# Equations of some embeddings of a projective space into another one

Marc Chardin  Institut de mathématiques de Jussieu, CNRS & Sorbonne
Université, 4 place Jussieu, 75005 Paris , France<EMAIL_ADDRESS>and
Navid Nemati  Université Côte d’Azur, Inria, 2004 route des Lucioles, 06902
Sophia Antipolis, France<EMAIL_ADDRESS>
###### Abstract.

In [8], Eisenbud, Huneke and Ulrich conjectured a result on the Castelnuovo-
Mumford regularity of the embedding of a projective space
$\mathbb{P}^{n-1}\hookrightarrow\mathbb{P}^{r-1}$ determined by generators of
a linearly presented $\operatorname{\mathfrak{m}}$-primary ideal. This result
implies in particular that the image is scheme defined by equations of degree
at most $n$. In this text we prove that the ideal of maximal minors of the
Jacobian dual matrix associated to the input ideal defines the image as a
scheme; it is generated in degree $n$. Showing that this ideal has a linear
resolution would imply that the conjecture in [8] holds. Furthermore, if this
ideal of minors coincides with the one of the image in degree $n$ \- what we
hope to be true - the linearity of the resolution of this ideal of maximal
minors is equivalent to the conjecture in [8].

###### Key words and phrases:

Birational map, blowup algebra, Castelnuovo-Mumford regularity, Jacobian dual
matrix

###### 2010 Mathematics Subject Classification:

13D02, 14E05, 13A30

##  1\. Introduction and Preliminaries

Let $\phi:\mathbb{P}^{n-1}\dashrightarrow\mathbb{P}^{r-1}$ be a rational map
defined by homogeneous forms $f_{1},\dots,f_{r}$ and $I$ be the ideal
generated by these forms. The Rees algebra associated to the ideal $I$,
$\mathcal{R}(I)$, gives the bihomogeneous coordinate ring of the closure of
the graph of $\phi$ inside $\mathbb{P}^{n-1}\times\mathbb{P}^{r-1}$. The
special fiber ring $\mathcal{F}(I)$ is the coordinate ring of the closed image
of $\phi$. The rings $\mathcal{R}(I)$ and $\mathcal{F}(I)$ are blowup rings
associated to $I$. It is a fundamental problem to find the defining equations
of the Rees ring from generators of $I$, for significant classes of ideals.
This problem has been studied extensively by commutative algebraists,
algebraic geometers, and, more recently, by applied mathematicians in
geometric modeling (see [1, 4, 10, 11, 12]). Rees algebras provide an
algebraic realization for the concept of blowing up a variety along a
subvariety.

Let $S=k[x_{1},\dots,x_{n}]$ be a polynomial ring over a field $k$ and $I$ be
an ideal of $S$. The Rees algebra $\mathcal{R}(I)$ of an ideal $I$ is defined
as

$\mathcal{R}(I)=S[It]=S\oplus It\oplus I^{2}t^{2}\oplus\cdots.$

The equations of the Rees algebra are elements in the kernel of the
epimorphism $\Phi:S[T_{1},\dots,T_{r}]\rightarrow\mathcal{R}(I)$ given by
$\Phi(T_{i})=f_{i}t$ where $I=(f_{1},\dots,f_{r})$. These equations do depend
on the choice of generators of $I$; however, their number and degrees do not,
if one chooses minimal generators of a graded ideal $I$.

An important object for the study of Rees algebras is the symmetric algebra,
$\text{Sym}(I)$, of an ideal $I$. It could be defined as

$\text{Sym}(I)=S[T_{1},\dots,T_{r}]/\mathcal{L},$

where $\mathcal{L}=([T_{1},\dots,T_{r}]\cdot M)$ and $M$ is a presentation
matrix of $I$. The map $\Phi$ above factors through the symmetric algebra. To
determine equations of $\mathcal{R}(I)$ of degrees at least two in the
$T_{i}$’s is equivalent to study the kernel of the map
$\text{Sym}(I)\rightarrow\mathcal{R}(I)$. Traditionally, techniques for
computing the defining ideal of $\mathcal{R}(I)$ often revolved around the
notion of Jacobian dual.

For a polynomial ring $S$ and an ideal $I$ with a presentation
$S^{N}\xrightarrow{M}S^{r}\rightarrow I\rightarrow 0$, the Jacobian dual of
$M$ is defined to be a matrix $\Theta(M)$ (we use $\Theta$ if the context
makes no confusion possible) with linear entries in $S[T_{1},\dots,T_{r}]$
such that

$[T_{1},\dots,T_{r}]\cdot M=[x_{1},\dots,x_{n}].\Theta(M).$

In this article, we focus on the Jacobian dual matrix of linearly presented
$\operatorname{\mathfrak{m}}$-primary ideals. The main theorem of this paper
is the following:

######  Main Theorem.

Let $S=k[x_{1},\dots,x_{n}]$ be a polynomial ring over a field $k$,
$I=(f_{1},\dots,f_{r})$ be a linearly presented
$\operatorname{\mathfrak{m}}$-primary ideal of $S$ and $\Theta$ be the
Jacobian dual matrix of a presentation matrix of $I$. Then the image of the
map $\mathbb{P}^{n-1}\rightarrow\mathbb{P}^{r-1}$ defined by the forms
$f_{1},\dots f_{r}$ is scheme-theoretically defined by the ideal of maximal
minors of $\Theta$.

It is worth mentioning that it is not hard to show that the image is defined
as a set by the ideal of maximal minors (see Lemma 1); the more delicate point
is to show that the image is scheme-theoretically defined by the ideal of
maximal minors. As a consequence, the image of $\Phi$ is scheme-theoretically
defined by equation of degree $n$. We finish the section by presenting two
conjectures about the ideal of maximal minors of $\Theta$.

In the last section, we present the connection between our work and a
conjecture by Eisenbud, Huneke, and Ulrich in [8].

######  Conjecture.

[8, Conjecture 1.1] Let $S=k[x_{1},\dots,x_{n}]$ and $I$ be an ideal of $S$.
If $I$ is a linearly presented $\operatorname{\mathfrak{m}}$-primary ideal
generated in degree $d$, then $I^{n-1}=\operatorname{\mathfrak{m}}^{d(n-1)}$.
Equivalently,

$\operatorname{reg}{(}I^{n-1})=(n-1)d.$

In particular, we show that this conjecture holds if the ideal of maximal
minors of $\Theta$ has a linear resolution (see Theorem 5 ).

##  2\. Jacobian dual matrix

For the rest of the article we adopt the following notations.

######  Notation 1.

Let $S=k[x_{1},\dots,x_{n}]$ be a polynomial ring over a field $k$ and
$I=(f_{1},\dots,f_{r})$ be a linearly presented
$\operatorname{\mathfrak{m}}$-primary ideal,
$\phi:\mathbb{P}^{n-1}\dashrightarrow\mathbb{P}^{r-1}$ be the map defined by
$\phi(\textbf{x})=(f_{1}(\textbf{x}),\cdots,f_{r}(\textbf{x}))$ and denote the
graph of $\phi$ by $\Gamma\subseteq\mathbb{P}^{n-1}\times\mathbb{P}^{r-1}$.
Set $\pi_{1}:\Gamma\rightarrow\mathbb{P}^{n-1}$ and
$\pi_{2}:\Gamma\rightarrow\mathbb{P}^{r-1}$ for the two natural projections
and $W:=\operatorname{Im}(\pi_{2})$. We write $B=k[T_{1},\dots,T_{r}]$. The
defining ideal $J$ of the symmetric algebra $\text{Sym}(I)$ is generated by
the bilinear forms $L_{i}(\textbf{x},\textbf{T})=\sum a_{i}(\mathbf{x})T_{i}$
where $\sum a_{i}(\mathbf{x})f_{i}=0$ is a linear syzygy of $I$. The Jacobian
dual $\Theta$ is given by

$\Theta\begin{bmatrix}x_{1}\\\ \vdots\\\ x_{n}\\\
\end{bmatrix}=\begin{bmatrix}L_{1}\\\ \vdots\\\ L_{N}\\\ \end{bmatrix},$

where the $L_{i}$’s are minimal generators of $J$ (equivalently, the
corresponding syzygies are minimal generators of the module of syzygies). Thus
$\Theta$ is a matrix whose entries are linear forms in the $T_{i}$’s and we
set $W^{\prime}:=\operatorname{proj}(\bar{B})$ where $\bar{B}=B/I_{n}(\Theta)$
and $I_{n}(\Theta)$ is the ideal of maximal minors of $\Theta$ (the $n\times
n$ minors). Finally, if $p\in W\subseteq W^{\prime}$ and $\mathfrak{p}$ is the
corresponding homogeneous prime in $\operatorname{proj}(B)$, we denote by
$M_{p}$ the image of a matrix $M$ by the natural map obtained by applying the
map $B\rightarrow B_{\mathfrak{p}}/I_{r}(M)_{\mathfrak{p}}$ to its entries,
where $r$ is the rank of $M$.

######  Definition 1.

Let $M$ be a $p\times q$ matrix over $B$. For a prime ideal
$\operatorname{\mathfrak{p}}\supseteq I_{q}(M)$, consider the natural map
$B\rightarrow(B/I_{q}(M))_{\operatorname{\mathfrak{p}}}$ and its natural
extension $\operatorname{Mat}_{p\times
q}(B)\rightarrow\operatorname{Mat}_{p\times
q}((B/I_{q}(M))_{\operatorname{\mathfrak{p}}})$. We write
$M_{\operatorname{\mathfrak{p}}}$ for the image of $M$ under this last map.

######  Lemma 1.

With the above notations, $\sqrt{I_{n}(\Theta)}=I(W)$.

