November

# Enhancing Room Security and Automating Class Attendance Using ID Cards

Shravan Bhat – 171EE240  Nithin R – 171EC131  Pranav S - 171EC135

(2019)

###### Abstract

With the rapid advancements in technology, automation has emerged as the
future of human endeavors. From simple tasks like attendance management to
complex security systems, automation has the potential to revolutionize
various aspects of our lives. This research paper explores the implementation
of a method aimed at enhancing room security in hostels and automating class
attendance using ID cards. In this study, we propose a system that utilizes
the unique identity information stored in ID cards for various security and
check-in tasks. By integrating RFID (Radio-Frequency Identification) reader
technology, GSM modules, Node MCU, and Arduino, we create a comprehensive
solution. The RFID reader scans the ID card, extracting the relevant
information and verifying the user’s identity. The data is then transmitted
via the GSM module to a central database, ensuring real-time monitoring and
security measures. Moreover, the system also enables the automation of class
attendance. By utilizing the same ID cards, students can simply tap their
cards on a reader placed in the classroom. This information is recorded
automatically, eliminating the need for manual attendance taking and reducing
errors and time consumption. This research project highlights the practical
implementation of ID card technology to enhance room security in hostels and
automate class attendance processes. By leveraging the power of automation, we
aim to streamline administrative tasks, improve security measures, and
optimize efficiency in educational institutions and other relevant settings.

ID card, RFID reader, GSM Module, Node MCU, Arduino

##  1 Introduction

Security and privacy is a basic need for any human being. India’s population
has been increasing exponentially since 19th century. Hence student intake for
colleges has been increasing every year. Automation would help in trivial
tasks like taking attendance, or making payments in a locality. Privacy and
security is also an issue in many colleges. Adding layers of security to rooms
and safe box would prevent petty theft from happening.

Main motivation of this project is to establish a attendance system within our
college campus, a cash-less payment system and also to implement safer and
key-less room locking systems in our university.

##  2  Literature Survey

###  2.1 Survey of State of Art

Smart card based door lock systems which are expensive and less secure are
currently available like the NFC (Near field communication) cards which are
used in the hotel rooms. Using these might be very expensive as it requires
complex hardware.

Automated attendance are available, which uses finger print as the ID, But
implementing that on a large scale like college is difficulty and would come
out to be rather expensive.

###  2.2 Features

  * •

RFID card and RFID reader is included in the door lock system. The door
unlocks only when the authorized card is scanned and corresponding pin in
entered using the keypad provided.

  * •

The locking and unlocking of the door latch is implemented using servo motors,
stepped motors and gears.

  * •

When a card is scanned an alert SMS is sent to the registered phone number and
also an alert notification is generated in the app. When an authorized card is
scanned without the user’s consent, the user can shut down the system by
sending a message from his phone.

  * •

The same RFID card can be used in classrooms as a check in attendance system

##  3  Details of implementation

###  3.1 Components Used

  * •

Sim900 GSM module

  * •

Arduino Uno

  * •

MFRC522 RFID reader and RFID cards

  * •

Servo motors, stepped motors and gears

  * •

4*4 keypad

  * •

Buzzer and power adaptor

  * •

Node MCU

  * •

LEDs and resistors

  * •

I2C LCD display

###  3.2 Working

Smart ID card is divided into 3 sub-systems
1) Security System
2) Payment System
3) Attendance System

  * •

Security System
The RFID reader communicates with the Arduino through the SPI protocol .The
I2C LCD communicates with the Arduino through the I2C protocol. The keypad is
connected to Arduino. The 4X4 keypad has 8 connections but the last column of
keypad is not required. We only require numbers for the password.
For powering the SIM900 module, 5V, 2A power adaptor is used. Once the SIM900
module is powered, the power light will light up and on pressing the power
key, the status led lights up. Then the phone is paired with the module.

GSM Module:
GSM is a mobile communication modem; it is stands for global system for mobile
communication (GSM). It is widely used mobile communication system in the
world. GSM is an open and digital cellular technology used for transmitting
mobile voice and data services.
GSM module is used here since it can communicate with a mobile and the data
which it receives can be processed and sent to the Arduino.

I2C Protocol:
I2C is a serial protocol for two-wire interface to connect low-speed devices
like microcontrollers, I/O interfaces and other similar peripherals in
embedded systems.

Figure 1: Security System Figure 2: Security System setup

  * •

Payment System:
The RFID reader communicates with the Node MCU through SPI protocol. The Node
MCU is connected to a web server where the data is stored. When the RFID card
is scanned and the pin is entered , the balance amount is displayed on the
screen.
Node MCU:
This device is used instead of only Arduino UNO because Node MCU has a wi-fi
module which can be connected to the web server. The ESP8266 can be controlled
from local Wi-Fi network or from the internet (after port forwarding). The
ESP-01 module has GPIO pins that can be programmed to control device/ execute
a code through the internet. The module can be programmed using an Arduino
through the serial pins (RX,TX).

Figure 3: Payment system Figure 4: Payment System setup Figure 5: Google docs
for payment system

  * •

Attendance system:
When the the ID is scanned on the RFID reader, the student name that is stored
in the RFID card is printed on the serial monitor. It is made sure that the
can’t be registered twice by comparing it with already registered IDs. An
external app is used to store the output from the serial monitor. The output
can be saved on to the computer.

Figure 6: Attendance System setup

##  4  Results and discussions

###  4.1 Security System

The Door Lock security system was successfully implemented. When an authorized
ID card is scanned onto the RFID reader and the correct password is entered
onto the keypad, only then the door unlocks when the servo motor turns.
Consequently a message is sent to the owner saying that the door is unlocked.
After few seconds the door locks back, turning the servo motor to the original
position
When the owner is inside the room, he/she can use a switch which is present
inside the room to unlock the door. Subsequently after few seconds the door
locks backs, turning the servo motor back to the original position
If in any case a wrong ID card or wrong password is entered. The whole system
locks down and an alarm is buzzed using a buzzer. A message is sent to the
owner saying that there was an attempt to breach the security system.
The security system fails to detect an intruder when RFID card’s ID is changed
to the owners ID. It will also fail if the owner is negligent, revealing the
password to others.

