# From Torch to Projector: Fundamental Tradeoff of Integrated Sensing and
Communications

Yifeng Xiong, Fan Liu, Kai Wan, Weijie Yuan, Yuanhao Cui, and Giuseppe Caire

###### Abstract

snc have been historically developed in parallel. In recent decade, they have
been evolving from separation to integration, giving rise to the integrated
sensing and communications (ISAC) paradigm, that has been recognized as one of
the six key 6G usage scenarios. Despite the plethora of research works
dedicated to ISAC signal processing, the fundamental performance limits of
sensing and communications (S&C) remain widely unexplored in an ISAC system.
In this tutorial paper, we attempt to summarize the recent research findings
in characterizing the performance boundary of ISAC systems and the resulting
S&C tradeoff from an information-theoretical viewpoint. We begin with a
folklore “torch metaphor” that depicts the resource competition mechanism of
S&C. Then, we elaborate on the fundamental capacity-distortion (C-D) theory,
indicating the incompleteness of this metaphor. Towards that end, we further
elaborate on the S&C tradeoff by discussing a special case within the C-D
framework, namely the Cramér-Rao bound (CRB)-rate region. In particular, S&C
have preference discrepancies over both the subspace occupied by the
transmitted signal and the adopted codebook, leading to a “projector metaphor”
complementary to the ISAC torch analogy. We also present two practical design
examples by leveraging the lessons learned from fundamental theories. Finally,
we conclude the paper by identifying a number of open challenges.

###### Index Terms:

Integrated sensing and communications, capacity-distortion theory, fundamental
limits, CRB-rate region.

##  I Introduction

ISAC has recently been recognized by the international telecommunication union
(ITU) as one of the vertices supporting the 6G hexagon of usage scenarios [1].
By relying on unified hardware platforms and radio waveforms, such integration
enables resource-efficient cooperation between S&C subsystems, supporting
various emerging applications including vehicular-to-everything (V2X)
networking, extended reality (XR), and digital twins.

Early endeavours on ISAC system design aim to extend the capability of
existing infrastructures, hence developed into _sensing-centric_ and
_communication-centric_ paradigms. Representative sensing-centric schemes
include communicating using radar sidelobes and the permutation degree of
freedom (DoF) in the waveform-antenna mapping of multiple-input multiple-
output (MIMO) radars [2]. Communication-centric schemes are exemplified by
sensing relying on orthogonal frequency-division multiplexing (OFDM) waveforms
[3]. These schemes do not target the optimal S&C performance. To push the
benefit of ISAC (termed as the “integration gain” [4]) to its limit, _joint
design_ schemes emerged more recently [5], which aim at conceiving novel joint
signaling strategies from the ground-up, capable of accomplishing both tasks
simultaneously. A natural, but easily overlooked question that arises is: What
is the fundamental limit of the integration gain?

If we focus back on conventional individual S&C systems, we see that their
fundamental performance limits originate from the resource budget. For
example, the celebrated Shannon capacity formula for the scalar Gaussian
channel exactly expresses the dependency of the communication performance on
the available power and bandwidth. However, the performance limit of ISAC
systems is vastly different. In its essence, the problem of ISAC system design
is a multi-objective optimization problem. To elaborate, separated S&C design
can be viewed as a special case of ISAC design, namely employing a time-
sharing strategy. By utilizing the synergies between S&C, more favorable
performance can be achieved. However, besides some rare occasions, it is
highly unlikely that S&C would achieve their optimal performance at the same
time, suggesting the existence of a fundamental S&C tradeoff in an ISAC
system. Such a tradeoff may be framed by the Pareto boundary of the multi-
objective ISAC system design problem.

For decades, S&C have been regarded as an information-theoretical “odd couple”
that are mutually intertwined in profound ways [6]. At large, the S&C tradeoff
may be understood from the perspective of a signal-and-system duality. For
instance, let us consider a simple linear Gaussian model given by

$\bm{\mathsfbr{Y}}=\bm{\mathsfbr{H}}\bm{\mathsfbr{X}}+\bm{\mathsfbr{Z}}$ (1)

with $\bm{\mathsfbr{X}}$, $\bm{\mathsfbr{Y}}$, $\bm{\mathsfbr{H}}$ and
$\bm{\mathsfbr{Z}}$ being the transmitted ISAC signal, the received signal,
the channel, and the noise, respectively. For communication, the task is to
decode the message encoded in $\bm{\mathsfbr{X}}$, whereas for sensing, the
task is to extract information from the channel $\bm{\mathsfbr{H}}$. In light
of this, one may view $\bm{\mathsfbr{H}}$ as the “transmitted signal” from the
environment to be sensed, while viewing $\bm{\mathsfbr{X}}$ as the “channel’
that this “signal” would pass through.

While the connection between estimation and information theories has been
well-studied in the context of, e.g., I-MMSE equation [7], the fundamental
tradeoff still remains widely open in general. To this end, this article will
summarize current understandings of different components in the S&C tradeoff
in a coherent manner, through the prism of information theory [8, 9, 10, 11,
12]. We shall commence with a folklore “torch metaphor” depicting the resource
competition between S&C subsystems, followed by the general capacity-
distortion theory suggesting the incompleteness of this metaphor. We then
introduce the CRB-rate region which clearly indicates that the S&C tradeoff is
two-fold: Apart from resources, S&C subsystems would also have different
preferences on the input distribution. For each component in this tradeoff, we
provide design examples illustrating its practical implications. Finally, we
conclude this paper with some open challenges.

##  II The Torch Metaphor

The fundamental S&C tradeoff in ISAC has been exemplified (implicitly or
explicitly) by the “torch metaphor” in some works [13, 5], as illustrated in
Fig. 1. In this picture, the role played by the ISAC base station (BS)
resembles a child holding a torch: If she points the torch towards the
communication user, the user would receive the message, while the sensing
target is left in the dark and hence can hardly be seen. On the other hand, if
the target is maximally illuminated, the user would receive very weak signals,
resulting in highly noise-corrupted messages. This metaphor is intuitive, and
it provides us with the following basic understandings of the S&C tradeoff:

  * •

Power allocation across orthogonal or quasi-orthogonal dimensions (in the case
of Fig. 1 it is space, or angle, but it could be time, or frequency) offers an
immediate way to tradeoff communication and sensing performance;

  * •

Since we are using a unified signal, when the communication user and the
sensing target are “close to each other”, the S&C tradeoff becomes less
prominent.

Inspired by these intuitions, most existing research contributions on ISAC
system design focus on power allocation in a generalized sense, including
beamforming in MIMO systems [5] and subcarrier power allocation in OFDM
systems [14]. Despite the effectiveness of these techniques, we would still
wonder, whether the torch metaphor depicts the full picture of the S&C
tradeoff.

Recently, it has been found that the torch metaphor does cover the full
picture, when the sensing task is to detect the presence of a potential
target, under some specific choice of sensing performance metrics [12].
Specifically, in this scenario, during the $n$-th channel use, the ISAC BS
would determine the absence/presence of the target (represented by a state
$\seta\in\\{0,1\\}$) based on an echo $\bm{\mathsfbr{Y}}_{{\rm s},n}$, while
the user aims for decoding the information conveyed in the transmitted ISAC
signal $\bm{\mathsfbr{X}}_{n}$, based on its received signal
$\bm{\mathsfbr{Y}}_{{\rm c},n}$. Both the echo and the user-received signal
are contaminated by circularly symmetric Gaussian noises, denoted by
$\bm{\mathsfbr{Z}}_{{\rm s},n}$ and $\bm{\mathsfbr{Z}}_{{\rm c},n}$,
respectively. Such a scenario may be characterized as

$\displaystyle\bm{\mathsfbr{Y}}_{{\rm c},n}$ $\displaystyle=\bm{H}_{\rm
c}\bm{\mathsfbr{X}}_{n}+\bm{\mathsfbr{Z}}_{{\rm c},n},$ (2a)
$\displaystyle\bm{\mathsfbr{Y}}_{{\rm s},n}$ $\displaystyle=\seta\bm{H}_{\rm
s}\bm{\mathsfbr{X}}_{n}+\bm{\mathsfbr{Z}}_{{\rm s},n},$ (2b)

where $n=1,2,\dotsc,N$, $N$ is the coding block length, $\bm{H}_{\rm c}$
denotes the communication channel, while $\bm{H}_{\rm s}$ represents the
target response matrix (also referred to as the sensing channel [10, 11, 12]).
For communication, a natural performance metric is the communication rate
given by

$R=\lim_{N\rightarrow\infty}\frac{1}{N}\log M_{N},$

under the assumption that the decoding error probability vanishes as $N$ tends
to infinity, and $M_{N}$ denotes the size of the communication codebook. For
sensing, the performance metric chosen in [12, 15] is the error exponent111The
error exponent is related to a class of more frequently used sensing
performance metrics, i.e. statistical divergences, via the large deviation
theory [16]. defined as

$E=\lim_{N\rightarrow\infty}\frac{1}{N}\log\frac{1}{\delta_{N}},$ (3)

where $\delta_{N}$ denotes the maximum detection error probability over all
codewords and target states, namely

$\delta_{N}=\max_{w\in[M_{N}]}\max_{s\in\\{0,1\\}}\mathbb{P}\left\\{\hat{S}\left(\bm{\mathsfbr{Y}}_{\rm
s}^{N},\bm{\mathsfbr{X}}^{N}(\mathsfbr{W})\right)\neq\seta|\seta=s,\mathsfbr{W}=w\right\\},$

with $\mathsfbr{W}\in[M_{N}]$ being the encoded message,
$\bm{\mathsfbr{Y}}_{\rm s}^{N}\\!=\\!\\{\bm{\mathsfbr{Y}}_{{\rm
s},1},\bm{\mathsfbr{Y}}_{{\rm s},2},\dotsc,\bm{\mathsfbr{Y}}_{{\rm s},N}\\}$,
and
$\bm{\mathsfbr{X}}^{N}(W)\\!=\\!\\{\bm{\mathsfbr{X}}_{1},\bm{\mathsfbr{X}}_{2},\dotsc,\bm{\mathsfbr{X}}_{N}\\}$.
Under the aforementioned setting, the set constituted by all achievable
$(R,E)$ pairs (termed as the rate-exponent region) is shown to be
characterized by [12]

$\displaystyle R$ $\displaystyle\leqslant\log\left|\bm{I}+\sigma_{\rm
c}^{-2}\bm{H}_{\rm c}\widetilde{\bm{R}}_{\bm{\mathsfbr{X}}}\bm{H}_{\rm c}^{\rm
H}\right|,$ (4a) $\displaystyle E$ $\displaystyle\leqslant\frac{1}{4}{\rm
Tr}\left\\{\sigma_{\rm s}^{-2}\bm{H}_{\rm
s}\widetilde{\bm{R}}_{\bm{\mathsfbr{X}}}\bm{H}_{\rm s}^{\rm H}\right\\},$ (4b)

where
$\widetilde{\bm{R}}_{\bm{\mathsfbr{X}}}=\mathbb{E}\left\\{\frac{1}{N}\sum_{n=1}^{N}\bm{\mathsfbr{X}}_{n}\bm{\mathsfbr{X}}_{n}^{\rm
H}\right\\}$ is the statistical covariance matrix of the transmitted ISAC
signal, satisfying a power budget constraint ${\rm
Tr}\\{\widetilde{\bm{R}}_{\bm{\mathsfbr{X}}}\\}\leqslant P$.

Figure 1: Graphical illustration of the torch metaphor.

