#  Eigenvalue contour lines of Kac–Murdock–Szegő matrices with a complex
parameter

George Fikioris and Christos Papapanos  School of Electrical and Computer
engineering, National Technical University of Athens, GR 157-73 Zografou,
Athens, Greece. Christos Papapanos is also with the Department of Electrical
Engineering and Computer Sciences, University of California, Berkeley, CA,
USA<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

A previous paper studied the so-called borderline curves of the
Kac–Murdock–Szegő matrix $K_{n}(\rho)=\left[\rho^{|j-k|}\right]_{j,k=1}^{n}$,
where $\rho\in\mathbb{C}$. These are the level curves (contour lines) in the
complex-$\rho$ plane on which $K_{n}(\rho)$ has a type-1 or type-2 eigenvalue
of magnitude $n$, where $n$ is the matrix dimension. Those curves have cusps
at all critical points $\rho=\rho_{c}$ at which multiple (double) eigenvalues
occur. The present paper determines corresponding curves pertaining to
eigenvalues of magnitude $N\neq n$. We find that these curves no longer
present cusps; and that, when $N<n$, the cusps have in a sense transformed
into loops. We discuss the meaning of the winding numbers of our curves.
Finally, we point out possible extensions to more general matrices.

###### Key words and phrases:

Toeplitz matrix, Kac–Murdock–Szegő matrix, eigenvalues, contour lines, level
curves, curve winding numbers

##  1\. Introduction and Preliminaries

This is the fourth in a series of papers [1, 2, 3] dealing with the
eigenvalues of the Toeplitz matrix

(1.1)
$K_{n}(\rho)=\left[\rho^{|j-k|}\right]_{j,k=1}^{n}=\begin{bmatrix}1&\rho&\rho^{2}&\ldots&\rho^{n-1}\\\
\rho&1&\rho&\ldots&\rho^{n-2}\\\ \rho^{2}&\rho&1&\ldots&\rho^{n-3}\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
\rho^{n-1}&\rho^{n-2}&\rho^{n-3}&\ldots&1\end{bmatrix}$

in which $n=3,4,5,\ldots$ and $\rho\in\mathbb{C}$. In the special case
$0<\rho<1$, $K_{n}(\rho)$ is usually called the Kac–Murdock–Szegő matrix. Ref.
[1] discusses its history and gives a number of applications. The matrix
$K_{n}(\rho)$ belongs to a number of frequently-investigated classes of
matrices. For any fixed $\rho\in\mathbb{C}$, for example, $K_{n}(\rho)$ is
complex-symmetric; for any fixed $\rho\in\mathbb{C}\setminus\mathbb{R}$ it is
non-Hermitian and nonnormal; and for any fixed $\rho\in\mathbb{C}$ with
$|\rho|>1$, the corresponding Laurent matrix (doubly infinite Toeplitz matrix)
does not have a well-defined and bounded symbol, thus complicating studies of
the spectral behavior of $K_{n}(\rho)$ [1, 3, 4]. Furthermore, the matrix
elements are complex-analytic functions of $\rho$, meaning that $K_{n}(\rho)$
belongs to the class of analytic matrix functions [3]. As noted in [1] and
[3], and as will be further suggested in the present work, studies of
$K_{n}(\rho)$ have initiated investigations pertaining to more general
matrices, and have served as means of illustrating the results of such
investigations. This has been true ever since the original work by Kac,
Murdock, and Szegő [5, 6] and remains true today [3, 7].

This work makes little use of [3] and can be considered a direct continuation
and generalization of [2]. It exploits the facts and associated terminology
that we list below.

  * •

Each eigenvalue of $K_{n}(\rho)$ is of one of two types, called type-1 and
type-2. The most notable distinguishing feature is that type-1 (type-2)
eigenvalues are associated with skew-symmetric (symmetric) eigenvectors. More
information on the two types can be found in Theorems 3.7 and 4.1 of [1], and
Remark 4.2 of [1].

  * •

For $k=1$ or $k=2$, the borderline curve $B_{n}^{(k)}$ is the level curve
(contour line) which consists of all $\rho$ in the complex plane for which
$K_{n}(\rho)$ has a type-$k$ eigenvalue whose magnitude is equal to the matrix
dimension $n$ [2]. The curves $B_{n}^{(1)}$ and $B_{n}^{(2)}$ are closed
curves [2] and are conjectured in [2] to be Jordan curves (with no self-
intersections).

  * •

For certain $n$-dependent values of $\rho$, $K_{n}(\rho)$ possesses repeated
eigenvalues. Apart from certain trivial cases, all repeated eigenvalues are
(algebraically) double eigenvalues equal to $-n$, and the critical points
$\rho=\rho_{c}$ for which such borderline/double eigenvalues occur satisfy
$\rho_{c}\in\mathbb{C}\setminus\mathbb{R}$ [2]. The aforementioned trivial
cases are described in Section 3 of [2]. As with any eigenvalue, each
borderline/double one is either of type-1 or of type-2. Any type-$k$ critical
point $\rho_{c}$, of course, belongs to the borderline curve $B_{n}^{(k)}$.

For $n=5$ the two borderline curves are shown in Fig. 1. The cusp-like curve
singularities discerned in the figure (at the points $\rho=\rho_{c}=\pm i2$ of
$B_{5}^{(1)}$ and $\rho=\rho_{c}\cong\pm 1.247\pm i1.456$ of $B_{5}^{(2)}$)
are true cusps [3] that signal the appearance of double eigenvalues. More
generally, the curve $B_{n}^{(k)}$ has a cusp singularity at $\rho=\rho_{c}$
iff $K_{n}(\rho)=K_{n}(\rho_{c})$ has a type-$k$ borderline/double eigenvalue
[2, 3]. As previously mentioned, any such eigenvalue equals $-n$.

For $n=5$, the matrix $K_{n}(\rho)=K_{5}(\rho)$ is special because two of its
eigenvalues can be found by means of explicit formulas. The two are
specifically given by

(1.2)
$\lambda^{(1)}=\frac{1}{2}\left[2-\rho^{2}-\rho^{4}\pm\rho(\rho^{2}-1)\sqrt{\rho^{2}+4}\right].$

Explicit computation of the two corresponding $\lambda^{(1)}$-eigenvectors is
also possible, showing that the two $\lambda^{(1)}$ in (1.2) are of type-$1$
(hence the superscript). Therefore, the solid line in Fig. 1 can be determined
by means of the simple formula (1.2). To illustrate, (1.2) immediately
verifies that both $K_{5}(i2)$ and $K_{5}(-i2)$ possess a double eigenvalue
equal to $-5$.

There seem to be very few cases for which such elementary and explicit
formulas can be found,111When $n=5$, for example, it does not seem possible to
explicitly determine the three type-2 eigenvalues (which give rise to
$B_{5}^{(2)}$). All known elementary cases are named in Section 1.1 of [3].
and the level curves in Fig. 1 were in fact obtained by more general methods
that were developed in [2]. The purpose of the present paper is to generalize
those methods to level curves (contour lines) pertaining to type-$k$
eigenvalues of magnitude $N$; to give a number of properties of the curves
thus obtained; to establish the significance of the curves’ winding numbers
[8]; and to point out possible extensions to more general matrices. We denote
the type-$k$ level curves of $K_{n}(\rho)$ by $L_{n,N}^{(k)}$, so that
$L_{n,n}^{(k)}=B_{n}^{(k)}$.

Figure 1. Introductory numerical results for $n=5$: Type-1 borderline curve
$B_{5}^{(1)}$ together with type-2 borderline curve $B_{5}^{(2)}$. The two
closed curves are depicted by solid and dashed lines, respectively. The curve
singularities at $\rho=\rho_{c}=\pm i2$ and $\rho=\rho_{c}\cong\pm 1.247\pm
i1.456$ are cusps. When $\rho$ is equal to any such $\rho_{c}$,
$K_{n}(\rho)=K_{5}(\rho)=K_{5}(\rho_{c})$ has a borderline/double (type-1 or
type-2) eigenvalue equal to $-5$.

Certain features of $L_{n,N}^{(k)}$ can be predicted beforehand. For any $N$
sufficiently near $n$, eigenvalue continuity makes one expect that
$L_{n,N}^{(k)}$ will be close to $B_{n}^{(k)}$. However, nontrivial multiple
eigenvalues can only have magnitude $n$; for this reason, we can further
anticipate that $L_{n,N}^{(k)}$ ($N\neq n$) will no longer exhibit cusps.

When $N$ is large, an additional prediction can be made. By Section 6.3 of
[2], for all sufficiently large $|\rho|$ the following hold: (i) There is
exactly one type-1 eigenvalue that has magnitude larger than $n$; (ii) there
is exactly one type-2 eigenvalue that has magnitude larger than $n$; and (iii)
the magnitudes of both these extraordinary eigenvalues are asymptotically
approximated by $|\rho|^{n-1}$ as $\rho\to\infty$ [2] . (Consistently with
this expectation, one of the two $\lambda^{(1)}$ in (1.2) satisfies
$|\lambda^{(1)}|\sim|\rho|^{4}$ as $|\rho|\to\infty$.) Accordingly, we can
further anticipate that both $L_{n,N}^{(1)}$ and $L_{n,N}^{(2)}$ approach the
circle

(1.3) $|\rho|=N^{\frac{1}{n-1}}$

when $N$ is large.

Apart from possible difficulties in numerical implementations, the methods to
be developed herein allow $N$ to be arbitrarily large. More interestingly, we
allow $N$ to become smaller than $n$. However $N$ must remain larger than a
certain threshold to be denoted by $N_{\mathrm{min}}(n)$. This parameter
equals $1$ for $n=3$, approximates $1.1$ for $n=4$, and approaches $0.21n$ for
$n=5,6,\ldots$. As we will see, $N_{\mathrm{min}}(n)$ is small enough to allow
$L_{n,N}^{(k)}$ to differ significantly from $B_{n}^{(k)}$.

The heart of this paper is Theorem 4.1 below, which stems from the following
lemma.

######  Lemma 1.1.

[2] Let $\rho\in\mathbb{C}$. $\lambda\in\mathbb{C}$ is a type-1 (or type-2)
eigenvalue of $K_{n}(\rho)$ iff

(1.4) $\lambda=-\frac{\sin(n\mu)}{\sin{\mu}}\quad\left({\rm
or}\,\,\lambda=\frac{\sin(n\mu)}{\sin{\mu}}\right),$

where $\mu\in\mathbb{C}$ satisfies

(1.5)
$\rho=\frac{\sin{\frac{(n+1)\mu}{2}}}{\sin{\frac{(n-1)\mu}{2}}}\quad\left({\rm
or}\,\,\rho=\frac{\cos{\frac{(n+1)\mu}{2}}}{\cos{\frac{(n-1)\mu}{2}}}\right).$

The statement of our Theorem 4.1 resembles that of Theorem 4.1 in [2]. As
already mentioned, there are essential differences in the two theorems’
predictions, especially when $n$ approaches $N_{\mathrm{min}}(n)$.
Furthermore, there are important differences in the theorems’ specifics, with
the theorem herein being more complicated. Accordingly, much of this paper is
devoted to an examination of certain auxiliary functions that appear in the
theorem.

