# Sensing using Coded Communications Signals

Sundar Aditya, , Onur Dizdar, , Bruno Clerckx, , and Xueru Li This work was
supported by Huawei Device Co., Ltd.S. Aditya, O. Dizdar and B. Clerckx are
with the Dept. of Electrical and Electronic Engineering, Imperial College
London, London SW7 2AZ, U.K. (e-mail: {s.aditya, o.dizdar,
b.clerckx}@imperial.ac.uk).X. Li is with Huawei Device Co., Ltd., Beijing,
China (email<EMAIL_ADDRESS>
###### Abstract

A key challenge for a common waveform for Integrated Sensing and
Communications (ISAC) – widely seen as an attractive proposition to achieve
high performance for both functionalities, while efficiently utilizing
available resources – lies in leveraging information-bearing channel-coded
communications signals (c.c.s) for sensing. In this paper, we investigate the
sensing performance of c.c.s in (multi-user) interference-limited operation,
and show that it is limited by _sidelobes_ in the range-Doppler map, whose
form depends on whether the c.c.s modulates a single-carrier or OFDM waveform.
While uncoded communications signals – comprising a block of $N$ i.i.d zero-
mean symbols – give rise to _asymptotically_ (i.e., as $N\rightarrow\infty$)
_zero sidelobes_ due to the law of large numbers, it is not obvious that the
same holds for c.c.s, as structured channel coding schemes (e.g., linear block
codes) induce dependence across codeword symbols. In this paper, we show that
c.c.s also give rise to asymptotically zero sidelobes – for both single-
carrier and OFDM waveforms – by deriving upper bounds for the tail
probabilities of the sidelobe magnitudes that decay as $\exp(-O(\mbox{code
rate}\times\mbox{block length}))$. This implies that for any code rate, c.c.s
are effective sensing signals that are robust to multi-user interference at
_sufficiently large_ block lengths, with negligible difference in performance
based on whether they modulate a single-carrier or OFDM waveform. We verify
the latter implication through simulations, where we observe the sensing
performance (characterized by the detection and false-alarm probabilities) of
a QPSK-modulated c.c.s (code rate $=120/1024$, block length $=1024$ symbols)
to match that of a comparable interference-free FMCW waveform even at high
interference levels (signal-to-interference ratio of $-11{\rm dB}$), for both
single-carrier and OFDM waveforms.

###### Index Terms:

Integrated Sensing and Communications (ISAC), Opportunistic sensing, Sensing
in beyond 5G networks, Sensing interference management, Correlation properties
of coded communications signals (c.c.s), Interference sidelobes

##  I Introduction

A combination of factors, such as (a) supporting emerging applications that
require both communications and sensing functionalities (e.g.,
augmented/virtual reality, autonomous vehicles, etc.) in beyond 5G networks,
and (b) the channel propagation characteristics at mmWave and THz frequency
bands, have provided the impetus for the ongoing research efforts on
integrated sensing and communications (ISAC). In this regard, the waveform
used for each of the functionalities is a crucial aspect of system design.
Broadly, there are two paradigms of waveform design for ISAC systems:

  * a)

_Separate waveforms_ : Such an approach allows commercial stakeholders, who
may have expertise in either communications or sensing but not both, to
continue to independently design their hardware and systems, based on the
desirable waveform properties for each functionality (e.g., FMCW for radar and
OFDM for communications). The individual waveforms can either be transmitted
orthogonally [1, 2] or superimposed [3], provided the mutual interference is
effectively mitigated.

  * b)

_Common waveform_ : The efficient utilization of valuable spectral, hardware
and energy resources is the driving factor behind this approach [4], with [5,
6, 7] being good examples that focus on optimizing the allocation of resources
(e.g., power, subcarriers etc.) between communications and sensing
functionalities for a common OFDM waveform. In contrast to these, a special
case of a common waveform involves sensing using _only_ the reference signals
of a primarily communications waveform, such as IEEE 802.11ad frames [8, 9,
10, 11]. While such a solution has the advantage of not requiring extensive
standardization efforts, it is sub-optimal as, in principle, even the
communications payload/signal within a common waveform can be used for sensing
in a monostatic configuration, since it is known apriori at the transmitting
node.

In this paper, we explore the sensing potential of (channel) coded
communications signals (c.c.s) – i.e., the communications payload – when
embedded in a common ISAC waveform. A key question w.r.t a common ISAC
waveform is whether to adopt a single- or multi-carrier waveform, with most
discussion of the benefits of one over the other focusing on issues like
signal processing complexity (both communications and sensing), easy
integration with existing standards, PAPR etc. [12, 13]. Our focus on the
single- versus multi-carrier question is restricted to the impact, if any, of
channel coding on the sensing performance of each waveform, especially in
interference-limited environments.

The principle of using known data symbols for sensing has been explored for a
variety of candidate waveforms, such as single-carrier [14], OFDM [15, 16, 17,
5, 18], and OTFS (Orthogonal Time Frequency and Space)[19]. Among these, [15,
5] and [19] consider uncoded communications signals – unrepresentative of
typical data payloads that are subject to channel coding – while the
assumptions around channel coding in [14, 16] and [17] are unclear. In
particular, the latter two references are experimental studies exploring the
sensing potential of 5G NR signals, and hence, even if channel coding was
implied, its impact on the sensing performance was not investigated. Finally,
none of the above references address the impact of interference on the sensing
performance of c.c.s.

TX$/$Radar 1⋮TX$/$Radar $M_{r}$RX 1⋮RX $M_{r}$Comm. link 1Comm. link
$M_{r}$Comm. intf.Target 1⋮Target $N_{\rm tar}$Desired radar returnsSensing
intf.(a)(b)TX/Radar 1TX/Radar 2TargetRX 1(RSU)RX 2(RSU)Comm. link 1Comm. link
2

Figure 1: (a) Using a common transmit waveform, TX $i$ communicates with RX
$i$ ($i=1,\cdots,M_{r})$, while all TXs/radars simultaneously sense a common
target scene non-cooperatively. This gives rise to distinct communications and
sensing interference, where the latter often dominates the former due to a
combination of link geometry and/or greater pathloss experienced by the
desired radar returns; e.g., in (b) a vehicular network that can be
interpreted as a special case of (a), corresponding to $M_{r}=2$ and $N_{\rm
tar}=1$.

To motivate the discussion on the sensing potential of c.c.s in interference-
limited operating environments, consider the scenario in Fig. 1a, where TX $i$
communicates with RX $i$ ($i=1,\cdots,M_{r}$), while all the TXs also act as
monostatic radars to simultaneously sense a scene without cooperation, using a
common ISAC waveform. Such a setup gives rise to _multi-user_ communications
and sensing interference (highlighted in Fig. 1a), where the former is the
interference experienced at RX $i$ due to transmissions of TX $j(\neq i)$, and
the latter is the interference at radar receiver $i$ (colocated with TX $i$)
due to the transmissions of TX/radar $j(\neq i)$111Note that our definition of
multi-user sensing interference is distinct from adversarial interference,
like jamming, which we do not consider in this paper. For brevity, we drop the
‘multi-user’ prefix when discussing sensing interference from here on.. In
general, the two types of interference are distinct in their profiles, and we
are especially interested in cases where the sensing interference _dominates_
the communications interference; an example of such a scenario could occur in
a vehicular network, as depicted in Fig. 1b, where a pair of vehicular
TXs/radars (i.e., $M_{r}=2$) are facing each other on opposite lanes and
sensing the road ahead of them, while also communicating with their RXs, which
may be the nearest road side units (RSUs) located on their respective
sidewalks. The geometry of this scene is such that even with large antenna
arrays, beamforming may be able to effectively suppress communications – but
not sensing – interference, as even with narrow beams, there could still be
significant signal leakage along the direct (line-of-sight) path linking the
two TXs. Geometry apart, the $R^{-4}$ attenuation of the desired radar returns
– in contrast to the $R^{-2}$ attenuation for the interference signal along
the direct path between two radars – also makes it more likely that sensing
interference dominates communications interference in other ISAC use cases.
Hence, at first glance, it is tempting to allocate the available
communications resources (e.g., time and frequency bands) orthogonally across
TXs to mitigate the sensing interference, at the expense of communications
spectral efficiency (measured in bits/s/Hz). However, if c.c.s happen to be
_good_ sensing signals – meaning, achieving high sensing performance, while
being robust to sensing interference – then, the communications spectral
efficiency in Fig. 1 can be significantly enhanced through a reuse factor of
one.

To provide intuition for why communications signals might be good sensing
signals, consider a collection of $M_{r}$ _uncoded_ communications signals –
$\\{v^{(i)}[n]:n=0,\cdots,N-1;~{}i=1,\cdots,M_{r}\\}$ – comprising i.i.d zero
mean, unit-energy symbols (e.g., QPSK). From the law of large numbers (LLN),
it can easily be shown that these signals have _asymptotically favourable
correlation properties_ (i.e., auto-correlation function tending to the
$\delta$-function, and mutual cross-correlation function tending to the zero
function, as $N\rightarrow\infty$), which – as shown in Section II-B1 – makes
them good sensing signals, when embedded in a single-carrier waveform.
Similarly, the IDFT of $v^{(q)}[n]/v^{(i)}[n]~{}(q\neq i)$ identically tends
to 0, as $N\rightarrow\infty$, which – as shown in Section II-B2 – makes the
collection of signals $\\{v^{(i)}[n]:i=1,\cdots,M_{r}\\}$ robust to sensing
interference, when embedded in an OFDM waveform. However, structured state-of-
the-art channel coding schemes (e.g., linear block codes, such as Polar and
LDPC codes) induce statistical dependence across symbols, whose impact on the
above functions of c.c.s has not yet been characterized, to the best of our
knowledge. Hence, we tackle these questions in this paper, with a view to
determine if (a) c.c.s can indeed achieve high sensing performance, while
being robust to sensing interference, and (b) whether the nature of the
waveform (i.e., single- v/s multi-carrier) influences the answer to (a). Our
contributions are as follows:

  * 1.

Single-Carrier waveform:

    * –

We derive an upper bound for the tail probability of (the non-zero lags of)
the auto-correlation function of c.c.s that decays as $\exp(-O(rN))$, where
$r$ is the code rate and $N$ the block length (Theorem 1). As a result, c.c.s
have an asymptotically ideal auto-correlation function that converges in
distribution to (a suitably scaled) $\delta$-function for large block lengths
(Corollary 1), which makes them good sensing signals, when embedded in a
single-carrier waveform. These results makes mild assumptions regarding c.c.s
and importantly, do not depend on the code structure.

    * –

For a pair of c.c.s with rates $r_{i}$ and $r_{q}$, generated from independent
message signals, we derive an upper bound for the tail probability of their
cross-correlation function that decays as $\exp(-O(\max(r_{i},r_{q})N))$
(Corollary 2). As a result, c.c.s generated by independent message signals
have a cross-correlation function that converges in distribution to the zero
function for large block lengths (Corollary 4), which makes them robust to
sensing interference, when embedded in a single-carrier waveform.

    * –

For the above setup, but with linear codes, we derive lower bounds for the
tail probabilities of the auto- and cross-correlation functions of c.c.s that
also decay as $\exp(-O(rN))$ (Theorem 2 and Corollary 3, respectively). These
results, along with the previous bullet points, imply that for linear codes, a
faster rate of decay for these functions, as a function of $N$, is not
possible.

  * 2.

OFDM waveform:

    * –

For a pair of c.c.s – $s^{(i)}[n]$ and $s^{(q)}[n]$, with rates $r_{i}$ and
$r_{q}$, respectively – generated from independent message signals, we derive
an upper bound for the tail probability of the IDFT of $s^{(q)}[n]/s^{(i)}[n]$
that decays as $\exp(-O(\min(r_{i},r_{q})N)$ (Corollary 5). As a result, the
above IDFT converges in distribution to the zero function for large block
lengths (Corollary 7), which makes c.c.s robust to sensing interference, when
embedded in an OFDM waveform.