From (4) we observe a concrete version of the torch metaphor. Apparently, the
communication-optimal $\widetilde{\bm{R}}_{\bm{\mathsfbr{X}}}$ would have its
eigenvectors matching those of $\bm{H}_{\rm c}$, while the eigenvalues
determined by the water-filling power allocation [16]. By contrast, the
sensing-optimal $\widetilde{\bm{R}}_{\bm{\mathsfbr{X}}}$ would be aligned with
the dominant eigenvector (possibly having multiplicity larger than $1$) of
$\bm{H}_{\rm s}$ [12]. Therefore, as long as the eigenspaces of $\bm{H}_{\rm
s}$ and $\bm{H}_{\rm c}$ do not match, or the water-filling strategy does not
concentrate the power on the dominant eigenvector, there would be a S&C
tradeoff. Furthermore, (4) also gives us an explicit sense in which the
communication user and the sensing target are “close to each other”: their
distance may be characterized by a measure of discrepancy between the
“communication subspace” ${\rm span}(\bm{H}_{\rm c})$ and the “sensing
subspace” – the dominant eigenvector of $\bm{H}_{\rm s}$. In light of this,
such a tradeoff is termed as the subspace tradeoff (ST) in [10], and thus the
torch metaphor may be represented concisely as the statement of

$\mathrm{S\&C}\leavevmode\nobreak\ {\rm tradeoff}=\mathrm{ST}.$ (5)

To further reveal the nature of the ST, let us consider a tangible example, in
which the BS is equipped with collocated transmit and receive uniform linear
antenna arrays (half-wavelength spacing) of the size $N_{\rm s}=M=10$, the
user has a single antenna (i.e. $N_{\rm c}=1$), while the sensing target is
point-like. In this scenario, the communication and sensing channels are given
by

$\bm{H}_{\rm c}=\alpha_{\rm c}\bm{a}^{\rm H}(\theta_{\rm
c}),\leavevmode\nobreak\ \leavevmode\nobreak\ \bm{H}_{\rm s}=\alpha_{\rm
s}\bm{a}(\theta_{\rm s})\bm{a}^{\rm H}(\theta_{\rm s}),$

where $\alpha_{\rm c}$ and $\alpha_{\rm s}$ denote the scalar channel
coefficients, $\theta_{\rm c}$ and $\theta_{\rm s}$ denote the bearing angle
of the user and the target relative to the BS, respectively, while
$\bm{a}(\theta)$ is the array steering vector given by

$\bm{a}(\theta)=[1,\leavevmode\nobreak\
e^{j\pi\sin(\theta)},\leavevmode\nobreak\
e^{j2\pi\sin(\theta)},\dotsc,e^{j\pi(N_{\rm s}-1)\sin(\theta)}]^{\rm T}.$

Correspondingly, the characterization of the rate-exponent region (4) can then
be simplified as

$\displaystyle\bm{r}(\lambda)$ $\displaystyle\\!=\\!\mathop{\rm
argmax}_{\bm{v}:\|\bm{v}\|=1}\leavevmode\nobreak\
(1\\!-\\!\lambda)|\bm{v}^{\rm H}\bm{a}(\theta_{\rm
c})|^{2}\\!+\\!\lambda|\bm{v}^{\rm H}\bm{a}(\theta_{\rm
s})|^{2},\leavevmode\nobreak\ \lambda\\!\in\\![0,1]$ (6a) $\displaystyle
R(\lambda)$ $\displaystyle\\!\leqslant\\!\log(1+P\sigma_{\rm
c}^{-2}|\alpha_{\rm c}|^{2}|\bm{r}^{\rm H}(\lambda)\bm{a}(\theta_{\rm
c})|^{2}),$ (6b) $\displaystyle E(\lambda)$
$\displaystyle\\!\leqslant\\!\frac{1}{4}{\rm Tr}\left\\{P\sigma_{\rm
s}^{-2}|\alpha_{\rm s}|^{2}|\bm{r}^{\rm H}(\lambda)\bm{a}(\theta_{\rm
s})|^{2}\right\\},$ (6c)

where $\lambda$ is a parameter controlling the S&C tradeoff. The communication
and sensing signal-to-noise ratios may be expressed as

${\rm SNR}_{\rm c}=P\|\bm{H}_{\rm c}\|_{\rm F}^{2}\sigma_{\rm
c}^{-2},\leavevmode\nobreak\ \leavevmode\nobreak\ {\rm SNR}_{\rm
s}=P\|\bm{H}_{\rm s}\|_{\rm F}^{2}\sigma_{\rm s}^{-2}.$

Figure 2: The rate-exponent region of the single-antenna user, point-like
target scenario, with communication and sensing SNRs equal to $10$dB and
$0$dB, respectively.

(a) $\theta_{\rm c}=60^{\circ},\leavevmode\nobreak\ \theta_{\rm s}=90^{\circ}$

(b) $\theta_{\rm c}=60^{\circ},\leavevmode\nobreak\ \theta_{\rm
s}=-30^{\circ}$

Figure 3: Beamspace illustration of the subspace tradeoff (corresponding to
the rate-exponent regions in Fig. 2). The “ISAC torch” can simultaneously
illuminate both the user and the target in case (a), while it has to apply a
power-splitting strategy in case (b).

Using (6), we may readily obtain the rate-exponent regions for given
configurations of $(\theta_{\rm c},\theta_{\rm s})$, as portrayed in Fig. 2.
Note that the closeness between the S&C subspaces in this scenario is
characterized by the intersection angle between $\bm{a}(\theta_{\rm c})$ and
$\bm{a}(\theta_{\rm s})$. We may observe from Fig. 2 that the S&C tradeoff
becomes more prominent as the angular separation between the target and the
user increases. More intuitively, as can be seen from Fig. 2, in the
$\theta_{\rm c}=60^{\circ}$, $\theta_{\rm s}=90^{\circ}$ scenario, the
sensing-optimal and communication-optimal beam patterns (corresponding to
$\lambda=0$ and $\lambda=1$, respectively) have a large overlap. By contrast,
in the $\theta_{\rm c}=60^{\circ}$, $\theta_{\rm s}=-30^{\circ}$ scenario, the
sensing- and communication-optimal beam patterns are almost orthogonal to each
other. This corroborates our intuition about the ST, that more synergies
between S&C tasks would be witnessed as the corresponding subspaces become
closer to each other. In the language of the torch metaphor, we may say that
the “ISAC torch” in the case of Fig. 3a can simultaneously illuminate both the
user and the target, while it has to apply a power-splitting strategy in the
case of Fig. 3b.

Now that we see (5) holds true for the target presence detection problem under
the sensing metric of error exponent, we would naturally ask in the sequel
that:

  * •

Does (5) hold in general?

  * •

If not, what is the key condition for (5) to hold? Is it the metric chosen
(error exponent), the specific type of the sensing task (target detection), or
both?

##  III Capacity-Distortion Theory

To answer these questions, one has to rely on a more general analytical
framework, that applies to both estimation and detection tasks and can handle
favorably all reasonable sensing performance metrics. The capacity-distortion
theory [8, 9, 10] is conceived in the hope of building such a universal
framework.

###  III-A Rate-Distortion Region

As the name suggests, the capacity-distortion theory investigates the tradeoff
between communication capacity and sensing distortion. Originally proposed by
Shannon in the context of rate-distortion theory for lossy data compression
[17], distortion refers to a wide range of functions taking the form of
$d(\bm{\seta},\hat{\bm{\seta}})$, whose inputs are the true value of some
quantity (in sensing problems, the sensing parameter) $\bm{\seta}$ and its
estimate $\hat{\bm{\seta}}$. Due to the randomness of the communication
message, the ISAC waveform (codeword) $\bm{\mathsfbr{X}}^{N}$ is also random.
To reflect the sensing performance over a relatively long period of time, a
common practice is to use the expectation of the distortion, instead of its
instantaneous values, as the performance metric. For example, in estimation
tasks, the squared Euclidean distance
$d(\bm{\seta},\hat{\bm{\seta}})=\|\bm{\seta}-\hat{\bm{\seta}}\|^{2}$ is a
widely applied distortion function, whose expectation is the mean-squared
error (MSE). In binary (e.g. target presence) detection problems in which the
task is to determine the value of a binary variable $\seta\in\\{0,1\\}$, a
valid distortion function is the Hamming distance given by
$d(\seta,\hat{\seta})=\seta\oplus\hat{\seta}$, whose expectation is related to
commonly used detection metrics, including the detection probability $P_{\rm
D}$ and the false alarm rate $P_{\rm FA}$, as follows:

$\displaystyle\mathbb{E}\\{\seta\oplus\hat{\seta}\\}$ $\displaystyle\hskip
8.53581pt=(1\oplus 1)\mathbb{P}\\{\hat{\seta}=1|\seta=1\\}+(0\oplus
0)\mathbb{P}\\{\hat{\seta}=0|\seta=0\\}$ $\displaystyle\hskip
19.91692pt+(1\oplus 0)\mathbb{P}\\{\hat{\seta}=1|\seta=0\\}+(0\oplus
1)\mathbb{P}\\{\hat{\seta}=0|\seta=1\\}$ $\displaystyle\hskip
8.53581pt=1-P_{\rm D}+P_{\rm FA}.$ (7)

Note that for constant false-alarm rate (CFAR) detectors based on the Neyman-
Pearson criterion with fixed $P_{\rm FA}$ [18], minimizing the Hamming
distance is equivalent to maximizing the detection probability $P_{\rm D}$.

Besides capacity and distortion, there is yet another important ingredient in
the capacity-distortion theory, namely the transmission cost. To elaborate,
not all ISAC waveforms (or codewords) cost equally in terms of wireless
resources. For example, the points in quadrature amplitude modulation (QAM)
constellations having different amplitudes would yield different power
consumptions. Apparently, as the overall resource budget increases, S&C
performances can be simultaneously enhanced. Therefore, one has to discuss the
capacity-distortion tradeoff under a specific resource budget. Similar to the
sensing distortion, in order to take into account the randomness of the
codewords, we typically use the expectation of the resource cost over all
possible codewords as the measure of transmission cost.

Once the resource budget is given and the sensing distortion metric is chosen,
the S&C tradeoff is expressed in terms of the largest achievable rate-
distortion region. Formally speaking, given an expected resource budget $B$,
the rate-distortion-cost triple $(R,D,B)$ is said to be achievable (in the
infinite block length regime), if there exists a sequence of $(2^{NR},R)$
codes $\\{\bm{\mathsfbr{X}}^{N}|N\in\mathbb{N}\\}$ encoding the message
$\mathsfbr{W}\in\\{1,\dotsc,2^{NR}\\}$, and a state estimator function
$\hat{\bm{\seta}}:(\bm{\mathsfbr{Y}}_{\rm
s},\bm{\mathsfbr{X}}^{N})\mapsto\hat{\bm{\seta}}$, such that the following
holds [10]

$\displaystyle\mathbb{E}\big{\\{}d\big{(}\bm{\seta}^{N},\hat{\bm{\seta}}^{N}\big{)}\big{\\}}\leqslant
D,$ (8a) $\displaystyle\mathbb{E}\\{b(\bm{\mathsfbr{X}}^{N})\\}\leqslant B,$
(8b) $\displaystyle P_{\rm
e}^{(N)}:=\frac{1}{2^{NR}}\sum_{i=1}^{2^{NR}}\mathbb{P}\\{\hat{\mathsfbr{W}}\neq
i|\mathsfbr{W}=i\\}\rightarrow 0,$ (8c)

as $N\rightarrow\infty$, where $b(\cdot)$ denotes the instantaneous cost of
single codeword. In addition, for generality, the values of the sensing
parameters are allowed to vary across time, denoted by $\bm{\seta}_{i}$ at the
$i$-th channel use, constituting a parameter sequence $\bm{\seta}^{N}$. The
capacity-distortion function given a specific resource budget $B$ is then
defined as

$C_{B}(D)=\sup\\{R|(R,D,B)\leavevmode\nobreak\ \textrm{is\leavevmode\nobreak\
achievable}\\}.$ (9)

\begin{overpic}[width=199.4681pt]{figures/general_CD_model2.eps}
\put(42.2,16.0){\footnotesize$P_{\bm{\mathsfbr{Y}}_{\rm
s},\bm{\mathsfbr{Y}}_{\rm c}|\bm{\mathsfbr{X}},\bm{\seta}}$}
\put(3.0,18.0){\footnotesize$\mathsfbr{W}$}
\put(29.5,17.5){\footnotesize$\bm{\mathsfbr{X}}_{i}$}
\put(61.0,25.0){\footnotesize$\bm{\mathsfbr{Y}}_{{\rm s},i}$}
\put(61.0,14.0){\footnotesize$\bm{\mathsfbr{Y}}_{{\rm c},i}$}
\put(93.0,7.5){\footnotesize$\hat{\mathsfbr{W}}$}
\put(93.5,28.0){\footnotesize$\hat{\bm{\seta}}_{i}$}
\put(48.0,2.0){\footnotesize$P_{\bm{\seta}}$}
\put(62.0,4.5){\footnotesize$\bm{\seta}_{i}$} \end{overpic} Figure 4: The
structure of a generic ISAC system with memoryless channels, often considered
in the capacity-distortion theory.

