# Hilbert multiplicity and irreducible multiplicity of idealizations

Tran Nguyen An  Tran Nguyen An: Thai Nguyen University of Education, Vietnam
<EMAIL_ADDRESS>
###### Abstract.

Let $(R,\mathfrak{m})$ be a Noetherian local ring and $M$ a finitely generated
$R$-module. In this paper, we study the relation among the Hilbert function,
Hilbert multiplicity, index of reducibility of an $\mathfrak{m}$-primary
ideal, and the irreducible multiplicity under idealization $R\ltimes M$. Some
applications to Cohen-Macaulayness and Cohen-Macaulay type of idealization are
given.

2020 Mathematics Subject Classification. 13A15, 13H10, 13H15

Key words and phrases. Hilbert multiplicity, irreducible multiplicity, Cohen-
Macaulay type, idealization

This research is funded by Thai Nguyen University and Thai Nguyen University
of Education under grant number DH2023-TN04-07

##  1\. Introduction

Let $(A,\mathfrak{n})$ be a Noetherian local ring and $I$ an
$\mathfrak{n}$-primary ideal of $A$. We denote by $\ell_{A}(*)$ the length of
an $A$-module. Let $L$ be a finitely generated $R$-module of dimension $t$. It
is well known that $\ell_{A}(L/I^{n+1}L)$ agrees with a polynomial function of
degree $t$ for all $n\gg 0$. That is, there exist integers
$\mathrm{e}_{I}^{0}(L),\dots,\mathrm{e}_{I}^{t}(L)$ such that

$\ell_{A}(L/I^{n+1}L)=\mathrm{e}_{I}^{0}(M)\binom{n+t}{t}-\mathrm{e}_{I}^{1}(L)\binom{n+t-1}{n-1}+\cdots+(-1)^{t}\mathrm{e}_{I}^{t}(L)$

for $n\gg 0$. We call $\mathrm{e}_{I}^{0}(L),\dots,\mathrm{e}_{I}^{t}(L)$ the
Hilbert coefficients of $L$ with respect to $I$. The leading coefficient
$\mathrm{e}_{I}^{0}(L)$ is called the (Hilbert) multiplicity of $L$ with
respect to $I$. The notions of Hilbert function, Hilbert multiplicity, and
Hilbert coefficients are well known and have attracted the interest of several
researchers [7, Chapter 4].

Next we consider another function
$\ell_{A}(I^{n+1}L:_{L}\mathfrak{n}/I^{n+1}L)$, the socle dimension of
$L/I^{n+1}L$. Since $I$ is $\mathfrak{n}$-primary, we also have

$\ell_{A}(I^{n+1}L:_{L}\mathfrak{n}/I^{n+1}L)=\operatorname{ir}_{L}(I^{n+1}L),$

where $\operatorname{ir}_{L}(I^{n+1}L)$ is the index of reducibility of
$I^{n+1}L$. Recall that a submodule $N$ of $L$ is called an irreducible
submodule if $N$ cannot be written as an intersection of two properly larger
submodules of $L$. The number of irreducible components of an irredundant
irreducible decomposition of $N$, which is independent of the choice of the
decomposition by Noether [13], is called the index of reducibility of $N$ and
denoted by $\mathrm{ir}_{L}(N)$. In 2015, N. T. Cuong, P. H. Quy, and H. L.
Truong proved that the function $\mathrm{ir}_{L}(I^{n+1}L)$ agrees with a
polynomial function of degree $t-1$ for $n\gg 0$ ([8, Theorem 4.1]). That is,
there exist integers $\mathrm{f}_{I}^{0}(L),\dots,\mathrm{f}_{I}^{t-1}(L)$
such that

$\displaystyle\begin{split}\mathrm{ir}_{L}(I^{n+1}L)=&\ell_{A}(I^{n+1}L:_{L}\mathfrak{n}/I^{n+1}L)\\\
=&\mathrm{f}_{I}^{0}(L)\binom{n+t-1}{t-1}-\mathrm{f}_{I}^{1}(L)\binom{n+t-2}{n-2}+\cdots+(-1)^{t-1}\mathrm{f}_{I}^{t-1}(L)\end{split}$

for $n\gg 0$. The numbers
$\mathrm{f}_{I}^{0}(L),\dots,\mathrm{f}_{I}^{t-1}(L)$ are called the
irreducibility coefficients of $L$ with respect to $I$ and the leading
coefficient $\mathrm{f}_{I}^{0}(L)$ is called the irreducible multiplicity of
$L$ with respect to $I$ (see [17], [3]). The index of reducibility and
irreducible multiplicity have a strong connection with the structure of rings
(see [3], [4], [8], [9], [11], [15]-[18]). Some relations between (Hilbert)
multiplicity and irreducible multiplicity are given (see [3], [17], [18]).

From now on, let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension
$d$ and $M$ be a finitely generated $R$-module of dimension $t$.

The purpose of this note is to study the Hilbert function, Hilbert
multiplicity, index of reducibility of powers of an ideal, and the irreducible
multiplicity under idealization $R\ltimes M$. Recall that the Cartesian
product $R\times M$ is a commutative ring concerning componentwise addition
and multiplication defined by

$(r_{1},m_{1})(r_{2},m_{2})=(r_{1}r_{2},r_{1}m_{2}+r_{2}m_{1}).$

and is called the idealization of $M$ or the trivial extension of $R$ by $M$,
denoted by $R\ltimes M$. Idealization of a module was introduced by Nagata
[12]. One can find many properties of idealizations in [5] and [12].
Idealization of a module is usually used to reduce results concerning
submodules to the ideal case, generalizing results from ring to modules and
constructing examples of commutative rings with zero divisors.

The first main result gives a relation among the Hilbert function, Hilbert
coefficients of idealization $R\ltimes M$, and those of $R$ and $M$ as
follows.

######  Theorem 1.1.

Let $I$ be an $\mathfrak{m}$-primary ideal of $R$. Set $J=I\times IM$. Then

$\ell_{R\ltimes M}(R\ltimes
M/J^{n+1})=\ell_{R}(R/I^{n+1})+\ell_{R}(M/I^{n+1}M).$

Furthermore, if $d=t$, then $\operatorname{e}_{J}^{i}(R\ltimes
M)=\operatorname{e}_{I}^{i}(R)+\operatorname{e}_{I}^{i}(M)$ for $i=0,\dots,d$.
If $d>t$, then $\operatorname{e}_{J}^{i}(R\ltimes
M)=\operatorname{e}_{I}^{i}(R)$ for $i=0,\dots,d-t-1,$ and
$\operatorname{e}_{J}^{i}(R\ltimes
M)=\operatorname{e}_{I}^{i}(R)+\operatorname{e}_{I}^{i}(M)$ for
$i=d-t,\dots,d.$

In the above theorem we assume that $J$ is a certain primary ideal of
$R\ltimes M$. We will also consider the multiplicity
$\operatorname{e}_{J}^{0}(R\ltimes M)$ of $R\ltimes M$ for arbitrary primary
ideal $J$ of $R\ltimes M$ in Corollary 2.1.

The second main result gives a relation among the index of reducibility of
powers of an ideal, and the irreducible multiplicity of idealization $R\ltimes
M$ and those of $R$ of $M$. The second main result is as follows.

######  Theorem 1.2.

Let $I$ be an $\mathfrak{m}$-primary ideal of $R$. Set $J=I\times IM$. Then

$\operatorname{ir}_{M}(I^{n+1}M)\leq\operatorname{ir}_{R\ltimes
M}(J^{n+1})\leq\operatorname{ir}_{R}(I^{n+1})+\operatorname{ir}_{M}(I^{n+1}M)$

for all $n>0$ and

$\mathrm{f}_{I}^{0}(M)\leq\mathrm{f}_{J}^{0}(R\ltimes
M)\leq\mathrm{f}_{I}^{0}(R)+\mathrm{f}_{I}^{0}(M).$

Furthermore for $n>0$, the following hold.

  1. (i)

$\operatorname{ir}_{M}(I^{n+1}M)=\operatorname{ir}_{R\ltimes M}(J^{n+1})$ if
and only if $M/I^{n+1}M$ is faithful.

  2. (ii)

$\operatorname{ir}_{R\ltimes
M}(J^{n+1})=\operatorname{ir}_{R}(I^{n+1})+\operatorname{ir}_{M}(I^{n+1}M)$ if
and only if $(I^{n+1}:_{R}\mathfrak{m})M=I^{n+1}M$.

In [10], S. Goto, S. Kumashiro, and N. T. H. Loan explored the Cohen-Macaulay
type of idealization. Suppose $M$ is a Cohen-Macaulay $R$-module and $Q$ is a
parameter ideal of $M$, then $\mathrm{f}_{Q}^{0}(M)$ is the Cohen-Macaulay
type of $M$ (see Lemma 2.4). With this observation, we can regard Theorem 1.2
as a generalization of [10, Theorem 2.2]. As shown in Example 2.6, the
inequalities in Theorem 1.2 can be strict. It seems to us that the behavior of
the irreducibility function and irreducible multiplicity are more mysterious
than those of Hilbert function and Hilbert multiplicity. Example 2.7 shows
that the irreducible multiplicity is not compatible with the reduction ideal.

In the next section we prove the main results.

##  2\. Proof of main results

Let $(R,\mathfrak{m})$ be a Noetherian local ring. Then $R\ltimes M$ is also a
Noetherian local ring with unique maximal ideal $\mathfrak{m}\times M$, and
$\dim R\ltimes M=\dim R$. Let $\rho:R\ltimes M\rightarrow R$, $\rho((a,m))=a$
be the canonical projection and $\sigma:R\rightarrow R\ltimes M$,
$\sigma(a)=(a,0)$ be the canonical inclusion. Then $\rho$ and $\sigma$ are
local ring homomorphisms. Let $L$ be an $R$-module (resp. an $R\ltimes
M$-module). Then $L$ is an $R\ltimes M$-module (resp. an $R$-module).
Furthermore a submodule of $L$ as an $R$-module (resp. an $R\ltimes M$-module)
is also a submodule of $L$ as an $R\ltimes M$-module (resp. an $R$-module).
This follows that $\ell_{R}(L)=\ell_{R\ltimes M}(L).$

###### Proof of Theorem 1.1.

Let $I$ be an $\mathfrak{m}$-primary ideal of $R$. Set $J=I\times IM$. Then
$J$ is an $\mathfrak{m}\times M$-primary ideal of $R\ltimes M$ by [5, Theorem
3.6]. By [5, Theorem 3.3 (2)] and by induction we have $J^{n+1}=(I\times
IM)^{n+1}=I^{n+1}\times I^{n+1}M$. Hence

$\displaystyle R\ltimes M/J^{n+1}$ $\displaystyle\cong(R\ltimes
M)/(I^{n+1}\ltimes I^{n+1}M)$
$\displaystyle\cong\big{(}R/I^{n+1}\big{)}\ltimes\big{(}M/I^{n+1}M\big{)}.$

These implies that

$\ell_{R\ltimes M}(R\ltimes
M/J^{n+1})=\ell_{R}(R/I^{n+1})+\ell_{R}(M/I^{n+1}M).$

So,

${\operatorname{e}}_{J}^{0}(R\ltimes
M)\binom{n+d}{d}-\operatorname{e}_{J}^{1}(R\ltimes
M)\binom{n+d-1}{d-1}+\cdots+(-1)^{d}{\operatorname{e}}_{J}^{d}(R\ltimes M)$
$={\operatorname{e}}_{I}^{0}(R)\binom{n+d}{d}-\operatorname{e}_{I}^{1}(R)\binom{n+d-1}{d-1}+\cdots+(-1)^{d}{\operatorname{e}}_{I}^{d}(R)+$
${\operatorname{e}}_{I}^{0}(M)\binom{n+t}{t}-\operatorname{e}_{I}^{1}(M)\binom{n+t-1}{t-1}+\cdots+(-1)^{t}{\operatorname{e}}_{I}^{t}(M).$

By comparing the coefficients, we have the second statement on multiplicity in
the theorem.