###  4.2 Payment System

when a ID is scanned in onto the RFID reader, the value that is stored in the
RFID, is sent to the server via WIFI module through internet on to the data
base with the date and time which is taken from the internet. This stored
value can be changed by the vendor or the shopkeeper to the new balance
amount. The changed balance amount is then updated in the ID card through the
WIFI module ESP8266
backdrop of this system is that the balance can be changed to a wrong value
giving a wrong balance

###  4.3 Attendance system

The attendance system was successfully implemented. When an registered ID card
is scanned onto the RFID reader, the ID card number is send to the database
through the wifi module Node MCU. The data base saves the student’s name, ID
number on the database. This present list can be retrieved from the database.
As a fail safe for the above implemented method, the RFID reader reads the ID
number of the card and compares it with the student register, if ID is
present, it prints the student’s name onto the serial monitor. An external app
saves the logs of the serial monitor as text.
This method would fail if some other student scans the card even if the owner
is not present in the class. So the scanner must be monitored while the
student is scanning on the RFID scanner

##  Acknowledgment

With immense pleasure we are presenting ”Enhancing Room Security and
Automating Class Attendance Using ID Cards”. As a part of the curriculum of
”Embedded Systems and Design” under the department of “Electronics and
Communication Engineering, National Institute of Technology, Karnataka”. We
wish to thank all people who gave us the unending support. We express my
profound thanks to our Professor, Dr. Ramesh Kini M., And all those who have
indirectly guided and helped us in the preparation of this project.

##

## References

  * [1] How RFID Works https://electronics.howstuffworks.com/gadgets/high-tech-gadgets/rfid.htm
  * [2] Specification of ESP8266 https://randomnerdtutorials.com/esp8266-adc-reading-analog-values-with-nodemcu/
  * [3] Data sheet ARDUINO UNO https://www.farnell.com/datasheets/1682209.pdf


# Cycle Integrals of the Parson Poincaré Series and Intersection Angles of
Geodesics on Modular Curves

Alessandro Lägeler and Markus Schwagenscheidt

###### Abstract.

We prove a geometric formula for the cycle integrals of Parson’s weight $2k$
modular integrals in terms of the intersection angles of geodesics on modular
curves. Our result is an analog for modular integrals of a classical formula
for the cycle integrals of certain hyperbolic Poincaré series, due to Katok.
On the other hand, it extends a recent geometric formula of Matsusaka and
Duke, Imamoḡlu, and Tóth for the cycle integrals of weight 2 modular
integrals.

##  1\. Introduction and statement of the main results

A classical result of Katok [6] states that for integers $k\geq 2$ the space
$S_{2k}(\Gamma)$ of cusp forms of weight $2k$ for a cofinite discrete subgroup
$\Gamma\subset\mathrm{SL}_{2}(\mathbb{R})$ is generated by a family of
Poincaré series associated with primitive hyperbolic matrices
$\gamma\in\Gamma$. Explicitly, these Poincaré series are defined by111We use a
slightly different normalization than Katok [6] to simplify our formulas.

(1.1) $\displaystyle
f_{k,\gamma}(z)=-\frac{D_{\gamma}^{k-\frac{1}{2}}}{\pi}\sum_{g\in\Gamma_{\gamma}\backslash\Gamma}\frac{1}{(Q_{\gamma}\circ
g)(z,1)^{k}},$

where $\Gamma_{\gamma}=\\{\pm\gamma^{n}:n\in\mathbb{Z}\\}$222We will assume
throughout that $-1\in\Gamma$., $Q_{\gamma}(x,y)=cx^{2}+(d-a)xy-by^{2}$ is the
binary quadratic form corresponding to $\gamma=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)$, $D_{\gamma}=\mathrm{tr}(\gamma)^{2}-4$ is the
discriminant of $Q_{\gamma}$, and $\Gamma$ acts on binary quadratic forms in
the usual way. These hyperbolic Poincaré series have many other interesting
applications, the most prominent one being Kohnen’s [7] construction of the
holomorphic kernel function for the Shimura correspondence.

Katok [6] also gave a beautiful geometric formula for (the imaginary part of)
the geodesic cycle integrals of the cusp forms $f_{k,\gamma}(z)$. If
$\gamma,\sigma\in\Gamma$ are primitive hyperbolic elements in $\Gamma$ which
have positive trace and which are not conjugacy equivalent, and $S_{\gamma}$
denotes the geodesic semi-circle in $\mathbb{H}$ connecting the two real fixed
points of $\gamma$, then Katok’s formula [6, Theorem 3] states that

(1.2)
$\displaystyle\mathrm{Im}\left(\int_{z_{0}}^{\sigma.z_{0}}f_{k,\gamma}(z)Q_{\sigma}(z,1)^{k-1}dz\right)=(D_{\gamma}D_{\sigma})^{\frac{k-1}{2}}\sum_{p\in[S_{\gamma}]\cap[S_{\sigma}]}\mu_{p}^{k}P_{k-1}(\cos\theta_{p}),$

where $z_{0}\in\mathbb{H}$ and the path of integration can be chosen
arbitrarily, the sum runs over the finitely many intersection points $p$ of
the closed geodesics $[S_{\gamma}]=\Gamma_{\gamma}\backslash S_{\gamma}$ and
$[S_{\sigma}]=\Gamma_{\sigma}\backslash S_{\sigma}$ in
$\Gamma\setminus\mathbb{H}$, and $P_{r}$ denotes the $r$-th Legendre
polynomial. Moreover, $\theta_{p}=\theta_{p}(\gamma,\sigma)\in[0,\pi]$ denotes
the _intersection angle_ at $p$, which is measured counterclockwise from the
tangent at $S_{\gamma}$ to the tangent at $S_{\sigma}$ at $p$, and
$\mu_{p}=\mu_{p}(\gamma,\sigma)\in\\{\pm 1\\}$ denotes the _sign of the
intersection_ at $p$, which is defined as follows: let $g\in\Gamma$ be chosen
such that the intersection point $p$ corresponds to the intersection point of
$S_{\gamma}$ and $S_{g\sigma g^{-1}}$ in $\mathbb{H}$, and suppose that
$S_{\gamma}$ and $S_{g\sigma g^{-1}}$ are oriented clockwise (which means that
the lower left entries of $\gamma$ and $g\sigma g^{-1}$ are positive). Then
$\mu_{p}(\gamma,\sigma)=+1$ if the left endpoint of $S_{g\sigma g^{-1}}$ lies
between the two endpoints of $S_{\gamma}$, and $\mu_{p}(\gamma,\sigma)=-1$
otherwise. The sign of $\mu_{p}$ changes if the orientation of either
$S_{\gamma}$ or $S_{g\sigma g^{-1}}$ is reversed. Note that
$\theta_{p}(\gamma,\sigma)$ does not depend on the orientation of $S_{\gamma}$
or $S_{\sigma}$, but it does depend on the order of $\gamma,\sigma$, that is,
we have $\theta_{p}(\sigma,\gamma)=\pi-\theta_{p}(\gamma,\sigma)$. Similarly,
we have $\mu_{p}(\gamma,\sigma)=-\mu_{p}(\sigma,\gamma)$.