##  2\. Useful properties of Chebyshev polynomials

We begin with some equalities and inequalities that are most concisely shown
when expressed in terms of $T_{k}(z)$ and $U_{k}(z)$ (i.e. the first- and
second-kind Chebyshev polynomials of degree $k$):

######  Lemma 2.1.

(2.1) $T_{k}(\cos u)=\cos(ku),\quad u\in\mathbb{C},\quad k=0,1,2,\ldots;$
(2.2) $U_{k}(\cos u)=\frac{\sin[(k+1)u]}{\sin u},\quad u\in\mathbb{C},\quad
k=0,1,2,\ldots;$ (2.3) $U_{k}(\cosh v)=\frac{\sinh[(k+1)v]}{\sinh v},\quad
v\in\mathbb{C},\quad k=0,1,2,\ldots;$ (2.4)
$U_{k}(z)=zU_{k-1}(z)+T_{k}(z),\quad z\in\mathbb{C},\quad k=1,2,3\ldots;$
(2.5) $-1\leq T_{k}(x)\leq 1,\quad-1\leq x\leq 1,\quad k=0,1,2,\ldots;$ (2.6)
$U_{k}(z)=\sum_{m=0}^{k}z^{m}\,T_{k-m}(z),\quad z\in\mathbb{C},\quad
k=0,1,2,\ldots;$ (2.7) $-(k+1)\leq U_{k}(x)\leq k+1,\quad-1\leq x\leq 1,\quad
k=0,1,2,\ldots;$ (2.8) $1\leq x_{1}<x_{2}\implies 1\leq
T_{k}(x_{1})<T_{k}(x_{2}),\quad k=1,2,3,\ldots;$ (2.9) $1\leq
x_{1}<x_{2}\implies k+1\leq U_{k}(x_{1})<U_{k}(x_{2}),\quad k=1,2,3,\ldots.$

###### Proof.

Eqns. (2.1) and (2.2) are definitions of the Chebyshev polynomials. Therefore
with $z=\cos u$, (2.4) is equivalent to a trigonometric identity. Set $u=iv$
($v\in\mathbb{C}$) in (2.2) to get (2.3). With $\cos u=x$ ($-1\leq x\leq 1$),
(2.1) implies (2.5). To verify (2.6), use $U_{0}(z)=T_{0}(z)=1$ and induction.
Eqn. (2.7) is trivial when $k=0$; and, for $k\geq 1$, (2.7) is a consequence
of (2.6), (2.5), and the triangle inequality. Setting $u=iv$ ($v\geq 0$) in
(2.1) shows that, for $k\geq 1$, $T_{k}(\cosh v)$ is a strictly increasing
function of $v$; this fact (together with $T_{k}(1)=1$) is tantamount to
(2.8). Setting $z=x$ ($x\geq 1$) in (2.6) and invoking (2.8), it is seen that
$U_{k}(x)$ is a sum of positive and strictly increasing functions, yielding
(2.9). ∎

######  Corollary 2.2.

The trigonometric/hyperbolic versions of (2.7)/(2.9) are

(2.10) $\bigg{|}\frac{\sin(ku)}{\sin u}\bigg{|}\leq k,\quad
u\in\mathbb{R},\quad k=1,2,3,\ldots,$ (2.11) $0\leq v_{1}<v_{2}\implies
k\leq\frac{\sinh(kv_{1})}{\sinh v_{1}}<\frac{\sinh(kv_{2})}{\sinh v_{2}},\quad
k=2,3,4,\ldots.$

We conclude this section with a lemma on the zeros and maxima of the positive
and even function $|U_{k}(z)|$.

######  Lemma 2.3.

For $k=2,3,4,\ldots$, let $\alpha_{k,m}$ be a zero of $U_{k}(z)$. The
$\alpha_{k,m}$ are all real, simple, and lie in the interval $(-1,1)$. They
are given by

(2.12) $\alpha_{k,m}=\cos\frac{(k+1-m)\pi}{k+1},\quad m=1,2,\ldots,k.$

Let $\beta_{k,m}$ be a zero of the derivative $U^{\prime}_{k}(z)$. The
$\beta_{k,m}$ are also real, with

(2.13) $\alpha_{k,m}<\beta_{k,m}<\alpha_{k,m+1},\quad m=1,2,\ldots,k-1.$

In (2.12), the indexing is such that $\alpha_{k,m}$ increases when $m$
increases, and ditto for the $\beta_{k,m}$ in (2.13).

The two numbers $|U_{k}(\beta_{k,1})|$ and $|U_{k}(\beta_{k,k-1})|$ are equal
and are the largest of the values $|U_{k}(\beta_{k,m})|$. Furthermore,

(2.14) $0<\max_{\alpha_{k,1}\leq
x\leq\alpha_{k,k}}|U_{k}(x)|=|U_{k}(\beta_{k,1})|=|U_{k}(\beta_{k,k-1})|<k+1,$

so that $|U_{k}(\beta_{k,1})|$ is the global maximum of the even function
$|U_{k}(x)|$ in the interval $[\alpha_{k,1},\alpha_{k,k}]$.

When $\alpha_{k,k}\leq x\leq 1$, the function $U_{k}(x)$ strictly increases
from $U_{k}(\alpha_{k,k})=0$ to $U_{k}(1)=k+1$. And when $-1\leq
x\leq\alpha_{k,1}$, $|U_{k}(x)|$ strictly decreases from $|U_{k}(-1)|=k+1$ to
$U_{k}(\alpha_{k,1})=0$.

###### Proof.

By (2.2), the $\alpha_{k,m}$ in (2.12) are zeros of $U_{k}(z)$. Since there
are $k$ distinct zeros, all are simple. By Rolle’s theorem and for
$m=1,2,\ldots,k-1$, $U^{\prime}_{k}(\beta_{k,m})=0$ for some $\beta_{k,m}$
satisfying (2.13). Therefore all zeros $\beta_{k,m}$ of $U^{\prime}_{k}(z)$
are real and are bracketed by the inequality in (2.13). By the extreme value
theorem and $U_{k}(\alpha_{k,m})=U_{k}(\alpha_{k,m+1})=0$, the nonnegative
function $|U_{k}(x)|$ assumes its maximum value within
$[\alpha_{k,m},\alpha_{k,m+1}]$ iff $x=\beta_{k,m}$. As $|U_{k-1}(x)|$ is
even, we have $|U_{k}(\beta_{k,1})|=|U_{k}(\beta_{k,k-1})|$. As discussed in
[9], the $|U_{k}(\beta_{k,m})|$ increase as $|\beta_{k,m}|$ increases away
from zero, and all $|U_{k}(\beta_{k,m})|$ are smaller than the endpoint values
$|U_{k}(\pm 1)|=k+1$, giving (2.14). Since no zero of $U^{\prime}_{k}(x)$ lies
in the interval $[\alpha_{k,k},1]$, $U_{k}(x)$ increases strictly in this
interval, from its value $0$ at the left endpoint to its value $k+1$ at the
right endpoint. The statement pertaining to the interval $[-1,\alpha_{1,k}]$
follows because $|U_{k}(x)|$ is even. ∎

##  3\. The functions $h_{n,N}(v)$ and $g_{n,N}(u)$

This section defines and gives useful properties of the functions $h_{n,N}(v)$
and $g_{n,N}(u)$, which will play an important role in our main theorem.

###  3.1. The function $h_{n,N}(v)$

######  Definition 3.1.

For $n=3,4,\ldots$ and $N\in\mathbb{R}$ with $N>0$, we define

(3.1) $h_{n,N}(v)=\sinh^{2}(nv)-N^{2}\sinh^{2}{v},\quad v\geq 0.$

In the case $N<n$, the behavior that interests us is straightforward. It is
given by the following proposition.

######  Proposition 3.2.

For $N<n$, the function $h_{n,N}(v)$ equals zero when $v=0$, and is positive
and strictly increasing when $v\in(0,+\infty)$.

###### Proof.

The equality $h_{n,N}(0)=0$ is obvious. For $v>0$, write

(3.2) $h_{n,N}(v)=[\sinh(nv)+N\sinh{v}][\sinh(nv)-N\sinh{v}].$

The first factor is positive and strictly increasing. The second factor equals
0 when $v=0$, equals $+\infty$ when $v=+\infty$ and has a derivative equal to
$n\cosh(nv)-N\cosh{v}$. As $\cosh(nv)>\cosh v>1$ and $n>N$, this derivative
cannot vanish. Thus the second factor is also positive and strictly
increasing, proving the lemma. ∎

In the case $N>n$, the behavior of $h_{n,N}(v)$ is not as simple. To describe
it, we introduce an auxiliary quantity $v_{0}(n,N)$. This is defined as the
solution to a transcendental equation as follows.

######  Lemma 3.3.

For $N>n$, the equation

(3.3) $\frac{\sinh(nv)}{\sinh v}=N,\quad v\geq 0,$

has a unique solution. We denote this solution, which is positive, by
$v=v_{0}(n,N)$. Its limiting value as $N\to n$ is

(3.4) $\lim_{N\to n+0}v_{0}(n,N)=0.$

###### Proof.

The continuous function on the left-hand side of (3.3) is strictly increasing
in $[0,+\infty)$ by Corollary 2.2, equals $n$ when $v=0$, and equals $+\infty$
when $v=+\infty$. Since the right-hand side $N$ is greater than $n$ there is a
unique solution, and it is positive. ∎

We can now give our results for $N>n$.

######  Proposition 3.4.

For $N>n$, the function $h_{n,N}(v)$ is negative in the interval
$(0,v_{0}(n,N))$, vanishes when $v=0$ or $v=v_{0}(n,N)$, and is positive and
strictly increasing in the interval $(v_{0}(n,N),+\infty)$.

###### Proof.

It is obvious that $h_{n,N}(0)=0$. By (3.2) and Lemma 3.3, $h_{n,N}(v)$
vanishes at exactly positive point, namely $v=v_{0}(n,N)$. The derivative
$h^{\prime}_{n,N}(v)$ is

$h^{\prime}_{n,N}(v)=n\sinh(2nv)-N^{2}\sinh{(2v)},\quad v>0.$

By Lemma 3.3, $h^{\prime}_{n,N}(v)$ also vanishes at exactly one point which,
by Rolle’s theorem, must lie in the interval $(0,v_{0}(n,N))$. (This point, in
fact, is $v=\frac{1}{2}v_{0}(n,\frac{N^{2}}{n})$.) Finally, we can easily show
that $h_{n,N}(v)$ is negative as $v\to 0+0$ and positive as $v\to+\infty$.
These aforementioned properties prove the lemma. ∎

###  3.2. The function $g_{n,N}(u)$

We now discuss the following function $g_{n,N}(u)$.

######  Definition 3.5.

For $n=3,4,\ldots$ and $N\in\mathbb{R}$ with $N>0$, we define

(3.5) $g_{n,N}(u)=N^{2}\sin^{2}{u}-\sin^{2}(nu),\quad-\pi\leq u\leq\pi.$

We first assume $N>n$. In this case, what interests us is uncomplicated:

######  Proposition 3.6.

For $N>n$, the function $g_{n,N}(u)$ is zero when $u=0$ and $|u|=\pi$, and
positive when $0<|u|<\pi$.

###### Proof.

The statement about zeros is trivial. In the case $0<|u|<\pi$ we have

(3.6) $g_{n,N}(u)>n^{2}\sin^{2}u-\sin^{2}(nu)\geq 0,$

where we used (3.5), the inequalities $N>n$ and $\sin u\neq 0$, and Corollary
2.2. ∎

When $N<n$, the behaviors of interest are best explained in terms of the
auxiliary quantities $x^{\prime}_{0}(n)$, $N_{\mathrm{min}}(n)$, $x_{0}(n,N)$,
and $u_{0}(n,N)$. These are defined via solutions to transcendental equations,
as described by the two lemmas that follow.

######  Lemma 3.7.

The equation

(3.7) $U^{\prime}_{n-1}(x)=0,\quad\cos\frac{2\pi}{n}<x<\cos\frac{\pi}{n},$

has a unique solution which we denote by $x^{\prime}_{0}(n)$. Further, the
$N_{\mathrm{min}}(n)$ defined by

(3.8)
$N_{\mathrm{min}}(n)=\Big{|}U_{n-1}\left(x^{\prime}_{0}(n)\right)\Big{|}$

satisfies

(3.9) $1\leq N_{\mathrm{min}}(n)<n,$

and is the global maximum attained by $|U_{n-1}(x)|$ in the interval
$[-\cos\frac{\pi}{n},\cos\frac{\pi}{n}]$.