###### Proof.

As $I$ is $\mathfrak{m}$-primary,
$\Gamma\subseteq\mathbb{P}^{n-1}\times\mathbb{P}^{r-1}$ is scheme defined by
the symmetric algebra, and is in particular, set defined by $J$. Consider the
projection
$\pi_{2}:\mathbb{P}^{n-1}\times\mathbb{P}^{r-1}\rightarrow\mathbb{P}^{r-1}$
and $p=(t_{1}:\dots:t_{r})\in\mathbb{P}^{r-1}$. Let $\Theta(p)$ be the image
of $\Theta$ by the specialization $T_{i}\mapsto t_{i}$. Then,

$\displaystyle p\in W$ $\displaystyle\Leftrightarrow\exists
x=(x_{1},\dots,x_{n});(x,p)\in\Gamma$ $\displaystyle\Leftrightarrow\exists
x;L_{i}(x,t)=0\,\forall i$ $\displaystyle\Leftrightarrow\exists
x;\Theta(p)\begin{bmatrix}x_{1}\\\ \vdots\\\ x_{n}\\\ \end{bmatrix}=0$
$\displaystyle\Leftrightarrow t\in I_{n}(\Theta).$

∎

######  Lemma 2.

Assume that $\psi:\mathbb{P}^{n-1}\setminus X\rightarrow\mathbb{P}^{r-1}$ is
defined by $I=(f_{1},\dots,f_{r})$ where $X=V(I)$. Let
$\Gamma\subseteq\mathbb{P}^{n-1}\times\mathbb{P}^{r-1}$ be the closure of the
graph of $\psi$. If $\pi:\Gamma\rightarrow\overline{\operatorname{Im}(\psi)}$
is the projection map, then $\forall p\in\overline{\operatorname{Im}(\psi)}$,
setting $\dim\emptyset:=-1$,

$\dim\pi^{-1}(p)\leq\dim X+1.$

###### Proof.

Let $p=(p_{1}:\dots:p_{r})\in\overline{\operatorname{Im}\psi}$ and let $V$ be
an irreducible component of $\pi^{-1}(p)$. If $V$ is contained in $X$ it has
dimension at most the one of $X$. Else $\psi|_{V}:V\setminus
X\rightarrow\mathbb{P}^{r-1}$ is a constant map with image $p$. Hence for all
$1\leq i,j\leq r$, $(f_{i}p_{j}-f_{j}p_{i})\in I(V)$. Choose $i$ with
$p_{i}\neq 0$, then $f_{j}-p_{j}f_{i}/p_{i}\in I(V)$ for all $j$, and
therefore $I\subseteq I(V)+(f_{i})$, and the height of $I(V)+(f_{i})$ is the
one of $I(V)$ plus one. ∎

######  Lemma 3.

Adopt Notation 1. For a point $p\in W\subseteq W^{\prime}$, the rank of
$\Theta_{\operatorname{\mathfrak{p}}}$ is $n-1$.

###### Proof.

By Lemma 2, $\pi^{-1}(p)$ is a zero dimensional scheme. The assertion follows
from the fact that this fiber is a linear space defined by the system of
equations

$\Theta_{\operatorname{\mathfrak{p}}}(p)\begin{bmatrix}x_{1}\\\ \vdots\\\
x_{n}\\\ \end{bmatrix}=0.$

∎

######  Remark 1.

Since $\operatorname{reg}{I}^{t}=dt+0$ for $t\gg 0$, the regularity of fibers
of the projection $\pi_{2}:\Gamma\rightarrow\mathbb{P}^{r-1}$ at any point is
zero [7]. The only finite set of point(s) with zero regularity is a single
reduced point. More generally, the regularity of stalks of $\pi_{2}$ at any
point is zero too [2].

######  Definition 2.

Let $N$ be a $m\times n$ matrix. For $1\leq r\leq n$ and $1\leq s\leq m$,
define $N^{(i_{1},\dots i_{r})}_{(j_{1},\dots,j_{s})}$ be the matrix obtained
by deleting $i_{1},\dots,i_{r}$-th columns and $j_{1},\dots j_{s}$-th rows of
$N$. Let $N$ be a $(n-1)\times n$ matrix, denote $\Delta_{i}(N)$ to be the
$(-1)^{i}\det N^{(i)}$, with $N^{(i)}$ the matrix obtained by deleting $i$-th
column of $N$. For simplicity we write $\Delta_{i}$ instead of $\Delta_{i}(N)$
if it is clear what matrix we consider.

######  Lemma 4.

Let $N$ be a $(n-1)\times n$ matrix, then

$(-1)^{i}\operatorname{adj}(N^{(i)})N=\left[\begin{array}[]{ccc|c|ccc}\Delta_{i}&&&-\Delta_{1}&&&\\\
&\ddots&&\vdots&&&\\\ &&\Delta_{i}&-\Delta_{i-1}&&&\\\
\hline\cr&&&-\Delta_{i+1}&\Delta_{i}&&\\\ &&&\vdots&&\ddots&\\\
&&&-\Delta_{n}&&&\Delta_{i}\\\ \end{array}\right],$

where only the non zero terms are displayed.

###### Proof.

The $(j,\ell)$-entry of $\operatorname{adj}(N^{(i)})N$ is
$\sum_{k}b_{j,k}a_{k,\ell}$ where $b_{j,k}$ and $a_{k,\ell}$ are the entries
of $\operatorname{adj}(N^{(i)})$ and $N$.

Notice that $b_{j,k}=(-1)^{j+k}\det N^{(i,j)}_{(k)}$ if $j<i$ and
$b_{j,k}=(-1)^{j+k}\det N^{(i,j+1)}_{(k)}$ else.

Hence then the $(j,\ell)$-entry of $\operatorname{adj}(N^{(i)})N$ is

$\sum_{k}(-1)^{j+k}\det N^{(i,j)}_{(k)}a_{k,\ell}\ {\rm if}\ j<i\quad{\rm
and}\ \sum_{k}(-1)^{j+k}\det N^{(i,j+1)}_{(k)}a_{k,\ell}\ {\rm if}\ j\geq i.$

First let $\ell<i$. If $\ell=j$, then it is equal to $\det
N^{(i)}=(-1)^{i}\Delta_{i}$. If $\ell\neq j<i$, replace the $j$-th column of
$N$ with its $\ell$-th column and call it $N^{\prime}$. By expanding
$\Delta_{i}(N^{\prime})$ along the $j$-th column we get

$0=\Delta_{i}(N^{\prime})=(-1)^{i}\sum_{k}(-1)^{j+k}\det
N^{(i,j)}_{(k)}a_{k,\ell},$

and similarly if $j\geq i$, by relacing the $(j+1)$-th column of $N$ with its
$\ell$-th column.

Second, if $\ell>i$, similar arguments as in the case $\ell<i$ show that if
$j=\ell-1$, the $(j,j+1)$-entry is equal to $\Delta_{i}$, and the
$(j,\ell)$-entry is equal to $0$ if $j\neq\ell-1$.

Thirdly, if $j<\ell=i$, the $(j,i)$-entry of $\operatorname{adj}(N^{(i)})N$ is

$\sum_{k}(-1)^{j+k}\det N^{(i,j)}_{(k)}a_{k,i}.$

By expanding $\Delta_{j}=(-1)^{j}\det N^{(j)}$ along the $i$-th column we get

$\Delta_{j}=(-1)^{j}\sum_{k}(-1)^{i-1+k}a_{k,i}\det
N^{(i,j)}_{(k)}=(-1)^{i-1}\sum_{k}(-1)^{j+k}a_{k,i}\det N^{(i,j)}_{(k)}.$

Finally, if $j\geq\ell=i$ then $b_{j,k}=(-1)^{j+k}\det N^{(i,j+1)}_{(k)}$ so
$(j,i)$-th entry $\operatorname{adj}(N^{(i)})N$ is

$\sum_{k}(-1)^{j+k}\det N^{(i,j+1)}_{(k)}a_{k,i}.$

By expanding $\Delta_{j+1}=(-1)^{j+1}\det N^{(j+1)}$ on the $i$-th column we
get

$\Delta_{j+1}=(-1)^{j+1}\sum_{k}(-1)^{i+k}a_{k,i}\det
N^{(i,j+1)}_{(k)}=(-1)^{i-1}\sum_{k}(-1)^{j+k}a_{k,i}\det N^{(i,j+1)}_{(k)}.$

∎

######  Lemma 5.

For a $1\times n$ row vector $R$ define $L$ as a product of $R$ and column
vector $(x_{1},\dots,x_{n}).$ Let $N$ be a $(n-1)\times n$ matrix with row
vectors $R_{1},\dots R_{n-1}$ and consider corresponding linear forms
$L_{1},\dots,L_{n-1}$. Let $M$ obtained by bordering $N$ with a row vector
$R_{n}$ and corresponding linear form $L_{n}$. Then for $1\leq i,j\leq n$

  * (1)

$\Delta_{i}L_{n}+(-1)^{n+1}\det(M)x_{i}\in(L_{1},\dots,L_{n-1})$

  * (2)

$(x_{i}\Delta_{j}-x_{j}\Delta_{i})\in(L_{1},\dots,L_{n-1})$.

  * (3)

If $j\neq n$,
$\Delta_{i}L_{j}\in(x_{1}\Delta_{i}-x_{i}\Delta_{1},\ldots,x_{n}\Delta_{i}-x_{i}\Delta_{n})$.

###### Proof.

(1) Consider

$\det(M)\begin{bmatrix}x_{1}\\\ \vdots\\\
x_{n}\end{bmatrix}=\operatorname{adj}(M)M\begin{bmatrix}x_{1}\\\ \vdots\\\
x_{n}\end{bmatrix}=\operatorname{adj}(M)\begin{bmatrix}L_{1}\\\ \vdots\\\
L_{n}\end{bmatrix}.$

The $(i,1)$-entries in the above matrix are $\det(M)x_{i}$. On the other, it
is equal to

$\sum_{1\leq k\leq n}(-1)^{k+i}\det M^{(i)}_{(k)}L_{k}=\sum_{1\leq k\leq
n-1}(-1)^{k+i}\det M^{(i)}_{(k)}L_{k}+(-1)^{n}\Delta_{i}L_{n}.$

(2) By choosing $L_{n}=e_{j}$ in (1) we get

$\displaystyle\Delta_{i}x_{j}+(-1)^{n+1}\det(M)x_{i}$
$\displaystyle=\Delta_{i}x_{j}+(-1)^{n+1}(-1)^{n+j}(-1)^{j}\Delta_{j}x_{i}$
$\displaystyle=\Delta_{i}x_{j}-x_{i}\Delta_{j}\in(L_{1},\dots,L_{n-1}).$

(3) By Lemma 4

$\Delta_{i}\begin{bmatrix}L_{1}\\\ \vdots\\\
L_{n-1}\end{bmatrix}=(-1)^{i}N^{(i)}\operatorname{adj}(N^{(i)})N\begin{bmatrix}x_{1}\\\
\vdots\\\
x_{n}\end{bmatrix}=N^{(i)}\begin{bmatrix}x_{1}\Delta_{i}-x_{i}\Delta_{1}\\\
\vdots\\\ x_{i-1}\Delta_{i}-x_{i}\Delta_{i-1}\\\
x_{i+1}\Delta_{i}-x_{i}\Delta_{i+1}\\\ \vdots\\\
x_{n}\Delta_{i}-x_{i}\Delta_{n}\end{bmatrix}.$

∎

######  Corollary 1.

For any $(n-1)\times n$ submatrix of $\Theta$ which corresponds to
$L_{i_{1}},\dots,L_{i_{n-1}}$ define $\Delta_{i}$ for $1\leq i\leq n$ as in
Definition 2. If $\Delta_{i}\neq 0$ for some $i$, then

  * (1)

$\Delta_{i}J\subseteq(L_{i_{1}},\dots,L_{i_{n-1}})$ modulo $I_{n}(\Theta)$,

  * (2)

$(L_{i_{1}},\dots,L_{i_{n-1}})B_{(\Delta_{i})}=I_{2}\begin{bmatrix}x_{1}&\cdots&x_{n}\\\
\Delta_{1}&\cdots&\Delta_{n}\end{bmatrix}B_{(\Delta_{i})}$.