More recently, Matsusaka [9] investigated the (homogenized) cycle integrals of
certain modular integrals of weight $2$ for $\mathrm{SL}_{2}(\mathbb{Z})$ with
rational period functions. These modular integrals were constructed by Duke,
Imamoḡlu, and Tóth in [1, 2], and are defined333Again, our normalization
differs from [9, 1, 2]. for primitive hyperbolic
$\gamma\in\mathrm{SL}_{2}(\mathbb{Z})$ by

(1.3) $\displaystyle
F_{\gamma}(z)=\frac{2\sqrt{D_{\gamma}}}{\pi}\sum_{n=0}^{\infty}\left(\int_{z_{0}}^{\gamma
z_{0}}j_{n}(\tau)\frac{d\tau}{Q_{\gamma}(\tau,1)}\right)e^{2\pi inz},$

where $j_{n}(\tau)$ denotes the unique weakly holomorphic modular function for
$\mathrm{SL}_{2}(\mathbb{Z})$ whose Fourier expansion has the shape
$j_{n}(\tau)=q^{-n}+O(q)$ with $q=e^{2\pi i\tau}$. It was shown in [1] that
the series defining $F_{\gamma}(z)$ converges to a holomorphic function on
$\mathbb{H}$, and satisfies for any $\sigma\in\mathrm{SL}_{2}(\mathbb{Z})$ the
transformation formula

(1.4) $\displaystyle
r_{\gamma}(\sigma,z)=(F_{\gamma}|_{2}\sigma)(z)-F_{\gamma}(z)=\frac{2\sqrt{D_{\gamma}}}{\pi}\sum_{\begin{subarray}{c}g\in\Gamma_{\gamma}\backslash\Gamma\\\
w_{g^{-1}\gamma g}^{\prime}<\sigma^{-1}.i\infty<w_{g^{-1}\gamma
g}\end{subarray}}\frac{\mathrm{sign}(Q_{\gamma}\circ g)}{(Q_{\gamma}\circ
g)(z,1)},$

where $w_{\gamma}^{\prime}<w_{\gamma}$ denote the two real fixed points of
$\gamma$, and we put $\mathrm{sign}(Q)=\mathrm{sign}(A)$ for a binary
quadratic form $Q(x,y)=Ax^{2}+Bxy+Cy^{2}$. Note that the sum on the right-hand
side is finite. The function $\sigma\mapsto r_{\gamma}(\sigma,z)$ defines a
holomorphic weight $2$ cocycle for $\mathrm{SL}_{2}(\mathbb{Z})$ with values
in the rational functions on $\mathbb{C}$, and the function $F_{\gamma}(z)$ is
called a _modular integral_ of weight $2$ for $r_{\gamma}(\sigma,z)$.
Matsusaka proved the remarkable formula

(1.5)
$\displaystyle\mathrm{Im}\left(\lim_{n\to\infty}\int_{\sigma^{n}.z_{0}}^{\sigma^{n+1}.z_{0}}F_{\gamma}(z)dz\right)=\sum_{p\in[S_{\gamma}]\cap[S_{\sigma}]}1.$

where $z_{0}\in\mathbb{H}$ and the path of integration can be chosen
arbitrarily, and $\gamma,\sigma\in\mathrm{SL}_{2}(\mathbb{Z})$ are primitive
hyperbolic matrices with positive trace which are not conjugacy equivalent
(compare [9, Corollary 3.8, Theorem 3.3]). Note that the homogenization of the
cycle integral on the left-hand side is necessary to make it independent of
$z_{0}$, and a conjugacy class invariant in $\sigma$. The right-hand side of
the formula (1.5) counts the number of intersections of $[S_{\gamma}]$ and
$[S_{\sigma}]$ in $\mathrm{SL}_{2}(\mathbb{Z})\backslash\mathbb{H}$, which by
[3, Theorem 3] can also be interpreted as the linking number of certain
modular knots associated with $\gamma$ and $\sigma$.

Notice that there is a striking similarity between Matsusaka’s formula (1.5)
and (the formal specialization to $k=1$ of) Katok’s formula (1.2). Motivated
by this observation, in the present work we extend Matsusaka’s formula (1.5)
to certain modular integrals of higher weight $2k$ (with $k\geq 2$) for
cofinite discrete subgroups $\Gamma\subset\mathrm{SL}_{2}(\mathbb{R})$, by
evaluating their homogenized cycle integrals in terms of intersection angles
of geodesics in $\Gamma\backslash\mathbb{H}$, much in the spirit of Katok’s
formula (1.2).

The modular integrals we consider here were introduced by Parson444In fact,
our definition is not precisely Parson’s, but differs from her original
Poincaré series by a cusp form. Moreover, we use a different normalization
than Parson to match our normalization of (1.1). in [10], and are defined for
integers $k\geq 2$ and primitive hyperbolic $\gamma\in\Gamma$ by

(1.6) $\displaystyle
F_{k,\gamma}(z):=-\frac{D_{\gamma}^{k-\frac{1}{2}}}{\pi}\sum_{g\in\Gamma_{\gamma}\backslash\Gamma}\frac{\mathrm{sign}(Q_{\gamma}\circ
g)}{(Q_{\gamma}\circ g)(z,1)^{k}}.$

A direct computation shows that the holomorphic function $F_{k,\gamma}(z)$
satsifies for any $\sigma\in\Gamma$ the transformation law

(1.7) $\displaystyle
r_{k,\gamma}(\sigma,z):=(F_{k,\gamma}|_{2k}\sigma)(z)-F_{k,\gamma}(z)=\frac{2D_{\gamma}^{k-\frac{1}{2}}}{\pi}\sum_{\begin{subarray}{c}g\in\Gamma_{\gamma}\backslash\Gamma\\\
w_{g^{-1}\gamma g}^{\prime}<\sigma^{-1}.i\infty<w_{g^{-1}\gamma
g}\end{subarray}}\frac{\mathrm{sign}(Q_{\gamma}\circ g)}{(Q_{\gamma}\circ
g)(z,1)^{k}}.$

The sum on the right hand side is finite. In particular, the map
$\sigma\mapsto r_{k,\gamma}(\sigma,z)$ is a holomorphic weight $2k$ cocycle
for $\Gamma$ with values in the rational functions on $\mathbb{C}$, and
$F_{k,\gamma}(z)$ is a modular integral for $r_{k,\gamma}(\sigma,z)$.