###### Proof.

In Lemma 2.3, set $k=n-1$ to obtain
$|\alpha_{k,1}|=\alpha_{k,k}=\cos\frac{\pi}{n}$,
$\alpha_{k,k-1}=\cos\frac{2\pi}{n}$,
$|\beta_{k,1}|=\beta_{k,k-1}=x^{\prime}_{0}(n)$, and
$N_{\mathrm{min}}(n)=\big{|}U_{n-1}\left(x^{\prime}_{0}(n)\right)\big{|}$.
Since $\cos\frac{3\pi}{2n}$ belongs to
$(\cos\frac{2\pi}{n},\cos\frac{\pi}{n})$ in which $N_{\mathrm{min}}(n)$ is the
maximum value, we have
$N_{\mathrm{min}}(n)\geq\big{|}U_{n-1}\left(\cos\frac{3\pi}{2n}\right)\big{|}$;
the first inequality in (3.9) then follows from (2.2). ∎

######  Remark 3.8.

As we will see, the value $N_{\mathrm{min}}(n)$ is the smallest level $N$ (of
the level curve $L_{n,N}^{(k)}$, for both $k=1$ and $k=2$) for which our main
theorem is valid. The previously-mentioned large-$n$ approximation
$N_{\mathrm{min}}(n)\cong 0.21n$ results by substituting
$x^{\prime}_{0}(n)\cong 3\pi/(2n)$ into (3.8), and using (2.2).

######  Lemma 3.9.

Let $N$ be such that $N_{\mathrm{min}}(n)<N<n$, where $N_{\mathrm{min}}(n)$ is
defined in Lemma 3.7. Then the equation

(3.10) $|U_{n-1}(x)|=N,\quad-1<x<1,$

has precisely one positive solution, to be denoted by $x_{0}(n,N)$; and
precisely one negative solution equal to $-x_{0}(n,N)$. It is also true that

(3.11) $|U_{n-1}(x)|\begin{cases}<N,\quad|x|<x_{0}(n,N),\\\
=N,\quad|x|=x_{0}(n,N),\\\ >N,\quad x_{0}(n,N)<|x|\leq 1.\end{cases}$

We further denote

(3.12) $u_{0}(n,N)=\arccos\left[x_{0}(n,N)\right].$

This quantity satisfies

(3.13) $0<u_{0}(n,N)<\frac{\pi}{n}\leq\frac{\pi}{3}$

and

(3.14) $\lim_{N\to n-0}u_{0}(n,N)=0.$

###### Proof.

Since $|U_{n-1}(x)|$ is even, it is sufficient to discuss solutions
$x_{0}(n,N)$ belonging to $[0,1)$, and to show (3.11) for $0\leq x\leq 1$.
Eqn. (3.10) can have no solution in the subinterval
$\big{[}0,\cos\frac{\pi}{n}\big{)}$ because
$N>N_{\mathrm{min}}(n)\geq|U_{n-1}(x)|$ for all $x$ in the subinterval, see
Lemma 3.7. There is a unique solution in $\left[\cos\frac{\pi}{n},1\right)$,
however, because the left-hand side of (3.10) increases strictly from $0$ to
$n$ (see the last assertion in Lemma 2.3), while the right-hand side $N$
belongs to $(0,n)$. We have thus shown existence, uniqueness, (3.11), as well
as the inequality

$\cos\frac{\pi}{n}<x_{0}(n,N)<1,$

which, via the definition (3.12), gives (3.13). ∎

We can now describe the desired behavior of $g$ when $N<n$. Besides $N<n$, we
also assume $N>N_{\mathrm{min}}(n)$.

######  Proposition 3.10.

Let $N$ be such that $N_{\mathrm{min}}(n)<N<n$, where $N_{\mathrm{min}}(n)$ is
defined in Lemma 3.7. Then the function $g_{n,N}(u)$ satisfies

(3.15) $g_{n,N}(u)\begin{cases}>0,\quad u_{0}(n,N)<|u|<\pi-u_{0}(n,N);\\\
=0,\quad u=0,\mathrm{\ or\ }|u|=\pi,\mathrm{\ or\ }|u|=u_{0}(n,N),\mathrm{\
or\ }|u|=\pi-u_{0}(n,N);\\\ <0,\quad 0<|u|<u_{0}(n,N),\mathrm{\ or\ }\pi-
u_{0}(n,N)<|u|<\pi.\end{cases}$

where $u_{0}(n,N)$ is defined in Lemma 3.9.

###### Proof.

Set $u=\arccos x$ in (3.5) and use (2.2) to get

(3.16) $g_{n,N}(\arccos
x)=(1-x^{2})\Big{(}N+|U_{n-1}(x)|\Big{)}\Big{(}N-|U_{n-1}(x)|\Big{)},\quad-1\leq
x\leq 1.$

By (3.12), the desired inequality (3.15) amounts to

(3.17) $g_{n,N}(\arccos x)\begin{cases}>0,\quad|x|<x_{0}(n,N);\\\
=0,\quad|x|=1,\mathrm{\ or\ }|x|=x_{0}(n,N);\\\ <0,\quad
x_{0}(n,N)<|x|<1.\end{cases}$

which holds because the first factor in (3.16) is zero when $|x|=1$ and
positive otherwise; the second is positive for all $|x|$; and the sign of the
third is given via (3.11). ∎

##  4\. Main theorem; some curve properties

This section presents a theorem that allows one to determine the level curves
$L_{n,N}^{(k)}$ of $K_{n}(\rho)$, and then gives certain rudimentary
properties of the said curves. Given $k$, $n$, and $N$ (and as long as
$N>N_{\mathrm{min}}(n)$), the theorem enables us to compute all complex values
$\rho$ that give rise to a type-$k$ eigenvalue with a magnitude equal to $N$.
The desired values of $\rho$ form the range of a complex-valued function
$f_{n,N}^{(1)}(u)$ [or $f_{n,N}^{(2)}(u)$], where $u$ belongs to $[-\pi,\pi]$,
or to an explicitly defined subinterval of $[-\pi,\pi]$. The functions
$f_{n,N}^{(1)}(u)$ and $f_{n,N}^{(2)}(u)$, finally, are defined via the unique
solution to a certain transcendental equation.

######  Theorem 4.1.

Let $N\in\mathbb{R}$ be such that $N_{\mathrm{min}}(n)<N<+\infty$, where
$N_{\mathrm{min}}(n)$ is given in Lemma 3.7. For $\rho\in\mathbb{C}$,
$K_{n}(\rho)$ possesses a type-1 eigenvalue $\lambda$ of magnitude
$|\lambda|=N$ iff $\rho=f_{n,N}^{(1)}(u)$ and $\lambda=b_{n,N}^{(1)}(u)$ where

(4.1)
$f_{n,N}^{(1)}(u)=\frac{\displaystyle\sin{\frac{(n+1)\mu(n,N,u)}{2}}}{\displaystyle\sin{\frac{(n-1)\mu(n,N,u)}{2}}},\qquad
b_{n,N}^{(1)}(u)=-\frac{\sin[n\mu(n,N,u)]}{\sin\mu(n,N,u)},$

in which

(4.2) $\mu(n,N,u)=u+iv(n,N,u).$

In (4.2), $u$ varies within $[-\pi,\pi]$ as specified below. With $g_{n,N}(u)$
defined in (3.5), the function $v(n,N,u)$ is the unique root of the
transcendental equation

(4.3) $\sinh^{2}(nv)-N^{2}\sinh^{2}{v}=g_{n,N}(u),$

with the desired root belonging to $(0,+\infty)$ or $[0,+\infty)$ as also
specified below.

Case 1: When $N>n$, the $v(n,N,u)$ is the unique positive root of the
transcendental equation (4.3), for all $u\in[-\pi,\pi]$. Furthermore,

(4.4) $v(n,N,u)\geq v_{0}(n,N),\quad u\in[-\pi,\pi],$

where $v_{0}(n,N)$ is defined in Lemma 3.3. In (4.4), equality occurs iff
$u=0$ or $|u|=\pi$.

Case 2: When $N_{\mathrm{min}}(n)<N\leq n$, the $v(n,N,u)$ is the unique
nonnegative root of the transcendental equation (4.3). Here, the values of $u$
are limited according to

(4.5) $u_{0}(n,N)\leq|u|\leq\pi-u_{0}(n,N),$

in which $u_{0}(n,N)$ is defined in Lemma 3.9 (when $N=n$, use the limiting
value (3.14)).

Similarly, $K_{n}(\rho)$ possesses a borderline type-2 eigenvalue $\lambda$
iff $\rho=f_{n,N}^{(2)}(u)$ and $\lambda=b_{n,N}^{(2)}(u)$ where

(4.6)
$f_{n,N}^{(2)}(u)=\frac{\cos{\displaystyle\frac{(n+1)\mu(n,N,u)}{2}}}{\cos{\displaystyle\frac{(n-1)\mu(n,N,u)}{2}}},\qquad
b_{n,N}^{(2)}(u)=\frac{\sin[n\mu(n,N,u)]}{\sin\mu(n,N,u)},$

in which $u$, $v(n,N,u)$, and $\mu(n,N,u)$ are exactly the same as in the
above-discussed type-1 case.

###### Proof.

For $N=n$, the theorem reduces to Theorem 4.1 of [2] and Lemma 4.2(ii) of [2].
We thus assume that $N\neq n$ throughout. Accordingly, Case 2 amounts to
$N_{\mathrm{min}}(n)<N<n$.

In Lemma 1.1, take $|\lambda|=N$ and set $u={\rm Re}\mu$ and $v={\rm Im}\mu$
to see that $K_{n}(\rho)$ has a borderline type-1 eigenvalue $\lambda$ iff
$\rho=\rho_{n,N}(u,v)$ and $\lambda=\lambda_{n,N}(u,v)$ where

(4.7)
$\rho_{n,N}(u,v)=\frac{\displaystyle\sin{\frac{(n+1)u}{2}}\cosh{\frac{(n+1)v}{2}}+i\cos{\frac{(n+1)u}{2}}\sinh{\frac{(n+1)v}{2}}}{\displaystyle\displaystyle\sin{\frac{(n-1)u}{2}}\cosh{\frac{(n-1)v}{2}}+i\cos{\frac{(n-1)u}{2}}\sinh{\frac{(n-1)v}{2}}},\quad
u,v\in\mathbb{R},$

and

(4.8) $\lambda_{n,N}(u,v)=-\frac{\sin[n(u+iv)]}{\sin(u+iv)},\quad
u,v\in\mathbb{R},$

where $u$, $v$, and $n$ are interrelated via

(4.9) $N^{2}=\frac{\sin^{2}(nu)+\sinh^{2}(nv)}{\sin^{2}{u}+\sinh^{2}{v}},\quad
u,v\in\mathbb{R}.$

Since the right-hand sides of (4.7)–(4.9) are $2\pi$-periodic in $u$, we
assume $u\in[-\pi,\pi]$ with no loss of generality. Since, also,
$\rho_{n,N}(-u,-v)=\rho_{n,N}(u,v)$ and
$\lambda_{n,N}(-u,-v)=\lambda_{n,N}(u,v)$, we further assume
$v\in[0,+\infty)$. We must now consider the two cases separately.

Case 1: When $N>n$, we limit ourselves to $v\in(0,+\infty)$: For $v=0$, the
right-hand side of the transcendental equation (4.9) is smaller than or equal
to $n^{2}$ by Corollary 2.2; and $n^{2}$ is, in turn, smaller than $N^{2}$.
Thus no $u\in[-\pi,\pi]$ can satisfy (4.9) and we take $v>0$.