###### Proof.

Choose for $N$ the submatrix of $\Theta$ such that

$N\begin{bmatrix}x_{1}\\\ \vdots\\\
x_{n}\end{bmatrix}=\begin{bmatrix}L_{i_{1}}\\\ \vdots\\\
L_{i_{n-1}}\end{bmatrix}.$

(1) follows from Lemma 5 (1), by choosing for $M$ the matrix corresponding to
add any minimal generator $L_{j}$ of $J$.

(2) follows from Lemma 5 (2) and (3). ∎

######  Theorem 1.

Let $S=k[x_{1},\dots,x_{n}]$ be a polynomial ring over a field $k$,
$I=(f_{1},\dots,f_{r})$ be a linearly presented
$\operatorname{\mathfrak{m}}$-primary ideal of $S$ and
$\Phi:\mathbb{P}^{n-1}\dashrightarrow\mathbb{P}^{r-1}$ be the map defined by
the forms $f_{1},\dots f_{r}$. Let $\Theta$ be the Jacobian dual matrix of a
presentation matrix of $I$, then

$I_{n}(\Theta)^{sat}=I(W).$

###### Proof.

By Lemma 1, $\sqrt{I_{n}(\Theta)}=I(W)$. Let
$\operatorname{\mathfrak{p}}\in\operatorname{spec}(B)$ containing
$I_{n}(\Theta)$. We need to show that
$(B/I_{n}(M))_{\operatorname{\mathfrak{p}}}\cong(B/I(W))_{\operatorname{\mathfrak{p}}}$.
For this to hold, it suffices to prove that
$\bar{B}_{\operatorname{\mathfrak{p}}}=(B/I_{n}(M))_{\operatorname{\mathfrak{p}}}$
is a domain. By Lemma 3, there exist $L_{i_{1}},\dots L_{i_{n-1}}$ and $i$
with $\Delta_{i}\notin\operatorname{\mathfrak{p}}$, with notations as in
Corollary 1. Now

$\displaystyle\bar{B}_{\operatorname{\mathfrak{p}}}[x_{1},\dots,x_{n}]/J\otimes_{B}\bar{B}_{\operatorname{\mathfrak{p}}}$
$\displaystyle=\bar{B}_{\operatorname{\mathfrak{p}}}[x_{1},\dots,x_{n}]/(L_{i_{1}},\dots
L_{i_{n-1}})\otimes_{B}\bar{B}_{\operatorname{\mathfrak{p}}}\qquad\mbox{by
Corollary \ref{corollary} (1)}$
$\displaystyle=\bar{B}_{\operatorname{\mathfrak{p}}}[x_{i}]\qquad\qquad\qquad\qquad\qquad\quad\qquad\qquad\mbox{by
Corollary \ref{corollary} (2).}$

As $J$ scheme defines $\Gamma$,
$\operatorname{proj}(\bar{B}_{\operatorname{\mathfrak{p}}}[x_{1},\dots,x_{n}]/J\otimes_{B}\bar{B}_{\operatorname{\mathfrak{p}}})$
is the stalk of the isomorphism $\pi:\Gamma\rightarrow W$ over
$V(\operatorname{\mathfrak{p}})$; in particular, it is reduced and
irreducible. This shows that
$\operatorname{proj}(\bar{B}_{p}[x_{i}])\cong\operatorname{spec}(\bar{B}_{\operatorname{\mathfrak{p}}})$
is reduced and irreducible. Hence $\bar{B}_{\operatorname{\mathfrak{p}}}$ is a
domain. ∎

We finish this section with two conjectures.

######  Conjecture 1.

Let $S=k[x_{1},\dots,x_{n}]$ be a polynomial ring and $I=(f_{1},\dots,f_{r})$
be a linearly presented $\operatorname{\mathfrak{m}}$-primary ideal of $S$.
Suppose that $\Phi:\mathbb{P}^{n-1}\dashrightarrow\mathbb{P}^{r-1}$ is a
rational map defined by forms $f_{1},\dots f_{r}$. Let $\Theta$ be the
Jacobian dual matrix of a presentation matrix of $I$, then

$I_{n}(\Theta)=I(W)_{\geq n}.$

######  Conjecture 2.

Let $S=k[x_{1},\dots,x_{n}]$ be a polynomial ring and $I=(f_{1},\dots,f_{r})$
be a linearly presented $\operatorname{\mathfrak{m}}$-primary ideal of $S$.
Let $\Theta$ be the Jacobian dual matrix of a presentation matrix of $I$ then
$I_{n}(\Theta)$ has a linear resolution.

##  3\. Asymptotic behavior of regularity

In this section, we study the conjecture of Eisenbud, Huneke, and Ulrich [8,
Conjecture 1.1] on the asymptotic behavior of regularity of linearly presented
$\operatorname{\mathfrak{m}}$-primary ideals. We will show the relation
between this conjecture and the ideal of maximal minors of Jacobian dual
matrices. We start this section by stating two equivalent definition of
Castelnuovo-Mumford regularity. Before that, we need to mention the definition
of graded Betti numbers. For a finitely generated graded $S$-module $M$, a
minimal graded free resolution of $M$ is an exact sequence

$0\rightarrow F_{p}\rightarrow F_{p-1}\rightarrow\cdots\rightarrow
F_{0}\rightarrow M\rightarrow 0,$

where every $F_{i}$ is a graded free $S$-module of the form
$F_{i}=\oplus_{j\in\mathbb{Z}}S(-j)^{\beta_{i,j}(M)}$ with the minimal number
of basis elements, and every map is graded. The value $\beta_{i,j(M)}$ is
called the $i$th graded Betti number of $M$ of degree $j$.

######  Theorem 2.

Let $M$ be a finitely generated $S$-module, the regularity of $M$ is given by:

$\operatorname{reg}{(}M)=\max_{i}\\{\operatorname{end}(H^{i}_{\operatorname{\mathfrak{m}}}(M))+i\\}.$

Although the regularity is a measure of vanishing of local cohomologies, it is
also a measure of vanishing of graded Betti numbers. Eisenbud and Goto in [6]
proved that

$\operatorname{reg}{(}M)=\max_{i}\\{j-i|\beta_{i,j}(M)\neq 0\\}.$

The most significant simple result on the regularity of powers of graded
ideals is the following one, due independently to Kodiyalam [9] and to
Cutkosky, Herzog and Trung [5].

######  Theorem 3.

Let $I$ be an ideal of $S=k[x_{1},\dots,x_{n}]$. There exists $t_{0}$ and $b$
such that

$\operatorname{reg}{(}I^{t})=td+b,\quad\forall t\geq t_{0}$

with

$d:=\min\\{\mu\,\,|\,\,\exists t\geq 1,\,\,(I_{\leq\mu})I^{t-1}=I^{t}\\}.$

Notice that when $I$ is equigenerated the number $d$ in the above theorem is
the degree of the generators of $I$. We call smallest such $t_{0}$ as the
stabilization index of $I$ and denote it by $\operatorname{Stab}(I)$. In [8],
authors showed that if $I$ is a linearly presented
$\operatorname{\mathfrak{m}}$-primary ideal, then the powers of $I$ eventually
have a linear resolution.

######  Theorem 4 ([8]).

Let $S=k[x_{1},\dots,x_{n}]$ and $I$ be an ideal of $S$. If $I$ is a linearly
presented $\operatorname{\mathfrak{m}}$-primary ideal generated in degree $d$,
then $\operatorname{reg}{(}I^{t})=td$ for $t\gg 0$.

In addition, they conjectured an upper bound for the stabilization index of
$I$.

######  Conjecture 3.

[8, Conjecture 1.1] Let $S=k[x_{1},\dots,x_{n}]$ and $I$ be an ideal of $S$.
If $I$ is a linearly presented $\operatorname{\mathfrak{m}}$-primary ideal
generated in degree $d$, then $I^{n-1}=\operatorname{\mathfrak{m}}^{d(n-1)}$.
In other word,

$\operatorname{reg}{(}I^{n-1})=(n-1)d.$

######  Remark 2.

Eisenbud, Huneke and Ulrich in [8] proved this conjecture when $n=3$ or $I$ is
a monomial ideal.

######  Remark 3.

Since the only $\operatorname{\mathfrak{m}}$-primary ideals with linear
resolution are the powers of $\operatorname{\mathfrak{m}}$, for an
$\operatorname{\mathfrak{m}}$-primary ideal $I$ generated in degree $d$,

$\operatorname{Stab}(I)=\min\\{t\,|h_{I^{t}}(dt)=h_{\operatorname{\mathfrak{m}}^{t}}(dt)\\},$

where $h_{M}(d):=\dim_{k}M_{d}$ is the Hilbert funtion of $M$ at degree $d$.

The following proposition provide a connection between regularity of powers of
an ideal $I=(f_{1},\dots,f_{r})$ and the image of
$\phi:\mathbb{P}^{n-1}\dashrightarrow\mathbb{P}^{r-1}$ defined by
$f_{1},\dots,f_{r}$.

######  Proposition 1.

Let $S=k[x_{1},\dots,x_{n}]$ and $I$ be an ideal of $S$. Adopting the Notation
1, If $I$ is a linearly presented $\operatorname{\mathfrak{m}}$-primary ideal
generated in degree $d$, then

  * (1)

$W$ is smooth of dimension $n-1$, and $e(W)=d^{r-1}$.

  * (2)

$\operatorname{reg}{I}(W)=\max\\{\operatorname{Stab}(I)+1,\operatorname{reg}{I}(V_{n}^{(d)})\\}=\max\\{\operatorname{Stab}(I)+1,n-\lceil\dfrac{n}{d}\rceil\\}$

where $V_{n}^{(d)}$ is the Veronese embedding of degree $d$.

###### Proof.

Part $(1)$ follows from [3, Proposition 1.7(b)]. For proving part $(2)$ we use
the definition of regularity via local cohomologies. Let $I(W)\subset
B=k[\mathbf{T}]$ and $I(V_{n}^{(d)})\subset B^{\prime}=k[\mathbf{T}^{\prime}]$
with $B\subset B^{\prime}$ and $I(W)=I(V_{n}^{(d)})\cap B$. As the natural map
$B/I(W)\rightarrow B^{\prime}/I(V_{n}^{(d)})$ is an isomorphism locally on the
punctured spectrum, $H^{i}_{\mathbf{T}}(B/I(W))\cong
H^{i}_{\mathbf{T}}(B^{\prime}/I(V_{n}^{(d)}))\cong
H^{i}_{\mathbf{T}^{\prime}}(B^{\prime}/I(V_{n}^{(d)}))$ for $i\geq 2$ and
there is an exact sequence:

$0\rightarrow B/I(W)\rightarrow B^{\prime}/I(V)\rightarrow
H^{1}_{\mathbf{T}}(B/I(W))\rightarrow 0.$

Since $h_{B/I(W)}(a)=h_{I^{a}}(da)$ and
$h_{B/I(V)}(a)=h_{\operatorname{\mathfrak{m}}^{da}}(da)$, by Remark 3 and
above exact sequence, $H^{1}_{\mathbf{T}}(B/I(W))=0$ if and only if
$i\geq\operatorname{Stab}(I)$. In other words,
$\text{end}H^{1}_{\mathbf{T}}(B/I(W))=\operatorname{Stab}(I)-1$. The assertion
follows from the definition of regularity and the fact that
$\operatorname{reg}{I}(W)=\operatorname{reg}{B}/I(W)+1$. ∎

Conjecture 3 was the initial motivation of this work, we end this paper by
showing that how our conjectures are related to Conjecture 3.