For the simplicity of discussion, let us consider memoryless channels,222It is
possible that ISAC channels have memory. However, the capacity-distortion
problem of ISAC channels with memory remains open in general, except for some
special cases [19]. exemplified by (2), and illustrated in Fig. 4. The effect
of such channels can be expressed as

$\displaystyle p_{\mathsfbr{W},\bm{\mathsfbr{Y}}_{\rm
c}^{N},\bm{\mathsfbr{Y}}_{\rm
s}^{N},\bm{\mathsfbr{X}}^{N},\bm{\seta}^{N}}(W,\bm{Y}_{\rm s},\bm{Y}_{\rm
c},\bm{X},\bm{\eta})=p_{\mathsfbr{W}}(W)$ $\displaystyle\hskip
8.53581pt\times\prod_{i=1}^{N}p_{\bm{\seta}}(\bm{\eta}_{i})p_{\bm{\mathsfbr{X}}|\mathsfbr{W}}(\bm{X}_{i}|\mathsfbr{W})p_{\bm{\mathsfbr{Y}}_{\rm
s},\bm{\mathsfbr{Y}}_{\rm c}|\bm{\mathsfbr{X}},\bm{\seta}}(\bm{Y}_{{\rm
s},i},\bm{Y}_{{\rm c},i}|\bm{X}_{i},\bm{\eta}_{i}).$ (10)

In particular, the channel $p_{\mathsfbr{W},\bm{\mathsfbr{Y}}_{\rm
c}^{N},\bm{\mathsfbr{Y}}_{\rm
s}^{N},\bm{\mathsfbr{X}}^{N},\bm{\seta}^{N}}(W,\bm{Y}_{\rm s},\bm{Y}_{\rm
c},\bm{X},\bm{\eta})$ produces a communication output $\bm{\mathsfbr{Y}}_{\rm
c}$ and a sensing output $\bm{\mathsfbr{Y}}_{\rm s}$, whereas their
relationship with the channel input $\bm{\mathsfbr{X}}$ can be different from
the linear model (2). Moreover, the way that the sensing parameter
$\bm{\seta}$ couples with the channel can also be different. For such
channels, it is natural to consider per-block resource budgets and distortion
metrics, given by

$\displaystyle\mathbb{E}\big{\\{}b\big{(}\bm{\mathsfbr{X}}^{N}\big{)}\big{\\}}$
$\displaystyle=\frac{1}{N}\sum_{n=1}^{N}\mathbb{E}\\{b(\bm{\mathsfbr{X}}_{i})\\},$
$\displaystyle\mathbb{E}\big{\\{}d\big{(}\bm{\seta}^{N},\hat{\bm{\seta}}^{N}\big{)}\big{\\}}$
$\displaystyle=\frac{1}{N}\sum_{n=1}^{N}\mathbb{E}\\{d(\bm{\seta}_{i},\hat{\bm{\seta}}_{i})\\}.$

The conditions (8) can then be concretized as

$\displaystyle\limsup_{N\rightarrow\infty}\leavevmode\nobreak\
\frac{1}{N}\sum_{n=1}^{N}\mathbb{E}\\{d(\bm{\seta}_{i},\hat{\bm{\seta}}_{i})\\}$
$\displaystyle\leqslant D,$ (11a)
$\displaystyle\limsup_{N\rightarrow\infty}\leavevmode\nobreak\
\frac{1}{N}\sum_{n=1}^{N}\mathbb{E}\\{b(\bm{\mathsfbr{X}}_{i})\\}$
$\displaystyle\leqslant B,$ (11b)
$\displaystyle\limsup_{N\rightarrow\infty}\leavevmode\nobreak\ P_{\rm
e}^{(N)}$ $\displaystyle=0.$ (11c)

In order to gain insights from (11), observe that when the sensing distortion
constraint (11a) is absent, the capacity is given by the classical result of
“capacity-with-cost” [20]

$C_{B}=\max_{p_{\bm{\mathsfbr{X}}}(\bm{X})\in{\cal{P}}_{B}}\leavevmode\nobreak\
\leavevmode\nobreak\ I(\bm{\mathsfbr{X}};\bm{\mathsfbr{Y}}_{\rm c}),$ (12)

where

${\cal{P}}_{B}=\\{p_{\bm{\mathsfbr{X}}}(\bm{X})|\mathbb{E}\\{b(\bm{\mathsfbr{X}})\\}\leqslant
B\\}.$

The result (12) implies that the capacity in the presence of cost budget has a
single-letter representation, which would extremely simplify further analysis.
Furthermore, since the sensing channel is memoryless, the optimal estimator
does not rely on historical observations, and hence the expected distortion
can be written as a function of the channel input, i.e.
$\mathbb{E}\\{d(\bm{\seta}_{i},\hat{\bm{\seta}}_{i})|\bm{\mathsfbr{X}}=\bm{X}\\}=c(\bm{X})$.
In light of this, the capacity-distortion function can now be viewed as a
capacity with two costs, namely [10]

$\displaystyle C_{B}(D)$
$\displaystyle=\sup_{p_{\bm{\mathsfbr{X}}}(\bm{X})\in{\cal{P}}_{B}\cap{\cal{P}}_{D}}\leavevmode\nobreak\
I(\bm{\mathsfbr{X}};\bm{\mathsfbr{Y}}_{\rm c}|\bm{\seta}),$ (13a)
$\displaystyle{\cal{P}}_{B}$
$\displaystyle=\\{p_{\bm{\mathsfbr{X}}}(\bm{X})|\mathbb{E}\\{b(\bm{\mathsfbr{X}})\\}\leqslant
B\\},$ (13b) $\displaystyle{\cal{P}}_{D}$
$\displaystyle=\\{p_{\bm{\mathsfbr{X}}}(\bm{X})|\mathbb{E}\\{c(\bm{\mathsfbr{X}})\\}\leqslant
D\\}.$ (13c)

Remarkably, the result (13) suggests that the expected sensing distortion may
be alternatively viewed as a kind of “sensing-induced communication cost”.
This understanding enables us to extend even further the range that the
capacity-distortion theory is applicable, to scenarios in which the sensing
performance cannot be described by a proper distortion function. For example,
the Cramér-Rao bound (CRB) widely used as the performance metric of estimation
problems is not dependent on any specific estimator $\hat{\bm{\seta}}$.
Sometimes it does not even depend on the true value of the parameter
$\bm{\seta}$. Therefore, CRB is not a distortion function, but it is related
to the transmitted codeword $\bm{\mathsfbr{X}}$ (as will be discussed later),
and hence the capacity-distortion theory can still be applied in the
generalized sense.

In this aforementioned scenario, we did not consider feedback. One may wonder
whether designing the ISAC codebook by relying on feedback can further enhance
the S&C performance. To this end, the state-dependent memoryless channel with
delayed feedback (SDMC-DF) model has been investigated [10]. The effect of an
SDMC-DF can be expressed as [8]

$\displaystyle
p_{\mathsfbr{W},\bm{\mathsfbr{Y}}^{N},\bm{\mathsfbr{Z}}^{N},\bm{\mathsfbr{X}}^{N},\bm{\seta}^{N}}(W,\bm{Y},\bm{Z},\bm{X},\bm{\eta})=p_{\mathsfbr{W}}(W)$
$\displaystyle\hskip
2.84526pt\times\\!\prod_{i=1}^{N}p_{\bm{\seta}}(\bm{\eta}_{i})p_{\bm{\mathsfbr{X}}|\mathsfbr{W},\bm{\mathsfbr{Z}}}(\bm{X}_{i}|\mathsfbr{W},\\!\bm{Z}_{i-1})p_{\bm{\mathsfbr{Y}},\bm{\mathsfbr{Z}}|\bm{\mathsfbr{X}},\bm{\seta}}(\bm{Y}_{i},\\!\bm{Z}_{i}|\bm{X}_{i},\\!\bm{\eta}_{i})$
(14)

where both sensing and communication rely on the same channel, which produces
the output $\bm{\mathsfbr{Y}}^{N}$. For sensing tasks, the channel input
$\bm{\mathsfbr{X}}_{i}$ at the $i$-th channel use cannot be designed based on
the real-time channel output $\bm{\mathsfbr{Y}}_{i}$ or the parameter value
$\bm{\seta}_{i}$, since the design process has to be causal. Rather, one can
only rely on a _delayed feedback_ $\bm{\mathsfbr{Z}}_{i-1}$, which may be a
function of the channel output $\bm{\mathsfbr{Y}}_{i}$.

It turns out that, for SDMC-DF models, (13) also applies, with the slight
modification that $\bm{\mathsfbr{Y}}_{\rm c}$ is replaced with
$\bm{\mathsfbr{Y}}$. To see this, first note that feedback does not improve
the capacity of memoryless channels. For the sensing performance, it has been
shown that the optimal sensing distortion is achievable by the simple letter-
wise minimum expected posterior distortion estimator [10] taking the form of

$\gamma(\bm{\mathsfbr{X}},\bm{\mathsfbr{Z}})=\left(\hat{\bm{\seta}}^{\ast}(\bm{\mathsfbr{X}}_{1},\bm{\mathsfbr{Z}}_{1}),\dotsc,\hat{\bm{\seta}}^{\ast}(\bm{\mathsfbr{X}}_{N},\bm{\mathsfbr{Z}}_{N})\right),$

where

$\hat{\bm{\seta}}^{\ast}(\bm{X},\bm{Z})=\mathop{\arg\min}_{\bm{\seta}^{\prime}}\leavevmode\nobreak\
\int
d(\bm{\seta},\bm{\seta}^{\prime})p_{\bm{\seta}|\bm{\mathsfbr{X}},\bm{\mathsfbr{Z}}}(\bm{\eta}|\bm{X},\bm{Z}){\rm
d}\bm{\eta}.$

We may now see that the expected distortion also admits a single-letter
representation

$c(\bm{X})=\mathbb{E}\big{\\{}d\big{(}\bm{\seta},\hat{\bm{\seta}}^{\ast}(\bm{\mathsfbr{X}},\bm{\mathsfbr{Z}})\big{)}|\bm{\mathsfbr{X}}=\bm{X}\big{\\}},$
(15)

and thus (13) still applies.

From the aforementioned discussion, we may conclude that a key condition for
(13) to hold is the memorylessness of channels. Indeed, for channels with
memory, even the capacity itself remains open. Furthermore, for such channels,
online and offline estimators may have very different performances. These
problems deserve further investigation.

###  III-B Computing the Capacity-Distortion Boundary

From a pure theorist’s perspective, the result (13) is a complete information
characterization (as opposed to the operational definition (11)) of the
capacity-distortion region. But wait! Remember that our expectation on the
capacity-distortion theory in the first place was to help understand the
nature of the S&C tradeoff, hopefully beyond the ST. However, we cannot even
dig ST itself out of (13), not to mention any further insight.

To serve our purpose, a possible approach is to compute explicitly the
capacity-distortion functions for some representative scenarios, in the hope
of obtaining useful intuitions. It turns out that the renowned Blahut-Arimoto
(B-A) algorithm [21] originally proposed for computing the unconstrained
capacity can be applied to compute capacity-distortion functions with some
modifications. To elaborate, given an initialization of the trail distribution
$r(\bm{X})$ for $p_{\bm{\mathsfbr{X}}}(\bm{X})$, the original B-A algorithm
solving $C=\max_{p_{\bm{\mathsfbr{X}}}(\bm{X})}\leavevmode\nobreak\
I(\bm{\mathsfbr{X}};\bm{\mathsfbr{Y}})$ is a fixed-point iteration repeating
the following two steps in each round:

  1. 1.

