# Input Distribution Optimization in OFDM Dual-Function Radar-Communication
Systems

Yumeng Zhang, Sundar Aditya, and Bruno Clerckx The authors are with the
Department of Electrical and Electronic Engineering, Imperial College London,
London SW7 2AZ, U.K. (e-mail: {yumeng.zhang19, s.aditya,
b.clerckx}@imperial.ac.uk).

###### Abstract

Orthogonal frequency division multiplexing (OFDM) has been widely adopted in
dual-function radar-communication (DFRC) systems. However, with random
communication symbols (CS) embedded in the DFRC waveform, the transmit signal
has a random ambiguity function that affects the radar’s delay-Doppler
estimation performance, which has not been well explored. This paper addresses
this gap by first characterizing the outlier probability (OP) – the
probability of incorrectly estimating a target’s (on-grid) delay-Doppler bin –
in OFDM DFRC w.r.t the CS. This subsequently motivates the OFDM DFRC waveform
design problem of minimizing the OP w.r.t the CS probability distribution
(i.e., the input distribution).  Conditioned on the CSs, the OP only depends
on the CS magnitudes. Hence, we consider the following two schemes for the
above optimization: CSs with (1) constant magnitude input distribution (phase
shift keying), and (2) variable magnitude input distribution (Gaussian). For
(1), minimizing the OP reduces to the familiar power allocation design across
OFDM’s subcarriers and symbols, with uniform power allocation across OFDM
subcarriers and a windowed power allocation across OFDM symbols being near-
optimal. For (2), the mean and variance of the Gaussian distribution at each
subcarrier is optimized, with an additional communication constraint to avoid
the zero-variance solution where no CSs are carried. We observe that
subcarriers with strong communication channels feature a large variance
(favour communications) while the others are characterized by a large mean
(favour radar). However, the overall power allocation (i.e., the sum of the
squared mean and variance) across the OFDM subcarriers and symbols is similar
to (1). Simulations for (2) show that while random CS magnitudes benefit
communications, they degrade radar performance, but this can be mitigated
using our optimized input distribution.

###### Index Terms:

OFDM signal design, dual-functional radar-communication, random communications
symbols, input distribution

##  I Introduction

Dual-function radar-communication (DFRC) has attracted considerable interest
as a solution to the problem of spectrum shortage arising from the increasing
use of wireless technologies [1, 2]. In DFRC, a common signal is used for
radar and communications simultaneously [3, 4]. Hence, of vital significance
in DFRC systems is the choice and design of a suitable signal that performs
both communications and radar satisfyingly [5, 6].

The orthogonal frequency division multiplexing (OFDM) signal is a strong
candidate for DFRC systems given its benefits for both communications and
radar. For the former, OFDM is well-known to provide robustness against
channel fading and combat inter-symbol interference to realize higher data
rates [5]. For the latter, a highly efficient 2D-FFT (2-dimensional fast
Fourier transform) based delay-Doppler bin estimator exists and has been shown
to achieve very promising estimation accuracy [6, 7, 8].

The performance of OFDM for communications and radar in DFRC can be further
enhanced through waveform optimization (e.g., optimizing the power allocation
across subcarriers) [9, 10, 11]. The communications/radar metrics adopted for
such optimization should be both intuitive and tractable. A widely accepted
metric for communications is the achievable rate. For radar, the most
recognized metrics include: (a) the detection probability (DP)/false alarm
probability (FAP) that describes the probability of correctly/falsely
detecting the existence of a target in a certain delay/Doppler/angle bin [12,
13, 14], and (b) the mean square error (MSE) of parameter estimation [15].
However, these radar metrics usually result in high complexity when adopted
for waveform optimization. As a substitute, indirect metrics based on
desirable properties that good sensing signals often share – such as the
ambiguity function (AF) [11, 16, 17, 18, 19] – have also been used. Other
indirect metrics that have been adopted for OFDM waveform optimization include
signal-to-noise-ratio (SNR) or the signal-to-interference-and-noise-ratio
(SINR) as a substitute for DP/FAP[20, 21]; radar mutual information (RMI) from
an information theory perspective [22, 10]; or the Cramer-Rao bound (CRB) as a
lower bound of MSE[23, 4, 9].

When using any radar metric for OFDM DFRC waveform optimization, it is
important to account for the inherent randomness in the waveform due to the
embedded information-bearing communications symbols (CS) – e.g., QAM. However,
previous literature on waveform optimization using the above metrics does not
fully capture the impact of the CS probability distribution – henceforth
termed the input distribution – on radar performance. For instance, maximizing
an OFDM waveform’s average SNR – an indirect metric for optimization w.r.t
DP/FAP – only optimizes power allocation across subcarriers (or the variance
of the input distribution while assuming zero means)[21, 17]. This ignores the
detrimental role that the  variance of the input distribution plays w.r.t
radar performance, as for radar, it is sufficient to have a deterministic
signal, represented by the mean of the input distribution with zero variance.
Hence, for a fixed total power, captured by the second moment of the input
distribution (equal to the sum of the squared mean and the variance), the
power allocation can be optimized between the mean and the variance to favour
both radar and communications. Similar issues exist in the literature for
other sensing metrics, such as the RMI where the average RMI of random CSs is
substituted by the RMI of the average CSs [10, 24].

The effect of input distribution on radar is worth considering, since there is
ample literature on the detrimental effects of different sources of randomness
in radar systems (e.g., the fluctuations of the target’s radar cross-section,
and signal-dependent clutter [12, 25, 26]), and how performance gains can be
achieved by accounting for such randomness through detector/waveform design
[12, 26]. Regarding the input distribution, [27] has demonstrated that the
phase randomness of CSs has only a trivial effect on the OFDM waveform’s AF,
which makes a compelling case for using constant magnitude phase-coded CSs
(e.g., PSK symbols) in OFDM waveforms for DFRC to achieve good radar
performance [28, 29, 30, 31]. However, to support higher communication rates,
input distributions with varying magnitudes (e.g., 256-QAM) may need to be
embedded into an OFDM waveform, and the impact of such distributions on radar
performance has not been explored.

Very recently, [32, 33] shed light on the radar-communication (R-C) trade-off
w.r.t. the randomness of the input distribution in DFRC. The authors
characterized the R-C region (the best achievable radar performance given
communication constraints), and examined the optimal input distributions at
the two corner points on the R-C region (i.e., radar-optimal and
communication-optimal respectively). The paper reveals the fundamental R-C
trade-off that the radar-optimal input distribution features CSs with
deterministic sample covariance (constant-magnitudes) while the communication-
optimal input distribution features CSs with zero-mean Gaussian (random-
magnitudes). However, for the practically relevant problem in DFRC systems of
on-grid delay-Doppler bin estimation of a point target, the metrics used in
[32] (radar MI) and [33] (CRB) may not be the most suitable. For instance, the
radar MI is best suited for extended targets, whereas the CRB is best suited
for continuous-valued parameters. In contrast, for on-grid delay-Doppler bin
estimation, the parameter space is discrete. Hence a more appropriate metric
is needed.

In this paper, we consider an OFDM DFRC waveform design via input distribution
optimization with the radar task being on-grid delay-Doppler bin estimation,
which is of considerable practical relevance. Our contributions are summarized
as follows:

(1) We start from why the outlier probability111OP is defined as the
probability that a wrong delay-Doppler bin is estimated. For the special case
where it is known apriori that a target is present in one of the delay-Doppler
bins, the OP coincides with the FAP, i.e., the ’false alarm’ refers to
deciding the target to any of the wrong bins after detection. (OP) is a
suitable metric for characterizing the on-grid delay-Doppler bin estimation
performance of the maximum likelihood (ML) estimator. We then derive an upper
bound for the OP (UBOP) that is a good approximation of the OP across the
whole SNR region. We show that the UBOP coincides with the signal’s AF
properties and is only a function of the CS magnitudes. The UBOP is adopted as
the radar metric throughout the paper.

(2) We minimize the UBOP w.r.t the input distributions of CSs with constant
magnitudes (e.g. PSK), to provide intuition for radar-favoured signals. In
this scheme, information is only conveyed through the CS phases, and the radar
performance of the optimized solution matches that of a deterministic constant
envelope signal in a pure radar system. The optimal input distribution
features uniform power allocation across the OFDM subcarriers (to favour delay
estimation) and a windowed power allocation across the OFDM symbols (to favour
Doppler estimation). This is also the first paper to design signals across
both the time and frequency domains because of a joint consideration of delay-
Doppler estimation. This scheme is hence used as a benchmark for the case of
CSs with variable magnitude input distributions.

(3) We consider the DFRC performance for input distributions of CSs with
variable magnitudes, by minimizing the average UBOP (aUBOP) w.r.t the non-zero
mean Gaussian input distribution. Deviating from the intuitive choice of a
_zero-mean_ Gaussian input distribution for communications, this is the first
paper to optimize a _non-zero_ Gaussian input distribution for DFRC systems. A
non-zero mean Gaussian gives rise to a trade-off in allocating the available
power budget between the symbol mean and the symbol variance, with
radar/communications favouring a less/more random input distribution through
greater power allocation to the symbol mean/variance. More specifically, the
simulations reveal that for subcarriers having high channel gain for
communications, power is concentrated on the symbol variance in order to
satisfy the communication constraints efficiently. On the other hand, for
subcarriers with weak channel gains for communications, power is concentrated
on the symbol means to favour radar performance. The power allocation (sum
over the squared symbol means and variance) across the OFDM subcarriers,
however, is similar to (2).

Organization: Section II introduces the signal and system model of a point-
target mono-static SISO DFRC scenario using OFDM waveform. Then, Section III
develops the ML estimator for joint delay-Doppler bin estimation, derives the
OP conditioned on CS magnitudes and its tractable upper bound (UBOP). Section
IV considers the problem of minimizing the UBOP for PSK input distribution to
shed insight into the patterns of power allocation across OFDM subcarriers and
OFDM symbols that yield good radar performance. Section V extends the scope of
input distributions to those symbols with variable magnitudes, by minimizing
the aUBOP subject to a communication achievable rate constraint for a non-zero
mean Gaussian input distribution. Section VI evaluates the OP performance of
the optimized input distributions from Sections IV and V and finally, Section
VII concludes the paper.

Notation: Throughout the paper, matrices and vectors are respectively denoted
in bold upper case and bold lower case. ${\mathbb{R}}$, ${\mathbb{C}}$ and
${\mathbb{Z}}$ denotes the set of real numbers, complex numbers and integers,
respectively. For $x\in{\mathbb{C}}$, $\mathfrak{R}\\{x\\}$,
$\mathfrak{I}\\{x\\}$, $\measuredangle x$ and $|{x}|$ represents the real
part, the imaginary part, the phase and the magnitude of ${x}$, respectively.
For a vector (matrix) $\mathbf{x}$ ($\mathbf{X}$), $\|\mathbf{x}\|$
($\|\mathbf{X}\|$) represents its $l_{2}$ (Frobenius) norm, $x_{n}$ refers to
the $n^{\mathrm{th}}$ entry of vector $\mathbf{x}$, and
$\mathrm{diag}\\{\mathbf{x}\\}$($\mathrm{diag}\\{\mathbf{X}\\}$) represents
the diagonal matrix (vector) formed by $\mathbf{x}$ (matrix $\mathbf{X}$).
$|\mathbf{x}|$ denotes the element-wise magnitude of $\mathbf{x}$, i.e.,
$|\mathbf{x}|_{k}=|{x}_{k}|$. $\mathbf{I}_{N_{t}}$ denotes $N_{t}\times N_{t}$
an identity matrix and $\mathbf{1}_{K}$ is the all-one column vector of length
$K$ (the size of the vector may not be explicitly specified where it is
obvious). $(\cdot)^{H}$, $(\cdot)^{*}$ and $(\cdot)^{T}$ represent the
Hermitian, the conjugate and the transpose operations respectively. $\odot$
represents the element-wise (Hadamard) product, $\oslash$ represents the
element-wise (Hadamard) division, $\otimes$ represents the Kronecker product.
Given a set $\kappa$, $|\kappa|$ refers to the cardinality of $\kappa$. For
$x\in{\mathbb{R}}$, $(x)^{+}\triangleq\max(x,0)$, and $\mathbbm{1}_{(\cdot)}$
is the indicator function. For a random variable $X$,
$\mathbb{E}_{X}\left\\{f(X)\right\\}$ represents the expected value of the
function $f(X)$ over $X$. ${\mathbb{P}}(\cdot)$ denotes probability.
$Q(\cdot)$ is the Marcum-Q function, and $I_{0}(\cdot)$ is the modified Bessel
function of the first kind with order 0. $\mathcal{N}(\mu,~{}\sigma^{2})$
($\mathcal{CN}(\mu,~{}\sigma^{2})$) refers to the normal (complex normal)
distribution with mean $\mu$ and variance $\sigma^{2}$ (for the complex
normal, the real and imaginary parts are independent with variance
$\sigma^{2}/2$).

##  II Signal Model

This section models a DFRC system where an OFDM transmit signal is used to
simultaneously communicate with a user and sense a point target. We consider
an OFDM block containing $M$ OFDM symbols. Each OFDM symbol has $K$
subcarriers, with an additional $K_{G}$ sub-pulses of cyclic prefix (CP). We
assume that a target’s delay and Doppler remain unchanged over one block.

Let $B$ denote the signal bandwidth, corresponding to a delay resolution of
$\Delta\tau=1/B$. We assume that the target round-trip delay is shorter than
the CP duration (the maximal delay that can be estimated is
$\tau_{\max}={K_{G}/B}$). Let $T_{\mathrm{OFDM}}=(K+K_{G})/B\triangleq
K_{T}/B$ denote the OFDM symbol duration. Hence, the radar coherent processing
interval (CPI) equals $MT_{\mathrm{OFDM}}$, corresponding to the Doppler
resolution of $\Delta f_{D}=1/(MT_{\mathrm{OFDM}})=B/(K_{T}M)$.

Consider the presence of a target at a distance $R$ and moving at velocity $v$
w.r.t the radar. For a monostatic configuration, $R$ and $v$ are captured on
the echo signal by its time delay $\tau=2R/c$ and Doppler frequency
$f_{\mathrm{D}}=f_{c}(2v/c)$, with $c$ being the speed of light. $\tau$ and
$f_{\mathrm{D}}$ can be represented as follows:

$\displaystyle\tau$ $\displaystyle=(n_{0}+n_{\epsilon})\Delta\tau,\hskip
8.53581ptn_{0}\in\\{0,~{}1,~{}\cdots,~{}K_{G}-1\\},$ (1) $\displaystyle
f_{\mathrm{D}}$ $\displaystyle=(v_{0}+v_{\epsilon})\Delta
f_{\mathrm{D}},\hskip
8.53581ptv_{0}\in\\{-{M}/{2},~{}\cdots,~{}0,~{}\cdots,~{}{(M-1)}/{2}\\},$ (2)

where $n_{\epsilon}\in(0,1)$ and $~{}v_{\epsilon}\in\left(-1/2,~{}1/2\right)$
are the fractional (off-grid) delay and Doppler parameters. $(n_{0},~{}v_{0})$
denotes the target’s _on-grid_ delay-Doppler bin, which is the parameter of
interest in this paper.

