text
stringlengths 1
1.3M
| id
stringlengths 2
2.39k
| metadata
dict |
|---|---|---|
\section{Introduction}
Evolution algebras were introduced by Tian in his book \cite{T1} (see also \cite{TV}) to model the self reproduction process in non-Mendelian genetics. As shown in \cite{T1}, the theory of evolution algebras is connected to many areas of Mathematics, like, for example, graph theory, group theory, stochastic processes, mathematical physics, among many others.
Since their introduction evolution algebras have attracted the attention to several researchers, who were eager to investigate them from an algebraic point of view; see \cite{BCS}-\cite{Imo} and references therein.
Here, we focus our attention on the evolution algebras whose square has dimension one; we classify them using the theory of inner product spaces and quadratic forms. We begin by introducing a few key ideas: any commutative algebra $A$ (not necessarily finite dimensional) over a field $K$ with $\dim(A^2)=1$ admits an inner product $\esc{\cdot,\cdot}$ such that the product in $A$ is given by $xy=\esc{x,y}a$, for some fixed element $a\in A$ (unique up to scalar multiples). From here, we obtain three (excluding) possibilities for $a$:
\begin{enumerate}
\item\label{one}
$a\in\mathrm{Ann}(A)$, which gives $A^3 = (A^2)^2 = 0$.
\item $a\notin \mathrm{Ann}(A)$ and $\esc{a,a} = 0$, which implies $(A^2)^2 = 0$ but $A^3\ne 0$.
\item $a\notin\mathrm{Ann}(A)$ and $\esc{a,a}\ne 0$, which yields $A^3 \ne 0$ and $(A^2)^2\ne 0$.
\end{enumerate}
Here $\mathrm{Ann}(A) = \{x \in A \, |\, xA = 0\}$. Thus, the algebras we will be dealing with come in three different flavours given by the trichotomy above. In any case, choosing a suplementary subspace $W$ of $\mathrm{Ann}(A)$ we obtain a decomposition $A = \mathrm{Ann}(A)\oplus W$ such that $\esc{\cdot, \cdot}\vert_W$ is nondegenerate. It is worth mentioning that we have quite a lot of flexibility to choose $W$, so depending on the flavour of our algebra we will require $W$ to satisfy certain conditions.
If $A$ is an evolution algebra of flavour \eqref{one}, we will show (in Theorem \ref{Iso1}) that the isomorphic class of $A$ is completely determined by the pair
$(\dim(A),[W])$, where $[W]$ denotes the isometry type of $W$. More precisely, if $A = \mathrm{Ann}(A)\oplus W_A$ and $B = \mathrm{Ann}(B) \oplus W_B$ are like in \eqref{one}, then we will prove that $A \cong B$ if and only if $\dim(A) = \dim(B)$, and $W_A$ and $W_B$ are isometric.
Similar results for flavours (2) and (3) will be explored. To do so, we will make used of the theory of inner products and/or quadratic forms.
The paper is organized as follows: in Section 2, we introduce the required background. Section 3 begins by noticing that our study of the evolution algebras $A$ with $\dim (A^2) = 1$ must be divided into two cases, attending on whether $(A^2)^2 \neq 0$ or $(A^2)^2 = 0$, which are treated in \S 3.1 and \S 3.2, respectively.
\section{Preliminaries}
Throughout the paper, $V$ denotes a vector space over a field $K$. An {\bf inner product space} is a pair $(V, \esc{\cdot, \cdot})$, where $\esc{\cdot, \cdot}: V \times V \to K$ is a symmetric bilinear form. Two inner product spaces $(V,\esc{\cdot, \cdot})$ and $(V',\esc{\cdot, \cdot}')$ are said to be {\bf isometric} or {\bf equivalent} if there is a (vector space) isomorphism $f: V \to V'$ such that $\esc{f(x),f(y)}'= \esc{x,y}$ for all $x, y \in V$.
\smallskip
A map $q: V\to K$ satisfying that
\begin{enumerate}
\item[\rm (1)] $q(\lambda v) = \lambda^2 q(v), \quad \forall \, \, \lambda \in K, \, \, v \in V$,
\item[\rm (2)] the map $\langle \cdot, \cdot \rangle_q: V \times V \to K$ given by $(x, y) \mapsto q(x + y) - q(x) - q(y)$, and called the {\bf polar form} of $q$, is bilinear,
\end{enumerate}
is called a {\bf quadratic form} on $V$, and $(V, q)$ is said to be a {\bf quadratic space}.
If the characteristic of $K$ is different from 2, the notions of inner product spaces and quadratic spaces are equivalent: the polar form of a quadratic form $q$ is now defined by
$\langle x, y \rangle_q = \frac{1}{2}\left(q(x + y) - q(x) - q(y)\right)$, and satisfies that $\langle x, x\rangle_q = q(x)$. And conversely, if $(V,\esc{\cdot, \cdot})$ is an inner product space, then $q(x):= \esc{x, x}$ for all $x \in V$, is a quadratic form, whose polar form is precisely $\esc{\cdot, \cdot}$. In such a case, we may write $(V, q)$ to refer to the quadratic space $(V, \langle \cdot, \cdot \rangle)$, and vice versa.
\smallskip
The {\bf radical} of an inner product space $(V, \langle \cdot, \cdot \rangle$) is the subspace of $V$ given by
\[
V^\bot = \{x \in V \, |\, \esc{x,V} = 0\},
\]
and $(V, \esc{\cdot, \cdot})$ is said to be {\bf nondegenerate} if $V^\bot = 0$; a {\bf subspace} $W$ of $V$ is called {\bf nondegenerate} if $(W, \esc{\cdot, \cdot}|_W)$ is nondegenerate.
Recall that $V$ can be written as
\begin{equation} \label{decomV}
V = V^\bot \oplus W,
\end{equation}
for $W$ a nondegenerate subspace of $V$.
If $V$ has finite dimension, then the matrix $M_B$ of $\langle \cdot, \cdot \rangle$ with respect to a basis $B$ of $V$ is called the {\bf Gram matrix} of $\esc{\cdot, \cdot}$ with respect to $B$. If $M_B$ and $M_{B'}$ are Gram matrices with respect to bases $B$ and $B'$ of $V$, then $M_B$ and $M_{B'}$ have the same rank; the {\bf rank} of $(V, \esc{\cdot, \cdot})$ is defined as the rank of a Gram matrix $M$ of $\esc{\cdot, \cdot}$, and $(V, \esc{\cdot, \cdot})$ is nondegenerate if and only if $M$ is nonsingular. The {\bf discriminant} of $(V, \esc{\cdot, \cdot})$ is defined as zero if $(V, \esc{\cdot, \cdot})$ is degenerate, and as the coset of $\det(M)$ in the factor group $K^*/{(K^*)}^2$, otherwise. The discriminant of two nondegenerate equivalent inner product spaces coincide.
If the characteristic of $K$ is not 2, then we can consider the associated quadratic form $q$ of $(V, \esc{\cdot, \cdot})$ and define its rank. Similarly, one can define the discriminant of $q$ provided that $(V, \esc{\cdot, \cdot})$ is nondegenerate.
If $V$ has finite dimension $n$, then a real quadratic form $q: V \to \mathbb R$ of rank $r$ can be expressed as
\[
q(x_1, \ldots, x_n) = x_1^2 + \ldots + x_p^2- x_{p+1}^2 - \ldots - x_r^2,
\]
with respect to a suitable basis of $V$; the {\bf signature} of $q$ is defined as $(p, \, r - p)$.
Recall that (finite dimensional) inner product spaces over algebraically closed fields are classified (up to congruence) by rank; over the reals they are classified according to their rank and signature (see \cite[Theorem 6.8]{BAI}); while over finite fields of odd characteristic
their rank and discriminant constitute a complete set of invariants (see \cite[Theorem 6.9]{BAI}). Over other types of fields many different invariants are available; for instance, over a local field those are the dimension, discriminant and the so-called Hasse invariant (see \cite[p. 39]{Serre}).
Lastly, diagonalizable inner product spaces over quadratically closed fields are classified by their rank.
\section{The first dichotomy}
An {\bf algebra} $A$ over $K$ is a vector space over $K$ endowed with a bilinear map $A \times A \to A$ written as $(a, b) \mapsto ab$, and called the {\bf product} of $A$. We say that $A$ is an {\bf evolution algebra} if there exists a basis $\{a_i\}_{i \in I}$, called a {\bf natural basis} of the underlying vector space of $A$ such that $a_ia_j = 0$ for all $i \neq j$.
\begin{remark} \label{product}
Let $A$ be an algebra such that $\dim (A^2) = 1$. Then we can find $0 \neq a \in A$ such that $A^2 = Ka$, and the product of any two elements $x, y \in A$ is given by $xy = \lambda_{xy} a$, where $\lambda_{xy} \in K$ depends linearly on both $x$ and $y$. In other words, the map $\langle \cdot, \cdot \rangle: A \times A \to K$ given by $\langle x, y \rangle = \lambda_{xy}$, for all $x, y \in A$, is an inner product in $A$. Clearly,
\begin{equation} \label{productA}
xy = \langle x, y \rangle a, \mbox{ for all } \, \, x, y \in A.
\end{equation}
In this case, notice that \eqref{decomV} becomes
\begin{equation} \label{decomA}
A = \mathrm{Ann}(A) \oplus W;
\end{equation}
in other words $\mathrm{Ann}(A) = A^\bot$.
\end{remark}
At this point, it is worth mentioning that if $A$ is an evolution algebra, then \eqref{productA} does not depend of the generator $a$ of $A^2$. We prove this fact in a more general way using pointed vector spaces.
\begin{lemma} \label{pointed}
Let $(S, s)$ be a pointed vector space, where $0 \neq s \in S$, and $(U, \langle \cdot, \cdot \rangle)$ a nondegenerate inner product space. Then the direct sum $A_s := S \oplus U$ becomes an algebra with 1-dimensional square under the product
\[
(s_1 + u_1) (s_2 + u_2) = \langle u_1, u_2 \rangle s,
\]
for all $s_1, s_2 \in S$ and $u_1, u_2 \in U$. Moreover, if $s'$ is another nonzero element of $S$,
then $A_s$ and $A_{s'}$ are isomorphic.
\end{lemma}
\begin{proof}
It is straightforward to check that $A_s$ is an algebra such that $\dim(A^2_s) = 1$. For $(S, s')$ another pointed vector space, take $\theta: S \to S$ a bijective linear map such that $\theta(s) = s'$. The map
$f:A_s \to A_{s'}$ given by
$f(s + u) = \theta(s) + u$, for all $s \in S$ and $u \in U$, is the desired isomorphism.
\end{proof}
\begin{proposition} \label{dicho}
Let $A$ be a commutative algebra with $\dim(A^2) = 1$. Then either $(A^2)^2 = 0$ or there is a unique nonzero idempotent in $A$.
\end{proposition}
\begin{proof}
From $\dim(A^2) = 1$ we can find $0 \neq a \in A$ such that $A^2 = Ka$. If $(A^2)^2 \neq 0$, then $0 \neq a^2 = \lambda a$, for some $0 \neq \lambda \in K$. We claim that $e := \lambda^{-1}a$ is the unique nonzero idempotent of $A$; in fact:
\[
e^2 = \lambda^{-2}a^2 = \lambda^{-2}(\lambda a) = \lambda^{-1}a = e.
\]
Clearly, $A^2 = Ke$. If $0 \neq u \in A$ is an idempotent, then $u = u^2 \in A^2 = Ke$, and so $u = \gamma e$ for some $0 \neq \gamma \in K$. But then
\[
\gamma e = u = u^2 = \gamma^2 e,
\]
which implies that $\gamma = 1$, and so $u = e$.
\end{proof}
\subsection{Case $(A^2)^2 \neq 0$} We first study the evolution algebras $A$ whose square is 1-dimensional and satisfy the condition $(A^2)^2 \neq 0$.
\smallskip
The following result follows from Remark \ref{product} and Proposition \ref{dicho}.
\begin{proposition} \label{inner}
Let $A$ be a commutative algebra such that $\dim(A^2) = 1$ and $(A^2)^2 \neq 0$. Then there exists a unique inner product $\langle \cdot, \cdot \rangle: A \times A \to K$ such that the product of $A$ is given by
\[
xy = \langle x, y \rangle e, \, \, \forall \, \, x, y \in A,
\]
where $e$ is the nonzero unique idempotent of $A$. Moreover, $\langle e, e \rangle = 1$.
\end{proposition}
\begin{definition}
The inner product defined in Proposition \ref{inner} is called the {\bf canonical inner product} of $A$, and $(A, \langle \cdot, \cdot \rangle)$ the {\bf canonical inner product space}.
\end{definition}
\begin{proposition}
Let $A$ be an evolution algebra such that $\dim(A^2) = 1$ and $(A^2)^2 \neq 0$. Then the canonical inner product of $A$ is diagonalizable, that is, there exists an orthogonal basis of $A$.
\end{proposition}
\begin{proof}
Suppose that $\{a_i\}_{i\in I}$ is a natural basis of $A$. Then, for all $i \neq j$ we have
\[
0 = a_ia_j = \langle a_i, a_j \rangle e,
\]
which implies $\langle a_i, a_j \rangle = 0$.
\end{proof}
\begin{conclu} \label{conclu1}
{\rm
We have proved that the product of an evolution algebra $A$ satisfying $(A^2)^2\ne 0$ and $\dim(A^2) = 1$,
is completely determined by a (unique) inner product $\langle \cdot, \cdot \rangle: A \times A \to K$ (diagonalizable with respect to a natural basis of $A$) and a (unique) nonzero idempotent $e$ of $A$ such that $\langle e, e \rangle = 1$. To be more precise, $xy = \langle x, y \rangle e$, for all $x, y \in A$. A kind of converse holds: if $(V, \langle \cdot, \cdot \rangle)$ is an inner product space with an orthogonal basis and a vector $v$ of norm one, then we can endow $V$ with an evolution algebra structure with the product
$xy = \langle x, y \rangle v$, for all $x, y \in V$. Moreover, $\dim(V^2) = 1$.}
\end{conclu}
We can reformulate our problem using categories:
\begin{itemize}
\item ${\mathcal A}_K$ denotes the category whose objects are the evolution $K$-algebras $A$ satisfying that $(A^2)^2 \ne 0$ and $\dim(A^2) = 1$, and morphisms the algebra homomorphisms; notice that ${\mathcal A}_K$ is a full subcategory of the category of all $K$-algebras;
\item ${\mathcal B}_K$ denotes the category whose objects are triples of the form $\big(V, \langle \cdot, \cdot \rangle, v \big)$, where $(V, \langle \cdot, \cdot \rangle)$ is a (diagonalizable) inner product space, and $v \in V$ a norm one vector; a morphism $f: \big(V, \langle \cdot, \cdot \rangle, v) \to \big(V', \langle \cdot, \cdot \rangle', v'\big)$ in ${\mathcal B}_K$ is a linear map $f: V \to V'$ such that $\langle x, y \rangle f(v) = \langle f(x), f(y)\rangle'v'$, for all $x, y\in V$;
\item ${\mathcal A}_K^0$ stands for the full subcategory of ${\mathcal A}_K$ consisting of
finite dimensional evolution algebras, while ${\mathcal B}_K^0$ denotes the full subcategory of ${\mathcal B}_K$ of triples
$\big(V, \langle \cdot, \cdot \rangle, v \big)$, where $V$ is finite dimensional.
\end{itemize}
We begin by characterizing the isomorphisms in ${\mathcal B}_K$:
\begin{theorem} \label{caractIso}
A morphism $f: \big(V, \langle \cdot, \cdot \rangle, v \big) \to \big(V', \langle \cdot, \cdot \rangle', v'\big)$ in ${\mathcal B}_K$ is an isomorphism if and only if $f$ is an isometry and $f(v) = v'$.
\end{theorem}
\begin{proof}
Suppose that $f$ is an isomorphism. Then for $\lambda := \langle f(v), f(v) \rangle' \in K$, we have that
\[
f(v) = \langle f(v), f(v)\rangle' v' = \lambda v',
\]
since $f$ is a morphism in ${\mathcal B}_K$ and $\langle v, v \rangle = 1$. From here we obtain that
\[
\lambda v' = \langle \lambda v', \lambda v' \rangle' v' = \lambda^2 v',
\]
which implies that $\lambda = 1$. Thus $f(v) = v'$, and $f$ is an isometry. The converse clearly holds.
\end{proof}
\begin{theorem}
The categories ${\mathcal A}_K$ and ${\mathcal B}_K$ are isomorphic.
\end{theorem}
\begin{proof}
The functors $F: {\mathcal A}_K \to {\mathcal B}_K$ and $G: {\mathcal B}_K \to {\mathcal A}_K$ mapping an evolution algebra $A$ to the triple $\big(A, \langle \cdot, \cdot \rangle, e\big)$ (where $e$ is the unique nonzero idempotent of $A$, and $\langle \cdot, \cdot \rangle$ is its canonical inner product), and a triple $\big(V, \langle \cdot, \cdot \rangle, v\big)$ to the evolution algebra $V$ whose product is $xy := \langle x, y \rangle v$, for all $x, y \in V$, respectively, are well-defined by Proposition \ref{inner} and Conclusion \ref{conclu1}. It is straightforward to check that
$FG = 1_{{\mathcal B}_K}$ and $GF = 1_{{\mathcal A}_K}$.
\end{proof}
A very important consequence of this theorem is the following:
\begin{corollary}
The problem of classifying (up to isomorphism) the evolution algebras in ${\mathcal A}_K$ is equivalent to classifying (up to isomorphism) the triples $\big(V, \langle \cdot, \cdot \rangle, v\big)$ in ${\mathcal B}_K$.
\end{corollary}
Keeping in mind that two objects in ${\mathcal B}_K$ are isomorphic in the presence of an isometry, what we are indeed doing here is transitioning from an algebraic classification problem to a geometric classification problem.
\begin{remark} \label{new}
If $K$ is quadratically closed and $(V, \esc{\cdot, \cdot})$ is finite dimensional, diagonalizable and nondegenerate, then there exists a basis $B$ of $V$ such that the Gram matrix $M_B$ is the identity. As a consequence, any two diagonalizable, nondegenerate inner product spaces of the same dimension are isometric. Moreover, if $(V_1,\esc{\cdot, \cdot}_1)$ and $(V_2,\esc{\cdot, \cdot}_2)$ are such that $\dim(V_1) = \dim(V_2)$ and $\dim(V^\bot_1) = \dim(V^\bot_2)$, then $(V_1,\esc{\cdot, \cdot}_1)$ and $(V_2,\esc{\cdot, \cdot}_2)$ are isometric.
\end{remark}
Using Witt's (Isometry) Extension Theorem, we can provide a way to construct isomorphic objects in ${\mathcal B}_K^0$. Before doing so, we remind the reader a trivial property of vector spaces very useful for our purposes.
\begin{remark} \label{complemento}
Let $V$ be a vector space and $U$ a subspace of $V$. If $v \in V$ is such that $v \notin U$, then there exists a subspace $U'$ of $V$ such that $v \in U'$ and $V = U \oplus U'$.
\end{remark}
\begin{lemma}\label{admunsen}
Let $K$ be a field of characteristic different from two or perfect of characteristic two and $(V, q, v) \in {\mathcal B}_K^0$. Suppose that $V = V^\bot \oplus V'$ for $V'$ a subspace of $V$ containing $v$. If $w \in V'$ satisfies that $q(w) = 1$, then $(V, q, v)$ and $(V, q, w)$ are isomorphic in ${\mathcal B}_K^0$.
\end{lemma}
\begin{proof}
Suppose first that the characteristic of $K$ is not two.
Then the linear map from $Kv'$ onto $Kw'$ mapping $v'$ onto $w'$ is an isometry, which can be extended to an isometry $\theta'$ of $V'$, by Witt's (Isometry) Extension Theorem. It is straightforward to check that the map $\theta: V \to V$ given by $\theta(t + x) = t + \theta'(x)$, for all $t \in V^\bot$ and $x \in V'$, is an isometry of $V$ mapping $v$ to $w$; the result follows from Theorem \ref{caractIso}.
Assume now that $K$ is perfect of characteristic two. In this case, $V$ can be written as orthogonal direct sums $V = Kv \oplus V' = Kw \oplus W'$ such that $V^\bot = V'^\bot = W'^\bot$.
Thus, $V' = V^\bot \oplus V''$ and $W'= V^\bot \oplus W''$, which imply that $V = Kv \oplus V^\bot \oplus V'' = Kw \oplus V^\bot \oplus W''$. Thus $(V'', \esc{\cdot, \cdot}|_{V''})$ and $(W'', \esc{\cdot, \cdot}|_{W''})$ are nondegenerate and have the same dimension, and so they are isometric by Remark \ref{new}. If $\theta'': V'' \to W''$ is an isometry, then we can easily construct an isometry $\theta: V \to W$ such that $\theta|_{V''} = \theta''$, $\theta(v) = w$.
\end{proof}
An immediate consequence of Lemma \ref{admunsen} in terms of isomorphisms of evolution algebras follows:
\begin{theorem} \label{uncaso}
Let $K$ be a field of characteristic different from two or perfect of characteristic two.
Two evolution algebras $A$ and $B$ in ${\mathcal A}_K$ are isomorphic if and only if their canonical inner product spaces are isometric. Moreover, if $K$ is algebraically closed, or of characteristic two and perfect, then $A$ and $B$ are isomorphic if and only if the rank of their canonical inner products coincide; if $K = \mathbb R$, then $A$ is isomorphic to $B$ if and only if the rank and signatures of their canonical inner products coincide.
\end{theorem}
We close this first case with some concrete examples:
\begin{example}
In dimension 4, Theorem \ref{uncaso} tells us that there are four different isomorphic classes in $\mathcal{A}_\mathbb C$, which correspond
to the triples $(\mathbb C^4, q_i, e_1)$, where $e_1 = (1, 0, 0, 0)$, and $q_1, q_2, q_3, q_4$ are as follows:
\begin{align*}
q_1(x_1, x_2, x_3, x_4) & = x_1^2,
\\
q_2(x_1, x_2, x_3, x_4) & = x_1^2 + x_2^2,
\\
q_3(x_1, x_2, x_3, x_4) & = x_1^2 + x_2^2 + x_3^2,
\\
q_4(x_1, x_2, x_3, x_4) & = x_1^2 + x_2^2 + x_3^2 + x_4^2,
\end{align*}
with respect to the canonical basis of $\mathbb C^4$.
\end{example}
\begin{example}
In dimension 3, Theorem \ref{uncaso} reveals that we have six different isomorphic classes in
$\mathcal{A}_\mathbb R$ corresponding to the triples $(\mathbb R^3, q_i, e_1)$, where $e_1 = (1, 0, 0)$ and $q_1, \ldots, q_6$ are displayed in the table below, where the coordinates are with respect to the canonical basis of $\mathbb R^3$.
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
$q_i$ & rank & signature\cr
\hline
$x_1^2$ & $1$ & $(1,0)$ \cr
$x_1^2 + x_2^2$ & $2$ & $(2,0)$ \cr
$x_1^2 - x_2^2$ & $2$ & $(1,1)$ \cr
$x_1^2 + x_2^2 + x_3^2$ & $3$ & $(3,0)$ \cr
$x_1^2 + x_2^2 - x_3^2$ & $3$ & $(2,1)$ \cr
$x_1^2 - x_2^2 - x_3^2$ & $3$ & $(1,2)$ \cr
\hline
\end{tabular}
\end{center}
\end{example}
\begin{example}
Let $\mathbf{F}_4$ be the field of four elements, which is perfect. In dimension 3, Theorem \ref{uncaso} tells us that there are three distinct classes in $\mathcal{A}_{\mathbf{F}_4}$, which correspond to the inner products (on the $\mathbf{F}_4$-vector space $\mathbf{F}^3_4$) whose matrices are $\mathrm{Id}$, and the diagonal matrices
$\mathrm{diag}(1, 1, 0)$ and $\mathrm{diag}(1, 0, 0)$.