Update the trail distribution $q(\bm{X}|\bm{Y})$ for the a posteriori
distribution $p_{\bm{\mathsfbr{X}}|\bm{\mathsfbr{Y}}}(\bm{X}|\bm{Y})$
according to

$q(\bm{X}|\bm{Y})\leftarrow\frac{p_{\bm{\mathsfbr{X}}}(\bm{X})p_{\bm{\mathsfbr{Y}}|\bm{\mathsfbr{X}}}(\bm{Y}|\bm{X})}{\int
r(\bm{X})p_{\bm{\mathsfbr{Y}}|\bm{\mathsfbr{X}}}(\bm{Y}|\bm{X}){\rm
d}\bm{X}};$ (16)

  2. 2.

Update $r(\bm{X})$ by

$r(\bm{X})\leftarrow\frac{e^{\int
p_{\bm{\mathsfbr{Y}}|\bm{\mathsfbr{X}}}(\bm{Y}|\bm{X})\log
q(\bm{X}|\bm{Y}){\rm d}\bm{Y}}}{\int e^{\int
p_{\bm{\mathsfbr{Y}}|\bm{\mathsfbr{X}}}(\bm{Y}|\bm{X})\log
q(\bm{X}|\bm{Y}){\rm d}\bm{Y}}{\rm d}\bm{X}},$ (17)

which is equivalent to solving the following constrained entropy maximization
problem:

$\displaystyle r(\bm{X})\leftarrow$
$\displaystyle\mathop{\arg\max}_{p(\bm{X})}-\int p(\bm{X})\log p(\bm{X}){\rm
d}\bm{X},$ $\displaystyle{\rm s.t.}\leavevmode\nobreak\ p(\bm{X})\geqslant
0\leavevmode\nobreak\ \forall\bm{X},\leavevmode\nobreak\ \int p(\bm{X}){\rm
d}\bm{X}=1,$ $\displaystyle\mathbb{E}_{p(\bm{X})}\Big{\\{}\int
p_{\bm{\mathsfbr{Y}}|\bm{\mathsfbr{X}}}(\bm{Y}|\bm{X})\log
q(\bm{X}|\bm{Y}){\rm d}\bm{Y}\Big{\\}}\geqslant t$

for some constant $t$.

Now that our objective function is the conditional mutual information
$I(\bm{\mathsfbr{X}};\bm{\mathsfbr{Y}}|\bm{\seta})$, we shall replace (16)
with

$q(\bm{X}|\bm{Y},\bm{\eta})\leftarrow\frac{p_{\bm{\mathsfbr{X}}}(\bm{X})p_{\bm{\mathsfbr{Y}}|\bm{\mathsfbr{X}},\bm{\seta}}(\bm{Y}|\bm{X},\bm{\eta})}{\int
p_{\bm{\mathsfbr{X}}}(\bm{X})p_{\bm{\mathsfbr{Y}}|\bm{\mathsfbr{X}},\bm{\seta}}(\bm{Y}|\bm{X},\bm{\eta}){\rm
d}\bm{X}}.$

In addition, since two extra constraints (13b) and (13c) are now enforced, we
should replace (17) with

$\displaystyle r(\bm{X})\leftarrow$ $\displaystyle\hskip
8.53581pt\frac{e^{\int[p_{\bm{\seta}}(\bm{\eta})p_{\bm{\mathsfbr{Y}}|\bm{\mathsfbr{X}},\bm{\seta}}(\bm{Y}|\bm{X},\bm{\eta})\log
q(\bm{X}|\bm{Y},\bm{\eta})-\lambda b(\bm{X})-\mu c(\bm{X})]{\rm
d}\bm{\eta}{\rm d}\bm{Y}}}{\int
e^{\int[p_{\bm{\seta}}(\bm{\eta})p_{\bm{\mathsfbr{Y}}|\bm{\mathsfbr{X}},\bm{\seta}}(\bm{Y}|\bm{X},\bm{\eta})\log
q(\bm{X}|\bm{Y},\bm{\eta})-\lambda b(\bm{X})-\mu c(\bm{X})]{\rm
d}\bm{\eta}{\rm d}\bm{Y}}{\rm d}\bm{X}},$

where $\lambda$ and $\mu$ are the Lagrangian multipliers corresponding to the
constraints (13b) and (13c), respectively.

From the above discussions, we may get a vague sense that the sensing
distortion requirements render the resulting ISAC signal $\bm{\mathsfbr{X}}$
“less random”, as they impose constraints on the entropy maximization problem.
Of course, this is not yet a rigorous statement with a well-defined meaning.
To validate our intuition, let us consider the toy example of a real-valued,
single-input single-output (SISO) Gaussian SDMC-DF channel model (with
Rayleigh fading)

$\mathsfbr{Y}_{i}=\seta_{i}\mathsfbr{X}_{i}+\mathsfbr{N}_{i},$

where $\seta_{i}$ and $\mathsfbr{N}_{i}$ are mutually independent zero-mean
Gaussian variables with unit variance, while $\mathsfbr{X}_{i}$ is the ISAC
waveform satisfying the power constraint
$\limsup_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^{N}\mathbb{E}\\{|\mathsfbr{X}_{i}|^{2}\\}\leqslant
B=10$dB. We consider the perfect feedback $\mathsfbr{Z}_{i}=\mathsfbr{Y}_{i}$
and the quadratic sensing distortion
$d(\eta,\hat{\eta})=(\eta-\hat{\eta})^{2}$. In this scenario, the sensing-
optimal estimator is the letter-wise minimum mean squared error (MMSE)
estimator given by $\hat{\seta}_{i,{\rm
MMSE}}=\mathbb{E}\\{\seta_{i}|\mathsfbr{X}_{i},\mathsfbr{Y}_{i}\\}$, whose MSE
can be calculated as

$\frac{1}{N}\sum_{i=1}^{N}\mathbb{E}\\{(\seta-\hat{\seta}_{i,{\rm
MMSE}})^{2}\\}=\mathbb{E}\bigg{\\{}\frac{1}{1+|\mathsfbr{X}|^{2}}\bigg{\\}},$

where $\mathsfbr{X}$ is the single-letter representation of the channel input
following the probability distribution $p_{\mathsfbr{X}}(X)$.

Under the aforementioned formulation, the capacity-distortion boundary may be
numerically computed using the modified B-A algorithm, as portrayed in Fig. 5.
At the communication-optimal point $\footnotesize1⃝$, the input distribution
is Gaussian, and we have

$\displaystyle C_{B}(D)=\frac{1}{2}\mathbb{E}\\{\log_{2}(1+B|\seta|^{2})\\}$
$\displaystyle\approx 1.213,$
$\displaystyle\mathbb{E}\\{c(\mathsfbr{X})\\}=\mathbb{E}\bigg{\\{}\frac{1}{1+|\mathsfbr{X}|^{2}}\bigg{\\}}$
$\displaystyle\approx 0.327.$

By contrast, at the sensing-optimal point $\footnotesize4⃝$, the input
distribution corresponds to binary phase-shift keying (BPSK) modulation, for
which we have

$\displaystyle C_{B}(D)$ $\displaystyle\approx 0.733,$
$\displaystyle\mathbb{E}\\{c(\mathsfbr{X})\\}=\frac{1}{1+B}$
$\displaystyle\approx 0.091.$

As we move along the capacity-distortion boundary from the communication-
optimal point $\footnotesize1⃝$ to the sensing-optimal point
$\footnotesize4⃝$, the corresponding input distribution $p_{\mathsfbr{X}}(X)$
exhibits a smooth transition from the Gaussian distribution to the BPSK
modulation, which agrees with our intuition that the ISAC signal becomes less
random as the sensing distortion requirement becomes more stringent.

(a) The capacity-distortion boundary

(b) $p_{\mathsfbr{X}}(X)$ at \tiny1⃝ (c) $p_{\mathsfbr{X}}(X)$ at \tiny2⃝

(d) $p_{\mathsfbr{X}}(X)$ at \tiny3⃝ (e) $p_{\mathsfbr{X}}(X)$ at \tiny4⃝

Figure 5: The capacity-distortion boundary of the real-valued SISO Gaussian
channel scenario with $B=10$dB, as well as the Pareto-optimal input
distributions $p_{\mathsfbr{X}}(X)$ along the boundary.

Note that there is not enough room to accommodate ST in the SISO scenario.
Therefore, the S&C tradeoff in the aforementioned example follows solely from
the difference in signal preference between communication and sensing tasks.
Intuitively, communication tasks would favor signals with a higher degree of
randomness, in order to pack more information into the signal. For example,
the capacity for AWGN channels is achieved by using Gaussian-distributed
signals, where the Gaussian distribution has the maximum entropy under power
constraint. By contrast, sensing tasks favor signals that are deterministic in
some sense, in order to better distinguish the received signals (echoes)
coupled with sensing parameters taking different values. Such a tradeoff is
termed as deterministic-random tradeoff (DRT) [11].

Naturally, we would then wonder whether the DRT exists in more general
scenarios, what effect it would exhibit, and how it would interact with the ST
when it exists. Unfortunately, although the modified B-A algorithm is
universal in principle, it is hardly applicable to the general settings in a
practical sense, due to its enormous computational complexity. To elaborate,
the modified B-A algorithm relies on numerical integration, which could be
computationally prohibitive when the dimensionality of the signals or the
sensing parameters is high. For example, if the number of samples per
dimension is $K$, the total number of samples $N_{\rm MC}$ that the widely
used Monte Carlo integration method would require will be on the order of
$N_{\rm
MC}=O\big{(}K^{N_{\bm{\mathsfbr{Y}}}+N_{\bm{\mathsfbr{X}}}+N_{\bm{\seta}}}\big{)}$,
where $N_{\bm{\mathsfbr{Y}}}$, $N_{\bm{\mathsfbr{X}}}$ and $N_{\bm{\seta}}$
are the dimensionalities of $\bm{\mathsfbr{Y}}$, $\bm{\mathsfbr{X}}$, and
$\bm{\seta}$, respectively. The exponentially increasing sample complexity
would easily surpass the capability of most computing devices, even for small-
scale MIMO systems. Furthermore, the modified B-A algorithm cannot provide
analytical solutions, which can be useful for practical system design.

Considering these difficulties, to push further our understanding of the S&C
tradeoff, we might need to sacrifice the generality of the capacity-distortion
theory to some degree. For example, we may try and find specific distortion
functions (or non-distortion sensing-induced communication costs, as implied
by (13)), that are both easy to analyze and sufficiently general to reflect ST
and DRT. In what follows, we shall consider one such example in detail.

##  IV CRB-Rate Region

We now focus our attention from the distortion measure
$d(\bm{\seta},\hat{\bm{\seta}})$ to a specific sensing metric, namely, CRB for
target parameter estimation. Unlike the MSE that specifically relies on the
employed estimator, CRB serves as a globally lower bound for all unbiased
estimators (satisfying the regularity condition), which usually leads to more
tractable analytical expressions, and is achievable at high-SNR regimes [18].
Recalling the single-letterization of the expected distortion in (15), one may
treat the CRB as a cost function of the transmitted codeword
$\bm{\mathsfbr{X}}$, and consider the interplay between the CRB and
communication rate as a special case of the C-D tradeoff.