Let $X[k,~{}m]$ denote the CS at the
$k^{\mathrm{th}}~{}(k=0,~{}\cdots,~{}K-1)$ subcarrier of the
$m^{\mathrm{th}}~{}(m=0,~{}\cdots,~{}M-1)$ symbol. After a $K$-point IDFT, let
$x[n,~{}m]~{}(n=0,~{}\cdots,~{}K-1)$ denote the resulting time domain signal.
After adding the CP, we obtain the following signals:

$x_{\mathrm{CP}}[n,~{}m]=\begin{cases}x[K-K_{G}+n,~{}m],&n\in\\{0,~{}\cdots,~{}K_{G}-1\\},\\\
x[n-K_{G},~{}m],&n\in\\{K_{G},~{}\cdots,~{}K_{T}-1\\},\end{cases}$ (3)

The (continuous-time) signal transmitted over the air is given by [34]:

$\displaystyle\color[rgb]{0,0,1}x(t)=\sum_{n,m}x_{\mathrm{CP}}[n,~{}m]\mathrm{rect}\left(t-(n+K_{T}m)/B\right).$
(4)

The radar return at the transceiver (after ADC) is given by:

$\displaystyle\color[rgb]{0,0,1}y_{\mathrm{CP}}[n,~{}m]=ax(t-\tau)e^{j2\pi
f_{\mathrm{D}}t}|_{t=(n+K_{T}m)/B}$ (5a) $\displaystyle=$ $\displaystyle
ax_{\mathrm{CP}}[n-n_{0},~{}m]e^{j2\pi\frac{f_{\mathrm{D}}(n+mK_{T})}{B}}+w[n,~{}m]$
(5b) $\displaystyle=$ $\displaystyle
ax_{\mathrm{CP}}[n-n_{0},~{}m]e^{j2\pi\frac{f_{\mathrm{D}}n}{B}}e^{j2\pi\frac{(v_{0}+v_{\epsilon})m}{M}}+w[n,~{}m]$
(5c) $\displaystyle\approx$ $\displaystyle
ax_{\mathrm{CP}}[n-n_{0},~{}m]e^{j2\pi\frac{(v_{0}+v_{\epsilon})m}{M}}+w[n,~{}m],$
(5d)

where (i) $a\in{\mathbb{C}}$ is the scattering coefficient of the target,
assumed to be unchanged throughout the OFDM block, (ii) in (5b), the received
samples after ADC lack information on the fractional $n_{\epsilon}$ because of
the rectangular basis in (4). Hence, $n_{\epsilon}$ is often disregarded in
OFDM radar and the target delay is assumed to be on-grid [35], (iii) $e^{j2\pi
f_{\mathrm{D}}n/B}\approx 1$, since typically $f_{\mathrm{D}}\ll B$, and (iv)
$w[n,~{}m]\sim\mathcal{CN}(0,~{}\sigma_{\mathrm{N}}^{2})$ is the AWGN. After
removing the CP, we get:

$\displaystyle y[n,~{}m]$ $\displaystyle\approx
ax[n-n_{0},~{}m]e^{j2\pi\frac{(v_{0}+v_{\epsilon})m}{M}}+w[n,~{}m],$ (6)

where $x[n-n_{0},~{}m]$ represents a circular shift of $n_{0}$ following (3).
After a $K$-point DFT, we get the frequency-domain signal:

$\displaystyle Y[k,~{}m]$
$\displaystyle=\mu_{k,m}(\mathrm{\boldsymbol{\theta}}_{0})+W[k,~{}m],$ (7a)
$\displaystyle\mbox{with}~{}\mu_{k,m}(\mathrm{\boldsymbol{\theta}}_{0})$
$\displaystyle:{=}aX[k,~{}m]e^{{-j2\pi\frac{n_{0}k}{K}}}e^{{j2\pi\frac{(v_{0}+v_{\epsilon})m}{M}}},$
(7b) $\displaystyle W[k,~{}m]$
$\displaystyle\sim\mathcal{CN}(0,~{}\sigma_{\mathrm{N}}^{2}).$ (7c)

In (7a)-(7c),
$\mathrm{\boldsymbol{\theta}}_{0}\triangleq[n_{0},~{}v_{0},~{}v_{\epsilon},~{}A_{0},~{}\phi_{0}]^{T}$
denotes the true values of all of the unknown parameters in the system, with
$A_{0}$ and $\phi_{0}$ denoting the amplitude and phase of $a$. $n_{0}$ and
$v_{0}$ are the parameters of interest, whereas the others are nuisance
parameters.

##  III Characterizing the ML estimator’s OP

This section begins by formulating the ML estimator for $(n_{0},~{}v_{0})$
over an OFDM block. We then derive the OP of this estimator. The OP is then
approximated by its upper bound to yield a tractable metric that can be
optimized w.r.t the CS input distribution.

###  III-A ML estimator

From (7a)-(7c), the joint probability density function (PDF) of $Y[k,~{}m]$,
conditioned on the unknown parameters
${\boldsymbol{\theta}}\triangleq[{n},~{}{v},~{}{v}^{\prime}_{\epsilon},~{}{A},~{}{\phi}]^{T}$,
is given by:

$\displaystyle
f(\mathbf{y}|{\boldsymbol{\theta}})={{(\pi\sigma_{\mathrm{N}}^{2})}^{-KM}}~{}e^{-\sum_{m=0}^{M-1}\sum_{k=0}^{K-1}\big{|}Y[k,~{}m]-\mu_{k,m}({\boldsymbol{\theta}})\big{|}^{2}\big{/}\sigma_{\mathrm{N}}^{2}},$
(8)

where
$\mathbf{y}\triangleq[\mathbf{y}^{T}_{0},~{}\cdots,~{}\mathbf{y}^{T}_{M-1}]^{T}$
with $\mathbf{y}_{m}\triangleq[Y[0,~{}m],~{}\cdots,~{}Y[K-1,~{}m]]^{T}$. The
ML estimate of ${\boldsymbol{\theta}}_{0}$, denoted by
$\hat{\boldsymbol{\theta}}$, is given by:

$\displaystyle\hat{\boldsymbol{\theta}}=$
$\displaystyle\arg\max_{{\boldsymbol{\theta}}}~{}\log
f({\mathbf{y}}|{\boldsymbol{\theta}})$ (9a) $\displaystyle=$
$\displaystyle\arg\max_{{\boldsymbol{\theta}}}~{}2{A}\mathfrak{R}\Big{\\{}e^{-j{\phi}}\sum_{m,k}Y[k,~{}m]X[k,~{}m]^{*}e^{j2\pi\frac{{n}k}{K}}e^{-j2\pi\frac{({v}+{v}^{\prime}_{\epsilon})m}{M}}\Big{\\}}$
$\displaystyle-\sum_{m,k}\left[|Y[k,~{}m]|^{2}+|{A}X[k,~{}m]|^{2}\right],$
(9b)

######  Remark 1.

In (9), estimating the off-grid parameter $v_{\epsilon}$ (embedded in
$Y[k,~{}m]$) typically involves exploiting sparsity (of targets), and is
attractive because it provides ”super-resolution”. In this paper however, we
restrict our focus to on-grid parameter estimation (with its well-defined
resolution limits [35, 36]), as it is a relatively simpler starting point to
investigate the relationship between the input distributions and sensing
performance. Hence, from the following ML estimator, we treat $v_{\epsilon}$
as a nuisance parameter affecting the estimation of on-grid parameters,
$(n_{0},~{}v_{0})$ (neglecting $v_{\epsilon}$), at the processor in (9).

In (9b), only the first term is a function of the target’s delay-Doppler bin.
Hence, the ML estimator of $(n_{0},~{}v_{0})$ is given by (Assume the true
delay-Doppler bin $(n_{0},~{}v_{0})=(0,~{}0)$ without loss of generality
[37]):

$\displaystyle(\hat{n},~{}\hat{v})=$
$\displaystyle\arg\underset{(n,~{}v)}{\max}~{}\Big{|}\sum_{m,~{}k}Y[k,~{}m]X[k,~{}m]^{*}e^{j2\pi\frac{nk}{K}}e^{-j2\pi\frac{vm}{M}}\Big{|}$
(10a) $\displaystyle=$
$\displaystyle\arg\underset{(n,~{}v)}{\max}~{}\Big{|}\sum_{m,~{}k}Ap_{k,m}e^{j2\pi\frac{nk}{K}}e^{-j2\pi\frac{vm}{M}}e^{j2\pi\frac{v_{\epsilon}m}{M}}+w_{\mathrm{o}}[n,~{}v]\Big{|}$
(10b) $\displaystyle=$
$\displaystyle\arg\underset{(n,~{}v)}{\max}\>\>|\gamma_{n,v,v_{\epsilon}}|,$
(10c) where $\displaystyle p_{k,m}$ $\displaystyle=|X[k,~{}m]|^{2},$ (10d)
$\displaystyle\gamma_{n,v,v_{\epsilon}}$
$\displaystyle=r_{n,v,v_{\epsilon}}+w_{\mathrm{o}}[n,~{}v],$ (10e)
$\displaystyle r_{n,v,v_{\epsilon}}$
$\displaystyle=\sum_{m,k}p_{k,m}e^{j2\pi\frac{nk}{K}}e^{-j2\pi\frac{vm}{M}}e^{j2\pi\frac{v_{{\epsilon}}m}{M}},$
(10f) $\displaystyle w_{\mathrm{o}}[n,~{}v]$
$\displaystyle={A}^{-1}\sum_{m,~{}k}W[k,~{}m]X[k,~{}m]^{*}e^{j2\pi\frac{nk}{K}}e^{-j2\pi\frac{vm}{M}},$
(10g)

for $n\in\\{0,~{}\cdots,~{}K_{G}-1\\},~{}v\in\\{-M/2,~{}\cdots,~{}M/2-1\\}$.

######  Remark 2.

$p_{k,m}$ in (10d) denotes the CS power at the $k^{\mathrm{th}}$ OFDM
subcarrier of the $m^{\mathrm{th}}$ OFDM symbol. $\gamma_{n,v,v_{\epsilon}}$
in (10e) represents the cross-correlation between the transmitted signals and
the received distorted signals, and $r_{n,v,v_{\epsilon}}$ in (10f) represents
the discrete AF of the transmitted signals. For $r_{n,v,v_{\epsilon}}$ in
(10f), the random generalization of CSs gives random $p_{k,m}$ and induces
sidebobes to $r_{n,v,v_{\epsilon}}$. Also, the fractional Doppler,
$v_{\epsilon}$, yields an $M$-DFT evaluated at a discrete frequency that is
_not_ an integer multiple of $1/M$. This induces sidelobes over all $M$
Doppler bins to $r_{n,v,v_{\epsilon}}$ in general.  As an illumination, Fig. 1
plots $|r_{n,v,v_{\epsilon}}|$ and $|\gamma_{n,v,v_{\epsilon}}|$ for: (a) BPSK
input (i.e., constant magnitude CSs) and $v_{{\epsilon}}=0$, (b) 256-QAM input
(i.e., variable magnitude CSs) and $v_{{\epsilon}}=0$, and (c) BPSK input and
$v_{{\epsilon}}\neq 0$. To successfully locate $(n_{0},~{}v_{0})$ from
$|\gamma_{n,v,v_{\epsilon}}|$, it is desirable for the peak lobe at
$|r_{0,0,v_{\epsilon}}|$ to be easily distinguishable from its sidelobes. Fig.
1b (Fig. 1c) show that CSs with variable magnitudes (non-zero $v_{\epsilon}$)
lead to AF sidelobes, which would increase the probability of incorrectly
estimating $(n_{0},~{}v_{0})$, i.e., the OP. As seen in Fig. 1, the sidelobe
levels are a function of both the CS magnitudes and $v_{\epsilon}$. Hence, the
overall sidelobe level can be suppressed (and along with it the OP) through
the careful design of CS input distributions (which is under our control),
accounting for the effect of $v_{\epsilon}$. This motivates the problem of
minimizing the OP w.r.t the input distribution.

Figure 1: The magnitudes of discrete AF ($|r_{n,v,v_{\epsilon}}|$ in (10f))
and the corresponding cross correlation ($|\gamma_{n,v,v_{\epsilon}}|$ in
(10e) averaging over random CSs for (b) and velocities for (c)) with uniform
power allocation across both OFDM subcarriers and symbols, with $K=32$, $M=16$
and $K_{G}=8$. The index, $q$, is related to delay-Doppler bin $(n,~{}v)$ as
follows: $q=n+(v+M/2)K_{G}$. In (b) and (c), we see that CSs with varying
magnitudes as well as $v_{\epsilon}\neq 0$ lead to AF sidelobes, which would
increase the OP.

###  III-B ML-based radar metric

A metric that directly characterizes the ML estimator’s sensing performance is
the OP – the probability of incorrectly estimating $(n_{0},~{}v_{0})$.

Figure 2: The overall MSE, the outlier MSE and the CRB relationship for
$K=32$, $K_{G}=8$, $M=8$.

######  Remark 3.

The popular CRB is not adopted as the radar metric. As a reason, the overall
MSE of $(n_{0},~{}v_{0},~{}v_{\epsilon})$ in (9) is composed by:
$\mathrm{MSE}\left(n,~{}v,~{}v_{\epsilon}\right)=\mathrm{MSE}\left(n,~{}v\right)+\mathrm{MSE}\left(v_{\epsilon}\right)$.
The first term on the right-hand side, $\mathrm{MSE}\left(n,~{}v\right)$,
captures the contribution to the MSE from incorrectly estimating the delay-
Doppler bin $(n_{0},~{}v_{0})$, and is known as the outlier MSE. The second
term captures the MSE w.r.t $v_{\epsilon}$, conditioned on having correctly
estimated the target’s delay-Doppler bin $(n_{0},~{}v_{0})$, and is bounded
below by the CRB. Fig. 2 plots their relationship. For the on-grid estimation
in our scenario, the outlier MSE should be our objective, and the outliers,
from Fig. 2, mostly occur at the threshold SNR region 222The threshold region
is a typical radar phenomenon and refers to the SNR region where radar’s
OP/MSE decreases sharply from a saturating high value to a saturating low
value, which is the region of interest. and below 333Hence, in the DFRC
scenario with variable magnitude input distributions in Section V-B, the
average UBOP is approximated in the low SNR region.. On the contrary, the
overall MSE only converges to the CRB in the high SNR region when
$v_{\epsilon}$ is also the estimation objective. Moreover, in our on-grid
delay-Doppler bin estimation, the CRB is not also suitable because the
parameters are discrete, breaking the regularity condition on which the CRB is
defined [37].