\end{example}
\begin{remark}
Recall that quadratic forms on a finite dimensional vector space over a finite field of odd characteristic are classified (up to congruence) by their rank and discriminant. Thus $(V_1,q_1,v_1) \cong (V_2, q_2, v_2)$ in ${\mathcal B}_K^0$ if and only if
$\dim(V_1) = \dim(V_2)$, $\dim(V_1^\bot) = \dim(V_2^\bot)$ and the discriminant of ${q_1}\vert_{V'_1}$ coincides with that of ${q_2}\vert_{V'_2}$, where $v_i\in V'_i$ and
$V'_i$ is the complement of $V_i^\bot$, for $i = 1, 2$.
\end{remark}
\begin{example}
Let $K = \mathbf{F}_3(i) = \{0, \pm 1, \pm i, \pm(1 + i), \pm(1 - i)\}$, where $i^2 = -1$, be the field of nine elements. Notice that $K$ can be seen as an extension of the field of three elements $\mathbf{F}_3$ by adjoining an element of square $-1$.
We have that $(K^\times)^2 = \{ \pm 1, \pm i\}$ is the cyclic group of order $4$, and the quotient group
$K^\times/(K^\times)^2$ is the cyclic group of order $2$. We can then write $K^\times/(K^\times)^2 = \{[1],[\omega]\}$, where $\omega = 1 + i$ and $[\cdot]$ denotes the corresponding equivalence class.
From here we obtain that the discriminant of a quadratic form over $K$ is either $[1]$ or $[\omega]$.
In particular, for $V = K^n$ and $q: V \to K$ nondegenerate, we have two possibilities: either $q$ is congruent to $x_1^2 +x_2^2+ \cdots + x_n^2$ or to $\omega x_1^2 + x_2^2 + \cdots + x_n^2$. A natural question arises: how many isomorphic classes of $3$-dimensional evolution $K$-algebras $A$ with $\dim(A^2)=1$ and $(A^2)^2\ne 0$ are there?
\newline
We obtain the three following types:
\begin{enumerate}
\item $\hbox{Ann}(A)=0$, then $A\cong K^3$ with product
\begin{align*}
(x,y,z)(x',y',z') & = (xx'+yy'+zz')(1,0,0), \mbox{ or }
\\
(x,y,z)(x',y',z') & = (\omega xx'+yy'+zz')(1,0,0).
\end{align*}
\item $\dim(\hbox{Ann}(A))=1$, then $A\cong K\times K^2$ with product
\begin{align*}
(x,y,z)(x',y',z') &= (yy'+zz')(0,1,0), \mbox{ or }
\\
(x,y,z)(x',y',z') &= (\omega yy'+zz')(0,1,0).
\end{align*}
\item $\dim(\hbox{Ann}(A))=2$, then $A\cong K^2\times K$ with product
\begin{align*}
(x,y,z)(x',y',z') & =zz'(0,0,1), \mbox{ or }
\\
(x,y,z)(x',y',z') &= \omega zz' (0,0,1).
\end{align*}
\end{enumerate}
\end{example}
\subsection{Case $(A^2)^2 = 0$} We treat now the remaining case: evolution algebras $A$ such that $\dim (A^2) = 1$ and $(A^2)^2 = 0$.
\smallskip
\begin{lemma} \label{lemmaanndecomp}
Let $A$ be an evolution $K$-algebra such that $\dim(A^2) = 1$ and $(A^2)^2 = 0$. Then
$A = \mathrm{Ann}(A) \oplus W$, where
$(W, \esc{\cdot, \cdot}\vert_W)$ is nondegenerate and has an orthogonal basis $\{w_i\}_{i \in I}$ of nonisotropic vectors.
Moreover, if $K$ is quadratically closed, then $\{w_i\}_{i \in I}$ is an orthonormal basis.
\end{lemma}
\begin{proof}
By \eqref{decomA} we have $A = \mathrm{Ann}(A) \oplus W$, for $W$ a subspace of $A$ such that $(W, \esc{\cdot, \cdot}\vert_W)$ is nondegenerate. Take $\{e_i\}_{i \in J}$ a natural basis of $A$, and express each $e_i$ as $e_i = r_i + w_i$, for $r_i \in \mathrm{Ann}(A)$ and $w_i \in W$.
Let $J_W = \{i \in J \mid w_i \neq 0\}$. We claim that $\{w_i\}_{i \in J_W}$ is a basis of $W$; in fact, it clearly spans $W$, and for $i \neq j$ we have that
\[
0 = e_ie_j = (r_i + w_i)(r_j + w_j) = r_ir_j + r_iw_j + w_ir_j + w_iw_j = w_iw_j,
\]
since $r_i, r_j \in \mathrm{Ann}(A)$. This shows that the $w_i's$ are pairwise orthogonal. Now if $w_i \neq 0$ and $w^2_i = 0$ for some $i$, then $\langle w_i, w_i \rangle = 0$, and so $\langle w_i, W \rangle = 0$, which implies that $w_i \in W^\bot \cap W = 0$, a contradiction. Thus $\langle w_i, w_j \rangle = 0$ if $i \neq j$, and $\langle w_i, w_i \rangle \neq 0$ (provided that $w_i \neq 0$), which implies that
the $w_i's$ are linearly independent, and so $\{w_i\}_{i \in J_W}$ is a basis of $W$.
To finish, notice that if $K$ is quadratically closed, then the set $\Big\{\frac1{\sqrt{\langle w_i, w_i\rangle}} w_i \Big\}_{i \in J_W}$ is an orthonormal basis of $W$. This finishes the proof.
\end{proof}
\smallskip
Let $A$ be an evolution algebra such that $\dim (A^2) = 1$ and $(A^2)^2 = 0$. We proceed like in Remark \ref{product}, and we write the product in $A$ as
\[
xy = \langle x, y \rangle a, \, \mbox{ for all } \, x, y \in A,
\]
where $a \in A$ satisfies $A^2 = Ka$.
Notice that $(A^2)^2 = 0$ implies $a^2 = 0$, and so $\langle a, a \rangle = 0$, which says that $a$ is isotropic. In what follows, we distinguish two sub cases depending on whether $a \in \mathrm{Ann}(A)$ or $a \notin \mathrm{Ann}(A)$.
\subsubsection{{\bf Sub case:} $a \in \mathrm{Ann}(A)$, or equivalently, $A^3 = 0$}
It turns out that $A$ is associative (and commutative) in this case.
\begin{remark}
Let $A$ be an evolution algebra as in Lemma \ref{lemmaanndecomp}. If $A^3 = 0$, then the basis $\{w_i\}$ of $W$ is such that $w^2_i \in \mathrm{Ann}(A)$ for all $i$.
\end{remark}
\begin{theorem} \label{Iso1}
Let $A$ and $B$ be evolution algebras satisfying that $\dim(A^2) = \dim(B^2) = 1$, $(A^2)^2 = A^3 = 0$ and $(B^2)^2 = B^3 = 0$.
Write $A = \mathrm{Ann}(A) \oplus W_A$ and $B = \mathrm{Ann}(B) \oplus W_B$ as in \eqref{decomA}.
Then $A$ and $B$ are isomorphic if and only if $\dim(\mathrm{Ann}(A)) = \dim(\mathrm{Ann}(B))$, and the spaces $W_A$ and $W_B$ are isometric.
\end{theorem}
\begin{proof}
Suppose first that $\theta: (W_A, \langle \cdot, \cdot \rangle \vert_{W_A}) \to (W_B, \langle \cdot, \cdot \rangle \vert_{W_B})$ is an isometry
and that $\dim(\mathrm{Ann}(A)) = \dim(\mathrm{Ann}(B))$. For a linear isomorphism $\xi: \mathrm{Ann}(A) \to \mathrm{Ann}(B)$, one can easily check that the map $F: A \to B$ given by $F(y + z) = \xi(y) + \theta(z)$, for all $y \in \mathrm{Ann}(A)$ and $z \in W_A$, is the desired isomorphism.
Conversely, suppose that $F: A \to B$ is an isomorphism. Then $B = \hbox{Ann}(B)\oplus F(W_A)$, which implies $W_B = F(W_A)$. If $a \in A$ is such that $A^2 = Ka$, then Lemma \ref{pointed} allows us to choose $b = F(a)$, as the generator of $B^2$.
It is clear that $F\vert_{\mathrm{Ann}(A)}$ is a linear isomorphism, and so $\dim(\mathrm{Ann}(A)) = \dim(\mathrm{Ann}(B))$. It remains to show that $\theta = F\vert_{W_A}$
is an isometry; in fact, for $z_1, z_2 \in W_A$ we have that
\[
z_1z_2 = \langle z_1, z_2 \rangle a,
\]
which implies that
\[
\langle z_1, z_2 \rangle b = F(z_1z_2) = F(z_1)F(z_2) = \theta(z_1)\theta(z_2) =
\langle \theta(z_1), \theta (z_2) \rangle b.
\]
Thus: $\langle z_1, z_2 \rangle = \langle \theta(z_1), \theta (z_2) \rangle$, proving that $\theta$ is an isometry, as desired.
\end{proof}
\begin{example}
Using Theorem \ref{Iso1} we can determine the 3-dimensional real evolution algebras $A$ such that $\dim(A^2) = 1$ and $(A^2)^2 = A^3 = 0$. In fact, for
$d = \dim(\mathrm{Ann}(A))$, we have the following cases:
\begin{itemize}
\item $d = 2$: We have that $A \cong \mathbb R^3$ with $\dim(W) = 1$. There are two nonisomorphic algebras with products given by
\[
(x_1, x_2, x_3)(y_1, y_2, y_3) = (\pm x_3y_3, 0, 0).
\]
\item $d = 1$: We have that $A \cong \mathbb R^3$ with $\dim(W) = 2$. There are three nonisomorphic algebras with products given by
\begin{align*}
(x_1, x_2, x_3)(y_1, y_2, y_3) &= (x_2y_2 + x_3y_3, 0, 0);
\\
(x_1, x_2, x_3)(y_1, y_2, y_3) &= (x_2y_2 - x_3y_3, 0, 0);
\\
(x_1, x_2, x_3)(y_1, y_2, y_3) &= (-x_2y_2 - x_3y_3, 0, 0).
\end{align*}
\item $d = 0$: We have that $A \cong \mathbb R^3$ with $\dim(W) = 3$. There are four nonisomorphic algebras with products given by
\begin{align*}
(x_1, x_2, x_3)(y_1, y_2, y_3) &= (x_1y_1 + x_2y_2 + x_3y_3, 0, 0);
\\
(x_1, x_2, x_3)(y_1, y_2, y_3) &= (x_1y_1 + x_2y_2 - x_3y_3, 0, 0);
\\
(x_1, x_2, x_3)(y_1, y_2, y_3) &= (x_1y_1 -x_2y_2 - x_3y_3, 0, 0);
\\
(x_1, x_2, x_3)(y_1, y_2, y_3) &= (-x_1y_1 -x_2y_2 - x_3y_3, 0, 0).
\end{align*}
\end{itemize}
\end{example}
We can improve Theorem \ref{Iso1}, provided that the field $K$ is quadratically closed.
\begin{theorem} \label{A3=0}
Let $K$ be a quadratically closed field.
Then the isomorphic class of an evolution algebra $A$ with $\dim(A^2) = 1$ and $(A^2)^2 = A^3 = 0$ is completely determined by $\dim (A)$ and $\dim \big(\mathrm{Ann}(A)\big)$.
\end{theorem}
\begin{proof}
Let $B$ be an evolution algebra with $\dim(B^2) = 1$ and $(B^2)^2 = B^3 = 0$. If $\dim(B) = \dim(A)$ and $\dim \big(\mathrm{Ann}(B)\big) = \dim \big(\mathrm{Ann}(A)\big)$, then $\dim(W_A) = \dim(W_B)$ by \eqref{decomA}. Thus, $W_A$ and $W_B$ are isometric by Lemma \ref{lemmaanndecomp}.
\noindent The converse follows from Theorem \ref{Iso1} and \eqref{decomA}.
\end{proof}
\begin{example}
Over $\mathbb C$, in dimension 4, Theorem \ref{A3=0} tells us that there are four different (isomorphic) classes of evolution algebras $A$ with $\dim(A^2) = 1$, $(A^2)^2 = A^3 = 0$ for $\dim(\mathrm{Ann}(A)) = 1$. Their quadratic forms are given by
\begin{align*}
q_1(x_1, x_2, x_3, x_4) & = x_2^2 + x_3^2 + x_4^2,
\\
q_2(x_1, x_2, x_3, x_4) & = x_3^2 + x_4^2,
\\
q_3(x_1, x_2, x_3, x_4) & = x_4^2,
\end{align*}
with respect to the canonical basis of $\mathbb C^4$.
\end{example}
\begin{example}
Let us now classify the $3$-dimensional evolution algebras $A$ with $\dim(A^2) = 1$ and $(A^2)^2 = A^3 = 0$ over the field $K$ of nine elements.
In this case the existence of a nonzero annihilator is compulsory, and we obtain two types:
\begin{enumerate}
\item $\dim(\hbox{Ann}(A))=1$, then $A\cong K\times K^2$ with product
\begin{align*}
(x,y,z)(x',y',z') & =(yy'+zz')(1,0,0), \mbox{ or }
\\
(x,y,z)(x',y',z') & =(\omega yy'+zz')(1,0,0).
\end{align*}
\item $\dim(\hbox{Ann}(A))=2$, then $A\cong K^2\times K$ with product
\begin{align*}(x,y,z)(x',y',z') &= zz'(1,0,0), \mbox{ or }
\\
(x,y,z)(x',y',z') &= \omega zz' (1,0,0).
\end{align*}
\end{enumerate}
\end{example}
\subsubsection{{\bf Sub case:} $A^3 \neq 0$} Suppose that $A = \mathrm{Ann}(A) \oplus W$ is as in \eqref{decomA}. If $a = x + w$, for $x \in \mathrm{Ann}(A)$ and $w \in W$, then $w$ is also isotropic; in fact:
\begin{equation} \label{isotropico}
0 = \langle a, a \rangle = \langle x, x \rangle + 2 \langle x, w \rangle + \langle w, w \rangle = \langle w, w \rangle.
\end{equation}
The proof of the next result is very similar to the proof of Theorem \ref{Iso1}. We provide a sketch of it and leave the details to the reader.
\begin{theorem} \label{Iso2}
Let $A$ and $B$ be evolution algebras such that $\dim(A^2) = \dim(B^2) = 1$, $(A^2)^2 = 0$, $(B^2)^2 = 0$ and that both $A^3$ and $B^3$ are nonzero. Let $a \in A$ and $b \in B$ such that $A^2 = Ka$ and $B^2 = Kb$. Suppose that $A = \mathrm{Ann}(A) \oplus W_A$ and $B = \mathrm{Ann}(B) \oplus W_B$ as in \eqref{decomA}, and let $a = x + w$, $b = x' + w'$, where $x \in \mathrm{Ann}(A)$, $x' \in \mathrm{Ann}(B)$, $w \in W_A$ and $w' \in W_B$.
Then $A$ and $B$ are isomorphic if and only if $\dim(\mathrm{Ann}(A)) = \dim(\mathrm{Ann}(B))$, and there exists an isometry $\theta: W_A \to W_B$ such that $\theta(w) = w'$.
\end{theorem}
\begin{proof}
If $\dim(\mathrm{Ann}(A)) = \dim(\mathrm{Ann}(B))$ and $\theta: W_A \to W_B$ is an isometry such that $\theta(w) = w'$, then choose a linear isomorphism $\xi: \mathrm{Ann}(A) \to \mathrm{Ann}(B)$ such that $\xi(x) = x'$ and proceed like in the proof of Theorem \ref{Iso1}.
For the converse, if $F: A \to B$ is an isomorphism, reason like in the proof of Theorem \ref{Iso1} by noticing that $F(x) = x'$ and $F(w) = w'$.
\end{proof}
\begin{example} \label{F4ejemplo2}
Let $K$ be the field of $4$ elements: $K = \hbox{\bf F}_4 = \{0, 1, \alpha, \beta\}$, where $\alpha + \beta = 1$, $\alpha^2 = \beta$, $\beta^2 = \alpha$ and $\alpha\beta = 1$. Since $K$ is perfect, any orthogonalizable nondegenerate inner product admits an orthonormal basis. This implies that (up to isometry) there is only one nondegenerate orthogonalizable inner product. Therefore, we can
consider the inner product $\esc{\cdot, \cdot}: K^2 \times K^2 \to K$ defined by $\esc{(x, y), (x', y')} = xx' + yy'$, for all $x, x', y, y' \in K$. Let us begin by computing the group $\mathcal{O}(K^2, \esc{\cdot,\cdot})$ of isometries of $(K^2, \esc{\cdot,\cdot})$. This can be easily done by identifying linear maps $K^2 \to K^2$ with their matrices with respect the canonical basis. In fact, a given linear map is an isometry if and only its matrix $M$ satisfies that $MM^t = M^2 = 1$. Proceeding in this way we obtain that
\[
\mathcal{O}(K^2, \esc{\cdot,\cdot}) =
\left \{
\left(\begin{array}{@{}cc@{}}
1 & 0
\\
0 & 1
\end{array} \right), \,
\left(\begin{array}{@{}cc@{}}
0 & 1
\\
1 & 0
\end{array} \right), \,
\left(\begin{array}{@{}cc@{}}
\alpha & \beta
\\
\beta & \alpha
\end{array} \right), \,
\left(\begin{array}{@{}cc@{}}
\beta & \alpha \\
\alpha & \beta
\end{array} \right)
\right \},
\]
which is isomorphic to the Klein group $\hbox{\bf F}_2 \times \hbox{\bf F}_2$. Next, notice that the nonzero isotropic vectors are
\[
K^\times(1, 1) = \big \{(1, 1), (\alpha, \alpha), (\beta, \beta) \big \} \cong \hbox{\bf F}_3.
\]
Thus, there are three singletons orbits under the natural action of the group $\mathcal{O}(K^2, \esc{\cdot,\cdot})$ on the set $K^\times(1, 1)$, namely: $\{(1, 1)\}$, $\{(\alpha, \alpha)\}$ and $\{(\beta, \beta)\}$.
We are now in a position to determine all the 3-dimensional evolution algebras $A$ such that
$\dim(A^2) = 1$, $(A^2)^2 = 0$, $A^3 \neq 0$ and $\dim(\mathrm{Ann}(A)) = 1$. Let us take $A$ one of these evolution algebras, and write $A = \mathrm{Ann}(A) \oplus W$ by \eqref{decomA}. Then $\dim(W) = 2$, and we can identify $(W, \langle \cdot, \cdot \rangle)\vert_W$ with $K^2$ endowed with the inner product space having the identity matrix (with respect to the canonical basis). Lemma \ref{pointed} allows us to choose the generator of $A^2$ of the form $a = (1, \lambda, \mu)$, where $(\lambda, \mu) \in K^\times(1, 1)$. In total, there are three possibilities for $(\lambda, \mu)$, which induce non-isomorphic evolution algebras. Theorem \ref{Iso2} allows us to conclude that (up to isomorphism) there are three evolution algebras with products:
\begin{align*}
(x, y, z)(x', y',z') &= (yy' + zz')(1, 1, 1),
\\
(x, y, z)(x', y',z') &= (yy' + zz')(1, \alpha, \alpha),
\\
(x, y, z)(x', y', z') &= (yy' + zz')(1, \beta, \beta).
\end{align*}
\end{example}
\begin{example}
The $3$-dimensional evolution algebras $A$ with $\dim(A^2)=1$ such that $(A^2)^2=0$ but $A^3\ne 0$ over the field $K$ of nine elements are of two types:
\begin{enumerate}
\item $\hbox{Ann}(A)=0$, then $A\cong K^3$ with product
\begin{align*}
(x,y,z)(x',y',z') &=(xx'+yy'+zz')(0,1,i), \mbox{ or }
\\
(x,y,z)(x',y',z') &=(\omega xx'+yy'+zz')(0,1,i).
\end{align*}
\item $\dim(\hbox{Ann}(A))=1$, then $A\cong K\times K^2$ with product
\[
(x,y,z)(x',y',z')=(yy'+zz')(0,1,i).
\]
\end{enumerate}
Notice that in (2), inner products of the form
\[
\langle (x, y, z), (x', y', z') \rangle = \omega yy' + zz',
\]
can not be considered since they do not have nonzero isotropic vectors.
\end{example}
\begin{remark}
Let $K$ be a field of characteristic two, and $\mathcal{C}_{nd}$ the class of $n$-dimensional evolution algebras $A$ satisfying that $\dim(A^2) = 1$, $(A^2)^2 = 0$, $A^3 \neq 0$, and
$d = \dim(\mathrm{Ann}(A))$. Notice that Example \ref{F4ejemplo2} shows us that the isomorphic clases of $\mathcal{C}_{nd}$ are in one to one correspondence with the orbits of the group $\mathcal{O}(K^{n-d}, \esc{\cdot,\cdot})$ (of isometries of $K^{n - d}$) on the set of isotropic vectors.
\end{remark}
As expected, we can improve Theorem \ref{Iso2} by requiring the field to be quadratically closed. Before doing so, we need to prove a few results, the first one is a reformulation of the famous result known as {\it Witt's Cancelation Theorem}.
\begin{proposition} \label{WCT2.0}
Let $V_1$ and $V_2$ be vectors spaces over the same field of characteristic not two, $q_i$ a nondegenerate quadratic form on $V_i$, and $U_i$ a nondegenerate subspace of $V_i$, for $i = 1, 2$. If the spaces $(V_1, q_1)$ and $(V_2, q_2)$ are isometric, and there is an isometry $U_1 \to U_2$, then
there is an isometry $U_1^\bot \to U_2^\bot$.
\end{proposition}
\begin{proof}
Suppose that $f: V_1 \to V_2$ and $g: U_1 \to U_2$ are isometries. Then the map $gf^{-1}: f(U_1)\to U_2$ is also an isometry. Witt's Cancellation Theorem tells us that there is an isometry between
$f(U_1)^\bot = f(U_1^\bot)$ and $U_2^\bot$, say $h$. To finish, notice that the composition
$h f\vert_{U_1^\bot}$ is the desired isometry.
\end{proof}
\begin{lemma} \label{Isometria}
Let $(W_1, q_1)$ and $(W_2, q_2)$ be nondegenerate isometric spaces over a field $K$ of characteristic not two. If $w_1 \in W_1$ and $w_2 \in W_2$ are nonzero
isotropic vectors, then there exists an isometry $f: W_1 \to W_2$ such that $f(w_1) = w_2$.
\end{lemma}
\begin{proof}
The result trivially holds in dimension 1. Suppose now that both $W_1$ and $W_2$ have dimension $\geq 2$. Write $\esc{\cdot,\cdot}_i$ to denote the polar form of $q_i$, for $i = 1, 2$. There exists $w'_i \in W_i$ such that $(w_i, w'_i)$ is a hyperbolic pair in $W_i$, for $i = 1, 2$. It is straightforward to check that the linear map $\xi: Kw_1\oplus Kw_1'\to Kw_2\oplus Kw_2'$ such
that $\xi(w_1) = w_2$ and $\xi(w_1') = w_2'$ is an isometry. If $\hbox{dim}(W_i) = 2$, then we are done. Otherwise, Proposition \ref{WCT2.0} gives
an isometry $\eta: (Kw_1\oplus Kw_1')^\bot \to (Kw_2\oplus Kw_2')^\bot$. Now, $V_i = (Kw_i \oplus Kw_i')\oplus (Kw_i\oplus Kw_i')^\bot$, for $i = 1, 2$, and we can easily construct an isometry $f: W_1 \to W_2$ such that $f(w_1) = w_2$, $f(w'_1) = w'_2$ and $f\vert_{(Kw_1\oplus Kw_1')^\bot} = \eta$, finishing the proof.
\end{proof}
\begin{theorem} \label{A3not0}
Let $K$ be a quadratically closed field of characteristic not two.
The isomorphic class of an evolution algebra $A$ with $\dim(A^2) = 1$, $(A^2)^2 = 0$ and $A^3 \neq 0$ is completely determined by $\dim (A)$ and $\dim \big(\mathrm{Ann}(A)\big)$.
\end{theorem}
\begin{proof}
Let $B$ be an evolution algebra such that $\dim(B^2) = 1$, $(B^2)^2 = 0$ and $B^3 \neq 0$.