######  Theorem 5.

Let $S=k[x_{1},\dots,x_{n}]$ and $I$ be an ideal of $S$. Adopting the Notation
1, If $I$ is a linearly presented $\operatorname{\mathfrak{m}}$-primary ideal
generated in degree $d$, then

  * $(1)$

Conjecture 3 follows from Conjecture 2.

  * $(2)$

If Conjecture 1 holds, then Conjecture 2 and Conjecture 3 are equivalent.

###### Proof.

$(1)$ Assume $I_{n}(\Theta)$ has a linear resolution, i.e.
$\operatorname{reg}{(}I_{n}(\Theta))=n$. By Theorem 1,
$I_{n}(\Theta)^{sat}=I(W)$. Hence

$\operatorname{reg}{I}(W)\leq\operatorname{reg}{(}I_{n}(\Theta))\leq n.$

As $\operatorname{reg}{(}I(V_{n}^{(d)})=n$, Proposition 1 implies that
$\operatorname{Stab}(I)\leq n-1$.

$(2)$ First we should note that by Proposition 1, $\operatorname{Stab}(I)\leq
n-1$ if and only if $\operatorname{reg}{(}I(W))\leq n$. As we assume
$I_{n}(\Theta)=I(W)_{\geq n}$, $\operatorname{reg}{(}I(W))\leq n$ if and only
if $I_{n}(\Theta)$ has a linear resolution. ∎

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# Demonstration of Frequency Stability
limited by Thermal Fluctuation Noise in Silicon Nitride Nanomechanical
Resonators

Chang Zhang, Raphael St-Gelais<EMAIL_ADDRESS>Department of
Mechanical Engineering, University of Ottawa, Ottawa, ON, Canada
<EMAIL_ADDRESS>
###### Abstract

The frequency stability of nanomechanical resonators (NMR) dictates the
fundamental performance limit of sensors that relate physical perturbations to
a resonance frequency shift. While the contribution of thermomechanical noise
to frequency stability was understood recently, thermal fluctuation noise has
attracted less attention despite being the ultimate performance limit of
temperature sensing. We provide a model for the frequency stability of NMR
considering both additive phase noise (i.e., thermomechanical and detection
noises) and thermal fluctuation noise. We then experimentally demonstrate
optimized NMR achieving frequency stability limited by thermal fluctuation
noise. Our work shows that current models for NMR frequency stability can be
incomplete. It also paves a way for NMR radiation detectors to reach the
unattained fundamental detectivity limit of thermal-based radiation sensing.

††preprint: APS/123-QED

Nanomechanical resonators (NMR) frequency stability is the fundamental
quantity dictating the performance of sensors measuring physical signals
through resonance frequency shifts (e.g., mass [1, 2, 3, 4], force [5, 6], and
thermal [7, 8, 9, 10, 11, 12] sensors). Hence, identifying the source of noise
that dominates frequency fluctuations in the absence of a signal is
fundamental for understanding the performance limit of NMR. Early theoretical
investigation of various sources of noise in NMR was proposed by Vig
$et~{}al.$ [13] and Cleland $et~{}al.$ [14]. These theoretical works provided
a comprehensive picture of multiple sources of noise in NMR, including surface
diffusion, absorption-desorption, thermomechanical, detection, and thermal
fluctuation noises. Thermomechanical noise was later identified as a dominant
source of noise in many cases. Substantial efforts were therefore devoted to
resolving the frequency stability limit imposed by thermomechanical noise [15,
16, 17, 18], and to predicting its effect after processing by various
frequency tracking schemes. Demir [19] recently provided a theoretical model
for predicting the frequency stability of resonators, which considers the
combined effect of thermomechanical and detection noise in commonly employed
phase-locked loop (PLL) frequency tracking. Such theoretical model was later
validated experimentally by Sadeghi $et~{}al.$ [20], confirming that
thermomechanical and detection noises can be the dominant noises in high-
stress silicon nitride string resonators.

Figure 1:  Additive phase noise vs. thermal fluctuation (a) Illustration of
the characteristic difference between additive phase noise and thermal
fluctuation. (b) Increase in temperature coefficient $\alpha$, resonance
frequency $f_{r}$, mechanical Q-factor, vibration amplitude $A_{rss}$, mass-
to-thermal-conductance ratio $m_{eff}/G$ of the nanomechanical resonator (NMR)
minimizes additive phase noise relative to thermal fluctuation noise.

While thermomechanical noise was proven to dominate frequency fluctuation in
many high-Q factor NMR, achieving regimes in which thermal fluctuation noise
would dominate remains of high interest, especially in temperature sensing
applications. In temperature sensors, the smallest temperature that
theoretically can be measured is ultimately dictated by thermal fluctuation
noise. This limit is of particular interest in the field of radiation
detection, as it dictates the fundamental detectivity limit that can
ultimately be reached by thermal-based radiation detectors [21, 22], or
bolometers. In mechanical resonator-based radiation sensors [7, 8, 9, 10, 11,
12], thermal fluctuation is typically not the dominant source of noise, which
means that their ultimate limit of performance has not been reached. Likewise,
the fundamental limit has not been reached in traditional (i.e., electrical)
bolometers since Johnson-Nyquist noise always dominates over thermal
fluctuation noise [21]. NMR are potentially interesting due to their immunity
to Johnson-Nyquist noise, but so far were found to be limited instead by
thermomechanical, detection [7, 9, 10, 11, 12], or flicker noise [8].

In the current work, we experimentally demonstrate low-stress, high Q-factor
SiN drum resonators in which frequency instability is minimized down to
fundamental thermal fluctuation noise. We also include thermal fluctuation
noise with recently proposed models for frequency stability within a closed-
loop frequency tracking scheme [19]. Our results and model, therefore, pave
the way for radiation sensors that could reach the fundamental detectivity
limit of physical radiation detectors.

We separate the sources of frequency fluctuation in a drum resonator into two
categories, namely, the additive phase noise $S_{y,add}(\omega)$ which
includes thermomechanical $S_{y,mech}(\omega)$ and detection noise
$S_{y,det}(\omega)$; and thermal fluctuation noise $S_{y,th}(\omega)$. Here
$y$ represents fractional frequency $\langle\delta f/f_{r}\rangle$
fluctuations. We also define the intrinsic noise spectral density
$S_{y}^{int}(\omega)$ before it is modified by the experimental frequency
measurement scheme (e.g., open-loop, phase-lock loop frequency tracking, or
self-sustained oscillation). For a given eigen frequency $f_{r}$, the
intrinsic frequency fluctuation caused by additive phase noise
$S_{y,add}^{int}(\omega)$ is therefore [19]:

$\displaystyle S_{y,add}^{int}(\omega)$
$\displaystyle=S_{y,mech}^{int}(\omega)+S_{y,det}(\omega)$ (1)
$\displaystyle=\frac{k_{B}T}{8\pi^{3}m_{eff}f_{r}^{3}QA_{rss}^{2}}|H_{mech}(j\omega)|^{2}$
$\displaystyle+\frac{k_{B}T}{8\pi^{3}m_{eff}f_{r}^{3}QA_{rss}^{2}}\kappa_{d}^{2}$
$\displaystyle=\frac{k_{B}T}{8\pi^{3}m_{eff}f_{r}^{3}QA_{rss}^{2}}(|H_{mech}(j\omega)|^{2}+\kappa_{d}^{2}),$

where $H_{mech}(j\omega)=1/(1+j\omega\tau_{mech})$ is a one-pole low pass
filter accounting for the mechanical time constant $\tau_{mech}=Q/(\pi f_{r})$
of the resonator, $k_{B}$ is the Boltzmann constant, $T$ is the eigenmode
temperature, $m_{eff}$ is the mode effective mass, $Q$ is the mechanical
quality factor, $A_{rss}$ is the vibration amplitude in steady-state and
$\kappa_{d}$ is a dimensionless parameter scaling the level of detection noise
relative to thermomechanical noise, as described in [19]. More specifically,
thermomechanical noise is resolved above detection noise when $\kappa_{d}<1$.

In turn, the intrinsic frequency fluctuation caused by thermal fluctuation
[13, 22] is:

$S_{y,th}^{int}(\omega)=\frac{2k_{B}T^{2}\alpha^{2}}{\pi
G}|H_{th}(j\omega)|^{2},$ (2)

where $G$ is the total thermal conductance [12] in W/K between the resonator
and its environment, $H_{th}(j\omega)=1/(1+j\omega\tau_{th})$ is a one-pole
filter accounting for the thermal response time $\tau_{th}=C_{th}/G$ of the
resonator, and $C_{th}$ is the heat capacity of the resonator in J/K. An
analytical model for $G$ (and hence $\tau_{th}$) in drum resonators is
developed in [12] and is used throughout this work. $\alpha$ is the
temperature coefficient of fractional frequency shifts, in $\mathrm{K^{-1}}$.
For a drum resonator, a reasonable approximation ($<20$% error relative to
finite element modeling [12]) of this parameter is:

$\alpha\approx\frac{E\alpha_{T}}{2\sigma(1-\nu)},$ (3)

in which $E$ is Young’s modulus, $\alpha_{T}$ is the drum material thermal
expansion coefficient, $\sigma$ is the built-in tensile stress, and $\nu$ is
the Poisson ratio of the drum resonator material. In Eq. (2), $T$ denotes the
drum resonator material temperature, whereas $T$ in Eq. (LABEL:eq:1) denotes
the temperature of the eigenmode of frequency $f_{r}$. We use the symbol $T$
interchangeably in this work since both are assumed to be $\approx 300$ K.

A key difference between the additive phase noise and thermal fluctuation
noise is schematized in Fig. 1(a), in which the additive phase noise manifests
itself as amplitude fluctuation, which then contributes to the frequency noise
via Robins’ formula [23]. Conversely, thermal fluctuations make the resonance
peaks of the NMR fluctuate directly in the frequency domain by affecting the
stiffness of the drum material. We note that one can change the level of
additive phase noise by utilizing different eigenmodes of the resonators [see
Eq. (LABEL:eq:1)] since $f_{r}$, $Q$ are mode-dependent. Likewise, additive
phase noise can be minimized by increasing the vibration amplitude $A_{rss}$
(within the linear actuation regime). On the contrary, mode and amplitude
changes do not affect the level of thermal fluctuation [see Eq. (2)], which
depends primarily on drum resonator geometric, material and heat transfer
properties (e.g., $\alpha$ and $G$).