###  IV-A Vector Gaussian System Model

We commence by re-examining the model in (2), which may be extended to a more
generic form as

$\displaystyle\bm{\mathsfbr{Y}}_{{\rm c}}$
$\displaystyle=\bm{\mathsfbr{H}}_{\rm
c}\bm{\mathsfbr{X}}+\bm{\mathsfbr{Z}}_{{\rm c}},$ (18a)
$\displaystyle\bm{\mathsfbr{Y}}_{{\rm s}}$
$\displaystyle=\bm{\mathsfbr{H}}_{\rm
s}(\bm{\seta})\bm{\mathsfbr{X}}+\bm{\mathsfbr{Z}}_{{\rm s}},$ (18b)

where the sensing channel $\bm{\mathsfbr{H}}_{\rm s}\in\mathbb{C}^{{N_{\rm
s}}\times M}$ is now defined as a deterministic, possibly nonlinear function
of the sensing parameter $\bm{\seta}\in\mathbb{R}^{K}$, e.g., angular MIMO
radar channel [22]. If not otherwise specified, the transmitted codeword
$\bm{\mathsfbr{X}}\in\mathbb{C}^{M\times T}$ will be referred to as an ISAC
signal matrix in this section. This model may be viewed as a special case of
the generic model shown in Fig. 4. Following the preceding memoryless channel
assumption, both the sensing parameter $\bm{\seta}$ and communication channel
$\bm{\mathsfbr{H}}_{\rm c}\in\mathbb{C}^{{N_{\rm c}}\times M}$ vary every $T$
symbols in an i.i.d. manner. The discussion on channels with memory (e.g.,
Markov channels) are designated as our future works. For convenience, we also
assume that the ISAC Tx has perfect knowledge on $\bm{\mathsfbr{H}}_{\rm c}$.
Finally, $\bm{\mathsfbr{Z}}_{{\rm c}}$ and $\bm{\mathsfbr{Z}}_{{\rm s}}$ are
zero-mean white Gaussian noise matrices with variances of $\sigma_{{\rm
c}}^{2}$ and $\sigma_{{\rm s}}^{2}$, respectively.

(a) Monostatic sensing

(b) Bistatic sensing

Figure 6: The ISAC scenarios described in (18), where the dual-functional
waveform $\bm{\mathsfbr{X}}$ is known to both the ISAC Tx and sensing receiver
(Rx).

At this point, it would be worthwhile to concretize the separate channel
models in (18) by illustrating the specific scenarios concerned in this
section. As shown in Fig. 6, the ISAC Tx emits a dual-functional signal
$\bm{\mathsfbr{X}}$ to simultaneously communicate with a single communication
Rx and sense one or more targets. The sensing Rx is either collocated with the
Tx (monostatic sensing), or connected with the Tx with a wired link (bistatic
sensing). In both cases, both the Tx and sensing Rx have perfect knowledge
about the ISAC signal $\bm{\mathsfbr{X}}$. Nevertheless, $\bm{\mathsfbr{X}}$
is unknown to the communication Rx as it conveys useful information intended
for the communication user. We are thus temped to model $\bm{\mathsfbr{X}}\sim
p_{\bm{\mathsfbr{X}}}(\bm{X})$ as a random matrix, whose realization is
perfectly known at both the ISAC Tx and sensing Rx, but is unknown at the
communication Rx.

###  IV-B CRB with Random but Known Nuisance Parameters

Since the communication performance of (18a) can be directly measured by the
mutual information $I(\bm{\mathsfbr{X}};\bm{\mathsfbr{Y}}_{\rm
c}|\bm{\mathsfbr{H}}_{\rm c})$, we are now in a position to rethink the
sensing performance evaluation in ISAC systems. Indeed, in conventional radar
systems, probing signals are typically deterministic, well-designed with good
ambiguity properties. In ISAC systems, on the other hand, the transmit signal
randomly varies from block to block given the communication data embedded in
the signal. This imposes a unique challenge in defining the CRB, which now
becomes a function of the random signal $\bm{\mathsfbr{X}}$.

One possible approach would be to treat $\bm{\mathsfbr{X}}$ as a nuisance
parameter, and either consider it as a part of the unknown sensing parameter,
or integrate it out of the likelihood function describing the observations,
leading to classical hybrid or marginal CRB expressions, respectively.
Nonetheless, neither of these methods grasps the fundamental feature of the
ISAC system, that the random signal $\bm{\mathsfbr{X}}$ is known to the
sensing Rx, and is thus loose in general. To that end, we resort to a Miller-
Chang type CRB [23] by computing the CRB for a given instance of
$\bm{\mathsfbr{X}}$, and take the expectation over $\bm{\mathsfbr{X}}$. For
any weakly unbiased estimator of $\bm{\seta}$, the MSE is lower bounded by the
Miller-Chang Bayesian CRB in the form of

${\rm MSE}_{\bm{\seta}}(\hat{\bm{\seta}})\geqslant\mathbb{E}\left({\rm
tr}\left\\{\bm{\mathsfbr{J}}_{\bm{\seta}|\bm{\mathsfbr{X}}}^{-1}\right\\}\right),$
(19)

where the expectation is taken with respect to $\bm{\mathsfbr{X}}$, and
$\bm{\mathsfbr{J}}_{\bm{\seta}|\bm{\mathsfbr{X}}}$ denotes the Bayesian Fisher
Information Matrix (BFIM) of $\bm{\seta}$ given by

$\displaystyle\bm{\mathsfbr{J}}_{\bm{\seta}|\bm{\mathsfbr{X}}}$
$\displaystyle\\!:=\\!\mathbb{E}\\!\left\\{\\!\frac{\partial\ln
p_{\bm{\mathsfbr{Y}}_{\rm s}|\bm{\mathsfbr{X}},\bm{\seta}}(\bm{Y}_{\rm
s}|\bm{X},\bm{\eta})}{\partial\bm{\eta}}\frac{\partial\ln
p_{\bm{\mathsfbr{Y}}_{\rm s}|\bm{\mathsfbr{X}},\bm{\seta}}(\bm{Y}_{\rm
s}|\bm{X},\bm{\eta})}{\partial\bm{\eta}^{\rm
T}}\bigg{|}\bm{\mathsfbr{X}}\right\\}$ (20) $\displaystyle\hskip
14.22636pt+\mathbb{E}\left\\{\frac{\partial\ln
p_{\bm{\seta}}(\bm{\eta})}{\partial\bm{\eta}}\frac{\partial\ln
p_{\bm{\seta}}(\bm{\eta})}{\partial\bm{\eta}^{\rm T}}\right\\}.$

More precisely, the BFIM $\bm{\mathsfbr{J}}_{\bm{\seta}|\bm{\mathsfbr{X}}}$
can be expressed as an affine map of the sample covariance matrix
$\bm{\mathsfbr{R}}_{\bm{\mathsfbr{X}}}=T^{-1}\bm{\mathsfbr{X}}\bm{\mathsfbr{X}}^{\rm
H}$ [11]

$\bm{\mathsfbr{J}}_{\bm{\seta}|\bm{\mathsfbr{X}}}=\frac{T}{\sigma_{\rm
s}^{2}}\bm{\varPhi}(\bm{\mathsfbr{R}}_{\bm{\mathsfbr{X}}}),$ (21)

where

$\bm{\varPhi}(\bm{A})=\sum_{i=1}^{r_{1}}\widetilde{\bm{F}}_{i}\bm{A}^{\rm
T}\widetilde{\bm{F}}_{i}^{\rm
H}+\sum_{j=1}^{r_{2}}\widetilde{\bm{G}}_{j}\bm{A}\widetilde{\bm{G}}_{j}^{\rm
H}+\widetilde{\bm{J}}_{\rm P},$ (22)

and $\widetilde{\bm{J}}_{\rm P}=\sigma_{\rm s}^{2}T^{-1}\bm{J}_{\rm P}$, with
the term $\bm{J}_{\rm P}$ being contributed by the prior distribution
$p_{\bm{\seta}}(\bm{\eta})$, i.e., the second term in (20). In particular, the
matrices $\widetilde{\bm{F}}_{i}$ and $\widetilde{\bm{G}}_{j}$ are partitioned
from the Jacobian matrix
$\bm{\mathsfbr{F}}:=\frac{\partial\operatorname{vec}(\bm{\mathsfbr{H}}_{\rm
s}^{\ast})}{\partial\bm{\seta}}$.

By noting (19), it turns out that the Miller-Chang CRB is nothing but an
equivalent “expected sensing distortion” discussed in Sec. III-A (though it is
not a real distortion measure), and may hence be viewed as a sensing-induced
cost imposed on signaling resources. Although with high dimensionality, one
may still deduce useful results on the CRB-rate tradeoff by exploiting the
affine structure of the BFIM, as discussed in the sequel.

Figure 7: CRB-rate region, Pareto boundary, and time-sharing inner bound.

###  IV-C CRB-Rate Tradeoff

The CRB-rate tradeoff can be characterized as the following Pareto
optimization problem

$\displaystyle\min_{p_{\bm{\mathsfbr{X}}}(\bm{X})}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\alpha\mathbb{E}\left({\rm
tr}\left\\{\left[\bm{\varPhi}({\bm{\mathsfbr{R}}}_{\bm{\mathsfbr{X}}})\right]^{-1}\right\\}\right)-(1-\alpha)I(\bm{\mathsfbr{X}};\bm{\mathsfbr{Y}}_{\rm
c}|\bm{\mathsfbr{H}}_{\rm c}),$ (23a) $\displaystyle{\rm s.t.}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \mathbb{E}\left({\rm
tr}\left\\{{\bm{\mathsfbr{R}}}_{\bm{\mathsfbr{X}}}\right\\}\right)=P_{\rm T},$
(23b)

where $\alpha\in\left[0,1\right]$ is a weighting factor controlling the
priority of S&C performance. We highlight here that by moving the CRB from
(23a) to (23b), (23) may be equivalently recast as a constrained capacity
characterization problem with two cost functions as in (13).

Needless to say, fully depicting the Pareto boundary of the CRB-rate region
would incur unaffordable computational overheads, as one has to numerically
seek for optimal $p_{\bm{\mathsfbr{X}}}(\bm{X})$ by leveraging the modified
B-A algorithm discussed in Sec. III-B. To reveal fundamental insights into the
CRB-rate tradeoff, we are interested in the two corner points over the Pareto
frontier as shown in Fig. 7, namely, $P_{\rm CS}$, the minimum achievable
sensing CRB constrained by the maximum communication capacity, and $P_{\rm
SC}$, the maximum achievable rate constrained by the minimized CRB. It is
obvious that the line segment linked to two points forms a time-sharing inner-
bound. In what follows, we briefly characterize the S&C performance at the two
points.

###  IV-D $P_{\rm CS}$ Performance Characterization

Let us first examine the point $P_{\rm CS}$. For point-to-point Gaussian
channel, it is well-known that Gaussian distribution is the unique capacity-
achieving input distribution (CAID) under an average power cost [24], in which
case (23) reduces to a capacity characterization problem with only power
constraint by simply letting $\alpha=0$. More specifically, at $P_{\rm CS}$,
each column of $\bm{\mathsfbr{X}}$ follows a circularly symmetric complex
Gaussian distribution
$\mathcal{CN}(\bm{0},\widetilde{\bm{R}}_{\bm{\mathsfbr{X}}}^{\rm CS})$ in an
i.i.d. manner, where the statistical covariance matrix
$\widetilde{\bm{R}}_{\bm{\mathsfbr{X}}}^{\rm CS}$ is obtained by solving the
following rate maximization problem

$\begin{gathered}{R_{\rm{max}}}=\mathop{\max}\limits_{\widetilde{\bm{R}}\succcurlyeq{\bm{0}},\widetilde{\bm{R}}={\widetilde{\bm{R}}^{\rm
H}}}\mathbb{E}\left\\{\log\left|{{\bm{I}}+\sigma_{c}^{-2}{{\bm{\mathsfbr{H}}}_{c}}\widetilde{\bm{R}}{\bm{\mathsfbr{H}}}_{c}^{\rm
H}}\right|\right\\}\hfill\\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm
s.t.}\;\;\;\;\;{\rm tr}\left\\{\widetilde{\bm{R}}\right\\}\leqslant{P_{\rm
T}}\hfill\\\
\;\;\;\;\;\;\;\;\;=\mathbb{E}\left\\{\log\left|{{\bm{I}}+\sigma_{\rm
c}^{-2}{{\bm{\mathsfbr{H}}}_{\rm
c}}\widetilde{\bm{R}}_{\bm{\mathsfbr{X}}}^{\rm CS}{\bm{\mathsfbr{H}}}_{c}^{\rm
H}}\right|\right\\}.\hfill\\\ \end{gathered}$ (24)

It is readily observed that the optimal solution of (24) has the following
eigenvalue decomposition structure

$\widetilde{\bm{R}}_{\bm{\mathsfbr{X}}}^{\rm CS}=\bm{U}_{\rm
c}\bm{\varLambda}_{\rm c}\bm{U}_{\rm c}^{\rm H},$ (25)

where $\bm{U}_{\rm c}$ contains the right singular vectors of
${{\bm{\mathsfbr{H}}}_{c}}$, and $\bm{\varLambda}_{\rm c}$ contains the
optimal eigenvalues that can be attained from the water-filling method.
Accordingly, the optimal ISAC signal structure at $P_{\rm CS}$ is

$\bm{\mathsfbr{X}}^{\rm CS}=\bm{U}_{\rm c}\bm{\varLambda}_{\rm
c}^{\frac{1}{2}}\bm{\mathsfbr{D}},$ (26)

where the entries of $\bm{\mathsfbr{D}}$ i.i.d. subject to
$\mathcal{CN}(0,1)$.