Theorem 1.1 considers Hilbert function and Hilbert coefficients of $R\ltimes
M$ with respect to a homogeneous $\mathfrak{m}\times M$-primary ideal
$J=I\times IM$. Recall that the homogeneous ideals of $R\ltimes M$ have the
form $I\times N$, where $I$ is an ideal of $R$, $N$ is a submodule of $M$ and
$IM\subseteq N$. Prime ideals of $R\ltimes M$ have the form
$\operatorname{\mathfrak{p}}\times M$, where
$\operatorname{\mathfrak{p}}\in\operatorname{Spec}(R)$. So they are
homogeneous. However, a primary ideal need not be homogeneous. For example, in
local ring $\mathbb{Z}/4\mathbb{Z}\ltimes\mathbb{Z}/2\mathbb{Z}$ ideal $J$
generated by $(\overline{2},\overline{1})$ is a primary ideal but $J$ is not
homogeneous. The following corollary consider the multiplicity of $J$ on
$R\ltimes M$ for arbitrary $\mathfrak{m}\times M$-primary ideal $J$ of
$R\ltimes M$.

######  Corollary 2.1.

Let $J$ be an $\mathfrak{m}\times M$-primary ideal of $R\ltimes M$. Set
$I=\sigma^{-1}(J)$, where $\sigma:R\rightarrow R\ltimes M$, $\sigma(a)=(a,0)$
is the canonical inclusion. Then the following statements are true.

(i) If $d=t$, then $\operatorname{e}_{J}^{0}(R\ltimes
M)=\operatorname{e}_{I}^{0}(R)+\operatorname{e}_{I}^{0}(M)$.

(ii) If $d>t$, then $\operatorname{e}_{J}^{0}(R\ltimes
M)=\operatorname{e}_{I}^{0}(R)$.

###### Proof.

Since $I=\sigma^{-1}(J)$, $I$ is an $\mathfrak{m}$-primary ideal of $R$. Set
$J^{\prime}=J+(0)\times M$. Then $J^{\prime}=I\times M$. We have $J\subseteq
J^{\prime}$. Since $((0)\times M)^{2}=0$, we get that

$(J^{\prime})^{n+1}=(J+(0))^{n+1}=J(J^{\prime})^{n},$

for all $n\geq 1$. Hence $J$ is a reduction of $J^{\prime}$. This implies that

$e_{J}^{0}(R\ltimes M)=e_{J^{\prime}}^{0}(R\ltimes M)=e_{I\times
M}^{0}(R\ltimes M)=e_{I\times IM}^{0}(R\ltimes M).$

The conclusion (i) and (ii) follow from Theorem 1.1.

Cohen-Macaulayness of idealizations was considered in [5] by considering the
depth and the dimension of idealizations. Using the above result on the
multiplicity of idealizations, we can give another proof.

######  Corollary 2.2.

The following statements hold true.

  1. (i)

Assume $d=t$. Then $R\ltimes M$ is Cohen-Macaulay if and only if $R$ is Cohen-
Macaulay and $M$ are Cohen-Macaulay.

  2. (ii)

Assume $d>t$. Then $R\ltimes M$ is Cohen-Macaulay if and only if $R$ is Cohen-
Macaulay and $M=0$.

###### Proof.

Let $Q=(r_{1},\dots,r_{d})$ be a parameter ideal of $R$. Then
$\ell_{R}(M/QM)<\infty.$ Let $\overline{Q}$ be an ideal generated by elements
$(r_{1},0),\dots,(r_{d},0)$. It is easy to check that $\overline{Q}=Q\times
QM$. Since

$R\ltimes M/\overline{Q}\cong R/Q\ltimes M/QM,$

we get that $\ell_{R\ltimes M}(R\ltimes
M/\overline{Q})=\ell_{R}(R/Q)+\ell_{R}(M/QM)<\infty$. Hence $\overline{Q}$ is
a parameter ideal of $R\ltimes M$.

(i) Assume $d=t$. By Theorem 1.1,

$\operatorname{e}(\overline{Q},R\ltimes M)=\operatorname{e}(Q\times
QM,R\ltimes M)=\operatorname{e}(Q,R)+\operatorname{e}(Q,M).$

Then

$\ell(R\ltimes M/\overline{Q})-\operatorname{e}(\overline{Q},R\ltimes
M)=\ell(R/Q)-\operatorname{e}(Q,R)+\ell(M/QM)-e(Q,M).$

This implies that $R\ltimes M$ is Cohen-Macaulay if and only if $R$ is Cohen-
Macaulay and $M$ are Cohen-Macaulay.

(ii) Assume that $d>t$. Then

$\ell(R\ltimes M/\overline{Q})-\operatorname{e}(\overline{Q},R\ltimes
M)=\ell(R/Q)-\operatorname{e}(Q,R)+\ell(M/QM).$

It follows that $R\ltimes M$ is Cohen-Macaulay if and only if $R$ is Cohen-
Macaulay and $M=0$.

An elementary property of colon ideal in idealization is given in the
following lemma (see [1]). It is a preliminary to prove Theorem 1.2.

######  Lemma 2.3.

Let $I\times N$ and $I^{\prime}\times N^{\prime}$ be homogeneous ideals of
$R\ltimes M$. Then

$(I\times N):_{R\ltimes M}(I^{\prime}\times
N^{\prime})=((I:_{R}I^{\prime})\cap(N:_{R}N^{\prime}))\times(N:_{M}I^{\prime}).$

Furthermore $(I\times N):_{R\ltimes M}(I^{\prime}\times N^{\prime})$ is a
homogeneous ideal of $R\ltimes M$.

###### Proof of Theorem 1.2.

By [5, Theorem 3.3 (2)] and by induction on $n$ we have

$J^{n+1}=(I\times IM)^{n+1}=I^{n+1}\times I^{n+1}M.$

By Lemma 2.3,

$J^{n+1}:_{R\ltimes M}(\mathfrak{m}\times
M)=((I^{n+1}:_{R}\mathfrak{m})\cap(I^{n+1}M:_{R}M))\times(I^{n+1}M:_{M}\mathfrak{m}).$

Then

$\displaystyle\Big{(}J^{n+1}$ $\displaystyle:_{R\ltimes M}(\mathfrak{m}\times
M)\Big{)}/J^{n+1}$
$\displaystyle=\Big{(}((I^{n+1}:_{R}\mathfrak{m})\cap(I^{n+1}M:_{R}M))\times(I^{n+1}M:_{M}\mathfrak{m})\Big{)}/\Big{(}I^{n+1}\times
I^{n+1}M\Big{)}$
$\displaystyle\cong\Big{(}\big{(}(I^{n+1}:_{R}\mathfrak{m})\cap(I^{n+1}M:_{R}M)\big{)}/I^{n+1}\Big{)}\times\Big{(}(I^{n+1}M:_{M}\mathfrak{m})/I^{n+1}M\Big{)}.$

So,

$\displaystyle\begin{split}\ell_{R}((I^{n+1}M&:_{M}\mathfrak{m})/I^{n+1}M)\leq\ell_{R\ltimes
M}(J^{n+1}:_{R\ltimes M}(\mathfrak{m}\times M)/J^{n+1})\\\
&=\ell_{R}((I^{n+1}:_{R}\mathfrak{m})\cap(I^{n+1}M:_{R}M)/I^{n+1})+\ell_{R}((I^{n+1}M:_{M}\mathfrak{m})/I^{n+1}M)\\\
&\leq\ell_{R}(I^{n+1}:_{R}\mathfrak{m}/I^{n+1})+\ell_{R}((I^{n+1}M:_{M}\mathfrak{m})/I^{n+1}M).\end{split}$

This proves the first statement. Dividing by $n^{t-1}$ and taking
$n\rightarrow\infty$, we get that

$\mathrm{f}_{I}^{0}(M)\leq\mathrm{f}_{J}^{0}(R\ltimes
M)\leq\mathrm{f}_{I}^{0}(R)+\mathrm{f}_{I}^{0}(M).$

(i) $\operatorname{ir}_{M}(I^{n+1}M)=\operatorname{ir}_{R\ltimes M}(J^{n+1})$
if and only if $(I^{n+1}:_{R}\mathfrak{m})\cap(I^{n+1}M:_{R}M)=I^{n+1}.$ This
is equivalent to
$(0:_{R/I^{n+1}}\mathfrak{m})\cap(0:_{R/I^{n+1}}M/I^{n+1}M)=0$. Since
$R/I^{n+1}$ is Artinian local ring, the above condition means
$\operatorname{Soc}(M/I^{n+1}M)=0$ or $M/I^{n+1}M$ is faithful.

(ii) $\operatorname{ir}_{R\ltimes
M}(J^{n+1})=\operatorname{ir}_{R}(I^{n+1})+\operatorname{ir}_{M}(I^{n+1}M)$ if
and only if $I^{n+1}:_{R}\mathfrak{m}\subseteq I^{n+1}M:_{R}M$. This is
equivalent to $(I^{n+1}:_{R}\mathfrak{m})M\subseteq I^{n+1}M$ or
$(I^{n+1}:_{R}\mathfrak{m})M=I^{n+1}M$.

The type [7, Definition 1.2.15] of a Cohen-Macaulay $R$-module $M$ is called
the Cohen-Macaulay type of $M$, defined by
$r_{R}(M)=\dim_{R/\mathfrak{m}}\big{(}\operatorname{Ext}_{R}^{t}(R/\mathfrak{m},M)\big{)}$.
The Cohen-Macaulay type, the index of reducibility of $QM$, and the dimension
of socle $\operatorname{Soc}(M/QM)$, where $Q$ is a parameter of a Cohen-
Macaulay $R$-module $M$ coincide as in the following lemma.

######  Lemma 2.4.

Suppose that $M$ is a Cohen-Macaulay $R$-module and $Q$ is a parameter ideal
of $M$. Then

$r_{R}(M)=\ell_{R}((Q:_{M}\mathfrak{m})/QM)=\operatorname{ir}_{M}(QM)=\dim_{R/\mathfrak{m}}(\operatorname{Soc}(M/QM))=\mathrm{f}_{Q}^{0}(M).$

###### Proof.

Since $Q$ is an $\mathfrak{m}$-primary ideal,

$\operatorname{ir}_{M}(QM)=\ell_{R}((Q:_{M}\mathfrak{m})/QM)=\dim_{R/\mathfrak{m}}(\operatorname{Soc}(M/QM)).$

Since $M$ is Cohen-Macaulay, $r_{R}(M)=\ell_{R}((Q:_{M}\mathfrak{m})/QM)$.
Furthermore, we also have that $Q$ is generated by $M$-regular sequence of $t$
element. This follows that
$Q^{n}M/Q^{n+1}M\cong\big{(}M/QM\big{)}^{\binom{n+t-1}{t-1}}.$ Since $Q$ is
generated by $M$-regular sequence,

$Q^{n+1}M:\mathfrak{m}\subseteq Q^{n+1}M:Q=Q^{n}M.$

We have

$\displaystyle\operatorname{ir}_{M}(Q^{n+1}M)$
$\displaystyle=\ell_{R}\big{(}(Q^{n+1}M:\mathfrak{m})/Q^{n+1}M\big{)}$
$\displaystyle=\ell_{R}\big{(}0:_{Q^{n}M/Q^{n+1}M}\mathfrak{m}\big{)}$
$\displaystyle=\ell_{R}((Q:_{M}\mathfrak{m})/QM)\binom{n+t-1}{t-1}.$

So, $\mathrm{f}_{Q}^{0}(M)=\ell_{R}((Q:_{M}\mathfrak{m})/QM)$. Furthermore, we
get that

(1)
$\displaystyle\operatorname{ir}_{M}(Q^{n+1}M)=\mathrm{f}_{Q}^{0}(M)\binom{n+t-1}{t-1}.$

Lemma 2.4 is not true if $Q$ is not a parameter ideal of $R$ (See Example
2.7).