By Proposition 3.6, the function $g_{n,N}(u)$ in the right-hand side of (4.3)
is nonnegative. By Proposition 3.4, when $v>0$ the function $h_{n,N}(v)$ in
the left-hand side of (4.3) is nonnegative only when
$v\in[v_{0}(n,N),\infty)$, and is strictly increasing there. We have thus
shown that (4.3) has a unique positive solution $v$—which we denote by
$v=v(n,N,u)$—and we have also shown (4.4). In (4.4), equality occurs iff
$g(n,N)(u)=0$ which, by Proposition 3.6, is equivalent to $u=0$ or $|u|=\pi$.

For Case 1, (4.9) is therefore equivalent to the definition (3.5) and the
transcendental equation (4.3), with $v\in(0,+\infty)$ and $u\in[-\pi,\pi]$.

Case 2: When $N_{\mathrm{min}}(n)<N<n$ our reasoning is similar. To begin
with, we exclude the values $u=0$ and $|u|=\pi$, for which the right-hand side
of (4.9) is larger than or equal to $n^{2}$ by Corollary 2.2.

By Proposition 3.2, the function $h_{n,N}(v)$ on the left-hand side of the
transcendental equation (4.3) is non-negative and strictly increasing. Thus,
(4.3) has a unique solution iff $g_{n,N}(u)\geq 0$ which, by Proposition 3.10,
is equivalent to $u=0$, $|u|=\pi$, or $u_{0}(n,N)\leq|u|\leq\pi-u_{0}(n,N)$.
As the first two cases have been excluded, we have shown (4.5). Thus in Case 2
too, (4.9) is equivalent to (3.5) and (4.3); but here, $v\in[0,\infty)$ and
$|u|$ varies in the interval specified in (4.5).

With $v=v(n,N,u)$ thus determined, the $\rho_{n,N}(u,v)$ of (4.7) and the
$\lambda_{n,N}(u,v)$ of (4.8) are no longer functions of $v$, and the
notations $f_{n,N}^{(1)}(u)=\rho_{n,N}(u,v)$ and
$b_{n,N}^{(1)}(u)=\lambda_{n,N}(u,v)$ prove (4.1) with (4.2). We have thus
shown all assertions pertaining to type-1 eigenvalues.

For the type-2 case, proceed as before with

$\rho_{n,N}(u,v)=\frac{\displaystyle\cos{\frac{(n+1)u}{2}}\cosh{\frac{(n+1)v}{2}}-i\sin{\frac{(n+1)u}{2}}\sinh{\frac{(n+1)v}{2}}}{\displaystyle\displaystyle\cos{\frac{(n-1)u}{2}}\cosh{\frac{(n-1)v}{2}}-i\sin{\frac{(n-1)u}{2}}\sinh{\frac{(n-1)v}{2}}},\quad
u,v\in\mathbb{R},$

∎

For brevity, denote by $R(n,N)$ the range in which $u$ is supposed to vary in
Cases 1 and 2, so that

(4.10) $R(n,N)=\begin{cases}[-\pi,\pi],\quad N>n\\\
[-\pi+u_{0}(n,N),-u_{0}(n,N)]\cup[u_{0}(n,N),\pi-u_{0}(n,N)],\
N_{\mathrm{min}}(n)<N\leq n\end{cases}$

We this notation, we can re-state the essence of Theorem 4.1 as follows.

######  Corollary 4.2.

Let $N\in\mathbb{R}$ with $N>N_{\mathrm{min}}(n)$ and let $k=1$ or $k=2$. The
matrix $K_{n}(\rho)$ possesses a type-$k$ eigenvalue of magnitude $N$ iff
$\rho\in L_{n,N}^{(k)}$, where $L_{n,N}^{(k)}$ is the curve given by

(4.11) $L_{n,N}^{(k)}=\left\\{\rho\in\mathbb{C}:\rho=f^{(k)}_{n,N}(u)\mathrm{\
for\ some\ }u\in R(n,N)\right\\},$

in which $f^{(k)}_{n,N}(u)$ is defined in Theorem 4.1.

The proposition that follows gives some elementary properties of the
$L_{n,N}^{(k)}$.

######  Proposition 4.3.

The level curves $L_{n,N}^{(1)}$ and $L_{n,N}^{(2)}$ exhibit the following
properties.
(i) For $k=1$ and $k=2$, $L_{n,N}^{(k)}$ intersects the real axis exactly
twice.
(ii) Both $L_{n,N}^{(1)}$ and $L_{n,N}^{(2)}$ are symmetric with respect to
the real $\rho$-axis.
(iii) The union $L_{n,N}^{(1)}\cup L_{n,N}^{(2)}$ is symmetric with respect to
the origin $\rho=0$.
(iv) For $n=3,5,7\ldots$, both $L_{n,N}^{(1)}$ and $L_{n,N}^{(2)}$ are
symmetric with respect to the imaginary $\rho$-axis.
(v) For $n=4,6,8\ldots$, $L_{n,N}^{(1)}$ and $L_{n,N}^{(2)}$ are mirror images
of one another with respect to the imaginary $\rho$-axis.

###### Proof.

The proof is very similar to the proof of Proposition 4.4 of [2]. ∎

We end this section with a lemma that we will use shortly. It states that
positive (negative) $u$-values correspond to points $L_{n,N}^{(k)}$ that lie
in the lower-half (upper-half) plane. It also gives the $u$-values for which
the intersections with the real axis occur.

######  Lemma 4.4.

Let $u\in R(n,N)$, let $N>N_{\mathrm{min}}(n)$, and let $k=1$ or $k=2$. Then

(4.12) $u>0\iff\mathrm{Im}f_{n,N}^{(k)}(u)<0$

Furthermore, the $u$-values for which $L_{n,N}^{(k)}$ intersects the real axis
are $u=0$ and $|u|=\pi$ in Case 1 ($N>n$), and $|u|=u_{0}(n,N)$ and $|u|=\pi-
u_{0}(n,N)$ in Case 2 ($N_{\mathrm{min}}<N<n$).

###### Proof.

See the proof of Lemma 4.2 of [2]. ∎

##  5\. Numerical results; further curve properties

We begin this section by stating our results in the form of an algorithm that
can generate $L^{(k)}_{n,N}$.

###  5.1. Algorithm

Given $n$, $N$, and a value $k=1$ or $k=2$, a point $\rho\in L^{(k)}_{n,N}$
can be determined as follows. Determine the required range $R(n,N)$ from
(4.10); pick $u\in R(n,N)$; compute $g_{n,N}(u)$ from (3.5); solve the
transcendental equation (4.3) for $v=v(n,N,u)$; set $\mu(n,N,u)=u+iv(n,N,u)$;
find $f_{n,N}^{(k)}(u)$ from (4.1) or (4.6); and set $\rho=f^{(k)}_{n,N}(u)$.
Repeat the above process for many $u\in R(n,N)$ until the continuous curve
$L^{(k)}_{n,N}$ is depicted.

###  5.2. Initial numerical results

Figure 2. Case $n=5$: Type-1 level curves $L^{(1)}_{5,N}$ for $N=5$ (solid
line), $N=3$ (dashed line), and $N=30$ (dot-dashed line).

For $n=5$ and $k=1$, Fig. 2 shows the $L^{(k)}_{n,N}$ generated by our
algorithm for three values of $N$, namely $N=5$, $N=30$, and $N=3$ (note that
$3>N_{\textrm{min}}(5)=1.25$). The $L^{(1)}_{5,5}$ curve is, of course, the
same as the type-1 borderline curve $B_{5}^{(1)}$ of Fig. 1. As anticipated
(see Introduction), singularities appear only in the $N=5$ curve; and the
$N=30$ curve is quite close to the circle $|\rho|=30^{1/4}\cong 2.3$. While
the $N=3$ curve presents no singularities, it self-intersects twice, and loops
around the two cusps of the $N=5$ curve. Clearly, a self-intersection point of
$L^{(k)}_{n,N}$ gives rise to two type-$k$ eigenvalues of magnitude $N$. In a
sense, the cusp of $B_{n}^{(k)}=L_{n,n}^{(k)}$ has transformed into the loop
of $L_{n,N}^{(k)}$. Conversely, it is illuminating to think of the cusp as a
degenerate loop.

Recall that the predictions of Fig. 2 can be checked via the elementary
formulas (1.2). When $\rho=i\sqrt{2}$, for example, (1.2) gives
$\lambda^{(1)}=\pm 3i$, meaning that the matrix $K_{5}(i\sqrt{2})$ has two
type-1 eigenvalues of magnitude $3$. This fact is in agreement with Fig. 2, in
which $\rho=i\sqrt{2}$ is seen to be a self-intersection point of the $N=3$
curve. As another example, the $N=5$ and $N=3$ curves are seen in the figure
to intersect once in the first quadrant. A close focus tells us that the
intersection point is $\rho\cong 0.139+i1.693$, meaning that the two
$\lambda^{(1)}$ must have magnitudes $5$ and $3$ when $\rho$ assumes the
aforementioned value. Eqn. (1.2) verifies this is indeed the case (the
respective phases are $1.87$ and $-2.12$ rads).

For $n=12$ and $k=2$, Fig. 3 shows the $L^{(k)}_{n,N}$ for $N=12$, $20$, $30$,
$70$, and $200$. An increase in $N$ is seen to result in a slightly larger
curve. Observe how the cusp singularities of the smallest ($N=12$) curve222The
number of cusps is $10$, corresponding to $10$ different borderline/double
type-2 eigenvalues. The number $10$ can be found a priori via Theorem 4.5 of
[1]. Note that no cusp lies on the real axis. This is always so because, for
$\rho\in\mathbb{R}$ with $|\rho|>1$, $K_{n}(\rho)$ has no multiple
eigenvalues, see Proposition 6.1 of [1]. gradually recede as $N$ grows, with
the largest ($N=200$) curve closely approaching the circle
$|\rho|=200^{1/11}\cong 1.62$, in accordance with (1.3).

Figure 3. Case $n=12$: Type-2 level curves $L^{(2)}_{12,N}$ for $N=12$,
$N=20$, $N=30$, $N=70$, and $N=200$. An increase in $N$ enlarges the curve.

###  5.3. On the existence of loops

As long as $N_{\mathrm{min}}(n)<N<n$, loops similar to the ones in Fig. 2 were
observed in all the $L_{n,N}^{(k)}$ generated by our algorithm. It is thus
worth proving that such loops always occur. A completely general proof seems
difficult, so we limit ourselves to the special case used in Section 6 of [3]
as a means of studying eigenvalue bifurcations. Our special case assumes that
$n=3,5,7,\ldots$ and that

(5.1) $k=\begin{cases}1,\mathrm{\ if\ }n=5,9,13,\ldots\\\ 2,\mathrm{\ if\
}n=3,7,11,\ldots\end{cases}$

Subject to the above assumptions, [3] shows that $B_{n}^{(k)}=L_{n,n}^{(k)}$
has a cusp on the positive-imaginary semi-axis, and one on the negative-
imaginary semi-axis, as in the example in Fig. 2. Theorem 5.3 will show that
the situation is different when $N<n$.

######  Lemma 5.1.

Let $N\in\mathbb{R}$ with $1\leq N<n$, where $n=3,5,7,\ldots$. Then the
equation

(5.2) $\cosh(nv)=N\cosh v,\quad v\geq 0$

has a unique nonnegative solution, to be denoted by $v_{\mathrm{im}}(n,N)$.

###### Proof.

The function $\cosh(nv)-N\cosh v$ equals $1-N<0$ when $v=0$, equals $+\infty$
when $v=+\infty$, and has the derivative $n\sinh(nv)-N\sinh v$, which is
positive by Corollary 2.2. ∎

The lemma that follows gives four points for which $L_{n,N}^{(k)}$ intersects
the imaginary axis. Two of the four are self-intersection points.