To compare the relative contributions of $S_{y,mech}^{int}(\omega)$ and
$S_{y,th}^{int}(\omega)$ with respect to the overall frequency fluctuation, we
define a dimensionless ratio $\gamma$:

$\displaystyle\gamma$
$\displaystyle=\frac{S_{y,th}^{int}(0)}{S_{y,mech}^{int}(0)}$ (4)
$\displaystyle=16\pi^{2}TQ\frac{m_{eff}}{G}\cdot\alpha^{2}A_{rss}^{2}f_{r}^{3}$

This ratio ignores, for simplicity, the effect of both thermal
$H_{th}(j\omega)$ and mechanical $H_{mech}(j\omega)$ response filters by
setting $\omega=0$. A value of $\gamma\approx 1$ indicates thermal fluctuation
and additive phase noise are at a similar level. If $\gamma$ is significantly
larger than 1 and detection noise is minimized (i.e., $\kappa_{d}<<1$), a
thermal-fluctuation-dominated noise profile can be achieved.

By examining $\gamma$, we find that thermal-fluctuation-dominated noise
profile can be achieved by maximizing the values of $Q$, $\frac{m_{eff}}{G}$,
$\alpha$, $A_{rss}$ and $f_{r}$. Among these listed parameters, the mass-to-
thermal-conductance ratio $\frac{m_{eff}}{G}$ is likely the most unintuitive.
Hence, in Fig. 2, we illustrate this ratio as a function of the dimensions of
a square SiN drum resonators of side length $L$ and thickness $t$. Values of
$G$ are presented in Fig. 2(a) and are calculated using the model developed in
[12] which includes conductive and radiative heat transfer. We note that for
relatively small drum resonators ($L<1$ mm) where heat transfer occurs mostly
by conduction, both $G$ and $m_{eff}$ scale with the thickness, such that $t$
cancels out in the $\frac{m_{eff}}{G}$ ratio [see Fig. 2(b)]. Conversely, as
$L>1$ mm, radiative heat transfer dominates, and $L$ now cancels, making the
curves plateau in Fig. 2(b).

In summary, considering this trend in $\frac{m_{eff}}{G}$ and the expression
of $\gamma$, we find that minimizing additive phase noise below thermal
fluctuation requires (i) a temperature sensitive resonator (i.e., high
$\alpha$) which can be maximized using a low-stress material as suggested in
Eq. (3), (ii) a large drum resonator (i.e., large $L$) to maximize
$\frac{m_{eff}}{G}$, (iii) a high order (i.e., high $f_{r}$) and high Q
eigenmode, and (iv) excitation at high amplitude $A_{rss}$ within the limits
of linear actuation. We finally note that, despite the trend observed in Fig.
2(b), increasing $t$ beyond $\approx 100~{}\mathrm{nm}$ to maximize
$\frac{m_{eff}}{G}$ is likely not a practical approach since high $t$ can also
be detrimental to Q-factor [24, 25].

Figure 2:  Effect of dimensions on heat transfer quantities in silicon nitride
drum resonators, namely (a) $G$ and (b) $\frac{m_{eff}}{G}$.

To evaluate the spectral density of the frequency fluctuations $S_{y}(\omega)$
in a practical experimental setting, we must also consider the effect of the
measurement scheme. A phase-locked loop (PLL) frequency tracking scheme, such
as in the current work, includes a proportional-integral controller with
proportional gain $K_{p}$ and integral gain $K_{i}$, and an input demodulator
filter of time constant $\tau_{demod}$. As shown in [19], the PLL frequency
tracking scheme imposes filters on the additive phase noises of the resonators
(i.e., the thermomechanical and detection noises):

$H_{mech}^{PLL}(j\omega)=\frac{(j\omega
K_{p}+K_{i})H_{L}(j\omega)}{-\omega^{2}+\frac{j\omega}{\tau_{mech}}+(j\omega
K_{p}+K_{i})H_{L}(j\omega)},$ (5)
$H_{det}^{PLL}(j\omega)=\frac{H_{mech}^{PLL}(j\omega)}{H_{mech}(j\omega)},$
(6)

where $H_{L}(j\omega)=1/(1+j\omega\tau_{demod})$ is the demodulator filter.
Since $S_{y,mech}(\omega)$ and $S_{y,th}(\omega)$ are both white noises, the
PLL frequency tracking scheme imposes the same filtering effect on them, with
the only difference being the time constant $\tau_{th}$:

$H_{th}^{PLL}(j\omega)=\frac{(j\omega
K_{p}+K_{i})H_{L}(j\omega)}{-\omega^{2}+\frac{j\omega}{\tau_{th}}+(j\omega
K_{p}+K_{i})H_{L}(j\omega)}.$ (7)

We then incorporate the PLL transfer functions [Eq. (5-7)] with
$S_{y,add}^{int}(0)$ and $S_{y,th}^{int}(0)$ to obtain the noise after
processing by the PLL frequency tracking:

$\begin{split}S_{y,add}^{PLL}(\omega)=S_{y,mech}^{int}&(0)|H_{mech}^{PLL}(j\omega)|^{2}+\\\
&S_{y,det}(0)|H_{det}^{PLL}(j\omega)|^{2},\end{split}$ (8)

$S_{y,th}^{PLL}(\omega)=S_{y,th}^{int}(0)|H_{th}^{PLL}(j\omega)|^{2}.$ (9)

Finally, the overall fractional frequency fluctuation $S_{y}(\omega)$ of a
resonator under the PLL frequency tracking scheme is simply
$S_{y}(\omega)=S_{y,add}^{PLL}(\omega)+S_{y,th}^{PLL}(\omega)$.

Figure 3:  Schematics of the experimental setup (a) Pictures of SiN drum
resonators of three different sizes characterized in this work. (b) Picture of
SiN drum resonator mounted on a steel plate using magnetic spheres inside our
vacuum chamber. (c) Overall schematics of the experimental setup which
includes the laser interferometer located outside of the high-vacuum chamber
and the actuation and mounting method of SiN drum resonator inside of the
chamber.

SiN drum resonators used in this work are fabricated using the process
provided in [22]. During characterization, the SiN drum resonators are mounted
magnetically onto a steel plate via three pairs of spherical magnets as shown
in Fig. 3(b) and exited mechanically via a ceramic piezo actuator mounted on
the other side of the steel plate [see Fig. 3(c)]. The magnet mounting method
provides minimum contact area between the mounts and the chip, thus minimizing
mechanical dissipation. We place the SiN drum resonators inside of a custom-
built, high-vacuum ($8\times 10^{-7}$ Torr typical operating pressure) chamber
to minimize air damping (i.e., maintaining high-Q factor) and convective heat
transfer. The chamber is left to thermally stabilize for two days (48 hours)
after pumping down before we perform frequency fluctuation measurements.

We detect the vibration signal of the SiN drum resonator, using a laser
interferometer that consists of a $1550~{}\mathrm{nm}$ $\mathrm{Orion^{TM}}$
laser with built-in optical isolation, a 90:10 optical fiber coupler, 5 dB
optical attenuator, optical isolator and a Thorlabs PDA20CS2 photodetector
which is shown schematically in Fig. 3(c). The isolator at the location in
Fig. 3(c) eliminates spurious optical cavities between the detector and the
sample chip. The laser power output is set at $3.7~{}\mathrm{mW}$ which is
then attenuated to $11.7~{}\mathrm{\mu W}$ prior to reaching the SiN drum
resonator, via the 5 dB optical attenuator and the 90:10 optical coupler
(i.e., 90% power attenuation). Laser power attenuation is critical to minimize
the effect of laser heating, as observed in a separate experiment in
Supplemental Material [26]. The detected signal is sampled by a lock-in
amplifier from Zurich Instrument Ltd. We use the phase-lock-loop (PLL) along
with a proportional-integral controller provided by the lock-in amplifier to
track the resonance frequency of the SiN drum resonators.

We quantify the frequency fluctuation of SiN drum resonators in this work
using Allan deviation $\sigma_{A}$ [27], a metrology standard widely used to
characterize the frequency fluctuation in nanoresonators. Based on the
theoretically expected spectral density of frequency fluctuation
$S_{y}(\omega)$, we can numerically compute the theoretical $\sigma_{A}$ via
[19]:

$\sigma_{A}(\tau)=\frac{2}{\sqrt{\pi}\tau}\left[\int_{-\infty}^{\infty}\frac{[\sin(\frac{\omega\tau}{2})]^{4}}{\omega^{2}}S_{y}(\omega)\,d\omega\right]^{\frac{1}{2}},$
(10)

where $\tau$ is the integration time. The asymptotic limit of $\sigma_{A}$
(i.e., excluding all intrinsic and PLL filters) for white noises can be
computed analytically as $\sqrt{S_{y}(0)/\tau}$, such that the asymptote
specifically for thermal fluctuation noise is $\sqrt{S_{y,th}^{PLL}(0)/\tau}$.

In order to observe additive phase noise and thermal fluctuation noise over a
broad range of $\tau$, the demodulation time constant in our experiment is set
to a high bandwidth ($\tau_{demod}=3.18\times 10^{-5}$ s) to minimize signal
filtering at the lock-in input. Likewise, the PLL bandwidth is set to $\approx
5$ times faster than the thermal fluctuation bandwidth
($\tau_{PLL}\approx\tau_{th}/5$) to prevent filtering of thermal fluctuation
noise $S_{y,th}^{int}$ by the PLL. The corresponding $K_{p}$ and $K_{i}$
values for achieving this bandwidth are determined using the relations given
in [19].

The geometric and material properties of the SiN drum resonators, and of the
eigenmodes chosen for actuation, are decided to minimize additive phase noise
related to thermal fluctuation noise according to Eq. (LABEL:eq:3). The drum
resonators large area [i.e., $L=1.5~{}\mathrm{mm}$, $3~{}\mathrm{mm}$,
$6~{}\mathrm{mm}$ shown in Fig. 3(a)] maximizes the mass-to-thermal-
conductance ratio $\frac{m_{eff}}{G}$ for the drum resonator thickness of 90
nm. The use of low-stress ($\sigma\approx 100$ MPa) SiN membrane maximizes the
temperature sensitivity $\alpha$ via Eq. (3). The values of $E$, $\alpha_{T}$
and $\nu$ for SiN are respectively 300 GPa, $2.2\times 10^{-6}$
$\mathrm{K^{-1}}$ and 0.28, which result in $\alpha\approx$ $4.6\times
10^{-3}$ $\mathrm{K^{-1}}$. We also pick high-order eigenmodes (high $f_{r}$)
having high Q-factors. Specifically, the values of $f_{r}$ and Q-factors, from
the smallest to largest SiN drum resonator are respectively 137.6, 90.2, 208.6
kHz and 0.86, 1.18, 1.24 million.