    * –

For the same setup as the previous bullet point, but with linear codes, we
derive a lower bound for the tail probability of the IDFT of
$s^{(q)}[n]/s^{(i)}[n]$, that also decays as $\exp(-O(rN))$ (Corollary 6). The
latter, along with the previous bullet point, implies that for linear codes, a
faster rate of decay for the IDFT, as a function of $N$, is not possible.

  * 3.

Implications:

    * –

The results under points 1 and 2 above suggest that for any code rate, c.c.s
can be effective sensing signals that are robust to sensing interference at
_sufficiently large_ block lengths, with negligible difference in performance
based on whether they modulate a single-carrier or OFDM waveform. We verify
this implication through simulations, where we observe that the sensing
performance (characterized by the detection and false-alarm probabilities) of
a QPSK-modulated c.c.s (code rate $=120/1024$, block length $=1024$ symbols)
matches that of a comparable interference-free FMCW waveform even at a signal-
to-interference ratio of $-11{\rm dB}$, for both single-carrier and OFDM
waveforms.

    * –

Thus, a common ISAC waveform (either single- or multi-carrier, but) largely
comprising coded data symbols is an effective sensing signal at large block
lengths that can also achieve high communications spectral efficiency.
Moreover, in multi-user ISAC scenarios with such a waveform, sensing
interference management essentially _takes care of itself_ for monostatic
radars, and is relatively simpler than communications interference management,
which is favourable for the evolution of existing wireless broadband networks
to support sensing applications.

This paper consists of five sections. In our system model in Section II, we
identify functions of c.c.s – depending on whether they modulate a single-
carrier or OFDM waveform – whose tail distribution needs to decay rapidly, for
c.c.s to be good sensing signals that are robust to sensing interference. In
Section III, we characterize the tail probabilities of these functions and
show that they decay rapidly at large block lengths. We verify the
effectiveness of c.c.s as sensing signals in interference-limited environments
through simulations in Section IV, where we see that the sensing performance
of c.c.s – for both single-carrier and OFDM waveforms – is at par with the
(interference-free) FMCW waveform, even at a signal-to-interference ratio
(SIR) of $-11{\rm dB}$. Finally, we conclude this paper in Section V with some
remarks on the implications of our results.

### Notation:

Vectors are represented by lower case bold letters;
${\mathcal{C}}{\mathcal{N}}(0,\sigma^{2})$ denotes the circularly symmetric
complex Gaussian distribution with mean zero and variance $\sigma^{2}$;
$\delta[\cdot]$ and $\mathbbm{1}(.)$ denote the discrete delta and the
indicator functions, respectively; ${\mathbb{P}}(\cdot)$ denotes probability;
${\mathbb{E}}[\cdot]$ denotes the expectation operator, with
${\mathbb{E}}_{X}[\cdot]$ explicitly denoting the expected value w.r.t the
random variable, $X$; $(\cdot)^{*}$ denotes complex conjugation, and
$\Re\\{\cdot\\}$ the real part. We assume that all discrete-time signals are
supported on ${\mathbb{Z}}$, the set of integers, with the understanding that
for a signal, $s[n]~{}(n=0,\cdots,N-1)$, with finite support, $s[n]=0$ for
$n\notin\\{0,\cdots,N-1\\}$. Finally, we introduce a few definitions that will
be used throughout the paper:

######  Definition 1 (Auto-correlation function).

For a discrete-time signal, $s[n]~{}(n=0,\cdots,N-1)$, its (aperiodic) auto-
correlation function at lag $l\in{\mathbb{Z}}$, denoted by $\chi(l;s[\cdot])$,
is given by:

$\displaystyle\chi(l;s[\cdot])$ $\displaystyle:=$
$\displaystyle\frac{1}{N}\sum_{n=-\infty}^{\infty}s[n]s^{*}[n-l].$ (1)

######  Definition 2 (Cross-correlation function).

For a pair of signals, $s_{1}[n]$ and $s_{2}[n]~{}(n=0,\cdots,N-1)$, their
(aperiodic) cross-correlation function at lag $l\in{\mathbb{Z}}$, denoted by
$\varrho(l;s_{1}[\cdot],s_{2}[\cdot])$, is given by:

$\displaystyle\varrho(l;s_{1}[\cdot],s_{2}[\cdot])$ $\displaystyle:=$
$\displaystyle\frac{1}{N}\sum_{n=-\infty}^{\infty}s_{1}[n]s_{2}^{*}[n-l].$ (2)

Clearly, by definition,
$\chi(l;s[\cdot])=\varrho(l;s_{1}[\cdot],s_{2}[\cdot])=0$ for $|l|\geq N$.
Hence, in this paper, we focus on the behavior of $\chi(l;s[\cdot])$ and
$\varrho(l;s_{1}[\cdot],s_{2}[\cdot])$ for $|l|\ll N$, for large $N$.
Alternately, for $N$-length signals, periodic auto- and cross-correlation
functions can be defined by replacing the index $n-l$ with $(n-l)\mod N$ in
(1) and (2), respectively. Our results in this paper are valid for these
functions, as well.

######  Definition 3 (Convergence in Distribution).

A sequence of random variables, $X_{1},\cdots,X_{n}$ converges in distribution
to a random variable, $X$ – denoted by
$\displaystyle\lim\limits_{n\rightarrow\infty}X_{n}\xrightarrow{d}X$ – if the
following condition is satisfied:

$\displaystyle\lim_{n\rightarrow\infty}{\mathbb{P}}(X_{n}>u)$ $\displaystyle=$
$\displaystyle{\mathbb{P}}(X>u),$ (3)

for all $u$ where the tail distribution, ${\mathbb{P}}(X>u)$, is continuous.

##  II System Model

Consider the scenario in Fig. 1a, where TX $i$ communicates with RX $i$
($=1,\cdots,M_{r}$), while all the TXs also act as monostatic radars to
simultaneously sense a scene comprising $N_{\rm tar}$ targets, without
cooperation. Let
${\mathcal{X}}^{(i)}_{M,N}:=\\{x^{(i)}[n,m]\in{\mathbb{C}}:m=1,\cdots,M;~{}n=0,\cdots,N-1\\}$
denote the (discrete) time-domain representation of the common ISAC waveform
transmitted by the $i$-th TX/radar, which can be interpreted as a collection
of $M$ signals, each of length $N$ – i.e.,
$\\{x^{(i)}[n,1],\cdots,x^{(i)}[n,M]:n=0,\cdots,N-1\\}$ – where $x^{(i)}[n,1]$
is transmitted first, followed by $x^{(i)}[n,2]$, and so on, as shown in Fig.
2a222Typically, in purely radar applications, $x^{(i)}[n,m]$ is identical
$\forall m\in\\{1,\cdots,M\\}$ (e.g., Fig. 2b). However, this is not a strict
requirement.. The indices, $n$ and $m$, represent two different time scales,
with the _fast time_ ($n$) capturing variations in the round-trip delays
associated with the target ranges, while the _slow time_ ($m$) is better
suited to capture variations in the Doppler frequencies associated with moving
targets. In general, for fixed $m$, each signal $x^{(i)}[n,m]$ is a function
of a c.c.s, $s^{(i)}[n,m]$, whose modeling is described below.

$0$$1$$\cdots$$N-1$$\cdots$$\cdots$Slow time$x^{(i)}[n,1]$Coherent Processing
Interval (CPI)$x^{(i)}[n,M]$Fast time(a) c.c.s. (transmitted by TX/radar
$i$)Time$\cdots$$\cdots$$\cdots$$x_{r}[n]$$x_{r}[n]$(b) FMCW

Figure 2: Sensing signal structure in time-domain for range-Doppler
estimation.

###  II-A c.c.s Model

${\mathbf{b}}_{m}^{(i)}$${\mathcal{G}}(\cdot)$Generator${\mathbf{c}}_{m}^{(i)}$$\pi(\cdot)$Interleaver$\tilde{{\mathbf{c}}}_{m}^{(i)}$
$m_{{\rm s}_{i}}$
---
bits/symb
$s^{(i)}[0,m],s^{(i)}[1,m],\cdots,s^{(i)}[N-1,m]$$u^{(i)}[0,m],u^{(i)}[1,m],\cdots,u^{(i)}[K_{i}-1,m]$

Figure 3: Model for a complex-valued c.c.s, that determines the signal
$x^{(i)}[n,m]$ transmitted by the $i$-th TX/radar in Fig. 2a in the $m$-th
slow time interval.

We model a c.c.s as shown in Fig. 3. At TX $i$, for each $m$, a binary message
vector, ${\mathbf{b}}_{m}^{(i)}\in\\{0,1\\}^{K_{i}m_{{\rm s}_{i}}}$, of length
$K_{i}m_{{\rm s}_{i}}$ is transformed into a binary codeword vector,
${\mathbf{c}}_{m}^{(i)}\in\\{0,1\\}^{Nm_{{\rm s}_{i}}}$, of length $Nm_{{\rm
s}_{i}}$ through a generator mapping, ${\mathcal{G}}(\cdot)$333For linear
block codes, ${\mathcal{G}}(\cdot)$ takes the form of either a generator
matrix or a parity check matrix.. The encoded bits are then interleaved and
mapped to a discrete, bounded constellation through a mapping that associates
$m_{{\rm s}_{i}}$ bits per symbol to produce a complex-valued c.c.s block,
$s^{(i)}[n,m]~{}(n=0,\cdots,N-1)$, of block length444In coding theory, the
block length typically refers to the number of bits in the codeword vector,
${\mathbf{c}}_{m}^{(i)}$ (equal to $Nm_{{\rm s}_{i}}$). Since the quantities
of interest in this paper take values in ${\mathbb{C}}$ (e.g., the auto-
correlation function), we slightly abuse the notation and refer to the number
of complex-valued symbols in the signal $s^{(i)}[n,m]$ (for fixed $m$) as the
block length., $N$. Let $u^{(i)}[n,m]~{}(n=0,\cdots,K_{i}-1)$ denote the
complex-valued message signal, of length $K_{i}$, obtained by mapping
${\mathbf{b}}_{m}^{(i)}$ to the same constellation. The code rate, $r_{i}$, of
$s^{(i)}[n,m]$ equals $K_{i}/N$.

In general, each c.c.s symbol, $s^{(i)}[n,m]$, is a _non-linear_ function of
one or more message symbols ($u^{(i)}[n_{1},m]$, $u^{(i)}[n_{2},m],\cdots$
etc.), which is difficult to characterize for most modulation and coding
schemes555Even for linear codes, the relationship between the message and
c.c.s symbols is non-linear, due to the symbol mapping operation.. Thus, for
tractability, we make some assumptions regarding the signals $u^{(i)}[n,m]$
and $s^{(i)}[n,m]$ below.

######  Assumption 1 (i.i.d. zero-mean message symbols).

The message symbols, $u^{(i)}[0,m],\cdots,u^{(i)}[K_{i}-1,m]$ are i.i.d with
zero mean and finite energy, as they are typically drawn uniformly from
bounded, symmetric constellations like QPSK, QAM, etc.

######  Assumption 2 (Systematic Encoding).

The first $K_{i}$ c.c.s symbols coincide with the message symbols, i.e.,
$s^{(i)}[n,m]=u^{(i)}[n,m]~{}(n=0,\cdots,K_{i}-1)$. This is referred to as
systematic encoding and any linear block code can be transformed into
systematic form through a linear mapping. Since widely used codes such as LDPC
and Polar Codes are linear block codes, we believe the systematic encoding
assumption666Note that we assume systematic encoding post-interleaving, as
illustrated in Fig. 3. Hence, $\pi(k)=k$ for $k\in\\{1,\cdots,Km_{{\rm
s}_{i}}\\}$, where $\pi(k)$ denotes the position of the $k$-th bit in
${\mathbf{c}}_{m}^{(i)}$ in the interleaved codeword,
$\tilde{{\mathbf{c}}}_{m}^{(i)}$. is reasonable.

