides a globally effective solution
that achieves low side-lobe levels under high-order QAM modulations across
diverse SNR conditions.

We provide an illustrative example in Fig. 3 to showcase the behavior of the
estimators in (15), (16) and (22) with $1024-$QAM modulation. The figure
demonstrates the range profiles obtained by 2-D FFT [13, 39, 52, 46] on
$\widehat{\mathbf{H}}_{i,b}$ under two different scenarios corresponding to
low and high SNR operation333Defining
${\rm{SNR}}_{\rm{mean}}\triangleq\sum_{b=1}^{B}{\rm{SNR}}_{b}/B$, the low and
high SNR scenarios have ${\rm{SNR}}_{\rm{mean}}=-29.7\,\rm{dB}$ and
${\rm{SNR}}_{\rm{mean}}=19.5\,\rm{dB}$, respectively.. We observe that the
results corroborate the interpretations in Sec. III-A4, i.e., the LMMSE
estimator converges to the MF and the RF estimators at low and high SNR
regimes, respectively. As seen from Fig. 3, the proposed LMMSE approach
emerges as a robust and globally effective solution, maintaining low side-lobe
levels under high-order QAM modulations across a range of SNR conditions. In
contrast, conventional RF and MF estimators struggle with the detection of
weak targets due to increased side-lobe levels at low and high SNRs,
respectively. Drawing from these findings, Table I offers general guidelines
for selecting estimators to achieve robust sensing performance across various
SNR levels.

Table I: Guidelines on Which Estimator to Use In OFDM Radar Sensing with High-Order Modulations Under Different SNR Regimes | RF | MF | LMMSE
---|---|---|---
Low SNR | ✗ | ✓ | ✓
High SNR | ✓ | ✗ | ✓

###  III-B Beam-Specific Delay-Doppler Estimation

We are now interested in estimating target delays and Dopplers from the output
of channel estimation $\widehat{\mathbf{H}}_{i,b}$ in (15), (16) or (22). The
channel estimates will have the form

$\displaystyle\widehat{\mathbf{H}}_{i,b}=\mathbf{H}_{i,b}+{\mathbf{Z}}_{i,b}\in\mathbb{C}^{N\times
M_{b}}\,,$ (26)

where $\mathbf{H}_{i,b}$ is given in (12) and ${\mathbf{Z}}_{i,b}$ represents
the channel estimation error term. Based on (12) and following a generalized
likelihood ratio test (GLRT) approach [46], the delay-Doppler images can be
computed simply via 2-D FFT, i.e.,

$\displaystyle
h_{i,b}(\tau,\nu)=\mathbf{b}^{H}(\tau)\widehat{\mathbf{H}}_{i,b}\mathbf{c}_{b}(\nu)\,,$
(27)

where $\mathbf{b}(\tau)$ in (13) and $\mathbf{c}_{b}(\nu)$ in (14) coincide
with (zero-padded) DFT matrix columns. In matrix form with sampled delay-
Doppler grid, (27) can be written as

$\displaystyle\widehat{\mathbf{H}}^{\rm{DD}}_{i,b}=\mathbf{F}_{N}^{H}\widehat{\mathbf{H}}_{i,b}\mathbf{F}_{M_{b}}\in\mathbb{C}^{N\times
M_{b}}\,,$ (28)

where $\mathbf{F}_{N}\in\mathbb{C}^{N\times N}$ is the unitary DFT matrix of
size $N$. Since target angles are a-priori unknown and 3-D delay-Doppler-angle
processing is computationally demanding, we propose to non-coherently
integrate over the antenna elements [53, 54] to obtain the delay-Doppler image
for each beam $b$ (i.e., without seeking phase-alignment with
$[\mathbf{a}_{\rm{R}}(\theta_{k})]_{i}$ in (12)):

$\displaystyle\widehat{\mathbf{H}}^{\rm{DD}}_{b}=\sum_{i=1}^{N_{\rm{R}}}\big{|}\widehat{\mathbf{H}}^{\rm{DD}}_{i,b}\big{|}^{2}\,.$
(29)

Multiple target detection and delay-Doppler estimation can be performed by
searching for peaks in (29), e.g., via a constant false alarm rate (CFAR)
detector [49, Ch. 6.2.4].

###  III-C Angle Estimation for Multiple Targets at the Same Delay-Doppler
Cell

For each delay-Doppler detection $({\widehat{\tau}},{\widehat{\nu}})$ obtained
via 2-D CFAR on $\widehat{\mathbf{H}}^{\rm{DD}}_{b}$ in (29), we estimate
angles of (possibly) multiple targets residing within the resolution cell of
$({\widehat{\tau}},{\widehat{\nu}})$. To this end, we construct the spatial
domain compressed observation
$\widehat{\mathbf{y}}_{b}\in\mathbb{C}^{N_{\rm{R}}\times 1}$ from the original
channel estimates in (26) as follows:

$\displaystyle[\widehat{\mathbf{y}}_{b}]_{i}$
$\displaystyle=\frac{1}{NM_{b}}\mathbf{b}^{H}({\widehat{\tau}})\widehat{\mathbf{H}}_{i,b}\mathbf{c}_{b}({\widehat{\nu}})\,,$
$\displaystyle=\sum_{k=0}^{K-1}\alpha_{b,k}\frac{\mathbf{b}^{H}({\widehat{\tau}})\mathbf{b}(\tau_{k})}{N}\frac{\mathbf{c}_{b}^{H}(\nu_{k})\mathbf{c}_{b}({\widehat{\nu}})}{M_{b}}[\mathbf{a}_{\rm{R}}(\theta_{k})]_{i}+[\mathbf{n}_{b}]_{i}\,,$
$\displaystyle\approx\sum_{k\in\mathcal{K}_{({\widehat{\tau}},{\widehat{\nu}})}}\alpha_{b,k}[\mathbf{a}_{\rm{R}}(\theta_{k})]_{i}+[\mathbf{n}_{b}]_{i}\,,$
(30)