Figure 4: (a) Schematic of changing noise profile via drive increase. (b)
Theoretical Allan deviations $\sigma_{A}$ comparison between the model that
solely considers additive phase noise (labelled ”ADD”), the model that solely
considers thermal fluctuation noise (labelled ”TH”) and the model that
considers both additive phase noise and thermal fluctuation noise (labelled
”ADD+TH”), for a $3\times 3$ mm SiN drum resonator at four levels of
$A_{rss}$. (c) Comparison of experimental Allan deviations $\sigma_{A}$ (black
solid lines) with respect to all theoretical $\sigma_{A}$ at two levels of
$A_{rss}$. Figure 5: Experimental and theoretical Allan deviations
$\sigma_{A}$ for SiN drum resonators of three different sizes, at multiple
vibration amplitudes $A_{rss}$.

Considering these drum resonator parameters, we compare, in Fig. 4(b), the
expected Allan deviation $\sigma_{A}$ plots for our model (labelled ”ADD+TH”)
with that of recent model [19] that solely includes additive phase noise
(labelled ”ADD”). In this case, we consider a low-stress $3\times 3$ mm SiN
drum resonator at different levels of actuation $A_{rss}$ and we set
$\kappa_{d}=0.012$ to account for typical detection noise in our experiment.
We note that at small $\tau\lesssim 0.005$ s, the two models overlap with each
other, which indicates that additive phase noise is dominant. More
specifically, $S_{y,det}$ dominates, since $S_{y,th}^{int}$ and
$S_{y,mech}^{int}$ are attenuated by their respective intrinsic filter
$H_{th}$ and $H_{mech}$ (with $\tau_{th}=\mathrm{0.09}$ s and
$\tau_{mech}=\mathrm{4.15}$ s). At intermediate integration time
$0.005\lesssim\tau\lesssim 0.07$ s, and when $A_{rss}$ is sufficiently high,
thermal fluctuation noise becomes non-negligible. In this case, theoretical
$\sigma_{A}$ plots that include thermal fluctuation converge towards an
$A_{rss}$-independent thermal fluctuation asymptote
($\sqrt{S_{y,th}^{PLL}(0)/\tau}$) as $\tau$ increases. On the contrary,
considering only additive phase noise in Fig. 4(b) do not exhibit amplitude-
independent-converging effect.

This difference between the models is confirmed experimentally in Fig. 4(c),
where we find that experimentally recorded $\sigma_{A}$ match closely with our
model. Conversely, the ”ADD” model fails at intermediate $\tau$ values (i.e.,
in the ”thermal fluctuation dominated region”) when $A_{rss}$ is sufficiently
high for thermal fluctuation noise to dominate. Eventually, drift occurs and
systematically dominates at $\tau\gtrsim 0.07$ s. Note that values presented
in Fig. 4(c) are a limited subset of the more complete data set presented in
Fig. 5. The experimental conditions and model fitting parameters are discussed
below for Fig. 5 therefore also applies to Fig. 4(c).

To further validate our theoretical model, we repeat Allan deviation
$\sigma_{A}$ measurements for resonators of three different sizes (i.e.,
$L=1.5$ mm, 3 mm, 6 mm) and for several drive amplitudes $A_{rss}$ in Fig. 5.
During experiments, we first excite the resonators at low $A_{rss}$, such that
the experimental $\sigma_{A}$ is overall slightly above the thermal
fluctuation asymptote—i.e., additive phase noise is marginally larger than
thermal fluctuation noise. We then increase $A_{rss}$ progressively to reduce
the additive phase noise. We find that all three drum resonators consistently
converge towards the thermal asymptote at intermediate $\tau$ values, and when
$A_{rss}$ is sufficiently large. The convergence is then rapidly shadowed by
drift at large $\tau$.

Correspondence between the model and experiment in FIG. 5 was obtained by
using only fit parameters that are expected from our experimental
uncertainties. Two fit parameters are used for all resonators, while a third
one is needed only for the largest resonator. The vibration amplitude of the
resonators $A_{rss}$ and the detection noise scaling factor $\kappa_{d}$ are
fitted for all three resonators to account for misalignment uncertainly
between the location of our optical fiber and the mode vibration anti-node.
This misalignment is found to underestimates $A_{rss}$ by fitted factors of
(1, 1.2, 2.5) for the three respective resonators sizes. Likewise, fitting
$\kappa_{d}$ yields increase factors of (50%, 30%, 10%) relative to the
$\kappa_{d}$ values expected from our measurement noise floor ($\approx 0.23$
$\mathrm{pm/\sqrt{Hz}}$) and from theoretically expected thermomechanical
fluctuations. Finally, we find that for the largest resonator, the
experimental thermal time constant $\tau_{th}$ must be reduced by a factor of
2 relative to the theoretically expected value. It is possible that the very
high order mode used in this case lead to a higher $G$ (lower $\tau_{th}$) due
to several mode anti-nodes being located close to the heat-dissipating silicon
frame.

We note that our measurement in Fig. 5 confirms important relations between
the level of thermal fluctuation relative to the membrane dimensions and
mechanical properties. We first observe the level of thermal fluctuation noise
(i.e., the TH asymptote vertical position) scales inversely with the drum
resonator side length (i.e., with $G$), as predicted by Eq. (2) and Fig. 2.
This as an important consequence in practice. While Eq. (2) and Fig. 2(b)
suggest that large drum resonators are always better (to maximize
$\frac{m_{eff}}{G}$), this is not entirely true in practice. As our membrane
size increases, it becomes increasingly difficult to identify the thermal
asymptote before drifts occur at higher $\tau$. A trade-off therefore exists
when designing sensors operating at the fundamental thermal fluctuation noise
limit; large drum resonator should be used to maximize the $\frac{m_{eff}}{G}$
ratio up to the plateau observed in Fig. 2(b). However, thermal fluctuation
noise limit will eventually be shadowed by drift if the resonator is too large
($L>>3$ mm).

In conclusion, we present and experimentally validate a model for computing
frequency stability of NMR considering the effect of both additive phase noise
and thermal fluctuation noise in close-loop frequency tracking scheme. We
demonstrated that by using SiN drum resonators of properly designed geometric
and material properties, one can minimize additive phase noise below thermal
fluctuation noise. We also identified which resonators inherent properties
(e.g., thermal conduction, thermal coefficient of frequency) most affect
thermal noise, in contrast with additive phase noise that can be minimized via
other parameters (e.g., drive amplitude). Our work therefore provides
fundamental guidance for building thermal sensors such as nanomechanical
bolometers. We provide a path for those to reach the never attained
fundamental detectivity limit [21] of thermal radiation detection, which
requires sensors limited only by fundamental thermal fluctuation noise [22].

The authors would like to acknowledge Prof. Silvan Schmid for useful
discussion on laser-induced heating. This work was funded by Natural Sciences
and Engineering Research Council of Canada (NSERC) Discovery Grants program.

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## Supplementary Information

###  .1 Finding the optimum laser power

Optimizing the laser power is critical for our experiment: too high power may
cause spurious heating that affects thermal fluctuation readings, while low
power increases detection noise and can therefore prevent observation of
fundamental thermal fluctuations. In this supplementary experiment, we
empirically identify a suitable range of laser power for observing the
dominant effect of thermal fluctuation noise in our SiN resonators. We replace
the fixed optical attenuator in Fig. 3(c) by a variable optical attenuator
(Thorlabs V1550A) such that the laser power incident on the SiN resonator can
be varied. We then drive the SiN resonator to an amplitude (40 nm) suitable
for observation of thermal fluctuations noise (see Fig. 4-5), and we record
Allan deviations for different laser powers. As shown in Fig. S1, in the
thermal fluctuation region, the Allan deviation varies with laser power when
the attenuation is the 0 – 3.1 dB range, meaning that laser at this intensity
heats up the membrane and adds to thermal fluctuation noise. In the optimal
3.1 – 5 dB attenuation range, the characteristic bump of thermal fluctuations
frequency noise is clearly noticeable and is now independent of laser power.
We therefore chose a fixed 5 dB attenuation in this work. Above 5 dB
attenuation, additive phase noise becomes excessive, such that the effect of
thermal fluctuation cannot be observed anymore.

Figure S1: Experimental Allan deviations $\sigma_{A}$ of a $3\times 3$ mm SiN
drum resonator at varying levels of laser power incident on the membrane.


# Patterned Beam Training: A Novel Low-Complexity and Low-Overhead Scheme for
ELAA

Hongkang Yu, Yuan Si, Shujuan Zhang, and Yijian Chen

###### Abstract

Extremely large antenna arrays (ELAAs) can provide higher spectral efficiency.
However, the use of narrower beams for data transmission significantly
increases the overhead associated with beam training. In this letter, we
propose a novel patterned beam training (PBT) scheme characterized by its low
overhead and complexity. This scheme requires only a single linear operation
by both the base station and the user equipment to determine the optimal beam,
reducing the training overhead to half or even less compared to traditional
exhaustive search methods. Furthermore, We discuss the pattern design
principles in detail and provide specific forms. Simulation results
demonstrate that the proposed scheme outperforms the compared methods in terms
of beam alignment accuracy and achieves a balance between signal-to-noise
ratio (SNR) conditions and training overhead, making it a promising
alternative.

###### Index Terms:

ELAA, beam training, pattern design, beam gain, overhead reduction.

##  I Introduction

Future mobile communication systems will operate at higher frequency bands to
provide greater bandwidth and higher data rates [1]. To combat severe
pathloss, the base station (BS) will be equipped with an extremely large
antenna array (ELAA), utilizing beamforming techniques to ensure coverage [2].
The increased array aperture leads to narrower beams, making beam alignment
between the BS and the user equipment (UE) crucial for system performance.

Typically, the BS performs beam training to achieve beam alignment [3]. One
common approach is to exhaustively sweep through all candidate beams and
determine the optimal one according to UE feedback. Unfortunately, the sharp
increase in training overhead has limited its application in ELAA scenarios.

To address this issue, hierarchical beam training scheme has been proposed [4,
5]. This scheme involves transmitting training beams with varying widths
across multiple stages. Nevertheless, due to the training beam cannot be
shared by all UEs, this approach exhibits limited efficiency in multi-user
scenarios. Moreover, [6, 7] leverage compressive sensing (CS) theory to
recover channel state information (CSI), and the optimal beam can be
identified naturally. However, this method needs iterative operations for
sparse signal recovery, leading to high complexity. Additionally, a multi-
armed beam training strategy is adopted in [8, 9, 10], where each training
beam incorporates multiple main lobes. Through cross-validation among these
training beams, the optimal beam can be rapidly obtained. Our previous
research [10] has analyzed the impact of training beam design on performance,
but there is still room for optimization.

In this letter, we propose a novel scheme called patterned beam training
(PBT). Based on the probe pattern and combining pattern designed in the
scheme, both BS and UE need to perform only a single linear processing over
training pilots to detect the optimal beam. The proposed scheme offers the
advantages of low overhead and complexity, while effectively supporting multi-
user scenarios. To the best of our knowledge, it can serve as a promising
alternative approach.