Notice that the cocycle $r_{\gamma}(\sigma,z)$ in (1.4) is the specialization
to $k=1$ of the cocycle $r_{k,\gamma}(\sigma,z)$ in (1.7). Hence, we may view
the weight $2$ modular integral $F_{\gamma}(z)$ defined in (1.3) as the $k=1$
analog of Parson’s modular integral $F_{k,\gamma}(z)$ defined in (1.6).
However, $F_{k,\gamma}(z)$ does not converge for $k=1$, although it is
probably possible to extend the definition (1.6) to $k=1$ using Hecke’s trick,
and to show that $F_{1,\gamma}(z)=F_{\gamma}(z)$.

Our main result is the following geometric formula for (the imaginary part of)
the cycle integrals of Parson’s modular integrals $F_{k,\gamma}(z)$. It is an
analog of Katok’s formula (1.2) for modular integrals, and a higher weight
analog of Matsusaka’s formula (1.5).

######  Theorem 1.1.

Let $k\geq 2$ be an integer and let $\gamma$, $\sigma$ be primitive hyperbolic
elements in $\Gamma$ with positive trace which are not conjugacy equivalent.
We have

(1.8)
$\mathrm{Im}\left(\lim_{n\to\infty}\int_{\sigma^{n}.z_{0}}^{\sigma^{n+1}.z_{0}}F_{k,\gamma}(z)Q_{\sigma}(z,1)^{k-1}dz\right)=(D_{\gamma}D_{\sigma})^{\frac{k-1}{2}}\sum_{p\in[S_{\gamma}]\cap[S_{\sigma}]}\mu_{p}^{k-1}P_{k-1}(\cos\theta_{p}),$

where the notation is as in (1.2).

The proof of Theorem 1.1 will be given in Section 2 below. We would like to
give a quick proof of the fact that the limit on the left-hand side exists,
and is independent of $z_{0}$. First, note that the limit of
$\sigma^{n}.z_{0}$ as $n\to\infty$ is independent of the choice of
$z_{0}\in\mathbb{H}$ and converges to $w_{\sigma}$ (if we assume for the
moment that $\sigma$ has positive trace and positive lower left entry; see [9,
Lemma 2.7]). Now the function
$x\mapsto\displaystyle\int_{x}^{\sigma.x}F_{k,\gamma}(z)Q_{\sigma}(z,1)^{k-1}dz$
is Lipschitz-continuous when $x$ is approaching $w_{\sigma}$. Indeed, note
that

(1.9)
$\displaystyle\begin{split}&\left|\int_{x_{0}}^{\sigma.x_{0}}F_{k,\gamma}(z)Q_{\sigma}(z,1)^{k-1}dz-\int_{x_{1}}^{\sigma.x_{1}}F_{k,\gamma}(z)Q_{\sigma}(z,1)^{k-1}dz\right|\\\
&=\left|\int_{x_{0}}^{x_{1}}r_{k,\gamma}(\sigma,z)Q_{\sigma}(z,1)^{k-1}\,dz\right|\leq
L_{k,\gamma,\sigma}|x_{0}-x_{1}|,\end{split}$

for some constant $L_{k,\gamma,\sigma}>0$ and $x_{0},x_{1}$ close to
$w_{\sigma}$, as $r_{k,\gamma}(\sigma,z)$ is holomorphic at $w_{\sigma}$ if
$\gamma$ and $\sigma$ are not conjugacy equivalent, so
$|r_{k,\gamma}(\sigma,z)Q_{\sigma}(z,1)^{k-1}|$ is bounded in a neighbourhood
of $w_{\sigma}$. This implies that
$\displaystyle\int_{\sigma^{n}.z_{0}}^{\sigma^{n+1}.z_{0}}F_{k,\gamma}(z)Q_{\sigma}(z,1)^{k-1}dz$
is a Cauchy sequence. Moreover, if we put $x_{0}=\sigma^{n}z_{0}$ and
$x_{1}=\sigma^{n}z_{1}$ in (LABEL:Lipschitz) and take the limit as
$n\to\infty$, we see that the left-hand side in Theorem 1.1 is independent of
$z_{0}$. Note that this also implies that the homogenized cycle integral of
$F_{k,\gamma}(z)$ is a conjugacy class invariant in $\sigma$. We would also
like to remark that the right-hand side in Theorem 1.1 is a finite sum, which
can be explicitly computed (numerically) as explained in [11].

Previous to Matsusaka’s formula (1.5), Duke, Imamoḡlu, and Tóth proved its
”parabolic version” which expresses the number of intersections of the net of
the geodesics equivalent to $S_{\gamma}$ with the non-compact geodesic
$S_{-d/c}$ from $-\frac{d}{c}$ to $i\infty$ as the central value of the
twisted $L$-function

$\displaystyle
L_{\gamma}\left(s,-\frac{d}{c}\right)=\sum_{n=1}^{\infty}\frac{a_{\gamma}(n)e^{-2\pi
i\frac{d}{c}n}}{n^{s}},\quad(c,d)=1,\;c>0,\quad\mathrm{Re}(s)\gg 1,$

of the modular integral $F_{\gamma}(z)=\sum_{n=1}^{\infty}a_{\gamma}(n)e^{2\pi
inz}$ in (1.3). The explicit formula reads

(1.10)
$\frac{1}{2\pi}\;\mathrm{Re}L_{\gamma}\left(1,-\frac{d}{c}\right)=\sum_{p\in[S_{\gamma}]\;\cap\;S_{-d/c}}1,$

where the equivalence class is over the action of
$\mathrm{SL}_{2}(\mathbb{Z})$ on the geodesic $S_{\gamma}$ (see [3, Theorem
5.2]). We generalize (1.10) to the higher weight case.

The Parson Poincaré series $F_{k,\gamma}(z)$ defined in (1.6) has a Fourier
expansion of the shape

$F_{k,\gamma}(z)=\sum_{\begin{subarray}{c}n\in\frac{1}{N}\mathbb{Z},n>0\end{subarray}}a_{k,\gamma}(n)e^{2\pi
inz},$

where $N$ denotes the width of the cusp $\infty$ with respect to the group
$\Gamma$. The Fourier coefficients $a_{k,\gamma}(n)$ can be explicitly
computed in terms of Kloosterman sums and Bessel functions as in [10, Theorem
3.1], or in terms of cycle integrals of weakly holomorphic modular forms as in
[2, Theorem 3]. We define the twisted $L$-function of $F_{k,\gamma}(z)$ by

$\displaystyle
L_{k,\gamma}\left(s,-\frac{d}{c}\right)=\sum_{n\in\frac{1}{N}\mathbb{Z},n>0}\frac{a_{k,\gamma}(n)e^{-2\pi
i\frac{d}{c}n}}{n^{s}},\quad(c,d)=1,\;c>0,\quad\mathrm{Re}(s)\gg 1,$

At its central value, the twisted $L$-function satisfies the following analog
of (1.10).