where
$[\mathbf{n}_{b}]_{i}\triangleq\frac{1}{NM_{b}}\mathbf{b}^{H}({\widehat{\tau}}){\mathbf{Z}}_{i,b}\mathbf{c}_{b}({\widehat{\nu}})$
is the noise component,
$\mathcal{K}_{({\widehat{\tau}},{\widehat{\nu}})}\subseteq\\{1,\ldots,K\\}$
contains the indices of the targets located within the resolution cell of
$({\widehat{\tau}},{\widehat{\nu}})$ (i.e.,
$\lvert\mathbf{b}^{H}({\widehat{\tau}})\mathbf{b}(\tau_{k_{1}})\rvert\gg\lvert\mathbf{b}^{H}({\widehat{\tau}})\mathbf{b}(\tau_{k_{2}})\rvert$
and
$\lvert\mathbf{c}_{b}^{H}({\widehat{\nu}})\mathbf{c}_{b}(\nu_{k_{1}})\rvert\gg\lvert\mathbf{c}_{b}^{H}({\widehat{\nu}})\mathbf{c}_{b}(\nu_{k_{2}})\rvert$
for
$k_{1}\in\mathcal{K}_{({\widehat{\tau}},{\widehat{\nu}})},k_{2}\notin\mathcal{K}_{({\widehat{\tau}},{\widehat{\nu}})}$).
Stacking (30) over the RX array yields

$\displaystyle\widehat{\mathbf{y}}_{b}=\sum_{k\in\mathcal{K}_{({\widehat{\tau}},{\widehat{\nu}})}}\alpha_{b,k}\mathbf{a}_{\rm{R}}(\theta_{k})+\mathbf{n}_{b}\,.$
(31)

Due to relatively small number of antenna elements $N_{\rm{R}}$ at typical
ISAC sensing receivers (e.g., in automotive scenarios [58, 46]), we propose to
retrieve multiple angles from (31) using the ESPRIT algorithm in order to
resolve targets closely spaced in the angular domain. In particular, we resort
to 1-D ESPRIT using spatial smoothing with Hankel matrix construction [59,
60]. Due to space limitations, the reader is referred to [59, Sec. IV-D] and
[60, Sec. III-A] for details on how to estimate multiple angles using (31).

###  III-D Beam-Specific Gain Estimation

Given the estimates
$\mathcal{E}_{b}=\\{{\widehat{\tau}}_{b,k},{\widehat{\nu}}_{b,k},{\widehat{\theta}}_{b,k}\\}_{k=1}^{K_{b}}$
from the beam-specific processing for the ${b^{\rm{th}}}$ beam, we estimate
the corresponding target gains $\bm{\alpha}_{b}\in\mathbb{C}^{K_{b}\times 1}$
via least-squares (LS) using the channel estimates
$\\{\widehat{\mathbf{H}}_{i,b}\\}_{i=1}^{N_{\rm{R}}}$ in (26). Based on the
channel structure in (26) and (12), the channel estimates over all RX antenna
elements can be expressed as

$\displaystyle\widehat{\mathbf{h}}_{b}\triangleq\begin{bmatrix}\widehat{\mathbf{h}}_{1,b}\\\
\vdots\\\
\widehat{\mathbf{h}}_{N_{\rm{R}},b}\end{bmatrix}=\underbrace{\begin{bmatrix}\mathbf{A}_{1,b}\\\
\vdots\\\
\mathbf{A}_{N_{\rm{R}},b}\end{bmatrix}}_{\triangleq\mathbf{A}_{b}\in\mathbb{C}^{NM_{b}N_{\rm{R}}\times
K_{b}}}\bm{\alpha}_{b}+\mathbf{z}_{b}\leavevmode\nobreak\ ,$ (32)

where
$\widehat{\mathbf{h}}_{i,b}\triangleq{\rm{vec}}\big{(}\widehat{\mathbf{H}}_{i,b}\big{)}\in\mathbb{C}^{NM_{b}\times
1}$ and

$\displaystyle\mathbf{A}_{i,b}$
$\displaystyle\triangleq[\mathbf{a}_{i,b}^{(1)}\leavevmode\nobreak\
\ldots\leavevmode\nobreak\
\mathbf{a}_{i,b}^{(K_{b})}]\in\mathbb{C}^{NM_{b}\times
K_{b}}\leavevmode\nobreak\ ,$ (33) $\displaystyle\mathbf{a}_{i,b}^{(k)}$
$\displaystyle\triangleq[\mathbf{a}_{\rm{R}}({\widehat{\theta}}_{b,k})]_{i}\,\mathbf{c}_{b}^{\ast}({\widehat{\nu}}_{b,k})\otimes\mathbf{b}({\widehat{\tau}}_{b,k})\in\mathbb{C}^{NM_{b}\times
1}\leavevmode\nobreak\ .$ (34)

Using (32), the gain estimates can be obtained as

$\displaystyle\bm{\widehat{\alpha}}_{b}=\mathbf{A}_{b}^{\dagger}\widehat{\mathbf{h}}_{b}\leavevmode\nobreak\
.$ (35)

###  III-E LMMSE Processing with A-Priori Unknown Gains

The LMMSE estimator expression in (22) involves the SNR term ${\rm{SNR}}_{b}$,
the evaluation of which requires the knowledge of the target channel gains
(i.e.,
$\sigma_{\alpha_{b,k}}^{2}=\mathbb{E}\\{\lvert\alpha_{b,k}\rvert^{2}\\}$).
Since the channel gains are a-priori unknown, we propose to first run the RF
estimator in (15) and follow the subsequent processing steps in Sec.
III-B–Sec. III-D. Then, we plug the resulting gain estimates from (35) into
the SNR expression (22), i.e.,

$\displaystyle{\rm{SNR}}_{b}=\left\lVert\bm{\widehat{\alpha}}_{b}\right\rVert^{2}/\sigma^{2}\,,$
(36)

which can now be inserted into (22) for LMMSE estimation.