To account for the arbitrariness of $v_{{\epsilon}}$, we assume it is
uniformly distributed over $(-1/2,~{}1/2)$, and consider the OP averaged over
$v_{\epsilon}$. This quantity, denoted by ${\mathbb{P}}_{\mathrm{o}}$, is the
probability that the sidelobe magnitudes ($|\gamma_{n,v,v_{\epsilon}}|$ for
$(n,~{}v)\neq(0,~{}0)$) is larger than the peak lobe magnitude
($|\gamma_{0,0,v_{\epsilon}}|$), which can be expressed as follows:

$\displaystyle{\mathbb{P}}_{\rm o}$
$\displaystyle\triangleq\mathbb{E}_{v_{{\epsilon}}}\Big{\\{}{\mathbb{P}}\Big{[}\bigcup_{n,v}\left\\{|\gamma_{n,v,v_{\epsilon}}|>|\gamma_{0,0,v_{\epsilon}}|\Big{\\}}\Big{]}\right\\}$
(11a)
$\displaystyle\leq\mathbb{E}_{v_{{\epsilon}}}\Big{\\{}\sum_{(n,~{}v)\neq(0,~{}0)}{\mathbb{P}}\left[|\gamma_{n,v,v_{\epsilon}}|>|\gamma_{0,0,v_{\epsilon}}|\right]\Big{\\}}$
(11b)
$\displaystyle\approx\mathbb{E}_{v_{{\epsilon}}}\Big{\\{}\sum_{(n,~{}v)\neq(0,~{}0)}\exp\Big{\\{}-\frac{|r_{0,0,v_{\epsilon}}|}{2\sigma^{2}}\Big{\\}}I_{0}\left(\frac{|r_{n,v,v_{\epsilon}}|}{2\sigma^{2}}\right)/2\Big{\\}}$
(11c)
$\displaystyle\leq\mathbb{E}_{v_{{\epsilon}}}\Big{\\{}\sum_{(n,~{}v)\neq(0,~{}0)}\exp\Big{\\{}-\frac{|r_{0,0,v_{\epsilon}}|-|r_{n,v,v_{\epsilon}}|}{2\sigma^{2}}\Big{\\}}/2\Big{\\}}$
(11d) $\displaystyle\overset{\triangle}{=}{\mathbb{P}}_{\rm UB},$ (11e)

where $\sigma^{2}=\sigma_{\mathrm{N}}^{2}/A^{2}$. The inequality in (11b)
stems from the union bound, while the upper bound in (11d), termed the UBOP,
is more tractable for optimization w.r.t the input distribution than
${\mathbb{P}}_{\mathrm{o}}$. Hence, we adopt the UBOP as the radar metric from
hereon. The derivation for (11c)-(11d) is provided in Appendix A and is shown
to capture the simulated OP soundly in Fig. 3.

Figure 3: The simulated and approximated OP for $K=32$, $K_{G}=8$, $M=8$.

######  Remark 4.

The expression in (11d) captures the relationship between the UBOP and the AF
[12, 13, 14]. Specifically, at low SNR,
$\sum_{(n,~{}v)\neq(0,~{}0)}\exp\left\\{-{(|r_{0,0,v_{\epsilon}}-r_{n,v,v_{\epsilon}})|}/{2\sigma^{2}}\right\\}\propto\sum_{(n,~{}v)\neq(0,~{}0)}-{(|r_{0,0,v_{\epsilon}}|-|r_{n,v,v_{\epsilon}}|)}/{2\sigma^{2}}$,
indicating that the UBOP is approximately inversely proportional to the
integrated-side-to-peak-lobe-difference (ISPLD) of the AF in this regime. On
the contrary, at high SNR, the maximal exponential term in (11d) dominates.
Hence, the UBOP in this regime is inversely proportional to the peak-to-the-
largest-side-lobe difference of the AF. The relationship with the AF further
demonstrates that the OP and UBOP are sound and well-motivated metrics.

######  Remark 5.

From (11d) (combining with (10f)), the UBOP is a function of the input
distribution through $p_{k,m}$, which is only related with the CS magnitudes.
This motivates us to consider the problem of minimizing the UBOP under two
broad classes of input distributions, (i) those with constant magnitudes
(e.g., PSK input distribution), and (ii) those with variable magnitudes (e.g.,
Gaussian input distribution), which we consider in the following sections.

##  IV Minimizing OP for Constant Magnitude Input Distributions

In this section, we minimize the UBOP (${\mathbb{P}}_{\rm UB}$ in (11d)) for
CSs with constant magnitude input distributions, such as PSK. In this scheme,
the UBOP is deterministic regardless of the random information carried by CSs’
phases. Hence, waveforms with constant magnitude input distributions are able
to achieve the same radar performance in DFRC as that of using deterministic
OFDM waveforms in a pure radar system.

###  IV-A Problem formulation

For the special case of constant CS magnitude, the UBOP is a deterministic
function of $p_{k,m}$ – i.e., the power allocation across subcarriers and OFDM
symbols – and is independent of the probability mass vector. Hence, we
formulate the following optimization problem: {p_k,m} P_UB in (11d), ∑_k, m
p_k,m ≤P^max p_k,m ≥0, ∀k, m, where $P^{\max}$ is the transmit power
constraint.

######  Remark 6.

In this section, our primary focus is on optimizing radar performance, while
still acknowledging that information can be conveyed by phases. The
communication constraint is disregarded because we implicitly assume a uniform
probability mass vector over the constellation symbols, which can achieve a
rate of up to $\log_{2}Q$ bits/symbol for $Q-$PSK [38]. Hence, this scheme of
constant magnitude input distributions provides a performance benchmark for
the scheme of variable magnitude input distributions (Gaussian input) in
Section V. This part will also inspire the simplified algorithm for the
Gaussian input scheme in section V-C.

For the above problem, we have the following result:

######  Lemma 1.

The optimal power allocation across subcarriers ($p_{k,m}$ across $k$ for
$\forall~{}m$) to minimize ${\mathbb{P}}_{\rm UB}$ is the uniform power
allocation.

###### Proof:

Let $p_{m}\triangleq\sum_{k}~{}p_{k,m}$ denote the total power of the
$m^{\mathrm{th}}$ OFDM symbol. The expectation over $v_{\epsilon}$ in (11d) is
approximated by taking multiple samples of $v_{{\epsilon}}$ from the set
$\\{{\epsilon}_{1},~{}\cdots,~{}{\epsilon}_{|\kappa|}\\}$ (denoted by set
$\kappa$, the larger the number of samples, the more accurate the
approximation) following its uniform distribution. Hence, ${\mathbb{P}}_{\rm
UB}$ becomes:

$\displaystyle{\mathbb{P}}_{\rm UB}$ $\displaystyle\approx$
$\displaystyle\frac{1}{|\kappa|}\sum_{{{\epsilon}}\in\kappa}\sum_{(n,~{}v)\neq(0,~{}0)}\exp\Big{\\{}-\frac{|r_{0,0,{\epsilon}}|-|r_{n,v,{\epsilon}}|}{2\sigma^{2}}\Big{\\}}\hskip
8.53581pt(\mbox{for large }|\kappa|)$ (12a) $\displaystyle=$
$\displaystyle\frac{1}{|\kappa|}\sum_{{{\epsilon}}\in\kappa}\sum_{n=0,~{}v\neq
0}\exp\Big{\\{}-\frac{|r_{0,0,{\epsilon}}|-|r_{n,v,{\epsilon}}|}{2\sigma^{2}}\Big{\\}}$
$\displaystyle+\frac{1}{|\kappa|}\sum_{{{\epsilon}}\in\kappa}\sum_{~{}n\neq
0}\exp\Big{\\{}-\frac{|r_{0,0,{\epsilon}}|-|r_{n,v,{\epsilon}}|}{2\sigma^{2}}\Big{\\}}$
(12b) $\displaystyle\geq$
$\displaystyle\frac{\sum_{{{\epsilon}}\in\kappa}}{|\kappa|}\Big{(}\sum_{v\neq
0}\exp\Big{\\{}-\frac{|r_{0,0,{\epsilon}}|-|r_{0,v,{\epsilon}}|}{2\sigma^{2}}\Big{\\}}+\sum_{n\neq
0}\exp\Big{\\{}-\frac{|r_{0,0,{\epsilon}}|}{2\sigma^{2}}\Big{\\}}\Big{)}$
(12c) $\displaystyle=$
$\displaystyle\frac{\mathbf{1}^{T}}{|\kappa|}\exp\Big{\\{}-\frac{{\mathbf{1}_{M-1}}\otimes|\mathbf{r}_{0}|-|\mathbf{r^{\prime}}|}{2\sigma^{2}}\Big{\\}}+\frac{(K_{G}-1)M}{|\kappa|}\mathbf{1}^{T}\exp\Big{\\{}-\frac{|\mathbf{r}_{0}|}{2\sigma^{2}}\Big{\\}}$
$\displaystyle\overset{\triangle}{=}$ $\displaystyle{\mathbb{P}}_{\rm
UB}(\mathbf{r}),$ (12d) where $\displaystyle\mathbf{r}_{v}=$
$\displaystyle\left[r_{0,v,{\epsilon}_{1}},~{}\cdots,~{}r_{0,v,{\epsilon}_{|\kappa|}}\right]^{T}\in\mathbb{C}^{|\kappa|},$
(12e) $\displaystyle\mathbf{r^{\prime}}=$
$\displaystyle\left[\mathbf{r}_{1}^{T},~{}\cdots,~{}\mathbf{r}_{M-1}^{T}\right]^{T}\in\mathbb{C}^{(M-1)|\kappa|},$
(12f) $\displaystyle\mathbf{r}=$
$\displaystyle\left[\mathbf{r}_{0}^{T},~{}\mathbf{r^{\prime}}^{T}\right]^{T}=\mathbf{F}^{\mathrm{D}}\mathbf{p}\in\mathbb{C}^{M|\kappa|},$
(12g) $\displaystyle\mathbf{p}=$
$\displaystyle\left[p_{0},~{}\cdots,~{}p_{M-1}\right]^{T}\in\mathbb{R}^{M},$
(12h) $\displaystyle\mathbf{F}^{\mathrm{D}}=$
$\displaystyle\Big{[}\mathbf{f}^{\mathrm{D}}_{1-M/2,{\epsilon}_{1}},~{}\mathbf{f}^{\mathrm{D}}_{1-{M}/2,{\epsilon}_{2}},~{}\cdots,~{}\mathbf{f}^{\mathrm{D}}_{M/2-1,{\epsilon}_{|\kappa|}}\Big{]}^{T}\in\mathbb{C}^{M|\kappa|\times
M},$ (12i) $\displaystyle{f}^{\mathrm{D}}_{v,{\epsilon}_{l},m}=$
$\displaystyle
e^{-j2\pi{m(v-{{\epsilon}_{l}})}/{M}},~{}l=(1,~{}\cdots,~{}|\kappa|),$ (12j)

where (12c) is achieved because $|r_{n,v,{\epsilon}}|\geq 0$ for $n\neq 0$
with the equality achieved if and only if $p_{k,m}=p_{m}/K$ since
$r_{n,v,{\epsilon}}=\sum_{m,k}p_{m}e^{j2\pi\frac{nk}{K}}e^{-j2\pi\frac{vm}{M}}e^{j2\pi\frac{{\epsilon}m}{M}}/K=\sum_{k}e^{j2\pi\frac{nk}{K}}\sum_{m}p_{m}e^{-j2\pi\frac{vm}{M}}e^{j2\pi\frac{{{\epsilon}}m}{M}}/K=0$.
Then in (12c), the only variables involved are the power allocation across
OFDM symbols, i.e., $p_{m}$
($r_{0,v,{\epsilon}}=\sum_{m}p_{m}e^{-j2\pi\frac{vm}{M}}e^{j2\pi\frac{{{\epsilon}}m}{M}}$).
Hence, the optimal power allocation would have uniform power allocation across
OFDM subcarriers, and we need to optimize the power allocation across OFDM
symbols, i.e., $p_{m}$ for $\forall m$ or equivalently $\mathbf{p}$. ∎

With uniform (optimal) power allocation across OFDM subcarriers, the
optimization problem (IV-A) is simplified into: p , rP_UB(r) in (12), 1^T p
≤P^max p⪰0 r=F^Dp . After obtaining the optimized power allocation across the
OFDM symbols from (1), i.e. $\mathbf{p}$, the optimal power allocation at each
OFDM symbol’s sub-carrier is $p_{k,m}=p_{m}/K$, $\forall k,~{}m$ (from Lemma
1).

###  IV-B Low complexity optimization

Solving the non-convex problem (1) can be very time-consuming with a large
number of OFDM symbols $M$, whereas in OFDM radar, $M$ shall be large, i.e.,
above $512$, to achieve high Doppler resolution [6]. Hence, in this section,
we adopt a low-complexity alternating direction method of multipliers (ADMM)
algorithm where problem (1) is approximately solved in an iterative manner
with closed-form solutions in each round.

Considering the linear constraint in (1), we adopt ADMM to iteratively update
$\mathbf{p}$ and $\mathbf{r}$ until the two successive solutions get close
[39]. Following the ADMM structure444With ADMM, we obtain local optimal
solutions (referred to as optimal in the following) as an approximation to the
global optimal solution in this paper, similarly to the BCD algorithm in
Section V-B., at the $l^{\mathrm{th}}$ iteration, problem (1) is iteratively
formulated as:

$\displaystyle\mathbf{r}^{(l+1)}$
$\displaystyle:=\underset{\mathbf{r}}{\arg\min}\left\\{{\mathbb{P}}_{\rm
UB}(\mathbf{r})+\frac{\rho}{2}\|\mathbf{r}-\mathbf{F}^{\mathrm{D}}\mathbf{p}^{(l)}+\mathbf{d}^{(l)}\|^{2}\right\\},$
(13) $\displaystyle\mathbf{p}^{(l+1)}$
$\displaystyle:=\underset{\mathbf{p}}{\arg\min}\left\\{\mathbbm{1}_{(\mathbf{1}^{T}\mathbf{p}\leq
P^{\max},~{}\mathbf{p}\succeq\mathbf{0})}+\frac{\rho}{2}\|\mathbf{r}^{(l+1)}-\mathbf{F}^{\mathrm{D}}\mathbf{p}+\mathbf{d}^{(l)}\|^{2}\right\\},$
(14) $\displaystyle\mathbf{d}^{(l+1)}$
$\displaystyle:=\mathbf{d}^{(l)}+\mathbf{r}^{(l+1)}-\mathbf{F}^{\mathrm{D}}\mathbf{p}^{(l+1)},$
(15)

where $\mathbf{d}^{(l)}$ is the ADMM stepsize at the $l^{\mathrm{th}}$
iteration and $\rho$ is the ADMM penalty coefficient.

####  IV-B1 Updating $\mathbf{r}$ in (13)

To solve subproblem (13), we adopt the successive convex programming (SCP) to
approximate the exponential function in (13) linearly by the first-order
Taylor expansion. The optimal solution of this approximated problem is then
used as a new operating point of the Taylor expansion for the next SCP round.
The procedure is repeated until two successive solutions are close enough.