######  Assumption 3 (Uncorrelated c.c.s symbols).

While the c.c.s symbols, $s^{(i)}[0,m],\cdots,s^{(i)}[N-1,m]$, are
statistically dependent, in general, we assume that they are mutually
uncorrelated. While Assumption 2 ensures that the first $K_{i}$ symbols are
uncorrelated, the encoding operation typically introduces correlation across
subsequent codeword bits (i.e., in ${\mathbf{c}}_{m}^{(i)}$ in Fig. 3), which
can be mitigated by interleaving; for instance, in Appendix A, we demonstrate
that for repetition codes, the interleaved codeword bits
($\tilde{{\mathbf{c}}}_{m}^{(i)}$ in Fig. 3) are asymptotically uncorrelated
for large $N$ (i.e., the correlation coefficient between a pair of arbitrarily
chosen bits in $\tilde{{\mathbf{c}}}_{m}^{(i)}$ tends to 0, as
$N\rightarrow\infty$). Hence, we believe this to be a reasonable assumption
concerning c.c.s symbols for most practical codes.

The relationship between the signals $s^{(i)}[n,m]$ and $x^{(i)}[n,m]$ depends
on whether the latter is a single- or multi-carrier waveform. We characterize
this relationship for a representative of each type, namely the conventional
single-carrier and the OFDM waveforms.

####  II-A1 Conventional Single-Carrier

The c.c.s symbols are transmitted as is in time-domain, i.e.,

$\displaystyle x^{(i)}[n,m]$ $\displaystyle=$ $\displaystyle s^{(i)}[n,m].$
(4)

####  II-A2 OFDM

The c.c.s symbols in $s^{(i)}[k,m]~{}(k=0,\cdots,N-1)$ are multiplexed in the
frequency domain over $N$ subcarriers to constitute the $m$-th OFDM symbol.
Thus, $x^{(i)}[n,m]$ and $s^{(i)}[k,m]$ are related by an IDFT, as follows:

$\displaystyle x^{(i)}[n,m]$ $\displaystyle=$
$\displaystyle\frac{1}{N}\sum_{k=0}^{N-1}s^{(i)}[k,m]\exp\left(j2\pi\frac{n}{N}k\right).$
(5)

The total number of OFDM symbols transmitted equals $M$, as per Fig. 2.

###  II-B Sensing Model

For the target scene in Fig. 1a, let
$(n_{t}^{(i)},m_{t}^{(i)})\in\\{1,\cdots,n_{\rm max}\\}\times\\{1,\cdots,M\\}$
denote the (range, Doppler)-bin in which the $t$-th target $(t=1,\cdots,N_{\rm
tar})$ is situated w.r.t the $i$-th TX/radar (i.e., corresponding to the path
TX $i\rightarrow$ Target $\rightarrow$ TX $i$), where $n_{\rm max}\ll N$
denotes the range bin associated with the maximum detectable range. Then, the
(discrete) time-domain radar return, $y^{(i)}[n,m]$, at the $i$-th TX/radar
corresponding to the transmission of $x^{(i)}[n,m]$ is given by:

$\displaystyle y^{(i)}[n,m]$ $\displaystyle=$ $\displaystyle\sum_{t=1}^{N_{\rm
tar}}\underbrace{\alpha_{t}^{(i)}x^{(i)}[n-n_{t}^{(i)},m]\exp\left(j2\pi\frac{m_{t}^{(i)}}{M}(m-1)\right)}_{\text{Desired
radar returns}}+\underbrace{y_{\rm intf}^{(i)}[n,m]}_{\text{Sensing
intf.}}+w^{(i)}[n,m],$ (6)

where

  * •

$\alpha_{t}^{(i)}\in{\mathbb{C}}$ is the gain, capturing the combined effects
of beamforming, path loss, and target reflection/scattering along the path TX
$i\rightarrow$ Target $t\rightarrow$ TX $i$;

  * •

$y_{\rm intf}^{(i)}[n,m]$ denotes the _sensing interference_ signal at the
$i$-th TX/radar; and,

  * •

$w^{(i)}[n,m]\sim{\mathcal{C}}{\mathcal{N}}(0,\sigma^{2})$ is the additive
noise signal, independent of the radar returns and also across $n,m$.

In the terms capturing the desired radar returns in (6), the delay shift
corresponding to the target ranges is assumed to be the same across all $m$,
and the Doppler shift is assumed to be constant over the duration of a slow-
time interval. Such a decoupling between the delay and Doppler shifts can be
assumed when the Doppler frequency is much smaller than the signal bandwidth.

We model $y^{(i)}_{\rm intf}[n,m]$ as follows: let $y_{\rm
intf}^{(q\rightarrow i)}[n,m]$ denote the sensing interference experienced at
the $i$-th TX/radar due to the transmissions of TX/radar $q~{}(\neq i)$, which
can be expressed as a superposition of $N_{\rm tar}+1$ components as follows:

$\displaystyle y_{\rm intf}^{(q\rightarrow i)}[n,m]$ $\displaystyle=$
$\displaystyle\sum_{l=0}^{N_{\rm tar}}\alpha_{l}^{(q\rightarrow
i)}x^{(q)}[n-n_{l}^{(q\rightarrow
i)},m]\exp\left(j2\pi\frac{m_{l}^{(q\rightarrow i)}}{M}(m-1)\right),$ (7)

where the $l$-th term in (7) represents the interfering signal along the path
TX $q\rightarrow$ Target $l\rightarrow$ TX $i$ for $l\neq 0$, while the term
corresponding to $l=0$ represents the signal leakage along the direct path TX
$q\rightarrow$ TX $i$. Similar to the desired radar returns in (6), each such
path is parameterized by a gain, delay and Doppler shift, captured by
$\alpha_{l}^{(q\rightarrow i)}\in{\mathbb{C}}$, $n_{l}^{(q\rightarrow
i)}\in\\{1,\cdots,n_{\rm max}\\}$ and $m_{l}^{(q\rightarrow
i)}\in\\{1,\cdots,M\\}$, respectively. Thus, $y^{(i)}_{\rm intf}[n,m]$ can be
expressed as follows:

$\displaystyle y^{(i)}_{\rm intf}[n,m]$ $\displaystyle=$
$\displaystyle\sum_{q\neq i}y^{(q\rightarrow i)}_{\rm intf}[n,m].$ (8)

We now explore the sensing signal processing for the conventional single-
carrier and the OFDM waveforms.

####  II-B1 Conventional Single-Carrier

For a single-carrier waveform, the target range bins can be estimated from (6)
using a bank of matched filters, followed by a DFT to estimate the Doppler
bins, as described below. Let $r^{(i)}[l,m]~{}(l=1,\cdots,n_{\rm max})$ denote
the (normalized) matched filter output, corresponding to $y^{(i)}[n,m]$ and
$x^{(i)}[n,m]$, given by:

$\displaystyle r^{(i)}[l,m]$ $\displaystyle:=$
$\displaystyle\frac{1}{N}\sum_{n=-\infty}^{\infty}y^{(i)}[n,m]x^{(i)^{*}}[n-l,m]$
(9) $\displaystyle=$ $\displaystyle\underbrace{\sum_{t=1}^{N_{\rm
tar}}\alpha_{t}^{(i)}\chi(l-n_{t}^{(i)};s^{(i)}[\cdot,m])\exp\left(j2\pi\frac{m_{t}^{(i)}}{M}(m-1)\right)}_{\text{Desired
signal component}}$ $\displaystyle+\underbrace{\sum_{q\neq
i}\sum_{t^{\prime}=0}^{N_{\rm tar}}\alpha_{t^{\prime}}^{(q\rightarrow
i)}\varrho(l-n_{t^{\prime}}^{(q\rightarrow
i)};s^{(q)}[\cdot,m],s^{(i)}[\cdot,m])\exp\left(j2\pi\frac{m_{t^{\prime}}^{(q\rightarrow
i)}}{M}(m-1)\right)}_{\text{Sensing interference component}}$
$\displaystyle+\frac{1}{N}\sum_{n=-\infty}^{\infty}w^{(i)}[n,m]s^{(i)^{*}}[n-l,m].$

Eqn. (9) is obtained from (4) and Definitions 1 and 2.

######  Remark 1 (Sensing Interference Management for a single-carrier
waveform).

The sensing interference in (9) can be completely eliminated if and only if
the collection of c.c.s transmitted by the TXs have mutually zero cross-
correlation across all lags [i.e.,
$\varrho(l;s^{(q)}[\cdot,m],s^{(i)}[\cdot,m])\equiv 0,~{}\forall q\neq i$],
which is unlikely for all realizations of random c.c.s, $s^{(i)}[n,m]$ and
$s^{(q)}[n,m]$. In particular, a large value of
$|\varrho(l;s^{(i)}[\cdot,m],s^{(q)}[\cdot,m])|$ at a non-target range bin,
$l$, gives rise to undesired sidelobes in $|r^{(i)}[l,m]|$ which, in turn,
affects the sensing performance by giving rise to false-alarms and missed
detections – the latter due to the near-far effect, wherein the weaker peaks
of farther targets are buried among the stronger interference sidelobes. To
minimize the occurrence of these outcomes, it is desirable for
${\mathbb{P}}(|\varrho(l;s^{(i)}[\cdot,m],s^{(q)}[\cdot,m])|>u)$ – the tail
probability of $|\varrho(l;s^{(i)}[\cdot,m],s^{(q)}[\cdot,m])|$ – to decay
rapidly w.r.t $u$. We characterize this quantity in terms of the c.c.s
parameters (i.e., $r_{i}$, $r_{q}$ and $N$) in Section III-B.

The single-carrier range-Doppler map, $R^{(i)}_{\rm sc}[l,\nu]$, whose
magnitude captures the strength of the radar return from the $l$-th range bin
and $\nu$-th Doppler bin, can be obtained from $r^{(i)}[l,m]$ through a
$M$-point DFT across the slow time index, $m$. Assuming the condition in
Remark 1 holds, the resulting expression for $R^{(i)}_{\rm sc}[l,\nu]$ is as
follows:

$\displaystyle R_{\rm sc}^{(i)}[l,\nu]$
$\displaystyle:=\sum_{m=1}^{M}r^{(i)}[l,m]\exp\left(-j2\pi\frac{\nu}{M}(m-1)\right)$
$\displaystyle=\sum_{t=1}^{N_{\rm
tar}}\alpha_{t}^{(i)}\sum_{m=1}^{M}|\chi(l-n_{t}^{(i)};s^{(i)}[\cdot,m])|\exp\left[j\left(2\pi\left(\frac{m_{t}^{(i)}-\nu}{M}\right)(m-1)+\underbrace{\arg\left(\chi(l-n_{t}^{(i)};s^{(i)}[\cdot,m])\right)}_{\text{autocorr.~{}phase
noise}}\right)\right]$ (10)

######  Remark 2 (Ideal Auto-correlation).

It is easy to see from (II-B1) that at a large enough SNR, if
$\chi(l;s^{(i)}[\cdot,m])\propto\delta[l]~{}\forall m$ (i.e., ideal auto-
correlation function), then $R_{\rm sc}^{(i)}[l,\nu]\propto\sum_{t=1}^{N_{\rm
tar}}\alpha_{t}^{(i)}\delta[l-n_{t}^{(i)}]\delta[\nu-m_{t}^{(i)}]$, resulting
in sharp peaks for $|R_{\rm sc}^{(i)}[l,\nu]|$ at
$(l,\nu)\in\\{(n_{t}^{(i)},m_{t}^{(i)}):t=1,\cdots,N_{\rm tar}\\}$ – the
targets’ range-Doppler bins w.r.t TX $i$. However, an ideal auto-correlation
function is unlikely for all realizations of random c.c.s, $s^{(i)}[n,m]$, and
when $\chi(l;s^{(i)}[\cdot,m])\not\propto\delta[l]$,

  * (a)

a large value of $|\chi(l;s^{(i)}[\cdot,m])|$ at $l\neq 0$ (i.e., sidelobe)
can give rise to false-alarms and the near-far effect, similar to Remark 1;
and,

  * (b)

the phase of $\chi(l;s^{(i)}[\cdot,m])$ impacts Doppler-bin estimation, as
seen in (II-B1).