We express $A$ and $B$ as in \eqref{decomA}:
\begin{equation} \label{ABDec}
A = \mathrm{Ann}(A) \oplus W_A, \quad
B = \mathrm{Ann}(B) \oplus W_B.
\end{equation}
We write $a = x + w$, $b = x' + w'$, where $x \in \mathrm{Ann}(A)$, $x' \in \mathrm{Ann}(B)$, and $w \in W_A$, $w' \in W_B$, and $a$ (respectively, $b$) generates $A^2$ (respectively, $B^2$). Notice that $w$ and $w'$ are both isotropic by \eqref{isotropico}.
Suppose first that $\dim(A) = \dim(B)$ and $\dim(\mathrm{Ann}(A)) = \dim(\mathrm{Ann}(B))$. Then $\dim(W_A) = \dim(W_B)$, and Lemma \ref{lemmaanndecomp} applies to get that the nondegenerate spaces $W_A$ and $W_B$ are isometric. Lemma \ref{Isometria} and Theorem \ref{Iso2} tell us that $A$ and $B$ are isomorphic.
The converse follows from \eqref{ABDec} and Theorem \ref{Iso2}.
\end{proof}
| proofpile-arXiv_065-3842 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
The most studied example of quantum field theory in curved spacetime is probably the theory of a scalar field in de Sitter space. Indeed the model is simple enough to be solved analytically and therefore the properties of the field can be studied in detail. In particular the thermal properties of the vacuum state, related to the phenomenon of particle creation, have been considered \cite{Gibbons:1977mu}. Moreover, the quantum dynamics of the field is of special importance for inflationary cosmological models where de Sitter space describes a universe in exponential expansion \cite{Mukhanov:2005sc}. De Sitter space has also attracted new interest in connection with the conjecture of the dS/CFT correspondence proposed almost a decade ago by Strominger \cite{Strominger:2001pn}.
In this letter we expose a new quantization scheme for a massive scalar field in de Sitter spacetime based on the general boundary formulation (GBF) of quantum field theory. The results presented here are mainly intended as a contribution to the GBF program, and represent indeed the first study of QFT in a curved spacetime within the GBF. Furthermore, the quantization scheme we introduce provides a new framework to analyze the results obtained or proposed so far in literature.
In a series of papers \cite{Oe:timelike,Oe:GBQFT,Oe:KGtl,CoOe:spsmatrix,CoOe:smatrixgbf,CoOe:smatrix2d} it has been shown that the GBF
provides a viable description of the dynamics of quantized fields. Not only this new formulation can recover known results of standard QFT, but more interesting the GBF can handle situations where the methods of standard QFT fail.
In QFT, dynamics is described in terms of the evolution of initial data from an initial Cauchy surface to a final Cauchy surface. Therefore, this standard picture
involves a spacetime region bounded by the disjoint union of two
Cauchy surfaces. In the GBF
evolution acquires a more general description: The boundary of the spacetime region where dynamics takes place can have arbitrary form and is not required to reduce to the disjoint union of two Cauchy surfaces.
The main novelty of the GBF resides in associating Hilbert spaces of states to arbitrary hypersurfaces in spacetime and amplitudes to spacetime regions and states living on their boundaries. For a region $M$ of spacetime, the amplitude is a map from the Hilbert space ${\cal H}_{\partial M}$ associated with the boundary $\partial M$ of the region to the complex numbers. The formal expression of the amplitude $\rho$, for a state $\psi \in {\cal H}_{\partial M}$, is given in terms of the Feynman path integral combined with the Schr\"odinger representation for quantum states,
\begin{equation}
\rho_{M}(\psi)= \int \xD \varphi \, \psi(\varphi) \int_{\phi|_{\partial M}=\varphi} \xD\phi\, e^{\im S_{M}(\phi)},
\label{eq:rho}
\end{equation}
where the outer integral is over all field configurations $\varphi$ on $\partial M$, and the inner integral is over all field configurations $\phi$ in the spacetime region $M$ matching $\varphi$ on the boundary.
Finally, a physical interpretation can be given to such amplitudes and an appropriate notion of probability can be extracted from them \cite{Oe:GBQFT,Oe:KGtl}.
So far this formalism has been applied only to the study of flat-spacetime-based QFT. There the standard $S$-matrix for an interacting scalar field has been shown to be equivalent to the one derived for free asymptotic quantum states at spatial infinity. The notion of spatial asymptotic state comes from the geometry considered: In particular, in Minkowski spacetime states were defined on an hypercylinder, namely the boundary of a threeball in space extended over all of time, and then the radius of the ball was sent to infinity.
The structure of this work follows that of \cite{CoOe:smatrixgbf}. So, in the following we will evaluate the $S$-matrix for coherent states on spacelike hypersurfaces of constant conformal de Sitter time. Then, the asymptotic amplitude will be derived for coherent states defined on an analogue of the hypercylinder in de Sitter space. Finally, by constructing an isomorphism between the respective state spaces, we prove the equivalence of these two types of amplitude.
We consider de Sitter spacetime with the metric
\begin{equation}
\mathrm{d} s^2 = \frac{1}{H^2t^2} \left( \mathrm{d} t^2 - \mathrm{d} \underline{x}^2 \right),
\label{eq:dSmetric}
\end{equation}
where $H$ is the Hubble constant, the conformal time $t$ takes values in the interval $]0,\infty[$ and $\underline{x} \in {\mathbb R}^3$ are coordinates on the equal time hypersurfaces. Such coordinates cover only half of de Sitter spacetime. The remaining half can be included by simply extending the domain of $t$ to $]-\infty,\infty[$, \cite{BiDa:qfcs}. We start with the derivation of the standard transition amplitude by computing the amplitude (\ref{eq:rho}) for a spacetime region $M$ bounded by two equal time hypersurfaces, $\Sigma_1$ and $\Sigma_2$, defined respectively by the values $t_1$ and $t_2$ of the conformal time $t$: $M = [t_1,t_2] \times \mathbb{R}^3$. Then the state space associated with the boundary $\partial M = \Sigma_1 \cup \Sigma_2$ results to be the tensor product ${\cal H}_1 \otimes {\cal H}_2^*$ of the Hilbert spaces defined on the hypersurfaces $\Sigma_1$ and $\Sigma_2$. Following \cite{CoOe:smatrixgbf} we introduce coherent states; their wave function at time t is parametrized by a complex function $\xi$ on momentum space, and in the interaction picture their form is
\begin{multline}
\psi_{t,\xi} (\varphi) = K_{t, \xi} \\
\times \exp \left( \int \frac{\mathrm{d} ^3 x \, \mathrm{d}^3 k}{(2 \pi)^3} \, \xi(\underline{k}) \, \frac{e^{\im \underline{k} \cdot \underline{x}}}{H_{\nu}(k t) t^{3/2}} \, \varphi(\underline{x}) \right)
\psi_{t,0}(\varphi),
\end{multline}
where $K_{t, \xi}$ is a normalization factor, $\psi_{t,0}(\varphi)$ is the vacuum wave function derived in \cite{Co:vacuum}, $H_{\nu}$ is the Hankel function of order $\nu = \sqrt{\frac{9}{4} - \frac{M^2}{H^2}}$, $M$ denotes the mass of the scalar field and we assume $\nu$ real.
Consider first the free theory. The amplitude (\ref{eq:rho}), denoted by the subscript $0$, associated with the tensor product of coherent states $\psi_{t_1,\xi_1} \otimes \overline{\psi_{t_2,\xi_2}}$ is independent of times $t_1$ and $t_2$ and can be written as
\begin{multline}
\rho_{M,0}(\psi_{\xi_1} \otimes \overline{\psi_{ \xi_2}}) =
\exp \left( \frac{\pi H^2}{4} \int \frac{\mathrm{d}^3 k}{(2 \pi)^3} \, \right. \\
\times \left. \left( \overline{\xi_{2}(\underline{k})} \xi_{1}(\underline{k}) - \frac{1}{2}|\xi_{1}(\underline{k})|^2 - \frac{1}{2} |\xi_{2}(\underline{k})|^2 \right) \right).
\label{eq:freeampl}
\end{multline}
Because of independence of initial and final time, the above expression represents the $S$-matrix describing the transition from the coherent state defined by $\xi_1$ (in the asymptotic past) to the coherent state defined by $\xi_2$ (in the asymptotic future).
As an intermediate step in the computation of the $S$-matrix for the general interacting theory, we consider the interaction of the scalar field with a source field $\mu$ of the form $\int \mathrm{d}^4 x \sqrt{-g} \, \phi(x) \, \mu(x)$, and we assume that $\mu$ vanishes outside the spacetime region considered here, namely $\mu |_{t \notin ]t_1,t_2[} = 0$. Adding such interaction term to the free action yields for the amplitude (\ref{eq:rho}), denoted by the subscript $\mu$, the result
\begin{multline}
\rho_{M,\mu}(\psi_{\xi_1} \otimes \overline{\psi_{\xi_2}}) = \\
\rho_{M,0}(\psi_{ \xi_1} \otimes \overline{\psi_{\xi_2}}) \exp \left( \int \mathrm{d} ^4 x \sqrt{-g} \mu(x) \hat{\xi}(x) \right) \\
\times \exp \left(\frac{\im}{2} \int \mathrm{d} ^4 x \sqrt{-g} \, \mu(x) \, \gamma(x) \right),
\label{eq:srcampl}
\end{multline}
where $g$ is the determinant of the metric (\ref{eq:dSmetric}), $\hat{\xi}$ is the complex solution of the Klein-Gordon equation determined by the initial and final coherent states,
\begin{multline}
\hat{\xi}(x) = \frac{\im \pi H^2}{4} \int \frac{ \mathrm{d}^3 k}{(2 \pi)^3} \left(\xi_1(\underline{k}) \, e^{\im \underline{k} \cdot \underline{x}} \, t^{3/2} \, \overline{H_{\nu}(k t)} \right. \\
\left. + \, \overline{\xi_2(\underline{k})} \, e^{-\im \underline{k} \cdot \underline{x}}\, t^{3/2} \, H_{\nu}(k t) \right).
\label{eq:hatxi}
\end{multline}
The function $\gamma$ in the last exponential of (\ref{eq:srcampl}) is the solution of the inhomogeneous Klein-Gordon equation,
\begin{equation}
(\Box + M^2) \gamma(x) = \mu(x),
\label{eq:inKG}
\end{equation}
with the following boundary conditions,
\begin{multline}
\gamma(t, \underline{x}) |_{t<t_1} = t^{3/2} \, H_{\nu}(k t) \, \frac{\im \pi H^2}{4 } \\
\times \int_{t_1}^{t_2} \mathrm{d} t' \sqrt{-g'} (t')^{3/2} \overline{H_{\nu}(k t')} \mu(t',\underline{x}), \label{eq:Fbc1}
\end{multline}
for early times $t$ before the source is switched on, and
\begin{multline}
\gamma(t, \underline{x}) |_{t>t_2} = t^{3/2} \, \overline{H_{\nu}(k t)} \, \frac{\im \pi H^2}{4 } \\
\times \int_{t_1}^{t_2} \mathrm{d} t' \sqrt{-g'} (t')^{3/2} H_{\nu}(k t') \mu(t',\underline{x}),\label{eq:Fbc2}
\end{multline}
for late times $t$ after the source is switched off. In the above expressions $g'$ is the determinant of the metric (\ref{eq:dSmetric}) expressed in the coordinates $(t', \underline{x})$, and the Bessel functions are to be understood as operators via the mode decomposition of the source field. It is convenient the write $\gamma$ in the form
\begin{equation}
\gamma(x) = \int \mathrm{d}^4 x' \sqrt{-g'} \, G(x,x') \, \mu(x'),
\label{eq:gamma}
\end{equation}
and $G$ is the Green function, solution of the equation $(\Box + M^2)G(x,x') = (-g)^{-1/2} \delta^4(x-x')$, given by
\begin{multline}
G(x,x')= \frac{H^2}{ 16 \pi} \left( \frac{1}{4} - \nu^2 \right) \sec(\nu \pi)\\
\times F \left( \frac{3}{2}-\nu , \frac{3}{2} + \nu ; 2 ; \frac{1+p- \im 0}{2}\right),
\label{eq:propagator}
\end{multline}
where $F$ is the hypergeometric function and $p= \frac{t^2+t'^2-|\underline{x}- \underline{x}'|^2}{2 t't}$. The above expression coincides with the expression of the Feynman propagator in de Sitter space computed in \cite{Schomblond:1976xc,Bunch:1978yq}, and we can therefore interpret the conditions (\ref{eq:Fbc1},\ref{eq:Fbc2}) has the Feynman boundary conditions. The form (\ref{eq:srcampl}) of the $S$-matrix in the presence of a source field is similar to the one obtained by the path integral in the holomorphic representation \cite{FaSl:gaugeqft}.
Finally we use functional methods to express a general interaction in terms of the source interaction,
\begin{multline}
\int \mathrm{d}^4 x \, \sqrt{-g} \, V(x, \phi(x)) = \\
\int \mathrm{d}^4 x \, V \left( x, \frac{\partial}{\partial \mu(x)}\right) \int \mathrm{d}^4 y \sqrt{-g} \, \phi(y) \mu(y) \bigg|_{\mu =0}.
\label{eq:genint}
\end{multline}
Assuming that the interaction vanishes for $t$ outside the interval $]t_1,t_2[$, the amplitude (\ref{eq:rho}), now indicated with the subscript $V$, takes the form
\begin{multline}
\rho_{M,V}(\psi_{\xi_1} \otimes \overline{\psi_{\xi_2}}) = \exp \left( \im \int \mathrm{d}^4 x \, V \left( x, - \im \frac{\partial}{\partial \mu(x)}\right) \right) \\
\times \rho_{M, \mu}(\psi_{\xi_1} \otimes \overline{\psi_{\xi_2}})\bigg|_{\mu =0}.
\label{eq:genampl}
\end{multline}
This expression is independent of $t_1$ and $t_2$ and consequently the restriction on $V$ introduced above can be removed. Moreover, the limit of asymptotic times is trivial and (\ref{eq:genampl}) is then interpreted as the $S$-matrix for the general interacting theory.
The second geometry we are interested in is conveniently described in terms of spherical coordinates, in which the metric (\ref{eq:dSmetric}) takes the form
\begin{equation}
\mathrm{d} s^2 = \frac{1}{H^2 t^2} \left( \mathrm{d} t^2 - \mathrm{d} r^2 - r^2 \mathrm{d} \Omega^2 \right),
\label{eq:dSmetric2}
\end{equation}
where $\mathrm{d} \Omega^2$ is the metric on a unit sphere.
We will now compute the amplitude (\ref{eq:rho}) associated with the spacetime region $M$ bounded by the hypersurface of constant radius, $r =R$. Hence, $M$ has one connected boundary that we call the \textit{hypercylinder} in analogy with the notion of the hypercylinder introduced in \cite{Oe:KGtl}. We proceed as before by considering coherent states defined in the Hilbert space ${\cal H}_R$ associated with the hypercylinder, and with wave function in the interaction picture given by
\begin{multline}
\psi_{R,\eta} (\varphi) = K_{R, \eta} \exp \left( \int \mathrm{d} t \, \mathrm{d} \Omega \, \mathrm{d} k \sum_{l,m} \right. \\
\left. \times \eta_{l,m}(k)\frac{t^{-1/2} Z_{\nu}(k t) Y_l^{-m}(\Omega)}{h_l(k R)} \varphi(t, \Omega) \right) \psi_{R,0}(\varphi),
\end{multline}
where $\eta$ is the complex function on momentum space that parametrizes the coherent state, $\psi_{R,0}$ the vacuum wave function on the hypercylinder of radius $R$ and $K_{R, \eta}$ a normalization factor. Here $Z_{\nu}$ denotes the Bessel function of the first or second kind, $Y_l^m$ the spherical harmonic and $h_l$ the spherical Bessel function of the third kind.
The free amplitude for such state reads,
\begin{multline}
\rho_{M,0}(\psi_{ \eta} ) = \exp \left( - \frac{H^2}{4} \int \mathrm{d} k \sum_{l,m} k^2 \right. \\
\times \left[ |\eta_{l,m}(k)|^2 - \eta_{l,m}(k) \, \eta_{l,-m}(k) \right] \Bigg),
\label{eq:freeamplhyp}
\end{multline}
and is independent of the radius $R$.
As before, we now look at the interaction with a source field $\mu$. Requiring this field to be confined in the interior of the hypercylinder, the amplitude for the coherent state $\psi_{\eta}$ turns out to be
\begin{multline}
\rho_{M,\mu}(\psi_{ \eta} ) =
\rho_{M,0}(\psi_{ \eta} ) \exp \left( \int \mathrm{d} ^4 x \sqrt{-g} \mu(x) \hat{\eta}(x) \right) \\
\times \exp \left(\frac{\im}{2} \int \mathrm{d} ^4 x \sqrt{-g} \, \gamma(x) \, \mu(x) \right).
\label{eq:srcamplhyp}
\end{multline}
$\hat{\eta}$ is the complex solution of the Klein-Gordon equation given by
\begin{multline}
\hat{\eta}(x) = \im H^2 \int \mathrm{d} k \, k \sum_{l,m} \, t^{3/2} Z_{\nu}(k t) \\
\times Y_l^m(\Omega) \, j_l(kr) \, \eta_{l,m}(k),
\label{eq:hateta}
\end{multline}
where $j_l$ is the spherical Bessel function of the first kind.
The function $\gamma$ in the last line of (\ref{eq:srcamplhyp}) satisfies the inhomogeneous Klein-Gordon equation (\ref{eq:inKG}), and can therefore be written via the Green function as in (\ref{eq:gamma}).
The Green function $G$, defined on the hypercylinder, turns out to be the same Green function that appears in (\ref{eq:gamma}), namely the Feynman propagator (\ref{eq:propagator}). The boundary condition satisfied by $\gamma$ can then be interpreted as the Feynman boundary condition on the hypercylinder, valid for large radius $r$ outside the source field,
\begin{multline}
\gamma(t,r, \Omega)\big|_{r>R} = k \,\im \, h_l(k r) \\
\times \int_0^R \mathrm{d} r' \, (r')^2 \sqrt{-g'} \,t^2 H^2 \, j_l(k r') \mu(t,r', \Omega).
\end{multline}
$g'$ denotes the determinant of the metric (\ref{eq:dSmetric2}) in the coordinates $(t,r', \Omega)$, and the Bessel functions are to be understood as operators acting on the mode expansion of $\mu$.
Again, we notice that no dependence on the radius $R$ is present in the amplitude (\ref{eq:srcamplhyp}).
To conclude, we apply the same functional techniques as before, expressing the general interaction as in (\ref{eq:genint}). Assuming that the interaction now vanishes outside the hypercylinder, we can write the amplitude for the general interacting theory as
\begin{multline}
\rho_{M,V}(\psi_{\eta}) = \exp \left( \im \int \mathrm{d}^4 x \, V \left( x, - \im \frac{\partial}{\partial \mu(x)}\right) \right) \\
\times \rho_{M, \mu}(\psi_{\eta})\bigg|_{\mu =0}.
\label{eq:genamplhyp}
\end{multline}
Since $R$ does not appear, the cutoff on the interaction can be dropped. Being the limit $R \rightarrow \infty$ trivial, we interpret (\ref{eq:genamplhyp}) as the asymptotic amplitude of the general interacting theory for the coherent state $\psi_{\eta}$.
Having computed the asymptotic amplitudes in the two geometries considered here, we now want to analyze their relation. To this aim we adopt an approach analogue to that used in \cite{CoOe:smatrixgbf}. We focus our attention on the expression of the amplitudes for the source interaction in both settings, i.e. (\ref{eq:srcampl}) and (\ref{eq:srcamplhyp}). Considering the same source field in the two cases, namely a source bounded in space and in time, we notice that the last terms of the amplitudes coincide because in the functions $\gamma$ the same propagator (\ref{eq:propagator}) appears. We turn to the second second terms in (\ref{eq:srcampl}) and (\ref{eq:srcamplhyp}): They coincide if and only if the complex solutions to Klein-Gordon equation, $\hat{\xi}$ and $\hat{\eta}$, coincide. It turns out that this equality, namely $\hat{\xi} = \hat{\eta}$, defines an isomorphic map between the state spaces of the two theories, i.e. the Hilbert space ${\cal H}_1 \otimes {\cal H}_2^*$ associated with the boundary of the spacetime region $M = [t_1,t_2] \times \mathbb{R}^3$ and the Hilbert space ${\cal H}_R$ associated with the hypercylinder. Hence, under the isomorphism we have: $\psi_{\xi_1} \otimes \overline{\psi_{\xi_2}} \cong \psi_{\eta}$. We are left with the first term appearing in (\ref{eq:srcampl}) and (\ref{eq:srcamplhyp}), the free amplitudes in the two settings given by (\ref{eq:freeampl}) and (\ref{eq:freeamplhyp}). It is not difficult to show that these free amplitudes are equal under the isomorphism. For example, expressing (\ref{eq:freeamplhyp}) in terms of the function $\hat{\eta}$, we substitute $\hat{\eta}$ with the expression (\ref{eq:hatxi}) of $\hat{\xi}$ and obtain (\ref{eq:freeampl}):
\begin{equation}
\rho_{M,0}(\psi_{\eta})\big|_{\hat{\eta} = \hat{\xi}} = \rho_{M,0}(\psi_{\xi_1} \otimes \overline{\psi_{ \xi_2}}).
\end{equation}
We can then conclude the equivalence of the asymptotic amplitudes, interpreted as $S$-matrices, for the general interacting theory, under the isomorphism. Such equivalence offers the possibility to study scattering processes in de Sitter space from a new perspective. Indeed the amplitude for a transition from an in-state with $m$ particles to an out-state with $n$ particles can be mapped to the amplitude for an $(m+n)$-particle state defined on the hypercylinder, and the physical probabilities extracted from the $S$-matrices of the two descriptions are the same. We recover here results analogous to those previously obtained in Minkowski \cite{CoOe:smatrixgbf} and Euclidean spacetime \cite{CoOe:smatrix2d}, and the conclusions discussed there can be exported, mutatis mutandi, to de Sitter space.
\begin{acknowledgments}
I am grateful to Robert Oeckl for helpful discussions and comments on an earlier draft of this letter. This work was supported in part by CONACyT grant 49093.
\end{acknowledgments}
\bibliographystyle{amsordx}
| proofpile-arXiv_065-6373 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
\section{Introduction}
Fifth generation (5G) communication networks will likely adopt \ac{mm-wave} and massive \ac{MIMO} technologies, thanks to a number of favorable properties. In particular, operating at carrier frequencies beyond 30 GHz, with large available bandwidths, \ac{mm-wave} can provide extremely high data rates to users through dense spatial multiplexing by using a large number of antennas \cite{Zhouyue,Rappaport}. While these properties are desirable for 5G services, \ac{mm-wave} communications also face a number of challenges. Among these, the severe path loss at high carrier frequencies stands out. The resulting loss in \ac{SNR} must be compensated through sophisticated beamforming at the transmitter and/or receiver side, leading to highly directional links \cite{Wang,Hur,Tsang}. However, beamforming requires knowledge of the propagation channel. Significant progress has been made in \ac{mm-wave} channel estimation, by exploiting sparsity and related compressed sensing tools, such as \ac{DCS-SOMP} \cite{Duarte}, \ac{CoSOMP} \cite{Duarte2}, and \ac{GCS} \cite{Bolcskei}. In particular, since at \ac{mm-wave} frequencies only the \ac{LOS} path and a few dominant multi-path components contribute to the received power,
\ac{mm-wave} channels are sparse in the angular domain \cite{BspaceSayeed,widebandbrady}. This is because in \ac{mm-wave} frequencies, the received power of diffuse scattering and multiple-bounce specular reflections are much lower than that in \ac{LOS} and single-bounce specular reflection \cite{Martinez-Ingles,Vaughan,mmMAGIC}. Different \ac{CS} methods for \ac{mm-wave} channel estimation are proposed in\cite{Marzi,LeeJ,AlkhateebA,ChoiJ,AlkhateebC,HanY,LeeJ2,Ramasamy,BerrakiD}. In \cite{Marzi}, a method for the estimation of \ac{AOA}, \ac{AOD}, and channel gains is proposed based on the compressive beacons on the downlink.