######  Lemma 5.2.

Let $N_{\mathrm{min}}(n)<N<n$ with $n=3,5,7,\ldots$, let $k$ be given by
(5.1), and let $u^{\prime}$ be any one of the six values

(5.3) $\pm\frac{\pi}{2},\ \pm\frac{\pi}{2}\pm u_{0}(n,N)$

where $u_{0}(n,N)$ is given in Lemma 3.9. Then for all six $u^{\prime}$,
$f_{n,N}^{(k)}(u^{\prime})$ is purely imaginary. Furthermore,

(5.4) $\rho_{\mathrm{upper}}\equiv
f_{n,N}^{(k)}\left(-\frac{\pi}{2}-u_{0}(n,N)\right)=f_{n,N}^{(k)}\left(-\frac{\pi}{2}+u_{0}(n,N)\right)$

and

(5.5) $\rho_{\mathrm{lower}}\equiv
f_{n,N}^{(k)}\left(\frac{\pi}{2}-u_{0}(n,N)\right)=f_{n,N}^{(k)}\left(\frac{\pi}{2}+u_{0}(n,N)\right)$

###### Proof.

With the function $v(n,N,u)$ defined in Theorem 4.1, we first show that

(5.6) $v(n,N,u^{\prime})=v_{\mathrm{im}}(n,N)$

for all six values $u^{\prime}$, where $v_{\mathrm{im}}(n,N)$ is defined in
Lemma 5.1. By Theorem 4.1 and eqn. (3.1), it suffices to show that

(5.7) $h_{n,N}\left(v_{\mathrm{in}}(n,N)\right)=g_{n,N}(u^{\prime})$

By Lemma 5.1 and (3.1), the left-hand side equals $N^{2}-1$. By (5.3), (3.5),
and (2.2), the right-hand side also equals $N^{2}-1$, thus proving (5.7) and
(5.6). With (5.6), (4.2) gives

(5.8) $\mu(n,N,u^{\prime})=u^{\prime}+iv_{\mathrm{im}}(n,N)$

for all six values $u^{\prime}$. We prove what remains upon substituting the
six $u^{\prime}$ into the expressions for $f_{n,N}^{(k)}(u)$ in (4.1) and
(4.6), and simplifying the resulting expressions using (5.8) and Lemma 5.1. ∎

######  Theorem 5.3.

Let $N_{\mathrm{min}}(n)<N<n$ with $n=3,5,7,\ldots$ and let $k$ be given by
(5.1). Then $L_{n,N}^{(k)}$ loops in both the upper- and the lower-half
planes. In the upper- (lower-) half plane, the self-intersection point is
given by the $\rho_{\mathrm{upper}}$ ($\rho_{\mathrm{lower}}$) in (5.4) or
(5.5). Both self-intersection points lie on the imaginary axis. In the limit
$N\to n-0$, furthermore, each loop shrinks to a single point, the point being
the corresponding cusp of $B_{n}^{(k)}=L_{n,n}^{(k)}$ on the imaginary axis.

###### Proof.

The point set

(5.9) $\left\\{\rho\in\mathbb{C}:\rho=f^{(k)}_{n,N}(u)\mathrm{\ for\ some\
}u\in\left[-\frac{\pi}{2}-u_{0}(n,N),-\frac{\pi}{2}+u_{0}(n,N)\right]\right\\},$

is a proper subset of $L_{n,N}^{(k)}$ by Corollary 4.2 and (3.13), and belongs
to the upper-half plane by Lemma 4.4. The endpoints of the interval in (5.9)
are different by (3.13). By (5.4), however, the two endpoints correspond to
the same point $\rho_{\mathrm{upper}}$ (on the positive imaginary semi-axis).
Therefore (5.9) is the parametric representation of a curve that is closed,
and the point $\rho$ is a self-intersection point of $L_{n,N}^{(k)}$. In other
words, our point set forms a loop of $L_{n,N}^{(k)}$. By (3.14), the loop
shrinks to a point (on the imaginary axis) in the limit $N\to n-0$, completing
our proof for the upper-half-plane loop. For a proof corresponding to the
lower-half plane, replace the two instances of $-\frac{\pi}{2}$ in (5.9) by
$\frac{\pi}{2}$ . ∎

Figure 4. Type-2 level curves $L^{(2)}_{11,N}$ for $N=11$ (solid line), $N=5$
(dashed line), and $N=2.8$ (dot-dashed line).

For $n=11$, Fig. 4 gives the three type-2 curves with $N=11$, $N=5$, and
$N=2.8$. The borderline curve $B_{11}^{(2)}=L_{11,11}^{(2)}$ exhibits a number
of cusps, two of which lie on the imaginary axis. The two other curves loop
around each of those cusps. As expected, the $N=5$ curve exhibits smaller
loops than does the $N=2.8$ curve; and the former curve is closer to the
borderline ($N=n=11$) curve.

As we always observed loops when $N_{\mathrm{min}}(n)<N<n$, proved the
existence of loops in a special case (Theorem 5.3), and never observed loops
when $N\geq n$, we are led to the conjecture that follows, which is a
generalization (to cases $N\neq n$) of Conjecture 4.5 of [2].

######  Conjecture 5.4.

The closed curves $L_{n,N}^{(1)}$ and $L_{n,N}^{(2)}$ are Jordan curves
whenever $N\geq n$, but non-Jordan whenever $N_{\mathrm{min}}(n)<N<n$.

##  6\. Eigenvalues with magnitudes larger than $N$; winding numbers

For $N>N_{\mathrm{min}}(n)$, and for a given $\rho\in\mathbb{C}$, this section
illustrates how a given curve $L_{n,N}^{(k)}$ can help one find the non-
negative integer $j_{n,N}^{(k)}(\rho)$ defined as follows.

######  Definition 6.1.

Let $\rho\in\mathbb{C}$ and $N>0$, and let $k=1$ or $k=2$. By
$j_{n,N}^{(k)}(\rho)$ we denote the number (counting multiplicities) of
type-$k$ eigenvalues of $K_{n}(\rho)$ whose magnitudes are larger than $N$.

Section 6 of [2] treats the special case $N=n$ and Conjecture 5.4 allows us to
extend that treatment to cases $N>n$. When $N_{\mathrm{min}}(n)<N<n$, however,
the self-intersections of $L_{n,N}^{(k)}$ render the determination of
$j_{n,N}^{(k)}(\rho)$ more involved. Although we use some of the principles
discussed in Section 6 of [2], what follows additionally involves the winding
numbers [8] associated with our curve. Accordingly, we provide $L_{n,N}^{(k)}$
with an orientation; this allows us to find winding numbers, and to
distinguish between points that are just to the left of the curve and points
that are just to the right. The orientation we choose is such that the curve
crosses the positive real axis from the lower- to the upper-half plane; this
initial orientation gives the one at any other curve point in a natural
manner. As we will see, the opposite orientation will do just as well.

Figure 5. Type-2 oriented level curve $L^{(2)}_{8,1.85}$ (solid line with
arrows), together with points showing the cusp locations of $B^{(2)}_{8}$. The
value $N=1.85$ is close to $N_{\mathrm{min}}(8)\cong 1.833$. (At first sight,
it might appear that there are self-intersection points in the second and
third quadrants, but a close focus shows this to be untrue.) The curve splits
the plane into components, and the numbers ($0$, $1$, $2$, or $3$) shown in
the figure are the $j_{8,1.85}^{(2)}(\rho)$ of each component.

Fig. 5 shows $L_{n,N}^{(k)}$ thus oriented for $k=2$, $n=8$, and $N=1.85$. The
six points pictured in Fig. 5 correspond to the six cusps of the borderline
curve $B_{n}^{(2)}=L_{n,n}^{(2)}$. (Recall that each cusp point gives rise to
a borderline/double eigenvalue equal to $-n$.) The value $N=1.85$ is slightly
larger than $N_{\mathrm{min}}(8)\cong 1.833$; as a result, there are
significant deviations from $B_{n}^{(2)}$.

The curve $L_{n,N}^{(k)}$ splits the complex-$\rho$ plane into a number of
disjoint components. We have labeled each component by its
$j_{n,N}^{(k)}(\rho)$ (which is constant within the component, see Section 6.1
of [2]). It is seen that $j_{n,N}^{(k)}(\rho)=0,1,2,$ or $3$, with the two
components labeled $3$ being very small. We see, also, that any component
containing a cusp point has $j_{n,N}^{(k)}(\rho)=2$; thus, at any cusp point
$\rho_{c}$, all type-$2$ eigenvalues of $K_{8}(\rho_{c})$ have magnitudes
smaller than $N=1.85$, with the exception of the borderline/double eigenvalue
(which has magnitude $n=8$ and is counted twice).

The reader is invited to show that the $j_{n,N}^{(k)}(\rho)$ in Fig. 5 can be
determined using the two rules that follow.

Rule 1: The unique component that extends to infinity has
$j_{n,N}^{(k)}(\rho)=j_{n,N}^{(k)}(\infty)=1$.

Rule 2: Crossing $L_{n,N}^{(k)}$ via a non-self-intersection point of
$L_{n,N}^{(k)}$ results in changing $j_{n,N}^{(k)}(\rho)$ by $1$, with the
larger $j_{n,N}^{(k)}(\rho)$ appearing: (i) on the right side of the curve for
our choice of orientation; but (ii) on the left side of the curve for the
opposite orientation.

Let us justify these rules. As $L_{n,N}^{(k)}$ is bounded, only one component
extends to infinity. Rule 1 is true because $N>N_{\mathrm{min}}(n)>1$ by
(3.9); and because, as $|\rho|\to\infty$, exactly one type-$k$ eigenvalue is
unbounded, while all other type-$k$ eigenvalues approach $-1$, see (6.6) of
[2].

The change by $1$ specified by Rule 2 is evident; see also Section 6.2 of [2].
If we start in the unbounded component and cross the curve via the positive
real axis, we enter the component containing the origin. This component has
$j_{n,N}^{(k)}(\rho)=j_{n,N}^{(k)}(0)=0$ because $K_{n}(0)$ is the $n\times n$
identity matrix, whose eigenvalues are all smaller than $N$ by (3.9) and
$N>N_{\mathrm{min}}(n)$. For the provided orientation, therefore, crossing
from right to left is accompanied by a decrease in $j_{n,N}^{(k)}(\rho)$, as
stated in Rule 2. The reverse would be true if we had chosen the opposite
orientation.

Evidently, our two rules are similar in nature to those used in computing
winding numbers of closed, bounded, and oriented curves [8]. The latter rules,
which appear in the literature in various contexts [10, 11], are often
referred to as Alexander numbering, the name originating from Alexander’s 1928
paper [12]. More precisely, for the orientation provided in Fig. 5, let
$\mathrm{wind}\left(L_{n,N}^{(k)},\rho\right)$ be the winding number of
$L_{n,N}^{(k)}$ with respect to the point $\rho\notin L_{n,N}^{(k)}$. At
infinity, in place of Rule 1 we have
$\mathrm{wind}\left(L_{n,N}^{(k)},\infty\right)=0$. Furthermore,
$\mathrm{wind}\left(L_{n,N}^{(k)},\rho\right)$ obeys Rule 2 but with the
larger of the two $\mathrm{wind}\left(L_{n,N}^{(k)},\rho\right)$ appearing on
the left side of the curve [8]–[11]. Consequently,
$-\mathrm{wind}\left(L_{n,N}^{(k)},\rho\right)$ obeys Rule 2. We have thus
arrived at the simple relation

(6.1) $j_{n,N}^{(k)}(\rho)=1-\mathrm{wind}\left(L_{n,N}^{(k)},\rho\right).$

Numerical experiments (specifically, numerical computations of the
eigenvalues) verified that our two rules—or their equivalent (6.1)—correctly
gave the $j_{n,N}^{(k)}(\rho)$ for all the $L_{n,N}^{(k)}$ we generated via
our algorithm. In particular, Rule 2 remains true when the non-self-
intersection point is a cusp (recall that cusps appear only when $N=n$): as
discussed in [3] and Section 6 of [2], any cusp is associated with eigenvalue
bifurcations, so that $j_{n,N}^{(k)}(\rho)$ changes by $1$ (rather than $2$,
even if there is a double eigenvalue at the cusp). It is also easy to
understand why $j_{n,N}^{(k)}(\rho)$ changes by $0$ or $2$ along a trajectory
that passes through a self-intersection point, as one can see from Fig. 5.