######  Corollary 2.5.

Let $M$ be a maximal Cohen-Macaulay $R$-module. Then

$\mathrm{r}_{R}(M)\leq\mathrm{r}(R\ltimes
M)\leq\mathrm{r}(R)+\mathrm{r}_{R}(M).$

Let $Q$ be a parameter ideal of $R$ and set
$\overline{R}=R/Q,\overline{M}=M/QM$. We then have the following.

(i) $\mathrm{r}(R\ltimes M)=\mathrm{r}_{R}(M)$ if and only if $\overline{M}$
is a faithful $\overline{R}$-module.

(ii) $\mathrm{r}(R\ltimes M)=\mathrm{r}(R)+\mathrm{r}_{R}(M)$ if and only if
$\left(Q:_{R}\mathfrak{m}\right)M=QM$.

###### Proof.

Assume $Q=(r_{1},\dots,r_{d})$ be a parameter ideal of $R$. Then $Q$ is a
parameter ideal of $M$. Let $\overline{Q}$ be an ideal of $R\ltimes M$
generated by $(r_{1},0),\dots,(r_{d},0)$. Then $\overline{Q}=Q\times QM$ is a
parameter ideal of $R\ltimes M$. Then (i) and (ii) follow from Theorem 1.2 and
Lemma 2.4.

######  Example 2.6.

([10, Example 2.3]) Let $k$ be a field and $\ell\geq 2$ be an integer. Set
$S=k[[X_{1},X_{2},\ldots,X_{\ell}]]$ the formal power series ring in variables
$X_{1},X_{2},\ldots,X_{\ell}$. Let $\mathfrak{a}=\mathbb{I}_{2}(\mathbb{M})$
denote the ideal of $S$ generated by the maximal minors of the matrix

$\mathbb{M}=\begin{pmatrix}X_{1}&X_{2}&\ldots&X_{\ell-1}&X_{\ell}\\\
X_{2}&X_{3}&\ldots&X_{\ell}&X_{1}^{q}\end{pmatrix},$

where $\mathfrak{q}\geq 2$. We set $R=S/\mathfrak{a}$. Then $R$ is a Cohen-
Macaulay local ring of dimension one. For each integer $2\leq p\leq\ell$, we
consider the ideal $I_{p}=(x_{1})+(x_{p},x_{p+1},\ldots,x_{\ell})$ of $R$,
where $x_{i}$ denotes the image of $X_{i}$ in $R$. Then by [10, Example 2.3],
$r(R)=\ell-1,$ $r(R\ltimes I_{p})=(l-p)+1$ and

$r_{R}(I_{p})=\begin{cases}\ell&\mbox{if}\,\,p=2\\\ \ell-1&\mbox{if}\,\,p\geq
3\end{cases}$

for each $2\leq p\leq\ell.$ Let $\ell=p=3$ and set $M=I_{3}$. we have

$r(M)<r(R\ltimes M)<r(R)+r(M).$

Note that $M$ is Cohen-Macaulay $R$-module of dimension 1. Let $Q=(r)$ be a
parameter ideal of $R$. Then it is a parameter ideal of $M$. Set
$\overline{Q}=(r,0)$. Then $\overline{Q}$ is a parameter ideal of $R\ltimes M$
and $\overline{Q}=Q\times QM$ (see Corollary 2.2). By Lemma 2.4,
$\mathrm{f}_{Q}^{0}(M)=r_{R}(M)$. So,

$\mathrm{f}_{Q}^{0}(M)<\mathrm{f}_{\overline{Q}}^{0}(R\ltimes
M)<\mathrm{f}_{Q}^{0}(R)+\mathrm{f}_{Q}^{0}(M).$

Since $R$, $M$ and $R\ltimes M$ are Cohen-Macaulay of dimension 1, by (1) in
proof of Lemm 2.4,

$\operatorname{ir}_{M}(Q^{n+1}M)<\operatorname{ir}_{R\ltimes
M}(\overline{Q}^{n+1})<\operatorname{ir}_{R}(Q^{n+1})+\operatorname{ir}_{M}(Q^{n+1}M).$

######  Example 2.7.

Consider the ring $R$ in Example 2.6, where $p=\ell=q=2.$ Then

$R=k[[x_{1},x_{2}]]/(x_{1}^{3}-x_{2}^{2})\cong k[[t^{2},t^{3}]],$

where $x_{1}\mapsto t^{2},x_{2}\mapsto t^{3}.$ Set $I=I_{2}$ and $J=(x_{1})$.
Then $I$ is the maximal ideal $\mathfrak{m}$ of $R$. Since $I^{2}=JI$, we get
that $J$ is a reduction ideal of $I$. By Example 2.6, $r(R)=\ell-1=1.$ Since
$J$ is a parameter ideal of the Cohen-Macaulay ring $R$,
$\mathrm{f}_{J}^{0}(R)=1$ by Lemma 2.4. Now, we compute
$\ell_{R}\big{(}(I^{n+1}:\mathfrak{m})/I^{n+1}\big{)}$. Since
$I^{n+1}=x_{1}^{n}I$,

$I^{n+1}:\mathfrak{m}=(t^{2n+2},t^{2n+3}):(t^{2},t^{3})=(t^{2n},t^{2n+1}).$

Hence

$\displaystyle\ell_{R}\big{(}(I^{n+1}:\mathfrak{m})/I^{n+1}\big{)}$
$\displaystyle=\ell_{R}\big{(}(t^{2n},t^{2n+1})/(t^{2n+2},t^{2n+3})\big{)}$
$\displaystyle=\dim_{k}(kt^{2n}+kt^{2n+1})$ $\displaystyle=2.$

So, $\mathrm{f}_{I}^{0}(R)=2$ and
$\mathrm{f}_{I}^{0}(R)\neq\mathrm{f}_{J}^{0}(R).$

Acknowledgment. The author wishes to express his thanks to S. Kumashiro for
useful discussion for this work.

## References

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#  Slotted ALOHA and CSMA Protocols for
FMCW Radar Networks

Haritha K1, Vineeth Bala Sukumaran2, Chandramani Singh1 2 Indian Institute of
Space Science and Technology, Trivandrum, India
<EMAIL_ADDRESS>1 Indian Institute of Science, Bangalore, India
{haritha<EMAIL_ADDRESS>
###### Abstract

We study medium access in FMCW radar networks. We assume that all the radars
use the same parameters, e.g., chirp duration, chirp slope, cutoff frequency,
number of chirps per packet, etc, and propose and analyze slotted ALOHA and
CSMA protocols to mitigate narrowband interference. We define a notion of
throughput to quantify the performance of the proposed protocols. In the case
of ALOHA, we analyze interference probability and throughput as functions of
the system parameters. We observe that interference probability and throughput
may behave differently than in wireless communication networks. For instance,
if the number of chirps per packet is larger than one, the interference
probabilities may be smaller for higher transmission rates. We define a medium
sensing procedure, referred to as clear channel assessment (CCA), as a part of
the proposed CSMA, and also define CCA success and failure events. In CSMA,
the radars transmit only after a successful CCA. We study, CCA success
probability, interference probability, and throughput as functions of the
system parameters. We observe that, unlike wireless communication networks,
using the highest possible attempt rates may maximize throughput in a few
network scenarios. We perform an extensive simulation to verify our analytical
results and to compare slotted ALOHA and CSMA. We observe that CSMA
outperforms ALOHA in all realistic scenarios.

##  1 Introduction

Self-driving cars, also referred to as driverless or autonomous cars are
poised to become a reality in the next five to ten years. To ensure the safety
and efficiency of transportation systems, self-driving cars must have a
$360$-degrees view of their surrounding in a way that is ecologically and
economically sustainable. These cars need various sensors such as optical
cameras, millimeter-wave radars, and laser imaging detection and ranging to
map the environment including cars, obstacles, and pedestrians surrounding
them.

Millimeter-wave Frequency Modulated Continuous Wave (FMCW) radars [1] are
long-range radars that can reliably detect distance, velocity, angle of other
vehicles and objects in the vicinity self-driving cars. These radars operate
in the $76$-$77$ GHz band [2] with a bandwidth of $1$ GHz. These can be made
with compact antennas while retaining sufficiently high gain and narrow
beamwidth to offer high range and angular resolution; they can offer
localization sensitivity up to $3$ cm. FMCW radar electronics are robust, and
radar frequencies are generally immune to adverse weather conditions
(snow/fog/rain or optical effects) [2].

For reliable operation of FMCW radars, it is imperative that received radar
signals be interference-free. Interference in $76$-$77$ GHz band will be a
crucial challenge in near future in view of the likely increase in: (a) the
number of self-driving cars from multiple manufacturers, (b) the number of
radars per car, and (c) the operating duty cycle per radar. There are two
kinds of inter-radar interference [3], [4]. $(1)$ Wideband interference: This
is applicable to heterogeneous radar networks in which different radars use
different chirp slopes. This manifests itself as an increase in noise floor,
leading to signal-to-noise-ratio (SNR) degradation, and therefore misdetection
of targets. $(2)$ Narrowband interference: this happens if a radar has another
(interfering) radar using the same chirp slope in its vicinity and the beat
signal frequency due to interfering radar is less than the cut-off frequency
of the Low Pass Filter (LPF). This causes ghost detections (or, false alarms)
[1]. Both types of interference deteriorate radar performance, but mitigation
of narrowband interference requires signal processing as well as medium access
techniques [5]. In this work, we focus on the mitigation of narrowband
interference using medium access techniques. In this work, we focus on the
mitigation of narrowband interference.

Medium access techniques have been widely studied in the context of wired and
wireless communication networks. In communication networks, if a node receives
signals from two simultaneously transmitting nodes, it cannot decode any of
the signals; this phenomenon is termed as collision. A number of centralized
(e.g., TDMA) and distributed (e.g., ALOHA, CSMA, CSMA-CA) multiple access
protocols have been proposed to minimize interference in communication
networks. In the context of FMCW radars, medium access control can reduce
false alarms but may also lead to missed targets. Design and analysis of
medium access control protocols is pivotal for large scale deployment of radar
networks.

###  1.1 Related work

Jin et al. [6] considered a network of FMCW radars with identical chirp
parameters and obtained statistical characterizations of interference signal
properties under an asynchronous pure random access scheme. A received chirp
is classified as a false alarm or a real target based on its received power.
Further, they considered continuous time setup and the radars chose
transmission start time randomly and after that, they transmit chirps
continuously. In our work, we consider slotted time structure and there is a
random time duration between the successive packets.

In [7] and [8] the authors use carrier sensing to mitigate the interference.
Ishikawa et al. [7] considered collocated radars with identical parameters.
The authors proposed a carrier sense multiple access (CSMA) algorithm to avoid
the interference of FMCW radar signals by shifting the transmission timing
after sensing the medium. But, they do not analyze the performance of the
algorithm when the radars are non-collocated. We propose a variant of CSMA and
discuss its performance in both collocated and non-collocated radar networks.

Ammen et al. [8] propose a technique that generates an interference replica of
the ghost signal during carrier sensing. This is subtracted from the received
signal in the frequency spectrum to mitigate narrowband interference.

To mitigate narrowband interference different techniques, e.g., TDM, FDM,
phase coding, frequency hopping, and chirp sequence are used to orthogonalize
the chirp transmissions in [9], [10], [11], [4], [12], [13], and [14].

Centralized medium access mechanisms for radars were proposed in [12] and
[13]. In [12], a control center was setup to receive speed and location
information from all the nearby radars. Using these the controller computes
the waveform parameters that provide orthogonal access to the medium and
dispatches them to each one to avoid interference. In [13], the authors
proposed a resource allocation strategy that relies on the existing
communication infrastructure to jointly assign slope directions and carrier
frequency offset to all the radar units operating in its vicinity.

Aydogdu et al. [9] proposed a medium access control protocol RadChat. It is a
coordinated framework among the nearby vehicles and uses time division and
frequency division techniques to mitigate interference caused by neighboring
radars. The schemes discussed in [9] and [12] use time synchronous orthogonal
techniques to avoid interference. The challenge in these time synchronous
schemes is propagation delay ($\approx$us) can deteriorate the efficacy of
orthogonality.