To minimize the occurrence of these outcomes, it is desirable for
${\mathbb{P}}(|\chi(l;s^{(i)}[\cdot,m])|>u)$ – the tail probability of
$|\chi(l;s^{(i)}[\cdot,m])|~{}(l\neq 0)$ – to decay rapidly w.r.t $u$. We
characterize this quantity in terms of the c.c.s parameters (i.e., $r_{i}$ and
$N$) in Section III-A.

####  II-B2 OFDM

The OFDM radar processing chain at the $i$-th TX involves an $N$-point DFT of
$y^{(i)}[n,m]$ in (6) over the fast time index, $n$, to obtain a frequency-
domain signal, $Y^{(i)}[k,m]~{}(k=0,\cdots,N-1)$, that has the following
expression:

$\displaystyle Y^{(i)}[k,m]$ $\displaystyle=$
$\displaystyle\underbrace{\sum_{t=1}^{N_{\rm
tar}}\alpha_{t}^{(i)}s^{(i)}[k,m]\exp\left(-j2\pi\frac{n_{t}^{(i)}}{N}k\right)\exp\left(j2\pi\frac{m_{t}^{(i)}}{M}(m-1)\right)}_{\text{Desired
signal returns}}$ (11) $\displaystyle+\underbrace{\sum_{q\neq
i}\sum_{l=0}^{N_{\rm tar}}\alpha_{l}^{(q\rightarrow
i)}s^{(q)}[k,m]\exp\left(-j2\pi\frac{n_{l}^{(q\rightarrow
i)}}{N}k\right)\exp\left(j2\pi\frac{m_{l}^{(q\rightarrow
i)}}{M}(m-1)\right)}_{\text{Interfering signal returns}}$
$\displaystyle+\sum_{n=0}^{N-1}w^{(i)}[n,m]\exp\left(-j2\pi\frac{k}{N}n\right).$

At the $i$-th TX, upon dividing (11) by the known $s^{(i)}[k,m]$, we
obtain777The LHS of (12) corresponds to zero-forcing equalization, which we
assume in our OFDM analysis in this paper. With minor modifications, our
analysis holds for other forms of equalization, as well (e.g., MMSE).:

$\displaystyle\frac{Y^{(i)}[k,m]}{s^{(i)}[k,m]}$ $\displaystyle=$
$\displaystyle\underbrace{\sum_{t=1}^{N_{\rm
tar}}\alpha_{t}^{(i)}\exp\left(-j2\pi\frac{n_{t}^{(i)}}{N}k\right)\exp\left(j2\pi\frac{m_{t}^{(i)}}{M}(m-1)\right)}_{\text{Desired
signal component}}$ (12) $\displaystyle+\underbrace{\sum_{q\neq
i}\sum_{l=0}^{N_{\rm tar}}\alpha_{l}^{(q\rightarrow
i)}\frac{s^{(q)}[k,m]}{s^{(i)}[k,m]}\exp\left(-j2\pi\frac{n_{l}^{(q\rightarrow
i)}}{N}k\right)\exp\left(j2\pi\frac{m_{l}^{(q\rightarrow
i)}}{M}(m-1)\right)}_{\text{Sensing interference component}}$
$\displaystyle+\frac{W^{(i)}[k,m]}{s^{(i)}[k,m]}.$

For the desired signal component in (12), each of the terms under the
summation sign is a scaled product of a pair of decoupled sinusoids – one
across the subcarriers (index $k$), whose frequency depends on the target
ranges, and another across OFDM symbols (index $m$), whose frequency depends
on the target velocity (Doppler). Hence, the target parameters can be
estimated from the OFDM range-Doppler map, $R^{(i)}_{\rm OFDM}[l,\nu]$,
obtained through a combination of an $N$-point IDFT (over $k$) and an
$M$-point DFT (over $m$), as follows [20]:

$\displaystyle R^{(i)}_{\rm OFDM}[l,\nu]$
$\displaystyle:=\frac{1}{N}\sum_{m=1}^{M}\sum_{k=0}^{N-1}\frac{Y^{(i)}[k,m]}{s^{(i)}[k,m]}\exp\left(j2\pi\frac{l}{N}k\right)\exp\left(-j2\pi\frac{\nu}{M}(m-1)\right)$
$\displaystyle=\underbrace{\sum_{t=1}^{N_{\rm
tar}}\alpha_{t}^{(i)}\delta\left[l-n_{t}^{(i)}\right]\delta\left[\nu-
m_{t}^{(i)}\right]}_{\text{Peaks at
$(l,\nu)=\\{(n_{t}^{(i)},m_{t}^{(i)}):t=1,\cdots,N_{\rm tar}\\}$}}$
$\displaystyle~{}+\underbrace{\sum_{q\neq i}\sum_{l=0}^{N_{\rm
tar}}\alpha_{l}^{(q\rightarrow
i)}\sum_{m=1}^{M}\exp\left(j2\pi\frac{(m_{l}^{(q\rightarrow
i)}-\nu)}{M}(m-1)\right).\underbrace{\frac{1}{N}\sum_{k=0}^{N-1}\frac{s^{(q)}[k,m]}{s^{(i)}[k,m]}\exp\left(j2\pi\frac{(l-n_{l}^{(q\rightarrow
i)})}{N}k\right)}_{\text{IDFT of
$s^{(q)}[k,m]/s^{(i)}[k,m]$}}}_{\text{Distortion induced by sensing intf.}}$
$\displaystyle~{}+\underbrace{\tilde{W}^{(i)}[l,\nu]}_{\text{Noise
component}}.$ (13)

From (II-B2), it is clear that in the absence of interference and a large
enough SNR, $|R^{(i)}_{\rm OFDM}[l,\nu]|$ would have sharp peaks only at
$(l,\nu)\in\\{(n_{t}^{(i)},m_{t}^{(i)}):t=1,\cdots,N_{\rm tar}\\}$ – the
targets’ range-Doppler bins w.r.t TX $i$. Furthermore, we also observe that
the distortion in $R^{(i)}_{\rm OFDM}[l,\nu]$ due to the interference from
$s^{(q)}[k,m]$ is governed by the $N$-point IDFT of
$s^{(q)}[k,m]/s^{(i)}[k,m]$ over the index $k$. We remark on this quantity
below.

######  Definition 4.

Let

$\displaystyle V^{(q\rightarrow i)}[l,m]$
$\displaystyle:=\frac{1}{N}\sum_{k=0}^{N-1}\frac{s^{(q)}[k,m]}{s^{(i)}[k,m]}\exp\left(j2\pi\frac{l}{N}k\right)$
(14)

denote the $N$-point IDFT of $s^{(q)}[k,m]/s^{(i)}[k,m]~{}(q\neq i)$ over the
index $k$.

######  Remark 3 (Distortion due to sensing interference in the OFDM range-
Doppler map).

We may reasonably assume that for fixed $m$, the signals $s^{(i)}[k,m]$ and
$s^{(q)}[k,m]$ are independent, as the c.c.s from different TXs are generated
from mutually independent message signals. Furthermore, since
$s^{(i)}[k,m]\neq 0$, it follows from Assumption 1 that
${\mathbb{E}}\left[\frac{s^{(q)}[k,m]}{s^{(i)}[k,m]}\right]=0$. Hence,
$V^{(q\rightarrow i)}[l,m]$ also has zero mean, and the interference term in
(II-B2) can be viewed as a zero-mean, additive non-Gaussian distortion.

######  Remark 4 (Sensing interference management for an OFDM waveform).

Despite being zero-mean, the distortion due to the sensing interference in
(II-B2) can give rise to sidelobes in $|R^{(i)}_{\rm OFDM}[l,\nu]|$,
accompanied by false-alarms and the near-far effect (similar to Remark 1), if
$|V^{(q\rightarrow i)}[l,m]|$ takes on a large value at a non-target range
bin, $l$. To minimize the occurrence of these outcomes, it is desirable for
${\mathbb{P}}(|V^{(q\rightarrow i)}[l,m]|>u)$ – the tail probability of
$|V^{(q\rightarrow i)}[l,m]|$ – to decay rapidly w.r.t $u$. We characterize
this quantity in terms of the c.c.s parameters (i.e., $r_{i}$, $r_{q}$ and
$N$) in Section III-C.

From Remarks 1 through 4, it is clear that the sensing potential of c.c.s is
intricately linked to the tail probabilities of
$|\varrho(l;s^{(i)}[\cdot,m],s^{(q)}[\cdot,m])|$, $|V^{(q\rightarrow
i)}[l,m]|$ and $|\chi(l;s^{(i)}[\cdot,m])|~{}(l\neq 0)$ – the first two
quantities determine the sensing interference suppression capabilities of
c.c.s in single-carrier and OFDM waveforms, respectively, while the last
impacts the sensing performance of c.c.s in a single-carrier waveform. We
characterize these tail probabilities in the following section.

##  III Sensing Potential of c.c.s

Consider a pair of c.c.s blocks, $s^{(i)}[n,m]$ and $s^{(q)}[n,m]~{}(i\neq
q)$, generated by message signals, $u^{(i)}[n,m]$ and $u^{(q)}[n,m]$,
respectively, according to Section II-A. Since $m$ is arbitrary, we omit this
index throughout this section to simplify the notation for all the quantities
of interest, such as $\chi(l;s^{(i)}[\cdot])$,
$\varrho(l,s^{(i)}[\cdot],s^{(q)}[\cdot])$, $V^{(q\rightarrow i)}[l]$, etc. In
deriving bounds for the tail probabilities of the latter, we restrict our
attention to their real parts; the analysis for the imaginary part follows
similarly. We now introduce a few lemmas that will be used in deriving our
main results later in this section.

######  Lemma 1 (Hoeffding’s Lemma).

Let $X$ be a (real) random variable, satisfying $|X|\leq b<\infty$ with
probability one, and ${\mathbb{E}}[X]=0$. Then, for $\lambda>0$,

$\displaystyle{\mathbb{E}}\left[\exp(\lambda X)\right]$ $\displaystyle\leq$
$\displaystyle\exp\left(\lambda^{2}b^{2}/2\right).$ (15)

###### Proof:

See [21, Lemma 2.19]. ∎

######  Lemma 2.

For $\lambda\in{\mathbb{R}}$, and a collection of (real) identically
distributed (but not necessarily independent) random variables,
$X_{1},\cdots,X_{n}$

$\displaystyle{\mathbb{E}}\left[\exp\left(\lambda X_{1}+\cdots+\lambda
X_{n}\right)\right]$ $\displaystyle\leq$
$\displaystyle{\mathbb{E}}\left[\exp\left(\lambda nX_{1}\right)\right].$ (16)

###### Proof:

From the AM-GM inequality, we have for $\lambda\in{\mathbb{R}}$,

$\displaystyle\exp\left(\lambda X_{1}+\cdots+\lambda X_{n}\right)$
$\displaystyle\leq$ $\displaystyle\frac{\exp\left(\lambda
nX_{1}\right)+\cdots+\exp\left(\lambda nX_{n}\right)}{n}$ (17)

Applying the ${\mathbb{E}}[\cdot]$ operator to both sides of the above
inequality, we obtain (16). ∎

###  III-A Tail Probability of $|\Re\\{\chi(l;s[\cdot])\\}|$

To further simplify the notation, we omit the index $i$ in this subsection,
and denote respectively by $\chi(l;s[\cdot])$ and $u[n]$, the auto-correlation
function and the message signal associated with c.c.s, $s[n]$.