A method for the continuous estimation of \ac{mm-wave} channel parameters is proposed in \cite{Ramasamy}, while \cite{LeeJ2} applies CS tools with refinement in the angular domain. In \cite{LeeJ}, a CS method is proposed based on the redundant dictionary matrices.
A two-stage algorithm with one-time feedback that is robust to noise is used in \cite{HanY}. In \cite{AlkhateebA}, an adaptive \ac{CS} method is proposed based on a hierarchical multi-resolution codebook design for the estimation of single-path and multi-path \ac{mm-wave} channels. In \cite{ChoiJ}, a beam selection procedure for the multiuser \ac{mm-wave} \ac{MIMO} channels with analog beamformers is proposed.
In \cite{AlkhateebC}, a \ac{CS} approach with reduced training overhead was considered.
Finally, \ac{CS} tools are used in \cite{BerrakiD} for the sparse estimation of power angle profiles of the \ac{mm-wave} channels and compared with the codebook designs in terms of overhead reduction. However, in all the aforementioned papers, a narrow-band \ac{mm-wave} channel model is used. When the bandwidths becomes larger, one needs to consider the effect of the delays of different paths in the \ac{mm-wave} channel model, i.e., the wide-band \ac{mm-wave} channel model.
Channel estimation provides information of the \ac{AOA}/\ac{AOD} and thus of the relative location of the transmitter and receiver. In addition, location information can serve as a proxy for channel information to perform beamforming: when the location of the user is known, the \ac{BS} can steer its transmission to the user, either directly or through a reflected path. This leads to synergies between localization and communication. The use of 5G technologies to obtain position and orientation was previously explored in \cite{sanchis2002novel,DenSaya,vari2014mmwaves} for mm-wave and in \cite{hu2014esprit,Dardari,savic2015fingerprinting} for massive MIMO. The early work \cite{sanchis2002novel} considered estimation and tracking of \ac{AOA} through beam-switching. User localization was treated in \cite{DenSaya}, formulated as a hypothesis testing problem, limiting the spatial resolution. A different approach was taken in \cite{vari2014mmwaves}, where meter-level positioning accuracy was obtained by measuring received signal strength levels. A
location-aided beamforming method was proposed in \cite{NGarcia} to speed up initial access between nodes. In the massive \ac{MIMO} case, \cite{hu2014esprit} considered the estimation of angles, \cite{NGarcia2} proposed a direct localization method by jointly processing the observations at the distributed
massive \ac{MIMO} \ac{BS}s, while \cite{Dardari} treated the joint estimation of delay, \ac{AOD}, and \ac{AOA}, in the \ac{LOS} conditions and evaluated the impact of errors in delays and phase shifters, and \cite{Arash} derived sufficient conditions for a nonsingular \ac{FIM} of delay, \ac{AOD}, \ac{AOA}, and channel coefficients. A hybrid \ac{TDOA}, \ac{AOA}, and \ac{AOD} localization was proposed in \cite{linhyb} using linearization. In \cite{savic2015fingerprinting}, positioning was solved using a Gaussian process regressor, operating on a vector of received signal strengths through fingerprinting. While latter this approach is able to exploit \ac{NLOS} propagation, it does not directly harness the geometry of the environment. Complementarily to the use of \ac{mm-wave} frequencies, approaches for localization using \ac{cm-wave} signals have been recently proposed as well. The combination of \ac{TDOA}s and \ac{AOA}s using an extended Kalman filter (EKF) was presented in \cite{DBLP:journals/twc/KoivistoCWHTLKV17,DBLP:journals/corr/KoivistoHCKLV16}, where the \ac{MS} has a single antenna, while the \ac{BS} employs an antenna array. This method assumes \ac{LOS} propagation thanks to the high density of access nodes and provides sub-meter accuracy even for moving devices.
In this paper, we show that \ac{mm-wave} and large \ac{MIMO} are enabling technologies for accurate positioning and device orientation estimation with only one \ac{BS}, even when the \ac{LOS} path is blocked. The limited scattering and high-directivity are unique characteristics of the \ac{mm-wave} channel and large \ac{MIMO} systems, respectively. We derive fundamental bounds on the position and orientation estimation accuracy, for \ac{LOS}\footnote{\ac{LOS} is defined as the condition where the \ac{LOS} path exists and there are no scatterers.}, \ac{NLOS}\footnote{\ac{NLOS} is defined as the condition where there are scatterers and the LOS path is not blocked.}, and \ac{OLOS}\footnote{\ac{OLOS} is referred to the condition where the LOS path is blocked and only the signals from the scatterers are received.} conditions. These bounds indicate that the information from the \ac{NLOS} links help to estimate the location and orientation of the \ac{MS}. We also propose a novel three-stage position and orientation estimation technique, which is able to attain the bounds at average to high \ac{SNR}. The first stage of the technique harnesses sparsity of the \ac{mm-wave} channel in the \ac{AOA} and \ac{AOD} domain \cite{BspaceSayeed,widebandbrady}. Moreover, the sparsity support does not vary significantly with frequency, allowing us to use \ac{DCS-SOMP} across different carriers. The delay can then be estimated on a per-path basis. As \ac{DCS-SOMP} limits the \ac{AOA} and \ac{AOD} to a predefined grid, we propose a refinement stage, based on the \ac{SAGE} algorithm. Finally, in the last stage, we employ a least-squares approach with \ac{EXIP} to recover position and orientation \cite{Stoicapp,Swindlehurstt}.
\begin{figure}
\psfrag{x}{\small $x$}
\psfrag{y}{\small $y$}
\psfrag{d0}{\small $d_{0}$}
\psfrag{dk1}{\small $d_{k,1}$}
\psfrag{dk2}{\small $d_{k,2}$}
\psfrag{dk}{\small $d_{k}=d_{k,1}+d_{k,2}$}
\psfrag{sk}{\hspace{-1mm} $\mathbf{s}_{k}$}
\psfrag{tt0}{\small \hspace{-1mm} $\theta_{\mathrm{Tx},0}$}
\psfrag{tt1}{\small \hspace{-4mm} $\theta_{\mathrm{Tx},k}$}
\psfrag{tt1b}{\small \hspace{-8mm} $\pi-(\theta_{\mathrm{Rx},k}+\alpha)$}
\psfrag{rr0}{\small \hspace{-8mm} $\pi-\theta_{\mathrm{Rx},0}$}
\psfrag{rr1}{\small \hspace{-8mm} $\pi-\theta_{\mathrm{Rx},k}$}
\psfrag{rr1b}{\small \hspace{-8mm} $\pi-(\theta_{\mathrm{Rx},k}+\alpha)$}
\psfrag{alphab}{\small \hspace{-4mm} $\alpha$}
\psfrag{q}{ \hspace{-1mm} $\mathbf{q}$}
\psfrag{qk}{ \hspace{-1mm} $\widetilde{\mathbf{q}}_{k}$}
\psfrag{p}{ \hspace{-4mm} $\mathbf{p}$}
\psfrag{BS}[][c]{BS}
\psfrag{VBS}[][c]{virtual BS}
\psfrag{MS}[][c]{MS}
\centering
\includegraphics[width=0.9\columnwidth]{VBST2.eps}
\caption{Two dimensional illustration of the \ac{LOS} (blue link) and \ac{NLOS} (red link) based positioning problem. The \ac{BS} location $\mathbf{q}$ and \ac{BS} orientation are known, but arbitrary. The location of the \ac{MS} $\mathbf{p}$, scatterer $\mathbf{s}_{k}$, rotation angle $\alpha$, \ac{AOA}s $\{\theta_{\textup{Rx},k}\}$, \ac{AOD}s $\{\theta_{\textup{Tx},k}\}$, the channels between \ac{BS}, \ac{MS}, and scatterers, and the distance between the antenna centers are unknown.}
\label{NLOS_Link}
\end{figure}
\section{System Model}\label{SEC:Formulation}
We consider a \ac{MIMO} system with a \ac{BS} equipped with $N_{t}$ antennas and a \ac{MS} equipped by $N_{r}$ antennas operating at a carrier frequency $f_c$ (corresponding to wavelength $\lambda_c$) and bandwidth $B$.
Locations of the \ac{MS} and \ac{BS} are denoted by $\mathbf{p}=[p_{x}, p_{y}]^{\mathrm{T}}\in\mathbb{R}^{2}$ and $\mathbf{q}=[q_{x}, q_{y}]^{\mathrm{T}}\in\mathbb{R}^{2}$ with the $\alpha\in[0, 2\pi)$ denoting the rotation angle of the \ac{MS}'s antenna array. The value of $\mathbf{q}$ is assumed to be known, while $\mathbf{p}$ and $\alpha$ are unknown.
\subsection{Transmitter Model}
We consider the transmission of \ac{OFDM} signals as in \cite{khateeb3}, where a \ac{BS} with hybrid analog/digital precoder communicates with a single \ac{MS}. At the \ac{BS}, $G$ signals are transmitted sequentially, where the $g$-th transmission comprises $M_{t}$ simultaneously transmitted symbols $\mathbf{x}^{(g)}[n]=[x_{1}[n],\ldots,x_{M_{t}}[n]]^{\mathrm{T}} \in \mathbb{C}^{M_{t}}$ for each subcarrier $n=0,\ldots,N-1$. The symbols
are first precoded and then transformed to the time-domain using $N$-point \acf{IFFT}. A \acf{CP} of length $T_{\mathrm{CP}}=DT_{s}$ is added before applying the \ac{RF} precoding where $D$ is the length of \ac{CP} in symbols. Here, $T_{s}=1/B$ denotes the sampling period and $T_{\mathrm{CP}}$ is assumed to exceed the delay spread of the channel. The transmitted signal over subcarrier $n$ at time $g$ can be expressed as $\mathbf{F}^{(g)}[n]\mathbf{x}^{(g)}[n]$. The beamforming matrix $\mathbf{F}[n] \in \mathbb{C}^{N_{t}\times M_{t}}$ is defined as $\mathbf{F}[n]=\mathbf{F}_{\mathrm{RF}}\mathbf{F}_{\mathrm{BB}}[n]$ where $\mathbf{F}_{\mathrm{RF}}$ is implemented using the analog phase shifters with the entries of the form $e^{j\phi_{m,n}}$, where $\{\phi_{m,n}\}$ are given phases, and $\mathbf{F}_{\mathrm{BB}}[n]$ is the digital beamformer, and overall they satisfy a total power constraint $\Vert\mathbf{F}_{\mathrm{RF}}\mathbf{F}_{\mathrm{BB}}[n]\Vert_{\mathrm{F}}=1$.
Considering the sparsity of the \ac{mm-wave} channels one usually needs much less beams $M_{t}$ than antenna elements $N_{t}$, i.e., $M_{t} \ll N_{t}$. Also, the presence of $\mathbf{F}[n]$ in the proposed model leads to the extension of system model to multi-user mm-wave downlink systems with a limited feedback channel from \ac{MS}s to the \ac{BS}.
Our work does not assume any specific beamformer. We will provide general expressions that permit the study of the impact on performance and optimization of different choices of beamformers $\mathbf{F}^{(g)}[n]$ and signals $\mathbf{x}^{(g)}[n]$, although this is out of the scope of the paper. Our approach is also compatible with beam reference signal (initial access) procedures, and it could be complemented with a Bayesian recursive tracker with user-specific precoding.
\subsection{Channel Model}
Fig.~\ref{NLOS_Link} shows the position-related parameters of the channel. These parameters include
$\theta_{\mathrm{Rx},k}$, $\theta_{\mathrm{Tx},k}$, and $d_{k}=c\tau_{k}$, denoting the \ac{AOA}, \ac{AOD}, and the path length (with \ac{TOA} $\tau_{k}$ and the speed of light $c$) of the $k$-th path ($k=0$ for the \ac{LOS} path and $k>0$ the \ac{NLOS} paths). For each NLOS path, there is a scatterer with unknown location $\mathbf{s}_{k}$, for which we define $d_{k,1}=\Vert\mathbf{s}_{k}-\mathbf{q}\Vert_{2}$ and $d_{k,2}=\Vert\mathbf{p}-\mathbf{s}_{k}\Vert_{2}$. We now introduce the channel model, under a frequency-dependent array response \cite{widebandbrady}, suitable for wideband communication (with fractional bandwidth $B/f_c$ up to $50\%$). Assuming $K+1$ paths and a channel that remains constant during the transmission of $G$ symbols, the $N_r \times N_t$ channel matrix associated with subcarrier $n$ is expressed as
\begin{equation}\label{Channel1}
\mathbf{H}[n]=\mathbf{A}_{\mathrm{Rx}}[n]\mathbf{\Gamma}[n]\mathbf{A}^{\mathrm{H}}_{\mathrm{Tx}}[n],
\end{equation}
for response vectors
\begin{align}
\mathbf{A}_{\mathrm{Tx}}[n]& =[\mathbf{a}_{\mathrm{Tx},n}(\theta_{\mathrm{Tx},0}),\ldots,\mathbf{a}_{\mathrm{Tx},n}(\theta_{\mathrm{Tx},K})], \\
\mathbf{A}_{\mathrm{Rx}}[n]& =[\mathbf{a}_{\mathrm{Rx},n}(\theta_{\mathrm{Rx},0}),\ldots,\mathbf{a}_{\mathrm{Rx},n}(\theta_{\mathrm{Rx},K})],
\end{align}
and
\begin{align}
\mathbf{\Gamma}[n]=\sqrt{N_{t}N_{r}}\mathrm{diag}\left\{ \frac{h_{0}}{\sqrt{\rho_0}}e^{-j2\pi n\tau_{0}/(NT_{s})},\ldots,\frac{h_{K}}{\sqrt{\rho_K}}e^{-j2\pi n\tau_{K}/(NT_{s})}\right\},
\end{align}
for path loss $\rho_k$ and complex channel gain $h_k$, respectively, of the $k$-th path. For later use, we introduce $\tilde{h}_{k}=\sqrt{(N_{t}N_{r})/\rho_{k}}h_{k}$ and $\gamma_{n}(h_{k},\tau_{k})=\tilde{h}_{k}e^{-j2\pi n\tau_{k}/(NT_{s})}$.
The structure of the frequency-dependent antenna steering and response vectors $\mathbf{a}_{\mathrm{Tx},n}(\theta_{\mathrm{Tx},k})\in \mathbb{C}^{N_t}$ and $\mathbf{a}_{\mathrm{Rx},n}(\theta_{\mathrm{Rx},k})\in \mathbb{C}^{N_r}$ depends on the specific array structure. For the case of a \ac{ULA}, which will be the example studied in this paper, we recall that (the response vector $\mathbf{a}_{\mathrm{Rx},n}(\theta_{\mathrm{Rx},k})$ is obtained similarly)
\begin{align}\label{steringvector1}
& \mathbf{a}_{\mathrm{Tx},n}(\theta_{\mathrm{Tx},k}) =\\
&\frac{1}{\sqrt{N_{t}}}[
e^{-j \frac{N_{t}-1}{2}\frac{2\pi}{\lambda_{n}}d\sin(\theta_{\mathrm{Tx},k})},\ldots,e^{j\frac{N_{t}-1}{2}\frac{2\pi}{\lambda_{n}}d\sin(\theta_{\mathrm{Tx},k})}
]^{\mathrm{T}}, \nonumber
\end{align}
where $\lambda_{n} = c/(n/(NT_s)+f_{c})$ is the signal wavelength at the $n$-th subcarrier and $d$ denotes the distance between the antenna elements (we will use $d=\lambda_{c}/2$). We note that when $B\ll f_c$, $\lambda_{n} \approx \lambda_c$, and $\eqref{steringvector1}$ reverts to the standard narrow-band model.
\subsection{Received Signal Model}
The received signal for subcarrier $n$ and transmission $g$, after \ac{CP} removal and \acf{FFT}, can be expressed as
\begin{equation}
\mathbf{y}^{(g)}[n]=\mathbf{H}[n]\mathbf{F}^{(g)}[n]\mathbf{x}^{(g)}[n]+\mathbf{n}^{(g)}[n],\label{Receivedb1}
\end{equation}
where $\mathbf{n}^{(g)}[n]\in\mathbb{C}^{N_r}$ is a Gaussian noise vector with zero mean and variance $N_{0}/2$ per real dimension. Our goal is now to estimate the position $\mathbf{p}$ and orientation $\alpha$ of the \ac{MS} from $\{\mathbf{y}^{(g)}[n]\}_{\forall n, g}$. We will first derive a fundamental lower bound on the estimation uncertainty and then propose a novel practical estimator.
\section{Position and Orientation Estimation: Fundamental Bounds}\label{SEC:FundamentalBound}
In this section, we derive the \ac{FIM} and the \acf{CRB} for the estimation problem of position and orientation of the \ac{MS} for \ac{LOS}, \ac{NLOS}, and \ac{OLOS}. To simplify the notation and without loss of generality, we consider the case of $G=1$, i.e., only 1 OFDM symbol is transmitted.
\subsection{FIM Derivation for Channel Parameters}
Let $\boldsymbol{\eta}\in\mathbb{R}^{5(K+1)}$ be the vector consisting of the unknown channel parameters
\begin{equation}\label{Parameters1}
\boldsymbol{\eta}=\begin{bmatrix}\boldsymbol{\eta}^{\mathrm{T}}_{0},\ldots,\boldsymbol{\eta}^{\mathrm{T}}_{K}\end{bmatrix}^{\mathrm{T}},
\end{equation}
in which $\boldsymbol{\eta}_{k}$ consists of the unknown channel parameters (delay, \ac{AOD}, \ac{AOA}, and channel coefficients) for the $k$-th path
\begin{equation}\label{Parameters2}
\boldsymbol{\eta}_{k}=\begin{bmatrix}
\tau_{k},\boldsymbol{\theta}_{k}^{\mathrm{T}},\tilde{\mathbf{h}}_{k}^{\mathrm{T}}
\end{bmatrix}^{\mathrm{T}},
\end{equation}
where $\tilde{\mathbf{h}}_{k}=[\tilde{h}_{\mathrm{R},k},\tilde{h}_{\mathrm{I},k}]^{\mathrm{T}}$ contains the real and imaginary parts defined as $\tilde{h}_{\mathrm{R},k}$ and $\tilde{h}_{\mathrm{I},k}$, respectively, and $\boldsymbol{\theta}_{k}=\begin{bmatrix}\theta_{\mathrm{Tx},k},\theta_{\mathrm{Rx},k}\end{bmatrix}^{\mathrm{T}}$.
Defining $\hat{\boldsymbol{\eta}}$ as the unbiased estimator of $\boldsymbol{\eta}$, the mean squared error (MSE) is bounded as \cite{Kay}
\begin{equation}\label{Parameters3}
\mathbb{E}_{\mathbf{y}\vert\boldsymbol{\eta}}\left[(\hat{\boldsymbol{\eta}}-\boldsymbol{\eta})(\hat{\boldsymbol{\eta}}-\boldsymbol{\eta})^{\mathrm{T}}\right]\succeq\mathbf{J}^{-1}_{\boldsymbol{\eta}},
\end{equation}
in which $\mathbb{E}_{\mathbf{y}\vert\boldsymbol{\eta}}[.]$ denotes the expectation parameterized by the unknown parameters $\boldsymbol{\eta}$, and $\mathbf{J}_{\boldsymbol{\eta}}$ is the $5(K+1)\times 5(K+1)$ FIM defined as
\begin{equation}\label{Parameters4}
\mathbf{J}_{\boldsymbol{\eta}}\triangleq\mathbb{E}_{\mathbf{y}\vert\boldsymbol{\eta}}\left[-\frac{\partial^{2} \ln f(\mathbf{y}\vert\boldsymbol{\eta})}{\partial\boldsymbol{\eta}\partial\boldsymbol{\eta}^{T}}\right],
\end{equation}
where $f(\mathbf{y}\vert\boldsymbol{\eta})$ is the likelihood function of the random vector $\mathbf{y}$ conditioned on $\boldsymbol{\eta}$. More specifically, $f(\mathbf{y}\vert\boldsymbol{\eta})$ can be written as \cite{Poor}
\begin{equation}\label{Parameters5b}
\!\!f(\mathbf{y}\vert\boldsymbol{\eta})\!\propto\! \exp\!\left\{\!\frac{2}{N_{0}}\!\!\sum^{N-1}_{n=0}\Re\{\boldsymbol{\mu}^{\mathrm{H}}[n]\mathbf{y}[n]\}\!-\!\frac{1}{N_{0}}\!\!\sum^{N-1}_{n=0}\Vert\boldsymbol{\mu}[n]\Vert_{2}^{2}\!\right\}\!,
\end{equation}
where $\boldsymbol{\mu}[n]\triangleq\mathbf{H}[n]\mathbf{F}[n]\mathbf{x}[n]$ and $\propto$ denotes equality up to irrelevant constants.
The \ac{FIM} in \eqref{Parameters4} can be structured as
\begin{equation}\label{Parameters6w}
\mathbf{J}_{\boldsymbol{\eta}}=\begin{bmatrix}
\mathbf{\Psi}(\boldsymbol{\eta}_{0},\boldsymbol{\eta}_{0})&\ldots&\mathbf{\Psi}(\boldsymbol{\eta}_{0},\boldsymbol{\eta}_{K})\\
\vdots&\ddots&\vdots\\
\mathbf{\Psi}(\boldsymbol{\eta}_{K},\boldsymbol{\eta}_{0})&\ldots&\mathbf{\Psi}(\boldsymbol{\eta}_{K},\boldsymbol{\eta}_{K})
\end{bmatrix},
\end{equation}
in which $\mathbf{\Psi}(\mathbf{x}_{r},\mathbf{x}_{s})$ is defined as
\begin{equation}\label{Parameters6ww}
\mathbf{\Psi}(\mathbf{x}_{r},\mathbf{x}_{s})\triangleq\mathbb{E}_{\mathbf{y}\vert\boldsymbol{\eta}}\left[-\frac{\partial^{2} \ln f(\mathbf{y}\vert\boldsymbol{\eta})}{\partial \mathbf{x}_{r}\partial \mathbf{x}^{\mathrm{T}}_{s}}\right].
\end{equation}
The $5 \times 5$ matrix $\mathbf{\Psi}(\boldsymbol{\eta}_{r},\boldsymbol{\eta}_{s})$ is structured as
\begin{equation}\label{Parameters8w}
\mathbf{\Psi}(\boldsymbol{\eta}_{r},\boldsymbol{\eta}_{s})=\begin{bmatrix}
\Psi(\tau_{r},\tau_{s})&\mathbf{\Psi}(\tau_{r},\boldsymbol{\theta}_{s})&\mathbf{\Psi}(\tau_{r},\mathbf{h}_{s})\\
\mathbf{\Psi}(\boldsymbol{\theta}_{r},\tau_{s})&\mathbf{\Psi}(\boldsymbol{\theta}_{r},\boldsymbol{\theta}_{s})&\mathbf{\Psi}(\boldsymbol{\theta}_{r},\mathbf{h}_{s})\\
\mathbf{\Psi}(\mathbf{h}_{r},\tau_{s})&\mathbf{\Psi}(\mathbf{h}_{r},\boldsymbol{\theta}_{s})&\mathbf{\Psi}(\mathbf{h}_{r},\mathbf{h}_{s})
\end{bmatrix}.
\end{equation}
The entries of $\mathbf{\Psi}(\boldsymbol{\eta}_{r},\boldsymbol{\eta}_{s})$ are derived in Appendix \ref{elements}.