Needless to say, any Jordan curve (see Conjecture 5.4) separates the complex
plane into two components, namely the interior and the exterior of
$L_{n,N}^{(k)}$, with $j_{n,N}^{(k)}(\rho)=j_{n,N}^{(k)}(0)=0$ and
$j_{n,N}^{(k)}(\rho)=j_{n,N}^{(k)}(\infty)=1$, respectively, with the value
$1$ corresponding to the extraordinary eigenvalue mentioned in our
Introduction.

##  7\. Extensions

###  7.1. An illustrative example

Via an application arising in physics [13], we now show that many of the
behaviors exhibited by $K_{n}(\rho)$ also occur elsewhere. Our example is
described by the cubic equation [13]

(7.1)
$\lambda(\lambda-\rho-\alpha^{2})^{2}+\frac{\pi^{2}\alpha^{4}}{4}=0,\quad\alpha>0$

whose unknown is $\lambda$. Consistent with our notation for the
Kac–Murdock–Szegő matrix $K_{n}(\rho)$, we consider $\rho\in\mathbb{C}$ to be
the varying parameter. As we will shortly discuss in more detail, our $\rho$
corresponds to the quantity $\epsilon_{a}\in\mathbb{R}$ of [13]. The notation
$\alpha$ (as well as the value $0.1$ to which we will soon fix $\alpha$) comes
from [13].

Conditions for multiple zeros follow from the familiar procedure [14] of
setting the discriminant of (7.1) equal to zero and solving for $\rho$. We
thus find that multiple zeros occur when $\rho$ is equal to one of the three
critical points $\rho_{c}(m)$, where

(7.2)
$\rho_{c}(m)=-\alpha^{2}-3\left(\frac{\pi\alpha^{2}}{4}\right)^{\frac{2}{3}}\exp\left({i\frac{2m\pi}{3}}\right),\quad
m=-1,0,1,$

and that the corresponding zeros are double zeros equal to

(7.3)
$\lambda_{c}(m)=-\left(\frac{\pi\alpha^{2}}{4}\right)^{\frac{2}{3}}\exp\left({i\frac{2m\pi}{3}}\right),\quad
m=-1,0,1,$

so that the magnitudes of all three double zeros are equal to $N_{0}$, with

(7.4)
$N_{0}=\left|\lambda_{c}(m)\right|=\left(\frac{\pi\alpha^{2}}{4}\right)^{\frac{2}{3}},\quad
m=-1,0,1.$

Figure 6. Level curves $L_{N}$ for the three zeros of the cubic equation (7.1)
with $\alpha=0.1$. The levels of the three curves are $N=N_{0}$ (solid line),
$N=0.5N_{0}$ (dashed line), and $N=1.4N_{0}$ (dot-dashed line). The arrows
indicate curve orientations.

Let $\lambda$ denote any of the three solutions of (7.1). The $N$-level curves
(contour lines on which $|\lambda|=N$) can be numerically generated via the
well-known closed-form expressions [14] for the zeros of cubic equations. For
$\alpha=0.1$, Fig. 6 shows contour lines $L_{N}$ thus obtained for $N=N_{0}$
(solid line), $N=0.5N_{0}$ (dashed line), and $N=1.4N_{0}$ (dot-dashed line).
The closed curves of Fig. 6, which can be compared to those in Fig. 2, have
the following features.

(i) The curve of level $N=N_{0}$ (but not the other two curves) exhibits three
cusp-like singularities. They occur at the double-eigenvalue positions, namely
at the $\rho=\rho_{c}(m)$ of (7.2). Therefore the borderline curve
$B_{n}^{(k)}=L_{n,n}^{(k)}$ of $K_{n}(\rho)$ is analogous to the $N=N_{0}$
curve of Fig. 6, with $n$ corresponding (in this respect) to the double-
eigenvalue level $N_{0}$.

(ii) The curve of level $N=0.5N_{0}$ envelops the one of level $N=N_{0}$.
Here, as opposed to Fig. 2, the $N$-value of the surrounding curve is smaller
than the $N$-value of the enclosed curve.

(iii) The non-Jordan curve (of level $N=1.4N_{0}$) loops around the three
cusps, similarly to the third curve (of level $N=3$) in Fig. 2.

(iv) As in Fig. 2, each curve in Fig. 6 separates the plane into components.
Let $j_{N}(\rho)\in\\{0,1,2,3\\}$ be the number of zeros of (7.1) whose
magnitudes exceed $N$. As $|\rho|\to\infty$, the three zeros of (7.1) approach
$\rho$, $\rho$, and $0$. Thus in place of Rule 1 of Section 6 we have

Rule 1′: For each $N$ in Fig. 6, the unbounded component has $j_{N}(\rho)=2$.

Let us provide the three curves with the orientations shown in Fig. 6. When
$\rho=-\alpha^{2}$, all three zeros of (7.1) have magnitudes
$2^{2/3}N_{0}\cong 1.59N_{0}$. Therefore for the three values of $N$ used in
Fig. 6 we have $j_{N}(-0.01)=3$. Since the component containing $\rho=-0.01$
is always adjacent to the unbounded one, we can replace Rule 2 of Section 6 by

Rule 2′: For the three curves of Fig. 6 (oriented as shown), crossing the
curve from right to left results in increasing $j_{N}(\rho)$ by $1$.

(v) For the three oriented curves in Fig. 6, the equation corresponding to
(6.1) is

(7.5) $j_{N}(\rho)=\mathrm{wind}(L_{N},\rho)+2$

where $\mathrm{wind}(L_{N},\rho)$ is the winding number of $L_{N}$ (oriented
as in the figure) about any point $\rho\notin L_{N}$.

When applied to the three $L_{N}$ of Fig. 6, eqn. (7.5) (which is tantamount
to Rules 1′ and 2′) easily gives any desired $j_{N}(\rho)$. For example, it is
apparent that any component of $L_{1.4N_{0}}$ that loops around a cusp-like
singularity has $j_{N}(\rho)=1$.

###  7.2. Physics of our example

Eqn. (7.1) pertains [13] to an open quantum system consisting of a discrete
quantum state of energy $\epsilon_{a}$ (the symbol $\epsilon_{a}$ of [13]
corresponds to our $\rho$), coupled with a one-dimensional continuum state.
Ref. [13], which assumes $\epsilon_{a}\in\mathbb{R}$, specifically studies the
Time-Symmetry Breaking Phase Transition (TSBPT) caused by the non-linearity
associated with a Van Hove singularity. Here, our generalization to complex
$\epsilon_{a}$ amounts to introducing a loss (or gain, depending on the sign
of $\mathrm{Im}\\{\epsilon_{a}\\}$) to the discrete quantum state, similarly
to a lossy optical cavity.

Contour lines such as the three in Fig. 6 consist of all complex-energy values
for which an eigenvalue maintains a constant magnitude $N$. TSBPT occurs along
any trajectory passing through one of the three critical values $\rho_{c}(m)$
given in (7.2). This happens at any of the three cusp-like singularities of
$L_{N_{0}}$; there, two coinciding eigenvalues equal $\lambda_{c}(m)$. Each
point $\rho_{c}(m)$ is a branch point of the double-valued function associated
with the two coalescing eigenvalues [3, 13], so it is impossible to
distinguish between the two when reversing the process (i.e., when one follows
the opposite trajectory of $\rho=\epsilon_{a}$ in the complex plane). Finally,
moving from the interior of $L_{N_{0}}$ to the exterior via a cusp-like
singularity is associated with eigenvalue bifurcations [3, 13].

###  7.3. An equivalent matrix problem; possible generalizations

The third-degree equation (7.1) is the characteristic equation
$\mathrm{det}[\lambda I-M(\rho)]=0$ of many $3\times 3$ matrices $M(\rho)$, an
example being

(7.6) $M(\rho)=\begin{bmatrix}\rho+\alpha^{2}&0&-\frac{\pi\alpha^{2}}{2}\\\
1&\rho+\alpha^{2}&1\\\ 1&\frac{\pi\alpha^{2}}{2}&0\\\ \end{bmatrix}$

The matrix $M(\rho)$ is an analytic matrix function in the sense that all
matrix elements are complex-analytic functions of $\rho$. Near any critical
point $\rho_{c}(m)$, furthermore, the two coalescing eigenvalues are
associated with a Puiseux series consisting of powers of
$[\rho-\rho_{c}(m)]^{1/2}$ [3]. For the critical point
$\rho_{c}(0)\in\mathbb{R}$, the Puiseux series is discussed in [13];
importantly, the coefficient of the square-root term (i.e., the coefficient of
$[\rho-\rho_{c}(0)]^{1/2}$) is nonzero.

For a general class of analytic matrix functions, eigenvalue behaviors near
such critical points are discussed in detail in Sections 3 and 4 of [3], with
the Puiseux series being the main tool of study. While the conclusions are
directly relevant to the example analyzed herein,333Corollary 3.5 of [3], for
example, tells us that the cusp-like singularities in Fig. 6 are true cusps.
ref. [3] mainly focuses on eigenvalue bifurcations.

We believe that the behaviors exhibited by the $N$-level curves of
$K_{n}(\rho)$ (and $M(\rho)$) are quite general. It might be possible to
extend the Puiseux-series techniques in Sections 3 and 4 of [3] so as to study
the local behaviors (near cusps, loops, and the like) of $N$-level curves of
the aforementioned class of analytic matrix functions. Preliminary work,
however, indicates that this is not a simple task.

The type-2 eigenvalue contour lines of $K_{8}(\rho)$ (Fig. 5) and the
eigenvalue contour lines of $M(\rho)$ (Fig. 6) are closed curves that share
two important common features: (i) they are continuous; and (ii) they can be
oriented in a natural manner. The $2\times 2$ Kac–Murdock–Szegő matrix

(7.7) $K_{2}(\rho)=\begin{bmatrix}1&\rho\\\ \rho&1\\\ \end{bmatrix}$

whose eigenvalues are $1-\rho$ and $1+\rho$, tells us that neither of these
features is necessarily true in more general cases. The eigenvalue contour
lines of $K_{2}(\rho)$ are two circles which intersect when $N>1$, but which
are disjoint when $N<1$. Therefore, (a) the eigenvalue level curves (contour
lines) $L_{N}$ are discontinuous whenever $N<1$, and (b) there seems to be no
straightforward and natural method of orienting the composite level curve
consisting of both circles. Note that difficulty (b) arises because here, we
are not distinguishing between type-1 and type-2 eigenvalues; the composite
level curve in Fig. 1 [consisting of both the solid (type-1) and the dashed
(type-2) curves] also presents difficulty (b).

For certain types of matrices (or polynomials) for which the aforementioned
conditions (i) and (ii) are satisfied, it is logical to expect that equations
such as (6.1) and (7.5)—which involve the winding numbers of oriented level
curves $L_{N}$ in the complex-$\rho$ plane—remain valid. We feel this matter
is worthy of a systematic and rigorous study: It would be interesting to find
matrix classes for which such equations remain true, and to state the
equations in general form. To this end, a good starting point is the argument
principle, see any text on functions of a single complex variable, or see the
introductory discussions [15] of degree theory in two dimensions. For any
fixed $\rho\notin L_{N}$, this principle can be used to express what we have
called $j_{N}(\rho)$ as a contour integral over the circle $|\lambda|=N$ of
the logarithmic derivative of the characteristic polynomial.