Consequently, assume that $\mathbf{r}^{(l+1,~{}t)}$ is the optimal solution at
the $t^{\mathrm{th}}$ iteration of SCP. Then $\mathbf{r}^{(l+1,~{}t+1)}$ is
updated as:

$\displaystyle\mathbf{r}^{(l+1,~{}t+1)}$
$\displaystyle=\underset{\mathbf{r}}{\arg\min}\left\\{{\boldsymbol{\alpha}^{(l+1,~{}t)}}^{T}|\mathbf{r}|+\frac{\rho}{2}\|\mathbf{r}-\mathbf{F}^{\mathrm{D}}\mathbf{p}^{(l)}\|^{2}\right\\}$
(16a)
$\displaystyle=\left(\mathbf{F}^{\mathrm{D}}\mathbf{p}^{(l)}-{\boldsymbol{\theta}^{(l)}\odot\boldsymbol{\alpha}^{(l+1,~{}t)}}/\rho\right)^{+},$
(16b)

where
$\boldsymbol{\theta}^{(l)}=\angle\mathbf{F}^{\mathrm{D}}\mathbf{p}^{(l)}$. The
closed-form expression in (16b) is derived in Appendix B. The first-order
Taylor coefficient
$\boldsymbol{\alpha}^{(l+1,~{}t)}=[{\boldsymbol{\alpha}_{0}^{(l+1,~{}t)}}^{T},~{}\cdots,~{}{\boldsymbol{\alpha}_{M-1}^{(l+1,~{}t)}}^{T}]^{T}$
is expressed in (17).

$\boldsymbol{\alpha}_{v}^{(l+1,t)}=\begin{cases}-\sum_{v=1}^{M-1}\exp\Big{\\{}-\frac{|{\mathbf{r}_{0}^{(l+1,t)}}|-|{\mathbf{r}_{v}^{(l+1,t)}}|}{2\sigma^{2}}\Big{\\}}/{\left(2\sigma^{2}{|\kappa|}\right)}-(K_{G}-1)M\exp\Big{\\{}-\frac{|{\mathbf{r}_{0}^{(l+1,t)}}|}{2\sigma^{2}}\Big{\\}}/{\left(2\sigma^{2}{|\kappa|}\right)},&\text{
for }v=0;\\\
\exp\Big{\\{}-\frac{|{\mathbf{r}_{0}^{(l+1,t)}}|-|{\mathbf{r}_{v}^{(l+1,t)}}|}{2\sigma^{2}}\Big{\\}}/{\left(2\sigma^{2}{|\kappa|}\right)},&\text{
otherwise.}\end{cases}$ (17)

####  IV-B2 Updating $\mathbf{p}$ in (14)

Given a fixed $\mathbf{r}^{(l+1)}$, the optimal $\mathbf{p}^{(l+1)}$ can be
directly solved by quadratic programming (QP) as: p ρ2∥r^(l+1)-F^Dp ∥^2, 1^T p
≤P^max p⪰0 . The algorithm is summarized in Algorithm 1.

Input: $l\leftarrow
0,~{}\mathbf{r}^{(0)},~{}\mathbf{p}^{(0)},~{}\rho,~{}{\epsilon}_{S}>0$

Repeat (Solving $\mathbf{p}$ in (1)):

  1. 1.

Update $\mathbf{r}^{(l+1)}$ in problem (13) by SCP

    1. Initialize $t\leftarrow 0$; $\mathbf{r}^{(l+1,~{}0)}\leftarrow\mathbf{r}^{(l)}$

    2. while $t\leq 10^{2}$

      1. Compute $\boldsymbol{\alpha}^{(l+1,~{}t)}$ in (17) at the operating point $\mathbf{r}^{(l+1,~{}t)}$

      2. Compute $\mathbf{r}^{(l+1,~{}t+1)}$ using (16b)

      3. $t\leftarrow t+1$; $\mathbf{r}^{(l+1)}\leftarrow\mathbf{r}^{(l+1,~{}t+1)}$

    3. Iteration complexity $\mathcal{O}\left(M^{2}|\kappa|\right)$

  2. 2.

Update $\mathbf{p}^{(l+1)}$ in (14) by QP (complexity
$\mathcal{O}\left(M^{3}\right)$)

  3. 3.

$l\leftarrow l+1$; ${\mathbf{p}}\leftarrow\mathbf{p}^{(l+1)}$

  4. 4.

Quit if
$\|\mathbf{p}^{(l+1)}-\mathbf{p}^{(l)}\|<{\epsilon}_{S}\|\mathbf{p}^{(l+1)}\|$

Output: ${\mathbf{p}}_{k,m}\leftarrow p_{m}/K$

Algorithm 1 ADMM for PSK input in (IV-A)

##  V Minimizing OP for variable magnitude Input Distributions (Gaussian)

This section evaluates the radar performance in OFDM DFRC with non-zero mean
Gaussian input, where the magnitude of the CSs becomes a random variable
obeying the Rician distribution. Consequently, we evaluate the aUBOP and then
optimize the Gaussian mean and variance at each symbol’s subcarriers to
minimize the aUBOP. In this scheme, an achievable communication constraint is
added to avoid a zero-variance optimization result where no information bits
are transmitted.

###  V-A Problem formulation

In this context, we first model the CS $X[k,~{}m]$ in (3) as a complex non-
zero mean asymmetric Gaussian variable:

$\displaystyle X[k,~{}m]={X}_{\mathrm{R}}[k,~{}m]+jX_{\mathrm{I}}[k,~{}m],$
(18)

where
${X}_{\mathrm{R/I}}[k,~{}m]\sim\mathcal{N}({\mu}_{\mathrm{R/I},~{}k,m},~{}{\sigma}_{\mathrm{R/I},~{}k,m}^{2})$.

Then, the average power (second-order moments) allocated to the real and
imaginary part of the $m^{\mathrm{th}}$ OFDM symbol at the $k^{\mathrm{th}}$
subcarrier are defined as:

$\displaystyle\color[rgb]{0,0,1}\overline{p}_{\mathrm{R},k,m}$
$\displaystyle={\mu}_{\mathrm{R},{k,m}}^{2}+{\sigma}_{\mathrm{R},{k,m}}^{2},$
(19a) $\displaystyle\overline{p}_{\mathrm{I},k,m}$
$\displaystyle={\mu}_{\mathrm{I},{k,m}}^{2}+{\sigma}_{\mathrm{I},{k,m}}^{2}.$
(19b)

On this basis, the AF $r_{n,v,v_{\epsilon}}$ in (11d) becomes random, making
the UBOP varying. Hence, we derive the aUBOP as follows (abbreviating the
expectation over $X[k,~{}m]$ as $X$, i.e.,
$\mathbb{E}_{X}\triangleq\mathbb{E}_{X[k,~{}m]}$ ):

$\displaystyle\overline{{\mathbb{P}}}_{\rm
UB}\left(\overline{\mathbf{p}},~{}\boldsymbol{\sigma}\right)$ (20a)
$\displaystyle=$
$\displaystyle\mathbb{E}_{v_{{\epsilon}},X}\Big{\\{}\sum_{(n,~{}v)\neq(0,~{}0)}\exp\Big{\\{}-\frac{|r_{0,0,v_{\epsilon}}|-|r_{n,v,v_{\epsilon}}|}{2\sigma^{2}}\Big{\\}}/2\Big{\\}}$
(20b) $\displaystyle\propto$
$\displaystyle\frac{1}{4\sigma^{2}}\sum_{(n,~{}v)\neq(0,~{}0)}\mathbb{E}_{{{\epsilon}},X}\left\\{-{|r_{0,0,{\epsilon}}|+|r_{n,v,{\epsilon}}|}\right\\}$
(20c) $\displaystyle\leq$
$\displaystyle\frac{1}{4\sigma^{2}}\sum_{(n,~{}v)\neq(0,~{}0)}\mathbb{E}_{{{\epsilon}}}\Big{\\{}-|\mathbb{E}_{X}\left\\{r_{0,0,{\epsilon}}\right\\}|+\sqrt{\mathbb{E}_{X}\left\\{|r_{n,v,{\epsilon}}|^{2}\right\\}}\Big{\\}}$
(20d) $\displaystyle=$
$\displaystyle\frac{1}{4|\kappa|\sigma^{2}}\sum_{{\epsilon}\in\kappa}\sum_{(n,~{}v)\neq(0,~{}0)}-\sqrt{\overline{\mathbf{p}}^{T}\mathbf{A}_{{\epsilon}}\overline{\mathbf{p}}}+g_{n,v,{\epsilon}}\left(\overline{\mathbf{p}},~{}\boldsymbol{\sigma}\right),$
(20e)
$\displaystyle\mbox{with}~{}g_{n,v,{\epsilon}}\left(\overline{\mathbf{p}},~{}\boldsymbol{\sigma}\right)=$
$\displaystyle\sqrt{\mathbf{\overline{p}}^{T}\mathbf{A}_{n,v,{\epsilon}}\mathbf{\overline{p}}-2\|\overline{\mathbf{p}}-\boldsymbol{\sigma}\|^{2}},$
(20f) $\displaystyle\overline{\mathbf{p}}=$
$\displaystyle\left[\overline{p}_{\mathrm{R},{0,0}},~{}\overline{p}_{\mathrm{R},{1,0}},~{}\cdots,,~{}\overline{p}_{\mathrm{I},{0,0}},~{}\cdots,~{}\overline{p}_{\mathrm{I},{K-1,M-1}}\right]^{T},$
(20g) $\displaystyle{\boldsymbol{\sigma}}=$
$\displaystyle\left[{\sigma}_{\mathrm{R},{0,0}},~{}{\sigma}_{\mathrm{R},{1,0}},~{}\cdots,~{}{\sigma}_{\mathrm{I},{0,0}},~{}\cdots,~{}{\sigma}_{\mathrm{I},{K-1,M-1}}\right]^{T},$
(20h) $\displaystyle\mathbf{A}_{{\epsilon}}=$
$\displaystyle\mathfrak{R}\left\\{{\mathbf{f}_{0,0,{{\epsilon}}}}\right\\}\mathfrak{R}\left\\{{\mathbf{f}_{0,0,{{\epsilon}}}}\right\\}^{T}+\mathfrak{I}\left\\{{\mathbf{f}_{0,0,{{\epsilon}}}}\right\\}\mathfrak{I}\left\\{{\mathbf{f}_{0,0,{{\epsilon}}}}\right\\}^{T},$
(20i) $\displaystyle\mathbf{A}_{n,v,{\epsilon}}=$
$\displaystyle{\mathfrak{R}\left\\{\mathbf{f}_{n,v,{\epsilon}}\right\\}}{\mathfrak{R}\left\\{\mathbf{f}_{n,v,{\epsilon}}\right\\}}^{T}+{\mathfrak{I}\left\\{\mathbf{f}_{n,v,{\epsilon}}\right\\}}{\mathfrak{I}\left\\{\mathbf{f}_{n,v,{\epsilon}}\right\\}}^{T}+2\mathbf{I}_{2KM},$
(20j) $\displaystyle\mathbf{f}_{n,v,{{\epsilon}}}=$
$\displaystyle\mathbf{1}_{2}\otimes\left(\mathbf{f}^{\mathrm{D}}_{v,~{}{{\epsilon}}}\otimes\mathbf{f}^{\mathrm{R}}_{n}\right)\in\mathbb{C}^{2KM},$
(20k) $\displaystyle{f}^{\mathrm{R}}_{n,k}=$ $\displaystyle
e^{j2\pi{nk}/{K}},$ (20l)

where (20c) assumes low SNR at the radar’s receiver555 In this paper, we only
discuss the statistic approximation and the optimization in the low SNR region
due to the space limit, while similar procedures can be done for the middle
SNR and for the high SNR scenarios. For the middle SNR region, (20c) can be
replaced by Taylor approximation with appropriate orders and coefficients
given its finite SNR region[40]. For the high SNR region, inspired by Remark
4, the objective function might be approximated by maximizing the minimal
average peak-to-side-lobe difference., giving $\exp\\{x\\}\rightarrow x+1$ for
$x\rightarrow 0$ ($r_{0,0,\epsilon}\geq r_{n,v,\epsilon}$ for any $(n,~{}v)$).
(20d) is due to the concavity of the square root function. (20e) is achieved
from Appendix C, whose first sum term refers to
$\mathbb{E}_{{{\epsilon}}}\left\\{-|\mathbb{E}_{X}\left\\{r_{0,0,{\epsilon}}\right\\}|\right\\}$
in (20d) representing the magnitude of average peak lobe, and whose second sum
term refers to
$\mathbb{E}_{{{\epsilon}}}\left\\{-|\mathbb{E}_{X}\left\\{r_{n,v,{\epsilon}}\right\\}|\right\\}$
at $(n,~{}v)\neq~{}(0,~{}0)$ in (20d) representing the magnitude of the
average sidelobes.

######  Remark 7.

(20d) indicates that the optimization objective (aUBOP),
$\overline{{\mathbb{P}}}_{\rm
UB}\left(\overline{\mathbf{p}},~{}\boldsymbol{\sigma}\right)$, finally becomes
a scaling term of the average ISPLD in low SNR, corresponding to Remark 4.
Also, (20e) indicates that, without communication constraints, the optimal
solution to minimize (20e) yields maximizing
$\|\mathbf{\overline{p}}-\boldsymbol{\sigma}\|$ for an arbitrary
$\mathbf{\overline{p}}$, i.e., $\sigma^{2}_{\mathrm{R/I},k,m}=0$ and hence
$\overline{p}_{\mathrm{R/I},k,m}=\mu_{\mathrm{R/I},k,m}^{2}$. In this context,
all the power is allocated to the deterministic component of the Gaussian
distributions for the radar-optimal purpose. In other words, the randomness of
the CS magnitudes degrades radar’s OP.

###  V-B Optimization

We then formulate the corresponding optimization problem in DFRC using the
developed metric in (20e). With the constraint on achievable rate, we have the
optimization problem (neglecting the constant multiplier $1/(4\sigma^{2})$):
¯p, σ∑_ϵ∈κ∑_(n, v)≠(0, 0)-¯p^TA_ϵ¯p+g_n,v,ϵ(¯p, σ), 1^T¯p≤P^max 12KMlog
det(I_2KM+Cdiag{σ}) ≥R^c ¯p ⪰σ σ⪰0, where (V-B) is the average rate constraint
over $M$ OFDM symbols. In (V-B),
$\mathbf{C}=2K\mathrm{diag}\left\\{{\mathbf{1}_{2M}\otimes\mathbf{h}/({B\sigma_{\mathrm{C}}^{2}})}\right\\}\in\mathbb{R}^{2KM\times
2KM}$ with the $k^{\mathrm{th}}$ entry of $\mathbf{h}$ being
${h}_{k}={|h_{\mathrm{C},k}|^{2}}$ for $k=[1,~{}\cdots,~{}K]$.
$h_{\mathrm{C},k}$ is the complex gain of the communication channel for the
$k^{\mathrm{th}}$ subcarrier. $\sigma_{\mathrm{C}}^{2}$ is the noise power
density at the communication receiver. $R^{c}$ is the minimum achievable rate
constraint for communications.

For efficiency, we use the method of block coordinate descent (BCD) to solve
problem (V-B). In BCD, given an initial point of
$\overline{\mathbf{p}}^{(l)}/\boldsymbol{\sigma}^{(l)}$ at the
$l^{\mathrm{th}}$ iteration, $\overline{\mathbf{p}}^{(l+1)}$ is optimized
first with fixed $\boldsymbol{\sigma}^{(l)}$. Then
$\boldsymbol{\sigma}^{(l+1)}$ is optimized with fixed
$\overline{\mathbf{p}}^{(l+1)}$. The iteration continues until two successive
solutions get close.