\subsection{FIM for Position and Orientation}\label{Trans_Convert}
We determine the FIM in the position space through a transformation of variables from $\boldsymbol{\eta}$ to $\tilde{\boldsymbol{\eta}}=\begin{bmatrix}\tilde{\boldsymbol{\eta}}^{\mathrm{T}}_{0},\ldots,\tilde{\boldsymbol{\eta}}^{\mathrm{T}}_{K}\end{bmatrix}^{\mathrm{T}}$, where $\tilde{\boldsymbol{\eta}}_{k}=\begin{bmatrix}\mathbf{s}^{\mathrm{T}}_{k},\tilde{\mathbf{h}}^{\mathrm{T}}_{k}\end{bmatrix}^{\mathrm{T}}$ for $k > 0$ and $\tilde{\boldsymbol{\eta}}_{0}=\begin{bmatrix}\mathbf{p}^{\mathrm{T}},\alpha,\tilde{\mathbf{h}}^{\mathrm{T}}_{0}\end{bmatrix}^{\mathrm{T}}$. If the \ac{LOS} path is blocked (i.e., \ac{OLOS}), we note that we must consider $\boldsymbol{\eta}_{\mathrm{olos}}=[\boldsymbol{\eta}^{\mathrm{T}}_{1},\ldots,\boldsymbol{\eta}^{\mathrm{T}}_{K}]^{\mathrm{T}}$ and $\tilde{\boldsymbol{\eta}}_{\mathrm{olos}}=[\mathbf{p}^{\mathrm{T}},\alpha,\tilde{\boldsymbol{\eta}}^{\mathrm{T}}_{1},\ldots,\tilde{\boldsymbol{\eta}}^{\mathrm{T}}_{K}]^{\mathrm{T}}$.
The \ac{FIM} of $\tilde{\boldsymbol{\eta}}$ is obtained by means of the $(4K+5)\times 5(K+1)$ transformation matrix $\mathbf{T}$ as
\begin{equation}\label{TFIM1}
\mathbf{J}_{\tilde{\boldsymbol{\eta}}}=\mathbf{T}\mathbf{J}_{\boldsymbol{\eta}}\mathbf{T}^{\mathrm{T}},
\end{equation}
where
\begin{equation}\label{TFIM2}
\mathbf{T}\triangleq\frac{\partial\boldsymbol{\eta}^{\mathrm{T}}}{\partial\tilde{\boldsymbol{\eta}}}.
\end{equation}
The entries of $\mathbf{T}$ can be obtained by the relations between the parameters in $\boldsymbol{\eta}$ and $\tilde{\boldsymbol{\eta}}$ from the geometry of the problem shown in Fig. \ref{NLOS_Link} as:
\begin{align}
\tau_{0} & = \Vert\mathbf{p}-\mathbf{q}\Vert_{2}/c, \label{TFIM2xy1}\\
\tau_{k}& = \Vert\mathbf{q}-\mathbf{s}_{k}\Vert_{2}/c+\Vert\mathbf{p}-\mathbf{s}_{k}\Vert_{2}/c,\: k>0 \label{TFIM2xy1b}\\
\theta_{\mathrm{Tx},0} & = \arccos((p_{x}-q_{x})/\Vert\mathbf{p}-\mathbf{q}\Vert_{2}),\label{TFIM2xy2}\\
\theta_{\mathrm{Tx},k} & = \arccos((s_{k,x}-q_{x})/\Vert\mathbf{s}_{k}-\mathbf{q}\Vert_{2}),\: k>0\label{TFIM2xy3}\\
\theta_{\mathrm{Rx},k} & = \pi -\arccos((p_{x}-s_{k,x})/\Vert\mathbf{p}-\mathbf{s}_{k}\Vert_{2})-\alpha ,\: k>0\label{TFIM2xy4}\\
\theta_{\mathrm{Rx},0} & =\pi+\arccos((p_{x}-q_{x})/\Vert\mathbf{p}-\mathbf{q}\Vert_{2})-\alpha.\label{TFIM2xy5}
\end{align}
Consequently, we obtain
\begin{equation}\label{TFIM3}
\mathbf{T}=\begin{bmatrix}
\mathbf{T}_{0,0}&\ldots&\mathbf{T}_{K,0}\\
\vdots&\ddots&\vdots\\
\mathbf{T}_{0,K}&\ldots&\mathbf{T}_{K,K}
\end{bmatrix},
\end{equation}
in which $\mathbf{T}_{k,k'}$ is defined as
\begin{equation}\label{TFIM4}
\mathbf{T}_{k,k'}\triangleq\frac{\partial\boldsymbol{\eta}^{\mathrm{T}}_{k}}{\partial\tilde{\boldsymbol{\eta}}_{k'}}.
\end{equation}
For $k'\neq 0$, $\mathbf{T}_{k,k'}$ is obtained as
\begin{equation}\label{TFIM5}
\mathbf{T}_{k,k'}=\begin{bmatrix}
\partial\tau_{k}/\partial\mathbf{s}_{k'}&\partial\boldsymbol{\theta}^{\mathrm{T}}_{k}/
\partial\mathbf{s}_{k'}&\partial\tilde{\mathbf{h}}^{\mathrm{T}}_{k}/
\partial\mathbf{s}_{k'}\\
\partial\tau_{k}/\partial\tilde{\mathbf{h}}_{k'}&\partial\boldsymbol{\theta}^{\mathrm{T}}_{k}/
\partial\tilde{\mathbf{h}}_{k'}&\partial\tilde{\mathbf{h}}^{\mathrm{T}}_{k}/\partial\tilde{\mathbf{h}}_{k'}
\end{bmatrix},
\end{equation}
and $\mathbf{T}_{k,0}$ is obtained as
\begin{equation}\label{TFIM6}
\mathbf{T}_{k,0}=\begin{bmatrix}
\partial\tau_{k}/\partial\mathbf{p}&\partial\boldsymbol{\theta}^{\mathrm{T}}_{k}/
\partial\mathbf{p}&\partial\tilde{\mathbf{h}}^{\mathrm{T}}_{k}/
\partial\mathbf{p}\\
\partial\tau_{k}/
\partial\alpha&\partial\boldsymbol{\theta}^{\mathrm{T}}_{k}/
\partial\alpha&\partial\tilde{\mathbf{h}}^{\mathrm{T}}_{k}/
\partial\alpha\\
\partial\tau_{k}/\partial\tilde{\mathbf{h}}_{0}&\partial\boldsymbol{\theta}^{\mathrm{T}}_{k}/
\partial\tilde{\mathbf{h}}_{0}&\partial\tilde{\mathbf{h}}^{\mathrm{T}}_{k}/\partial\tilde{\mathbf{h}}_{0}
\end{bmatrix},
\end{equation}
where
\begin{align*}
\partial\tau_{0}/\partial\mathbf{p} & =\frac{1}{c}\begin{bmatrix}\cos(\theta_{\mathrm{Tx},0}),\sin(\theta_{\mathrm{Tx},0})\end{bmatrix}^{\mathrm{T}} ,\\
\partial\theta_{\mathrm{Tx},0}/\partial\mathbf{p} & =\frac{1}{\Vert\mathbf{p}-\mathbf{q}\Vert_{2}}\begin{bmatrix}-\sin(\theta_{\mathrm{Tx},0}), \cos(\theta_{\mathrm{Tx},0})\end{bmatrix}^{\mathrm{T}},\\
\partial\theta_{\mathrm{Rx},0}/\partial\mathbf{p} & =\frac{1}{\Vert\mathbf{p}-\mathbf{q}\Vert_{2}}\begin{bmatrix}-\sin(\theta_{\mathrm{Tx},0}), \cos(\theta_{\mathrm{Tx},0})\end{bmatrix}^{\mathrm{T}},\\
\partial\theta_{\mathrm{Rx},k}/\partial\alpha & =-1, k\ge 0\\
\end{align*}
\begin{align*}
\partial\tau_{k}/\partial\mathbf{p} & = \frac{1}{c}\begin{bmatrix}\cos(\pi-\theta_{\mathrm{Rx},k}),
-\sin(\pi-\theta_{\mathrm{Rx},k})\end{bmatrix}^{\mathrm{T}},\:k> 0\\
\partial\tau_{k}/\partial\mathbf{s}_{k} & = \frac{1}{c}\begin{bmatrix}\cos(\theta_{\mathrm{Tx},k})+\cos(\theta_{\mathrm{Rx},k}),
\sin(\theta_{\mathrm{Tx},k})+\sin(\theta_{\mathrm{Rx},k})\end{bmatrix}^{\mathrm{T}},\:k> 0\\
\partial\theta_{\mathrm{Tx},k}/\partial\mathbf{s}_{k}& =\frac{1}{\Vert\mathbf{s}_{k}-\mathbf{q}\Vert_{2}}\begin{bmatrix}-\sin(\theta_{\mathrm{Tx},k}),\cos(\theta_{\mathrm{Tx},k})\end{bmatrix}^{\mathrm{T}},\:k> 0\\
\partial\theta_{\mathrm{Rx},k}/\partial\mathbf{p}& = \frac{1}{\Vert\mathbf{p}-\mathbf{s}_{k}\Vert_{2}}\begin{bmatrix}\sin(\pi-\theta_{\mathrm{Rx},k}), \cos(\pi-\theta_{\mathrm{Rx},k})\end{bmatrix}^{\mathrm{T}},\: k > 0\\
\partial\theta_{\mathrm{Rx},k}/\partial\mathbf{s}_{k}& =-\frac{1}{\Vert\mathbf{p}-\mathbf{s}_{k}\Vert_{2}}\begin{bmatrix}\sin(\pi-\theta_{\mathrm{Rx},k}), \cos(\pi-\theta_{\mathrm{Rx},k})\end{bmatrix}^{\mathrm{T}},\:k> 0
\end{align*}
and $\partial\tilde{\mathbf{h}}^{\mathrm{T}}_{k}/\tilde{\mathbf{h}}_{k}=\mathbf{I}_{2}$ for $k \ge 0$. The rest of entries in $\mathbf{T}$ are zero.
\subsection{Bounds on Position and Orientation Estimation Error}
The \ac{PEB} is obtained by inverting $\mathbf{J}_{\tilde{\boldsymbol{\eta}}}$, adding the diagonal entries of the $2\times 2$ sub-matrix, and taking the root square as:
\begin{equation}\label{PEBexpr}
\mathrm{PEB}=\sqrt{\mathrm{tr}\left\{[\mathbf{J}^{-1}_{\tilde{\boldsymbol{\eta}}}]_{1:2,1:2}\right\}},
\end{equation}
and the \ac{REB} is obtained as:
\begin{equation}\label{REBexpr}
\mathrm{REB}=\sqrt{[\mathbf{J}^{-1}_{\tilde{\boldsymbol{\eta}}}]_{3,3}},
\end{equation}
where the operations $[.]_{1:2,1:2}$ and $[.]_{3,3}$ denote the selection of the first $2\times 2$ sub-matrix and the third diagonal entry of $\mathbf{J}^{-1}_{\tilde{\boldsymbol{\eta}}}$, respectively.
\subsection{The Effect of Multi-Path Components on Position and Orientation Estimation Error}
In this subsection, we discuss the effect of adding \ac{MPCs} for localization under different conditions.
As the number of antennas in the MS increases, the scalar product between steering vectors corresponding to different receive directions tends to vanish, i.e. $\vert\mathbf{a}^{\mathrm{H}}_{\mathrm{Rx},n}(\theta_{\mathrm{Rx},r})\mathbf{a}_{\mathrm{Rx},n}(\theta_{\mathrm{Rx},s})\vert\ll 1$ for $\theta_{\mathrm{Rx},r} \neq \theta_{\mathrm{Rx},s}$.
Also, increasing the number of antenna elements in the transmitter results in narrower beams and the spatial correlation between different beams is reduced. Moreover, as the system bandwidth
increases, the different \ac{MPCs} coming from different scatterers can be more easily resolved. In other words, the MPCs can be considered to be orthogonal \cite{LeitingerJSAC2015,WitrisalSPM2016}. Consequently, large $N_{t}$, $N_{r}$, and bandwidth lead to very small multipath cross-correlation terms in the \ac{FIM} \cite{DBLP:journals/corr/Abu-ShabanZASW17}.
Ignoring those terms, the approximate expression for the \ac{EFIM} of position and rotation angle $\mathbf{J}_{e}(\mathbf{p},\alpha)$ with large $N_{t}$, $N_{r}$, and bandwidth is\footnote{In computing \eqref{ans_com1}, we used the fact that the last two rows of $\mathbf{T}_{k,0}$ are zero for $k\neq 0$.}
\begin{equation}\label{ans_com1}
\mathbf{J}_{e}(\mathbf{p},\alpha)\approx\tilde{\mathbf{T}}_{0,0}\mathbf{\Lambda}_{e,0}\tilde{\mathbf{T}}^{\mathrm{T}}_{0,0}+
\sum_{k=1}^{K}\left[\mathbf{\Upsilon}_{e,k}\right]_{1:3,1:3},
\end{equation}
where
\begin{equation}\label{ans_com2}
\mathbf{\Upsilon}_{e,k}=\mathbf{T}_{k,0}\mathbf{\Psi}(\boldsymbol{\eta}_{k},\boldsymbol{\eta}_{k})\mathbf{T}^{\mathrm{T}}_{k,0}-\mathbf{T}_{k,0}\mathbf{\Psi}(\boldsymbol{\eta}_{k},\boldsymbol{\eta}_{k})\mathbf{T}^{\mathrm{T}}_{k,k}\left(\mathbf{T}_{k,k}\mathbf{\Psi}(\boldsymbol{\eta}_{k},\boldsymbol{\eta}_{k})\mathbf{T}^{\mathrm{T}}_{k,k}\right)^{-1}
\mathbf{T}_{k,k}\mathbf{\Psi}(\boldsymbol{\eta}_{k},\boldsymbol{\eta}_{k})\mathbf{T}^{\mathrm{T}}_{k,0},
\end{equation}
in which $\tilde{\mathbf{T}}_{0,0}$ is the $3\times 3$ sub-matrix in the transformation matrix $\mathbf{T}_{k,0}$ for $k=0$ in \eqref{TFIM6} containing the derivatives with respect to $\mathbf{p}$ and $\alpha$, $[.]_{1:3,1:3}$ denotes the selection of the first $3\times 3$ sub-matrix, and $\mathbf{\Lambda}_{e,0}$ denotes the EFIM of the delay, AOD, and AOA from LOS, i.e., $\{\tau_{0},\theta_{\mathrm{Tx},0},\theta_{\mathrm{Rx},0}\}$. From simulations, it is observed that the exact and approximate FIM lead to nearly identical PEBs, under the mentioned conditions.
Hence, greedy techniques from compressed sensing, which extract path after path, are a natural tool for such scenarios.
In the LOS case, \eqref{ans_com1} only contains the term corresponding to $k=0$, i.e., the first term. When MPCs are present, the terms corresponding to $k\geq 1$ appear, i.e., the second summand in \eqref{ans_com1}, which contains terms that are added and others that are subtracted (because the scatterer location is an additional parameter that has to be estimated for each MPC \cite[eq. (3.59)]{LeitingerPhD2016}). The additive terms imply that the presence of MPCs help in the estimation of the MS localization, as they add information to the EFIM. In general the contribution of the MPCs results in a positive contribution to the FIM, and hence in a reduction of the CRB as shown in papers \cite{LeitingerJSAC2015,WitrisalSPM2016}. It is only in the cases where the MPCs heavily overlap, specially with the LOS, in the directional and time domains that the negative terms are dominant, and then the presence of MPCs degrades the MS localization.
\section{Position and Orientation Estimation: Estimator in Beamspace}
Next, we propose the use of a beamspace channel transformation in order to estimate the channel parameters in \eqref{Receivedb1}. The considered beamspace representation of the channel reduces the complexity by exploiting the sparsity of the \ac{mm-wave} MIMO channel. If the fractional bandwidth and the number of antennas are not violating the condition for the small array dispersion \cite{widebandbrady}, there exists a common sparse support across all subcarriers. Consequently, the \ac{DCS-SOMP} method from \cite{Duarte2} can be applied for the estimation of \ac{AOA}, \ac{AOD}, and \ac{TOA}. As the estimates of \ac{AOA} and \ac{AOD} are limited to lie on a grid defined by the transformation, we apply a refinement of the estimates of all parameters using the \ac{SAGE} algorithm. Finally, we invoke the \ac{EXIP} to solve for the position $\mathbf{p}$ and orientation $\alpha$.
\subsection{Beamspace Channel Representation}
We introduce the $N_t \times N_t$ transformation matrix, uniformly sampling the virtual spatial angles \cite{BspaceSayeedx}
\begin{align*}
\mathbf{U}_{\mathrm{Tx}}&\triangleq\left[\mathbf{u}_{\mathrm{Tx}}({-(N_{t}-1)/2}),\ldots,\mathbf{u}_{\mathrm{Tx}}({(N_{t}-1)/2})\right],\\
\mathbf{u}_{\mathrm{Tx}}(p)&\triangleq \begin{bmatrix}
e^{-j2\pi \frac{N_{t}-1}{2} \frac{p}{{N_{t}}}},\ldots,e^{j2\pi \frac{N_{t}-1}{2} \frac{p}{N_{t}}}
\end{bmatrix}^{\mathrm{T}},
\end{align*}
where we assumed $N_t$ to be even.
Similarly, we define the $N_r \times N_r$ matrix $\mathbf{U}_{\mathrm{Rx}}$. Both $\mathbf{U}_{\mathrm{Tx}}$ and $\mathbf{U}_{\mathrm{Rx}}$ are unitary matrices. The partial virtual representation of the channel with respect to the angular domain can be written as
\begin{align}
\check{\mathbf{H}}[n]&=\mathbf{U}_{\mathrm{Rx}}^{\mathrm{H}}\mathbf{H}[n]\mathbf{U}_{\mathrm{Tx}}\label{BWTransceiver1a}\\
& = \sum_{k=0}^{K}\gamma_{n}(h_{k},\tau_{k})\mathbf{U}_{\mathrm{Rx}}^{\mathrm{H}}\mathbf{a}_{\mathrm{Rx},n}(\theta_{\mathrm{Rx},k})\mathbf{a}^{\mathrm{H}}_{\mathrm{Tx},n}(\theta_{\mathrm{Tx},k})\mathbf{U}_{\mathrm{Tx}}.
\end{align}
It is readily verified that \cite{widebandbrady}
\begin{align}
[\check{\mathbf{H}}[n]]_{i,i'}&=\sum_{k=0}^{K}\gamma_{n}(h_{k},\tau_{k})\chi_{r}\big(\frac{d}{\lambda_{n}}\sin(\theta_{\mathrm{Rx},k})-\frac{i}{N_{r}}\big)\chi_{{t}}\big(\frac{d}{\lambda_{n}}\sin(\theta_{\mathrm{Tx},k})-\frac{i'}{N_{t}}\big),\label{BWTransceiver1b}
\end{align}
for $-(N_r-1)/2 \le i \le (N_r-1)/2$ and $-(N_t-1)/2 \le i' \le (N_t-1)/2$. We have introduced
\begin{align}
\chi_t(\phi) & = \frac{\sin(\pi N_{t}\phi)}{\sqrt{N_{t}}\sin(\pi\phi)},\\
\chi_r(\phi) & = \frac{\sin(\pi N_{r}\phi)}{\sqrt{N_{r}}\sin(\pi\phi)}.
\end{align}
From \eqref{BWTransceiver1b}, it is observed that $\check{\mathbf{H}}[n]$ is approximately sparse, since `strong' components are only present in the directions of $\{\theta_{\mathrm{Tx},k}\}$ and $\{\theta_{\mathrm{Rx},k}\}$.
Stacking the observation $\mathbf{y}^{(g)}[n]$ from \eqref{Receivedb1}, we obtain
\begin{equation}\label{BWTransceiver2x}
\check{\mathbf{y}}[n]=\mathbf{\Omega}[n]\check{\mathbf{h}}[n]+\check{\mathbf{n}}[n],
\end{equation}
where
\begin{align}
\mathbf{\Omega}[n]&=\begin{bmatrix}
\mathbf{\Omega}^{(1)}[n]\\\vdots\\\mathbf{\Omega}^{(G)}[n]
\end{bmatrix},\\
\mathbf{\Omega}^{(g)}[n]&=(\mathbf{Z}^{(g)}_{\mathrm{Tx}}[n])^{\mathrm{T}}\otimes\mathbf{U}_{\mathrm{Rx}},\\
\mathbf{Z}^{(g)}_{\mathrm{Tx}}[n]&=\mathbf{U}^{\mathrm{H}}_{\mathrm{Tx}}\mathbf{F}^{(g)}[n]\mathbf{x}^{(g)}[n],\\
\check{\mathbf{h}}[n] &= \mathrm{vec}(\check{\mathbf{H}}[n]).
\end{align}
Hence, since $\check{\mathbf{h}}[n]$ is an approximately sparse vector, we can interpret solving \eqref{BWTransceiver2x} for $\check{\mathbf{h}}[n]$ as a {CS} problem, allowing us to utilize tools from that domain. In principle, the columns of $\mathbf{U}_{\mathrm{Tx}}$ and $\mathbf{U}_{\mathrm{Rx}}$ corresponding to non-zero entries of the sparse vector $\check{\mathbf{h}}[n]$ correspond to coarse estimates of the AOA/AOD, while the entries in $\check{\mathbf{h}}[n]$ are estimates of $\gamma_{n}(h_{k},\tau_{k})$ (including the effect of the functions $\chi_t(\cdot)$ and $\chi_r(\cdot)$). The latter values can then be used to estimate $\tau_{k}$ for each path. Since the vectors $\check{\mathbf{h}}[n]\in\mathbb{C}^{N_{r}N_{t}\times 1}$, for $i=1,\ldots,N$, corresponding to the sensing matrix $\mathbf{\Omega}[n]$ in \eqref{BWTransceiver2x} are approximately jointly $(K+1)$-sparse, i.e., the support of $\check{\mathbf{h}}[n]$ does not vary significantly from subcarrier to subcarrier, we can use specialized techniques, such as \ac{DCS-SOMP} for estimating all $\check{\mathbf{h}}[n]$ jointly in an efficient manner.
Based on the above discussion, we propose to use the following approach:
\begin{enumerate}
\item Coarse estimation of AOA/AOD using a modified \ac{DCS-SOMP} algorithm.
\item Fine estimation using the SAGE algorithm, initialized by the coarse estimates.
\item Estimation of the position and orientation.
\end{enumerate}
\subsubsection*{Remark}
The above sparse representation is not unique. Another representation could rely on a sparse vector of length $N_{t}\times N_{r}\times N$, where each entry would then correspond to an AOA/AOD/TOA triplet. However, the complexity of such an approach would be significantly higher, since $N$ is generally a large number.
\subsection{Step 1: Coarse Estimation of Channel Parameters using DCS-SOMP}\label{EstRef}
The first stage of the algorithm involves calling the DCS-SOMP algorithm, providing estimates of the number of paths, the AOA/AOD, and estimates of $\check{\mathbf{h}}[n]$. For the sake of completeness, the steps of DCS-SOMP can be found in Algorithm \ref{algor0_det}. We note that the algorithm is rank-blind as it does not assume knowledge of the number of the paths (i.e., $K+1$) \cite{Davies}. Since $K+1$ is unknown, we use the change of residual fitting error $\sum_{n=0}^{N-1}\Vert\mathbf{r}_{t-1}[n]-\mathbf{r}_{t-2}[n]\Vert^{2}_{2}$ at each iteration $t$ to a threshold $\delta$. The value for $\delta$ is obtained using a similar procedure as in \cite{Marzi}:
\begin{equation}\label{reqproof3}
\delta=N_0\gamma^{-1}\left(N,\Gamma(N)(1-{P}_{\mathrm{fa}})^{{1}/({N_{r}N_{t}})}\right),
\end{equation}
in which $\gamma^{-1}\left(N,x\right)$ denotes the inverse of the incomplete gamma distribution, $\Gamma(N)$ is the gamma function, and ${P}_{\mathrm{fa}}$ is the false alarm probability.
\begin{algorithm}[h!]
\caption{Modified DCS-SOMP\label{algor0_det}}
\textbf{Input:} Recieved signals $\check{\mathbf{y}}[n]$, sensing matrix $\mathbf{\Omega}[n]$, and the threshold $\delta$.\\
\textbf{Output:} estimates of $K$, ${\theta}_{\mathrm{Tx},k}$, ${\theta}_{\mathrm{Rx},k}$, $\check{\mathbf{h}}[n]$, $n=0,\ldots,N-1.$
\begin{algorithmic}[1]
\STATE For $n=0,\ldots,N-1$, the residual vectors are set to $\mathbf{r}_{-1}[n]=\mathbf{0}$ and $\mathbf{r}_{0}[n]=\check{\mathbf{y}}[n]$, the
orthogonalized coefficient vector $\hat{\boldsymbol{\beta}}_{n}=\mathbf{0}$, $\mathcal{K}_{0}$ is chosen to be an empty set, and iteration index $t=1$. $\boldsymbol{\omega}_{m}[n]$ is the $m$-th column of measurement matrix $\mathbf{\Omega}[n]$.