Furthermore, we conduct a detailed analysis of the pattern design principles
and propose multiple specific forms of patterns, where exhaustive and multi-
armed beam training schemes can be regarded as special cases of PBT.
Simulation results indicate that the PBT can significantly reduce training
overhead and achieve a balance between overhead and signal-to-noise ratio
(SNR) conditions compared to the exhaustive method. Additionally, the proposed
scheme exhibits superior beam alignment accuracy compared to other approaches.

##  II System Model

Consider a multi-input single-output (MISO) system, where the BS is equipped
with a uniform linear array (ULA) with $N$ antenna elements. This letter
focuses on the beam training process between the BS and a single-antenna UE.
Given that the proposed scheme could be shared by all UEs in the cell, it can
be easily extended to multi-user scenarios. Assuming that the UE remains
stationary or moves at a low speed, the channel is considered fixed during the
beam training process. We adopt a geometry-based model to represent the
sparsity of high-frequency channels, which can be expressed as

${\mathbf{h}}={\alpha_{0}}{\mathbf{a}}\left({{\theta_{0}}}\right)+\sum\limits_{p=1}^{P}{{\alpha_{p}}{\mathbf{a}}\left({{\theta_{p}}}\right)},$
(1)

where
${\mathbf{a}}\left(\theta\right)={\left[{1,{e^{-j\pi\theta}},\ldots,{e^{-j\pi\left({N-1}\right)\theta}}}\right]^{\text{T}}}/\sqrt{N},\theta\in\left[{-1,1}\right]$
denotes the steering vector of the ULA, $\alpha_{0}[\alpha_{p}]$ and
$\theta_{0}[\theta_{p}]$ represent the gain and the angle of the line-of-sight
(LoS) [$p$th non-LoS (NLoS)] path, respectively. Here, $P$ is the total number
of NLoS paths.

In high-frequency channels, ${\alpha_{0}}\gg{\alpha_{p}}$, and the goal of
beam training is to select the optimal beam from candidate beams for data
transmission, maximizing the beam gain in $\theta_{0}$. We suppose that the
set of candidate beams consists of $N$ discrete Fourier transform (DFT) beams.
The beamforming weight of the $n$-th ($n=1,2,\dots,N$) beam is denoted as $\bf
d_{n}$, corresponding to the $n$-th column of the $N$-dimensional DFT matrix
$\bf D$, where
${d_{n,m}}={\exp({-j2\pi\left({n-1}\right)\left({m-1}\right)/N})}/\sqrt{N}$.

In traditional exhaustive beam training schemes, the BS sequentially uses the
aforementioned DFT beams to transmit pilot signals. The symbols received by
the UE under $N$ training beams can be expressed as

${\mathbf{y}}={{\mathbf{D}}^{\text{H}}}{\mathbf{h}}+{\mathbf{v}},$ (2)

where
${\mathbf{v}}\sim\mathcal{C}\mathcal{N}\left({{\mathbf{0}},\sigma_{\text{v}}^{2}{\mathbf{I}}}\right)$
denotes the noise vector, and $\sigma_{\text{v}}^{2}$ is the normalized noise
power.

To estimate the optimal beam index $n^{*}$, the UE compares the received
signal power ${\left|{{y_{n}}}\right|^{2}}$ of each training beam and feeds
the estimation result
$\hat{n}=\arg\mathop{\max}\limits_{n}{\left|{{y_{n}}}\right|^{2}}$ back to the
BS. Subsequently, the BS will use the corresponding beam
${{\mathbf{d}}_{\hat{n}}}$ to transmit data. According to the above
description, the overhead of exhaustive beam training is proportional to the
number of antennas. Therefore, this approach faces significant challenges in
future ELAA scenarios.

##  III Proposed Patterned Beam Training Scheme

To reduce the overhead of beam training, this section introduces the proposed
PBT scheme, which offers the advantages of low overhead and low complexity.
First, we describe the overall framework of PBT, followed by the principles
for pattern design within the scheme. Finally, several specific forms of
patterns are proposed.

###  III-A Framework of Patterned Beam Training

In PBT, the BS transmits $M<N$ training beams. We define the _probe pattern_
${\mathbf{B}}=\left[{{{\mathbf{b}}_{1}},{{\mathbf{b}}_{2}},\ldots,{{\mathbf{b}}_{M}}}\right]$,
which is composed of the weights $\mathbf{b}_{m}$ of the training beam. The
received signal at the UE side can be represented as

${\mathbf{r}}={{\mathbf{B}}^{\text{H}}}{\mathbf{h}}+{\mathbf{z}},$ (3)

where $\bf z$ is an $M$-dimensional noise vector, having the same distribution
as $\bf v$.

To select the optimal beam from $N$ candidate beams, a key operation in PBT is
the linear dimensionality expansion of $\bf r$. Specifically, the BS requires
the UE to process $\bf r$ using the corresponding _combining pattern_
${\mathbf{\tilde{B}}}\in{\mathbb{C}^{N\times M}}$ according to the selected
$\bf B$, which results in111This letter focuses on the processing of complex
signals $\bf r$, requiring coherence between different pilot symbols. For
certain pattern designs, processing the magnitude $|\bf r|$ can also identify
the optimal beam. Generally, coherent processing methods offer better
performance.

${\mathbf{\tilde{y}}}={\mathbf{\tilde{B}r}}={\mathbf{\tilde{B}}}{{\mathbf{B}}^{\text{H}}}{\mathbf{h}}+{\mathbf{\tilde{z}}},$
(4)

where ${\mathbf{\tilde{z}}}={\mathbf{\tilde{B}z}}$ is colored noise.
Similarly, by comparing the magnitude of each element in
${\mathbf{\tilde{y}}}$, the UE can provide feedback to the BS regarding the
estimated optimal beam.

By comparing (2) and (4), it can be implied that through sophisticated pattern
design to make
${\mathbf{\tilde{B}}}{{\mathbf{B}}^{\text{H}}}\approx{{\mathbf{D}}^{\text{H}}}$,
PBT can achieve functionality similar to the exhaustive beam training.
Unfortunately, due to $M<N$, we have
${\text{rank}}({{\mathbf{\tilde{B}}}{{\mathbf{B}}^{\text{H}}}})<N$ and
${\mathbf{\tilde{B}}}{{\mathbf{B}}^{\text{H}}}$ does not strictly equal $\bf
D^{\text{H}}$. Therefore, PBT cannot achieve the same performance as
exhaustive beam training. Nonetheless, pattern design has a significant impact
on the accuracy of beam alignment in the proposed scheme, which will be
detailed in the following subsection.

Remark 1 (_Comparison of PBT and existing schemes_): The primary innovation of
the proposed approach lies in obtaining the variable ${\mathbf{\tilde{y}}}$
for determining the optimal beamforming solely through a single linear
transformation ${\mathbf{\tilde{B}}}$. In contrast, for CS approaches like the
orthogonal matching pursuit (OMP) algorithm, iterative processing of $\bf r$
is necessary. Meanwhile, AI-based beam training schemes require the
computation of ${f_{{\text{NN}}}}\left({\mathbf{r}}\right)$, where
${f_{{\text{NN}}}}\left(\cdot\right)$ represents a neural network. Both
approaches involve nonlinear processing of the received signals. Supported by
simulations, we believe that if the goal is solely to identify the optimal
beam rather than to accurately estimate the channel vector $\bf h$, the
proposed scheme relying on linear transformations already exhibits
sufficiently good performance.

###  III-B Pattern Design Principles

This subsection analyzes the principles for designing probe and combining
patterns from the perspective of beam gain. Consider a normalized beamforming
weight $\bf w$, which induces the beam gain
${\left|{{{\mathbf{w}}^{\text{H}}}{\mathbf{a}}\left(\theta\right)}\right|^{2}}$
at $\theta$. In our analysis, we focus on the beam gain at discrete grid
points $\theta=-1+2i/N,i=0,1,\dots,N-1$, which can be calculated using the DFT
matrix, i.e.,
${\mathbf{g}}={\left|{{{\mathbf{w}}^{\text{H}}}{\mathbf{D}}}\right|^{2}}$. For
exhaustive beam training schemes, we have $\mathbf{g}=\mathbf{e}_{i}$, where
$\mathbf{e}_{i}$ is the selection vector. This implies that the power of DFT
training beams is concentrated in the main lobe direction.

In the case of PBT, according to (4), after the combining operation, it can be
considered to have $N$ equivalent training beams with weights
${{\mathbf{w}}_{n}}={\mathbf{B}}{{\mathbf{\tilde{b}}}_{n}}$, where
${{\mathbf{\tilde{b}}}_{n}}$ is the $n$-th column of
${{\mathbf{\tilde{B}}}^{\text{H}}}$. To ensure a fair comparison, we constrain
${\|{{{\mathbf{b}}_{m}}}\|_{2}}=1$ and
${\|{{{{\mathbf{\tilde{b}}}}_{n}}}\|_{2}}=1$, thereby limiting the
transmission power and the noise amplification factor. Analogous to the beam
gain of DFT training beams, we define the gain matrix
${\mathbf{G}}={\left|{{\mathbf{\tilde{B}}}{{\mathbf{B}}^{\text{H}}}{\mathbf{D}}}\right|^{2}}$
of PBT and propose the following principles for pattern design.

  * •

_P1: Equal Main Lobe Gain_

We aim for the main lobe gains of each equivalent training beam in PBT to be
equal, ensuring that UEs covered by different beams can select the optimal
beam with equal probability, which corresponds to

$\min\;{\left\|{{\text{diag}}\left({\mathbf{G}}\right)-g{\mathbf{1}}}\right\|_{2}},$
(5)

where $g={\left\|{{\text{diag}}\left({\mathbf{G}}\right)}\right\|_{1}}/N$.

  * •

_P2: Minimize Power Leakage of Sidelobe_

Ideal training beams have low sidelobes. Therefore, it is desired that all
sidelobe leakage power be minimized, and this can be represented by the sum of
off-diagonal elements of matrix $\bf G$, i.e.,

$\min\;\frac{1}{{{{\left({N-1}\right)}^{2}}}}\sum\limits_{n}{\sum\limits_{n^{\prime}\neq
n}{{{\left[{\mathbf{G}}\right]}_{n,n^{\prime}}}}}.$ (6)

  * •

_P3: Suppress Maximum Side Lobe_

The last principle focuses on maximum side lobe suppression, corresponding to

$\min\;\mathop{\max}\limits_{n,n^{\prime}\neq
n}\;{\left[{\mathbf{G}}\right]_{n,n^{\prime}}}.$ (7)

Figure 1: Relationship between gain matrix $\bf G$ and (equivalent) training
beams with $N=16$, $M=8$. (a). $\bf G$ for the ZC pattern; (b). $\bf G$ for
the proposed pattern that considers _P3_ ; (c). Comparison of gains for
different training beams.