######  Theorem 1.2.

Let $k\geq 2$ be an odd integer, $\gamma$ be a hyperbolic element with
positive trace and $d,c$ be comprime integers with $c>0$. We have

(1.11)
$\displaystyle(-1)^{\frac{k-1}{2}}\frac{(k-1)!}{(2\pi)^{k}}\;\mathrm{Re}L_{k,\gamma}\left(k,-\frac{d}{c}\right)=D_{\gamma}^{\frac{k-1}{2}}\sum_{p\in[S_{\gamma}]\;\cap\;S_{-d/c}}P_{k-1}\left(\cos\theta_{p}\right),$

where $\theta_{p}$ denotes the angle of intersection at the point $p$ between
the geodesic in the equivalence class of $S_{\gamma}$ and the geodesic
$S_{-d/c}$.

The left-hand side of (1.11) can be written as the imaginary part of the cycle
integral of $F_{k,\gamma}(z)$ along the non-compact geodesic from $-d/c$ to
$i\infty$, so Theorem 1.2 can be viewed as a ”parabolic” analog of Theorem
1.1.

##  2\. The proofs of Theorem 1.1 and Theorem 1.2

###  2.1. Proof of Theorem 1.1

The proof is similar to the proof of Katok’s formula (1.2), compare [6,
Theorem 3]. The main difference is that we can’t unfold the Parson Poincaré
series as such.

Up to interchanging $\gamma$ with $\gamma^{-1}$ and $\sigma$ with
$\sigma^{-1}$ we may assume that the lower left entries of $\gamma$, $\sigma$
are positive. First, we rewrite

$\int_{\sigma^{n}.z_{0}}^{\sigma^{n+1}.z_{0}}F_{k,\gamma}(z)Q_{\sigma}(z,1)^{k-1}dz=\int_{z_{0}}^{\sigma.z_{0}}(c_{n}z+d_{n})^{-2k}F_{k,\gamma}(\sigma^{n}.z)Q_{\sigma}(z,1)^{k-1}dz,$

where we put $\sigma^{n}=\left(\begin{smallmatrix}*&*\\\
c_{n}&d_{n}\end{smallmatrix}\right)$. A direct computation shows that

(2.1)
$\mathrm{sign}(Q_{\gamma}\circ\sigma^{-n}(1,0))\to\mathrm{sign}(Q_{\gamma}(w_{\sigma}^{\prime},1))\;\;\mathrm{as}\;n\to+\infty;$

see [9, Lemma 2.7]. If the two geodesics $S_{\gamma}$ and $S_{\sigma}$
intersect, we have
$\mathrm{sign}(Q_{\gamma}(w_{\sigma}^{\prime},1))=-\mu_{p}(\gamma,\sigma)$.
Now the Parson Poincaré series becomes

$\displaystyle(c_{n}z+d_{n})^{-2k}F_{k,\gamma}(\sigma^{n}.z)$
$\displaystyle=-\frac{D_{\gamma}^{k-1/2}}{\pi}\sum_{g\in\Gamma_{\gamma}\setminus\Gamma}\frac{\mathrm{sign}(Q_{\gamma}\circ
g)}{(c_{n}z+d_{n})^{2k}(Q_{\gamma}\circ g)(\sigma^{n}.z,1)^{k}}$
$\displaystyle=-\frac{D_{\gamma}^{k-1/2}}{\pi}\sum_{g\in\Gamma_{\gamma}\setminus\Gamma}\frac{\mathrm{sign}(Q_{\gamma}\circ
g\sigma^{-n}))}{(Q_{\gamma}\circ g)(z,1)^{k}}$
$\displaystyle\to-\frac{D_{\gamma}^{k-1/2}}{\pi}\sum_{g\in\Gamma_{\gamma}\setminus\Gamma}\frac{\mathrm{sign}((Q_{\gamma}\circ
g)(w_{\sigma}^{\prime},1))}{(Q_{\gamma}\circ
g)(z,1)^{k}},\;\mathrm{as}\;n\to+\infty,$

where in the last line we applied (2.1). The resulting series is modular of
weight $2k$ for the group
$\Gamma_{\sigma}=\\{\pm\sigma^{n}:n\in\mathbb{Z}\\}$. Hence, the cycle
integral

$\int_{z_{0}}^{\sigma.z_{0}}\sum_{g\in\Gamma_{\gamma}\setminus\Gamma}\frac{\mathrm{sign}((Q_{\gamma}\circ
g)(w_{\sigma}^{\prime},1))}{(Q_{\gamma}\circ
g)(z,1)^{k}}Q_{\sigma}(z,1)^{k-1}dz$

is independent of the choice of $z_{0}$.

Now a typical unfolding argument yields

$\displaystyle\int_{z_{0}}^{\sigma.z_{0}}\sum_{g\in\Gamma_{\gamma}\setminus\Gamma}\frac{\mathrm{sign}((Q_{\gamma}\circ
g)(w_{\sigma}^{\prime},1))}{(Q_{\gamma}\circ
g)(z,1)^{k}}Q_{\sigma}(z,1)^{k-1}dz$
$\displaystyle=\sum_{g\in\Gamma_{\gamma}\setminus\Gamma/\Gamma_{\sigma}}\sum_{m\in\mathbb{Z}}\int_{\sigma^{m}.z_{0}}^{\sigma^{m+1}.z_{0}}\frac{\mathrm{sign}((Q_{\gamma}\circ
g)(w_{\sigma}^{\prime},1))}{(Q_{\gamma}\circ
g)(z,1)^{k}}Q_{\sigma}(z,1)^{k-1}dz$
$\displaystyle=\sum_{g\in\Gamma_{\gamma}\setminus\Gamma/\Gamma_{\sigma}}\int_{S_{\sigma}}\frac{\mathrm{sign}((Q_{\gamma}\circ
g)(w_{\sigma}^{\prime},1))}{(Q_{\gamma}\circ
g)(z,1)^{k}}Q_{\sigma}(z,1)^{k-1}dz.$

Subtracting the complex conjugate from the last equation, gives the sum of
integrals