###  III-F Clustering Detections over All Beams

Since the same target might be detected in multiple beams during beam
sweeping, we need a clustering algorithm to merge detections from all beams.
To this end, given the estimates $\\{\mathcal{E}_{b}\\}_{b=1}^{B}$, we resort
to the density-based spatial clustering of applications with noise (DBSCAN)
algorithm [61, 62] to cluster detections over $B$ beams. In DBSCAN, we set the
minimum number of points in a cluster to $1$ and define the distance measure
(used to characterize the $\epsilon$-neighbourhood of a point [61]) in the
range-velocity-angle space as the weighted Euclidean distance:

$\displaystyle
d_{\mathbf{W}}(\widehat{\mathbf{s}}_{p},\widehat{\mathbf{s}}_{q})=[(\widehat{\mathbf{s}}_{p}-\widehat{\mathbf{s}}_{q})^{T}\mathbf{W}(\widehat{\mathbf{s}}_{p}-\widehat{\mathbf{s}}_{q})]^{1/2}\,,$
(37)

where
$\widehat{\mathbf{s}}_{p}=[\widehat{R}_{p},\widehat{v}_{p},{\widehat{\theta}}_{p}]^{T}$
contains the range, velocity and angle estimates of the ${p^{\rm{th}}}$
detection in $\\{\mathcal{E}_{b}\\}_{b=1}^{B}$, and
$\mathbf{W}\in\mathbb{R}^{3\times 3}$ is a diagonal matrix to account for
scaling due to unit differences. The $\epsilon$ parameter [61] is set to
$\epsilon=(\mathbf{s}_{\Delta}^{T}\mathbf{W}\mathbf{s}_{\Delta})^{1/2}$, where
$\mathbf{s}_{\Delta}=[\Delta R,\Delta v,\Delta\theta]^{T}$ with $\Delta
R=c/(2N\Delta f)$ denoting the range resolution, $\Delta v=\lambda
B/(2MT_{\rm{sym}})$ the velocity resolution per beam and444Since the ESPRIT
algorithm in Sec. III-C offers higher angular resolution than the standard
Rayleigh resolution of $\mathbf{a}_{\rm{R}}(\theta)$ in (8), we set the
proximity criterion in angle to a lower value than the standard resolution.
$\Delta\theta=2^{\circ}$.

###  III-G Summary of the Proposed Algorithm

The summary of the proposed LMMSE based radar sensing algorithm is provided in
Algorithm 2, which uses Algorithm 1 as a subroutine.

Algorithm 1 Beam-Specific Parameter Estimation from Unstructured Radar Channel
Estimates

1:Input: Frequency/slow-time radar channel estimates
$\\{\widehat{\mathbf{H}}_{i,b}\\}_{i=1}^{N_{\rm{R}}}$ for the ${b^{\rm{th}}}$
beam in (15), (16) or (22), probability of false alarm $P_{\rm{fa}}$.

2:Output: Delay, Doppler, angle and gain estimates
$\\{{\widehat{\tau}}_{b,k},{\widehat{\nu}}_{b,k},{\widehat{\theta}}_{b,k},\widehat{\alpha}_{b,k}\\}_{k=1}^{K_{b}}$
of multiple targets.

3: Obtain the noncoherently integrated delay-Doppler image
$\widehat{\mathbf{H}}^{\rm{DD}}_{b}$ for the ${b^{\rm{th}}}$ beam via (28) and
(29).

4: Run a CFAR detector on $\widehat{\mathbf{H}}^{\rm{DD}}_{b}$ with the
specified $P_{\rm{fa}}$ for target detection in the delay-Doppler domain.

5: For each delay-Doppler detection $({\widehat{\tau}},{\widehat{\nu}})$,
compute the spatial domain observation $\widehat{\mathbf{y}}_{b}$ in (30).

6:Estimate angles from $\widehat{\mathbf{y}}_{b}$ via 1-D ESPRIT.

7:Using the channel estimates
$\\{\widehat{\mathbf{H}}_{i,b}\\}_{i=1}^{N_{\rm{R}}}$ and the parameter
estimates
$\\{{\widehat{\tau}}_{b,k},{\widehat{\nu}}_{b,k},{\widehat{\theta}}_{b,k}\\}_{k=1}^{K_{b}}$,
estimate the gains via (32)–(35).

Algorithm 2 LMMSE-Based MIMO-OFDM Radar Sensing Algorithm

1:Input: Space/frequency/slow-time MIMO-OFDM radar data cube
$\\{\mathbf{y}_{n,m}\\}$ in (5), transmit symbols $\mathbf{X}$, number of
distinct beams $B$, probability of false alarm $P_{\rm{fa}}$.

2:Output: Delay-Doppler-angle estimates
$\\{{\widehat{\tau}}_{k},{\widehat{\nu}}_{k},{\widehat{\theta}}_{k}\\}_{k=0}^{K-1}$
of multiple targets.

3:for $b=1,\ldots,B$

4: Compute the RF channel estimate $\widehat{\mathbf{H}}_{i,b}$ in (15) for
each RX element $i=1,\ldots,N_{\rm{R}}$.

5: Run Algorithm 1 to obtain the gain estimates
$\bm{\widehat{\alpha}}_{b}=[\widehat{\alpha}_{b,1},\ldots\,\widehat{\alpha}_{b,K_{b}}]^{T}$.

6: Compute ${\rm{SNR}}_{b}$ in (36) using $\bm{\widehat{\alpha}}_{b}$.

7: Compute the LMMSE channel estimate $\widehat{\mathbf{H}}_{i,b}$ in (22) for
each RX element $i=1,\ldots,N_{\rm{R}}$.

8: Run Algorithm 1 to obtain the delay, Doppler and angle estimates
$\mathcal{E}_{b}=\\{{\widehat{\tau}}_{b,k},{\widehat{\nu}}_{b,k},{\widehat{\theta}}_{b,k}\\}_{k=1}^{K_{b}}$.

9:end for

10:Perform DBSCAN clustering of the resulting detections
$\\{\mathcal{E}_{b}\\}_{b=1}^{B}$ using the settings in Sec. III-F.

##  IV Communications Data Rate Evaluation

In this section, we provide a methodology to evaluate the data rate of the
communications subsystem of the considered ISAC system under different
constellations, following a similar approach to those in [63, 17].