Son et al. [14] proposed an algorithm where they assign a chirp sequence for
FMCW radar. They design a chirp sequence set such that the slope of each
vehicle’s chirp sequence will not overlap within the set. By assigning one of
the chirp sequences to each vehicle, interference can be mitigated.

Jin et al. [10] proposed asynchronous non-cooperative protocols to mitigate
interference in radars. They quantify the network performance using cross
layer performance metrics such as multiple access capacity and target
misdetection probability. They explore different combination of frequency
hopping, phase coding, and random frequency division multiple access
techniques to avoid interference.

Luo et al. [11] proposed a frequency-hopping random chirp technique that
reconfigures the chirp sweep frequency and time in every cycle. This results
in a noise-like frequency response after the received signal is demodulated at
the receiver. Rao et al. [4] used clock drift that changes the slope of
interfering radar with respect to the tagged radar. Further, they employ
binary phase encoding to mitigate interference.

Al-Hourani et al. [15] considered a Poisson point process and a Bernoulli
lattice process as models for the spatial distribution of radars. Using
stochastic geometry they compute interference statistics and obtain analytical
expressions for the probability of successful range estimation.

###  1.2 Our Contribution

We consider slotted ALOHA protocol, wherein the radar transmits for a fixed
amount of time and enters into backoff state. We propose CSMA, in which each
radar senses the medium before transmitting the packet.

  * •

We derive the probability of interference in FMCW radar networks assuming that
all the radars are synchronized and collocated in ALOHA.

  * •

We also derive the probability of interference in the scenario where the
radars are not collocated and their clocks are not synchronized, i.e., each
pair of radars is associated with a non-zero propagation time in ALOHA.

  * •

In both the cases, as expected, the interference probability increases with
medium access probability if the number of chirps per packet is one. However,
if packets consist of more than one chirp, the interference probability first
increases and then decreases with medium access probability.

  * •

We define a notion of throughput and derive throughput for both of the above
scenarios. If each packet consists of exactly one chirp, the throughput first
increases and then decreases with medium access probability. In this case, we
derive the expression for the optimal medium access probability. When the
packets consist of more than one chirp the throughput first increases and
reaches the maximum, then decreases and increases with medium access
probability.

  * •

We propose a CSMA protocol that senses the medium before transmitting a
packet. we define a medium sensing procedure called clear channel assessment
(CCA) and CCA success and failure events. Further, we analyze the CSMA
protocol using fixed point analysis for a restricted class of radar networks.

  * •

We perform extensive simulation to verify our analytical results and to
compare slotted ALOHA and CSMA.

##### Organization of the paper

We explain the operation of a FMCW radar in Section 2.1 and introduce the
system model in Section 2.2. In Section 3 we analyze slotted ALOHA and derive
the probability of interference and throughput. In Section 4 we propose CSMA
protocol. In Sections 4.1 and 4.2 we analyze CSMA in collocated and non-
collocated radar networks respectively. In Section 5 we verify our analytical
results obtained and compare the performance of slotted ALOHA and CSMA
protocols.

##  2 System Model

###  2.1 FMCW Radars

In this section we explain the operation of a FMCW radar and narrowband
interference using radar transmit/receive block Figure 1 [6]. The FMCW radar
is used to measure the range, velocity, and angle of arrival of a target in
front of it. The radar generates a sinusoid signal called chirp whose
frequency increases linearly with time. The chirp is characterized by
following parameters, starting frequency $(f_{\min})$, ending frequency
$(f_{\max})$, bandwidth $(B)$, chirp duration $(T_{c})$, and slope
$(h=\frac{B}{T_{c}})$ as shown in Figure 2.

To explain the operation of FMCW radar we consider a single target in front of
the radar at distance $d$. The MAC scheduler schedules the chirps
transmissions and the synthesizer generates the chirps at these scheduled
times. The transmitting antenna transmits them. The reflected chirp from the
target is received by receiving antenna with a delay $\tau=\frac{2d}{c}$. The
reflected and transmitted chirps are given as inputs to a mixer. The mixer is
a device that takes two sinusoids as inputs and generates another sinusoid
whose instantaneous frequency is equal to the absolute difference of
instantaneous frequencies of the input signals. Note that the reflected chirp
is delayed by $h\tau$ from the transmitter chirp as shown Figure 2. The target
at distance $d$ generates an IF signal with frequency $\frac{2dh}{c}$ at the
output of the mixer. This is further passed through a LPF with cut-off
frequency $f_{H}$. The cut-off frequency is a function of the sampling
frequency of the ADC. The IF signal is further digitized using ADC and
converted into frequency domain using DFT. The detector detects a peak at
frequency $\frac{2dh}{c}$. Note that the maximum frequency that can be
observed is limited by the cut-off frequency of the LPF. This leads to,

$\displaystyle h\tau$ $\displaystyle<f_{H}$
$\displaystyle\tau<\frac{f_{H}}{h}$ $\displaystyle\text{ and
}d<\frac{cf_{H}}{2h}.$ (1)

##### Target range $(d_{\max})$

The maximum distance between the radar and the target such that the
corresponding IF signal is passed through the LPF. This is given by
$\frac{cf_{H}}{2h}$.

##### Slot $(\Delta)$

The maximum round trip delay between the radar and the target such that the
corresponding IF signal is passed through LPF and is given by $f_{H}/h$.

Figure 1: TX/RX block diagram for a FMCW radar Figure 2: Illustration of the
transmitted chirp, received chirp, and the IF signal obtained by mixing

##### Multiple targets in front of radar

Now consider the scenario where there are two targets in front of the radar
with in the target range and we want to compute their distance from the radar.
Let $d_{1},d_{2}$ be the distances from the radar. Ideally we should be able
to observe two peaks in the frequency spectrum at
$\frac{2hd_{1}}{c},\frac{2hd_{2}}{c}$ corresponding to the targets at distance
$d_{1},d_{2}$ respectively. The maximum amount of time available to observe
the reflected chirps is $T_{c}$ (see Figure 2). This observation window
$T_{c}$ can distinguish frequency components that are separated by more than
$\frac{1}{T_{c}}$Hz [1]. So, to observe two peaks corresponding to the
distances $d_{1}$ and $d_{2}$ in the frequency spectrum,

$\displaystyle\frac{2hd_{1}}{c}-\frac{2hd_{2}}{c}$
$\displaystyle>\frac{1}{T_{c}}$ $\displaystyle d_{1}-d_{2}>\frac{c}{2B}$ (2)

where $B$ is the bandwidth of the chirp.

##### Range Resolution $(\Delta d)$

Range resolution is the minimum distance between the two targets such that the
radar can distinguish them and it is given by $\Delta d=\frac{c}{2B}$.

######  Remark 2.1.

If the distance between two targets is less than the range resolution then we
observe only a single peak in the DFT spectrum and radar detects them as a
single target.

##### False alarm or interference

Now consider two FMCW radars with identical parameters and separated by a
distance $d$. Let us label these as first radar and second radar respectively.
Each radar receives chirps transmitted by the other radar as shown in Figure
3.

Let $t_{1},t_{2}$ be the start times of chirps at the first radar and the
second radar respectively. Then at time $t_{3}=t_{2}+\frac{d}{c}$ the first
radar receives the chirp transmitted by the second radar. If
$\tau=|t_{1}-t_{3}|<\Delta$ (see Figure 3a) a signal at frequency
$f_{IF}=h\tau$ is observed at the output of LPF. Note that the first radar
assumes the chirp received from the second radar as reflected version of its
own chirp and computes the distance. The first radar assumes that there is a
target at distance $d_{g}=\frac{cf_{IF}}{2h}$. We call this a ghost target
because there is no real target at this distance. The phenomenon of a radar
detecting a ghost target is called a false alarm. We also call this
interference. Note that the first radar does not observe a false alarm for
every chirp received from the second radar. The following time condition has
to be satisfied to observe a false alarm at the first radar.

$\displaystyle t_{1}-\Delta<t_{2}+\frac{d}{c}<t_{1}+\Delta.$ (3)

We define the Interference range $(d_{I,\max})$ as the maximum distance
between two radars such that the interference is observed. It is a function of
chirp power transmitted by the second radar. Any radar outside the
interference range contributes no interference.

Figure 3: Illustration of a "ghost target" being detected at radar $1$. In (a) and (b) we illustrate the transmitted and received chirps of radars $1$ and $2$. If a radar (radar $1$) receives a transmission within $(t_{1}-\Delta,t_{1}+\Delta)$ then we have interference and a ghost target is detected. Symbol | Description
---|---
$f_{\min}$ | Chirp starting frequency
$f_{\max}$ | Chirp ending frequency
$B$ | Chirp bandwidth
$h$ | Chirp slope
$T_{c}$ | Chirp duration in seconds
$M$ | Number of radars
$K$ | Chirp duration in number of slots
$L$ | Number of chirps per packet
$W$ | Number of backoff slots
TABLE I:  Notations and symbols used

###  2.2 FMCW Radar Networks

We consider a network of $M$ radars with identical chirp parameters. The
radars operate using a slotted time structure with slot size $\Delta$. We
assume that the chirp duration is an integer multiple of slot size, i.e.,
$T_{c}=K\Delta,$ where $K$ is a positive integer. Further, we define the clock
offset $\delta_{k,l}$ of radar $l$ with respect to radar $k$ as the time
duration between a radar $k$’s clock tick and a radar $l$’s clock tick that
comes immediately after radar $k$’s clock. Note that
$\delta_{k,l}\in[0,\Delta)$. We call a network synchronous if
$\delta_{k,l}=0,\forall k,l\in\\{1,2,\dots M\\}$ else we call it asynchronous.
As described in Section 2.1, radars’ transmissions interfere with each other.
The radars employ a MAC protocol to schedule their transmissions for the
possible reduction of interference. In general, in a MAC protocol, each radar
interleaves transmissions with silence periods, termed as backoffs. The
backoff durations are random whereas a fixed number of, say $L$, chirps are
contiguously transmitted before going for the next backoff. We refer to the
contiguous sequence of $L$ chirps as a packet.

We introduce two MAC protocols, namely slotted ALOHA and CSMA, in Sections 3
and 4, respectively. Here, we define a metric, called throughput, to quantify
the performance of a MAC protocol. It is the fraction of time slots spent by
all the radars in transmitting packets successfully without interference.

##### Throughput $(\Theta)$

Let $S(T)$ be the total number of packets transmitted successfully by all the
radars in a network in $T$ slots. Then the throughput of the network is given
by,

$\displaystyle\Theta=\lim_{T\rightarrow\infty}KL\frac{S(T)}{T},$ (4)

where $K$ is the chirp length in slots and $L$ is the number of chirps per
packet.

##### Effect of backoff on target detection

Let us define a backoff and the subsequent packet transmission as a
transmission cycle (see Figure 4). The length of a transmission cycle is
typically much shorter than $d_{\max}/v_{\max}$ where $v_{\max}$ is the
maximum relative speed of the target; it is the time derivative of the
distance between the radar and the target. For instance, consider the
following example with typical radar parameters and vehicle speeds [7]. The
slot $\Delta=\frac{4}{3}\text{x}10^{-6}$ s and $d_{\max}=200$m. Let $W$ be the
number of backoff slots. The change in distance between target and radar in
this duration is $\Delta Wv_{\max}$ meters. We consider typical values,
$v_{\max}=50$m/s ($180$kmph), $W=100$. The distance traveled by the target in
$W$ slots is $0.67$cm. We have used typical values for radar parameters and
vehicle speeds in this calculation, the actual values may differ but would be
of the same order. Consider a target just entered into the radar range when
the radar is in backoff state. Let $d$ be the distance between them. The
change in $d$ before and after backoff is at most in the order of centimeters
and very small compared to $d_{\max}\approx 200$m. So, it takes several
transmission cycles for a target to come alarmingly close to the radar after
it enters the transmission range of the radar. Therefore, the radars are
unlikely to miss the targets because of backoffs.