$\sum_{g\in\Gamma_{\gamma}\setminus\Gamma/\Gamma_{\sigma}}\displaystyle\int_{C(\sigma)}\frac{\mathrm{sign}((Q_{\gamma}\circ
g)(w_{\sigma}^{\prime},1))}{(Q_{\gamma}\circ
g)(z,1)^{k}}Q_{\sigma}(z,1)^{k-1}dz$

over the circle $C(\sigma)$ through the roots of $Q_{\sigma}(z,1)=0$. The
integrands are all meromorphic, with poles only at the real roots of
$(Q_{\gamma}\circ g)(z,1)$. Hence, by Cauchy’s theorem, they vanish if the
geodesic connecting the roots of $(Q_{\gamma}\circ g)(z,1)$ does not intersect
$S_{\sigma}$. Thus, we are left with

$\frac{D_{\gamma}^{k-1/2}}{\pi}\sum_{p\in[S_{\gamma}]\cap[S_{\sigma}]}\mu_{p}(\gamma,\sigma)\displaystyle\int_{C(\sigma)}\frac{1}{(Q\circ
g)(z,1)^{k}}Q_{\sigma}(z,1)^{k-1}dz,$

as $\mu_{p}(\gamma,\sigma)=-\mathrm{sign}((Q\circ g)(w_{\sigma}^{\prime},1))$
if the geodesics $S_{\sigma}$ and $S_{g^{-1}\gamma g}$ intersect.

The integrals evaluate to

$\displaystyle\int_{C(\sigma)}\frac{1}{(Q_{\gamma}\circ
g)(z,1)^{k}}Q_{\sigma}(z,1)^{k-1}dz=2\pi
iD_{\gamma}^{-k/2}D_{\sigma}^{\frac{k-1}{2}}\mu_{p}^{k}P_{k-1}(\cos\theta_{p});$

see [6, p. 478]. This finishes the proof of Theorem 1.1.

###  2.2. Proof of Theorem 1.2

The proof is a careful application of [6, Lemma 2], which asserts that for odd
$k\in\mathbb{Z},k\geq 3$, and any $A,B,C\in\mathbb{R}$ with $D=B^{2}-4AC>0$,
we have

(2.2)
$\int_{-\infty}^{\infty}\frac{t^{k-1}}{(-At^{2}+Bit+C)^{k}}dt=\begin{cases}0,&AC>0,\\\
(-1)^{\frac{k+1}{2}}\mathrm{sign}(A)2\pi
D^{-k/2}P_{k-1}\left(\frac{B}{\sqrt{D}}\right),&AC<0.\end{cases}$

Consider the integral
$\displaystyle\int_{-\frac{d}{c}}^{i\infty}F_{k,\gamma}(z)(cz+d)^{k-1}dz$. A
standard argument gives that

$\int_{-\frac{d}{c}}^{i\infty}F_{k,\gamma}(z)(cz+d)^{k-1}dz=\left(\frac{c}{2\pi}\right)^{k}\Gamma(k)\frac{i^{k}}{c}L_{\gamma,k}\left(k,-\frac{d}{c}\right).$

The imaginary part of the integral is equal to

$\displaystyle\mathrm{Im}\int_{-\frac{d}{c}}^{i\infty}F_{k,\gamma}(z)(cz+d)^{k-1}dz$
$\displaystyle=c^{k-1}\mathrm{Im}\int_{0}^{i\infty}F_{k,\gamma}\left(z-\frac{d}{c}\right)z^{k-1}dz$
$\displaystyle=-c^{k-1}\frac{D_{\gamma}^{k-\frac{1}{2}}}{\pi}\sum_{g\in\Gamma_{\gamma}\setminus\Gamma}\mathrm{Im}\left(\int_{0}^{i\infty}\frac{\mathrm{sign}(Q_{\gamma}\circ
g)z^{k-1}}{(Q_{\gamma}\circ g)(z-d/c,1)^{k}}dz\right),$

where we can exchange sum and integral by Fubini’s theorem, as the integrand
is absolutely convergent.

Fix $g\in\Gamma_{\gamma}\setminus\Gamma$ for the moment and write
$Az^{2}+Bz+C$ for $(Q_{\gamma}\circ g)(z,1)$. The quadratic form

$(Q_{\gamma}\circ
g)(z-d/c,1)=Az^{2}+\left(B-2Ad/c\right)z+\left(A\left(d/c\right)^{2}-Bd/c+C\right)=A^{\prime}z^{2}+B^{\prime}z+C^{\prime}$

intersects with the non-compact geodesic $S_{-d/c}$ if and only if

$A^{\prime}C^{\prime}=A\left(A\left(d/c\right)^{2}-Bd/c+C\right)<0.$

With (2.2), we get

$\displaystyle\mathrm{Im}\left(\int_{0}^{i\infty}\frac{\mathrm{sign}(A^{\prime})z^{k-1}}{\left(A^{\prime}z^{2}+B^{\prime}z+C^{\prime}\right)^{k}}dz\right)$
$\displaystyle=\frac{(-1)^{\frac{k-1}{2}}}{2}\int_{-\infty}^{\infty}\frac{\mathrm{sign}(A^{\prime})t^{k-1}}{\left(-A^{\prime}t^{2}+B^{\prime}it+C^{\prime}\right)^{k}}dt$
$\displaystyle=-\pi
D_{\gamma}^{-k/2}P_{k-1}\left(\frac{B^{\prime}}{\sqrt{D_{\gamma}}}\right)$

if and only if the geodesic $S_{g\gamma g^{-1}}$ intersects the non-compact
geodesic $S_{-d/c}$ associated to the quadratic form $cz+d$. Otherwise, the
integral evaluates to zero by (2.2). By [11, Proposition 2.2], the
intersection angle $\theta_{p}$ between these two geodesics is given by
$\cos\theta_{p}=\frac{Bc-2Ad}{c\sqrt{D_{\gamma}}}$. This finishes the proof.

##  3\. Additional Remarks

In this section, we present some further properties and possible applications
of the homogenized cycle integrals and periods of Parson’s modular integrals
$F_{k,\gamma}(z)$.

###  3.1. Explicit representation of the cycle integral

As the Parson Poincaré series is no longer modular, the cycle integral
$\displaystyle\int_{z_{0}}^{\sigma.z_{0}}F_{k,\gamma}(z)Q_{\sigma}(z,1)^{k-1}dz$
depends on the choice of the point $z_{0}$ and is a complicated function in
terms of $z_{0}$. Only when taking the homogenization we get a conjugacy class
invariant object independent of the choice of $z_{0}$, which has the nice
representation on the right hand side of (1.8).