###  IV-A Calculation of Mutual Information

The received signal at the communications RX in (9) can be recast as

$\displaystyle y^{{\rm{com}}}_{n,m}=h_{n,m}x_{n,m}+z_{n,m}\,,$ (38)

where $h_{n,m}\triangleq(\mathbf{h}^{{\rm{com}}}_{n,m})^{T}\mathbf{f}_{m}$. As
we accumulate the values of $y^{{\rm{com}}}_{n,m}$ over the entire OFDM frame,
we obtain

$\displaystyle\mathbf{Y}^{{\rm{com}}}=\mathbf{H}\odot\mathbf{X}+\mathbf{Z}\,,$
(39)

where $\mathbf{Y}^{{\rm{com}}}\in\mathbb{C}^{N\times M}$ with
$[\mathbf{Y}^{{\rm{com}}}]_{n,m}=y^{{\rm{com}}}_{n,m}$, the effective channel
matrix $\mathbf{H}\in\mathbb{C}^{N\times M}$ with
$[\mathbf{H}]_{n,m}=h_{n,m}$, and $\mathbf{Z}\in\mathbb{C}^{N\times M}$ is a
noise matrix with independently and identically distributed (i.i.d.) entries
$z_{n,m}\sim{\mathcal{CN}}(0,\sigma^{2})$. Hence, assuming perfect knowledge
of the channel at the RX, the mutual information (MI) of the entire frame can
be represented as

$\displaystyle
I(\mathbf{X};\mathbf{Y}^{{\rm{com}}}|\mathbf{H})=\sum_{n,m}I(x_{n,m};y^{{\rm{com}}}_{n,m}\,|\,h_{n,m})\,,$
(40)

where the MI term is given by

$\displaystyle I(x_{n,m};y^{{\rm{com}}}_{n,m}\,|\,h_{n,m})$
$\displaystyle=h(y^{{\rm{com}}}_{n,m}\,|\,h_{n,m})-h(y^{{\rm{com}}}_{n,m}\,|\,h_{n,m},x_{n,m})\,,$
(41)

with $h(\cdot)$ representing the (conditional) entropy of a random variable.
The separability of the MI in (40) results from the independence of
$y^{{\rm{com}}}_{n,m}$ across subcarriers and symbols.

###  IV-B Evaluation of Entropy

The entropy on the right-hand side of (41) is given by

$\displaystyle h(y^{{\rm{com}}}_{n,m}\,|\,h_{n,m},x_{n,m})$ (42)
$\displaystyle=\int_{\mathbb{C}}-f(y^{{\rm{com}}}_{n,m}\,|\,h_{n,m},x_{n,m})\log
f(y^{{\rm{com}}}_{n,m}\,|\,h_{n,m},x_{n,m})\,\;\mathrm{d}y^{{\rm{com}}}_{n,m}\,,$
$\displaystyle=\log\left(\pi e\sigma^{2}\right)\,,$

which is due to
$f\\!\left(y^{{\rm{com}}}_{n,m}\,|\,h_{n,m},x_{n,m}\right)={\mathcal{CN}}(h_{n,m}x_{n,m},\sigma^{2})$.
We now calculate $h(y^{{\rm{com}}}_{n,m}\,|\,h_{n,m})$ in (41) via

$\displaystyle h(y^{{\rm{com}}}_{n,m}\,|\,h_{n,m})$
$\displaystyle=\mathbb{E}\left[-\log
f(y^{{\rm{com}}}_{n,m}\,|\,h_{n,m})\right]\,,$
$\displaystyle\approx-\frac{1}{N_{\mathrm{s}}}\sum_{i=1}^{N_{\mathrm{s}}}\log
f(y_{i}|\,h_{n,m})\,,$ (43)

where the approximation leverages the law of large numbers and becomes exact
as $N_{\mathrm{s}}$ increases. Assuming that $x_{n,m}$ is uniformly
distributed over a finite alphabet
$\mathcal{X}=\\{x_{1},x_{2},\ldots,x_{L}\\}$, (43) can be evaluated via

$\displaystyle f(y^{{\rm{com}}}_{n,m}\,|\,h_{n,m})$
$\displaystyle=\frac{1}{L}\sum_{\ell=1}^{L}f(y^{{\rm{com}}}_{n,m}\,|\,h_{n,m},x_{\ell})$
(44)
$\displaystyle=\frac{1}{L}\sum_{\ell=1}^{L}\frac{1}{\pi\sigma^{2}}\exp\left(-\frac{|y^{{\rm{com}}}_{n,m}-h_{n,m}x_{\ell}|^{2}}{\sigma^{2}}\right).$

###  IV-C General Procedure to Evaluate MI in (40)

Drawing from Sec. IV-A and Sec. IV-B, we outline the MI evaluation process
from (40) in Algorithm 3. To reduce computational demands for large $NM$, we
utilize the near-constant nature of the frequency/time channel $\mathbf{H}$
within certain regions (e.g., within a beam $b$) under a dominant LOS path.
This approach allows for evaluating (40) using a limited number of $h_{n,m}$’s
instead of all subcarriers and symbols.

Algorithm 3 Rate Evaluation under Different Constellations

1:Input: Frequency/slow-time communication channel $\mathbf{H}$ in (39),
alphabet $\mathcal{X}=\\{x_{1},x_{2},\ldots,x_{L}\\}$, number of samples for
entropy approximation $N_{\mathrm{s}}$.

2:Output: MI in (40).

3:for $n=0,\ldots,N-1$, $m=0,\ldots,M-1$

4: for $i=1,\ldots,N_{\mathrm{s}}$

5: Select a symbol $x_{i}$ from the alphabet $\mathcal{X}$ in a uniformly
random manner.

6: Generate a realization of $n_{n,m}\sim{\mathcal{CN}}(0,\sigma^{2})$.

7: Calculate the output via $y_{i}=h_{n,m}x_{i}+n_{n,m}$.

8: Compute $f(y_{i}|\,h_{n,m})$ via (44).

9: end for

10: Compute $h(y^{{\rm{com}}}_{n,m}\,|\,h_{n,m})$ via (43).