One would then raise the natural question: What is the sensing performance if
a Gaussian signal is transmitted? From (19) to (22), it is obvious that the
CRB is determined by the sample covariance matrix
$\bm{\mathsfbr{R}}_{\bm{\mathsfbr{X}}}=T^{-1}\bm{\mathsfbr{X}}\bm{\mathsfbr{X}}^{\rm
H}$ rather than the statistical covariance matrix
$\widetilde{\bm{R}}_{\bm{\mathsfbr{X}}}=\mathbb{E}(\bm{\mathsfbr{R}}_{\bm{\mathsfbr{X}}})$.
At $P_{\rm CS}$, since each column of $\bm{\mathsfbr{X}}$ is i.i.d. Gaussian
distributed, $\bm{\mathsfbr{R}}_{\bm{\mathsfbr{X}}}$ follows a complex Wishart
distribution. Note the fact that
${{\text{tr}}\left({{{\left[{{\bm{\varPhi}}\left({{\bm{\mathsfbr{R}}_{\bm{\mathsfbr{X}}}}}\right)}\right]}^{-1}}}\right)}$
is a convex function in $\bm{\mathsfbr{R}}_{\bm{\mathsfbr{X}}}$. Upon denoting
the CRB at $P_{\rm CS}$ as $\epsilon_{\rm CS}$, and based on Jensen’s
inequality, we have

$\begin{gathered}{\epsilon_{{\rm{CS}}}}\triangleq\frac{T}{\sigma_{\rm
s}^{2}}{\mathbb{E}}\left\\{{{\text{tr}}\left({{{\left[{{\bm{\varPhi}}\left({{\bm{\mathsfbr{R}}_{\bm{\mathsfbr{X}}}}}\right)}\right]}^{-1}}}\right)}\right\\}\geqslant\frac{T}{\sigma_{\rm
s}^{2}}{\text{tr}}\left\\{{{{\left({{\bm{\varPhi}}\left[{\mathbb{E}\left({\bm{\mathsfbr{R}}_{\bm{\mathsfbr{X}}}}\right)}\right]}\right)}^{-1}}}\right\\}\hfill\\\
\;\;\;\;\;\;=\frac{T}{\sigma_{\rm
s}^{2}}{\text{tr}}\left\\{{{{\left[{{\bm{\varPhi}}\left(\widetilde{\bm{R}}_{\bm{\mathsfbr{X}}}^{\rm
CS}\right)}\right]}^{-1}}}\right\\},\hfill\\\ \end{gathered}$ (27)

which holds for arbitrarily distributed
$\bm{\mathsfbr{R}}_{\bm{\mathsfbr{X}}}$. This suggests there is certain
performance loss for sensing due to the Wishart distributed
$\bm{\mathsfbr{R}}_{\bm{\mathsfbr{X}}}$, since the Jensen lower bound is
attained when
$\bm{\mathsfbr{R}}_{\bm{\mathsfbr{X}}}=\mathbb{E}(\bm{\mathsfbr{R}}_{\bm{\mathsfbr{X}}})=\widetilde{\bm{R}}_{\bm{\mathsfbr{X}}}$,
which holds for Wishart matrix only when $T\to\infty$.

A more non-trivial result in [11] depicts the upper bound of
${\epsilon_{{\rm{CS}}}}$, given by

${\mathbb{E}}\left\\{{{\text{tr}}\left({{{\left[{{\bm{\varPhi}}\left({{\bm{\mathsfbr{R}}_{\bm{\mathsfbr{X}}}}}\right)}\right]}^{-1}}}\right)}\right\\}\leqslant\frac{T\cdot{\rm
tr}\left\\{\bm{\varPhi}(\widetilde{\bm{R}}_{\bm{\mathsfbr{X}}}^{\rm
CS})^{-1}\right\\}}{T-\min\\{K,\mathsfbr{M}_{\rm CS}\\}},$ (28)

with $\mathsfbr{M}_{\rm CS}={\rm
rank}(\widetilde{\bm{R}}_{\bm{\mathsfbr{X}}}^{\rm CS})$, which clearly
indicates that the maximum sensing performance loss at $P_{\rm CS}$ is jointly
determined by the number of sensing parameters $K$ and the rank333In high-SNR
regime we have ${\rm rank}(\widetilde{\bm{R}}_{\bm{\mathsfbr{X}}}^{\rm
CS})={\rm rank}(\bm{\mathsfbr{H}}_{\rm c})$. of
$\widetilde{\bm{R}}_{\bm{\mathsfbr{X}}}^{\rm CS}$. Note again that when
$T\to\infty$, the sensing performance is lossless since the upper bound
converges to its lower counterpart.

###  IV-E $P_{\rm SC}$ Performance Characterization

The performance characterization at $P_{\rm SC}$ becomes more challenging
compared to that of $P_{\rm CS}$, as the achieving strategy remains unknown in
general. By denoting the achievable CRB at $P_{\rm SC}$ as $\epsilon_{\min}$,
and using the Jensen’s inequality again, we have

$\begin{gathered}\frac{T}{\sigma_{\rm
s}^{2}}{\mathbb{E}}\left\\{{{\text{tr}}\left({{{\left[{{\bm{\varPhi}}\left({{\bm{\mathsfbr{R}}_{\bm{\mathsfbr{X}}}}}\right)}\right]}^{-1}}}\right)}\right\\}\geqslant\frac{T}{\sigma_{\rm
s}^{2}}{\text{tr}}\left\\{{{{\left({{\bm{\varPhi}}\left[{\mathbb{E}\left({\bm{\mathsfbr{R}}_{\bm{\mathsfbr{X}}}}\right)}\right]}\right)}^{-1}}}\right\\}\hfill\\\
=\frac{T}{\sigma_{\rm
s}^{2}}{\text{tr}}\left\\{{{{\left[{{\bm{\varPhi}}\left(\widetilde{\bm{R}}_{\bm{\mathsfbr{X}}}^{\rm
SC}\right)}\right]}^{-1}}}\right\\}\triangleq\epsilon_{\min}\hfill\\\
\end{gathered}$ (29)

holds for any complex semidefinite matrix
${\bm{\mathsfbr{R}}_{\bm{\mathsfbr{X}}}}$ satisfying the average power
constraint, where $\widetilde{\bm{R}}_{\bm{\mathsfbr{X}}}^{\rm SC}$ is the
solution of the deterministic CRB minimization problem

${{\widetilde{\bm{R}}_{\bm{\mathsfbr{X}}}^{\rm{SC}}}}=\mathop{\arg\min}\limits_{{\widetilde{\bm{R}}}\succcurlyeq{\mathbf{0}},\;{\widetilde{\bm{R}}}={{\widetilde{\bm{R}}}^{\rm
H}}}\;{\text{tr}}\left\\{\left[{{{\bm{\varPhi}}}\left({\widetilde{\bm{R}}}\right)}\right]^{-1}\right\\}\;\;\operatorname{s.t.}\;{\text{tr}}\left({\widetilde{\bm{R}}}\right)\leqslant{P_{\rm
T}}.$ (30)

While problem (30) is a convex semidefinite programming (SDP), it is not
strictly convex. As a result, the optimal solution is not unique. In fact, all
the solutions of an SDP belongs to the subspace spanned by the maximum-rank
solution, hence may be parameterized as

$\widetilde{\bm{R}}_{\bm{\mathsfbr{X}}}^{\rm{SC}}=\bm{U}_{\rm
s}\bm{\varLambda}_{\rm s}\bm{U}_{\rm s}^{\rm H},$ (31)

where $\bm{U}_{\rm s}$ consists of the eigenvectors of the maximum-rank
sensing-optimal solution corresponding to the non-zero eigenvalues, while
$\bm{\varLambda}_{\rm s}$ is a positive semidefinite Hermitian matrix.

Provably, (30) admits an unique solution in most situations [11], where the
equality in (29) holds iff

$\bm{\mathsfbr{R}}_{\bm{\mathsfbr{X}}}=\mathbb{E}(\bm{\mathsfbr{R}}_{\bm{\mathsfbr{X}}})=\widetilde{\bm{R}}_{\bm{\mathsfbr{X}}}^{\rm{SC}}.$
(32)

That is, the sample covariance matrix $\bm{\mathsfbr{R}}_{\bm{\mathsfbr{X}}}$
becomes a deterministic matrix when the global minimum of the CRB
$\epsilon_{\min}$ is attained. One may then wonder whether there is any
communication DoF left in the ISAC signal at $P_{\rm SC}$. The answer is non-
trivially yes. This is because a deterministic
$\bm{\mathsfbr{R}}_{\bm{\mathsfbr{X}}}$ does not necessarily imply a
deterministic $\bm{\mathsfbr{X}}$. The latter may still be a random signal
conveying information, given by

$\bm{\mathsfbr{X}}^{\rm
SC}=\sqrt{T}(\widetilde{\bm{R}}_{\bm{\mathsfbr{X}}}^{\rm{SC}})^{\frac{1}{2}}\bm{\mathsfbr{Q}}=\sqrt{T}\bm{U}_{\rm
s}\bm{\varLambda}_{\rm s}^{\frac{1}{2}}\bm{\mathsfbr{Q}},$ (33)

where $\bm{\mathsfbr{Q}}\in\mathbb{C}^{\mathsfbr{M}_{\rm SC}\times T}$ is a
random semi-unitary matrix such that $\bm{\mathsfbr{Q}}\bm{\mathsfbr{Q}}^{\rm
H}=\bm{I}$, where $\mathsfbr{M}_{\rm SC}={\rm
rank}(\widetilde{\bm{R}}_{\bm{\mathsfbr{X}}}^{\rm{SC}})$. Since
$(\widetilde{\bm{R}}_{\bm{\mathsfbr{X}}}^{\rm{SC}})^{\frac{1}{2}}$ is
deterministic, the communication DoFs at $P_{\rm SC}$ are only contributed by
the randomness of $\bm{\mathsfbr{Q}}$.

We are now ready to characterize the achievable communication rate at $P_{\rm
SC}$. That is, seeking for the optimal distribution
$p_{\bm{\mathsfbr{Q}}}({\bm{Q}})$ over the set of all $\mathsfbr{M}_{\rm
SC}\times T$ semi-unitary matrices, namely, the Stiefel manifold
$\mathcal{S}(T,\mathsfbr{M}_{\rm SC})$, such that the mutual information
$I(\bm{\mathsfbr{Q}};\bm{\mathsfbr{Y}}_{\rm c}|\bm{\mathsfbr{H}}_{\rm c})$ is
maximized. In the high-SNR regime, this is equivalent to solving the sphere
packing problem over the Stiefel manifold, where the optimal
$p_{\bm{\mathsfbr{X}}}(\bm{X})$ is uniform distribution, leading to the
following asymptotic achievable rate [11]

$R_{\rm SC}=\mathbb{E}\Big{\\{}\Big{(}1-\frac{\mathsfbr{M}_{\rm
SC}}{2T}\Big{)}\log|\sigma_{\rm c}^{-2}\bm{\mathsfbr{H}}_{\rm
c}\widetilde{\bm{R}}_{\bm{\mathsfbr{X}}}^{\rm SC}\bm{\mathsfbr{H}}_{\rm
c}^{\rm H}|+\mathsfbr{c}_{0}\Big{\\}}+O(\sigma_{\rm c}^{2}),$ (34)

where

$\mathsfbr{c}_{0}=\frac{\mathsfbr{M}_{\rm
SC}}{T}\Big{[}\Big{(}T\\!-\\!\frac{\mathsfbr{M}_{\rm
SC}}{2}\Big{)}\log\frac{T}{e}\\!-\\!\log\Gamma(T)\\!+\\!\log(2\sqrt{\pi})\Big{]}$
(35)

converges to zero as $T\rightarrow\infty$.