Explicitly, we have

$\displaystyle\int_{z_{0}}^{\sigma.z_{0}}F_{k,\gamma}(z)Q_{\sigma}(z,1)^{k-1}dz=\int_{i\infty}^{\sigma.i\infty}F_{k,\gamma}(z)Q_{\sigma}(z,1)^{k-1}dz$
$\displaystyle+\sum_{n=0}^{2k-2}\sum_{\begin{subarray}{c}g\in\Gamma_{\gamma}\setminus\Gamma,\\\
w_{g\gamma g^{-1}}^{\prime}<\sigma^{-1}.i\infty<w_{g\gamma
g^{-1}}\end{subarray}}\rho_{n,g,\gamma,\sigma}(z_{0})\;_{2}F_{1}\left(k,2k-1-n;2k;1-\frac{z_{0}-w_{Q}}{z_{0}-w_{Q}^{\prime}}\right),$

where $\rho_{n,g,\gamma,\sigma}(z_{0})$ is a rational function given by
$\rho_{n,g,\gamma,\sigma}(z_{0})=\frac{\Gamma(2k-n-1)}{\Gamma(2k)}\frac{\partial_{z}^{n}Q_{\sigma}(z,1)^{k-1}|_{z=z_{0}}}{(z_{0}-w_{g\gamma
g^{-1}}^{\prime})^{n-2k-1}}$. One can also see from this representation that
the integral converges as $z_{0}\to w_{\sigma}$.

To prove this, we use

$\int_{z_{0}}^{\sigma.z_{0}}F_{k,\gamma}(z)Q_{\sigma}(z,1)^{k-1}dz=\int_{i\infty}^{\sigma.i\infty}F_{k,\gamma}(z)Q_{\sigma}(z,1)^{k-1}dz+\int_{i\infty}^{z_{0}}r_{k,\gamma}(\sigma,z)Q_{\sigma}(z,1)^{k-1}dz.$

This follows from differentiating both sides in $z_{0}$ and observing that
both sides are equal at $z_{0}\to i\infty$. Rewriting the polynomial
$Q_{\sigma}(z,1)^{k-1}=\sum_{n=1}^{2k-2}a_{n,\sigma}(z_{0})(z-z_{0})^{n}$ in
its Taylor expansion about $z_{0}$, we obtain a finite sum of integrals

$\int_{i\infty}^{z_{0}}r_{k,\gamma}(\sigma,z)Q_{\sigma}(z,1)^{k-1}dz=\sum_{n=1}^{2k-2}a_{n,\sigma}(z_{0})\sum_{\begin{subarray}{c}Q\sim
Q_{\gamma},\\\
w_{Q}^{\prime}<\sigma^{-1}.i\infty<w_{Q}\end{subarray}}\int_{z_{0}}^{i\infty}\frac{(z-z_{0})^{n}}{(z-w_{Q})^{k}(z-w_{Q}^{\prime})^{k}}dz,$

which can be solved. Standard integral transformations give

(3.1)
$\displaystyle\begin{split}&\int_{z_{0}}^{i\infty}\frac{(z-z_{0})^{n}}{(z-w_{Q})^{k}(z-w_{Q}^{\prime})^{k}}dz=\\\
&\frac{\Gamma(2k-n-1)\Gamma(n+1)}{\Gamma(2k)}(z-w_{Q}^{\prime})^{n-2k+1}\
_{2}F_{1}\left(k,2k-n-1,2k;1-\frac{z-w_{Q}}{z-w_{Q}^{\prime}}\right).\end{split}$

That the right hand side of (LABEL:integralevaluation) is symmetric in
$w_{Q}^{\prime}$ and $w_{Q}$ also follows from the identity $\
{}_{2}F_{1}(c-a,b,c;z/(z-1))=(1-z)^{b}\ _{2}F_{1}(a,b,c;z)$.

The integral evaluation (LABEL:integralevaluation) can also be used to prove
that the weight $2-2k$ cocycle

$\displaystyle R_{\gamma}(\sigma,z)$ $\displaystyle=\frac{(-2\pi
i)^{2k-1}D^{k-1/2}}{\pi(2k-1)!\binom{2k-2}{k-1}}\sum_{w_{Q}^{\prime}<-\frac{d}{c}<w_{Q}}\frac{1}{|Q(1,0)|(z-w_{Q}^{\prime})}\
_{2}F_{1}\bigg{(}k,1,2k;1-\frac{z-w_{Q}}{z-w_{Q}^{\prime}}\bigg{)}$
$\displaystyle+\frac{(-2\pi
i)^{2k-1}}{(2k-2)!}\frac{i}{c^{2k-1}}\sum_{n=0}^{2k-2}\binom{2k-2}{n}i^{n}\left(\frac{c}{2\pi}\right)^{n+1}\Gamma(n+1)L_{k,\gamma}(n+1,a/c)(cz+d)^{n}$

for $\sigma=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\Gamma$ and $Q$ running over the equivalence
class $[Q_{\gamma}]$ is a $(2k-1)$-th primitive of $r_{\gamma}(g,z)$, i.e.
$\mathcal{D}^{2k-1}R_{\gamma}(g,z)=r_{\gamma}(g,z)$ with
$\mathcal{D}=\frac{1}{2\pi i}\frac{\partial}{\partial z}$. The Fourier
coefficients $a_{k,\gamma}(n)$ can be explicitly calculated as in [10, Theorem
3].

###  3.2. Periods of modular integrals

In this subsection we let $\Gamma=\mathrm{SL}_{2}(\mathbb{Z})$. Kohnen and
Zagier [8] studied the periods of the cusp forms $f_{k,\gamma}(z)$ defined in
(1.1), and showed that certain linear combinations of these periods are
rational numbers.