11: Compute the MI in (41) via (42) and (43).

12:end for

13:Compute the MI of the entire frame in (40) via (41).

##  V Numerical Results

In this section, we assess the performance of the proposed sensing algorithm
in Algorithm 2 as well as the accompanying ISAC trade-offs using the data rate
evaluation procedure in Algorithm 3. The default simulation parameters are
provided in Table II. For sensing beam sweeping in Sec. II-E, we set the
number of beams $B=N_{\rm{T}}$ and $\theta_{\rm{max}}=70^{\circ}$. To evaluate
the detection performance, we set $P_{\rm{fa}}=10^{-4}$ in Algorithm 2 and
consider $100$ independent Monte Carlo noise realizations in (5), where each
realization corresponds to a 3-D noise tensor of size $N_{\rm{R}}NM$. For
benchmarking purposes, we compare the sensing and ISAC performance of the
following algorithms:

  * •

LMMSE: The proposed MIMO-OFDM radar sensing algorithm in Algorithm 2.

  * •

LMMSE (ideal): The genie-aided version of Algorithm 2 where the true value of
${\rm{SNR}}_{b}$ is inserted into (22) on Line 7 instead of estimating it via
Lines 4–6. This will serve as an upper bound on the performance of LMMSE.

  * •

RF: The standard RF-based sensing algorithm that executes Algorithm 2 by
skipping Lines 4–6 and replacing (22) by (15) on Line 7.

  * •

MF: The standard MF-based sensing algorithm that executes Algorithm 2 by
skipping Lines 4–6 and replacing (22) by (16) on Line 7.

For the communication channel in (10), we consider the presence of
$\widetilde{K}=4$ paths to an RX located at $[43,-25]^{T}\,\rm{m}$ with
respect to the ISAC TX, including a LOS path and $3$ non-line-of-sight (NLOS)
paths. The corresponding scatterers are located at $[40,-20]^{T}\,\rm{m}$,
$[42,-27]^{T}\,\rm{m}$ and $[38,-30]^{T}\,\rm{m}$, with the RCSs
$\\{-5,-10,-10\\}\,\rm{dBsm}$, respectively. Considering stationary RX and
scatterers, we set $\widetilde{\nu}_{k}=0$ for all the paths555Under this
setting, the magnitude of the channel $\mathbf{H}$ in (39) within a beam $b$
(i.e., $\lvert[\mathbf{H}]_{:,{\mathcal{M}}_{b}}\rvert^{2}$) exhibits
variation less than $1\,\rm{dB}$ across frequency and remains constant over
time. Hence, for the rate evaluation in Algorithm 3, we select one subcarrier
per beam. This approach reduces the execution of the outer loop to only $B$
iterations, as opposed to $NM$ times..

In the following subsections, we first explore the time-frequency and spatial
domain ISAC trade-offs under concurrent transmission. Then, we compare
concurrent and time-sharing strategies as described in Sec. II-B.

Table II: Simulation Parameters Carrier frequency, $f_{c}$ | $28\,\rm{GHz}$
---|---
Subcarrier spacing, $\Delta f$ | $120\,\rm{kHz}$
Number of subcarriers, $N$ | $3330$
Bandwidth, $B$ | $400\,\rm{MHz}$
Number of symbols, $M$ | $1120$
Transmit power, ${P_{\rm{T}}}$ | $20\,\rm{dBm}$
ULA size at the ISAC TX, $N_{\rm{T}}$ | $8$
ULA size at the sensing RX, $N_{\rm{R}}$ | $8$
Noise PSD | $-174\,\rm{dBm/Hz}$
Noise figure | $8\,\rm{dB}$

###  V-A Time-Frequency Domain Trade-offs under Concurrent Transmission

To investigate time-frequency domain ISAC trade-offs, the performance of the
algorithms under concurrent transmission, as outlined in Sec. II-B1, is
evaluated using different modulation orders for transmit symbols $\mathbf{X}$
in (5). We set $\rho=0.8$ in (3) and consider two sensing scenarios, low-SNR
and high-SNR (quantified by ${\rm{SNR}}_{\rm{mean}}$ defined in Footnote 3),
which are crucial for highlighting the distinct characteristics of the
algorithms, as extensively demonstrated earlier in Fig. 3.

####  V-A1 Low-SNR Sensing Scenario

In the low-SNR scenario, we consider a single target with range $40\,\rm{m}$,
velocity $15\,\rm{m/s}$, angle $10^{\circ}$ and RCS $5\,\rm{dBsm}$, leading to
${\rm{SNR}}_{\rm{mean}}=-31.6\,\rm{dB}$. The goal is to investigate whether
the target is drowned out by its own interference under high-order modulations
due to increased side-lobe levels as depicted in Fig. 2 and Fig. 3. In Fig. 4,
we show the probability of detection ($P_{\rm{d}}$) of various algorithms
relative to target RCS, employing $1024-$QAM modulation for $\mathbf{X}$. It
is observed that the proposed LMMSE estimator significantly outperforms the RF
estimator, achieving gains in $P_{\rm{d}}$ as high as $0.8$ for a constant RCS
and enabling the detection of targets with RCS values smaller by up to
$8\,\rm{dBsm}$ for a specified $P_{\rm{d}}$. Furthermore, consistent with the
discussions in Sec. III-A4 and findings in Fig. 3, LMMSE and MF achieve very
similar performances. Thus, in compliance with Table I, LMMSE and MF are
suitable for low-SNR sensing, whereas RF leads to severe degradations in
probability of detection. Moreover, LMMSE exhibits detection performance very
close to its genie-aided counterpart with perfect knowledge of
${\rm{SNR}}_{b}$ in (22), which validates the effectiveness of the SNR
estimation approach in Algorithm 1.