####  V-B1 Updating $\mathbf{\overline{p}}$

Given $\boldsymbol{\sigma}^{(l)}$, the sub-problem for $\overline{\mathbf{p}}$
is: ¯p∑_ϵ∈κ∑_(n, v)≠(0, 0)-¯p^TA_ϵ¯p+g_n,v,ϵ(¯p, σ^(l)), 1^T¯p≤P^max ¯p
⪰σ^(l), whose objective function in (V-B1) is non-convex. As a solution, we
adopt SCP similarly to Section IV-B, where the square root function in (V-B1)
is approximated by its first-order Taylor expansion:
$\sqrt{x}\approx(x-x_{0})/(2\sqrt{x_{0}})+\sqrt{x_{0}}=x/(2\sqrt{x_{0}})+\sqrt{x_{0}}/2$.

Assume $\overline{\mathbf{p}}^{(t)}$ is the operating point at the beginning
of the $t^{\mathrm{th}}$ iteration when solving problem (V-B1). Then, (V-B1)
at the $t^{\mathrm{th}}$ iteration is linearly approximated as:

$\displaystyle\sum_{{\epsilon}\in\kappa}\sum_{(n,~{}v)\neq(0,~{}0)}-\sqrt{\overline{\mathbf{p}}^{T}\mathbf{A}_{{\epsilon}}\overline{\mathbf{p}}}+g_{n,v,{\epsilon}}\left(\overline{\mathbf{p}},~{}\boldsymbol{\sigma}^{(l)}\right)$
$\displaystyle\approx$
$\displaystyle\overline{\mathbf{p}}^{T}\mathbf{B}^{(l,~{}t)}\overline{\mathbf{p}}+{\boldsymbol{\alpha}^{(l,~{}t)}}^{T}\overline{\mathbf{p}}+{\alpha_{c}}^{(l,~{}t)},$
(21a) with $\displaystyle\mathbf{B}^{(l,~{}t)}=$
$\displaystyle\sum_{{\epsilon}\in\kappa}\sum_{(n,~{}v)\neq(0,~{}0)}\mathbf{A}_{n,v,{\epsilon}}/g_{n,v,{\epsilon}}\big{(}\overline{\mathbf{p}}^{(t)},~{}\boldsymbol{\sigma}^{(l)}\big{)}/2$
(21b) $\displaystyle=$ $\displaystyle
2\|1\oslash\mathbf{g}^{(l,~{}t)}\|^{2}\mathbf{I}_{2KM}+\big{(}\mathfrak{R}\left\\{\mathbf{F}\right\\}\oslash\mathbf{g}^{(l,~{}t)}\big{)}\big{(}\mathfrak{R}\left\\{\mathbf{F}\right\\}\oslash\mathbf{g}^{(l,~{}t)}\big{)}^{T}$
$\displaystyle+\big{(}\mathfrak{I}\left\\{\mathbf{F}\right\\}\oslash\mathbf{g}^{(l,~{}t)}\big{)}\big{(}\mathfrak{I}\left\\{\mathbf{F}\right\\}\oslash\mathbf{g}^{(l,~{}t)}\big{)}^{T},$
(21c) $\displaystyle\boldsymbol{\alpha}^{(l,~{}t)}=$
$\displaystyle-(K_{G}M-1)\sum_{{\epsilon}\in\kappa}\mathbf{A}_{{\epsilon}}\overline{\mathbf{p}}^{(t)}/\sqrt{{\mathbf{p}^{(t)}}^{T}\mathbf{A}_{{\epsilon}}{\mathbf{p}^{(t)}}}$
$\displaystyle-4\big{(}\overline{\mathbf{p}}^{(t)}-\boldsymbol{\sigma}^{(t)}\big{)}\|1\oslash\mathbf{g}^{(l,~{}t)}\|^{2},$
(21d) $\displaystyle\mathbf{F}=$
$\displaystyle\begin{bmatrix}\mathbf{F}_{0,-M/2},~{}\mathbf{F}_{1,-M/2},~{}\cdots,~{}\mathbf{F}_{K_{G}-1,M/2-1}\end{bmatrix},~{}(\text{w/o
}\mathbf{F}_{0,0})$ (21e) $\displaystyle\mathbf{F}_{n,v}=$
$\displaystyle\begin{bmatrix}\mathbf{f}_{n,v,{\epsilon}_{1}},~{}\cdots,~{}\mathbf{f}_{n,v,{\epsilon}_{|\kappa|}}\end{bmatrix}\in\mathbb{R}^{2KM\times|\kappa|},$
(21f) $\displaystyle\mathbf{g}^{(l,~{}t)}=$
$\displaystyle\left[4{\mathbf{q}^{(t)}+8\|\overline{\mathbf{p}}^{(t)}\|^{2}-8\|\overline{\mathbf{p}}^{(t)}-\boldsymbol{\sigma}^{(l)}\|^{2}}\right]^{1/4},$
(21g) $\displaystyle\mathbf{q}^{(t)}=$
$\displaystyle{\mathbf{\overline{p}}^{(t)}}^{T}{\mathfrak{R}\left\\{\mathbf{F}\right\\}}\odot\big{(}{\mathbf{\overline{p}}^{(t)}}^{T}{\mathfrak{R}\left\\{\mathbf{F}\right\\}}\big{)}^{*}+{\mathbf{\overline{p}}^{(t)}}^{T}\mathfrak{I}\left\\{\mathbf{F}\right\\}\odot\big{(}{\mathbf{\overline{p}}^{(t)}}^{T}\mathfrak{I}\left\\{\mathbf{F}\right\\}\big{)}^{*},$
(21h)

where ${\alpha_{c}}^{(l,~{}t)}$ is the Taylor constant that can be omitted.

Hence, we form the sub-problem towards solving problem (V-B1) as: ¯p¯p^TB^(l,
t)¯p+α^(l, t)^T¯p,  (V-B1)- (V-B1), which is a QP problem.

####  V-B2 Updating $\boldsymbol{\sigma}$

When fixing $\mathbf{\overline{p}}$ as $\mathbf{\overline{p}}^{(l)}$, we
observe that the objective function in problem (V-B) is a monotonically
increasing function with
$2{\mathbf{\overline{p}}^{(l)}}^{T}\boldsymbol{\sigma}-\|\boldsymbol{\sigma}\|^{2}$.
Hence, the sub-problem for $\boldsymbol{\sigma}$ becomes: σ2¯p^(l)^Tσ-∥σ∥^2,
12KMlog det(I_2KM+Cdiag{σ}) ≥R^c ¯p^(l) ⪰σ σ⪰0, which is convex. Again, for
lower complexity to handle a large number of variables, we solve problem
(V-B2) iteratively by SCP with closed-form solutions. At the $t^{\mathrm{th}}$
iteration of the SCP sub-problem, the concave objective function is
approximated by the first order Taylor expansion, and the sub-problem becomes:
σβ^(l, t)^Tσ, (V-B2)- (V-B2), with
$\boldsymbol{\beta}^{(l,~{}t)}=2\left(\mathbf{\overline{p}}^{(l)}-\boldsymbol{\sigma}^{(t)}\right)$.
The closed-form solution for problem (V-B2), after KKT, is given by:

$\displaystyle\color[rgb]{0,0,1}{\sigma}_{q}=\begin{cases}{\overline{p}}^{(l)}_{q},&~{}u>\frac{2KM{\beta}^{(l,~{}t)}_{q}\left(1+C_{q,q}{{\overline{p}}}^{(l)}_{q}\right)}{C_{q,q}};\\\
0,&~{}0<u<\frac{2KM{\beta}^{(l,~{}t)}_{q}}{C_{q,q}};\\\
\frac{u}{2KM{\beta}^{(l,~{}t)}_{q}}-\frac{1}{C_{q,q}},&~{}\text{otherwise},\end{cases}$
(22)

where $u$ is the Lagrangian multiplier of constraint (V-B2), obtained by bi-
section search.

The entire algorithm to solve problem (V-B) is summarized in Algorithm 2.

Input: $l\leftarrow
0,~{}\overline{\mathbf{p}}^{(0)},~{}{\boldsymbol{\sigma}}^{(0)}_{R},~{}{\epsilon}_{S}>0$;

Output: $\overline{\mathbf{p}},~{}{\boldsymbol{\sigma}}$;

Repeat:

  1. 1.

Compute $\overline{\mathbf{p}}^{(l+1)}$ in problem (V-B1) using SCP

    1. Initialize $t\leftarrow 0$; $\overline{\mathbf{p}}^{(l+1,~{}0)}\leftarrow\overline{\mathbf{p}}^{(l)}$

    2. while $t\leq 10^{2}$

      1. Compute $\mathbf{B}^{(l+1,~{}t)}$ and $\boldsymbol{\alpha}^{(l+1,~{}t)}$ in (21c)-(21)

      2. Compute $\overline{\mathbf{p}}^{(l+1,~{}t+1)}$ using (V-B1)

      3. $t\leftarrow t+1$; $\overline{\mathbf{p}}^{(l+1)}\leftarrow\overline{\mathbf{p}}^{(l+1,~{}t+1)}$

    3. Iteration complexity: $\mathcal{O}\left(K^{3}M^{3}|\kappa|\right)$

  2. 2.

Compute ${\boldsymbol{\sigma}}^{(l+1)}$ in problem (V-B2) using SCP

    1. Initialize $t\leftarrow 0$; ${\boldsymbol{\sigma}}^{(l+1,~{}0)}\leftarrow{\boldsymbol{\sigma}}^{(l)}$

    2. while $t\leq 10^{2}$

      1. Compute $\boldsymbol{\beta}^{(l+1,~{}t)}$ in (V-B2) and ${\boldsymbol{\sigma}}^{(l+1,~{}t+1)}$ in (22)

      2. $t\leftarrow t+1$; ${\boldsymbol{\sigma}}^{(l+1)}\leftarrow{\boldsymbol{\sigma}}^{(l+1,~{}t+1)}$

    3. Iteration complexity: $\mathcal{O}\left(KM\right)$

  3. 3.

$l\leftarrow l+1$;
$\overline{\mathbf{p}}\leftarrow\overline{\mathbf{p}}^{(l+1)}$;
${\boldsymbol{\sigma}}\leftarrow{\boldsymbol{\sigma}}^{(l+1)}$

  4. 4.

Quit if $|\overline{{\mathbb{P}}}^{(l+1)}_{\rm
UB}\left(\overline{\mathbf{p}},~{}\boldsymbol{\sigma}\right)-\overline{{\mathbb{P}}}^{(l)}_{\rm
UB}\left(\overline{\mathbf{p}},~{}\boldsymbol{\sigma}\right)|<{\epsilon}_{S}$

Algorithm 2 BCD for Gaussian input in (V-B)

###  V-C Decoupling optimization

Algorithm 2 can be time-consuming given large $K$ and $M$ because of the
large-dimensional variables and coefficients. However, inspired by the optimal
power allocation in section IV, we give a simplified method by decoupling the
optimization of Gaussian input distribution across time domain (OFDM symbols)
and frequency domain (OFDM subcarriers), namely the decoupling method.
Specifically, we assume that the distribution of CSs on each OFDM symbol is a
scaled version of a common distribution
$\overline{\mathbf{p}}_{\mathrm{K}}\in\mathbb{R}^{2K}$ and
$\boldsymbol{\sigma}_{\mathrm{K}}\in\mathbb{R}^{2K}$ (accounting for the real
and imaginary part). Mathematically, the final distribution of $X[k,~{}m]$ in
(18) has the form:

$\displaystyle\overline{\mathbf{p}}=\mathbf{F}_{\mathrm{M}}\overline{\mathbf{p}}_{\mathrm{K}},~{}{\boldsymbol{\sigma}}=\mathbf{F}_{\mathrm{M}}\boldsymbol{\sigma}_{\mathrm{K}},$
(23)

where
$\mathbf{F}_{\mathrm{M}}=\begin{bmatrix}\mathbf{p}\otimes\mathbf{I}_{K}&\mathbf{0}_{KM\times
K}\\\ \mathbf{0}_{KM\times
K}&\mathbf{p}\otimes\mathbf{I}_{K}\end{bmatrix}\in\mathbb{R}^{2KM\times 2K}$
with $\mathbf{p}\in\mathbb{R}^{M}$ being the optimal power allocation across
OFDM symbols in Section IV. Correspondingly, for the power allocation to be
optimized across OFDM subcarriers, we have
${\mathbf{1}}^{T}\mathbf{\overline{p}}_{\mathrm{K}}\leq 1$ as a substitute of
the power constraint and adopt the algorithm in Section V-B. Consequently,
compared with optimizing $\overline{\mathbf{p}}$ and $\boldsymbol{\sigma}$ of
totally $4KM$ variables in section V-B, after decoupling, we only need to
optimize ${\mathbf{p}}$, $\overline{\mathbf{p}}_{\mathrm{K}}$ and
$\boldsymbol{\sigma}_{\mathrm{K}}$ which are totally $M+4K$ variables.

On this basis, $\overline{{\mathbb{P}}}_{\rm
UB}\left(\overline{\mathbf{p}},~{}\boldsymbol{\sigma}\right)$ in (20e) as a
function of $\overline{\mathbf{p}}_{\mathrm{K}}$ and
$\boldsymbol{\sigma}_{\mathrm{K}}$ becomes:

$\displaystyle\overline{{\mathbb{P}}}_{\rm
UB}\left(\overline{\mathbf{p}}_{\mathrm{K}},~{}\boldsymbol{\sigma}_{\mathrm{K}}\right)=$
$\displaystyle\sum_{{\epsilon}\in\kappa}\left[-\sqrt{{\overline{\mathbf{p}}_{\mathrm{K}}}^{T}{\mathbf{F}_{\mathrm{M}}}^{T}\mathbf{A}_{{\epsilon}}\mathbf{F}_{\mathrm{M}}\overline{\mathbf{p}}_{\mathrm{K}}}\right.$
$\displaystyle\left.+\sum_{(n,~{}v)\neq(0,~{}0)}g_{n,v,{\epsilon}}\left(\mathbf{F}_{\mathrm{M}}\overline{\mathbf{p}}_{\mathrm{K}},~{}\mathbf{F}_{\mathrm{M}}\boldsymbol{\sigma}_{\mathrm{K}}\right)\right]$
(24a) $\displaystyle=$
$\displaystyle{A}_{\mathrm{D}}^{0}\mathbf{1}^{T}\overline{\mathbf{p}}_{\mathrm{K}}+\sum_{(n,~{}v)\neq(0,~{}0),~{}{\epsilon}}g^{0}_{n,v,{\epsilon}}\left(\overline{\mathbf{p}}_{\mathrm{K}},~{}\boldsymbol{\sigma}_{\mathrm{K}}\right),$
(24b) $\displaystyle\mbox{with}~{}{A}_{\mathrm{D}}^{0}=$
$\displaystyle\sum_{{\epsilon}\in\kappa}{\sqrt{{\left(\mathbf{p}^{T}\mathfrak{R}\left\\{\mathbf{f}^{\mathrm{D}}_{0,~{}{\epsilon}}\right\\}\right)^{2}}+{\left(\mathbf{p}^{T}\mathfrak{I}\left\\{\mathbf{f}^{\mathrm{D}}_{0,~{}{\epsilon}}\right\\}\right)^{2}}}},$
(24c) $\displaystyle g^{0}_{n,v,{\epsilon}}$
$\displaystyle\left(\overline{\mathbf{p}}_{\mathrm{K}},~{}\boldsymbol{\sigma}_{\mathrm{K}}\right)$
$\displaystyle=$
$\displaystyle\sqrt{{\mathbf{\overline{p}}_{K}}^{T}{\mathbf{A}}^{0}_{n,v,{\epsilon}}\mathbf{\overline{p}}_{K}-2\|\mathbf{p}\|^{2}\|\overline{\mathbf{p}}_{\mathrm{K}}-\boldsymbol{\sigma}_{\mathrm{K}}\|^{2}},$
(24d) $\displaystyle{\mathbf{A}}^{0}_{n,v,{\epsilon}}=$
$\displaystyle\gamma_{v,{\epsilon}}\left[{\mathfrak{R}\left\\{\mathbf{f}_{n}^{\mathrm{R_{0}}}\right\\}}{\mathfrak{R}\left\\{\mathbf{f}_{n}^{\mathrm{R_{0}}}\right\\}}^{T}+{\mathfrak{I}\left\\{\mathbf{f}_{n}^{\mathrm{R_{0}}}\right\\}}{\mathfrak{I}\left\\{\mathbf{f}_{n}^{\mathrm{R_{0}}}\right\\}}^{T}\right]$
$\displaystyle+2\|\mathbf{p}\|^{2},$ (24e)
$\displaystyle\gamma_{v,{\epsilon}}=$
$\displaystyle|\mathbf{p}^{T}\mathbf{f}^{\mathrm{D}}_{v,~{}{\epsilon}}|^{2},~{}\mathbf{f}_{n}^{\mathrm{R_{0}}}=\mathbf{1}_{2}\otimes\mathbf{f}_{n}^{\mathrm{R}},$
(24f)

where (24b) comes from Appendix D.