Observe immediately that when $T\rightarrow\infty$, the communication DoFs are
lossless, since even a Gaussian matrix would have asymptotically orthogonal
rows with the increase of $T$, making it asymptotically equivalent to a semi-
unitary matrix.

Figure 8: Graphical illustration of the projector metaphor.

###  IV-F The Two-fold S&C Tradeoff: A Projector Metaphor

The above results clearly demonstrate the effect of both ST and DRT in an ISAC
system. By comparing the left-most parts of $\bm{\mathsfbr{X}}^{\rm CS}$ and
$\bm{\mathsfbr{X}}^{\rm SC}$, we see that the communication- and sensing-
optimal ISAC signals should be aligned to $\bm{U}_{\rm c}$ and $\bm{U}_{\rm
s}$, respectively, which may be regarded as the orthogonal bases of
communication and sensing subspaces. Then the ST is nothing but to flow the
signal power from $\operatorname{span}(\bm{U}_{\rm c})$ to
$\operatorname{span}(\bm{U}_{\rm s})$, perfectly fitting the picture of the
“ISAC torch” metaphor. More interestingly, by comparing the right-most parts
of $\bm{\mathsfbr{X}}^{\rm CS}$ and $\bm{\mathsfbr{X}}^{\rm SC}$, e.g.,
$\bm{\mathsfbr{D}}$ and $\bm{\mathsfbr{Q}}$, we see that communication- and
sensing-optimal signals adopt Gaussian and semi-unitary codebooks,
respectively, which again reflects the DRT. That is, when the ISAC system
moves along the Pareto frontier from $P_{\rm CS}$ to $P_{\rm SC}$,
$p_{\bm{\mathsfbr{X}}}(\bm{X})$ gradually changes from Gaussian to less random
distribution, and eventually becomes uniform distribution over semi-unitary
matrices. In that sense, the DRT considered in Fig. 5 is simply a one-
dimensional special case, since the semi-unitary matrix reduces to constant-
modulus signals in its scalar form, namely, the BPSK modulation in Fig. 5(e).

In addition to the tradeoff between input distributions, the DRT may also be
observed in the attainable communication and sensing DoF at the two corner
points. At $P_{\rm CS}$, the communication subsystem apparently acquires the
full DoF of the Gaussian channel, namely, $\mathsfbr{M}_{\rm CS}$, which is
the rank of $\widetilde{\bm{R}}_{\bm{\mathsfbr{X}}}^{\rm CS}$. As shown in
(28), due to the Gaussian signaling, the sensing subsystem suffers from the
DoF loss, or, equivalently, the reduction in the number of individual
observations of the target, which is up to $\min\\{K,\mathsfbr{M}_{\rm
CS}\\}$. At $P_{\rm SC}$, the sensing subsystem attains the full DoF of $T$
thanks to the deterministic signal sample covariance matrix. In contrast, the
semi-unitary signaling incurs a communication DoF loss of
$\frac{\mathsfbr{M}_{\rm SC}^{2}}{2T}$, as indicated by (34).

To achieve the S&C performance tradeoff, it is critical to determine the
steering direction of the ISAC signal. More importantly, what kind of codebook
is transmitted through that direction also matters. With the above
understanding, we may now correct the “ISAC torch” metaphor with a more
comprehensive picture, i.e., the “ISAC projector”. As shown in Fig. 8: A child
(the ISAC Tx) holds a projector, and wishes to simultaneously illuminate a
target (sensing) while sending an image to a receiver (communication). To form
an image, the brightness of each pixel may be used to convey information.
Nevertheless, those dark pixels result in imperfect illumination of the
target.

##  V DRT and ST in Practical ISAC Systems

###  V-A DRT: Sensing with Random Signals

The lessons learned from the DRT inspired us to rethink the practical design
philosophy of ISAC systems. In particular, one has to take the randomness of
communication data into account while conceiving a sensing strategy, which is
a unique challenge emerged in the context of ISAC. In this subsection, we
investigate a novel precoding design for sensing with random signals.

Let us consider again the sensing model (18b), with the sensing parameters
being the entries of the channel matrix $\bm{\mathsfbr{H}}_{\rm s}$, i.e.,
$\bm{\seta}=\operatorname{vec}(\bm{\mathsfbr{H}}_{\rm s})$. For a given
instance of $\bm{\mathsfbr{X}}$, the linear minimum MSE (LMMSE) estimator
reads

$\hat{\bm{\mathsfbr{H}}}_{\rm{LMMSE}}=\bm{\mathsfbr{Y}}_{\rm
s}\left(\bm{\mathsfbr{X}}^{\rm
H}\bm{R}_{\bm{\mathsfbr{H}}}\bm{\mathsfbr{X}}+\sigma_{\rm s}^{2}N_{\rm
s}\bm{I}\right)^{-1}\bm{\mathsfbr{X}}^{\rm H}\bm{R}_{\bm{\mathsfbr{H}}},$ (36)

where $\bm{R}_{\bm{\mathsfbr{H}}}=\mathbb{E}(\bm{\mathsfbr{H}}_{\rm s}^{\rm
H}\bm{\mathsfbr{H}}_{\rm s})$ represents the channel correlation matrix. The
resulting estimation error is given as

$\xi_{\bm{\mathsfbr{H}}_{\rm
s}|\bm{\mathsfbr{X}}}=\operatorname{tr}\Big{[}\Big{(}\bm{R}_{\bm{\mathsfbr{H}}}^{-1}+\frac{1}{\sigma_{\rm
s}^{2}N_{\rm s}}\bm{\mathsfbr{X}}\bm{\mathsfbr{X}}^{\rm
H}\Big{)}^{-1}\Big{]}.$ (37)

Once again, the estimation error relies on the instantaneous realization of
the random ISAC signal $\bm{\mathsfbr{X}}$. To depict the average sensing
performance, we take the expectation of (37) over $\bm{\mathsfbr{X}}$,
yielding

$\xi_{\bm{\mathsfbr{H}}_{\rm
s}}=\mathbb{E}\left\\{\operatorname{tr}\Big{[}\Big{(}\bm{R}_{\bm{\mathsfbr{H}}}^{-1}+\frac{1}{\sigma_{\rm
s}^{2}N_{\rm s}}\bm{\mathsfbr{X}}\bm{\mathsfbr{X}}^{\rm
H}\Big{)}^{-1}\Big{]}\right\\}.$ (38)

We refer the term (38) to as the ergodic LMMSE (E-LMMSE) [25], as it may be
understood as a time average over different realizations of
$\bm{\mathsfbr{X}}$. Obviously, it is lower-bounded by the Miller-Chang CRB in
(19).

We now investigate a specific form of the ISAC signal by letting
$\bm{\mathsfbr{X}}=\bm{W}\bm{\mathsfbr{S}}$, where
$\bm{W}\in\mathbb{C}^{M\times M}$ is a precoding matrix, and
$\bm{\mathsfbr{S}}=\left[{\bm{\mathsfbr{s}}_{1}},{\bm{\mathsfbr{s}}_{2}},\ldots,{\bm{\mathsfbr{s}}_{T}}\right]\in\mathbb{C}^{M\times
T}$ contains column-wise i.i.d. data symbols, satisfying
$\mathbb{E}(\bm{\mathsfbr{s}}_{i})=\bm{0}$ and
$\mathbb{E}({\bm{\mathsfbr{s}}_{i}}\bm{\mathsfbr{s}}_{i}^{\rm H})=\bm{I}$. A
fundamental question to ask is: What is the optimal precoder $\bm{W}$ that
minimizes the E-LMMSE? In classical MIMO radar waveform design, strictly
orthogonal signals are typically employed, namely,
$\frac{1}{T}\bm{\mathsfbr{S}}\bm{\mathsfbr{S}}^{\rm H}=\bm{I}$, where
$\bm{\mathsfbr{S}}$ is a semi-unitary matrix, corresponding to the $P_{\rm
SC}$ point discussed in Sec. IV. In such a case, it is known that the LMMSE-
optimal precoder has a water-filling structure given by [26]

$\bm{W}_{\mathsf{WF}}=\sqrt{\frac{\sigma_{\rm s}^{2}{N_{\rm
s}}}{T}}\bm{Q}\Big{[}\Big{(}\mu_{0}\bm{I}-\bm{\varLambda}^{-1}\Big{)}^{+}\Big{]}^{\frac{1}{2}},$
(39)

where $\bm{Q}$ and $\bm{\varLambda}$ contain the eigenvectors and eigenvalues
of $\bm{R}_{\bm{\mathsfbr{H}}}$, respectively, and $\mu_{0}$ is a constant
meeting the power constraint $\|\bm{W}_{\mathsf{WF}}\|_{F}^{2}=P_{\rm T}$.
However, in the ISAC scenario, the water-filling solution (39) may not be
optimal due to the randomness in $\bm{\mathsfbr{S}}$. Evidently, applying the
Jensen’s inequality to (38) yields

$\displaystyle\xi_{\bm{\mathsfbr{H}}_{\rm s}}$
$\displaystyle=\mathbb{E}\left\\{\operatorname{tr}\Big{[}\Big{(}\bm{R}_{\bm{\mathsfbr{H}}}^{-1}+\frac{1}{\sigma_{\rm
s}^{2}N_{\rm s}}\bm{W}\bm{\mathsfbr{S}}\bm{\mathsfbr{S}}^{\rm H}\bm{W}^{\rm
H}\Big{)}^{-1}\Big{]}\right\\}$
$\displaystyle\overset{(a)}{\geqslant}\mathrm{tr}\Big{[}\Big{(}\bm{R}_{H}^{-1}+\frac{1}{\sigma_{\rm
s}^{2}N_{\rm
s}}\mathbb{E}_{\bm{S}}\Big{\\{}\bm{W}\bm{\mathsfbr{S}}\bm{\mathsfbr{S}}^{\rm
H}\bm{W}^{\rm H}\Big{\\}}\Big{)}^{-1}\Big{]}$
$\displaystyle\overset{(b)}{=}\mathrm{tr}\Big{[}\Big{(}\bm{R}_{H}^{-1}+\frac{T}{\sigma_{\rm
s}^{2}N_{\rm s}}\bm{W}\bm{W}^{H}\Big{)}^{-1}\Big{]},$ (40)

where (a) is due to the convexity of $\xi_{\bm{\mathsfbr{H}}_{\rm
s}|\bm{\mathsfbr{X}}}$ with respect to
$\bm{\mathsfbr{S}}\bm{\mathsfbr{S}}^{\rm H}$, and (b) holds from the fact
$\mathbb{E}(\bm{\mathsfbr{S}}\bm{\mathsfbr{S}}^{\rm H})=T\bm{I}$. The equality
in (a) holds only asymptotically444To the best of our knowledge, most existing
ISAC precoding literature overlooked the data randomness by assuming
$\frac{1}{T}\bm{\mathsfbr{S}}\bm{\mathsfbr{S}}^{\rm H}\approx\bm{I}$. for
$\frac{T}{M}\to\infty$, since
$\frac{1}{T}\bm{\mathsfbr{S}}\bm{\mathsfbr{S}}^{\rm H}\neq\bm{I}$ due to the
i.i.d. assumption in the columns of $\bm{\mathsfbr{S}}$.

It turns out that the water-filling solution (39) minimizes the Jensen lower
bound in (V-A), rather than $\xi_{\bm{\mathsfbr{H}}_{\rm s}}$ itself.
Specifically, when $T$ is comparable to $M$, and when non-unitary codebooks,
e.g., Gaussian codebook (where $\bm{\mathsfbr{S}}\bm{\mathsfbr{S}}^{\rm H}$
follows Wishart distribution), are employed, the orthogonality of
$\bm{\mathsfbr{S}}$ breaks and the Jensen bound is no longer tight. To see
this, we show an example in Fig. 9a with $M=64,N_{\rm s}=32$, where the water-
filling precoder (39) is applied to both Gaussian distributed and semi-unitary
data matrix. The Jensen bound (semi-unitary sensing performance) is attained
when $T\geqslant 2048$. To account for the random nature of the ISAC signal,
one needs to develop novel precoders that directly minimize the E-LMMSE,
rather than its Jensen lower bound. Here we briefly introduce two possible
methodologies, namely, data-dependent and data-independent designs, both of
which yield better performance than that of the water-filling precoder [25].