## References

  * [1] G. Fikioris, Spectral properties of Kac–Murdock–Szegő matrices with a complex parameter, Linear Algebra Appl. 553 (2018) 182–210.
  * [2] G. Fikioris and Th. K. Mavrogordatos, Double, borderline, and extraordinary eigenvalues of Kac–Murdock–Szegő matrices with a complex parameter, Linear Algebra Appl. 575 (2019) 314–333.
  * [3] G. Fikioris, Eigenvalue bifurcations in Kac–Murdock–Szegő matrices with a complex parameter, Linear Algebra Appl. 607 (2020) 118–150.
  * [4] J. M. Bogoya, A. Böttcher, S. M. Grudsky, Eigenvalues of Hermitian Toeplitz matrices with polynomially increasing entries, J. Spectr. Theory 2 (2012) 267–292.
  * [5] M. Kac, W. L. Murdock, G. Szegő, On the eigen-values of certain Hermitian forms, Arch. Ration. Mech. Anal. 2 (1953) 767–800.
  * [6] U. Grenander, G. Szegő, Toeplitz forms and their applications, Chelsea Publishing Co., New York, 1958, chapt. 5.
  * [7] J. M. Bogoya, A. Böttcher, S. M. Grudsky, E. A. Maximenko, Eigenvectors of Hermitian Toeplitz matrices with smooth simple-loop symbols, Linear Algebra Appl. 493 (2016) 606–637.
  * [8] J. Roe, Winding around: The winding number in topology, geometry, and analysis, American Mathematical Society, 2015, §3.3.
  * [9] J. C. Mason, D. C. Handscomb, Chebyshev polynomials, Chapman & Hall/CRC, Boca Raton, FL, 2003, §2.2.
  * [10] H-C Chang and J. Erickson, Electrical reduction, homotopy moves, and defect, arXiv: 1510.00571 [cs.CG].
  * [11] J. S. Carter, S. Kamada, and M. Saito, Alexander numbering of knotted surface diagrams, Proc. Amer. Math. Soc. 128 (2000) 3761–3771.
  * [12] J. W. Alexander, Topological invariants of knots and links, Trans. Amer. Math. Soc. 30 (1928) 275–306.
  * [13] S. Tanaka, S. Garmon, K. Kanki, and T. Petrosky, Higher-order time-symmetry-breaking phase transition due to meeting of an exceptional point and a Fano resonance, Phys. Rev. A 94 (2016) 022105, 1–11.
  * [14] F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, Digital Library of Mathematical Functions, National Institute of Standards and Technology from http://dlmf.nist.gov/. §1.11.
  * [15] J. A. Baker, Plane curves, polar coordinates and winding numbers, Mathematics Magazine, 64 (2018) 75–91.


# Voices Obscured in Complex Environmental Settings (VOICES) corpus

###### Abstract

This paper introduces the Voices Obscured In Complex Environmental Settings
(VOICES) corpus, a freely available dataset under Creative Commons BY 4.0.
This dataset will promote speech and signal processing research of speech
recorded by far-field microphones in noisy room conditions. Publicly available
speech corpora are mostly composed of isolated speech at close-range
microphony. A typical approach to better represent realistic scenarios, is to
convolve clean speech with noise and simulated room response for model
training. Despite these efforts, model performance degrades when tested
against uncurated speech in natural conditions. For this corpus, audio was
recorded in furnished rooms with background noise played in conjunction with
foreground speech selected from the LibriSpeech corpus. Multiple sessions were
recorded in each room to accommodate for all foreground speech-background
noise combinations. Audio was recorded using twelve microphones placed
throughout the room, resulting in 120 hours of audio per microphone. This work
is a multi-organizational effort led by SRI International and Lab41 with the
intent to push forward state-of-the-art distant microphone approaches in
signal processing and speech recognition.

Index Terms: corpus, speech recognition, speaker recognition, data collection,
LibriSpeech

##  1 Introduction

SRI International and Lab41, In-Q-Tel, are proud to release the Voices
Obscured in Complex Environmental Settings (VOICES) corpus, a collaborative
effort that brings speech data in acoustically challenging reverberant
environments to the researcher. Clean speech was recorded in rooms of
different sizes, each having distinct room acoustic profiles, with background
noise played concurrently. These recordings provides audio data that better
represent real-use scenarios. The intended purpose of this corpus is to
promote acoustic research including, but not limited to:

Speech Processing

Speaker Identification, speech recognition, speaker detection

Audio Classification

Event and background classification, speech/non-speech

Acoustic Signal Processing

source separation and localization, noise reduction, general enhancement,
acoustic quality metrics

The corpus contains the source audio, the retransmitted audio, orthographic
transcriptions, and speaker labels. The ultimate goal of this corpus is to
advance acoustic research by providing access to complex acoustic data. The
corpus will be released as open source, Creative Commons BY 4.0, free for
commercial, academic, and government use.

Datasets for speech research are typically expensive, limited in scope, and
behind paywalls. Synthetic data can be created by superimposing audio samples
from datasets of isolated speech and noise and using software to generate
reverberation[1]. Unfortunately, these techniques do not accurately represent
the acoustics of real-world environments and dynamic noise. Datasets collected
in real environments often use few speakers[2].

Successfully deploying speech and acoustic signal processing algorithms in the
field hinges on access to realistic data. To this end, audio for the VOICES
corpus was recorded under realistic, noisy conditions, that better represent
real-use situations. In the remainder of this paper, a detailed description of
the VOICES corpus is provided, including model baselines for automatic speech
recognition and speaker identification. Section 2 describes the collection
effort itself, Section 3 provides some insight into the statistics of the
dataset, and Section 4 outlines model baselines that were run on the dataset.
The corpus will be available on Amazon Web Services, where details on use
cases and a download link will be provided.

##  2 Dataset Collection

The main focus when developing the VOICES corpus was to provide an open-source
dataset centered on distant microphone collection under realistic conditions.
Pre-recorded foreground speech and background noise were played in two
furnished rooms with different acoustic profiles (reverberation, HVAC
background, echo, etc.) and recorded by 12 distant microphones. Recording
rooms were windowed and carpeted, with mostly bare walls and bare ceiling,
furnished with tables and chairs. Four recording sessions were held in each
room: one for each distractor noise type (television, radio, or babble) played
concurrently with the foreground speech, and one session with foreground
speech only. One hour of only distractor noise or ambient room background
noise was recorded at the end of each session. This resulted in over 120 hours
of recorded speech per microphone, for a total of 374,688 audio files.

###  2.1 Audio Sources

The audio for foreground speech and distractor noise was selected from sources
either in the public domain or under a creative commons attribution license
that permits data derivatives and commercial use.

####  2.1.1 Foreground Speech

A total of 15 hours (3,903 audio files) were selected from LibriSpeech[3], a
corpus of audiobooks in the public domain. All audio contains English read
speech. Audio was taken from 300 speakers in the ”clean” data subsets, with an
even split between females and males. At least three minutes of speech were
selected from each speaker, with at least one minute from three different book
chapters - an amount sufficient for speaker identification tasks. LibriSpeech
files use a sample rate of 16kHz, 16-bit precision, and Free Lossless Audio
Codec (FLAC) encoding. Selected files were corrected for DC offset, normalized
based on their peak amplitude, and converted to WAV format. The selected audio
files were concatenated together with 2 seconds of intervening silence into a
continuous audio file. The loudspeaker playing the foreground speech was on a
motorized rotating platform. The order of the individual audio files was
randomized, to guarantee that there was no correlation between a particular
human speaker and a position of the loudspeaker. Signals to evaluate the room
response were added at the beginning of each session. These included a steady
tone, a rising tone, and a transient noise. The final concatenated source file
was 19 hours long.

Figure 1: Microphone and loudspeaker configuration (not to scale) used for
recording sessions in (a) room 1 (146” x 107”) and (b) room 2 (225” x 158”).
The foreground loudspeaker (shown here at its $90^{\circ}$ position), orange
rectangle, was placed in a corner of the room, and speakers playing noise,
blue squares, were placed with their cones directed toward the center of the
room. Studio and lavalier microphones are shown as large (dark) and small
(light) green circles; microphone number labeling corresponds to those
outlined in Table 1.

####  2.1.2 Distractor Noise

Audio was recorded under four different noise conditions: one without any
added noise (ambient room noise only) and three with a distractor noise played
simultaneously with the foreground speech. The distractor noises were
television, music, or overlapping speech from multiple speakers (referred to
here as babble). During recording sessions, the audio for television or music
was played from a single loudspeaker; babble was played from three noise-
dedicated loudspeakers. An extra hour of just distractor noise was recorded at
the end of each session.

Television noise was selected from movies and television shows in the public
domain[4, 5]. Audio from 76 videos was extracted in M4A format and converted
to WAV with a 16kHz sample rate and 16-bit precision. Five-minute excerpts
were chosen from each audio file and each excerpt was normalized to its peak
amplitude. Depending on the length of the source audio, 5 to 8 excerpts were
taken from each movie or show, randomized, and concatenated into a single
20-hour audio file.

Music noise was selected from the MUSAN corpus[6]. All music files are in the
public domain or under a Creative Commons license. Any music files having no
derivative (ND) or non-commercial (NC) license restrictions were omitted from
the sample set. The music files were randomized and concatenated into a single
20-hour audio file. Due to the large variability in signal amplitudes for
different genres of music, the concatenated audio file was run through the
compander tool in the SoX audio utility, combining compression and expansion
of the signal dynamic range. This ensured a more uniform music volume
throughout the recording sessions and a more consistent signal-to-noise ratio.

Babble noise was constructed using the ”us-gov” subset of the MUSAN corpus[6].
This subset contains audio recording excerpts of various US government
meetings; all are in the public domain. Each excerpt is about 5 minutes long
and was normalized to its peak amplitude. Babble tracks were constructed by
randomizing and concatenating together meeting excerpts into 20-hour audio
files and then mixing three audio files into one. Three babble tracks were
created and were played out of three noise-dedicated loudspeakers (i.e., nine
overlapping voices) simultaneously with the foreground speech.

###  2.2 Recording Setup

Two different rooms were used for recording: room-1 with dimensions 146” x
107” (x 107” height) and room-2 with dimensions 225” x 158” (x 109” height).
Twelve microphones were placed in strategic locations throughout the room: 7
cardioid dynamic studio microphones (SHURE SM58), 4 omnidirectional condenser
lavalier microphones (AKG 417L), and 1 omnidirectional dynamic lavalier
microphone (SHURE SM11). Paired studio and lavalier microphones were placed at
four different positions: (1) Behind the foreground loudspeaker, (2) on a
table directly in front of the foreground loudspeaker, (3) on a table in front
of the foreground loudspeaker at a farther distance than (2), and (4) across
the room from the foreground loudspeaker. The remaining four lavalier
microphones were placed in other locations in the room, fully or partially
obstructed by a physical barrier. Distances between the foreground loudspeaker
and microphones are listed in Table 1. All audio was played on high-quality
speakers; one speaker was reserved for foreground speech, and three others
were used to play distractor noise. A schematic of speaker and microphone
placement in both rooms in shown in Figure 1.

Table 1: Microphone type, location, distance from foreground loudspeaker (s) and height (h) for room-1 and -2 configurations. Mic ID (type) | Location | Room-1 (s, h) | Room-2 (s, h)
---|---|---|---
01 (studio), 02 (lavalier) | near on table | (38”, 42”) | (80”, 39”)
03 (studio), 04 (lavalier) | far on table | (72”, 42”) | (131”, 39”)
05 (studio), 06 (lavalier) | across room | (119”, 70”) | (228”, 70”)
07 (studio), 08 (lavalier) | behind loudspeaker | (29”, 70”) | (29”, 70”)
09 (lavalier) | partially obstructed, table | (58”, 28”) | (109”, 25”)
10 (lavalier) | on ceiling, clear | (75”, 105”) | (128”, 105”)
11 (lavalier) | on ceiling, fully obstructed | (75”, 106”) | (128”, 106”)
12 (lavalier) | fully obstructed, wall | (130”, 12”) | (116”, 10”)

The foreground speaker was placed 43” from the floor on a robotic platform
that automatically rotated the position of the foreground speaker by ten
degrees every hour, spanning a total of 180 degrees. The rotating platform’s
step motor was sufficiently shielded to prevent recording background noise
from the motor movement. The motivation to have a non-static audio source was
to emulate common human behavior that occurs during conversations such as head
movement or walking, that is not captured in other datasets.