Given (24b), the optimal $\overline{\mathbf{p}}_{\mathrm{K}}$ and
$\boldsymbol{\sigma}_{\mathrm{K}}$ using the optimal ${\mathbf{p}}$ from
Algorithm 1 are obtained by: ¯p_K, σ_K¯P_UB(¯p_K, σ_K)  in (24b) , 1^T¯p_K≤1
12KM∑_mlog det(I_2K+p_mC’diag{σ_K}) ≥R^c ¯p_K⪰σ_K σ_K ⪰0, where
$\mathbf{C}^{\prime}=2K\mathrm{diag}\left\\{{\mathbf{1}_{2}\otimes\mathbf{h}/({B\sigma^{2}})}\right\\}\in\mathbb{R}^{2K\times
2K}$. (V-C) can be solved in the same way as that in (V-B) in Section V-B
where $\overline{\mathbf{p}}_{\mathrm{K}}$ and
$\boldsymbol{\sigma}_{\mathrm{K}}$ are updated iteratively following a BCD
structure, with only slight modifications on the coefficients to couple with
the optimal $\mathbf{p}$. The details are:

####  V-C1 Updating $\mathbf{p}_{\mathrm{K}}$

Then, we follow the same SCP procedure as (V-B1) in Section V-B1 to update
$\mathbf{p}_{K}$, with the only difference being the coefficients in (21c) and
(21) updated by (similarly to (21)):

$\displaystyle\mathbf{B}_{0}^{(l,~{}t)}=$ $\displaystyle
2{\|\mathbf{p}\|^{2}}\|1\oslash\mathbf{g}_{0}^{(l,~{}t)}\|^{2}\mathbf{I}_{2K}+\big{(}\mathbf{F}_{\mathrm{R_{0}}}\oslash\mathbf{g}_{0}^{(l,~{}t)}\big{)}\big{(}\mathbf{F}_{\mathrm{R_{0}}}\oslash\mathbf{g}_{0}^{(l,~{}t)}\big{)}^{T}$
$\displaystyle+\big{(}\mathbf{F}_{\mathrm{I_{0}}}\oslash\mathbf{g}_{0}^{(l,~{}t)}\big{)}\big{(}\mathbf{F}_{\mathrm{I_{0}}}\oslash\mathbf{g}_{0}^{(l,~{}t)}\big{)}^{T},$
(25a) $\displaystyle\boldsymbol{\alpha}_{0}^{(l,t)}=$
$\displaystyle{A}_{D}^{0}\mathbf{1}_{2K}-4\|\mathbf{p}\|^{2}\|1\oslash\mathbf{g}_{0}^{(l,~{}t)}\|^{2}\big{(}\overline{\mathbf{p}}_{\mathrm{K}}^{(t)}-\boldsymbol{\sigma}_{\mathrm{K}}^{(t)}\big{)},$
(25b) with $\displaystyle\mathbf{g}_{0}^{(l,~{}t)}=$
$\displaystyle\left[4{{\mathbf{q}_{0}^{(t)}}+8\|\mathbf{p}\|^{2}\|{\overline{\mathbf{p}}_{\mathrm{K}}^{(t)}}\|^{2}-8\|\mathbf{p}\|^{2}\|{\overline{\mathbf{p}}_{\mathrm{K}}^{(t)}}-\boldsymbol{\sigma}_{\mathrm{K}}^{(l)}\|^{2}}\right]^{1/4},$
(25c) $\displaystyle{\mathbf{q}_{0}^{(t)}}=$
$\displaystyle{\mathbf{\overline{p}}_{\mathrm{K}}^{(t)}}^{T}\big{[}{\mathbf{F}_{\mathrm{R_{0}}}}\odot\big{(}{\mathbf{\overline{p}}_{\mathrm{K}}^{(t)}}^{T}{\mathbf{F}_{\mathrm{R_{0}}}}\big{)}^{*}+{\mathbf{F}_{\mathrm{I_{0}}}}\odot\big{(}{\mathbf{\overline{p}}_{\mathrm{K}}^{(t)}}^{T}{\mathbf{F}_{\mathrm{I_{0}}}}\big{)}^{*}\big{]},$
(25d) $\displaystyle\color[rgb]{0,0,1}\mathbf{F}_{\mathrm{R_{0}/I_{0}}}=$
$\displaystyle\color[rgb]{0,0,1}\sqrt{\boldsymbol{\gamma}}\otimes\mathfrak{R/I}\left\\{\begin{bmatrix}\mathbf{f}^{\mathrm{R_{0}/I_{0}}}_{0},~{}\cdots,~{}\mathbf{f}^{\mathrm{R_{0}/I_{0}}}_{K_{G}-1}\end{bmatrix}\right\\},$
$\displaystyle\color[rgb]{0,0,1}\text{removing
}\left[\gamma_{0,{\epsilon}_{1}},~{}\cdots,~{}\gamma_{0,{\epsilon}_{|\kappa|}}\right]\otimes\mathbf{f}^{\mathrm{R_{0}/I_{0}}}_{0},$
(25e) $\displaystyle\color[rgb]{0,0,1}\boldsymbol{\gamma}\triangleq$
$\displaystyle\color[rgb]{0,0,1}\big{[}\gamma_{-M/2,{\epsilon}_{1}},~{}\cdots,~{}\gamma_{-M/2,{\epsilon}_{|\kappa|}},~{}\cdots,~{}\gamma_{M/2-1,{\epsilon}_{|\kappa|}}\big{]}.$
(25f)

####  V-C2 Updating $\boldsymbol{\sigma}_{\mathrm{K}}$

Each entry of $\boldsymbol{\sigma}_{\mathrm{K}}$ in (V-C) is involved in $M$
$\log\det(\cdot)$ terms with different coefficients $p_{m}$, which requires
solving an $M$-order polynomial function if following the same KKT procedure
as when handling problem (V-B2). However, the closed form solution in (22) for
problem (V-B2) comes from solving a linear function after KKT (only $1$
$\log\det(\cdot)$ term is involved as in (V-B2)). As a solution, we lower
bound the constraint in (V-C) by a function in the same form as (V-B2), and
update the lower bound to approach the real constraint through iterations.
Specifically, we have:

$\displaystyle\frac{1}{2KM}$
$\displaystyle\sum_{m}\log~{}\det\left(\mathbf{I}_{2K}+p_{m}\mathbf{C}^{\prime}\mathrm{diag}\\{\boldsymbol{\sigma}_{\mathrm{K}}\\}\right)$
$\displaystyle\geq$
$\displaystyle{{\mathbf{f}_{1}^{(t)}}^{T}}\log~{}\left(\mathbf{I}_{2K}+\mathbf{G}\boldsymbol{\sigma}_{\mathrm{K}}\right)+f_{2}^{(t)},$
(26a) $\displaystyle\mbox{with}~{}\mathbf{f}_{1}^{(t)}=$
$\displaystyle\frac{1}{2KM}\sum_{m}\frac{{p}_{m}\mathbf{C}^{\prime}\big{(}\mathbf{I}_{2K}+\mathbf{G}\mathrm{diag}\big{\\{}\boldsymbol{\sigma}_{\mathrm{K}}^{(t)}\big{\\}}\big{)}}{\big{(}\mathbf{G}+{p}_{m}\mathbf{C}^{\prime}\mathbf{G}\mathrm{diag}\big{\\{}\boldsymbol{\sigma}_{\mathrm{K}}^{(t)}\big{\\}}\big{)}},$
(26b) $\displaystyle f_{2}^{(t)}=$
$\displaystyle\frac{1}{2KM}\sum_{m}\log~{}\det\big{(}\mathbf{1}_{2K}+{p}_{m}\mathbf{C}^{\prime}\mathrm{diag}\big{\\{}\boldsymbol{\sigma}_{\mathrm{K}}^{(t)}\big{\\}}\big{)}-$
$\displaystyle{\mathbf{f}_{1}^{(t)}}^{T}\log\big{(}\mathbf{1}_{2K}+\mathbf{G}\boldsymbol{\sigma}_{\mathrm{K}}^{(t)}\big{)},$
(26c) $\displaystyle\color[rgb]{0,0,1}\mathbf{G}=$
$\displaystyle\color[rgb]{0,0,1}\max\\{\mathbf{p}\\}\mathbf{C}^{\prime},$
(26d)

whose inequality comes from $\log(1+c_{1}x)\geq c_{2}\log(1+c_{3}x)+c_{4}$
with $c_{2}=c_{1}(1+c_{3}x_{0})/(c_{3}+c_{1}c_{3}x_{0})$ and
$c_{4}=\log(1+c_{1}x_{0})-c_{2}\log(1+c_{3}x_{0})$ (by making the Talyor
approximation of $\log(1+c_{1}x)$ w.r.t the term $\log(1+c_{3}x)$, i.e.,
$\log(1+c_{1}x)=\log\left(1+c_{1}/c_{3}\left(e^{t}-1\right)\right)|_{t=\log(1+c_{3}x)}$).
Specifically, in (26), each diagonal entry in
$\log\left(\mathbf{I}_{2K}+{p}_{m}\mathbf{C}^{\prime}\mathrm{diag}\left\\{\boldsymbol{\sigma}_{\mathrm{K}}\right\\}\right)$
for $\forall$ $m$ is approximated at the point
$\log\left(\mathbf{I}_{2K}+\mathbf{G}\boldsymbol{\sigma}_{\mathrm{K}}\right)$,
combining the $M$ $\log\det(\cdot)$ terms in (V-C) into one $\log\det(\cdot)$
term. This will give (V-C) the same format as in (V-B2) in terms of the
communication rate constraint. As a result, the closed-form solution for
updating $\boldsymbol{\sigma}_{\mathrm{K}}$ in (22) can be adopted directly
with coefficient modifications.

The entire decoupling method is summarized in algorithm 3.

Input: $l\leftarrow
0,~{}\overline{\mathbf{p}}_{\mathrm{K}}^{(0)},~{}\boldsymbol{\sigma}_{\mathrm{K}}^{(0)},~{}{\epsilon}_{S}>0$;

Output: $\overline{\mathbf{p}},~{}\boldsymbol{\sigma}$;

  1. 1.

Obtain ${\mathbf{p}}$ by Algorithm 1; complexity
$\mathcal{O}\left(M^{2}\max\left\\{M,|\kappa|\right\\}\right)$

  2. 2.

Obtain
$\overline{\mathbf{p}}_{\mathrm{K}},~{}\boldsymbol{\sigma}_{\mathrm{K}}$ by
Algorithm 2, with updated coefficients in (25) and $P^{\max}=1$; complexity
$\mathcal{O}\left(K^{3}\right)$

  3. 3.

Update
$\overline{\mathbf{p}}\leftarrow\mathbf{F}_{\mathrm{M}}\overline{\mathbf{p}}_{\mathrm{K}}$,
$\boldsymbol{\sigma}\leftarrow\mathbf{F}_{\mathrm{M}}\boldsymbol{\sigma}_{\mathrm{K}}$

Algorithm 3 Decoupling Algorithm

##  VI Simulations

This section investigates the radar performance of the OFDM DFRC waveform
optimized w.r.t our proposed metrics - UBOP for PSK input distribution and
aUBOP for Gaussian input distribution. For comparison, we also consider OFDM
DFRC waveforms optimized w.r.t two widely used radar metrics, the CRB [9, 4]
and the RMI [41, 42]. Simulations are performed based on the 802.11p standard
(wireless access in vehicular environments) following a similar setup as in
[6], with a center frequency of $24$ GHz and bandwidth of $90.909$ MHz. In
most simulations, we select $K=1024$, $K_{G}=256$ and $M=512$, which gives a
range resolution of $1.67{\rm m}$ and a velocity resolution of $2.14{\rm
m/s}$. We also assume a noise power density of $-208{\rm dBW/Hz}$ at both
communications and radar receivers, with a communication path loss of $108$ dB
and an average radar path loss of $130$ dB (target at 106.7 m) for
optimization. At the target, assume an average cross section power loss of
$-10{\rm dBm^{2}}$. For the Gaussian input, the communication channel is an
NLOS model with $18$ taps from Model B in [43]. The range(velocity) of the
target is chosen uniformly within the maximum values that can be unambiguously
estimated (213m (218m/s)).

###  VI-A OFDM with PSK input distribution (constant magnitude)

Figure 4: The grey-scale map of the proposed optimal OFDM waveform for
$K=1024$, $K_{G}=256$, $M=512$ at $P^{\max}=-15$ dBW (a low SNR region), with
the red curve illuminating the total power (sum over subcarriers) at each OFDM
symbol.

This section investigates the radar performance and characteristics of an OFDM
DFRC waveform optimized w.r.t UBOP for PSK input. As it has been mentioned in
Section IV, the PSK input distribution is intended to shed insight into the
characteristics of an OFDM signal well-suited for on-grid delay-Doppler bin
estimation, and hence the communication rate constraint is not explicitly
enforced.