\WHILE{ $\sum_{n=0}^{N-1}\Vert\mathbf{r}_{t-1}[n]-\mathbf{r}_{t-2}[n]\Vert^{2}_{2}>\delta$}
\STATE Find AOA/AOD pair
\begin{align}
\tilde{n}_{t}& =\underset{m=1,\ldots,N_{r}N_{t}}
{\mathrm{argmax}} \:\sum_{n=0}^{N-1}\frac{\vert
\boldsymbol{\omega}_{m}^{\mathrm{H}}[n]\mathbf{r}_{t-1}[n]\vert}{\Vert\boldsymbol{\omega}_{m}[n]\Vert_{2}},\label{tinex1ee}\\
n_{\mathrm{Tx},t}&=\lceil \tilde{n}_{t}/N_{r}\rceil, ~~
n_{\mathrm{Rx},t}=\mathrm{mod}(\tilde{n}_{t}-1,N_{r})+1,\\
\hat{\theta}^{(0)}_{\mathrm{Tx},t} & =\arcsin\left((\lambda_{c}/d)(n_{\mathrm{Tx},t}-{(N_{t}-1)/2}-1)/N_{t}\right),\label{tinex1}\\
\hat{\theta}^{(0)}_{\mathrm{Rx},t}& =\arcsin\left((\lambda_{c}/d)(n_{\mathrm{Rx},t}-{(N_{r}-1)/2}-1)/N_{r}\right).\label{rinex1}
\end{align}
\STATE Update AOA/AOD set of indices $\mathcal{K}_{t}=\mathcal{K}_{t-1}\cup\{\tilde{n}\}$.
\STATE Orthogonalize the selected basis vector:
\begin{equation}\label{BWTransceiver2zfh}
\boldsymbol{\rho}_{t}[n]=\boldsymbol{\omega}_{\tilde{n}_{t}}[n]-\sum_{\tilde{t}=0}^{t-1}
\frac{\boldsymbol{\omega}^{\mathrm{H}}_{\tilde{n}_{t}}[n]\boldsymbol{\rho}_{\tilde{t}}[n]}{\Vert\boldsymbol{\rho}_{\tilde{t}}[n]\Vert_{2}}\boldsymbol{\rho}_{\tilde{t}}[n].
\end{equation}
\STATE Update the residual vector $\mathbf{r}_{t}[n]$ by subtracting the effect of chosen columns from $\mathbf{r}_{t-1}[n]$: $\mathbf{r}_{t}[n]=\mathbf{r}_{t-1}[n]-\hat{\beta}_{n}(t)\boldsymbol{\rho}_{t}[n]$,
where
\begin{equation}\label{BWTransceiver2zdyb}
\hat{\beta}_{n}(t)=\frac{\boldsymbol{\rho}^{\mathrm{H}}_{t}[n]\mathbf{r}_{t-1}[n]}{\Vert\boldsymbol{\rho}_{t}[n]\Vert^{2}_{2}}.
\end{equation}
\STATE $t=t+1$.
\ENDWHILE
\STATE Perform QR factorization of the mutilated basis $\mathbf{\Omega}_{\mathcal{K}_{t}}[n]=[\boldsymbol{\omega}_{\tilde{n}_{1}}[n],\ldots,\boldsymbol{\omega}_{\tilde{n}_{\hat{K}+1}}[n]]=\mathbf{\Upsilon}[n]\mathbf{R}[n]$ where $\mathbf{\Upsilon}[n]=[\boldsymbol{\rho}_{1}[n],\ldots,\boldsymbol{\rho}_{\hat{K}+1}[n]]$ and $\mathbf{R}[n]$ is an upper triangular matrix. Since $\mathbf{\Omega}_{\mathcal{K}_{t}}[n]\hat{\check{\mathbf{h}}}[n]=\mathbf{\Upsilon}[n]\mathbf{R}[n]\hat{\check{\mathbf{h}}}[n]=\mathbf{\Upsilon}[n]\hat{\boldsymbol{\beta}}_{n}$, we obtain
\begin{equation}\label{BWTransceiver2z}
\hat{\check{\mathbf{h}}}[n]=\mathbf{R}^{-1}[n]\hat{\boldsymbol{\beta}}_{n}.
\end{equation}
\end{algorithmic}
\end{algorithm}
For each path $k = 0,\ldots, \hat{K}$, we can now write
\begin{equation}\label{BWTransceiver2tz0}
\hat{\check{\mathbf{h}}}^{(k)}=\tilde{h}_{k}\mathbf{A}(\tau_{k})\mathbf{z}^{(k)}+\mathbf{v}^{(k)},
\end{equation}
where $\hat{\check{\mathbf{h}}}^{(k)}=[\hat{\check{h}}^{(k)}[0],\ldots,\hat{\check{h}}^{(k)}[N-1]]^{\mathrm{T}}$ in which $\hat{\check{h}}^{(k)}[n]$ is the entry on subcarrier $n$, related to the $k$-th path found in Algorithm \ref{algor0_det}, $\mathbf{A}(\tau_{k})=\mathrm{diag}\{1,\ldots,e^{-j2\pi (N-1)\tau_{k}/(NT_{s})}\}$, $\mathbf{v}_{k}$ is the $N\times 1$ noise vector, and $\mathbf{z}^{(k)}$ has entries
\begin{equation}\label{Refine2}
z_{n}(k)\triangleq\mathbf{u}^{\mathrm{H}}_{\mathrm{Rx}}(\frac{n_{\mathrm{Rx},k}-{(N_{r}-1)/2}-1}{N_{r}})\mathbf{a}_{\mathrm{Rx},n}(\hat{\theta}^{(0)}_{\mathrm{Rx},k})\mathbf{a}^{\mathrm{H}}_{\mathrm{Tx},n}(\hat{\theta}^{(0)}_{\mathrm{Tx},k})\mathbf{u}_{\mathrm{Tx}}(\frac{n_{\mathrm{Tx},k}-{(N_{t}-1)/2}-1}{N_{t}}).
\end{equation}
For the purpose of coarse estimation, we ignore the dependence on $n$ in \eqref{Refine2}, leading to the simple model
\begin{align}
\hat{\check{\mathbf{h}}}^{(k)}=\tilde{h}_{k}{z}^{(k)}\mathbf{a}(\tau_k)+\mathbf{v}^{(k)},
\end{align}
where $\mathbf{a}(\tau_k) = [1,\ldots,e^{-j2\pi (N-1)\tau_{k}/(NT_{s})}]^{\mathrm{T}}$ and ${z}^{(k)}$ is as in \eqref{Refine2}, but considering only $\lambda_c$ instead of $\lambda_n$. From this model, we can recover $\tau_{k}$ and $\tilde{h}_{k}$ by solving a \ac{LS} problem
\begin{equation}\label{BWTransceiver2tzsf}
[\hat{\tau}^{(0)}_{k},\hat{\tilde{h}}^{(0)}_{k}]=\underset{\tau_{k},\tilde{h}_{k}}{\mathrm{argmin}}\:\:\Vert\hat{\check{\mathbf{h}}}^{(k)}-\tilde{h}_{k}{z}^{(k)}\mathbf{a}(\tau_k)\Vert_{2}^{2}.
\end{equation}
Solving for ${\tilde{h}}_{k}$ yields
\begin{align}\label{BWTransceiver2tzsfzxe2}
\hat{\tilde{h}}^{(0)}_{k}= \frac{\mathbf{a}^{\mathrm{H}}(\tau_k)\hat{\check{\mathbf{h}}}^{(k)}}{{z}^{(k)}N}.
\end{align}
Substituting \eqref{BWTransceiver2tzsfzxe2} into \eqref{BWTransceiver2tzsf} and expanding the square allows us to solve for ${\tau}_{k}$:
\begin{align}\label{BWTransceiver2tzsfzxe3}
\hat{\tau}^{(0)}_{k}=\underset{\tau_{k}}{\mathrm{argmax}}\:\: |\mathbf{a}^{\mathrm{H}}(\tau_k)\hat{\check{\mathbf{h}}}^{(k)}|^2.
\end{align}
\subsection{Step 2: Fine Estimation of Channel Parameters using SAGE}
Channel parameter estimates are refined in an iterative procedure, which is initialized by the estimates from step 1. In principle, we can perform an iterative ascent algorithm directly on the log-likelihood function associated with the model \eqref{BWTransceiver2x}. However, this requires a multi-dimensional minimization and computationally complex solutions. A more practical approach is to use the \ac{SAGE} algorithm with the incomplete data space in \eqref{BWTransceiver2x} as the superposition of $K+1$ complete data space $\check{\mathbf{y}}_{k}[n]$ as:
\begin{equation}\label{Refine1}
\check{\mathbf{y}}[n]=\sum_{k=0}^{\hat{K}}\underbrace{\mathbf{\Omega}[n]\check{\mathbf{h}}_{k}[n]+\check{\mathbf{n}}_{k}[n]}_{\check{\mathbf{y}}_{k}[n]},
\end{equation}
where $\check{\mathbf{h}}_{k}[n]$ denotes the vectorized form of $\check{\mathbf{H}}_{k}[n]=\mathbf{U}_{\mathrm{Rx}}^{\mathrm{H}}\mathbf{H}_{k}[n]\mathbf{U}_{\mathrm{Tx}}$ with $\mathbf{H}_{k}[n]$ being the corresponding term for the $k$-th path in the channel frequency response $\mathbf{H}[n]$ in \eqref{Channel1}. Writing \eqref{Refine1} for all the subcarriers results in:
\begin{equation}\label{Refine1x}
\check{\mathbf{y}}=\sum_{k=0}^{\hat{K}}\underbrace{\check{\mathbf{\Omega}}\check{\mathbf{h}}_{k}
+\check{\mathbf{n}}_{k}}_{\check{\mathbf{y}}_{k}},
\end{equation}
where
\begin{align*}
\check{\mathbf{\Omega}}&=\mathrm{diag}\left\{\mathbf{\Omega}[0],\ldots,\mathbf{\Omega}[N-1]\right\},\\
\check{\mathbf{y}}&=\left[\check{\mathbf{y}}^{\mathrm{T}}[0],\ldots,\check{\mathbf{y}}^{\mathrm{T}}[N-1]\right]^{\mathrm{T}},\\
\check{\mathbf{h}}_{k}&=\left[\check{\mathbf{h}}^{\mathrm{T}}_{k}[0],\ldots,\check{\mathbf{h}}^{\mathrm{T}}_{k}[N-1]\right]^{\mathrm{T}},\\
\check{\mathbf{n}}_{k}&=\left[\check{\mathbf{n}}_{k}^{\mathrm{T}}[0],\ldots,\check{\mathbf{n}}_{k}^{\mathrm{T}}[N-1]\right]^{\mathrm{T}}.
\end{align*}
In the $(m+1)$-th iteration where $m$ is the iteration index, the expectation and maximization steps are performed as described below. For the initialization of the iterative procedure, we use the AOA/AOD, TOA, and channel coefficients from the detection phase using $\hat{\theta}^{(0)}_{\mathrm{Tx},k}$ and $\hat{\theta}^{(0)}_{\mathrm{Rx},k}$ obtained from \eqref{tinex1} and \eqref{rinex1}, respectively, $\hat{\tau}^{(0)}_{k}$ computed from \eqref{BWTransceiver2tzsfzxe3}, and the corresponding coefficient obtained from \eqref{BWTransceiver2tzsfzxe2}.
\subsubsection*{Expectation step} We compute the conditional expectation of the hidden data space $\check{\mathbf{y}}_{k}$ log-likelihood function based on the previous estimation $\hat{\boldsymbol{\eta}}^{(m)}$ and the incomplete data space $\check{\mathbf{y}}$ as:
\begin{equation}\label{ExpectationS1}
Q(\boldsymbol{\eta}_{k}\vert\hat{\boldsymbol{\eta}}^{(m)})\triangleq \mathbb{E}\left[\ln f(\check{\mathbf{y}}_{k}\vert\boldsymbol{\eta}_{k},\{\hat{\boldsymbol{\eta}}^{(m)}_{l}\}_{l\neq k})\vert\check{\mathbf{y}},\hat{\boldsymbol{\eta}}^{(m)}\right].
\end{equation}
For $k=0,\ldots,\hat{K}$, we obtain
\begin{equation}\label{ExpectationS2}
Q(\boldsymbol{\eta}_{k}\vert\hat{\boldsymbol{\eta}}^{(m)})\propto -\Vert\hat{\mathbf{z}}^{(m)}_{k}-\check{\boldsymbol{\mu}}(\boldsymbol{\eta}_{k})\Vert_{2}^{2},
\end{equation}
where $\check{\boldsymbol{\mu}}(\boldsymbol{\eta}_{k})=\check{\mathbf{\Omega}}\check{\mathbf{h}}_{k}$, and
\begin{equation}\label{ExpectationS3}
\hat{\mathbf{z}}^{(m)}_{k}=\check{\mathbf{y}}-\sum_{l\neq k, l=0}^{\hat{K}}\check{\boldsymbol{\mu}}(\hat{\boldsymbol{\eta}}^{(m)}_{l}).
\end{equation}
\subsubsection*{Maximization step} The goal is to find $\boldsymbol{\eta}_{k}$ such that \eqref{ExpectationS2} is maximized. In other words, we have
\begin{equation}\label{MaximizationS1}
\hat{\boldsymbol{\eta}}^{(m+1)}_{k}=\underset{\boldsymbol{\eta}_{k}}{\mathrm{argmax}}\:\:Q(\boldsymbol{\eta}_{k}\vert\hat{\boldsymbol{\eta}}^{(m)}).
\end{equation}
Solving \eqref{MaximizationS1} directly for $\boldsymbol{\eta}_{k}$ is analytically complex due to the fact that it is hard to compute the gradient and Hessian with respect to $\boldsymbol{\eta}_{k}$. Instead, we update the parameters $\hat{\theta}^{(m+1)}_{\mathrm{Tx},k}$, $\hat{\theta}^{(m+1)}_{\mathrm{Rx},k}$, $\hat{\tau}^{(m+1)}_{k}$, and $\hat{\tilde{h}}^{(m+1)}_{k}$ sequentially using Gauss-Seidel-type iterations \cite{Ortega}.
\subsection{Step 3: Conversion to Position and Rotation Angle Estimates}
As a final step, based on the refined estimates of AOA/AOD/TOA from step 2, here we show how the position and orientation of the MS is recovered. Four scenarios are considered: LOS, NLOS, OLOS, and unknown condition.
\begin{itemize}
\item \textbf{LOS:} When $\hat{K}=1$ and we are in LOS condition, the expressions \eqref{TFIM2xy1}, \eqref{TFIM2xy2}, and \eqref{TFIM2xy5} describe a mapping $\boldsymbol{\eta} = \boldsymbol{f}_{\mathrm{los}}(\tilde{\boldsymbol{\eta}})$. The classical invariance principle of estimation theory is invoked to prove the equivalence of minimizing the \ac{ML} criterion in terms of either $\boldsymbol{\eta}_{0}$ or $\tilde{\boldsymbol{\eta}}_{0}$ \cite{Zacks}. Consequently, the estimated values of $\hat{\mathbf{p}}$ and $\hat{\alpha}$ are obtained directly from
\begin{align}\label{EXIP2}
\hat{\mathbf{p}}&=\mathbf{q}+c\hat{\tau}_{0}[\cos(\hat{\theta}_{\mathrm{Tx},0}),\sin(\hat{\theta}_{\mathrm{Tx},0})]^{\mathrm{T}},\\
\hat{\alpha}&=\pi+\hat{\theta}_{\mathrm{Tx},0}-\hat{\theta}_{\mathrm{Rx},0}.
\end{align}
\item \textbf{NLOS:} For the case with $\hat{K}$ scatterers and a \ac{LOS} path, the \ac{EXIP} can be used, as \eqref{TFIM2xy1}--\eqref{TFIM2xy5} describe a mapping $\boldsymbol{\eta} = \boldsymbol{f}_{\mathrm{nlos}}(\tilde{\boldsymbol{\eta}})$. Consequently, the estimated $\hat{\tilde{\boldsymbol{\eta}}}$ obtained as
\begin{equation}\label{EXIP4}
\hat{\tilde{\boldsymbol{\eta}}}=\underset{\tilde{\boldsymbol{\eta}}}{\mathrm{argmin}}\:
\underbrace{\left(\hat{\boldsymbol{\eta}}-\boldsymbol{f}_{\mathrm{nlos}}(\tilde{\boldsymbol{\eta}})\right)^{\mathrm{T}}\mathbf{J}_{\hat{\boldsymbol{\eta}}}\left(\hat{\boldsymbol{\eta}}-\boldsymbol{f}_{\mathrm{nlos}}(\tilde{\boldsymbol{\eta}})\right)}_{v_{\mathrm{nlos}}(\tilde{\boldsymbol{\eta}})},
\end{equation}
is asymptotically (w.r.t.~$G\times N$) equivalent to the ML estimate of the transformed parameter $\tilde{\boldsymbol{\eta}}$ \cite{Stoicapp,Swindlehurstt}. Note that $\mathbf{J}_{\boldsymbol{\eta}}$ could be replaced by the identity matrix, leading also to a meaningful estimator of $\tilde{\boldsymbol{\eta}}$, although with probably slightly larger \ac{RMSE}.
The \ac{LMA} can be used to solve \eqref{EXIP4} \cite{Levenberg,Marquardt}, initialized as follows:
we first estimate $\hat{\mathbf{p}}$ and $\hat{\alpha}$ from the \ac{LOS} path (i.e., the path with the smallest delay). Then, for the first-order reflection $\hat{\mathbf{s}}_{k}$
can be obtained by the intersection of the following two lines: $\tan(\pi-(\hat{\theta}_{\mathrm{Rx},k}+\hat{\alpha}))=(\hat{p}_{y}-s_{1,y})/(\hat{p}_{x}-s_{1,x})$ and $\tan(\hat{\theta}_{\mathrm{Tx},k})=(s_{1,y}-q_{y})/(s_{1,x}-q_{x})$.
\item \textbf{OLOS:} For the case with $\hat{K}$ scatterers and no LOS path, the \ac{EXIP} could be used, as \eqref{TFIM2xy1b}, \eqref{TFIM2xy3}, and \eqref{TFIM2xy4} describe a mapping $\boldsymbol{\eta}_{\mathrm{olos}} = \boldsymbol{f}_{\mathrm{olos}}(\tilde{\boldsymbol{\eta}}_{\mathrm{olos}})$.
Consequently, the estimated $\hat{\tilde{\boldsymbol{\eta}}}_{\mathrm{olos}}$ obtained as
\begin{equation}\label{EXIP4b}
\hat{\tilde{\boldsymbol{\eta}}}_{\mathrm{olos}}=\underset{\tilde{\boldsymbol{\eta}}_{\mathrm{olos}}}{\mathrm{argmin}}\:
\underbrace{\left(\hat{\boldsymbol{\eta}}_{\mathrm{olos}}-\boldsymbol{f}_{\mathrm{olos}}(\tilde{\boldsymbol{\eta}}_{\mathrm{olos}})\right)^{\mathrm{T}}\mathbf{J}_{\hat{\boldsymbol{\eta}}_{\mathrm{olos}}}\left(\hat{\boldsymbol{\eta}}_{\mathrm{olos}}-\boldsymbol{f}_{\mathrm{olos}}(\tilde{\boldsymbol{\eta}}_{\mathrm{olos}})\right)}_{v_{\mathrm{olos}}(\tilde{\boldsymbol{\eta}}_{\mathrm{olos}})},
\end{equation}
is asymptotically equivalent to the ML estimate of the transformed parameter $\tilde{\boldsymbol{\eta}}_{\mathrm{olos}}$ where $\mathbf{J}_{\hat{\boldsymbol{\eta}}_{\mathrm{olos}}}$ denotes the \ac{FIM} of $\boldsymbol{\eta}_{\mathrm{olos}}$. The estimated parameters from the \ac{NLOS} links could be used to initialize $\tilde{\boldsymbol{\eta}}_{\mathrm{olos}}$ for the application of the \ac{LMA} algorithm. The process is slightly more involved than under NLOS. We consider different trial values of $\alpha$, with a resolution $\Delta \alpha$ over a range $[-\alpha_m,+\alpha_m]$ of possible rotation values. For each trial value $\hat{\alpha}_{\mathrm{trial}}$, we can find a corresponding estimate of $\mathbf{p}$. For instance, by solving a set of linear equations for two paths:
\begin{equation}\label{EXIP5b}
\mathbf{p}=\mathbf{q}+d_{k,1}\begin{bmatrix}
\cos(\hat{\theta}_{\mathrm{Tx},k})\\\sin(\hat{\theta}_{\mathrm{Tx},k})
\end{bmatrix}+(c\hat{\tau}_{k}-d_{k,1})\begin{bmatrix}
\cos(\hat{\theta}_{\mathrm{Rx},k}+\hat{\alpha}_{\mathrm{trial}})\\-\sin(\hat{\theta}_{\mathrm{Rx},k}+\hat{\alpha}_{\mathrm{trial}})
\end{bmatrix},\: k \in \{ k_1,k_2 \}
\end{equation}
where $d_{k,1}$ was introduced in Fig.~\ref{NLOS_Link}. After solving \eqref{EXIP5b} for $[\mathbf{p},d_{1,1},d_{2,1}]$, it is straightforward to determine the scatterer locations (as was done in the NLOS case). For each trial value $\hat{\alpha}_{\mathrm{trial}}$, we can then apply the \ac{LMA} to \eqref{EXIP4b} to obtain $\hat{\tilde{\boldsymbol{\eta}}}_{\mathrm{olos}}$. The solution $\hat{\tilde{\boldsymbol{\eta}}}_{\mathrm{olos}}$ with the smallest $v_{\mathrm{olos}}(\tilde{\boldsymbol{\eta}}_{\mathrm{olos}})$ (with respect to all possible trial value $\hat{\alpha}_{\mathrm{trial}}$) is then retained. Clearly, there is a performance/complexity trade-off based on the choice of $\Delta \alpha$. It is readily seen that to obtain estimates of all parameters, at least three scatterers are needed, since then we have 9 available estimated parameters (1 AOA, 1 AOD, 1 TOA per path) and 9 unknowns (6 scalars for the scatterer locations $\mathbf{s}_k$, 3 scalars for $\mathbf{p}$ and $\alpha$).
\item \textbf{Unknown:} For the case that the receiver does not know whether it operates in \ac{NLOS} or \ac{OLOS},
the receiver could apply the technique above under \ac{NLOS} and under \ac{OLOS}, separately. This will give two solutions with different cost (measured in terms of \eqref{EXIP4} and \eqref{EXIP4b}). The best solution (the one with lowest cost) can then be retained.
\end{itemize}
The complexity analysis for each step of the aforementioned algorithm is presented in Appendix \ref{comprep}.
\section{Simulation Results}
In this section, we present simulation results show the values of the bounds and the performance of the proposed estimators for different parameters.
\subsection{Simulation Setup}
We consider a scenario representative of indoor localization in a small conference room with the maximum distance between MS and BS of 4 meters \cite{Maltsevx}. We set $f_c = 60 \:\mathrm{GHz}$, $B=100 \:\mathrm{MHz}$, $c=0.299792\:\mathrm{m}/\mathrm{ns}$, and $N=20$. The geometry-based statistical path loss is used with path length $d_{k}$ and the number of reflectors in each path is set to one, i.e., it is assumed that there is one reflector in each \ac{NLOS} path \cite{geomted}. The path loss $\rho_{k}$ between \ac{BS} and \ac{MS} for the $k$-th path is computed based on geometry statistics \cite{Qian1,Qian2}. We set
\begin{equation}\label{pathgeom1}
1/\rho_{k}=\sigma^{2}_{0}\mathbb{P}_{0}(d_{k,2})\xi^{2}(d_{k})\left(\frac{\lambda_{c}}{4\pi d_{k}}\right)^{2},
\end{equation}
where $\sigma^{2}_{0}$ is the reflection loss, $\mathbb{P}_{0}(d_{k,2})=(\gamma_{r}d_{k,2})^{2}e^{-\gamma_{r}d_{k,2}}$ denotes the Poisson distribution of environment geometry with density $\gamma_{r}$ (set to $1/7$ \cite{geomted}), $\xi^{2}(d_{k})$ denotes the atmospheric attenuation over distance $d_{k}$, and the last term is the free space path loss over distance $d_{k}$. For the \ac{LOS} link, we obtain
\begin{equation}\label{pathgeom2}
1/\rho_{0}=\xi^{2}(d_{0})\left(\frac{\lambda_{c}}{4\pi d_{0}}\right)^{2}.