######  Definition 5.

Let $M_{l}:=\left\lceil\frac{N-l}{K-l}\right\rceil$. Then,
$\Re\\{\chi(l;s[\cdot])\\}$ can be expressed as follows :

$\displaystyle\Re\\{\chi(l;s[\cdot])\\}$ $\displaystyle:=$
$\displaystyle\frac{1}{N}\left(Y_{1}^{(l)}+\cdots+Y_{M_{l}}^{(l)}\right),$
$\displaystyle\mbox{where}~{}Y_{i}^{(l)}$ $\displaystyle:=$
$\displaystyle\Re\left\\{\sum_{k^{\prime}=0}^{K-1-l}s[(i-1)(K-l)+k^{\prime}+l]s^{*}[(i-1)(K-l)+k^{\prime}]\right\\}$
(18) $\displaystyle(i=1,\cdots,M_{l}-1),$
$\displaystyle\mbox{and}~{}Y_{M_{l}}^{(l)}$ $\displaystyle:=$
$\displaystyle\Re\left\\{s\left[\left(M_{l}-1\right)(K-l)+l\right]s^{*}\left[\left(M_{l}-1\right)(K-l)\right]\right.$
(19)
$\displaystyle\left.+s\left[\left(M_{l}-1\right)(K-l)+l+1\right]s^{*}\left[\left(M_{l}-1\right)(K-l)+1\right]\right.$
$\displaystyle\left.+\cdots+s[N-1]s^{*}[N-1-l]\right\\}.$

According to (18),
$Y_{1}^{(l)}:=\Re\left\\{s[l]s^{*}[0]+s[l+1]s^{*}[1]+\cdots+s[K-1]s^{*}[K-1-l]\right\\}$
denotes the sum of the first $(K-l)$ non-trivial terms in the RHS of (1),
$Y_{2}^{(l)}$ denotes the sum of the next $(K-l)$ non-trivial terms, and so
on. The last such quantity, $Y_{M_{l}}^{(l)}$, may contain fewer terms, but
for simplicity, we assume that it contains $K-l$ non-trivial terms, as well.
Decomposing $\Re\\{\chi(l;s[\cdot])\\}$ into partial sums in this manner,
along with Assumption 2, ensures that $Y_{1}^{(l)}$ is a function of i.i.d
random variables (i.e., the message signal $u[n]$), whose distribution can be
characterized using concentration inequalities, as shown in Lemma 3 below. On
the other hand, $Y_{i}^{(l)}~{}(i\neq 1)$, is much harder to characterize, as
it is a function of a mixture of i.i.d and dependent random variables, in
general. Thus, to help characterize $Y_{i}^{(l)}~{}(i\neq 1)$, we make the
following assumption.

######  Assumption 4.

Due to their similarity in form, we assume that
$Y_{1}^{(l)},\cdots,Y_{M_{l}}^{(l)}$ are identically distributed. This is
certainly true for $(N,K)$-linear MDS (Maximum Distance Separable) codes
(e.g., the Reed-Solomon code) since any collection of $K$ c.c.s symbols are
mutually i.i.d. for such codes [22, Theorem 5.4.5], and each $Y_{i}^{(l)}$ is
a function of $K$ c.c.s symbols, namely,
$s[(i-1)(K-l)],s[(i-1)(K-l)+1],\cdots,s[i(K-l)+l-1)]$. Note that
$Y_{1}^{(l)},\cdots,Y_{M_{l}}^{(l)}$ are statistically dependent, in general.

######  Lemma 3.

For $\lambda\in{\mathbb{R}}$,

$\displaystyle{\mathbb{E}}\left[\exp\left(\lambda Y_{1}^{(l)}\right)\right]$
$\displaystyle\leq$ $\displaystyle\exp\left(\lambda^{2}b^{2}(K-l)/2\right),$
(20)

where $b$ satisfies $|\Re\\{s[n]s^{*}[n-l]\\}|\leq b<\infty$, for any $n,l$.

###### Proof:

Let ${\mathbf{u}}^{(K-2)}:=\left[u[0],\cdots,u[K-2]\right]$. From Assumption 2
and (18), we have

$\displaystyle Y_{1}^{(l)}$ $\displaystyle=$
$\displaystyle\Re\left\\{u[l]u^{*}[0]+u[l+1]u^{*}[1]+\cdots+u[K-1]u^{*}[K-1-l]\right\\}$
(21) $\displaystyle=$ $\displaystyle A({\mathbf{u}}^{(K-2})+B(u[K-1]),$
$\displaystyle\mbox{where}~{}A({\mathbf{u}}^{(K-2})$ $\displaystyle:=$
$\displaystyle\Re\left\\{u[l]u^{*}[0]+u[l+1]u^{*}[1]+\cdots+u[K-2]u^{*}[K-2-l]\right\\},$
(22) $\displaystyle\mbox{and}~{}B(u[K-1])$ $\displaystyle:=$
$\displaystyle\Re\left\\{u[K-1]u^{*}[K-1-l]\right\\}.$ (23)

Conditioning on ${\mathbf{u}}^{(K-2)}$, ${\mathbb{E}}\left[\exp\left(\lambda
Y_{1}^{(l)}\right)\right]$ can be written as:

$\displaystyle{\mathbb{E}}\left[\exp\left(\lambda Y_{1}^{(l)}\right)\right]$
$\displaystyle={\mathbb{E}}_{{\mathbf{u}}^{(K-2)}}\left[{\mathbb{E}}_{u[K-1]}\left[\exp\left(\lambda
Y_{1}^{(l)}\right)|{\mathbf{u}}^{(K-2)}\right]\right]$
$\displaystyle={\mathbb{E}}_{{\mathbf{u}}^{(K-2)}}\left[\exp\left(\lambda
A({\mathbf{u}}^{(K-2)})\right){\mathbb{E}}_{u[K-1]}\left[\exp\left(\lambda
B(u[K-1])\right)|{\mathbf{u}}^{(K-2)}\right]\right].$ (24)

Since $u[K-1]$ is independent of ${\mathbf{u}}^{(K-2)}$, we have

$\displaystyle{\mathbb{E}}_{u[K-1]}\left[\exp\left(\lambda
B(u[K-1])\right)|{\mathbf{u}}^{(K-2)}\right]={\mathbb{E}}_{u[K-1]}\left[\exp\left(\lambda
B(u[K-1])\right)\right]\leq\exp\left(\frac{\lambda^{2}b^{2}}{2}\right),$ (25)

where the latter inequality stems from Lemma 1, as
$|\Re\\{u[K-1]u^{*}[K-1-l]\\}|=|\Re\\{s[K-1]s^{*}[K-1-l]\\}|\leq b$. Thus,
from (III-A) and (25),

$\displaystyle{\mathbb{E}}\left[\exp\left(\lambda
Y_{1}^{(l)}\right)\right]\leq\exp\left(\frac{\lambda^{2}b^{2}}{2}\right){\mathbb{E}}_{{\mathbf{u}}^{(K-2)}}\left[\exp\left(\lambda
A({\mathbf{u}}^{(K-2)})\right)\right].$ (26)

Repeating the above steps recursively by conditioning on
${\mathbf{u}}^{(K-3)},{\mathbf{u}}^{(K-4)}$ and so on, we get,

$\displaystyle{\mathbb{E}}\left[\exp\left(\lambda
Y_{1}^{(l)}\right)\right]\leq\exp\left(\frac{\lambda^{2}b^{2}}{2}(K-l)\right)$
(27)

∎

######  Theorem 1 (Auto-correlation Upper Bound).

For $u>0$, the tail probability of $|\Re\\{\chi(l;s[\cdot])\\}|~{}(l\neq 0)$
satisfies the following upper bound:

$\displaystyle{\mathbb{P}}\left(|\Re\\{\chi(l;s[\cdot])\\}|>u\right)\leq
2\exp(-O(rN)u^{2}),$ (28)

as $K,N\rightarrow\infty$ and $K/N=r$.

###### Proof:

We use the Chernoff bound to prove our result. For $u>0$,

$\displaystyle{\mathbb{P}}\left(\Re\\{\chi(l;s[\cdot])\\}>u\right)$
$\displaystyle={\mathbb{P}}\left(Y_{1}^{(l)}+\cdots+Y_{M_{l}}^{(l)}>Nu\right)\hskip
8.53581pt(\mbox{from Definition~{}\ref{def:Ys}})$
$\displaystyle={\mathbb{P}}\left(\exp\left[\lambda Y_{1}^{(l)}+\cdots+\lambda
Y_{M_{l}}^{(l)}\right]>\exp(\lambda Nu)\right)\hskip
8.53581pt(\mbox{for}~{}\lambda>0)$ $\displaystyle\leq\exp(-\lambda
Nu)~{}{\mathbb{E}}\left[\exp\left(\lambda Y_{1}^{(l)}+\cdots+\lambda
Y_{M_{l}}^{(l)}\right)\right]\hskip 8.53581pt(\mbox{Markov ineq.})$ (29)
$\displaystyle\leq\exp(-\lambda Nu)~{}{\mathbb{E}}\left[\exp\left(\lambda
M_{l}Y_{1}^{(l)}\right)\right]\hskip 8.53581pt(\mbox{Lemma~{}\ref{lem:am-
gm}})$ (30) $\displaystyle\leq\exp\left(-\lambda
Nu+\frac{\lambda^{2}M_{l}^{2}b^{2}}{2}(K-l)\right)\hskip
8.53581pt(\mbox{Lemma~{}\ref{lem:charY1}}).$ (31)

The tightest upper bound in (31) is obtained at
$\lambda=Nu/(b^{2}M_{l}^{2}(K-l))$, by equating the derivative of the argument
of the exponent to zero. Substituting this value in (31), we obtain

$\displaystyle{\mathbb{P}}\left(\Re\\{\chi(l;s[\cdot])\\}>u\right)\leq\exp\left(-\frac{N^{2}u^{2}}{2b^{2}M_{l}^{2}(K-l)}\right).$
(32)

Similarly,

$\displaystyle{\mathbb{P}}\left(\Re\\{\chi(l;s[\cdot])\\}<-u\right)$
$\displaystyle={\mathbb{P}}(Y_{1}^{(l)}+\cdots+Y_{M_{l}}^{(l)}<-Nu)$
$\displaystyle={\mathbb{P}}\left(\exp(\lambda
Y_{1}^{(l)}+\cdots+Y_{M_{l}}^{(l)})>\exp(-\lambda Nu)\right)\hskip
14.22636pt(\mbox{for}~{}\lambda<0).$ (33)

The rest of the analysis follows along the same lines as (29)-(31), with
$\lambda=-Nu/(b^{2}M_{l}^{2}(N-l))$, resulting in the following bound

$\displaystyle{\mathbb{P}}\left(\Re\\{\chi(l;s[\cdot])\\}<-u\right)$
$\displaystyle\leq\exp\left(-\frac{N^{2}u^{2}}{2b^{2}M_{l}^{2}(K-l)}\right).$
(34)

Combining (32) and (34), we get

$\displaystyle{\mathbb{P}}(|\Re\\{\chi(l;s[\cdot])\\}|>u)\leq
2\exp\left(-\frac{N^{2}u^{2}}{2b^{2}M_{l}^{2}(K-l)}\right).$ (35)

As $K,N\rightarrow\infty$ with $K/N=r$, $N^{2}/(2b^{2}M_{l}^{2}(K-l))=O(rN)$.
This completes the proof. ∎

######  Remark 5 (“Extent of i.i.d-ness”).

To provide intuition for the result in Theorem 1, consider $r=1$,
corresponding to an uncoded communications signal comprising i.i.d zero-mean
symbols. For this special case, (28) can be interpreted as a concentration
inequality closely related to the Hoeffding inequality [21, Theorem 2.16], for
which the order of the exponential decay is determined by the number of i.i.d
random variables present (i.e., $N$). A c.c.s can be viewed as a mixture of
i.i.d and dependent random variables/symbols; in particular, the maximal
number of i.i.d symbols over a block length $N$ equals888This is true for any
linear code, and does not require systematic encoding (i.e., Assumption 2).
$K=rN$, which, in turn, governs the $O(rN)$ exponential decay in (28).