First, Fig. 4 sheds light on an OFDM DFRC waveform optimized to minimize the
UBOP metric for PSK input distribution at $P^{\max}=-15$ dBW (a low SNR
region). Fig. 4 demonstrates an interesting observation that, the optimal
power allocation has uniform power allocation across subcarriers, as proved in
Section IV. However, in the time domain, the power allocation shapes like a
window (similar to Hamming or Kaiser in the low SNR), as illuminated by the
red curve in Fig. 4, which plots the sum power on each OFDM symbol in the time
domain. This is reasonable from the perspective of windowing to handle the
problem of leakage frequencies giving fractional $v_{{\epsilon}}$. With a low
receive SNR, all the sidelobes arising from the frequency leakage might lead
to outlier (Remark 2 and Remark 4), in which case a window with high dynamic
delay (such as Hamming and Kaiser) is beneficial since those dissimilar
frequency components that are not aligned to the desired frequency can all be
suppressed satisfyingly [44]. In contrast, given a high receive SNR, the
outlier mainly originates from the aligned highest sidelobe around the peak
lobe, in which occasion the optimal window turns back to uniform power
allocation across OFDM symbols since the rectangular window provides higher
sensitivity. The windowing shift tendency from low SNR to high SNR is
consistent with Remark 4.

(a)

(b)

(c)

Figure 5: The empirical OP and MSE of delay-Doppler estimation between the
proposed metric (OPT), RMI, CRB and two windowed power allocation (Hamming and
Kaiser) across the OFDM symbols, with BPSK input for $K=1024$, $K_{G}=256$,
$M=512$. The worst SNR (the farthest range bin) at the radar’s receiver and
the SNR at the communication’s receiver respectively range from $-51$ dB to
$-41$ dB and from $1.4$ dB to $16.4$ dB respectively. For the subfigures, (a)
OP, (b) Range MSE (representing delay), and (c) Velocity MSE (representing
Doppler).

Given the optimal waveform intuition from Fig. 4, Fig. 5a compares the
empirical OP between OFDM waveforms optimized w.r.t UBOP (red, ’OPT’), RMI
(purple) and CRB (black) as a function of the transmit power. Two windowed
power allocation strategies are also introduced as an optimization-free
strategy inspired by the optimal ’OPT’ in Fig. 4, where uniform power
allocation is adopted across subcarriers and a window (green for the ’Hamming’
window and yellow for the ’Kaiser’ window) is added for power allocation
across symbols. Besides the OP, we also plot the MSE of the range-velocity
(delay-Doppler) estimation in Fig. 5b and Fig. 5c, to provide more insight
into the characteristics of different power allocation strategies.

Fig. 5a demonstrates that our proposed power allocation ’OPT’ achieves the
best OP, followed by the two windowed power allocation strategies, ’Kaiser’
and ’Hamming’, and then ’RMI’ and ’CRB’. The optimality of the ’OPT’ power
allocation originates from a thorough consideration of a joint delay-Doppler
bin estimation, especially the frequency leakage problem as it has been
analyzed. Using Hamming and Kaiser windows achieves sub-optimal performance in
Fig. 5a since these two windows both help suppress the sidelobes in the low
SNR. Also noteworthy, between ’RMI’ and ’CRB’, ’RMI’ shows relatively better
performance on delay estimation in Fig. 5b, because its optimal power
allocation gives uniform spectrum density (the same as the optimal power
allocation across subcarriers) 666Given a pure radar system, the RMI metric
intuitively benefits the received power., which is coincident with the
waveform preference for delay estimation. In contrast, ’CRB’ achieves
relatively better Doppler MSE (even exceeds that of the ’OPT’ power allocation
in low SNR) in Fig. 5c, because the ’CRB’ 777The CRB intuitively maximizes the
effective bandwidth, whose optimal power allocation naturally concentrates on
the edge frequencies [9]. power allocation obtains very reliable Doppler
estimation from the first and the last subcarriers by concentrating the power
upon them. However, this results in very poor delay estimation performance on
the other hand.

###  VI-B OFDM with Gaussian input distribution (variable magnitude)

Figure 6: The empirical OP given $R^{c}=2.12$ bits/s/Hz with $K=32$,
$K_{G}=8$, $M=16$. The red curve refers to the optimal input distribution
proposed in section V-B, shortened as ’OPT’. The green curve refers to the
simplified decoupling method in section V-C, shortened as ’Decouple’. The cyan
curve is the radar benchmark from the optimal PSK signal in Section IV,
shortened as ’Deter’. Similarly, the purple curve refers to the ’RMI’ metric
and the black curve refers to the ’CRB’ metric. Figure 7: The empirical OP
with Gaussian input, with $K=1024$, $K_{G}=256$, $M=512$ (Same simulation
parameters as in Fig. 5.)

(a)

(b)

(c)

Figure 8: The R-C trade-off in OFDM DFRC for $P^{\max}=0$ dBW, $K=32$,
$K_{G}=8$ and $M=16$: (a) The R-C region; (b) The power ratio of the squared
symbol means (deterministic components, red) and the symbol variance (random
components, blue) corresponding to the ’OPT’ input distribution (red) in Fig.
8a; (c) Examples of the mean power (red) and the variance power (blue) at each
subcarrier (sum over OFDM symbols) in one channel realization for the ’OPT’
input distribution at points ’A’, ’B’ and ’C’ in Fig. 8a. The sum of the mean
power (red) and the variance power (blue) is the total power at the
subcarrier.

In this section, we evaluate the OP performance given Gaussian input
distribution (simulated as 256-QAM). In this part, since there is no closed-
form expression for the average RMI and CRB with communication constraints, we
substitute these two metrics with their sub-optimal solutions888Specifically,
we first decide the minimal power that is required by the Gaussian variance to
satisfy the communication constraint. The rest power of ’RMI’ and ’CRB’ are
then allocated to their Gaussian mean for the benefit of radar, following
their criterion to maximize the ’RMI’ and minimize the ’CRB’.. In this
section, we first verify the effectiveness of the proposed aUBOP metric by
comparing its radar performance with other metrics given a fixed communication
constraint (Fig. 6 and Fig. 7). Later, we focus on the R-C trade-off from the
aspect of the R-C region, the randomness trade-off and the optimal waveform
tendency (Fig. 8).

Fig. 6 simulates the radar performance with a smaller number of OFDM
subcarriers and symbols to first compare the proposed two inputs, the ’OPT’
(red) in Algorithm 2 and the ’Decouple’ (green) in Algorithm 3, together with
the sub-optimal ’RMI’ (purple) and ’CRB’ (black) solutions. We also plot the
optimal ’PSK’ input as a benchmark (’Deter’, cyan) to show the effect of the
variable magnitudes of Gaussian CSs. Firstly, by comparing with the ’Deter’
input, Fig. 6 shows that variable magnitudes degrade the radar performance.
Then, when comparing between the four Gaussian inputs, Fig. 6 verifies that
our proposed ’OPT’ input achieves the best overall performance, followed by
the ’Decouple’ input which has a close OP performance but with much lower
complexity. To further verify the efficiency of the ’Decouple’ input, we also
plot in Fig. 7 with a normal OFDM set-up of $1024$ subcarriers and $512$
symbols, where the ’Decouple’ input still outperforms the ’RMI’ input and the
’CRB’ input significantly.

Fig. 8 highlights the R-C trade-off in OFDM DFRC from different aspects: (1)
the R-C region; (2) the R-C trade-off w.r.t. the randomness of the ’OPT’ input
distribution; (3) a general trend of the ’OPT’ input distribution, e.g., the
power allocation strategy in the spectrum domain given different communication
constraints. Specifically:

For (1), Fig. 8a depicts the average R-C region over $300$ channel
realizations. It again shows that our proposed metrics achieve a larger R-C
region than the other metrics, with the optimal being ’OPT’, closely followed
by the simplified ’Decouple’ method. The ’CRB’ is the worst since it does not
fit our scenario of on-grid parameter estimation (as in Remark 3).

For (2), given the R-C region in Fig. 8a, Fig. 8b depicts the power ratio
occupied by the symbol means (red) and symbol variance (blue) respectively,
corresponding to the ’OPT’ input distribution in Fig. 8a. It shows that, with
stricter constraints on the achievable rate, the power budget is shifting from
the symbol means to the symbol variance. This illuminates the R-C trade-off
from the perspective of statistics, i.e., the deterministic-random trade-off,
with the former being favoured by radar and the latter being favoured by
communications.

For (3), Fig. 8c sheds light on a general trend in the power allocation
strategy in the spectrum domain, taking examples at points ’A’ (no
communication constraint), ’B’ (a relatively strict communication constraint)
and ’C’ (the strictest achievable communication constraint in the channel
realization), from the top to the bottom. From ’A’ to ’C’, giving increasing
achievable rate constraints, the total power is transferred from symbol means
(red) to symbol variance (blue), which is coincident with the deterministic-
random trade-off in (2). Specifically, ’A’ emphasizes that the best radar
performance is achieved by uniform power allocation across the subcarriers to
the symbol mean, and ’C’ is the input distribution that achieves the highest
communication rate for the selected channel realization (hence whose power is
all allocated to the symbol variance in a water-filling way). ’C’ shows that
subcarriers $17$, $19$, etc, have strong channel gains for communications and
hence are allocated more power on their variances, whereas subcarriers $7$,
$28$ have weak channel gains for communications on the contrary.
Interestingly, if comparing ’B’ with ’C’ then, we observe a tendency that,
subcarriers with strong communication channel gains (e.g., $17$, $19$, etc)
tend to have large variances to satisfy the achievable rate constraint in a
power-efficient way, with the rest power budget allocated to the symbol means
prior to the remaining subcarriers (e.g., $7$, $28$). This roughly guarantees
a relatively uniform power allocation across OFDM subcarriers, and is similar
to the power allocation strategy for the constant magnitude input distribution
(PSK).

##  VII Conclusions

This paper investigates the optimal input distributions of random CSs in OFDM
DFRC. We introduce the OP as the radar metric in OFDM DFRC and show its
appropriateness through analysis and simulations. The OP is then upper-bounded
for optimization tractability and is a function of the magnitude of CSs, which
inspires us to consider two classes of CS input distributions, (1) constant
magnitude input distribution (i.e., PSK), and (2) variable magnitude input
distribution (i.e., non-zero mean Gaussian). The OP is optimized under both
scenarios, and low-complexity algorithms with closed-form solutions are
proposed. Interestingly, through analysis and simulations, we show that for
both scenarios, the optimized power allocation features a uniform power
allocation across OFDM subcarriers and a windowed power allocation across OFDM
symbols. For (2), the optimized input distribution exhibits a randomness
trade-off, i.e., a higher achievable rate constraint for communications forces
a transfer of the available power budget from the symbol means (to favour
radar) to the symbol variance (to favour communications). More specifically,
for subcarriers having high channel gain for communications, power is
concentrated on the symbol variance in order to satisfy the communication
constraints efficiently. On the other hand, for subcarriers with weak channel
gains for communications, power is concentrated on the symbol means to favour
radar performance.

##  Appendix A Derivation of the OP

The probability of
${\mathbb{P}}\left[|\gamma_{n,v,{\epsilon}}|>|\gamma_{0,0,{\epsilon}}|\right]$
in (11b) is derived based on [45]. Given a pair of complex Gaussian random
variables $[\gamma_{0,0,{\epsilon}},\gamma_{n,v,{\epsilon}}]$ with the mean
$[r_{0,0,{\epsilon}},r_{n,v,{\epsilon}}]$, we have:

$\displaystyle{\mathbb{P}}\left[A_{n,v,{\epsilon}}\right]={\mathbb{P}}\left[|\gamma_{n,v,{\epsilon}}|>|\gamma_{0,0,{\epsilon}}|\right]={\mathbb{P}}\left[|\gamma_{0,0,{\epsilon}}|^{2}-|\gamma_{n,v,{\epsilon}}|^{2}<0\right]$
(27a) $\displaystyle\approx$ $\displaystyle
Q\left\\{\nu_{1}(n,~{}v),\nu_{2}(n,~{}v)\right\\}$ $\displaystyle-$
$\displaystyle\frac{1}{2}I_{0}\left(\nu_{1}(n,~{}v)\nu_{2}(n,~{}v)\right)\exp\left\\{-\left(\nu_{1}(n,~{}v)^{2}+\nu_{2}(n,~{}v)^{2}\right)/2\right\\}$
(27b) $\displaystyle=$ $\displaystyle
Q\left\\{\nu_{1}(n,~{}v),\nu_{2}(n,~{}v)\right\\}-\frac{1}{2}\exp\left\\{-\frac{|r_{0,0,{\epsilon}}|}{2\sigma^{2}}\right\\}I_{0}\left(\frac{|r_{n,v,{\epsilon}}|}{2\sigma^{2}}\right)$
(27c) $\displaystyle\approx$
$\displaystyle\frac{1}{2}\exp\big{\\{}-\frac{|r_{0,0,{\epsilon}}|}{2\sigma^{2}}\big{\\}}I_{0}\big{(}\frac{|r_{n,v,{\epsilon}}|}{2\sigma^{2}}\big{)},$
(27d) $\displaystyle\mbox{with}~{}\nu_{1}(n,~{}v)=$
$\displaystyle\sqrt{(r_{0,0,{\epsilon}}-R_{n,v,{\epsilon}}^{1/2})/(2\sigma^{2})},$
(27e) $\displaystyle\nu_{2}(n,~{}v)=$
$\displaystyle\sqrt{(r_{0,0,{\epsilon}}+R_{n,v,{\epsilon}}^{1/2})/(2\sigma^{2})},$
(27f) $\displaystyle R_{n,v,{\epsilon}}=$
$\displaystyle|r_{0,0,{\epsilon}}|^{2}-|r_{n,v,{\epsilon}}|^{2},$ (27g)

where (27b) comes from [45], and (27d) is a further simplification adopting
Marcum-Q lower bound from [46].

The Bessel function involved in (27d) will introduce high complexity when
being used for optimization in the following. Hence, we upper bound (27d) as:

$\displaystyle{\mathbb{P}}\left[A_{n,v,{\epsilon}}\right]\approx\frac{1}{2}\exp\left\\{-\frac{r_{0,0,{\epsilon}}}{2\sigma^{2}}\right\\}I_{0}\left(\frac{|r_{n,v,{\epsilon}}|}{2\sigma^{2}}\right)$
(28a) $\displaystyle=$
$\displaystyle\frac{1}{2}\exp\left\\{-\frac{r_{0,0,{\epsilon}}}{2\sigma^{2}}\right\\}\sum_{p=0}^{\infty}\frac{1}{p!p!}\left(\frac{|r_{n,v,{\epsilon}}|}{4\sigma^{2}}\right)^{2p}$
(28b) $\displaystyle\leq$
$\displaystyle\frac{1}{2}\exp\big{\\{}-\frac{r_{0,0,{\epsilon}}}{2\sigma^{2}}\big{\\}}\big{[}\sum_{p=0}^{\infty}\frac{1}{p!}\left(\frac{|r_{n,v,{\epsilon}}|}{4\sigma^{2}}\right)^{p}\big{]}^{2}=\frac{1}{2}\exp\big{\\{}-\frac{r_{0,0,{\epsilon}}-|r_{n,v,{\epsilon}}|}{2\sigma^{2}}\big{\\}},$
(28c)

where the inequality in (28c) comes from
$\sum_{k}a_{k}^{2}\leq\left(\sum_{k}a_{k}\right)^{2}$ for $a_{k}\geq 0$
$\forall\>k$, and (28c) comes from the Taylor expansion of exponential
functions.