\end{equation}
The average reflection loss for the first-order reflection $\sigma^{2}_{0}$ is set to $-10$ dB with the \ac{RMS} deviation equal to $4$ dB \cite{newref5gsix}, and the atmospheric attenuation over distance $d_{k}$ is set to $16$ dB/km \cite{Rappaport}. The number of transmit and receive antennas are set to $N_{t}=65$ and $N_{r}=65$, respectively. The number of simultaneous beams is $M_{t}=1$, and the number of sequentially transmitted signals is $G=32$, unless otherwise stated. The \ac{BS} is located at $\mathbf{q}\:[\mathrm{m}]=[0, 0]^{\mathrm{T}}$ and the \ac{MS} is located at $\mathbf{p}\:[\mathrm{m}]=[4, 0]^{\mathrm{T}}$ with the rotation angle $\alpha=0.1\:\mathrm{rad}$. The elements of the analog beamformers are generated as random values uniformly distributed on the unit circle. The sequences $\tilde{\mathbf{x}}^{(g)}[n]=\mathbf{F}^{(g)}_{\mathrm{BB}}[n]\mathbf{x}^{(g)}[n]$ are obtained as complex exponential terms $e^{j\phi_{g,n}}$ with uniform random phases in $[0, 2\pi)$ along different subcarriers, indexed by $n$, and sequentially transmitted symbols, indexed by $g$.
The values of the \ac{CRB} for $\sqrt{\mathrm{CRB}(\tau_{k})}$, $\sqrt{\mathrm{CRB}(\theta_{\mathrm{Rx},k})}$, and $\sqrt{\mathrm{CRB}(\theta_{\mathrm{Tx},k})}$ are defined similar to PEB and REB in \eqref{PEBexpr} and \eqref{REBexpr}, that is, by inverting the \ac{FIM} $\mathbf{J}_{\tilde{\boldsymbol{\eta}}}$ from \eqref{TFIM1}, choosing the corresponding diagonal entries and taking the square root. Finally, the received \acf{SNR} is defined as
\begin{align}
\mathrm{SNR}\triangleq \frac{\mathbb{E}[\Vert \mathrm{diag}\{\mathbf{\Omega}[0],\ldots \mathbf{\Omega}[N-1]\} \mathrm{vec}\{\check{\mathbf{h}}[0],\ldots,\check{\mathbf{h}}[N-1]\}\Vert^{2}_{2}]}{\mathbb{E}[\Vert \mathrm{vec}\{\check{\mathbf{n}}[0],\ldots,\check{\mathbf{n}}[N-1]\} \Vert^{2}_{2}]},
\end{align}
in which $\mathrm{diag}\{\cdot \}$ creates a block diagonal matrix from its arguments and $\mathrm{vec}\{ \cdot\}$ creates a tall column vector from its arguments.
The performance of the \ac{RMSE} of the estimation algorithm was assessed from $1000$ Monte Carlo realizations. The false alarm probability was set to ${P}_{\mathrm{fa}}=10^{-3}$ to determine the threshold $\delta$.
\subsection{Results and Discussion} \label{results}
\subsubsection*{The Performance versus number of sequential beams}
Fig. \ref{Geffect} shows the \ac{CDF} of the PEB and the $\mathrm{RMSE}(\hat{\mathbf{p}})$ as a function of the number of beams for LOS conditions. The MS can be anywhere in a rectangle with vertices at the coordinates (in meters): $(2,0)$, $(4,0)$, $(2,0.3)$, and $(4,0.3)$.
The signal is scaled so that the total transmit power is kept constant.
By increasing the number of beams $G$, the probability of covering the target location in the specified area with a certain accuracy increases. In other words, due to the ergodicity of the process localization accuracy with a certain number of randomly selected sequential beams in each step converges to a constant value for sufficient number of beams $G$. The reason is that for a larger number of beams, the bound decreases thanks to the better spatial coverage. But this effect vanished when the number of beams is sufficient to cover the area where the \ac{MS} may be located, and then increasing the number of beams only translated into an increased complexity. In principle, the 3 dB beam width for the ULA is approximately $2/N_{t}$, thus reducing when increasing the number of transmit antennas $N_{t}$. Consequently, the number of required beams $G$ to cover the target location in the specified area with the same probability increases. Similarly, by reducing the number of transmit antennas $N_{t}$, the number of required beams $G$ to cover the area decreases. However, the localization accuracy is improved for the case with larger number of transmit antennas $N_{t}$ with the cost of transmitting more beams $G$ for the same coverage. It is observed that for the aforementioned system parameters, $G\geq 20$ randomly selected beams approximately provides the same localization accuracy with $\mathrm{CDF}=0.9$. Note that fewer beams would be needed under a well-chosen deterministic strategy.
The same behavior has been observed in NLOS conditions.
\begin{figure}
\centering
\psfrag{G}[c][]{\footnotesize number of beams}
\psfrag{PEB}[c][]{\scriptsize PEB}
\psfrag{RMSE}[c][]{\scriptsize \qquad$\mathrm{RMSE}(\hat{\mathbf{p}})$}
\psfrag{PEB,G=24}[c][]{\scriptsize \qquad PEB, $24$ beams}
\psfrag{RMSE, G=4}[c][]{\scriptsize \:\:\:\qquad$\mathrm{RMSE}(\hat{\mathbf{p}})$, $4$ beams}
\psfrag{RMSE, G=8}[c][]{\scriptsize \:\:\:\qquad$\mathrm{RMSE}(\hat{\mathbf{p}})$, $8$ beams}
\psfrag{RMSE, G=20}[c][]{\scriptsize \qquad$\mathrm{RMSE}(\hat{\mathbf{p}})$, $20$ beams}
\psfrag{RMSE, G=24}[c][]{\scriptsize \qquad$\mathrm{RMSE}(\hat{\mathbf{p}})$, $24$ beams}
\psfrag{RMSE, G=26}[c][]{\scriptsize \qquad$\mathrm{RMSE}(\hat{\mathbf{p}})$, $26$ beams}
\psfrag{CDF}[c][]{\footnotesize CDF}
\psfrag{Localization error [m]}[c][]{\footnotesize Localization error [m]}
\includegraphics[width=0.7\columnwidth]{PEBCDFvsG10.eps}
\caption{The effect of increasing the number of beams on (top) PEB and $\mathrm{RMSE}(\hat{\mathbf{p}})$ at $\mathrm{CDF}=0.9$ and (bottom) CDF plots for LOS conditions.}
\label{Geffect}
\end{figure}
\subsubsection*{Performance in LOS}
Fig.~\ref{ParamvsIter} shows the evolution of the \ac{RMSE} of TOA and AOA/AOD in the LOS conditions. The Cram\'{e}r-Rao bounds are shown by the red lines with the corresponding markers. It is observed that after a few iterations of Algorithm 2 the \ac{RMSE} of TOA and AOA/AOD converges to the corresponding bounds even for $\mathrm{SNR}=-20\:\mathrm{dB}, -10\:\mathrm{dB}, 0\:\mathrm{dB}$. The performance of the \ac{RMSE} of the estimation algorithm with respect to different values of the received SNR is shown in Fig.~\ref{ParamvsSNR}--\ref{PREBvsSNR_losss}.
It is observed that after $\mathrm{SNR}\approx -20$ dB the \ac{RMSE} of the TOA, AOA/AOD, rotation angle, and position converge to their corresponding bounds (red dashed lines). Moreover, the proposed algorithm performs well even for very low values of the received SNR, which is the typical case at \ac{mm-wave} systems before beamforming.
We observe that at $\mathrm{SNR}\approx -20\:\mathrm{dB}$ the TOA, AOA/AOD, rotation angle, and position approach the corresponding bounds.
\begin{figure}
\centering
\psfrag{snr2=-20dB}[c][]{\footnotesize \qquad\qquad$\mathrm{SNR}=-20$ dB}
\psfrag{snr3=-10dB}[c][]{\footnotesize \qquad\qquad$\mathrm{SNR}=-10$ dB}
\psfrag{snr3=0dB}[c][]{\footnotesize \qquad\qquad$\mathrm{SNR}=0$ dB}
\psfrag{TOA0 at x}[c][]{\footnotesize \qquad\qquad$\:\:\mathrm{TOA}_{0}$ at $-20$ dB}
\psfrag{TOA0 at y}[c][]{\footnotesize \qquad\qquad$\:\:\mathrm{TOA}_{0}$ at $-10$ dB}
\psfrag{TOA0 at z}[c][]{\footnotesize \qquad\qquad$\:\:\mathrm{TOA}_{0}$ at $0$ dB}
\psfrag{AOA0 at x}[c][]{\footnotesize \qquad\qquad$\:\:\mathrm{AOA}_{0}$ at $-20$ dB}
\psfrag{AOA0 at y}[c][]{\footnotesize \qquad\qquad$\:\:\mathrm{AOA}_{0}$ at $-10$ dB}
\psfrag{AOA0 at z}[c][]{\footnotesize \qquad\qquad$\:\:\mathrm{AOA}_{0}$ at $0$ dB}
\psfrag{AOD0 at x}[c][]{\footnotesize \qquad\qquad$\:\:\mathrm{AOD}_{0}$ at $-20$ dB}
\psfrag{AOD0 at y}[c][]{\footnotesize \qquad\qquad$\:\:\mathrm{AOD}_{0}$ at $-10$ dB}
\psfrag{AOD0 at z}[c][]{\footnotesize \qquad\qquad$\:\:\mathrm{AOD}_{0}$ at $0$ dB}
\psfrag{RMSETOA1}[c][]{\footnotesize $\mathrm{RMSE}(\hat{\tau}_{0})$ [ns]}
\psfrag{RMSEAOA1}[c][]{\footnotesize $\mathrm{RMSE}(\hat{\theta}_{\mathrm{Rx},0})$ [rad]}
\psfrag{RMSEAOD1}[c][]{\footnotesize $\mathrm{RMSE}(\hat{\theta}_{\mathrm{Tx},0})$ [rad]}
\psfrag{iteration index}[c][]{\footnotesize iteration index}
\includegraphics[width=0.58\columnwidth]{RMSE_LOS_IterFfU.eps}
\caption{The evolution of \ac{RMSE} of TOA and AOA/AOD for the LOS for $\mathrm{SNR}=-20\:\mathrm{dB}, -10\:\mathrm{dB}, 0\:\mathrm{dB}$. The red lines with the same markers show the bounds for the same value of \ac{SNR} corresponding to the \ac{RMSE} of TOA and AOA/AOD.}
\label{ParamvsIter}
\end{figure}
\begin{figure}
\centering
\psfrag{RMSE0}[c][]{\footnotesize $\mathrm{RMSE}(\hat{\tau}_{0})$ [ns]}
\psfrag{RMSE1}[c][]{\footnotesize $\mathrm{RMSE}(\hat{\theta}_{\mathrm{Rx},0})$ [rad]}
\psfrag{RMSE2}[c][]{\footnotesize $\mathrm{RMSE}(\hat{\theta}_{\mathrm{Tx},0})$ [rad]}
\psfrag{SNR}[c][]{\footnotesize SNR (in dB)}
\psfrag{RMSEx1}[c][]{\tiny \qquad$\:\:\mathrm{RMSE}(\hat{\tau}_{0})$}
\psfrag{RMSEx2}[c][]{\tiny \qquad$\:\:\mathrm{RMSE}(\hat{\theta}_{\mathrm{Rx},0})$}
\psfrag{RMSEx3}[c][]{\tiny \qquad$\:\:\mathrm{RMSE}(\hat{\theta}_{\mathrm{Tx},0})$}
\psfrag{RMSEy1}[c][]{\tiny \qquad$\:\:\sqrt{\mathrm{CRB}(\tau_{0})}$}
\psfrag{RMSEy2}[c][]{\tiny \qquad$\:\:\sqrt{\mathrm{CRB}(\theta_{\mathrm{Rx},0})}$}
\psfrag{RMSEy3}[c][]{\tiny \qquad$\:\:\sqrt{\mathrm{CRB}(\theta_{\mathrm{Tx},0})}$}
\psfrag{iteration index}[c][]{\small iteration index}
\includegraphics[width=0.58\columnwidth]{ParamvsSNRFff.eps}
\caption{RMSE in dB scale plotted against received SNR for TOA and AOA/AOD in the LOS conditions. The red lines show the corresponding bounds.}
\label{ParamvsSNR}
\end{figure}
\begin{figure}
\centering
\psfrag{SNR}[c][]{\footnotesize SNR (in dB)}
\psfrag{REBb}[c][]{\footnotesize \qquad REB}
\psfrag{PEBb}[c][]{\footnotesize \qquad PEB}
\psfrag{RMSExx}[c][]{\footnotesize $\mathrm{RMSE}(\hat{\alpha})$ [rad]}
\psfrag{RMSEyy}[c][]{\footnotesize $\mathrm{RMSE}(\hat{\mathbf{p}})$ [m]}
\psfrag{RMSEx}[c][]{\footnotesize \qquad$\mathrm{RMSE}(\hat{\alpha})$}
\psfrag{RMSEy}[c][]{\footnotesize \qquad$\mathrm{RMSE}(\hat{\mathbf{p}})$}
\includegraphics[width=0.57\columnwidth]{PREB_LOSFf.eps}
\caption{RMSE in dB scale plotted against received SNR for rotation angle (top) and position (bottom) in the \ac{LOS}. The red lines show the corresponding bounds.}
\label{PREBvsSNR_losss}
\end{figure}
\subsubsection*{Performance in NLOS}
Fig.~\ref{ParamvsIterNLOS} shows the evolution of the \ac{RMSE} of TOA and AOA/AOD for $1000$ Monte Carlo realizations in the presence of a scatterer located at $\mathbf{s}_{k}\:[\mathrm{m}]=[1.5, 0.4]^{\mathrm{T}}$. It can be observed that the \ac{RMSE} of the TOA and the AOA/AOD obtained with the proposed algorithm for both the parameters of the LOS and the reflected signals converges to the theoretical also in this case, even at very low received SNR.
At $\mathrm{SNR}\approx -5\:\mathrm{dB}$ the TOA, AOA/AOD, rotation angle, and position approach the corresponding bounds.
\begin{figure}
\centering
\psfrag{snr2=-5dB}[c][]{\footnotesize $\qquad\mathrm{SNR}=-5$ dB}
\psfrag{snr3=0dB}[c][]{\footnotesize $\qquad\mathrm{SNR}=0$ dB}
\psfrag{RMSETOA1}[c][]{\footnotesize $\mathrm{RMSE}(\hat{\tau}_{0})$ [ns]}
\psfrag{RMSETOA2}[c][]{\footnotesize $\mathrm{RMSE}(\hat{\tau}_{1})$ [ns]}
\psfrag{RMSEAOA1}[c][]{\footnotesize $\mathrm{RMSE}(\hat{\theta}_{\mathrm{Rx},0})$ [rad]}
\psfrag{RMSEAOA2}[c][]{\footnotesize $\mathrm{RMSE}(\hat{\theta}_{\mathrm{Rx},1})$ [rad]}
\psfrag{RMSEAOD1}[c][]{\footnotesize $\mathrm{RMSE}(\hat{\theta}_{\mathrm{Tx},0})$ [rad]}
\psfrag{RMSEAOD2}[c][]{\footnotesize $\mathrm{RMSE}(\hat{\theta}_{\mathrm{Tx},1})$ [rad]}
\psfrag{iteration index}[c][]{\footnotesize iteration index}
\includegraphics[width=0.9\columnwidth]{RMSE_NLOS_IterFfU.eps}
\caption{The evolution of \ac{RMSE} of TOA and AOA/AOD for the LOS (left column) and the NLOS (right column) paths at $\mathrm{SNR}=-5\:\mathrm{dB}, 0\:\mathrm{dB}$. The red lines with the same markers show the bounds.}
\label{ParamvsIterNLOS}
\end{figure}
\begin{figure}
\centering
\psfrag{RMSE0}[c][]{\footnotesize $\mathrm{RMSE}(\hat{\tau}_{k})$ [ns]}
\psfrag{RMSE1}[c][]{\footnotesize $\mathrm{RMSE}(\hat{\theta}_{\mathrm{Rx},k})$ [rad]}
\psfrag{RMSE2}[c][]{\footnotesize $\mathrm{RMSE}(\hat{\theta}_{\mathrm{Tx},k})$ [rad]}
\psfrag{SNR}[c][]{\footnotesize SNR (in dB)}
\psfrag{RMSEx1}[c][]{\footnotesize \:$\qquad\mathrm{RMSE}(\hat{\tau}_{0})$}
\psfrag{RMSEy1}[c][]{\footnotesize \:$\qquad\mathrm{RMSE}(\hat{\tau}_{1})$}
\psfrag{RMSEx2}[c][]{\footnotesize \:\:$\qquad\mathrm{RMSE}(\hat{\theta}_{\mathrm{Rx},0})$}
\psfrag{RMSEy2}[c][]{\footnotesize \:\:$\qquad\mathrm{RMSE}(\hat{\theta}_{\mathrm{Rx},1})$}
\psfrag{RMSEx3}[c][]{\footnotesize \:\:$\qquad\mathrm{RMSE}(\hat{\theta}_{\mathrm{Tx},0})$}
\psfrag{RMSEy3}[c][]{\footnotesize \:\:$\qquad\mathrm{RMSE}(\hat{\theta}_{\mathrm{Tx},1})$}
\psfrag{TOA0}[c][]{\footnotesize \:$\qquad\sqrt{\mathrm{CRB}(\tau_{0})}$}
\psfrag{TOA1}[c][]{\footnotesize \:$\qquad\sqrt{\mathrm{CRB}(\tau_{1})}$}
\psfrag{AOA0}[c][]{\footnotesize \:\:$\qquad\sqrt{\mathrm{CRB}(\theta_{\mathrm{Rx},0})}$}
\psfrag{AOA1}[c][]{\footnotesize \:\:$\qquad\sqrt{\mathrm{CRB}(\theta_{\mathrm{Rx},1})}$}
\psfrag{AOD0}[c][]{\footnotesize \:\:$\qquad\sqrt{\mathrm{CRB}(\theta_{\mathrm{Tx},0})}$}
\psfrag{AOD1}[c][]{\footnotesize \:\:$\qquad\sqrt{\mathrm{CRB}(\theta_{\mathrm{Tx},1})}$}
\includegraphics[width=0.7\columnwidth]{ParamvsSNRnlosFf.eps}
\caption{RMSE in dB scale for the \ac{NLOS} plotted against received SNR for TOA and AOA/AOD in the presence of a scatterer located at $\mathbf{s}_{k}\:[\mathrm{m}]=[1.5, 0.4]^{\mathrm{T}}$. The red lines show the corresponding bounds.}
\label{ParamvsSNRnlos}
\end{figure}
\begin{figure}
\centering
\psfrag{SNR}[c][]{\footnotesize SNR (in dB)}
\psfrag{REBb}[c][]{\footnotesize REB}
\psfrag{PEBb}[c][]{\footnotesize PEB}
\psfrag{RMSExx}[c][]{\footnotesize $\mathrm{RMSE}(\hat{\alpha})$ [rad]}
\psfrag{RMSEyy}[c][]{\footnotesize $\mathrm{RMSE}(\hat{\mathbf{p}})$ [m]}
\psfrag{RMSEx}[c][]{\footnotesize $\qquad\mathrm{RMSE}(\hat{\alpha})$}
\psfrag{RMSEy}[c][]{\footnotesize $\qquad\mathrm{RMSE}(\hat{\mathbf{p}})$}
\includegraphics[width=0.57\columnwidth]{PREB_NLOSFfz.eps}
\caption{RMSE in dB scale for the \ac{NLOS} plotted against received SNR for rotation angle (top) and position (bottom) in the presence of a scatterer located at $\mathbf{s}_{k}\:[\mathrm{m}]=[1.5, 0.4]^{\mathrm{T}}$. The red lines show the corresponding bounds.}
\label{PREBvsSNR}
\end{figure}
\begin{figure}
\centering
\psfrag{SNR}[c][]{\small SNR (in dB)}
\psfrag{REBb}[c][]{\scriptsize REB}
\psfrag{PEBb}[c][]{\scriptsize PEB}
\psfrag{RMSEa}[c][]{\small $\mathrm{RMSE}(\hat{\alpha})$ [rad]}
\psfrag{RMSEp}[c][]{\small $\mathrm{RMSE}(\hat{\mathbf{p}})$ [m]}
\psfrag{RMSEy}[c][]{\scriptsize $\qquad\qquad\qquad\qquad\mathrm{RMSE}(\hat{\alpha}),\Delta\alpha\:[\mathrm{rad}]=0.01$}
\psfrag{RMSEyy}[c][]{\scriptsize $\qquad\qquad\qquad\qquad\mathrm{RMSE}(\hat{\alpha}),\Delta\alpha\:[\mathrm{rad}]=0.05$}
\psfrag{RMSEx}[c][]{\scriptsize $\qquad\qquad\qquad\qquad\mathrm{RMSE}(\hat{\mathbf{p}}),\Delta\alpha\:[\mathrm{rad}]=0.01$}
\psfrag{RMSExx}[c][]{\scriptsize $\qquad\qquad\qquad\qquad\mathrm{RMSE}(\hat{\mathbf{p}}),\Delta\alpha\:[\mathrm{rad}]=0.05$}
\includegraphics[width=0.57\columnwidth]{PREB_OLOSfzz.eps}
\caption{RMSE in dB scale plotted against received SNR for rotation angle (top) and position (bottom) in the \ac{OLOS} with three scatterers located at $\mathbf{s}_{k}\:[\mathrm{m}]=[1.5, 0.4+0.5(k-1)]^{\mathrm{T}}$ for $k=1, 2, 3$ and $\Delta\alpha\:[\mathrm{rad}]=\{0.01,0.05\}$. The red lines show the corresponding bounds.}
\label{PREBvsSNRolos}
\end{figure}
\subsubsection*{Performance in OLOS}
Finally, the performance in the \ac{OLOS} case for three scatterers located at $\mathbf{s}_{k}\:[\mathrm{m}]=[1.5, 0.4+0.5(k-1)]^{\mathrm{T}}$ for $k=1, 2, 3$ is investigated in this section using two different initializations of the rotation angle: one with grid resolution $\Delta\alpha\:[\mathrm{rad}]=0.01$ and one with $\Delta\alpha\:[\mathrm{rad}]=0.05$. For both, we set
$\alpha_{m}\:[\mathrm{rad}]=0.5$. Fig.~\ref{PREBvsSNRolos} shows the performance of the \ac{RMSE} with respect to the received \ac{SNR} for position and rotation angle estimation. The proposed estimation method approaches the bound even for the initialization with the resolution $\Delta\alpha\:[\mathrm{rad}]=0.05$. However, the performance of the estimation algorithm is dependent on the resolution of the grid of points $\Delta\alpha$. In particular, a finer grid for the rotation angle leads to better initial estimates and thus a lower final RMSE.
For $\mathrm{SNR}\approx -10\:\mathrm{dB}$ the \ac{RMSE} of position and rotation angle approach the corresponding bounds. We note that the OLOS values, for a fixed SNR, are significantly higher in the OLOS than in the NLOS case.