To investigate ISAC trade-offs, we explore the impact of modulation order on
the sensing and communication performances. To this end, in Fig. 4 we depict
the probability of detection and achievable rate across various modulations,
from QPSK to $1024-$QAM, for a constant RCS of $5\,\rm{dBsm}$. The detection
curves reveal the robustness of the proposed LMMSE approach against increasing
modulation order, showcasing its capability to mitigate target masking effects
under high-order modulations. Conversely, the RF estimator suffers from
substantial loss in probability of detection with increasing modulation order
due to rising side-lobe levels, corroborating the findings in Fig. 2.
Furthermore, the achievable rate improves as modulation order increases, as
expected, which suggests that LMMSE (along with its genie-aided, ideal version
and MF) achieves much more favorable ISAC trade-offs compared to RF. Hence,
employing LMMSE in the sensing receiver enables significant enhancements in
communication rates through high-order QAM signaling without compromising
detection performance.

Figure 4: ISAC performances in the low-SNR sensing scenario under concurrent
transmission. 4 Probability of detection with respect to target RCS using
$1024-$QAM data. 4 Probability of detection and achievable rate with respect
to modulation order for target RCS of $5\,\rm{dBsm}$.

####  V-A2 High-SNR Sensing Scenario

We now turn our attention to a high-SNR scenario which contains three targets
with ranges of $(75,5,5)\,\rm{m}$, velocities of $(15,-10,-10)\,\rm{m/s}$,
angles of $(10^{\circ},10^{\circ},18^{\circ})$ and RCSs of
$(20,20,5)\,\rm{dBsm}$, resulting in ${\rm{SNR}}_{\rm{mean}}=19.6\,\rm{dB}$.
In Fig. 5, we show the probability of detection of Target-$1$ relative to its
RCS with $1024-$QAM modulation. It is seen that the proposed LMMSE estimator
outperforms both RF and MF benchmarks, indicating that it can effectively
reduce side-lobe levels to prevent target masking when using varying-amplitude
data. In accordance with Table I and Fig. 3, at high sensing SNRs, MF
experiences significant loss in detection performance, while RF leads to
slight degradations in performance compared to LMMSE.

####  V-A3 Summary of Low-SNR and High-SNR Scenarios

Fig. 4 and Fig. 5 reveal that LMMSE provides consistently superior detection
performance across different SNR regimes over existing benchmarks when using
high-order QAM signaling. This establishes LMMSE as a universally effective
strategy for OFDM radar sensing across a spectrum of SNR levels and modulation
schemes.

Figure 5: Probability of detection with respect to target RCS using $1024-$QAM
data in the high-SNR sensing scenario under concurrent transmission.

###  V-B Spatial Domain Trade-offs under Concurrent Transmission

To evaluate spatial domain ISAC trade-offs, we study the impact of the trade-
off weight $\rho$ in (3) on sensing and communication performances with
varying modulation orders for $\mathbf{X}$, considering the same low-SNR and
high-SNR scenarios as in Sec. V-A. The low-SNR scenario yields
${\rm{SNR}}_{\rm{mean}}=(-31.3,-20.9)\,\rm{dB}$ for $\rho=(0,1)$,
respectively, while the high-SNR scenario leads to
${\rm{SNR}}_{\rm{mean}}=(9.9,20.3)\,\rm{dB}$ for $\rho=(0,1)$. In the
considered setting, the UE is located at an angle of $-30.2^{\circ}$, while
the targets lie at $10^{\circ}$ and $18^{\circ}$. Hence, we expect to observe
a trade-off between sensing and communications as $\rho$ varies. In Fig. 6, we
plot the ISAC trade-off curves obtained by the algorithms under consideration
in the low-SNR and high-SNR scenarios as $\rho$ sweeps the interval $[0,1]$,
using both QPSK and $1024-$QAM modulations666Since all estimators are
equivalent under QPSK signaling, as discussed in Sec. III-A4, a single curve
is plotted for the case of QPSK..

####  V-B1 Low-SNR Sensing Scenario

Looking at the low-SNR results, LMMSE significantly improves ISAC trade-offs
over RF when utilizing $1024-$QAM signaling, aligning with insights from Table
I and Fig. 4. In addition, as anticipated, the performance of LMMSE mirrors
that of MF in low-SNR environments. When comparing QAM and QPSK signaling, we
observe that LMMSE can achieve substantially better ISAC trade-offs with QAM
signaling through its robustness in detection performance against increasing
modulation order (as illustrated in Fig. 4). This robustness enables the use
of high-order QAM to boost rates without sacrificing sensing performance777The
steep trade-off curve for QPSK, in contrast to QAM, is attributed to the fact
that the rate for QAM reaches its maximum at a higher SNR level than QPSK,
leading to a more gradual decrease in $P_{\rm{d}}$ as the rate increases. Note
that as $\rho$ in (3) approaches $0$, the sensing SNR decreases and the
communication SNR increases.. Moreover, in the low rate regime (i.e., until
the rate reaches the QPSK plateau), RF experiences a significant reduction in
$P_{\rm{d}}$ with QAM compared to QPSK. However, in the high rate regime, RF
can attain more favorable ISAC trade-offs with QAM signaling (i.e., leading to
higher rates for a given $P_{\rm{d}}$).

####  V-B2 High-SNR Sensing Scenario

As observed in Sec. V-A, the trends for MF and RF become opposite when
transitioning from low-SNR to high-SNR scenarios. Specifically, in high-SNR
environments, MF fails to provide satisfactory ISAC trade-off performance
whereas RF surpasses MF, albeit with a considerable performance gap compared
to LMMSE. Consequently, employing QAM signaling at the transmit side, in
conjunction with the LMMSE algorithm at the sensing receiver, yields superior
ISAC trade-offs over both RF and MF benchmarks, regardless of whether QAM or
QPSK signaling is used.

####  V-B3 Summary of Low-SNR and High-SNR Scenarios

In alignment with Table I and the previous findings in Fig. 4 and Fig. 5, the
spatial domain trade-off results in Fig. 6 demonstrate that LMMSE consistently
outperforms the traditional OFDM radar sensing benchmarks MF and RF across
various SNRs, modulation schemes and sensing-communication beamforming
weights. These results further indicate the possibility of achieving and
adjusting flexible trade-offs between sensing and communications to satisfy
specific OFDM ISAC system requirements. Based on the findings in Fig. 6, Table
III provides rough guidelines on the choice of transmit signaling strategies
and sensing channel estimators under different sensing SNRs and ISAC
requirements.