##  Appendix B Proof of the Closed-form Solution for (16b)

For
$\mathbf{r}^{(l+1,~{}t+1)}=\arg\min\left\\{{\boldsymbol{\alpha}^{(l+1,~{}t)}}^{T}\mathbf{r}^{|\cdot|}+\frac{\rho}{2}\|\mathbf{r}-\mathbf{F}^{\mathrm{D}}\mathbf{p}^{(l)}\|^{2}\right\\}$,
we have:

$\displaystyle{\boldsymbol{\alpha}^{(l+1,~{}t)}}^{T}\mathbf{r}^{|\cdot|}+\frac{\rho}{2}\|\mathbf{r}-\boldsymbol{\beta}^{(l)}\|^{2}=\sum_{q}\alpha^{(l+1,~{}t)}_{q}|r_{q}|+\frac{\rho}{2}|r_{q}-\beta^{(l)}_{q}|^{2}$
(29a) $\displaystyle=$
$\displaystyle\sum_{q}\alpha^{(l+1,~{}t)}_{v}\sqrt{\mathfrak{R}\left\\{r_{q}\right\\}^{2}+\mathfrak{I}\left\\{r_{q}\right\\}^{2}}+\frac{\rho}{2}\left(\mathfrak{R}\left\\{r_{q}\right\\}^{2}+\mathfrak{I}\left\\{r_{q}\right\\}^{2}\right)$
$\displaystyle-\rho\left(\mathfrak{R}\left\\{r_{q}\right\\}\mathfrak{R}\left\\{\beta^{(l)}_{q}\right\\}+\mathfrak{I}\left\\{r_{q}\right\\}\mathfrak{I}\left\\{\beta^{(l)}_{q}\right\\}\right)+\frac{\rho}{2}|\beta^{(l)}_{q}|^{2}$
(29b) $\displaystyle\geq$
$\displaystyle\sum_{q}\frac{\rho}{2}|r_{q}|^{2}+\left(\alpha^{(l)}_{q}-\rho|\beta^{(l)}_{q}|\right)|r_{q}|+\frac{\rho}{2}|\beta^{(l)}_{q}|^{2},$
(29c)

where $\boldsymbol{\beta}^{(l)}=\mathbf{F}^{\mathrm{D}}\mathbf{p}^{(l)}$.
(29c) applies the Cauchy-Schwarz inequality and reaches the minimal if and
only if
$\mathfrak{R}\left\\{r_{q}\right\\}/\mathfrak{I}\left\\{r_{q}\right\\}=\mathfrak{R}\left\\{\beta^{(l)}_{q}\right\\}/\mathfrak{I}\left\\{\beta_{q}\right\\}$.
Hence, the minimal value of (29c) is achieved by
$\left[\mathbf{F}^{\mathrm{D}}\mathbf{p}^{(l)}-{\angle\left(\mathbf{F}^{\mathrm{D}}\mathbf{p}^{(l)}\right)\odot\boldsymbol{\alpha}^{(l)}}/{\rho}\right]^{+}$.

##  Appendix C Proof of (20e)

For (20d), we have for the first term (peak lobe related):

$\displaystyle\mathbb{E}_{X[k,~{}m]}\left\\{r_{0,0,{\epsilon}}\right\\}$
$\displaystyle=\sum_{k,~{}m}\mathbb{E}_{X[k,~{}m]}\left\\{p_{k,~{}m}\exp\left\\{{j2\pi
mv_{{\epsilon}}}/{M}\right\\}\right\\}$ (30)
$\displaystyle=\sum_{k,~{}m}\overline{p}_{k,~{}m}\exp\left\\{{j2\pi
mv_{{\epsilon}}}/{M}\right\\}.$ (31)

Then, we take expectation over $v_{{\epsilon}}$ by sampling, similarly to the
solution in Section IV:

$\displaystyle\mathbb{E}_{{{\epsilon}}}\left\\{\left|\mathbb{E}_{X}\left\\{r_{0,0,{\epsilon}}\right\\}\right|\right\\}=\frac{1}{|\kappa|}\sum_{v_{{\epsilon}}}|\sum_{k,~{}m}\overline{p}_{k,~{}m}\exp\left\\{{j2\pi
mv_{{\epsilon}}}/{M}\right\\}|$
$\displaystyle=\frac{1}{|\kappa|}\sum_{v_{{\epsilon}}}\sqrt{\overline{\mathbf{p}}^{T}\mathbf{A}_{{\epsilon}}\overline{\mathbf{p}}},$
(32)

with $\mathbf{A}_{{\epsilon}}$ in (20i).

For $(n,~{}v)\neq(0,~{}0)$ in (20d), we first take average over $X[k,~{}m]$
(denote $f^{\mathrm{D}}_{v,{\epsilon},m}f^{\mathrm{R}}_{n,k}$ by
$F^{(n,v,{\epsilon})}_{k,m}$):

$\displaystyle\color[rgb]{0,0,1}\mathbb{E}_{X}\left\\{|r_{n,v,{\epsilon}}|^{2}\right\\}=\mathbb{E}_{X}\Big{\\{}|\sum_{k,~{}m}p_{k,~{}m}F^{(n,v,{\epsilon})}_{k,m}|^{2}\Big{\\}}$
$\displaystyle=$
$\displaystyle\color[rgb]{0,0,1}\sum_{k_{1},~{}m_{1},~{}k_{2},~{}m_{2}}\mathbb{E}_{X}\left\\{p_{k_{1},m_{1}}F^{(n,v,{\epsilon})}_{k_{1},m_{1}}p_{k_{2},m_{2}}{F^{(n,v,{\epsilon})}_{k_{2},~{}m_{2}}}^{*}\right\\}$
$\displaystyle\color[rgb]{0,0,1}+\sum_{k,~{}m}\mathbb{E}_{X}\left\\{|p_{k,m}|^{2}\right\\}$
(33a) $\displaystyle=$
$\displaystyle\color[rgb]{0,0,1}\sum_{k_{1},~{}m_{1},~{}k_{2},~{}m_{2}}\mathbb{E}_{X}\left\\{p_{k_{1},m_{1}}\right\\}\mathbb{E}_{X}\left\\{p_{k_{2},m_{2}}\right\\}F^{(n,v,{\epsilon})}_{k_{1},m_{1}}{F^{(n,v,{\epsilon})}_{k_{2},m_{2}}}^{*}$
$\displaystyle\color[rgb]{0,0,1}-\sum_{k,~{}m}\left[\mathbb{E}_{X}\left\\{p_{k,m}\right\\}\mathbb{E}_{X}\left\\{p_{k,m}\right\\}-\mathbb{E}_{X}\left\\{|p_{k,m}|^{2}\right\\}\right]$
(33b) $\displaystyle=$
$\displaystyle\color[rgb]{0,0,1}\sum_{k_{1},~{}m_{1},k_{2},~{}m_{2}}\overline{p}_{k_{1},m_{1}}\overline{p}_{k_{2},m_{2}}F^{(n,v,{{\epsilon}})}_{k_{1},m_{1}}{F^{(n,v,{\epsilon})}_{k_{2},m_{2}}}^{*}-\sum_{k,~{}m}\overline{p}^{2}_{k,m}+\sum_{k,~{}m}Q_{k,m}$
(33c) $\displaystyle=$
$\displaystyle\color[rgb]{0,0,1}\left(\mathfrak{R}\left\\{{\mathbf{f}^{(n,v,{\epsilon})}}^{T}\right\\}\overline{\mathbf{p}}\right)^{2}+\left(\mathfrak{I}\left\\{{\mathbf{f}^{(n,v,{\epsilon})}}^{T}\right\\}\overline{\mathbf{p}}\right)^{2}$
$\displaystyle\color[rgb]{0,0,1}+2\sum_{k,~{}m}\left[{\overline{p}_{\mathrm{R},k,m}}^{2}+{\overline{p}_{\mathrm{I},k,m}}^{2}-{\mu_{\mathrm{R},k,m}}^{4}-{\mu_{\mathrm{I},k,m}}^{4}\right]$
(33d) $\displaystyle=$
$\displaystyle\mathbf{\overline{p}}^{T}\mathbf{A}_{n,v,{\epsilon}}\mathbf{\overline{p}}-2\|\overline{\mathbf{p}}-\boldsymbol{\sigma}\|^{2},$
(33e)

with $\mathbf{A}_{n,v,{\epsilon}}$ in (20j), and where (33c) is achieved
since:

$\displaystyle Q_{k,~{}m}$
$\displaystyle=\mathbb{E}_{X[k,~{}m]}\left\\{|p_{k,m}|^{2}\right\\}$
$\displaystyle=3\overline{p}^{2}_{k,m}-2({\mu_{\mathrm{R},k,m}}^{4}+{\mu_{\mathrm{I},k,m}}^{4})-4{\overline{p}_{\mathrm{R},k,m}}{\overline{p}_{\mathrm{I},k,m}}$
$\displaystyle=\overline{p}^{2}_{k,m}+2\left[{\overline{p}_{\mathrm{R},k,m}}^{2}+{\overline{p}_{\mathrm{I},k,m}}^{2}-{\mu_{\mathrm{R},k,m}}^{4}-{\mu_{\mathrm{I},k,m}}^{4}\right].$
(34a)

##  Appendix D Proof of (24b) in the Decoupling Method

For the first term in (24a):

$\displaystyle\sum_{{\epsilon}\in\kappa}\sqrt{{\overline{\mathbf{p}}_{\mathrm{K}}}^{T}{\mathbf{F}_{\mathrm{M}}}^{T}\mathbf{A}_{{\epsilon}}\mathbf{F}_{\mathrm{M}}\overline{\mathbf{p}}_{\mathrm{K}}}$
$\displaystyle=$
$\displaystyle\sum_{{\epsilon}\in\kappa}\footnotesize{\sqrt{{\overline{\mathbf{p}}_{\mathrm{K}}}^{T}{\mathbf{F}_{\mathrm{M}}}^{T}\left[\mathfrak{R}\left\\{{\mathbf{f}_{0,0,{\epsilon}}}\right\\}\mathfrak{R}\left\\{{\mathbf{f}_{0,0,{\epsilon}}}\right\\}^{T}+\mathfrak{I}\left\\{{\mathbf{f}_{0,0,{\epsilon}}}\right\\}\mathfrak{I}\left\\{{\mathbf{f}_{0,0,{\epsilon}}}\right\\}^{T}\right]\mathbf{F}_{\mathrm{M}}\overline{\mathbf{p}}_{\mathrm{K}}}}$
(35a) $\displaystyle=$
$\displaystyle\sum_{{\epsilon}\in\kappa}{\sqrt{{\left(\mathbf{p}^{T}\mathfrak{R}\left\\{\mathbf{f}^{\mathrm{D}}_{0,~{}{\epsilon}}\right\\}\right)^{2}}+{\left(\mathbf{p}^{T}\mathfrak{I}\left\\{\mathbf{f}^{\mathrm{D}}_{0,~{}{\epsilon}}\right\\}\right)^{2}}}}\mathbf{1}^{T}\overline{\mathbf{p}}_{\mathrm{K}}=\color[rgb]{0,0,1}A_{0}^{\mathrm{D}}\mathbf{1}^{T}\overline{\mathbf{p}}_{\mathrm{K}},$
(35b) where (35b) comes from:
$\displaystyle{\mathbf{F}_{\mathrm{M}}}^{T}\mathfrak{R}\left\\{{\mathbf{f}_{0,0,{\epsilon}}}\right\\}\mathfrak{R}\left\\{{\mathbf{f}_{0,0,{\epsilon}}}\right\\}^{T}\mathbf{F}_{\mathrm{M}}$
$\displaystyle=$
$\displaystyle\begin{bmatrix}\left(\mathbf{p}^{T}\otimes\mathbf{I}_{K}\right){\left(\mathfrak{R}\left\\{\mathbf{f}^{\mathrm{D}}_{0,~{}{\epsilon}}\right\\}\otimes{\mathbf{f}^{\mathrm{R}}_{0}}\right)}\\\
\left(\mathbf{p}^{T}\otimes\mathbf{I}_{K}\right){\left(\mathfrak{R}\left\\{\mathbf{f}^{\mathrm{D}}_{0,~{}{\epsilon}}\right\\}\otimes{\mathbf{f}^{\mathrm{R}}_{0}}\right)}\end{bmatrix}\begin{bmatrix}\left(\mathbf{p}^{T}\otimes\mathbf{I}_{K}\right){\left(\mathfrak{R}\left\\{\mathbf{f}^{\mathrm{D}}_{0,~{}{\epsilon}}\right\\}\otimes{\mathbf{f}^{\mathrm{R}}_{0}}\right)}\\\
\left(\mathbf{p}^{T}\otimes\mathbf{I}_{K}\right){\left(\mathfrak{R}\left\\{\mathbf{f}^{\mathrm{D}}_{0,~{}{\epsilon}}\right\\}\otimes{\mathbf{f}^{\mathrm{R}}_{0}}\right)}\end{bmatrix}^{T}$
(35c) $\displaystyle=$
$\displaystyle{\left(\mathbf{p}^{T}\mathfrak{R}\left\\{\mathbf{f}^{\mathrm{D}}_{0,~{}{\epsilon}}\right\\}\right)^{2}}\mathbf{1}_{2K\times
2K}.$ (35d)

For the second term in (24a):

$\displaystyle{\mathbf{A}}^{0}_{n,v,{\epsilon}}={\mathbf{F}_{\mathrm{M}}}^{T}\mathbf{A}_{n,v,{\epsilon}}\mathbf{F}_{\mathrm{M}}$
$\displaystyle=$
$\displaystyle\mathbf{F}_{\mathrm{M}}^{T}\left[{\mathfrak{R}\left\\{\mathbf{f}_{n,v,{\epsilon}}\right\\}}{\mathfrak{R}\left\\{\mathbf{f}_{n,v,{\epsilon}}\right\\}}^{T}+{\mathfrak{I}\left\\{\mathbf{f}_{n,v,{\epsilon}}\right\\}}{\mathfrak{I}\left\\{\mathbf{f}_{n,v,{\epsilon}}\right\\}}^{T}+2\mathbf{I}_{2KM}\right]\mathbf{F}_{\mathrm{M}}$
(36a) $\displaystyle{=}$
$\displaystyle|\gamma_{v,{\epsilon}}|^{2}\left[{\mathfrak{R}\left\\{\mathbf{f}_{n}^{\mathrm{R_{0}}}\right\\}}{\mathfrak{R}\left\\{\mathbf{f}_{n}^{\mathrm{R_{0}}}\right\\}}^{T}+{\mathfrak{I}\left\\{\mathbf{f}_{n}^{\mathrm{R_{0}}}\right\\}}{\mathfrak{I}\left\\{\mathbf{f}_{n}^{\mathrm{R_{0}}}\right\\}}^{T}\right]+2\|\mathbf{p}\|^{2},$
(36b)

where
$\gamma_{v,{\epsilon}}=|\mathbf{p}^{T}\mathbf{f}^{\mathrm{D}}_{v,~{}{\epsilon}}|^{2}$
and
$\mathbf{f}_{n}^{\mathrm{R_{0}}}=\mathbf{1}_{2}\otimes\mathbf{f}_{n}^{\mathrm{R}}\in\mathbb{C}^{2K}$.

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