\subsubsection*{Unknown Conditions}
To analyze the application of the algorithm when the propagation conditions are unknown, we consider the case where there are three scatterers and the LOS path is blocked, that is, the OLOS condition. Starting with the wrong assumption that the path with the shortest delay is the LOS path (i.e., the NLOS condition) leads to very large values of the cost function \eqref{EXIP4} compared to the actual value of the cost function \eqref{EXIP4b}. The results are summarized in Table. \ref{tab:assumptions} for the average value of the ratio ${\Delta v}\triangleq v_{\mathrm{nlos}}(\hat{\tilde{\boldsymbol{\eta}}})/v_{\mathrm{olos}}(\hat{\tilde{\boldsymbol{\eta}}}_{\mathrm{olos}})$ between the cost function with the wrong and true assumptions. The values in Table \ref{tab:assumptions} are obtained by averaging 100 realizations, and with a grid resolution of $\Delta\alpha=0.05 [\mathrm{rad}]$. The slight difference in the ratio for different values of SNR is due to the limited number of trials.
\begin{table}[!t]
\centering
\caption{Unknown conditions} \label{tab:assumptions}
\begin{tabular}{|c|c|c|c|c|}
\hline
SNR (in dB) & -20 & -10 & 0 &10
\\\hline
${\Delta v}$& 5.5&5.2&5&5.3\\\hline
\end{tabular}
\end{table}
It is clear that using the wrong assumption about the path with the shortest delay leads to much larger values of the cost function, i.e., the mean value of the ratio ${\Delta v}$ between the cost function with the wrong and true assumptions is on the order of $5$. The main reason for the increase of the cost function using the wrong assumption about the shortest path is that the estimate of \ac{MS} rotation angle obtained from the AOA and AOD of this path is heavily erroneous. When the shortest path is considered to be a LOS but it is really a reflection, there is a clear mismatch between the geometry of the propagation and the model equations, since there is a scatterer that breaks the direct relation between AOA and AOD existing with the LOS. This mismatch causes a large error in the initial position that is propagated to the final solution. Therefore, observing the ratio of cost functions, we can identify that the path with the shortest delay is related to the scatterer and the LOS path does not exist, that is to say, the OLOS condition is correctly recognized.
\subsubsection*{Comparison of LOS versus NLOS Performance}
Fig. \ref{CDFdiff} compares the performance of the positioning algorithm in LOS and NLOS for $\mathrm{SNR}= -5$ dB and $G=20$. The MS is anywhere in the same rectangle described at the beginning of Sec.~\ref{results}. The scatterers are located at coordinates (in meters) $\mathbf{s}_{1}=(1.5,0.4)$ and $\mathbf{s}_{2}=(1.5,0.6)$. The accuracy and robustness of the localization algorithm is improved by adding the scatterers compared to the case when only LOS is used. Moreover, the performance in the OLOS is much worse than in LOS or NLOS due to the severe effect of path loss as shown already in the paper by comparing Figs. \ref{PREBvsSNR_losss} and \ref{PREBvsSNR} with Fig. \ref{PREBvsSNRolos}.
\begin{figure}
\centering
\psfrag{CDF}[c][]{\small CDF}
\psfrag{Localization error [m]}[c][]{\small Localization error [m]}
\psfrag{NLOS with 1 scatterer}[c][]{\footnotesize \qquad NLOS with 1 scatterer}
\psfrag{NLOS with 2 scatterers}[c][]{\footnotesize \qquad NLOS with 2 scatterers}
\psfrag{LOS}[c][]{\footnotesize \qquad LOS}
\includegraphics[width=0.57\columnwidth]{CDFvsError3b.eps}
\caption{CDF of the localization error in LOS and NLOS with one and two scatterers for $\mathrm{SNR}=-5$ dB and $G=20$.}
\label{CDFdiff}
\end{figure}
\section{Conclusion}\label{SEC:Conclusion}
We have studied the determination of a receiver position and orientation using a single transmitter in a MIMO system. Our study includes \ac{LOS}, as well as \ac{NLOS} and \ac{OLOS} conditions, shedding insight into the potential of locating a receiver even when the \ac{LOS} is blocked. We have derived fundamental performance bounds on the estimation uncertainty for delay, angle of arrival, angle of departure, and channel gain of each path, as well as the user position and orientation angle. We also proposed a novel three stage algorithm for the estimation of the user position and orientation angle. This algorithm determines coarse estimates of the channel parameters by exploiting the sparsity of the \ac{mm-wave} in beamspace, followed by an iterative refinement, and finally a conversion to position and orientation. Through simulation studies, we demonstrate the efficiency of the proposed algorithm, and show that even in \ac{OLOS} conditions, it is possible to estimate the user's position and orientation angle, by exploiting the information coming from the multipath, though at a significant performance penalty.
\appendices
\section{Elements in \eqref{Parameters8w} }\label{elements}
Replacing $\mathbf{y}[n]$ from (\ref{Receivedb1}) in (\ref{Parameters5b}), using (\ref{Parameters6ww}), and considering $\mathbb{E}_{\mathbf{y}\vert\boldsymbol{\eta}}[\mathbf{n}[n]]=\mathbf{0}$, we obtain
\begin{equation}\label{appa0}
\Psi(x_{r},x_{s})=\frac{2}{N_{0}}\sum_{n=0}^{N-1}\Re\left\{\frac{\partial\boldsymbol{\mu}^{\mathrm{H}}[n]}{\partial x_{r}}\frac{\partial\boldsymbol{\mu}[n]}{\partial x_{s}}\right\}.
\end{equation}
The elements of the \ac{FIM} are obtained based on \eqref{appa0}. The entry associated with the $n$-th subcarrier is denoted as $\Psi_n(x_{r},x_{s})$, and given by (for $\left\lbrace\tau_{r}, \tau_{s}\right\rbrace$ and $\left\lbrace\boldsymbol{\theta}_{r}, \boldsymbol{\theta}_{s}\right\rbrace$)
\begin{IEEEeqnarray}{rCl}
\Psi_{n}(\tau_{r},\tau_{s})& = &\frac{2}{N_{0}}\Re\{\tilde{h}^{*}_{r}\tilde{h}_{s} A_{\mathrm{Rx},n}(\theta_{\mathrm{Rx,r}},\theta_{\mathrm{Rx,s}})A^{(2)}_{\mathrm{Tx},\mathbf{F},n}(\tau_{r},\tau_{s},\theta_{\mathrm{Tx,s}},\theta_{\mathrm{Tx,r}})\},\label{Parameters9w}\\
\Psi_{n}(\tau_{r},\theta_{\mathrm{Tx},s})& = &\frac{2}{N_{0}}\Re\{j\tilde{h}^{*}_{r}\tilde{h}_{s}A_{\mathrm{Rx},n}(\theta_{\mathrm{Rx,r}},\theta_{\mathrm{Rx,s}})A^{(1)}_{\mathbf{D}_{\mathrm{Tx},s},\mathbf{F},n}(\tau_{r},\tau_{s},\theta_{\mathrm{Tx,s}},\theta_{\mathrm{Tx,r}})\},\label{Parameters10w}\\
\Psi_{n}(\tau_{r},\theta_{\mathrm{Rx},s})& = &\frac{2}{N_{0}}\Re\{j\tilde{h}^{*}_{r}\tilde{h}_{s}A_{\mathbf{D}_{\mathrm{Rx,s}},n}(\theta_{\mathrm{Rx,r}},\theta_{\mathrm{Rx,s}})A^{(1)}_{\mathrm{Tx},\mathbf{F},n}(\tau_{r},\tau_{s},\theta_{\mathrm{Tx,s}},\theta_{\mathrm{Tx,r}})\},\label{Parameters11w}\\
\Psi_{n}(\theta_{\mathrm{Tx},r},\theta_{\mathrm{Tx},s})& = &\frac{2}{N_{0}}\Re\{\tilde{h}^{*}_{r}\tilde{h}_{s}A_{\mathrm{Rx},n}(\theta_{\mathrm{Rx,r}},\theta_{\mathrm{Rx,s}})A_{\mathbf{Dd}_{\mathrm{Tx}},\mathbf{F},n}(\tau_{r},\tau_{s},\theta_{\mathrm{Tx,s}},\theta_{\mathrm{Tx,r}})\}, \label{Parameters12w}\\
\Psi_{n}(\theta_{\mathrm{Tx},r},\theta_{\mathrm{Rx},s})& = &\frac{2}{N_{0}}\Re\{\tilde{h}^{*}_{r}\tilde{h}_{s}A_{\mathbf{D}_{\mathrm{Rx,s}},n}(\theta_{\mathrm{Rx,r}},\theta_{\mathrm{Rx,s}})A^{(0)}_{\mathbf{D}_{\mathrm{Tx},r},\mathbf{F},n}(\tau_{r},\tau_{s},\theta_{\mathrm{Tx,s}},\theta_{\mathrm{Tx,r}})\}, \label{Parameters13w}\\
\Psi_{n}(\theta_{\mathrm{Rx},r},\theta_{\mathrm{Rx},s})& = &\frac{2}{N_{0}}\Re\{\tilde{h}^{*}_{r}\tilde{h}_{s}A_{\mathbf{D}_{\mathrm{Rx,r,s}},n}(\theta_{\mathrm{Rx,r}},\theta_{\mathrm{Rx,s}})A^{(0)}_{\mathrm{Tx},\mathbf{F},n}(\tau_{r},\tau_{s},\theta_{\mathrm{Tx,s}},\theta_{\mathrm{Tx,r}})\}.\label{Parameters14w}
\end{IEEEeqnarray}
The following notations are introduced:
\begin{align}
A^{(k)}_{\mathrm{Tx},\mathbf{F},n}(\tau_{r},\tau_{s},\theta_{\mathrm{Tx,s}},\theta_{\mathrm{Tx,r}})&\triangleq\mathbf{a}_{\mathrm{Tx},\mathbf{F},n}^{\mathrm{H}}(\theta_{\mathrm{Tx,s}})\mathbf{A}_{k,n}(\tau_{r},\tau_{s})\mathbf{a}_{\mathrm{Tx},\mathbf{F},n}(\theta_{\mathrm{Tx,r}}),\label{Parameters15w}\\
A^{(l)}_{\mathbf{D}_{\mathrm{Tx},s},\mathbf{F},n}(\tau_{r},\tau_{s},\theta_{\mathrm{Tx,s}},\theta_{\mathrm{Tx,r}})&\triangleq\mathbf{a}_{\mathbf{D}_{\mathrm{Tx}},\mathbf{F},n}^{\mathrm{H}}(\theta_{\mathrm{Tx,s}})\mathbf{A}_{l,n}(\tau_{r},\tau_{s})\mathbf{a}_{\mathrm{Tx},\mathbf{F},n}(\theta_{\mathrm{Tx,r}}),\label{Parameters16w}\\
A^{(l)}_{\mathbf{D}_{\mathrm{Tx},r},\mathbf{F},n}(\tau_{r},\tau_{s},\theta_{\mathrm{Tx,s}},\theta_{\mathrm{Tx,r}})&\triangleq\mathbf{a}_{\mathrm{Tx},\mathbf{F},n}^{\mathrm{H}}(\theta_{\mathrm{Tx,s}})\mathbf{A}_{l,n}(\tau_{r},\tau_{s})\mathbf{a}_{\mathbf{D}_{\mathrm{Tx}},\mathbf{F},n}(\theta_{\mathrm{Tx,r}}),\label{Parameters17w}\\
A_{\mathbf{Dd}_{\mathrm{Tx}},\mathbf{F},n}(\tau_{r},\tau_{s},\theta_{\mathrm{Tx,s}},\theta_{\mathrm{Tx,r}})&\triangleq\mathbf{a}_{\mathbf{D}_{\mathrm{Tx}},\mathbf{F},n}^{\mathrm{H}}(\theta_{\mathrm{Tx,s}})\mathbf{A}_{0,n}(\tau_{r},\tau_{s})\mathbf{a}_{\mathbf{D}_{\mathrm{Tx}},\mathbf{F},n}(\theta_{\mathrm{Tx,r}}),\label{Parameters18w}
\end{align}
where $l \in \{0,1\}$, and $\mathbf{A}_{k,n}(\tau_{r},\tau_{s}), k \in \{0,1,2 \}$, is given by
\vspace{-2mm}
\begin{equation}\label{Parameters19w}
\mathbf{A}_{k,n}(\tau_{r},\tau_{s})\triangleq (2\pi n/(NT_{s}))^{k}~\mathbf{x}[n]\mathbf{x}^{\mathrm{H}}[n]e^{-j2\pi n(\tau_{r}-\tau_{s})/(NT_{s})}.
\end{equation}
The vectors $\mathbf{a}_{\mathrm{Tx},\mathbf{F},n}(\theta_{\mathrm{Tx},r})$ and $\mathbf{a}_{\mathbf{D}_{\mathrm{Tx}},\mathbf{F},n}(\theta_{\mathrm{Tx},r})$ are given by $\mathbf{a}_{\mathrm{Tx},\mathbf{F},n}(\theta_{\mathrm{Tx},r}) = \mathbf{F}^{\mathrm{H}}[n]\mathbf{a}_{\mathrm{Tx},n}(\theta_{\mathrm{Tx},r})$ and $\mathbf{a}_{\mathbf{D}_{\mathrm{Tx}},\mathbf{F},n}(\theta_{\mathrm{Tx},r})=\mathbf{F}^{\mathrm{H}}[n]\mathbf{D}_{\mathrm{Tx},r}[n]\mathbf{a}_{\mathrm{Tx},n}(\theta_{\mathrm{Tx},r})$. The matrix $\mathbf{D}_{\mathrm{Tx},r}[n]$ is defined as
\begin{equation}\label{Parameters19ww}
\mathbf{D}_{\mathrm{Tx},r}[n]\triangleq j\frac{2\pi}{\lambda_{n}}d\cos(\theta_{\mathrm{Tx},r})\mathrm{diag}\{0,\ldots,N_{\mathrm{t}}-1\}.
\end{equation}
The scalars $A_{\mathrm{Rx},n}(\theta_{\mathrm{Rx,r}},\theta_{\mathrm{Rx,s}})$, $A_{\mathbf{D}_{\mathrm{Rx,s}},n}(\theta_{\mathrm{Rx,r}},\theta_{\mathrm{Rx,s}})$, and $A_{\mathbf{D}_{\mathrm{Rx,r,s}},n}(\theta_{\mathrm{Rx,r}},\theta_{\mathrm{Rx,s}})$ are defined as
\begin{align}
A_{\mathrm{Rx},n}(\theta_{\mathrm{Rx,r}},\theta_{\mathrm{Rx,s}})&\triangleq\mathbf{a}^{\mathrm{H}}_{\mathrm{Rx},n}(\theta_{\mathrm{Rx},r})\mathbf{a}_{\mathrm{Rx},n}(\theta_{\mathrm{Rx},s}),
\label{Parameters20w}\\
A_{\mathbf{D}_{\mathrm{Rx,s}},n}(\theta_{\mathrm{Rx,r}},\theta_{\mathrm{Rx,s}})&\triangleq\mathbf{a}^{\mathrm{H}}_{\mathrm{Rx},n}(\theta_{\mathrm{Rx},r})\mathbf{D}_{\mathrm{Rx},s}[n]\mathbf{a}_{\mathrm{Rx},n}(\theta_{\mathrm{Rx},s}),
\label{Parameters21w}\\
A_{\mathbf{D}_{\mathrm{Rx,r,s}},n}(\theta_{\mathrm{Rx,r}},\theta_{\mathrm{Rx,s}})&\triangleq\mathbf{a}^{\mathrm{H}}_{\mathrm{Rx},n}(\theta_{\mathrm{Rx},r})\mathbf{D}^{\mathrm{H}}_{\mathrm{Rx},r}[n]\mathbf{D}_{\mathrm{Rx},s}[n]\mathbf{a}_{\mathrm{Rx},n}(\theta_{\mathrm{Rx},s}),
\label{Parameters22w}
\end{align}
where $\mathbf{D}_{\mathrm{Rx},r}[n]$ has the same expression as \eqref{Parameters19ww} by replacing the subscript $\mathrm{Tx}$ by $\mathrm{Rx}$ and $N_{t}$ by $N_{r}$. The terms including channel coefficients are summarized as:
\begin{multline}\label{appa1}
\mathbf{\Psi}_{n}(\tau_{r},\tilde{\mathbf{h}}_{s})= \frac{2}{N_{0}}[\Re\{j\tilde{h}^{*}_{r}A_{\mathrm{Rx},n}(\theta_{\mathrm{Rx,r}},\theta_{\mathrm{Rx,s}})A^{(1)}_{\mathrm{Tx},\mathbf{F},n}(\tau_{r},\tau_{s},\theta_{\mathrm{Tx},s},\theta_{\mathrm{Tx},r})\},\\\Re\{-\tilde{h}^{*}_{r}A_{\mathrm{Rx},n}(\theta_{\mathrm{Rx,r}},\theta_{\mathrm{Rx,s}})A^{(1)}_{\mathrm{Tx},\mathbf{F},n}(\tau_{r},\tau_{s},\theta_{\mathrm{Tx},s},\theta_{\mathrm{Tx},r})\}],
\end{multline}
\begin{multline}\label{appa2}
\mathbf{\Psi}_{n}(\theta_{\mathrm{Tx},r},\tilde{\mathbf{h}}_{s})= \frac{2}{N_{0}}[\Re\{\tilde{h}^{*}_{r}A_{\mathrm{Rx},n}(\theta_{\mathrm{Rx,r}},\theta_{\mathrm{Rx,s}})A^{(0)}_{\mathbf{D}_{\mathrm{Tx},r},\mathbf{F},n}(\tau_{r},\tau_{s},\theta_{\mathrm{Tx,s}},\theta_{\mathrm{Tx,r}})\},\\\Re\{j\tilde{h}^{*}_{r}A_{\mathrm{Rx},n}(\theta_{\mathrm{Rx,r}},\theta_{\mathrm{Rx,s}})A^{(0)}_{\mathbf{D}_{\mathrm{Tx},r},\mathbf{F},n}(\tau_{r},\tau_{s},\theta_{\mathrm{Tx,s}},\theta_{\mathrm{Tx,r}})\}],
\end{multline}
\begin{multline}\label{appa3}
\mathbf{\Psi}_{n}(\theta_{\mathrm{Rx},r},\tilde{\mathbf{h}}_{s})= -\frac{2}{N_{0}}[\Re\{\tilde{h}^{*}_{r}A_{\mathbf{D}_{\mathrm{Rx},r},n}(\theta_{\mathrm{Rx,r}},\theta_{\mathrm{Rx,s}})A^{(0)}_{\mathrm{Tx},\mathbf{F},n}(\tau_{r},\tau_{s},\theta_{\mathrm{Tx},s},\theta_{\mathrm{Tx},r})\},\\\Re\{j\tilde{h}^{*}_{r}A_{\mathbf{D}_{\mathrm{Rx},r},n}(\theta_{\mathrm{Rx,r}},\theta_{\mathrm{Rx,s}})A^{(0)}_{\mathrm{Tx},\mathbf{F},n}(\tau_{r},\tau_{s},\theta_{\mathrm{Tx},s},\theta_{\mathrm{Tx},r})\}],
\end{multline}
\begin{multline}\label{appa4}
\Psi_{n}(\Re\{\tilde{h}_{r}\},\Re\{\tilde{h}_{s}\})=\Psi_{n}(\Im\{\tilde{h}_{r}\},\Im\{\tilde{h}_{s}\})=\\ \frac{2}{N_{0}}\Re\{A_{\mathrm{Rx},n}(\theta_{\mathrm{Rx,r}},\theta_{\mathrm{Rx,s}})A^{(0)}_{\mathrm{Tx},\mathbf{F},n}(\tau_{r},\tau_{s},\theta_{\mathrm{Tx},s},\theta_{\mathrm{Tx},r})\},
\end{multline}
\begin{multline}\label{appa5}
\Psi_{n}(\Re\{\tilde{h}_{r}\},\Im\{\tilde{h}_{s}\})=-\Psi_{n}(\Im\{\tilde{h}_{r}\},\Re\{\tilde{h}_{s}\})=\\ \frac{2}{N_{0}}\Re\{jA_{\mathrm{Rx},n}(\theta_{\mathrm{Rx,r}},\theta_{\mathrm{Rx,s}})A^{(0)}_{\mathrm{Tx},\mathbf{F},n}(\tau_{r},\tau_{s},\theta_{\mathrm{Tx},s},\theta_{\mathrm{Tx},r})\}.
\end{multline}
\section{Complexity Analysis}\label{comprep}
We analyze the complexity of different stages of the proposed algorithm.\begin{itemize}\item Coarse Estimation: The complexity in performing \eqref{tinex1ee} is on the order of $O(N^{2}_{r}N^{2}_{t}GN_{\mathrm{sub}})$ where $N_{\mathrm{sub}}$ denotes the few subcarriers sufficient to detect the dominant path. The QR factorization of the mutilated basis $\mathbf{\Omega}_{\mathcal{K}_{t}}[n]$ approximately requires $O(GN_{r}\hat{K}^{2})$ operations for each subcarrier, and matrix inversion to obtain the channel coefficients in \eqref{BWTransceiver2z} approximately takes $O(N\hat{K}^{3})$ operations for all the subcarriers. The complexity in computing \eqref{BWTransceiver2tzsfzxe3} is on the order of $O(ND_{o}\hat{K})$ where $D_{o}$ denotes the number of delay grid points, and \eqref{BWTransceiver2tzsfzxe2} requires $O(N\hat{K})$ operations. Consequently, the maximum complexity from coarse estimation of the channel parameters is dominated by the term $\hat{K}\times O(N^{2}_{r}N^{2}_{t}GN_{\mathrm{sub}})$.
\item Fine Estimation: In the refinement phase, the complexity is mainly affected by Gauss-Seidel-type iterations with first and second order derivatives of a vector $\mathbf{a}(x)$ of length $L_{\mathrm{x}}$ with respect to a variable $x$ that can be delay, AOA, and AOD. These operations lead to a complexity on the order of $O(L^{2}_{\mathrm{x}}N)$ for each path. Given the subsequent path refinement, the maximum complexity of fine estimation is on the order of $O(\hat{K}^{2})\times O(L^{2}_{\mathrm{x}}N)$.\item Conversion to Position and Orientation: The conversion to position and orientation in the LOS case is easy to implement since it involves only some basic operations. For the NLOS and OLOS scenarios, the LMA algorithm is applied. It is not considered the complexity driver, since it combines the advantages of gradient-descent and Gauss-Newton methods. The LMA algorithm can be effectively applied by implementing delayed gratification, which leads to higher success rate and fewer Jacobian evaluations.
\end{itemize}
\bibliographystyle{IEEEtran}
| proofpile-arXiv_065-7442 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
"\\section{Introduction} \nOne of the most intriguing and mysterious objects in the Universe is the(...TRUNCATED) | proofpile-arXiv_065-9775 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
"\\section*{Abstract}\n{\\bf\nThe abstract is in boldface, and should fit in 8 lines.\nIt should be (...TRUNCATED) | proofpile-arXiv_065-10081 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
"\\section{\\label{sec:introduction}Introduction}\nImagining oneself attempting to swim in a pool of(...TRUNCATED) | proofpile-arXiv_065-10753 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
"\\section{Introduction}\r\nThere is extensive experimental evidence, eg.\\ in Refs.~\\cite{Adams:20(...TRUNCATED) | proofpile-arXiv_065-11401 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
"\\section{\\label{sec:Introduction} Introduction}\n{\nAs the temperature is increased, strongly int(...TRUNCATED) | proofpile-arXiv_065-14643 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
"\\section{Introduction}\n\n\\setcounter{equation}{0}\n\n\\renewcommand{\\mathcal X}{{X}}\n\\newcomm(...TRUNCATED) | proofpile-arXiv_066-681 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
"\\section{Introduction}\n\nFor a metric space $(X,d)$ with a notion of volume and boundary, an isop(...TRUNCATED) | proofpile-arXiv_066-1310 | {
"file_path": "/home/ubuntu/dolma-v1_7/algebraic-stack-train-0000.json.gz"
} |
End of preview. Expand
in Dataset Viewer.
This dataset is a 0.1% sample of Dolma v1.7, equating to around ~3B tokens and uploaded directly as a Hugging Face dataset.
As a pure sample, it maintains the ODC-BY license.
- Downloads last month
- 306