A PreSonus StudioLive RML32AI digital mixer and PreSonus Capture recording
software were used to play and record the audio. A sound pressure meter,
placed close to microphone 01, was used to measure the playback audio and
adjust volume levels on the PreSonus mixer for both the foreground audio (~65
dB) and distractor noise (~50 dB). All channels were sample synchronous. Each
recording session lasted 20 hours (19 hours of foreground speech and 1 hour of
only distractor or ambient noise). The recording sessions were segmented
according to the source files from LibriSpeech, yielding 1440 hours of audio
(347,688 audio files) across all microphones and sessions. Audio was recorded
with a 48kHz sample rate and 24-bit precision in WAV format with PCM encoding,
and is also available in 16kHz and 16-bit precision in WAV format. The corpus
also contains the source audio files (16kHZ sample rate, 16-bit precision, WAV
format).

##  3 Data Statistics

To obtain an assessment of the statistics of the corpus, the duration, minimum
and maximum amplitude, root mean square (RMS) energy, and signal-to-noise
ratio (SNR) were calculated for all audio files in the corpus. Statistics were
calculated using a combination of the SoX utility and SRI’s in-house
utilities.

The average and median duration for all data subsets is 15.62s and 15.97s,
respectively, with a standard deviation of 1.91s. This is evidence that the
automatic audio segmentation worked correctly and that we can directly compare
noisy files with source files.

The RMS, measuring the amplitude of the audio file relative to the digital
system’s maximum level (with maximum value at 0 decibels relative to full
scale - dBFS), was consistent across the various subsets. Average values were
measured between -27.0 and -27.5 dBFS, indicating the playback volume was
consistently set for all recordings.

The minimum and maximum amplitudes represent the lowest and highest amplitude
for samples in a given audio file, on a normalized scale of $\pm 1$. These
were measured to be between -0.5 to 0.5 across all data subsets, showing
reasonable use of the digital recording system’s levels. The average minimum
and maximum amplitude levels for the source audio were -0.93 and 0.91.

The SNR measures the strength of a primary signal relative to the background
noise. Differences in SNR were evident between rooms and distractor noises,
and in general degraded with increasing distance between the foreground
loudspeaker and microphone. The average SNR for audio recorded in room-1 and
room-2 was 22.19 dB and 19.50 dB, respectively.

Table 2 shows the calculated SNR for audio recorded under different noise
conditions as compared to the source audio’s SNR. The SNR significantly
degrades for audio recorded at a distance in a real acoustic environment, even
without distractor noise. A decrease of 18 dB was observed for this case. The
addition of noise further decreases the SNR, the worst is with babble noise.
The SNR for microphones close to and behind the foreground loudspeaker was
22.3 dB, and for those at mid- and far-distance, it was 20.5 dB.

Table 2: Measured signal-to-noise ratio for the source audio and audio recorded at a distance with and without distractor noise. | Source | No distractor | Music | TV | Babble
---|---|---|---|---|---
SNR | 41.7 | 23.6 | 19.2 | 22.2 | 18.4

##  4 Model Baselines

SRI’s in-house automatic speech recognition (ASR) and speaker identification
(SID) systems were used to examine the recorded data. This provides data
validation for analytics and a point of reference for future model
implementations.

###  4.1 Automatic speech recognition (ASR)

The ASR system was run on a subset of data: audio from lavalier microphones
when the foreground loudspeaker was positioned at $90^{\circ}$ (directly
aligned with microphones on table). The ASR system was built using the Kaldi
Speech Recognition Toolkit [7]. It uses filterbank features and a time delay
neural network (TDNN) and was trained on 500 hours of segmented English
speech, which included data collected under DARPA’s Translation Systems for
Tactical Use (TRANSTAC) program and SRI proprietary data. Training audio is
included twice, once in its original form and a second with artificially added
reverberation. Because no full test or development subset of data from
LibriSpeech is included in the VOICES corpus, a direct comparison with
published ASR results using LibriSpeech is not possible. It is possible,
however, to make a rough comparison with results using the dev-clean
LibriSpeech dataset. Published results for this subset achieved 4.9% and 7.8%
word error rate (WER), for models trained on LibriSpeech and on the Wall
Street Journal data, respectively[3]. The SRI system achieved a 9.3% WER.

Table 3 shows the WER when the foreground speaker is at $90^{\circ}$
(centered) as a function of distractor noise. Results show a sharp increase in
WER for data recorded in realistic acoustic environments. The WER for audio
recorded by distance microphones with no added distractor noise is 19.0% -
more than double the WER on the source audio. Added distractor noise degrade
the performance further. The worst performance is on audio with babble noise,
as this type of noise contains only speech and easily confuses the ASR system.

Table 3: WER as a function of distractor noise type for room-1 with foreground loudspeaker centered at $90^{\circ}$, obtained from in-house SRI ASR system. | Source | No distractor | TV | Music | Babble
---|---|---|---|---|---
WER | 9.3 | 19.0 | 27.6 | 29.3 | 33.0

In general the ASR performance is dependent on the distance between the
foreground loudspeaker and microphone, and on individual room acoustics, as
depicted in Figure 2. Results are shown for microphones 02, 04, 06 in both
rooms when the foreground loudspeaker is at $90^{\circ}$. There is an increase
in WER with increased distance between the microphone and foreground
loudspeaker. Differences in WER for microphones in room-1 and room-2 that are
at comparable distances show the effect of each room’s acoustic environment.

Figure 2: The WER performance is affected by distance from the foreground
loudspeaker, as well as room acoustic profile.

###  4.2 Speaker identification (SID)

A state-of-the-art SID system from SRI was run on the VOICES corpus. The model
used is a Universal Background Model (UBM) identity vector (i-vector) based
system [8, 9], with a probabilistic linear discriminant analysis (PLDA) [10]
as back-end classifier. A gender-independent PLDA was used to compute the
scores of the speaker recognition system. The model was trained using the
PRISM dataset [11]. The equal error rate (EER), describing the value where
false positives equal false negatives, is used as the metric for the SID
system performance. For our experimental setup, we ensured enroll and test
audio segments corresponded to different book chapters from the original
corpus. Speech segments were on average 14s long for both enroll and test
subsets. Results are shown for microphones 01 and 02 (Close), microphones 03
and 04 (Mid), and microphones 05 and 06 (Far).

Table 4 shows the impact of microphone distance on the SID performance. In
this experiment, enrollment was performed on clean source data, and the EER is
shown when testing on a variety of distant conditions. In order to highlight
the effect of distance alone, no distractor noises were used. We observe that
the EER of this UBM-IV system doubles when comparing the source audio
($5.72\%$) to audio from the close microphones for both rooms
($10.7\%$-$10.9\%$), and it almost triples for the far room microphone
($15.1\%$-$16.6\%$).

Table 4: Impact of microphone distance and placement on the performance of the UBM-IV speaker recognition systems in terms of EER (%). Enrollment was done on source audio, and testing performed on distant microphones with no distractors. Mics | Source | Close | Mid | Far
---|---|---|---|---
Rm1 | 5.72 | 10.7 | 13.0 | 15.1
Rm2 | 5.72 | 10.9 | 13.2 | 16.6

Table 5 shows the effect of distractor noise on SID performance. In order to
mimic a realistic test case, speakers were enrolled using a recording from the
close lavalier microphone (Close) in room-1 with no distractor noise. The test
segments originate from all microphones and were recorded in room-2 with
different types of background noise. We observe that distractor noise degrades
the EER by $2\%$ absolute for music and television and $3.5\%$ absolute for
babble. This is perhaps because it is very speech-like, but also possibly
because babble was the only distractor played out of three separate
loudspeakers.

Table 5: Impact of distractor noise on the performance of the UBM-IV speaker recognition systems in terms of EER (%). Each condition has above $18k/2.8M$ target$/$impostor trials. Distractor | No distractor | TV | Music | Babble
---|---|---|---|---
UBM-IV | 17.2 | 19.3 | 19.2 | 20.9

##  5 Conclusions and Future Work

The VOICES corpus provides audio data that closely resemble acoustic
conditions found in real recording environments - distant microphones,
background noise, and reverberant room acoustics. The corpus can serve as a
test and development set for research in the areas of speech and acoustics. It
will enable the development of robust acoustic models that can better perform
in the wild. By making the corpus publicly available, SRI International and
Lab41 hope to promote and advance acoustic research on event and background
detection, source separation, speech enhancement, source distance and sound
localization, speech activity detection, as well as speaker and speech
recognition. Data presented here correspond to phase I data collection. The
corpus will be augmented with further data collection in phase II, that will
include additional rooms and more challenging distractor noise profiles.

## References

  * [1] T. H. Falk and W.-Y. Chan, “Modulation spectral features for robust far-field speaker identification,” in _2010 IEEE Transactions on Audio, Speech, and Language Processing_ , vol. 18, no. 1, 2010, pp. 90–100.
  * [2] Q. Jin, T. Schultz, and A. Waibel, “Far-field speaker recognition,” in _IEEE Transactions on Audio, Speech, and Language Processing_ , vol. 15, no. 7, 2007, pp. 2023–2032.
  * [3] V. Panayotov, G. Chen, D. Povey, and S. Khudanpur, “Librispeech: An asr corpus based on public domain audio books,” in _2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_ , April 2015, pp. 5206–5210.
  * [4] publicdomainmovies.net. (2018) Public domain movies. [Online]. Available: http://publicdomainmovies.net/
  * [5] P. S. E. LLC. (2018) Public domain movies and tv shows. [Online]. Available: http://publicdomainmovies.org/
  * [6] D. Snyder, G. Chen, and D. Povey, “Musan: A music, speech, and noise corpus,” _CoRR_ , vol. abs/1510.08484, 2015. [Online]. Available: http://arxiv.org/abs/1510.08484
  * [7] D. Povey, A. Ghoshal, G. Boulianne, L. Burget, O. Glembek, N. Goel, M. Hannemann, P. Motlicek, Y. Qian, P. Schwarz, J. Silovsky, G. Stemmer, and K. Vesely, “The kaldi speech recognition toolkit,” in _IEEE 2011 Workshop on Automatic Speech Recognition and Understanding_. IEEE Signal Processing Society, Dec. 2011, iEEE Catalog No.: CFP11SRW-USB.
  * [8] N. Dehak, P. J. Kenny, R. Dehak, P. Dumouchel, and P. Ouellet, “Front-end factor analysis for speaker verification,” _IEEE Transactions on Audio, Speech, and Language Processing_ , vol. 19, no. 4, pp. 788–798, May 2011.
  * [9] D. Garcia-Romero and C. Espy-Wilson, “Analysis of i-vector length normalization in speaker recognition systems.” in _Proc. Interspeech_ , 01 2011, pp. 249–252.
  * [10] S. J. D. Prince and J. H. Elder, “Probabilistic linear discriminant analysis for inferences about identity,” in _2007 IEEE 11th International Conference on Computer Vision_ , Oct 2007, pp. 1–8.
  * [11] L. Ferrer, H. Bratt, L. Burget, H. Cernockyy, O. Glembeky, M. Graciarena, A. Lawson, Y. Lei, P. Matejkay, O. Plchoty, and N. Scheffer, “Promoting robustness for speaker modeling in the community: the prism evaluation set,” in _Proceedings of NIST 2011 workshop_ , 2011.