Remark 2 (_The necessity of P3_): It’s worth noting that _P3_ is not redundant
compared to _P2_. Here’s an example to illustrate this. If we constrain
$\bf{b}_{m}$ to be a constant modulus sequence and require
${\mathbf{\tilde{B}}}=\sqrt{N/M}{{\mathbf{D}}^{\text{H}}}{\mathbf{B}}$,222The
coefficient is used to ensure ${\|{{{{\mathbf{\tilde{b}}}}_{n}}}\|_{2}}=1$. it
can be verified that using Zadoff–Chu (ZC) sequences as the probe pattern can
simultaneously optimize _P1_ and _P2_. The ZC sequences is given as
${b_{1,n}}={\exp({{{j\pi{{\left({n-1}\right)}^{2}}}}/{N}})}$, and
$\mathbf{b}_{m+1}$ is obtained by cyclically shifting $\mathbf{b}_{m}$.
Unfortunately, as shown in Fig. 1, the main lobe of the equivalent training
beams corresponding to the ZC pattern widens compared to the DFT beams,
leading to a significant decrease in beam alignment accuracy. In contrast,
considering _P3_ additionally ensures that the beam width of the proposed
pattern is closer to that of the DFT beams.

In addition, the above example also illustrates the challenge in finding the
optimal pattern that simultaneously minimizes the three proposed objective
functions. Therefore, in the next subsection, we will design patterns based on
a heuristic method and compare the performance of different patterns through
Monte Carlo simulations.

###  III-C Specific Pattern Designs

This subsection introduces the specific designs of five patterns. The first
two correspond existing schemes, used to demonstrate that PBT is a general
framework. The subsequent three are our proposed designs with novelty.

_Pattern 1: Exhaustive beam training scheme_

It is evident that the exhaustive method can be considered as a special case
of PBT, corresponding to ${\mathbf{B}}={\mathbf{D}}$ and
${\mathbf{\tilde{B}}}={\mathbf{I}}$.

_Pattern 2: Traditional multi-armed beam training scheme_

In the traditional multi-armed training scheme, each training beam has
multiple main lobes, with its weight being the superposition of $K$ DFT
vectors, i.e.,
${{\mathbf{b}}_{m}}=\sum\nolimits_{n\in{\mathcal{C}_{m}}}{{{\mathbf{d}}_{n}}}/\sqrt{K}$,
where ${\mathcal{C}_{m}}$ denotes the set of indices of $K$ selected DFT
vectors for the $m$-th training beam. To select the optimal DFT beam, the UE
performs a voting mechanism, with details available in [10].

From the perspective of PBT scheme, the probe pattern is derived as
${\mathbf{B}}={\mathbf{DC}}$, where $\bf C$ is the hash codebook described in
[10], which can be expressed as

${c_{n,m}}=\left\\{\begin{gathered}1/\sqrt{K},\;n\in{\mathcal{C}_{m}},\hfill\\\
0,\;{\text{otherwise}}.\hfill\\\ \end{gathered}\right.$ (8)

Moreover, the corresponding combining pattern for the voting mechanism is
given by ${\mathbf{\tilde{B}}}=\sqrt{N/M}{\mathbf{C}}$.

_Pattern 3: Proposed multi-armed beam training scheme with improvement_

According to the pattern design principle _P3_ , an improved multi-armed
training scheme can be easily identified by adding a random phase shift factor
before $c_{n,m}$, i.e., ${\tilde{c}_{n,m}}={e^{j{\phi_{n,m}}}}{c_{n,m}}$,
where $\phi_{n,m}$ follows a uniform distribution over $[0,2\pi]$. The probe
pattern and combining pattern remain ${\mathbf{B}}={\mathbf{D\tilde{C}}}$ and
${\mathbf{\tilde{B}}}=\sqrt{N/M}{\mathbf{\tilde{C}}}$. In the improved scheme,
we have
${\mathbf{G}}=\sqrt{N/M}{\mathbf{\tilde{C}}}{{\mathbf{\tilde{C}}}^{\text{H}}}$,
with its off-diagonal elements

$\begin{split}\left|{{{\left[{{\mathbf{\tilde{C}}}{{{\mathbf{\tilde{C}}}}^{\text{H}}}}\right]}_{p,q}}}\right|=&\left|{\sum\limits_{m=1}^{M}{{e^{j\left({{\phi_{p,m}}-{\phi_{q,m}}}\right)}}{c_{p,m}}{c_{q,m}}}}\right|\\\
\leqslant&\left|{\sum\limits_{m=1}^{M}{{c_{p,m}}{c_{q,m}}}}\right|=\left|{{{\left[{{\mathbf{C}}{{\mathbf{C}}^{\text{H}}}}\right]}_{p,q}}}\right|,p\neq
q,\end{split}$ (9)

which proves that adding a random phase shift can effectively suppress side
lobes.

_Pattern 4: Proposed random pattern for PBT_

We typically desire training beam weights to have a constant modulus to ensure
efficient power utilization. However, _Pattern 2_ and _Pattern 3_ do not
satisfy this requirement. A simple probe pattern with constant modulus can be
realized using a random phase distribution, i.e.,
${b_{n,m}}={e^{j{\varphi_{n,m}}}}/\sqrt{N}$, where $\varphi_{n,m}$ follows a
uniform distribution over $[0,2\pi]$. Furthermore, by computing
${{\mathbf{D}}^{\text{H}}}{\mathbf{B}}$ and normalizing each row, we can
obtain the combining pattern ${\mathbf{\tilde{B}}}$. The qualitative
explanation for the effectiveness of the random pattern is that we can
approximate ${\mathbf{B}}{{\mathbf{B}}^{\text{H}}}\approx{\mathbf{I}}$,
leading to
${\mathbf{\tilde{B}}}{{\mathbf{B}}^{\text{H}}}\approx{{\mathbf{D}}^{\text{H}}}$,
thereby achieving functionality similar to the exhaustive method.

_Pattern 5: Proposed pattern based on P1/P2/P3 for PBT_

_Pattern 4_ has the advantage of being easy to implement, but there is still a
certain gap between the gain of the equivalent training beam and the DFT beam.
Based on the proposed design principle, an optimization problem is formulated
to find a local optimal pattern design. We still require the training beam to
have constant modulus, i.e., ${b_{n,m}}={e^{j{\varphi_{n,m}}}}$. Furthermore,
to reduce the search space of optimization variables, let
${\mathbf{\tilde{B}}}$ be the normalized
${{\mathbf{D}}^{\text{H}}}{\mathbf{B}}$, then the optimization problem for
pattern design is given as

$\begin{split}\mathop{\min}\limits_{\left\\{{{\varphi_{n,m}}}\right\\}}&\,{\lambda_{1}}f_{1}+\lambda_{2}f_{2}+{\lambda_{3}}f_{3}\\\
{\text{s}}{\text{.t}}{\text{.}}&\;{\left\|{{{\mathbf{b}}_{m}}}\right\|_{2}}=1,{\left\|{{{{\mathbf{\tilde{b}}}}_{n}}}\right\|_{2}}=1,\end{split}$
(10)

where $f_{i}$ and $\lambda_{i}$ represents the objective function and weight
of each principle, respectively. To solve this problem, automatic
differentiation tools like TensorFlow can be utilized [11]. Through gradient
descent, a local optimal solution $\left\\{{\varphi_{n,m}^{*}}\right\\}$ can
be obtained. In addition, to expedite convergence, _Pattern 4_ can be used as
the initial value.

##  IV Simulation Results

This section verifies the PBT scheme and compares it with other existing
solutions. The simulation assumes that the BS is equipped with $N=256$ antenna
elements, corresponding to 256 candidate DFT beams. For the geometry-based
channel model, there are $P=3$ NLoS paths, where the LoS path gain
${\alpha_{0}}=1$, the NLoS path gain
${\alpha_{p}}\sim\mathcal{C}\mathcal{N}\left({0,0.1}\right)$, and the path
angles ${\theta_{p}}$ obey a uniform distribution in $\left[{-1,1}\right]$.
For _Pattern 5_ , the optimization weight is set to
${\lambda_{1}}={\lambda_{2}}={\lambda_{3}}=1$. In the scheme based on CS, the
OMP algorithm is used to achieve sparse signal recovery, and the number of
pilot symbols is kept the same as PBT.

TABLE I: Comparison of Different Pattern Designs | Pattern
---
Design
Overhead $M$ | _P1_ | _P2_ | _P3_ |  | Constant
---
Modulus
1 | 256 | 0 | 0 | 0 | Yes
2 | 128 | 0 | 0.0081 | 0.297 | No
64 | 0 | 0.0073 | 0.250
3 | 128 | 0 | 0.0069 | 0.198 | No
64 | 0 | 0.0069 | 0.248
4 | 128 | 0.703 | 0.0078 | 0.078 | Yes
64 | 0.497 | 0.0078 | 0.079
5 | 128 | $2.30\times 10^{-5}$ | 0.004 | 0.039 | Yes
64 | $1.16\times 10^{-5}$ | 0.006 | 0.051

Table I compares the characteristics of different pattern designs based on the
proposed principles, with _Pattern 1_ serving as the ideal case. We focus our
analysis on the other pattern designs. Compared to _Pattern 2_ , _Pattern 3_
successfully reduces energy leakage and suppresses the maximum side lobe.
However, both pattern designs fail to ensure the constant modulus property of
the training beams. _Pattern 5_ , which comprehensively considers all design
principles, does not guarantee strictly equal main lobe gain but offers the
best performance in terms of side lobe power leakage and maximum side lobe
suppression, while also maintaining the constant modulus advantage.

Figure 2: Beam alignment performance of different pattern designs.

Fig. 2 compares the beam alignment error probability
$\mathbb{P}\left\\{{\hat{n}\neq{n^{*}}}\right\\}$ under different pattern
designs. The simulation results show a similar trend to the data in Table I.
_Pattern 3_ exhibits better beam alignment performance compared to _Pattern 2_
, highlighting the significant impact of sidelobe suppression on performance.
Moreover, _Pattern 5_ outperforms the existing CS scheme and other pattern
designs in terms of beam alignment accuracy. Using only half the number of
pilots compared to the exhaustive method, it shows about a 3 dB performance
gap. This is because, under _Pattern 5_ , the power of the training signals is
halved, which is also reflected in the beam gain shown in Fig. 1.
Nevertheless, the proposed PBT scheme can achieve a balance between overhead
and SNR conditions.

##  V Conclusion

To address the issue of excessive beam training overhead in ELAA, this paper
proposes a novel PBT scheme. With only a single linear processing of the
pilots at both the BS and the UE, the proposed scheme can achieve the
functionality of traditional exhaustive methods while requiring only half, or
even less, of the overhead. Additionally, we provide a detailed discussion on
the design principles of probing patterns and combining patterns, along with
specific examples. Simulation results demonstrate that the PBT scheme can
achieve beam alignment with high accuracy, making it a promising alternative.

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