We tabulate the symbols and notations used in TABLE I. Further, we use
${\mathbb{E}}[.]$ for expectation and Geo $(p)$ for geometric distribution
with parameter $p$.

##  3 Slotted ALOHA

##### Slotted ALOHA

We consider a slotted ALOHA protocol wherein each radar transmits $L$ chirps
contiguously and enters into backoff for $W$ slots (see Figure 4) and repeats
the same. Successive backoffs constitute a sequence of independent and
identically distributed (i.i.d) random variables, each having cumulative
distribution function $F$. Let $W$ denote a generic random variable with
distribution $F$. We assume that ${\mathbb{E}}[W]$ is finite.

Figure 4: Transmission cycle

In ALOHA each radar transmits packets irrespective of the other radars’
transmissions in the network. So, each radar’s packets can be interfered by
other radars’ packets.

In this section, we analytically characterize the throughput of slotted ALOHA
using the probability of interference. We assume all radars in the network
have identical chirp parameters. Further, the clock offset of radar $l$ with
respect to radar $k$ is given by $\delta_{k,l}$ and the distance between them
is given by $d_{k,l}$. We compute the probability of interference in the
following network scenarios.

  1. 1.

We assume that all the $M$ radars are collocated, i.e., we ignore the
propagation delays and that their clocks are synchronized i.e.,
$\delta_{k,l}=0,\text{ for all }k,l$. We use the insights from this scenario
to analyze the second scenario, described below.

  2. 2.

We assume that all the $M$ radars are non-collocated and each radar
$k,k\in\\{1,\dots,M\\}$ can potentially interfere with all the other radars
$l,l\in\\{1,\dots,M\\}\setminus k$ in the network. Further, we assume clocks
are not synchronized.

In order to derive the probability of interference, we define the state of a
radar in a slot as follows. We say that a radar is in state $0$ if it is in
backoff. Further, if a radar is transmitting $i$-th slot in the packet then we
say that it is in state $i,i\in\\{1,\dots KL\\}$. Recall that in ALOHA
protocol each radar transmits packets irrespective of other radars’ packet
transmissions. So, the state evolution of each radar is independent of other
radars. Let us define $\pi=(\pi_{0},\pi_{1},\pi_{2},\dots\pi_{KL})$ where
$\pi_{i},i\in\\{0,1,\dots KL\\}$ is long term fraction of slots the radar
spends in state $i$. We evaluate $\pi$ in the following and use it in the
Propositions 1, 2 and 3 to compute the probability of interference and
throughput.

The radar’s state evolves according to a renewal process with the slots in
which the radar state is $1$ being the renewal epochs. The duration of a
transmission cycle $C_{i},i\in\\{1,2,\dots\\}$ is random and independent
across $i$. Further, $\mathbb{E}[C_{i}]=KL+\mathbb{E}[W]$. With this we define
renewal epochs at a radar

$Z_{j}=\sum_{i=1}^{j}C_{i},\text{ and }Z_{0}=0$

and renewal process

$\phi(T)={\sup\\{j\geq 0:Z_{j}\leq T\\}},$

where $\phi(T)$ denotes the number of radar transmission cycles occurred until
time slot $T$. Let $R_{i}$ be the reward associated with $i$-th transmission
cycle. The total reward earned in $\phi(T)$ cycles is given by

$R(T)=\sum_{i=1}^{\phi(T)}R_{i}(T).$

We define this reward as the amount of time the radar has spent in
transmitting chirps i.e., $R_{i}=KL,i\in\\{1,2,\dots\phi(T)\\}$. From the
Elementary Renewal theorem and Renewal Reward theorem [16]

$\displaystyle\lim_{T\rightarrow\infty}$
$\displaystyle\frac{\phi(T)}{T}=\frac{1}{KL+\mathbb{E}[W]}\text{ almost
surely},$ $\displaystyle\lim_{T\rightarrow\infty}$
$\displaystyle\frac{R(T)}{T}=\frac{KL}{KL+\mathbb{E}[W]}\text{ almost surely
}.$ (5)

We defined

$\pi_{i}=\lim_{T\rightarrow\infty}\frac{\text{The amount time spent in state
}i\text{ until slot }T}{T},$

using (3) in above we write,

$\displaystyle\pi_{0}$ $\displaystyle=\frac{\mathbb{E}[W]}{KL+\mathbb{E}[W]},$
$\displaystyle\pi_{1}$
$\displaystyle=\pi_{2}=\dots=\pi_{KL}=\frac{1}{KL+\mathbb{E}[W]}.$ (6)

Now we consider the scenarios described at the beginning of this section to
compute interference probability. First, we compute the probability of
interference in two radar network and using this result we compute in $M$
radar network. In two radar network we label the radar at which we are
observing as tagged radar and the other one as interfering radar. Let $d$ and
$\delta$ be the distance between the two radars and clock offset of
interfering radar with respect to the tagged radar respectively. We first
consider synchronous and collocated radar network. We consider asynchronous
and non-collocated radar network subsequently.

###  3.1 The case $\delta=0,d=0$

In this case the time condition (3) for interference at the tagged radar in
the slotted system reduces to

$t_{1}-1<t_{2}<t_{1}+1,$

where $t_{1},t_{2}$ indicate chirp start time slot indices at the tagged and
interfering radars respectively. As $t_{1},t_{2}$ are integers the above
condition gives $t_{1}=t_{2}$, i.e., if two chirps at the tagged and
interfering radars start at the same slot then tagged radar suffers
interference. Recall that (see Figure 4) each radar transmits $L$-chirps
contiguously in one transmission cycle. So, the interference can be caused by
any one of the $L$-chirps from interfering radar to any one of the $L$-chirps
at the tagged radar as shown in Figure 5. Further, we say that a packet is
successfully transmitted if none of its chirps are interfered by the packets
from other radars.

Without loss of generality let us assume a packet transmission started at the
tagged radar at slot $t_{1}$. Let us define following events at the
interfering radar $\\{E_{-(L-1)},\dots E_{0},E_{1}\dots E_{L-1}\\}$ where
event $E_{i},i\in\\{-(L-1),\dots L-1\\}$ is starting packet transmission at
slot $t_{1}+iL$. Further, observe that these are the only events that can
cause interference at the tagged radar.

This is explained for the case $L=2$ in Figure 5. The interfering events
possible in are $\\{E_{-1},E_{0},E_{1}\\}$. We have shown events
$E_{-1},E_{1}$ starting packet transmission at slots $t_{1}-2,t_{1}+2$
respectively. In the case of $E_{-1}$ first and second chirps from the tagged
and interfering radars started at the same slot. Similarly, in $E_{1}$, second
and first chirps from tagged and interfering radars started at the same slot.

Figure 5: Illustrating the interference events $E_{-1},E_{1}$ for $K=2,L=2$,
a. The tagged radar transmitting a packet, b. Interfering events at the
interfering radar

Given the tagged radar started transmission cycle at slot $t_{1}$ the only
events that can cause interference are $\\{E_{-(L-1)},\dots E_{0},E_{1}\dots
E_{L-1}\\}$. From (3) we have the probability of starting packet transmission
in any slot is
${\mathbb{P}}(E_{i})=\frac{1}{KL+\mathbb{E}[W]},i\in\\{-(L-1)\dots L-1\\}$.
Using this we define the probability of interference at the tagged radar as

$\displaystyle p_{I2}=\mathbb{P}\Big{(}\cup_{i=-(L-1)}^{L-1}E_{i}\Big{)}.$

In Proposition 1, we derive the probability of interference at the tagged
radar in two radar network and extend that to the case of a $M$-radar network.

######  Proposition 1.

  1. 1.

The interference probability at the tagged radar in two radar network is given
by,

$\displaystyle p_{I2}=$ $\displaystyle\frac{2L-1}{KL+\mathbb{E}[W]}$
$\displaystyle-\frac{1}{KL+\mathbb{E}[W]}\sum_{i=1}^{L-1}(L-i)\mathbb{P}(W=(i-1)K).$

  2. 2.

The interference probability at the tagged radar in $M$ radar network is given
by,

$\displaystyle p_{I}=1-(1-p_{I2})^{M-1}.$

###### Proof:

We observe that $\\{E_{-(L-1)}\dots E_{0}\\}$ are mutually exclusive and
$\\{E_{0},\dots E_{(L-1)}\\}$ are mutually exclusive. Further,
$E_{i},i\in\\{1,\dots L-1\\}$ and $E_{-(L-j)},j\in\\{1,\dots i\\}$ are not
mutually exclusive. Now consider,

$\displaystyle p_{I2}=\mathbb{P}$
$\displaystyle\big{(}\cup_{i=-(L-1)}^{L-1}E_{i}\big{)}$ $\displaystyle=$
$\displaystyle\sum_{i=-(L-1)}^{L-1}\mathbb{P}\big{(}E_{i}\setminus(\cup_{j=-(L-1)}^{i-1}E_{j})\big{)}$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}$
$\displaystyle\sum_{i=-(L-1)}^{0}\mathbb{P}(E_{i})+\sum_{i=1}^{L-1}\mathbb{P}\big{(}E_{i}\setminus(\cup_{j=-(L-1)}^{i-1}E_{j})\big{)}$
$\displaystyle=$
$\displaystyle\sum_{i=-(L-1)}^{(L-1)}\mathbb{P}(E_{i})-\mathbb{P}\Big{(}E_{i}\cap(\cup_{j=-(L-1)}^{i-1}E_{j})\Big{)}$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{=}}$
$\displaystyle\sum_{i=-(L-1)}^{L-1}\mathbb{P}(E_{i})-\sum_{i=1}^{L-1}\sum_{j=1}^{i}\mathbb{P}(E_{i}\cap
E_{-(L-j)}).$ (7)

The $(a)$ and $(b)$ follows from mutually exclusive nature of
$\\{E_{-(L-1)}\dots E_{0}\\}$ and $\\{E_{1},\dots E_{(L-1)}\\}$ respectively.
Now consider second term from the above.

$\displaystyle\sum_{i=1}^{L-1}$
$\displaystyle\sum_{j=1}^{i}\mathbb{P}(E_{i}\cap E_{-(L-j)})$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{L-1}\sum_{j=1}^{i}\mathbb{P}(E_{-(L-j)})\mathbb{P}(E_{i}|E_{-(L-j)})$
$\displaystyle=$
$\displaystyle\frac{1}{KL+\mathbb{E}[W]}\sum_{i=1}^{L-1}\sum_{j=1}^{i}\mathbb{P}(W=(i-j)K)$
$\displaystyle=$
$\displaystyle\frac{1}{KL+\mathbb{E}[W]}\sum_{i=1}^{L-1}(L-i)\mathbb{P}(W=(i-1)K).$

In the second equality follows from (3) and given $E_{-(L-j)}$, $E_{i}$ takes
place if backoff duration ($W$) is exactly $(i-j)K$ slots. Using above in
(3.1) gives

$\displaystyle p_{I2}=$ $\displaystyle\frac{2L-1}{KL+\mathbb{E}[W]}$
$\displaystyle-\frac{1}{KL+\mathbb{E}[W]}\sum_{i=1}^{L-1}(L-i)\mathbb{P}(W=(i-1)K).$
(8)

The probability of tagged radar suffering no interference in two radar network
is given by $1-p_{I2}$, and in $M$ independent radar network is given by
$(1-p_{I2})^{M-1}$. So, the probability of interference at the tagged radar in
$M$ radar network is,

$p_{I}=1-(1-p_{I2})^{M-1}.$

∎

######  Remark 3.1.

If the distance between the tagged and interfering radars is $d=j\Delta
c,j\in\\{0,1,\dots\\}$, we have the same expression for $p_{I2}$.