######  Theorem 2 (Auto-correlation Lower Bound).

For linear codes, the tail probability of
$|\Re\\{\chi(l;s[\cdot])\\}|~{}(l\neq 0,~{}l\ll N)$ satisfies the following
lower bound for sufficiently small $u>0$:

$\displaystyle{\mathbb{P}}\left(|\Re(\chi(l;s[\cdot]))|>u\right)\geq\exp(-O(rN)),$
(36)

where $K/N=r$.

###### Proof:

Let ${\mathcal{S}}$ denote the set of feasible values of the signal $s[n]$
(i.e., the codebook). In particular, let $a_{0}[n]$ denote the signal
corresponding to the all-zero codeword (i.e.,
${\mathbf{c}}=\mathbf{0}=\tilde{{\mathbf{c}}}$ in Fig. 3), which occurs with
probability $2^{-m_{s}K}$ for linear codes. It is easily seen from (1) that
when all the symbols in $s[n]$ are identical, as is the case with $a_{0}[n]$,
then $|\chi(l;s[\cdot])|>u$ for $l\ll N$ and sufficiently small $u$. Thus,
conditioning on ${\mathcal{S}}$, we have

$\displaystyle{\mathbb{P}}\left(|\Re(\chi(l;s[\cdot]))|>u\right)$
$\displaystyle=\sum_{a[n]\in{\mathcal{S}}}\mathbbm{1}\left(|\Re\\{\chi(l;a[\cdot])\\}|>u\right){\mathbb{P}}(s[n]=a[n])$
$\displaystyle\geq\mathbbm{1}\left(|\Re\\{\chi(l;a_{0}[\cdot])\\}|>u\right){\mathbb{P}}(s[n]=a_{0}[n])$
$\displaystyle={\mathbb{P}}(s[n]=a_{0}[n])~{}(\mbox{for sufficiently small
$u$})$ $\displaystyle=2^{-m_{s}K}=\exp(-(m_{\rm s}\ln 2)rN)$

∎

######  Remark 6 (Implication of Theorems 1 and 2).

The combination of results from Theorems 1 and 2 implies that for linear
codes, a faster order of decay of the auto-correlation function, in terms of
the block length, is not possible. Importantly, the structure of a linear code
(i.e., its generator/parity-check matrices) does not impact the order of the
auto-correlation decay.

######  Corollary 1 (Asymptotically Ideal Auto-Correlation).

As $N\rightarrow\infty$, $|\Re\\{\chi(l;s[\cdot])\\}|$ converges in
distribution to $\delta[l]$, i.e.,

$\displaystyle\lim_{N\rightarrow\infty}|\Re\\{\chi(l;s[\cdot])\\}|\xrightarrow{d}\delta[l].$
(37)

###### Proof:

For $l\neq 0$ and $u>0$, we have from Theorem 1,

$\displaystyle
0\leq\lim_{N\rightarrow\infty}{\mathbb{P}}\left(|\Re\\{\chi(l;s[\cdot])\\}|>u\right)$
$\displaystyle\leq\lim_{N\rightarrow\infty}2\exp(-O(rN)u^{2})=0$
$\displaystyle\therefore~{}\lim_{N\rightarrow\infty}{\mathbb{P}}\left(|\Re\\{\chi(l;s[\cdot])\\}|>u\right)$
$\displaystyle=0$ (38)

From (38) and Definition 3, it follows that $|\Re\\{\chi(l;s[\cdot])\\}|$
converges in distribution to the deterministic $\delta[l]$, as
$N\rightarrow\infty$. ∎

###  III-B Tail probability of
$|\Re\\{\varrho(l;s^{(i)}[\cdot],s^{(q)}[\cdot])\\}|$

Assuming the symbols in $u^{(i)}[n]$ and $u^{(q)}[m]$ are mutually
independent, we can derive the following bounds for
$\Re\\{\varrho(l;s^{(i)}[\cdot],s^{(q)}[\cdot])\\}$, similar to Theorems 1 and
2.

######  Corollary 2 (Cross-correlation Upper Bound).

For $u>0$, the tail probability of
$|\Re\\{\varrho(l;s^{(i)}[\cdot],s^{(q)}[\cdot])\\}|$ satisfies the following
upper bound:

$\displaystyle{\mathbb{P}}(|\Re\\{\varrho(l;s^{(i)}[\cdot],s^{(q)}[\cdot])\\}|>u)\leq
2\exp(-O(\max(r_{i},r_{q})N)u^{2}),$ (39)

as $K_{i},K_{q},N\rightarrow\infty$, $K_{i}/N=r_{i}$, and $K_{q}/N=r_{q}$.

###### Proof:

Let $\tilde{K}:=\max(K_{i}-l,K_{q})$. Similar to Definition 5,
$\Re\\{\varrho(l;s^{(i)}[\cdot],s^{(q)}[\cdot])\\}$ can be expressed as
follows:

$\displaystyle\Re\\{\varrho(l;s^{(i)}[\cdot],s^{(q)}[\cdot])\\}$
$\displaystyle:=\frac{1}{N}\left(\tilde{Y}_{1}^{(l)}+\cdots+\tilde{Y}_{\tilde{M}_{l}}^{(l)}\right),$
(40) $\displaystyle\mbox{where}~{}\tilde{M}_{l}$
$\displaystyle:=\left\lceil\frac{N-l}{\tilde{K}}\right\rceil,$
$\displaystyle\tilde{Y}_{a}^{(l)}$
$\displaystyle:=\Re\left\\{\sum_{k^{\prime}=0}^{\tilde{K}-1}s^{(i)}[(a-1)\tilde{K}+l+k^{\prime}]s^{(q)^{*}}[(a-1)\tilde{K}+k^{\prime}]\right\\},~{}a=1,\cdots,\tilde{M}_{l}-1$
$\displaystyle\tilde{Y}_{\tilde{M}_{l}}^{(l)}$
$\displaystyle:=\Re\left\\{s^{(i)}\left[\left(\tilde{M}_{l}-1\right)\tilde{K}+l\right]s^{(q)^{*}}\left[\left(\tilde{M}_{l}-1\right)\tilde{K}\right]\right.$
$\displaystyle\hskip
28.45274pt\left.+s^{(i)}\left[\left(\tilde{M}_{l}-1\right)\tilde{K}+l+1\right]s^{(q)^{*}}\left[\left(\tilde{M}_{l}-1\right)\tilde{K}+1\right]\right.$
$\displaystyle\hskip
28.45274pt\left.+\cdots+s^{(i)}[N-1]s^{(q)^{*}}[N-1-l]\right\\}$ (41)

The remainder of the proof follows along the same lines as Theorem 1. ∎

######  Corollary 3 (Cross-correlation Lower Bound).

For linear codes, the tail probability of
$|\Re\\{\varrho(l;s^{(i)}[\cdot],s^{(q)}[\cdot]))|$ satisfies the following
lower bound for $l\ll N$ and sufficiently small $u>0$:

$\displaystyle{\mathbb{P}}(|\Re\\{\varrho(l;s^{(i)}[\cdot],s^{(q)}[\cdot])\\}|>u)$
$\displaystyle\geq$ $\displaystyle\exp(-O((r_{i}+r_{q})N)),$ (42)

where $K_{i}/N=r_{i}$ and $K_{q}/N=r_{q}$.

###### Proof:

For linear codes, the joint probability of the all-zero codeword for both
$s^{(i)}[n]$ and $s^{(q)}[n]$ equals $2^{-(m_{{\rm s}_{i}}K_{i}+m_{{\rm
s}_{q}}K_{q})}$. The remainder of the proof follows along the same lines as
Theorem 2.∎

######  Remark 7 (Implications of Corollary 2 and 3).

Similar to Remark 6, the results from Corollaries 2 and 3 imply that for
linear codes, a faster rate of decay of the cross-correlation function, in
terms of the block length, is not possible.

######  Corollary 4 (Asymptotically Zero Cross-Correlation).

As $N\rightarrow\infty$, $|\Re\\{\varrho(l;s^{(i)}[\cdot],s^{(q)}[\cdot])\\}|$
converges in distribution to 0, for any $l$, i.e.,

$\displaystyle\lim_{N\rightarrow\infty}|\Re\\{\varrho(l;s^{(i)}[\cdot],s^{(q)}[\cdot])\\}|\xrightarrow{d}0,~{}\mbox{
any $l$}.$ (43)

###### Proof:

Similar to Corollary 1. ∎

######  Remark 8 (Communications versus Sensing trade-off w.r.t favourable
correlation properties).

Broadly, there are two contrasting mechanisms for obtaining signals with
(nearly) favourable correlation properties: (a) through deterministic
construction (e.g., m-sequences, Zadoff-Chu sequences etc.), or (b) through
LLN, exploiting (information-bearing) randomness (e.g., c.c.s). Signals
belonging to the first class typically have better correlation properties
relative to c.c.s (i.e., lower sidelobes for a given $N$), and therefore, have
excellent sensing performance, but without any communication value999Some
communication value can be embedded to the deterministic sensing signals by
modulating them with information-bearing symbols [23], but the data rates that
can be achieved by this method are very low. On the other hand, the randomness
from the information-bearing symbols contributes to reducing the sidelobe
levels for such signals [24]. (e.g., data rate). At the other end of the
spectrum, the information-bearing randomness in c.c.s offers high
communication value at the expense of potentially larger sidelobes, that in
turn, may restrict its effectiveness as sensing signals to specific scenarios
(e.g., small-range applications, where the near-far effect is less prevalent).

###  III-C Tail Probability of $|\Re\\{V^{(q\rightarrow i)}[l]\\}|$

Similar to Corollaries 2 and 3, we can derive the following bounds for
$\Re\\{V^{(q\rightarrow i)}[l]\\}$, for mutually independent $u^{(i)}[n]$ and
$u^{(q)}[n]$.

######  Corollary 5 (OFDM Interference Sidelobe Upper Bound).

For $u>0$, the tail probability of $\Re\\{V^{(q\rightarrow
i)}[l]\\}~{}(l=0,\cdots,N-1)$ satisfies the following upper bound:

$\displaystyle{\mathbb{P}}\left(|\Re\\{V^{(q\rightarrow
i)}[l]\\}|>u\right)\leq 2\exp(-O(\min(r_{i},r_{q})N)u^{2}),$ (44)

as $K_{i},K_{q},N\rightarrow\infty$, $K_{i}/N=r_{i}$, and $K_{q}/N=r_{q}$.

###### Proof:

Let $K_{0}:=\min(K_{i},K_{q})$. Similar to Definition 5,
$\Re\\{V^{(q\rightarrow i)}[l]\\}$ can be expressed as follows :

$\displaystyle\Re\\{V^{(q\rightarrow i)}[l]\\}$
$\displaystyle:=\frac{1}{N}\left(T_{1}[l]+\cdots+T_{M_{0}}[l]\right),$
$\displaystyle\mbox{where}~{}M_{0}$
$\displaystyle:=\left\lceil\frac{N}{K_{0}}\right\rceil,$ $\displaystyle
T_{i}[l]$
$\displaystyle:=\Re\left\\{\sum_{k=(i-1)K_{0}}^{iK_{0}-1}\frac{s^{(q)}[k]}{s^{(i)}[k]}\exp\left(j2\pi\frac{l}{N}k\right)\right\\};~{}i=1,\cdots,M_{0}-1,$
(45) $\displaystyle\mbox{and}~{}T_{M_{0}}[l]$
$\displaystyle:=\Re\left\\{\sum_{k=(M_{0}-1)K_{0}}^{N-1}\frac{s^{(q)}[k]}{s^{(i)}[k]}\exp\left(j2\pi\frac{l}{N}k\right)\right\\}.$
(46)

The remainder of the proof follows along the same lines as Theorem 1. ∎

######  Corollary 6 (OFDM Interference Sidelobe Lower Bound).

For linear codes, the tail probability of $\Re\\{V^{(q\rightarrow i)}[0]\\}$
satisfies the following lower bound for sufficiently small $u>0$:

$\displaystyle{\mathbb{P}}(|\Re\\{V^{(q\rightarrow
i)}[0]\\}|>u)\geq\exp(-O((r_{i}+r_{q})N)),$ (47)

where $K_{i}/N=r_{i}$ and $K_{q}/N=r_{q}$.