Figure 6: ISAC trade-off curves under concurrent transmission for QPSK and $1024-$QAM modulations as the trade-off weight $\rho$ in (3) varies over $[0,1]$. 6 Low-SNR sensing scenario. 6 High-SNR sensing scenario.  Table III: Guidelines on Which Transmit Signaling Strategy (QAM or QPSK) and Channel Estimator to Use Under Different SNR Regimes and ISAC Requirements ISAC | High $P_{\rm{d}}$ | Medium $P_{\rm{d}}$ | Low $P_{\rm{d}}$
---|---|---|---
Requirement | Low Rate | Medium Rate | High Rate
Low | QPSK/ | QAM/ | QAM/
SNR | All Estimators | LMMSE+MF | LMMSE+MF
High | QPSK/ | QAM/ | QAM/
SNR | All Estimators | LMMSE | LMMSE+RF

###  V-C Concurrent vs. Time-Sharing Transmission

We now carry out a comparative analysis of Concurrent and Time-Sharing
strategies using the same scenarios in Sec. V-B. Fig. 7 shows the trade-off
curves achieved by Time-Sharing888Since only QPSK pilots are used for sensing
in the Time-Sharing strategy (see Sec. II-B2 and Fig. 1), all sensing
algorithms are equivalent. as the time-sharing ratio
$\lvert\mathcal{S}\rvert/M$ in (4) sweeps the interval $[0,1]$, along with
those belonging to the Concurrent strategy (same as in Fig. 6)999We show only
the results of LMMSE and RF in the concurrent case to declutter the figures..

The comparison between Concurrent and Time-Sharing strategies reveals a subtle
dynamic, shaped by the interaction of two counteracting factors, each
prevailing under distinct operational conditions:

  * •

Factor-1: Full Data Utilization for Sensing: Concurrent strategy leverages the
entire OFDM frame for sensing, offering a potential advantage in data
utilization over Time-Sharing, which limits sensing to dedicated pilots,
possibly reducing detection capabilities.

  * •

Factor-2: Impact of QAM Data on Sensing: While Concurrent transmission
benefits from using QAM data for sensing, this does not guarantee enhanced
detection over Time-Sharing. The presence of high side-lobe levels from QAM
can be detrimental if the sensing algorithm is ill-equipped to handle them,
potentially placing Concurrent at a disadvantage against Time-Sharing that
relies on QPSK pilots for sensing.

It is worth emphasizing that the influence of these factors on sensing
performance depends on the specific circumstances, suggesting that observed
performance patterns may vary with changes in sensing and communication
channel characteristics, time/power allocations between the two
functionalities and the choice of sensing algorithm, as subsequent discussions
will illustrate.

For the low-SNR scenario depicted in Fig. 7, Concurrent transmission with QAM,
in combination with LMMSE at the sensing receiver, yields significantly better
ISAC trade-offs than Time-Sharing. This indicates that the full utilization of
data for sensing (Factor-1) outweighs the challenges associated with using QAM
data (Factor-2) in this scenario, confirming the effectiveness of the proposed
LMMSE approach. The allocated sensing pilots in Time-Sharing do not suffice to
achieve $P_{\rm{d}}$ as high as that achieved by Concurrent for an equivalent
rate (corresponding to a certain $\lvert\mathcal{S}\rvert/M$ in Time-Sharing
and a certain $\rho$ in Concurrent). Conversely, a scenario where Factor-2
dominates Factor-1 is seen when comparing the performance of the RF estimator
in Concurrent transmission to Time-Sharing. Despite the utilization of the
full OFDM frame with QAM data by the RF estimator in Concurrent transmission,
it experiences a noticeable reduction in $P_{\rm{d}}$ compared to Time-
Sharing, which relies solely on QPSK pilots for sensing. This disadvantage for
RF in Concurrent mode results from increased side-lobe levels at low SNRs, as
previously shown in Fig. 2 and Fig. 3.

The high-SNR scenario in Fig. 7 reveals trends in ISAC trade-offs that differ
considerably from those observed in the low-SNR scenario. This distinction
underscores the scenario-specific nature of the relative weights of Factor-1
and Factor-2. In this scenario, the high-SNR environment renders the use of
dedicated QPSK pilots sufficiently effective for achieving a detection
probability ($P_{\rm{d}}$) of 1, making the full utilization of OFDM data for
sensing (Factor-1) less critical up to a certain rate threshold. This effect
is particularly noticeable until the rate approaches $1.4\,\rm{Gbps}$, where
$P_{\rm{d}}$ of 1 is attainable with only a fraction of the transmit symbols
dedicated to pilots. A key insight emerges when rates surpass
$1.5\,\rm{Gbps}$: Concurrent begins to outperform Time-Sharing as the reduced
proportion of pilots no longer suffices for maintaining high $P_{\rm{d}}$ in
Time-Sharing. In contrast, exploitation of data in Concurrent provides
additional SNR benefits, enhancing detection performance, thereby indicating
the dominance of Factor-1 over Factor-2 in this rate regime. We note that such
gains are possible with the use of the proposed LMMSE estimator, which can
successfully counteract target masking effects, whereas RF (and, thus MF) with
Concurrent transmission is outperformed by Time-Sharing. The threshold at
which Concurrent gains an advantage over Time-Sharing varies with the
scenario, influenced by the interplay between Factor-1 and Factor-2. For
instance, this crossover occurs at rates exceeding $0.3\,\rm{Gbps}$ in the
low-SNR scenario, highlighting the scenario-dependent dynamics between the two
strategies.

Figure 7: ISAC trade-off curves under concurrent and time-sharing transmission
strategies, where dedicated QPSK pilots are used for sensing in time-sharing
and $1024-$QAM data are used for communications (see Fig. 1). The time-sharing
curves are obtained by sweeping $\lvert\mathcal{S}\rvert/M$ in (4) over
$[0,1]$. 7 Low-SNR sensing scenario. 7 High-SNR sensing scenario.