###  3.2 The case $\delta>0,d\geq 0$

Consider the two radar system with $\delta\in[0,\Delta),d>0$. The time
condition for interference (3) at the tagged radar in this scenario reduces to

$(t_{1}-1)\Delta<t_{2}\Delta+\delta+\frac{d}{c}<(t_{1}+1)\Delta,$

where $t_{1},t_{2}$ are chirp start time slot indices of tagged and
interfering radars, $d$ is distance and $\delta$ is the clock offset at
interfering radar with respect to the tagged radar. Now consider above time
condition, and let $\delta_{1}=\delta+\frac{d}{c}$.

$\displaystyle(t_{1}-1)\Delta<$ $\displaystyle
t_{2}\Delta+\delta_{1}<(t_{1}+1)\Delta$
$\displaystyle(t_{1}-1)-\frac{\delta_{1}}{\Delta}<$ $\displaystyle
t_{2}<(t_{1}+1)-\frac{\delta_{1}}{\Delta}.$

As $t_{1},t_{2}$ are integers,

$\displaystyle
t_{2}=t_{1}-1-\Big{\lfloor}\frac{\delta_{1}}{\Delta}\Big{\rfloor}\text{ or
}t_{1}+1-\Big{\lceil}\frac{\delta_{1}}{\Delta}\Big{\rceil}.$ (9)

Observe that irrespective of the exact values of $d,\delta$ chirp start slot
index $t_{2}$ can take two successive integers values. If the chirp at the
interfering radar starts at any of these two slot indices then tagged radar
suffers interference. Note that in the first scenario $d=0,\delta=0$ (Section
3.1) the number of possible slot indices that can cause interference was only
one.

Now we define events similar to the one defined in Section 3.1 that can cause
interference at the tagged radar. Without loss of generality let us assume a
packet transmission started at the tagged radar at time slot $t_{1}$. Let us
define events $\\{E_{-(L-1)},\dots,E_{L-1}\\}$ at the interfering radar, where
event $E_{i},i\in\\{-(L-1)\dots L-1\\}$ is starting packet transmission at
slot

$t_{1}+iK-1-\Big{\lfloor}\frac{\delta_{1}}{\Delta}\Big{\rfloor}\text{ or
}t_{1}+iK+1-\Big{\lceil}\frac{\delta_{1}}{\Delta}\Big{\rceil}.$

These are the only events that can cause interference at the tagged radar.

For example consider $L=2,K=2,\delta=0.5\Delta,d=0.2c\Delta,\text{ so
}\delta_{1}=0.7\Delta$. Consider $E_{0}$, transmission cycle at interfering
starts at slot $t_{2}=t_{1}-1$ or $t_{2}=t_{1}$. Similarly for $E_{1}$, the
transmission cycle starts at $t_{2}=t_{1}+1\text{ or }t_{1}+2$ as explained in
Figure 6.

Figure 6: Illustrating the interference events $E_{0},E_{1}$ for $K=2,L=2$, a.
The tagged radar transmitting a packet, b. Interfering events at the
interfering radar

We compute the probability of these events in Lemma 3.1.

######  Lemma 3.1.

The probability of event $E_{i},i\in\\{-(L-1)\dots L-1\\}$ is given by

$\displaystyle\mathbb{P}(E_{i})=\begin{cases}\frac{2}{KL+{\mathbb{E}}[W]}\text{
for }KL>1\\\
\frac{1}{1+{\mathbb{E}}[W]}+\frac{1-{\mathbb{P}}(W=0)}{L+{\mathbb{E}}[W]}\text{
for }KL=1.\end{cases}$ (10)

###### Proof:

First we consider the case $KL=1$. In this case we have only one chirp of
length one slot in each transmission cycle. Let $t_{1}$ be the start of packet
transmission at the tagged radar. From (9) we have two successive slots in
which if the packet transmission starts at the interfering radar the tagged
radar suffers interference. Let us call these $t_{2},t_{2}^{\prime}$. Observe
that when $KL=1$, two consecutive packets can start in $t_{2},t_{2}^{\prime}$
if the backoff duration is zero between them as shown in Figure 7b. Note that
if $KL>1$ two consecutive packets can not start in any two successive slots.
The event $E_{i}$ is defined as the packet starting in any one of these slots.

Figure 7:  Illustrating the interference event $E_{0}$ for $KL=1$, a. The
tagged radar transmitting a packet, b. Interfering events at the interfering
radar

In order to compute ${\mathbb{P}}(E_{i})$ let us define the state of a radar
in a slot as follows. We say that the radar is in state $0$ if it is
transmitting. Further, we say that it is in state $j$ if it is in $j$-th
consecutive backoff slot. Note that the latter case arises only if the radar
is in the middle of a backoff of length greater than or equal to $j$. Also,
notice that these definitions of states are different from those in Section 3.
Let us define $\alpha_{j}$ to be the long term fraction of slots the radar
spends in state $j$. The radar’s state evolves according to a renewal process
with the slot in which the radar is in state $0$ being the renewal epochs. We
can therefore derive $\alpha_{j},\text{ for }j\geq 1$s as follows.

$\displaystyle\alpha_{j}$
$\displaystyle=\lim_{T\rightarrow\infty}\frac{\text{No. of transmission cycles
until slot }T\text{ }}{T}{\mathbb{P}}(W\geq j)$
$\displaystyle=\frac{{\mathbb{P}}(W\geq j)}{1+{\mathbb{E}}[W]},$

the last equality follows from Elementary Renewal theorem [16].

Now consider the event $E_{i}$ at interfering radar i.e., starting the chirp
at slot $t_{2}$ or radar is in the last time slot of the backoff at slot
$t_{2}$, (this automatically makes $t^{\prime}_{2}=t_{2}+1$ chirp starting
slot). Probability of starting chirp in $t_{2}$ is given by
$\frac{1}{1+{\mathbb{E}}[W]}$. Now the probability of $t_{2}$ being the last
slot of backoff is given by

$\sum_{j\geq 1}\alpha_{j}\frac{{\mathbb{P}}(W=j)}{1-F(j-1)}.$

Using these two we compute ${\mathbb{P}}(E_{i})$.

$\displaystyle{\mathbb{P}}(E_{i})=$
$\displaystyle{\mathbb{P}}(\text{transmission cycle starting at slot
}t_{2}\text{ or }t_{2}+1)$ $\displaystyle=$
$\displaystyle\frac{1}{1+{\mathbb{E}}[W]}+\sum_{i\geq
1}\alpha_{j}\frac{{\mathbb{P}}(W=j)}{1-F(j-1)}$ $\displaystyle=$
$\displaystyle\frac{1}{1+{\mathbb{E}}[W]}+\frac{1-{\mathbb{P}}(W=0)}{1+{\mathbb{E}}[W]}$

Now consider $KL>1$. In this case the packet transmission starting slots at
$t_{2}$ and $t_{2}+1$ are mutually exclusive events and from (3) we write
${\mathbb{P}}(E_{i})=\frac{2}{KL+E[W]}$. ∎

Using the ${\mathbb{P}}(E_{i}),i\in\\{-(L-1)\dots L-1\\}$ derived in Lemma 3.1
we derive the probability of interference at the tagged radar in two radar
network and $M$ radar network in Proposition 2.

######  Proposition 2.

  1. 1.

The probability of interference at the tagged radar in two radar network is
given by,

$\displaystyle p_{I2}=$ $\displaystyle(2L-1)\mathbb{P}(E_{1})$
$\displaystyle-\frac{1}{KL+\mathbb{E}[W]}\sum_{i=0}^{L-2}(L-i-1)\Big{(}2{\mathbb{P}}\big{(}W=iK\big{)}$
$\displaystyle+{\mathbb{P}}\big{(}W=iK+1\big{)}+{\mathbb{P}}\big{(}W=iK-1\big{)}\Big{)}.$

where $\mathbb{P}(E_{i})$ is given by (10).

  2. 2.

The probability of interference at the tagged radar in $M$ radar system is
given by,

$\displaystyle p_{I}=1-(1-p_{I2})^{M-1}.$

###### Proof.

Using similar arguments as in the proof of Proposition 1 we have,

$\displaystyle p_{I2}=$
$\displaystyle\mathbb{P}\big{(}\cup_{i=-(L-1)}^{L-1}E_{i}\big{)}$
$\displaystyle=$
$\displaystyle\sum_{i=-(L-1)}^{L-1}\mathbb{P}(E_{i})-\sum_{i=1}^{L-1}\sum_{j=1}^{i}\mathbb{P}(E_{i}\cap
E_{-(L-j)}).$ (11)

Now consider the second term from above,

$\displaystyle\mathbb{P}$ $\displaystyle(E_{i}\cap E_{-(L-j)})$
$\displaystyle=$
$\displaystyle\mathbb{P}(E_{-(L-j)})\mathbb{P}(E_{i}|E_{-(L-j)})$
$\displaystyle=$
$\displaystyle\frac{1}{KL+\mathbb{E}[W]}\Big{(}2\mathbb{P}\big{(}W=(i-j)K\big{)}$
$\displaystyle+\mathbb{P}\big{(}W=(i-j)K-1\big{)}+\mathbb{P}\big{(}W=(i-j)K+1\big{)}$

Using above in (3.2), we get $p_{I2}$.

Using the similar arguments as in the proof of Proposition 1 we have,

$p_{I}=1-(1-p_{I2})^{M-1}.$

∎

###  3.3 Throughput

We compute the throughput we defined in Section 2.2 in the following
Proposition 3.

######  Proposition 3.

Throughput of $M$ radar network is given by,

$\displaystyle\Theta=MLK\pi_{1}(1-\mathbb{P}_{I2})^{M-1}$ (12)

###### Proof:

From the definition of throughput (see 2.2), we have

$\Theta=\lim_{T\rightarrow\infty}KL\frac{S(T)}{T},$

where $S(T)$ total number of packets transmitted successfully by all radars in
$T$ time slots. Now consider a radar in the network. Let the total number of
transmissions cycles until time slot $T$ is given by $\phi(T)$. The average
number of successful packets transmission cycles is given by
$\phi(T)(1-p_{I2})^{M-1}$ and number of chirps transmitted is given by
$L\phi(T)(1-p_{I2})^{M-1}$. In $M$ independent radars network the average
number of successful transmission cycles is given by
$M\phi(T)(1-p_{I2})^{M-1}$. Using this we can write,

$\displaystyle\Theta$
$\displaystyle=\lim_{T\rightarrow\infty}\frac{KML\phi(T)(1-p_{I2})^{M-1}}{T}$
$\displaystyle=\frac{1}{KL+{\mathbb{E}}[W]}MKL(1-p_{I2})^{M-1}$
$\displaystyle=MLK\pi_{1}(1-p_{I2})^{M-1}$

where the first and second equalities follow from Elementary Renewal and
Renewal Reward theorems [16]. ∎

###  3.4 $p_{I}$ and $p_{opt}$ for geometric backoff

In this section we present results obtained for geometric backoff
distribution, $W\sim\text{Geo}(p)$ where $p$ is the attempt probability and
supported on the set $\\{0,1,2,\dots\\}$.

  1. 1.

The probability of interference at the tagged radar in two radar network is
given by,

$\displaystyle p_{I2}=$ $\displaystyle(2L-1)\mathbb{P}(E_{i})$
$\displaystyle-\frac{p^{2}}{pKL+1-p}\sum_{i=1}^{L-1}(L-i)\Big{(}2(1-p)^{(i-1)K}$
$\displaystyle+(1-p)^{(i-1)K+1}+(1-p)^{(i-1)K-1}\Big{)},$

where

$\displaystyle\mathbb{P}(E_{i})=\begin{cases}\frac{2}{KL+{\mathbb{E}}[W]}\text{
for }KL>1\\\
\frac{1}{1+{\mathbb{E}}[W]}+\frac{p{\mathbb{E}}[W]}{1+{\mathbb{E}}[W]}\text{
for }KL=1.\end{cases}$

and ${\mathbb{E}}[W]=\frac{1-p}{p}$.

  2. 2.

Further, we consider number of chirps per packet equal to one i.e., $L=1$ and
evaluate the attempt probability $p_{opt}$ at which the network has the
maximum throughput.