###### Proof:

From (14), we observe that when all the symbols in $s^{(i)}[n]$ and
$s^{(q)}[n]$ are identical – e.g., the all-zero codeword for both signals –
then, $|\Re\\{V^{(q\rightarrow i)}[0]\\}|>u$, for sufficiently small $u$. The
remainder of the proof follows along the same lines as Corollary 3 and Theorem
2. ∎

######  Remark 9 (Implications of Corollary 5 and 6).

Similar to Remarks 6 and 7, the results from Corollaries 5 and 6 imply that
for linear codes, a faster rate of decay of $V^{(q\rightarrow i}[l]$, in terms
of the block length, is not possible.

######  Corollary 7 (Asymptotic Interference Suppression).

As $N\rightarrow\infty$, $|\Re\\{V^{(q\rightarrow i)}[l]\\}|$ converges in
distribution to 0, for any $l$, i.e.,

$\displaystyle\lim_{N\rightarrow\infty}|\Re\\{V^{(q\rightarrow
i)}[l]\\}|\xrightarrow{d}0,~{}\mbox{ any $l$}.$ (48)

###### Proof:

Similar to Corollary 1. ∎

######  Remark 10 (Single-Carrier v/s OFDM (Common) Waveform for ISAC – Impact
of c.c.s on Sensing Performance).

The similarity in the asymptotic behavior of the the tail probabilities of
$\chi(l;s[\cdot])~{}(l\neq 0)$, $\varrho(l;s^{(i)}[\cdot],s^{(q)}[\cdot])$ and
$V^{(q\rightarrow i)}[l]$ – i.e., the $\exp(-O(rN))$ decay – implies that for
any code rate, c.c.s are effective sensing signals that are robust to sensing
interference at sufficiently large block lengths, with negligible difference
in performance based on whether they modulate a single-carrier or OFDM
waveform. We investigate this claim in the following section.

##  IV Simulation Results

In this section, we first verify the bounds derived in Section III, before
comparing the range-Doppler sensing performance of c.c.s in interference-
limited operation with that of a benchmark interference-free FMCW waveform.

###  IV-A Peak-to-Sidelobe Ratio and Sensing Interference Suppression of c.c.s

The sidelobes101010The sidelobes of $\chi(l;s[\cdot])$ have two main
contributors: (a) the correlation properties of $s[n]$, and (b) timing errors.
These two factors are independent, in the sense that even if
$\chi(l;s[\cdot])=\delta[l]$, sidelobes can still be present if there is a
timing offset. In this paper, we focus on the sidelobes due to (a) only,
assuming no timing errors. The sidelobes due to (b) can be minimized using a
pulse shape with a fast roll-off (e.g., root-raised cosine or Gaussian
pulses). of $\chi(l;s[\cdot])$, can be measured using the _peak-to-sidelobe
ratio_ (PSLR) metric, defined as follows:

$\displaystyle{\rm PSLR~{}(dB)}$ $\displaystyle:=-20\log_{10}\left(\max_{l\neq
0}|\chi(l;s[\cdot])|\right),$ (49)

where a large value signifies a closer approximation to $\delta[l]$.
Similarly, replacing $\chi(l;s[\cdot])$ with
$\varrho(l;s^{(i)}[\cdot],s^{(q)}[\cdot])$ and $V^{(q\rightarrow i)}[l]$ in
(49) yields a metric that measures the extent of interference suppression
provided by c.c.s for single-carrier and OFDM waveforms, respectively.

(a) PSLR of c.c.s

(b) Interference Suppression of c.c.s

Figure 4: The median PSLR and mutual interference suppression provided by
c.c.s increases by approximately $3{\rm dB}$ when the block length is doubled,
mirroring the bounds in Theorem 1, Corollary 2 and Corollary 5.

Figs. 4a and 4b respectively plot the median PSLR and interference suppression
(obtained over 10000 simulation instances) as a function of $N$, for LDPC and
Polar coded signals111111Unlike the analysis in Section III, we have not
imposed systematic encoding (Assumption 2) in our simulations. at two rates –
$120/1024$ and $682.5/1024$ – corresponding to the smallest code rates for
which QPSK and 256QAM are used in 5G NR, respectively [25, Table 5.1.3.1-2].
We make the following remarks, based on the plotted curves:

######  Remark 11 ($3{\rm dB}$ increase in (median) PSLR and Interference
Suppression with a doubling of the block length).

Theorem 1, Corollary 2 and Corollary 5 each provide lower bounds121212A lower
bound is obtained for a suitable scaling factor of the $O(rN)$ term. We have
assumed a value of $1/2b^{2}$, where $b$ is defined in Lemma 3 and depends on
the symbol constellation., which suggests that the median PSLR and
interference suppression provided by c.c.s should eventually increase by at
least $3{\rm dB}$ when the block length is doubled; for the values of $N$
considered, we observe that this holds true, as the curves corresponding to
c.c.s are nearly parallel to those obtained from the bounds.

######  Remark 12 (Impact of Code structure).

The convergence behaviour of $\chi(l;s[\cdot])~{}(l\neq 0)$,
$\varrho(l;s^{(i)}[\cdot],s^{(q)}[\cdot])$ and $V^{(q\rightarrow i)}[l]$ is
governed by LLN and the “extent of i.i.d-ness” among the codeword symbols
(Remark 5), which does not depend on the code structure (i.e., Polar or LDPC)
for fixed $r$, $N$ and modulation scheme. Hence, the code structure has
negligible impact on the median PSLR and interference suppression provided by
c.c.s.

######  Remark 13 (The Effectiveness of Interleaving).

A key component of the c.c.s model in Fig. 3 is the interleaver, and in Fig.
4, the codeword bits are interleaved according to a uniformly distributed
random permutation. To underline the effectiveness of interleaving, Fig. 5
compares the median PSLR with and without interleaving for Polar coded c.c.s,
where we observe the following:

  * •

From the bound in Theorem 1, we observe that the median PSLR (in dB) is
proportional to $10\log_{10}(rN)$. Hence, for code rates sufficiently close to
one, the block length has a much greater influence on the median PSLR. This is
reflected in the curves for uninterleaved c.c.s, where (a) the median PSLR
increases with $r$ for fixed $N$, but the performance at $r=682.5/1024$
practically coincides with that of an uncoded signal (i.e., $r=1$); and (b)
the difference in the median PSLR between low and high rate c.c.s diminishes
with $N$.

  * •

Regardless of the modulation and coding scheme, the interleaved c.c.s has the
same median PSLR as that of an uncoded signal. This suggests that the
interleaved c.c.s – comprising mutually uncorrelated symbols – is ergodic,
despite exhibiting statistical dependence over a block. The seeming ergodicity
of the interleaved c.c.s may also explain why the bounds derived in Section
III are conservative in Fig. 4 – the bounds are based on the extent of
i.i.d-ness within c.c.s (see Remark 5) and not on their ergodic properties.

In summary, interleaving considerably improves the PSLR (and similarly, the
interference suppression) of c.c.s, especially for small code rates and block
lengths.

######  Remark 14 (Effective Sensing Interference Suppression).

Higher interference suppression at larger block lengths is a noteworthy
feature of c.c.s that is shared with other well-known deterministic sensing
signals (e.g., m-sequences), but is not a universal feature among the latter,
in general; for instance, a pair of FMCW chirp signals with different chirp
slopes gives rise to a distinctive interference pattern that is not
straightforward to eliminate [26, 27]. However, we note that this robustness
is restricted to multi-user interference, and not any form of adversarial
interference, like jamming.

######  Remark 15 (Near-far effect).

The PSLR of a sensing waveform is a measure of its robustness to the near-far
effect. To illustrate this, consider two identical targets – one near and one
far from the radar – and suppose $N=1024$, for which the median PSLR is
$\approx 25{\rm dB}$. Due to $R^{-4}$ attenuation, the radar return from the
far target is $>25~{}{\rm dB}$ weaker than that of the near one, when the
former’s range exceeds the latter’s by a factor $>4.2$. Thus, c.c.s are
effective sensing signals only up to a certain maximum range that depends on
the block length. We explore this in more detail in Section IV-B.

Figure 5: Interleaving improves the PSLR of c.c.s, especially for small code
rates and block lengths.

###  IV-B Sensing Performance of c.c.s and the Near-Far Effect

In this subsection, we explore the sensing performance of c.c.s in an
interference-limited scenario featuring the near-far effect, by considering
the radar scene in Fig. 1b with TX $1$ as the desired radar, TX $2$ as the
interfering radar, and two (point) targets (i.e., $M_{r}=N_{\rm tar}=2$). We
model this scene in terms of the notation in Section II-B below (see Table I
for a full list of simulation parameters):

  * (i)

With respect to TX $1$, suppose the near target is at $d_{1}=2{\rm m}$ and
moving at $v_{1}=10{\rm mph}$, while the far target is at $d_{2}=4{\rm m}$ and
moving at $v_{2}=15{\rm mph}$ in the same direction as the near target.
Assuming a signal bandwidth of 1 GHz, a carrier frequency of $140~{}{\rm
GHz}$, and $M=N=1024$, the targets’ (range, Doppler) bins are
$(n_{1}^{(1)},m_{1}^{(1)})=(14,516)$ and $(n_{2}^{(1)},m_{2}^{(1)})=(27,518)$.
Note that the Doppler bins can vary for a different choice of $M$.

  * (ii)

We assume that the interference experienced at TX $1$ is dominated by the
signal received along the direct path, TX $2\rightarrow$ TX $1$, corresponding
to $l=0$ in the RHS of (7). Hence, for simplicity, we consider only this
component and assume $(n_{0}^{(2\rightarrow 1)},m_{0}^{(2\rightarrow
1)})=(29,518)$, which corresponds to TX $2$ situated at $d_{\rm intf}=4.3{\rm
m}$ away from TX $1$ and moving at $v_{\rm intf}=15{\rm mph}$ – i.e., close to
and moving as fast as the far target in (i) above.

  * (iii)

For the amplitudes of the desired radar returns, we assume
$\alpha_{1}^{(1)}=1$ and $\alpha_{2}^{(1)}=10^{-12/10}$, resulting in a
$24{\rm dB}$ difference in the return energy between the near and far targets.
Of this, $12{\rm dB}$ is due to $R^{-4}$ attenuation [i.e.,
$\propto(d_{1}/d_{2})^{4}$], while the rest is accounted for by assuming a
weaker radar cross section for the far target.

  * (iv)

Unlike the radar returns, the interference signal experiences $R^{-2}$
attenuation, and therefore, $\frac{|\alpha_{0}^{(2\rightarrow
1)}|^{2}}{|\alpha_{1}|^{2}}\propto\frac{d_{1}^{4}}{d_{\rm intf}^{2}}$. Hence,
we assume $\alpha_{0}^{(2\rightarrow 1)}=\sqrt{10^{11/10}}$ – based on a unit
radar cross section for the near target – resulting in an SIR of $-11{\rm dB}$
w.r.t the near target. With these parameter values, a couple of key question
are whether (a) the strong signal from the _interference bin_ is suppressed,
and (b) the weak return from the far target bin is visible in the range-
Doppler maps? We explore these in Fig. 7.

  * (iv)

The c.c.s blocks, $s^{(1)}[n,m]$ and $s^{(2)}[n,m]$, are obtained from a rate
$120/1024$ Polar code, followed by QPSK modulation. The corresponding message
blocks, $u^{(1)}[n,m]$ and $u^{(2)}[n,m]$, are mutually independent. Finally,
the SNR w.r.t to the near target is $0{\rm dB}$.