##  VI Concluding Remarks

In this paper, we have investigated two fundamental trade-offs in monostatic
OFDM ISAC systems: (i) the time-frequency trade-off, resulting from the choice
of modulation scheme for random data, and (ii) the spatial trade-off, stemming
from how ISAC transmit beamforming is applied to balance sensing and
communication objectives. A novel LMMSE based sensing algorithm has been
proposed to deal with the increased side-lobe levels induced by high-order QAM
data, showcasing substantial improvements over existing OFDM radar sensing
benchmarks. Moreover, two ISAC transmission strategies have been considered:
(i) Concurrent, utilizing all data for sensing with power allocated between
sensing and communication beams, and (ii) Time-Sharing, where sensing relies
solely on dedicated pilots with time-multiplexing for sensing and
communication beams. Extensive simulations demonstrate the superiority of
LMMSE over the conventional RF and MF estimators under a wide range of
operating conditions, while unveiling key insights into the ISAC trade-offs
achieved by these strategies at different SNRs and modulation schemes.
Moreover, two factors have been identified that influence the performance
comparison between Concurrent and Time-Sharing, leading to scenario-dependent
ISAC trade-off dynamic depending on the relative impact of these factors. As
future research, we plan to focus on the same fundamental trade-offs in
bistatic ISAC scenarios.

##  Appendix A Proof of Lemma 1

Using (17), (III-A3) and the definitions in Lemma 1, one can readily obtain

$\displaystyle\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{\mathbf{h}}_{i,b}=\sum_{k=0}^{K-1}\mathbb{E}\\{\alpha_{b,k}\\}\mathbb{E}\\{\mathbf{c}_{b}^{\ast}(\nu_{k})\otimes\mathbf{b}(\tau_{k})[\mathbf{a}_{\rm{R}}(\theta_{k})]_{i}\\}={{\bm{0}}}\,.$
(61)

As to the covariance, we have that

$\displaystyle\mathbf{C}_{i,b}=\sum_{k_{1}=1}^{K}\sum_{k_{2}=1}^{K}\mathbb{E}\\{\alpha_{b,k_{1}}\alpha^{\ast}_{b,k_{2}}\bm{\phi}_{k_{1},b,i}\bm{\phi}^{H}_{k_{2},b,i}\\}\,,$
(62)

where
$\bm{\phi}_{k,b,i}\triangleq\mathbf{c}_{b}^{\ast}(\nu_{k})\otimes\mathbf{b}(\tau_{k})[\mathbf{a}_{\rm{R}}(\theta_{k})]_{i}$.
Due to the independence across the different parameters and the different
targets, and using the definitions in Lemma 1, (62) simplifies to

$\displaystyle\mathbf{C}_{i,b}$
$\displaystyle=\sum_{k=0}^{K-1}\mathbb{E}\\{\lvert\alpha_{b,k}\rvert^{2}\\}\mathbb{E}\\{\bm{\phi}_{k,b,i}\bm{\phi}^{H}_{k,b,i}\\}\,,$
(63)
$\displaystyle=\sum_{k=0}^{K-1}\sigma_{\alpha_{b,k}}^{2}\mathbb{E}\\{\bm{\phi}_{k,b,i}\bm{\phi}^{H}_{k,b,i}\\}\,,$
(64)
$\displaystyle=\sum_{k=0}^{K-1}\sigma_{\alpha_{b,k}}^{2}\mathbb{E}\Big{\\{}\big{|}[\mathbf{a}_{\rm{R}}(\theta_{k})]_{i}\big{|}^{2}\Big{\\}}\mathbb{E}\Big{\\{}\big{[}\mathbf{c}_{b}^{\ast}(\nu_{k})\otimes\mathbf{b}(\tau_{k})\big{]}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\times\big{[}\mathbf{c}_{b}^{\ast}(\nu_{k})\otimes\mathbf{b}(\tau_{k})\big{]}^{H}\Big{\\}}\,,$
(65)
$\displaystyle=\sum_{k=0}^{K-1}\sigma_{\alpha_{b,k}}^{2}\mathbb{E}\big{\\{}\mathbf{c}_{b}^{\ast}(\nu_{k})\mathbf{c}_{b}^{T}(\nu_{k})\big{\\}}\otimes\mathbb{E}\big{\\{}\mathbf{b}(\tau_{k})\mathbf{b}^{H}(\tau_{k})\big{\\}}\,,$
(66)

where the transition from (65) to (66) stems from (8) and the mixed-product
property of the Kronecker product. Using (13), it follows that

$\displaystyle\mathbb{E}\big{\\{}[\mathbf{b}(\tau_{k})\mathbf{b}^{H}(\tau_{k})]_{n,n}\big{\\}}$
$\displaystyle=1\,,$
$\displaystyle\mathbb{E}\big{\\{}[\mathbf{b}(\tau_{k})\mathbf{b}^{H}(\tau_{k})]_{n_{1},n_{2}}\big{\\}}$
$\displaystyle=\int_{0}^{\frac{1}{\Delta f}}e^{j2\pi(n_{2}-n_{1})\Delta
f\tau_{k}}\;\mathrm{d}\tau_{k}=0$

for $n_{1}\neq n_{2}$, which yields
$\mathbb{E}\big{\\{}\mathbf{b}(\tau_{k})\mathbf{b}^{H}(\tau_{k})\big{\\}}={\bm{\mathrm{I}}}_{N}$.
Similarly, using (14), we obtain
$\mathbb{E}\big{\\{}\mathbf{c}_{b}^{\ast}(\nu_{k})\mathbf{c}_{b}^{T}(\nu_{k})\big{\\}}={\bm{\mathrm{I}}}_{M_{b}}$.
Plugging these into (66) gives
$\mathbf{C}_{i,b}=\sigma_{\alpha_{b}}^{2}{\bm{\mathrm{I}}}$.

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  *[S&C]: sensing and communication
  *[ULA]: uniform linear array
  *[AWGN]: additive white Gaussian noise
  *[AOA]: angle-of-arrival
  *[AOD]: angle-of-departure
  *[LOS]: line-of-sight
  *[i.i.d.]: independently and identically distributed
  *[MI]: mutual information
  *[NLOS]: non-line-of-sight