$\displaystyle p_{opt}=\begin{cases}\frac{1}{2M+1-K},\text{ if }2M+1-K>0\\\
1,\text{ otherwise. }\end{cases}$

We observe that a centralized transmission scheduling can achieve a throughput
close to $\min\\{K,M\\}$ which is much more than the throughput achieved by
ALOHA, e.g., see Figure 12b. This motivates us to explore other MAC protocols.

##  4 CSMA

In slotted ALOHA, the radars initiate transmissions irrespective of the
activities of other radars. Consequently, in most of the scenarios, slotted
ALOHA achieves a throughput far below the best possible values (see Figure
12b). This motivates us exploring other MAC protocols that account for other
radars’ activities. In this section, we study a MAC protocol in which a radar
senses the medium following each backoff, and initiates a transmission only if
the medium is sensed idle.

##### CSMA

In CSMA, a radar after each backoff, senses the medium for one slot which is
referred to as clear channel assessment (CCA) slot. In the CCA slot, the radar
sets the Tx chirp frequency input to the mixer to $f_{\min}$ and observes the
output of its LPF at the end of the CCA slot. If the radar observes a signal
(of frequency less than $f_{H}$), it assumes the CCA to have failed and enters
into another random backoff. If the radar does not observe a signal at the end
of the CCA slot, it assumes the CCA to have succeeded and starts a packet
transmission on completion of the CCA slot. We illustrate the flow chart of
our CSMA protocol in Figure 8.

######  Remark 4.1.

Note that if the radar observes a signal at the end of the CCA slot and still
starts a packet transmission in the next slot, this packet would interfere
with the packet that caused the observed signal. In other words a transmission
following a CCA failure would certainly lead to interference. On the other
hand, a transmission following a CCA success need not be interference free as
we explain in Section 4.1.2.

Recall that in ALOHA each radar transmits packets irrespective of other
radars’ transmissions. But in CSMA all the radars’ backoffs, CCAs and packet
transmissions are coupled. In the following subsections we analyze CSMA in
various cases.

Figure 8: CSMA, After every backoff radar senses the medium for $f_{\min}$
(CCA), if CCA is successful then it transmits else enter into backoff

###  4.1 Collocated radars

We consider a collocated and asynchronous radar network. Further, we first
consider number of chirps per packet, $L$ to be one. We consider $L>1$
subsequently.

####  4.1.1 $L=1$

In this case we observe that the radars do not suffer interference in CSMA.
Observe that, in CSMA, for a CCA success at a radar, the immediately preceding
chirp transmission by any of the other radars must have started at least
$\Delta$ time before the end of the CCA slot. Consequently, start times of
packet transmissions by the radars are separated by at least $\Delta$, and so
the packets will not interfere with each other. We illustrate it by showing
sample CCA failure and CCA success events in Figure 9. Let $t_{1}$ be the CCA
slot at the tagged radar. The tagged radar senses the $f_{\min}$ of the chirp
transmitted (dotted red color) by the interfering radar, experiences CCA
failure as shown in Figure 9. The next CCA attempt is successful and tagged
radar starts transmission immediately after the CCA slot.

Figure 9: Illustrating CCA success, CCA failure and packet transmission at the
tagged radar in a collocated radar network for $L=1,K=2$

In the following we compute the throughput. Let us consider a tagged radar. We
assume that the backoffs are sampled from Geo$(p)$ distribution. Let
$I_{s}(T)$ be the number of successful transmissions at the tagged radar until
slot $T$. Further, let $p_{s}$ be the conditional CCA success probability at
the tagged radar i.e., the probability of CCA success given the tagged radar
is in a CCA. We define the tagged radar’s transmission rate and the aggregate
throughput, $p_{r}$ and $\Theta$, respectively, as follows.

$\displaystyle p_{r}:=\lim_{T\rightarrow\infty}\frac{I_{s}(T)}{T}.$ (13)

Using above and $L=1$ in the definition of throughput 2.2 we have

$\displaystyle\Theta=MKp_{r}.$ (14)

######  Remark 4.2.

Observe that the maximum throughput that can be achieved by any algorithm for
$L=1$ is $\min\\{M,K\\}$. Using this in (14) gives
$p_{r}\leq\frac{1}{\max\\{M,K\\}}$.

In the following we compute the throughput using decoupling approximation and
fixed point analysis [17],[18]. Towards this we find the relation between
$p_{r},p_{s}$ in Lemma 4.1.

######  Lemma 4.1.

$\displaystyle p_{r}=\frac{1}{K+\frac{1}{pp_{s}}}$

###### Proof:

Let us define the state of a radar in a slot as follows. We say that the radar
is in state $-1$ if it is sensing the medium, state $0$ if it is in backoff.
Further, we say that radar is in state $i,1\leq i\leq K$ if it is transmitting
$i$-th slot of the packet. The radar’s state evolves according to a renewal
process with the slots in which the radar state is $1$ being the renewal
epochs. Let the duration between any two renewal epochs $i$ and $i+1$ is given
by $D_{i}$. Observe that $D_{i},i\in\\{1,2,\dots\\}$ is random and independent
across $i$. Further, ${\mathbb{E}}[D_{i}]=K+\frac{1}{pp_{s}}$.

Using Elementary Renewal theorem [16],

$\displaystyle p_{r}=$
$\displaystyle\lim_{T\rightarrow\infty}\frac{I_{s}(T)}{T}$ $\displaystyle=$
$\displaystyle\frac{1}{K+\frac{1}{pp_{s}}}$

almost surely. ∎

To compute $p_{s}$ we use the decoupling approximation [17], [18]. The
decoupling approximation assumes that the aggregate attempt process of the
other radars is independent of the backoff process of the tagged radar. Recall
in any $\Delta$ interval of tagged radar’s timeline, at most one radar can
start packet transmission. The unconditional probability of any one among
$M-1$ radars starting transmission in one slot is given by $(M-1)p_{r}$. The
unconditional probability of no radar starting transmission is given by
$1-(M-1)p_{r}$. From Remark 4.2 we observe that $1-(M-1)p_{r}>0$. We want to
calculate the conditional probability, $p_{s}$ at the tagged radar i.e., given
the tagged radar is in a CCA slot the probability of its CCA success. But we
use the unconditional CCA success probability $1-(M-1)p_{r}$ for $p_{s}$ in
Lemma 4.1 to compute $p_{r}$. Further, we substitute $p_{r}$ in (14) to
compute the throughput.

$\displaystyle p_{r}$
$\displaystyle=\frac{1}{K+\frac{1}{p\big{(}1-(M-1)p_{r}\big{)}}}$

For a given $M,p$ we can write above as

$\displaystyle p_{r}=f(p_{r}).$

In Lemma 4.2 we show that $p_{r}$ can be computed using fixed point analysis.

######  Lemma 4.2.

For a $p_{r}=f(p_{r})$ has a unique fixed point.

###### Proof:

  * a.

$f(.)$ is a continuous map from $[0,\frac{1}{\max\\{M,K\\}}]$ to
$[f(0),f\big{(}\frac{1}{\max\\{M,K\\}}\big{)}]$ and decreasing in $p_{r}$

  * b.

$f\big{(}\frac{1}{\max\\{M,K\\}}\big{)}<\frac{1}{\max\\{M,K\\}}$

Hence by intermediate value theorem there exists a unique fixed point. ∎

We use $p_{r}$ obtained from Lemma 4.2 in (14) to compute the throughput. We
find from Section 5 (see Figure 13b) that the throughput obtained from the
fixed point analysis matches the simulation with less than $10$ percent error
for $M>K$. Further, from Figure 14 we observe that in CSMA the throughput
increases in attempt probability, and for all values of $p$ throughput of CSMA
is better than ALOHA.

####  4.1.2 $L>1$

We notice that when $L>1$, packets may interfere under the proposed CSMA. We
illustrate this using Figure 10. Let us assume that the tagged radar starts
CCA at slot $t_{1}$, while other radar transmitting a packet (dotted red
color). The tagged radar CCA is successful as the starting time of other radar
is earlier than $t_{1}$. However the subsequent packet (solid blue color)
transmission of the tagged radar is interfered by the the second chirp of the
packet from the other radar. Observe that during interference one of the radar
should be transmitting the first chirp in a packet and other must be
transmitting chirp $j,1<j\leq L$ in a packet.

Figure 10: Illustrating interference at the tagged radar after CCA success in
a collocated radar network for $L=2,K=2$

From simulation (see Figure 17) we observe that CSMA’s throughput is higher
than that of slotted ALOHA for all values of $p$. We also observe that the
throughput is a non-monotonic function of $p$.

###  4.2 Non-collocated radars

We consider a non-collocated and asynchronous radar network. In Section 4.1.1
we saw that in a collocated radar network radars do not suffer interference
for $L=1$. But in non-collocated setting radars suffer interference for $L=1$
also. We explain this in the following using Figure 11. Let $d$ be the
distance between the tagged and interfering radars and $d<d_{I,\max}$. The
tagged radar receives the packets transmitted by the interfering radar with a
delay of $\frac{d}{c}$. Let $d=c\Delta$. This causes a delay of one slot as
shown in Figure 11. Let us assume that the tagged radar started CCA at slot
$t_{1}$. In Figure 11 observe that though the interfering radar started a
packet transmission during the CCA slot of the tagged radar, the tagged radar
can not sense it because of the delay caused by the distance. The packet
received at the tagged radar is shown in dotted lines (see Figure 11a). The
packet transmitted by the interfering radar reached the tagged radar after the
CCA slot. So, the CCA was successful and the tagged radar started the packet
transmission at slot $t_{1}+1$. The tagged radar received the packet during
the first slot of packet transmission and suffered interference.

Now consider another CCA slot at $t_{2}$. The tagged radar senses the packet
from the other radar, it is a CCA failure and packet transmission did not
start at slot $t_{2}+1.$

Figure 11: Illustrating interference in a non-collocated radar network for
$L=1,K=2$, a. Transmitted and received packets at the tagged radar, b.
Transmitted packets at the tagged radar

So, the CSMA avoids interference caused by packets that are received at the
tagged radar in the CCA slot. From Section 5 (see Figure 20) we observe that
for $L=1$ in CSMA the throughput increases in $p$, and for all values of $p$
throughput of CSMA is better than ALOHA.

##  5 Numerical Results

In this section, we simulate an FMCW radar network with $M$ radars. The
radars’ clocks are not synchronized. Assume that there is a hypothetical
global clock, the clock offsets for each radar is assumed to be uniformly
distributed random variable in $[0,\Delta)$. More precisely, we consider the
cases of collocated and non-collocated radars. In the case of non-collocated
radars, we assume that the radars are placed on a line segment
$[0,d_{I,\max}]$. Radar locations are chosen from i.i.d.
$\text{Unifrom}[0,d_{I,\max}]$ distributions. We set $d_{I,\max}=4d_{\max}$.
We assume that backoffs are sampled from Geo($p$) distribution.

We first verify the probability of interference ($p_{I}$) and throughput
($\Theta$) obtained in Proposition 2 and Proposition 3, respectively for
different values of packet length $(L)$, attempt probability $(p)$ and number
of radars $(M)$ in ALOHA. Then we verify the fixed point solution obtained in
Lemma 4.2 for collocated radar networks by varying $L,p,M$ in CSMA. Finally,
we compare the performance of ALOHA and CSMA by varying $L,p,M$.

In Figure 12 we demonstrate how $p_{I}$ and $\Theta$ vary with the attempt
probability $p$ and $L$ in ALOHA. Towards this we consider a network with
$M=50$ radars with chirp length $K=40$. Further, we consider $L=1,4,5$.

Figure 12:  Verifying the analytical results obtained in Propositions 2 and 3
for ALOHA, a. Probability of interference vs attempt probability, b.
Throughput vs attempt probability for $M=50,K=40,L=1,4,5$

We observe in Figure 12 that $p_{I},\Theta$ obtained from simulation and
analysis match very closely (less than $2$ percent error). Further, we also
observe that at $p=1$, $p_{I}$ and $\Theta$ do not vary with $L$. This is
because, at $p=1$, each radar generates a continuous train of packets with no
backoff between them. We also observe that for $L>1$ interference probability
increases and then decreases with $p$ which is different from wireless
networks. Further, we find from Figure 12b that the maximum throughput for
$L=1$ is achieved at $p_{opt}=\frac{1}{2M+1-K}=0.016$ (see Section 3.4).