(a) $M=64,N_{\rm s}=32$

(b) $M=64,N_{\rm s}=32,T=32$

Figure 9: Sensing with random signals. (a). Tightness of the Jensen bound with
an increasing block length. (b). Estimation performance of the DRT-aware
precoding designs.

####  V-A1 Data-Dependent Precoding

Despite its randomness, the fact that $\bm{\mathsfbr{S}}$ is known at the ISAC
Tx enables a precoding design based on given instances of $\bm{\mathsfbr{S}}$.
Let us denote the $n$th realization of $\bm{\mathsfbr{S}}$ as $\bm{S}_{n}$,
and the to-be-designed precoder as $\bm{W}_{n}$. One may then directly
minimize $\xi_{\bm{\mathsfbr{H}}_{\rm s}|\bm{\mathsfbr{X}}}$ based on the
known $\bm{S}_{n}$ by solving the following problem

$\min_{\|\bm{W}_{n}\|_{F}^{2}=P_{\rm
T}}\operatorname{tr}\Big{[}\Big{(}\bm{R}_{\bm{\mathsfbr{H}}}^{-1}+\frac{1}{\sigma_{\rm
s}^{2}N_{\rm s}}{\bm{W}_{n}}{\bm{S}_{n}}{\bm{S}_{n}^{\rm H}}{\bm{W}_{n}^{\rm
H}}\Big{)}^{-1}\Big{]}.$ (41)

While problem (41) is non-convex, it is provable that it admits an optimal
closed-form solution [27]. Consequently, one may minimize the E-LMMSE
$\xi_{\bm{\mathsfbr{H}}_{\rm s}}$ by minimizing every instance of
$\xi_{\bm{\mathsfbr{H}}_{\rm s}|\bm{\mathsfbr{X}}}$.

####  V-A2 Data-Independent Precoding

The optimality of the data-dependent precoder is achieved at the price of high
complexity, as one has to solve for $\bm{W}_{n}$ for every instance
$\bm{S}_{n}$. To ease the computational burden, an alternative option would be
to conceive a data-independent precoder, where a single $\bm{W}$ is leveraged
for every instance of $\bm{\mathsfbr{S}}$, leading to the following stochastic
optimization problem

$\min_{\|\bm{W}_{n}\|_{F}^{2}=P_{\rm
T}}\mathbb{E}\left\\{\operatorname{tr}\Big{[}\Big{(}\bm{R}_{\bm{\mathsfbr{H}}}^{-1}+\frac{1}{\sigma_{\rm
s}^{2}N_{\rm s}}\bm{W}\bm{\mathsfbr{S}}\bm{\mathsfbr{S}}^{\rm H}\bm{W}^{\rm
H}\Big{)}^{-1}\Big{]}\right\\},$ (42)

which can be solved via stochastic gradient descent (SGD) algorithm in an
offline manner, where massive training data samples may be locally generated
based on the adopted communication codebook.

To validate the performance of the proposed sensing precoding design for
random signals, we show their corresponding estimation errors in Fig. 9b with
$M=64,T=32$ and $N_{\rm s}=32$, where a Gaussian codebook is employed again
for generating $\bm{\mathsfbr{S}}$. As expected, both precoding designs
significantly outperform the classical water-filling approach (39). In
particular, the computationally expensive data-dependent design achieves
better average estimation performance (0.4-1.4 dB gain) compared to its data-
independent counterpart, while the latter attains a favorable performance-
complexity tradeoff. This provides strong evidence that the data randomness is
non-negligible in ISAC signaling. To achieve the S&C performance boundary,
DRT-aware ISAC precoding techniques are yet to be implemented in practical
ISAC systems.

###  V-B Frequency-domain ST: Valuating Sensing Resources

Let us now turn our attention to the ST. In previous sections, we have
illustrated the ST using spatial-domain examples. In these examples, roughly
speaking, the ST is manifested as the preference for direct-illuminating
beams, which holds for both the communication user and the sensing target. By
contrast, when we consider non-spatial scenarios, the ST might take a less
intuitive form, but it would also be more interesting and non-trivial.

To see this, let us consider the example of ranging waveform design,
characterized by the simple observation model

$y(t)=s(t-\tau)+n(t),$

where $y(t)$, $s(t)$ and $n(t)$ represent the received signal, the transmitted
signal, and the noise with constant power spectral density (PSD) $N_{0}$,
respectively. The term $\tau=d/c$ denotes the propagation delay, with $c$
being the propagation speed, and $d$ being the distance to be estimated. For
this model, the CRB reads

$\mathbb{E}\\{(d-\hat{d})^{2}\\}\geqslant c^{2}(8\pi^{2}\beta^{2}{\rm
SNR})^{-1},$ (43)

where $\beta=\int_{-\infty}^{\infty}f^{2}|S(f)|^{2}{\rm
d}f/\int_{-\infty}^{\infty}|S(f)|^{2}{\rm d}f$ is referred to as the “ root-
mean-square (RMS) bandwidth” [28], while ${\rm
SNR}=\frac{1}{N_{0}}\int_{-\infty}^{+\infty}s^{2}(t){\rm d}t$.

What does (43) imply? Upon assuming that the signal is constrained to reside
in the frequency interval of $[0,f_{\rm high}]$, we find immediately that the
CRB-optimal signal is in fact a sinusoidal signal with frequency $f_{\rm
high}$, which maximizes the RMS bandwidth. The intuition behind this result is
that, as long as the integer ambiguity can be resolved, ranging methods based
on carrier phase sensing would yield the optimal performance, as has been
recognized in the literature of global navigation satellite system
(GNSS)-based positioning [29].

Figure 10: The PSDs of ZZB-optimal waveforms at different SNRs.

Of course, if not supplemented by further information, the integer ambiguity
of a single-tone signal can never be resolved. However, CRB is known to be
unable to capture the ambiguity phenomenon. To this end, we may use the Ziv-
Zakai bound (ZZB) given by [30]

$\mathbb{E}\left\\{(d\\!-\\!\hat{d})^{2}\right\\}\\!\geqslant\\!\int_{0}^{\epsilon_{\max}}xQ\Big{(}\sqrt{2^{-1}{\rm
SNR}(1\\!-\\!\widetilde{R}(x))}\Big{)}{\rm d}x,$ (44)

where $\epsilon_{\max}$ is the maximum possible ranging error, $Q(\cdot)$
denotes the Marcum Q-function, $\widetilde{R}(x)$ denotes the normalized
autocorrelation function (ACF) defined as $\widetilde{R}(x)=R(x/c)/R(0)$, with
$R(\tau)=\int_{-\infty}^{\infty}s(t-\tau)s(t){\rm d}t$. Although it does not
admit a closed-form expression, we may observe from (44) that the ZZB can
reflect the ambiguity phenomenon: It is a decreasing function with respect to
$(1-\widetilde{R}(x))$. Since the normalized ACF $\widetilde{R}(x)$ achieves
its maximum at $\widetilde{R}(0)=1$, $(1-\widetilde{R}(x))$ can be viewed as a
measure of the sidelobe level. Intuitively, given a fixed noise level, a
higher sidelobe level would make it less distinguishable from the mainlobe,
and hence cause larger errors.

With the aid of ZZB, we are now able to understand the behaviour of waveforms
that achieve (near-) optimal sensing performance. Numerically computed PSDs of
ZZB-optimal waveforms (using the method in [30]) are plotted in Fig. 10.
Observe that as the total SNR increases, the ZZB-optimal waveform would
refocus its power from the low-frequency band to the high-frequency band. The
reason is that when the total SNR is sufficiently high, one may effectively
resolve the ambiguity caused by relatively high sidelobes, and hence the power
is focused on the high-frequency band. By contrast, when the total SNR is
lower, one needs to lower the sidelobe level to fight against the ambiguity
issue, which would inevitably widen the mainlobe, leading to low-frequency
waveforms.555The curvature of the ACF mainlobe at $\tau=0$ is proportional to
the RMS bandwidth [28]. Therefore, wider mainlobes correspond to lower-
frequency waveforms.

(a) Communication Subspace

(b) Sensing Subspace

Figure 11: Frequency-domain communication and sensing subspaces (for the
ranging task) in low- and high-SNR regimes.

Note that this is a remarkable observation suggesting that sensing tasks have
a unique preference on the subspaces where the ISAC signal resides. This is in
stark contrast to communication tasks, for which the optimal frequency-domain
power allocation scheme is the water-filling strategy. In particular, the
water-filling strategy assigns more power to frequency bands having higher
SNR, and might even abandon some low-SNR bands when the total power constraint
is stringent, or equivalently, when the total SNR is relatively low. In the
language of ST, we may say that in the frequency domain, the communication
subspace corresponds to those frequency bands with high SNR. By contrast, the
sensing subspace is not solely dependent on the SNR; rather, as the total SNR
increases, the sensing subspace would move from the low-frequency band to the
high-frequency band, as portrayed in Fig. 11.

We may obtain further insights from the perspective of resource allocation.
One of the resources of communication tasks is the DoF, which is manifested as
the bandwidth in the frequency domain. For communication tasks, the value of
the frequency bands only depends on their _quality_ (in terms of SNR).
However, for sensing tasks, the value of frequency bands depends not only on
their quality, but on their _location_ as well. In light of this, we may say
that the DoF is not, in its nature, a sensing resource. We are thus motivated
to ask the following question: _What do we really mean when we say “sensing
resources”?_

##  VI Concluding Remarks and Open Challenges

Unfortunately, at the moment of writing, we do not have a well-stated answer
to this question. After all, in contrast to communication tasks always aiming
to deliver information, sensing tasks have vastly diverse purposes, and hence
may rely on different resources. Apart from this question, many important
problems remain open in the context of S&C tradeoffs. To name a few:

  1. 1.

How does the DRT manifest itself under generic sensing performance metrics?

  2. 2.

How do we characterize the S&C tradeoff when the channels are not memoryless?
What are the performances of online and offline estimators in such scenarios?

  3. 3.

How do we design ISAC systems that are capable of achieving the entire
capacity-distortion boundary (not just the corner points)?

  4. 4.

Can we unify S&C performance metrics?

These challenging questions remind us that there is still a long way to go
before the merit of ISAC can be fully understood and utilized. Nevertheless,
the ST-DRT decomposition (i.e. the “projector metaphor”) is likely to be a
useful meta-intuition in future investigations of ISAC systems: the
fundamental tradeoff in ISAC is manifested as the preference discrepancies
between S&C tasks, concerning both the resources (ST) and the signal patterns
(DRT).

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  *[ISAC]: integrated sensing and communications
  *[S&C]: sensing and communications
  *[ITU]: international telecommunication union
  *[DoF]: degree of freedom
  *[MIMO]: multiple-input multiple-output
  *[OFDM]: orthogonal frequency-division multiplexing
  *[BS]: base station
  *[ST]: subspace tradeoff
  *[SNRs]: signal-to-noise ratio
  *[MSE]: mean-squared error
  *[CFAR]: constant false-alarm rate
  *[QAM]: quadrature amplitude modulation
  *[CRB]: Cramér-Rao bound
  *[SDMC-DF]: state-dependent memoryless channel with delayed feedback
  *[B-A]: Blahut-Arimoto
  *[SISO]: single-input single-output
  *[MMSE]: minimum mean squared error
  *[BPSK]: binary phase-shift keying
  *[DRT]: deterministic-random tradeoff
  *[Tx]: transmitter
  *[Rx]: receiver
  *[PSD]: power spectral density
  *[RMS]: root-mean-square
  *[GNSS]: global navigation satellite system
  *[PSDs]: power spectral density
  *[ZZB]: Ziv-Zakai bound
  *[ACF]: autocorrelation function
  *[SNR]: signal-to-noise ratio