Bandwidth, $B$ | $1~{}{\rm GHz}$ | Target Ranges $(d_{1},d_{2})$ |  $2{\rm m}$, $4{\rm m}$
---|---|---|---
Range resolution, $\Delta r$ | $0.15{\rm m}$ | Target Range bins $(n_{1}^{(1)},n_{2}^{(1)})$ |  $14$, $27$
Block length, $N$ |  $1024$ (symbols) | Target speeds $(v_{1},v_{2})$ |  $10{\rm mph}$, $15{\rm mph}$
Largest range bin, $n_{\rm max}$ | $32$ | Target Doppler bins $(m_{1}^{(1)},m_{2}^{(1)})$ |  $516$, $518$
Maximum detectable range | $4.65{\rm m}$ | Target return gains $(\alpha_{1}^{(1)},\alpha_{2}^{(1)})$ |  $1$, $10^{-12/10}$
Carrier frequency, $f_{c}$ | $140~{}{\rm GHz}$ | Intf. range ($d_{\rm intf}$) | $4.3{\rm m}$
No. of blocks, $M$ | $1024$ | Intf. range bin ($n_{0}^{(2\rightarrow 1)}$) | $29$
Velocity resolution, $\Delta v$ | $2.28{\rm mph}$ | Intf. speed ($v_{\rm intf}$) | $15{\rm mph}$
Coding | Polar Code ($r=120/1024$) | Intf. Doppler bin ($m_{0}^{(2\rightarrow 1)}$) | 518
Modulation | QPSK | Intf. gain ($\alpha_{0}^{(2\rightarrow 1)}$) | $\sqrt{10^{11/10}}$
SNR (w.r.t near target) | $0{\rm dB}$ |  |
SIR (w.r.t near target) | $-11{\rm dB}$ |  |
TABLE I: List of simulation parameters for the near-far target scenario in
Section IV-B.

The presence/absence of a target in (range, Doppler) bin $(l,\nu)$ is decided
using a threshold rule, as follows:

$\displaystyle H_{1}(H_{0})$ $\displaystyle:~{}\mbox{Target present (absent)}$
$\displaystyle|R_{w}^{(1)}[l,\nu]|$
$\displaystyle\underset{H_{0}}{\overset{H_{1}}{\gtrless}}\eta,~{}w\in\\{{\rm
sc,OFDM}\\}.$ (50)

where $R_{w}^{(1)}[l,\nu]$ denotes the range-Doppler map at TX $1$ for
$w\in\\{\rm sc,OFDM\\}$, defined in (II-B1) and (II-B2). For a given
threshold, $\eta$, the sensing performance of c.c.s is characterized by the
detection and false alarm probabilities – denoted by $P^{w}_{d}(\eta)$ and
$P^{w}_{f}(\eta)$, respectively – and defined as follows:

$\displaystyle P^{w}_{d}(\eta)$ $\displaystyle:=$
$\displaystyle\underbrace{{\mathbb{P}}\left(\left|R_{w}^{(1)}[n_{1}^{(1)},m_{1}^{(1)}]\right|>\eta;~{}\left|R_{w}^{(1)}[n_{2}^{(1)},m_{2}^{(1)}]\right|>\eta\right)}_{\text{Both
targets detected}}$ (51)
$\displaystyle+\underbrace{\frac{1}{2}{\mathbb{P}}\left(\left|R_{w}^{(1)}[n_{1}^{(1)},m_{1}^{(1)}]\right|>\eta;~{}\left|R_{w}^{(1)}[n_{2}^{(1)},m_{2}^{(1)}]\right|\leq\eta\right)}_{\text{Only
near target detected (hence, the 1/2 scaling factor)}}$
$\displaystyle+\underbrace{\frac{1}{2}{\mathbb{P}}\left(\left|R_{w}^{(1)}[n_{1}^{(1)},m_{1}^{(1)}]\right|\leq\eta;~{}\left|R_{w}^{(1)}[n_{2}^{(1)},m_{2}^{(1)}]\right|>\eta\right)}_{\text{Only
far target detected (hence, the 1/2 scaling factor)}},$ $\displaystyle
P_{f}^{w}(\eta)$ $\displaystyle:=$
$\displaystyle{\mathbb{P}}\left(\left|R_{w}^{(1)}(l,\nu)\right|>\eta,~{}\mbox{where}~{}(l,\nu)\notin\\{(n_{1}^{(1)},m_{1}^{(1)}),(n_{2}^{(1)},m_{2}^{(1)})\\}\right).$
(52)

Figure 6: Continued in the next page Figure 7: For the simulation setup in
Table I, the magnitude of the range-Doppler maps at the target bin locations –
$(14,516)$ and $(27,518)$ – are similar for both FMCW and c.c.s waveforms,
even at $-11{\rm dB}$ SIR for the latter.

Fig. 7 shows the range-Doppler maps for a given realization of c.c.s and FMCW
waveforms. We observe that:

  * •

For $N=1024$, the median PSLR [$\approx 25{\rm dB}$ in Fig. 4(a)] is nearly at
par with the difference in return strengths between the near and far targets.
Despite this, the far target is clearly “visible” because the targets are
situated in different Doppler bins, which provides nearly $30{\rm dB}$ of
additional sidelobe suppression for $N=1024$ through the Doppler DFT.

  * •

For $N=1024$, the nearly $25{\rm dB}$ median interference suppression in Fig.
4b seems insufficient, at first glance, to detect the weaker target in the
face of strong interference from a nearby bin. However, from (49), we see that
the interference suppression is defined in terms of the maximum sidelobe level
_across all_ bins, which can be a conservative estimate of the sidelobe levels
at a specific interference bin. In this case, we see that the strong
interference signal from a nearby bin is adequately suppressed and the weaker
target is clearly visible. However, the essence of Remark 15 – i.e., a limit
on the maximum target range imposed by the sidelobes of c.c.s – remains valid.

  * •

The magnitude of the c.c.s range-Doppler maps at the target bin locations is
nearly the same as that for the FMCW waveform – both with and without
interference and for both single-carrier and OFDM waveforms. This is
consistent with the results from Section III that c.c.s yield asymptotically
ideal range-Doppler maps (i.e., $\delta[\cdot]$ spikes at the target bins) at
large block lengths, despite the presence of sensing interference.

  * •

The large sidelobes across the range bins in the FMCW range-Doppler map are a
consequence of rectangular windowing assumed for the range FFT. Lower sidelobe
levels can be achieved through a better choice of windowing function, which we
do not pursue as it is beyond the scope of this paper.

Figure 8: Due to the similarity of the range-Doppler maps in Fig. 7, the
curves corresponding to the detection and false-alarm probabilities for all
the waveforms also exhibit considerable overlap.

Fig. 8 compares the detection performance of c.c.s, where we observe the
following:

  * •

For a threshold-based detection rule of the form given by (IV-B), the
detection probability exhibits a step-like behavior with increasing $\eta$,
where for small $\eta$, both targets are detected; followed by a regime where
only the near target is detected; before none of the targets is detected.

  * •

For a small set of thresholds, the FMCW waveform has a higher false-alarm
probability than c.c.s without interference (see inset), due to the large
sidelobes induced by rectangular windowing, as discussed w.r.t Fig. 7. Thus,
window functions that result in lower sidelobes would also cause the false-
alarm probability curve in Fig. 8 to shift leftward.

  * •

Despite strong interference ($-11{\rm dB}$ SIR), the detection performance of
c.c.s matches that of an interference-free FMCW waveform, and is only
marginally worse – in terms of the behavior of the false-alarm probability –
than c.c.s without interference.

  * •

The _sweet-spot_ threshold(s) for a given waveform are those that result in
both targets being detected, without any false-alarms. From the previous
bullet point, it follows that the sweet spot regions for c.c.s and FMCW mostly
coincide.

We conclude this section by noting that the results from Figs. 4 through 8
corroborate the bounds derived in Section III, thereby demonstrating that
c.c.s are effective sensing signals even in the presence of strong sensing
interference, with negligible difference in performance based on whether they
modulate a single-carrier or OFDM waveform.

##  V Concluding Remarks

In this paper, we explored the sensing potential of c.c.s in interference-
limited ISAC scenarios featuring both multi-user communications and sensing
interference, with the latter dominating the former. We started by identifying
– based on whether the waveform was single-carrier or OFDM – functions of
c.c.s that could adversely affect its sensing performance through large
sidelobes. We then derived upper bounds for the tail probabilities of the
sidelobe levels that decayed exponentially in terms of the code rate-block
length product, which suggested that c.c.s were effective sensing signals that
were robust to sensing interference at sufficiently large block lengths for
any fixed code rate, with negligible difference in performance based on
whether they modulate a single-carrier or OFDM waveform. The latter
implication was verified through simulations, where we observed that the
sensing performance of c.c.s – in terms of detection and false-alarm
probabilities – was at par with that of the interference-free FMCW waveform
for both single-carrier and OFDM waveforms, even at an SIR of $-11~{}{\rm
dB}$. Thus, our results imply that (a) a common ISAC waveform (either single-
or multi-carrier, but) largely comprising coded data symbols is an effective
sensing signal at large block lengths, and (b) in multi-user ISAC scenarios
with such a waveform, sensing interference management essentially _takes care
of itself_ for monostatic radars, and is relatively simpler than
communications interference management. These are highly favourable outcomes
in terms of the evolution of existing wireless networks to support sensing
applications while also maximizing the communications spectral efficiency, as
the sensing functionality does not impose additional constraints on the
available resources, either in the form of needing more reference signals or
orthogonal resource allocation across users.

##  Appendix A Repetition Codes: Interleaving mitigates correlation across
codeword bits

Let ${\mathbf{c}}=[c_{0}~{}\cdots~{}c_{\gamma
K-1}]=[\underbrace{b_{0}~{}\cdots~{}b_{K-1}~{}\cdots~{}b_{0}~{}\cdots~{}b_{K-1}}_{\text{$\gamma$
times}}]$ denote the codeword corresponding to the i.i.d. Bernoulli($1/2)$
message vector ${\mathbf{b}}=[b_{0}~{}\cdots b_{K-1}]$ for a rate $1/\gamma$
repetition code, where $\gamma\in\\{1,2,\cdots\\}$. Then, for $i\neq j$ and
$j>i$, the correlation coefficient of $c_{i}$ and $c_{j}$ – denoted by
$\rho(c_{i},c_{j})$ – is given by:

$\displaystyle\rho(c_{i},c_{j})=\begin{cases}1,~{}|j-i|\mod K=0\\\
0,~{}\mbox{else},\end{cases}$ (53)

which captures the fact that codeword bits that are separated by a multiple of
$K$ are fully correlated.

Let $\tilde{{\mathbf{c}}}=\pi({\mathbf{c}})$ denote the interleaved codeword,
where $\pi(.)$ is a permutation of bit locations such that: (a) $\pi(i)=i$,
for $i=0,\cdots,K-1$ (i.e., systematic encoding from Assumption 2), and (b)
$[\pi(K),~{}\cdots~{},\pi(\gamma K-1)]$ is a uniformly distributed random
permutation of $\\{K,K+1\cdots,\gamma K-1\\}$. Then, for $i\neq j$ and $j>i$

$\displaystyle\rho(\tilde{c}_{i},\tilde{c}_{j})$
$\displaystyle={\mathbb{P}}(|\pi^{-1}(i)-\pi^{-1}(j)|\mod K=0)$
$\displaystyle=\begin{cases}0,\hskip 8.53581pti,j\in\\{0,\cdots,K-1\\}\\\
\frac{\gamma-1}{(\gamma-1)K},\hskip
8.53581pti\in\\{0,\cdots,K-1\\},j\in\\{K,\cdots,\gamma K-1\\}\\\
\frac{\gamma-2}{(\gamma-1)K-1},\hskip 8.53581pti,j\in\\{K,\cdots,\gamma
K-1\\}.\end{cases}$ (54)

From (A), we see that $\rho(\tilde{c}_{i}\tilde{c}_{j})\rightarrow 0$ as
$K\rightarrow\infty$. Thus, the interleaved codeword bits are asymptotically
uncorrelated.

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