In Figure 13 we demonstrate the fixed point solution obtained in Lemma 4.2 for
CSMA and compare it with the simulation. Towards this we consider a radar
network with $M=50$ radars each using chirp length $K=40$ and $L=1$. In Figure
13a we plot fixed point solution for different attempt probabilities
$p=0.2,0.4,0.6$. In Figure 13b we plot throughput obtained from fixed point
analysis and the simulation by varying $p$ for $M=30,40,50$. We find that the
error in throughput obtained from the fixed point analysis matches the
simulation with less than $10$ percent error.

Figure 13: a. Fixed point analysis for $M=50,K=40$, b. Throughput vs attempt
probability for a collocated radar network $K=40,L=1$

To compare the performance of ALOHA and CSMA we consider the following two
network scenarios separately.

  * •

collocated and asynchronous radar network,

  * •

non-collocated and asynchronous radar network.

###  5.1 Collocated and asynchronous radar network

In this section, we simulate a collocated radar network. We first consider the
case where the number of chirps per packet $L=1$. Subsequently, we consider
$L>1$ case.

####  5.1.1 $L=1$

To begin with we demonstrate how the throughput varies with $p$. Towards this
we fix $K=40$ and vary $M$ as shown in Figure 14. We observe that in CSMA the
throughput increases in $p$. We observe that the network achieves maximum
throughput at $p=1$. We explain this in the following. Recall from Section
4.1.1 that in a collocated and asynchronous network radars suffer no
interference. At $p=1$ each radar continuously senses the medium until its CCA
is successful i.e., the radar is either transmitting the packet or sensing the
medium. Further, we observe that for $L=1$ the maximum throughput that can be
achieved is $\min\\{M,K\\}$. We find that in Figure 14 for $M=30,40,50$ and
$K=40$ CSMA achieves $29.3,39.1,39.25$ at $p=1$ respectively. In ALOHA the
throughput decreases with the increase in the number of radars after maximum
throughput. This is because each radar transmits packets irrespective of other
radar packet transmissions. So, at a higher value of attempt probability every
radar transmits packets often and suffers interference. Further, we observe
that CSMA has better throughput than ALOHA.

Figure 14: Throughput vs attempt probability for collocated and asynchronous
radar network, $L=1,K=40,M=30,40,50$

In Figure 15 we demonstrate how $p_{opt}$ and $\Theta_{opt}$ vary with $M$ for
CSMA and ALOHA. Towards this, we fix $K=40$. From Figure 14 we observed that
in CSMA throughput increases in $p$, so observe $p_{opt}=1$ for all values of
$M$ in Figure 15a. Further, in Figure 15b we observe that $\Theta_{opt}$
increases in $M$ in CSMA though the increase is small. In ALOHA we observe
that $p_{opt}$ decreases with the increase in the number of radars. This is
because for a fixed $p$, as the number of radars increase interference
probability increases. So, radar has to transmit packets less often to avoid
interference, i.e., $p$ has to decrease with the increase in $M$. From Figure
15b we find that $\theta_{opt}$ decreases with $M$ in ALOHA though the
decrease is small. Further, the optimum throughput achieved by CSMA is always
more than the optimum throughput achieved by ALOHA.

Figure 15: a. $p_{opt}$ vs number of radars, b. $\Theta_{opt}$ vs number of
radars for collocated and asynchronous radar network, $L=1,K=40$

In Figure 16 we demonstrate how $p_{opt}$ and $\Theta_{opt}$ vary with $K$.
Towards this we fix $M=40$. From Figure 14 we observed $p_{opt}=1$ for CSMA.
From Figure 16b we observe that $\Theta_{opt}$ increases in $K$ though the
increase is small. From Figure 16b we observe that in ALOHA $p_{opt}$
increases in $K$. This is because for a fixed $p$ the probability of
interference decreases with the increase in chirp length, $K$. So, we can
increase $p$ to achieve optimum throughput.

Figure 16: a. $p_{opt}$ vs chirp length, b. $\Theta_{opt}$ vs chirp length for
collocated and asynchronous radar network, $L=1,M=40$

####  5.1.2 $L>1$

To begin with, we demonstrate how the throughput varies with $p$. Towards
this, we fix $K=40,L=4$ and vary $M$ as shown in Figure 17. In both the
protocols we observe that throughput is less compared with $L=1$ (Figure 14).
We explain this in the following. In CSMA for $L=1$ the radars suffer no
interference (see Section 4.1.1). For $L>1$ radars suffer interference and
this deteriorates the throughput. In ALOHA for a given $p,M$, interference
probability increases in $L$. So, the throughput is less compared to the case
$L=1$.

Figure 17: Throughput vs attempt probability for collocated and asynchronous
radar network, $L=4,K=40,M=30,40,50$

In Figure 18 we demonstrate how $p_{opt}$ and $\Theta_{opt}$ vary with $M$ in
the network for CSMA and ALOHA. In Figure 18a we observe that $p_{opt}$
decreases with the increase in $M$ in both CSMA and ALOHA. This is because,
for a given $p$, the interference probability increases with the increase in
$M$. So, the radar has to participate less aggressively in transmitting the
packets. Further, in Figure 18b we observe a decrease in throughput with the
increase in $M$ for both CSMA and ALOHA. This is because for a fixed $p$
interference probability increases in $M$ so the throughput decreases.

Figure 18: a. $p_{opt}$ vs number of radars, b. $\Theta_{opt}$ vs number of
radars for collocated and asynchronous radar network, $L=4,K=40$

In Figure 19 we demonstrate how $p_{opt}$ and $\Theta_{opt}$ vary with chirp
length for CSMA and ALOHA. In Figure 19a we observe that $p_{opt}$ increases
in $K$ for both CSMA and ALOHA. This is because for a given $p,M$ interference
probability decrease with chirp length, $K$. So, we can increase $p$ to
achieve optimum throughput. In Figure 19b we observe that $\Theta_{opt}$
increases in $K$ for both CSMA and ALOHA.

Figure 19: a. $p_{opt}$ vs chirp length, b. $\Theta_{opt}$ vs chirp length for
collocated and asynchronous radar network, $L=4,M=40,$

###  5.2 Non-collocated and asynchronous radar netwrok

In this section, we simulate a collocated radar network. We first consider the
case where the number of chirps per packet $L=1$. Subsequently, we consider
$L>1$ case.

####  5.2.1 $L=1$

To begin with we demonstrate how the throughput varies with $p$. Towards this
we fix $K=40$ and vary $M$ as shown in Figure 20. We observe that in CSMA the
throughput increases in $p$. However, it is less compared to the case $L=1$
collocated radar network at all the values of $p$ (see Figure 14). This is
because in CSMA, the radars do not suffer interference when they are
collocated but they suffer interference when they are not collocated. In ALOHA
the throughputs are the same in collocated and non-collocated radar networks.
This is because the radars do not sense the medium in ALOHA and each radar
transmits packets irrespective of other radars’ transmissions.

Figure 20: Throughput vs attempt probability for non-collocated and
asynchronous radar network, $L=1,K=40,M=30,40,50$

In Figure 21 we demonstrate how $p_{opt}$ and $\Theta_{opt}$ vary with $M$ in
CSMA and ALOHA. In Figure 20 we observed that in CSMA throughput increases in
$p$. So, in Figure 21a we observe $p_{opt}=1$ for CSMA. In Figure 21b we
observe that throughput increases in $M$, in CSMA. In ALOHA, given network
parameters $L,K,M,p$ throughputs are the same for collocated and non-
collocated networks. So, the same observations made for collocated network in
ALOHA apply here (see Section 5.1.1).

Figure 21: a. $p_{opt}$ vs number of radars, b. $\Theta_{opt}$ vs number of
radars for non-collocated and asynchronous radar network, $L=1,K=40$

In Figure 22 we demonstrate how $p_{opt}$ and $\Theta_{opt}$ vary with chirp
length in CSMA and ALOHA. In Figure 22a we observe that $p_{opt}=1$ for CSMA.
In Figure 22b we observe that throughput increases in $K$ in CSMA. This is
because the probability of interference decreases with the increase in $K$.
So, the throughput increases.

Figure 22: a. $p_{opt}$ vs chirp length, b. $\Theta_{opt}$ vs chirp length for
non-collocated and asynchronous radar network, $L=1,M=40,$

####  5.2.2 $L>1$

To begin with, we demonstrate how the throughput varies with $p$. Towards
this, we fix $K=40,L=4$ and vary $M$ as shown in Figure 23. In Figure 23 we
observe that the throughput in CSMA is less compared to the case of collocated
radar network $L=4$ at all the values of $p$(see Figure 17). This is because
the probability of interference increases in $L$. So, the throughput
decreases. In ALOHA, given network parameters, $L,K,M,p$ throughputs are the
same for collocated and non-collocated networks. So, the same observations
made for collocated networks in ALOHA are hold here (see Section 4.1.2.

Figure 23: Throughput of slotted CSMA and slotted ALOHA for non-collocated
radar network with asynchronous clock, $L=4,K=40,M=30,40,50$

In Figure 24 we demonstrate how $p_{opt}$ and $\Theta_{opt}$ vary with $M$ in
CSMA and ALOHA. In Figure 24a we observe that $p_{opt}$ decreases with $M$ in
both CSMA and ALOHA. This is because as the number of radars increases in the
network the interference probability also increases. So, the radar has to
participate less aggressively in transmitting the packets. Further, in Figure
18b we observe a decrease in throughput with the increase in the number of
radars, $M$ for both CSMA and ALOHA. This is because for a given $p$ the
interference probability increases in $M$. So, the throughput decreases.

Figure 24: a. $p_{opt}$ vs number of radars, b. $\Theta_{opt}$ vs number of
radars for non-collocated and asynchronous radar network, $L=4,K=40$

In Figure 25 we demonstrate how $p_{opt}$ and $\Theta_{opt}$ vary with chirp
length for CSMA and ALOHA. In Figure 25a we observe that $p_{opt}$ increases
in $K$ for both CSMA and ALOHA. This is because, for a given $p,M$
interference probability decreases with the increase in $K$. So, radars can
aggressively participate in the packet transmission and we can improve the
throughput. So, in Figure 25b we observe the increase in $\Theta_{opt}$ with
chirp length for both CSMA and ALOHA.

Figure 25: a. $p_{opt}$ vs chirp length, b. $\Theta_{opt}$ vs chirp length for
non-collocated and asynchronous radar network, $L=4,M=40$

##  6 Conclusion

We studied medium access in FMCW radar networks in which all the radars used
the same parameters, e.g., chirp duration, chirp slope, cutoff frequency,
number of chirps per packet, etc. We proposed and analyzed slotted ALOHA and
CSMA protocols in terms of interference probability and throughput. In either
case, we observed that interference probability and throughput may behave
differently than in wireless communication networks. For instance, in the case
of ALOHA, if the number of chirps per packet is larger than one, the
interference probabilities may be smaller for higher transmission rates (see
Figure 12a) and throughputs may be maximum at the highest possible
transmission rates (see Figure 12b). In the case of CSMA also, using the
highest possible attempt rates may maximize throughput (see Figure 14). We
observed that CSMA outperformed ALOHA in all the realistic scenarios.

If the radars in a FMCW network use different chirp slopes, they may also be
subject to wideband interference from other radars. Designing medium access
protocols that also consider wideband interference is a challenging open
problem. Usually, the received signals at the radars undergo further
processing for estimation of object locations, velocities, etc (see [6]). The
notions of interference probability and throughput can be refined to account
for the detection and estimation errors. Designing medium access protocols
that optimize the refined notion of throughput is another potential future
direction.

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