A natural follow-up question to Theorem 1.1 would be to study the periods of
the Parson Poincaré series, i.e.
$p_{n}(F_{k,\gamma})=\displaystyle\int_{0}^{\infty}F_{k,\gamma}(it)t^{n}\;dt$
for $0\leq n\leq 2k-2$. As in [8] we define the symmetrizations

$F_{k,\gamma}^{+}=F_{k,\gamma}+F_{k,\gamma^{\prime}}(z),\qquad
F_{k,\gamma}^{-}=i(F_{k,\gamma}(z)-F_{k,\gamma^{\prime}}(z)),$

where $\gamma^{\prime}=\left(\begin{smallmatrix}-1&0\\\
0&1\end{smallmatrix}\right)\gamma\left(\begin{smallmatrix}-1&0\\\
0&1\end{smallmatrix}\right)$. We split the period polynomial

$p(F_{k,\gamma})(z)=\displaystyle\int_{0}^{i\infty}F_{k,\gamma}(z)(x-z)^{2k-2}dz=\sum_{n=0}^{2k-2}i^{-n+1}\binom{2k-2}{n}p_{n}(F_{k,\gamma})x^{2k-2-n}$

of $F_{k,\gamma}$ into its even and odd part as
$p(F_{k,\gamma})=ip^{+}(F_{k,\gamma})+p^{-}(F_{k,\gamma})$ with

$\displaystyle p^{+}(F_{k,\gamma})(x)$
$\displaystyle=\sum_{\begin{subarray}{c}0\leq n\leq 2k-2\\\ n\text{
even}\end{subarray}}(-1)^{n/2}\binom{2k-2}{n}p_{n}(F_{k,\gamma})x^{2k-2-n},$
$\displaystyle p^{-}(F_{k,\gamma})(x)$
$\displaystyle=\sum_{\begin{subarray}{c}0<n<2k-2\\\ n\text{
odd}\end{subarray}}(-1)^{(n-1)/2}\binom{2k-2}{n}p_{n}(F_{k,\gamma})x^{2k-2-n}.$

By closely following the proof of [8, Theorem 5], one obtains

(3.2)
$\displaystyle\begin{split}&p^{+}(F_{k,\gamma}^{+})(x)+p^{-}(F_{k,\gamma}^{-})(x)\\\
&\quad\doteq-2\sum_{\begin{subarray}{c}[a,b,c]\in[Q_{\gamma}]\\\
a<0<c\end{subarray}}(ax^{2}-bx+c)^{k-1}-\frac{2D^{k-1/2}\zeta_{Q_{\gamma}}(k)}{\binom{2k-2}{k-1}(2k-1)\zeta(2k)}(x^{2k-2}-1),\end{split}$

where $\zeta_{Q_{\gamma}}(s)$ is the $\zeta$-function associated with
$Q_{\gamma}$ as in [8, p. 222], $\zeta(s)$ is the Riemann $\zeta$-function,
and $\doteq$ means equality up to a non-zero multiplicative constant555The
formulas (3.2) and (3.3) are correct if we normalize $F_{k,\gamma}$ as in [8].
Since our normalization of $F_{k,\gamma}$ is different, we get some simple but
unpleasant extra factors.. In particular, the periods
$p_{n}(F_{k,\gamma}^{+})$ for even $0<n<2k-2$ and the periods
$p_{n}(F_{k,\gamma}^{-})$ for odd $0<n<2k-2$ are rational.

We let $F_{k,D}(z)=\sum_{D_{\gamma}=D}F_{k,\gamma}(z)$ where the sum ranges
over a system of representatives $\gamma$ of the conjugacy classes of
primitive hyperbolic elements in $\mathrm{SL}_{2}(\mathbb{Z})$ with
discriminant $D_{\gamma}=D$. From (3.2) it follows that for odd
$k\in\mathbb{Z}$ we have

(3.3)
$\displaystyle\begin{split}p^{+}(F_{k,D})&\doteq-\sum_{\begin{subarray}{c}[a,b,c]\in\mathcal{Q}_{D}\\\
a<0<c\end{subarray}}(ax^{2}+bx+c)^{k-1}-\frac{D^{k-1/2}\zeta(k)L_{D}(k)}{\binom{2k-2}{k-1}(2k-1)\zeta(2k)}(x^{2k-2}-1),\end{split}$

where $L_{D}(s)$ is the Dirichlet $L$-function associated to the Kronecker
symbol $\left(\frac{D}{\cdot}\right)$. Formula (3.3) is the analog of [8,
Theorem 4]. In particular, the periods $p_{n}(F_{k,D})$ for even $0<n<2k-2$
are rational and satisfy the symmetry $p_{2k-2-n}(F_{k,D})=p_{n}(F_{k,D})$.

###  3.3. The homogenized cycle integral as an ”inner product”

By unfolding and [7, Proposition 7], one can also show that

$\displaystyle\lim_{n\to\infty}\int_{\sigma^{n}.z_{0}}^{\sigma^{n+1}.z_{0}}F_{k,\gamma}(z)Q_{\sigma}(z,1)^{k-1}dz$
$\displaystyle\qquad\doteq\int_{\Gamma\setminus\mathbb{H}}\sum_{g\in\Gamma_{\gamma}\backslash\Gamma}\sum_{h\in\Gamma_{\sigma}\backslash\Gamma}\frac{\mathrm{sign}((Q_{\gamma}\circ
g)(w_{h\sigma h^{-1}}^{\prime},1))}{(Q_{\gamma}\circ
g)(z,1)^{k}(Q_{\sigma}\circ h)(\overline{z},1)^{k}}y^{2k}\frac{dxdy}{y^{2}}.$

Since $w_{h\sigma h^{-1}}^{\prime}=h^{-1}.w_{\sigma}^{\prime}$, the integrand
is $\Gamma$-invariant.

One should compare this with the fact that the cycle integral
$\displaystyle\int_{z_{0}}^{\sigma.z_{0}}f_{k,\gamma}(z)Q_{\sigma}(z,1)^{k-1}dz$
is up to constants equal to the Petersson inner product $\langle
f_{k,\sigma},f_{k,\gamma}\rangle$ of the hyperbolic Poincaré series (1.1).

###  3.4. Equidistribution of intersection angles

A possible application of Theorem 1.1 could be to give another proof the fact
that the intersection angles $\theta_{p}$ of a fixed geodesic $[S_{\gamma}]$
with the geodesics of discriminant $D$ (for $\mathrm{SL}_{2}(\mathbb{Z})$)
equidistribute to the measure $\frac{1}{2}\sin\theta\;d\theta$ as
$D\to+\infty$. This was conjectured by Rickards [11, Conjecture 4.2] and
recently proved by Jung and Sardari [5]. To show this, it suffices to prove
that $\cos\theta_{p}$ equidistribute to the Lebesgue measure on $[-1,1]$. The
Legendre polynomials $P_{k-1}$ with $k\geq 1$ form a complete orthonormal
system of the space of continuous functions on $[-1,1]$. For $k>1$ even, the
Weyl sum on the right hand side of (1.2) can be estimated using the Shimura-
theory of cusp forms [7] and non-trivial bounds on the Fourier coefficients of
half-integral weight cusp forms [4]. A Siegel-type bound for the number of
intersections can be obtained using (1.5), [9, Theorems 3.3 and 4.7, Remark
4.8], and some elementary considerations on continued fractions. For $k>1$
odd, this approach does not work due to the appearance of the
$\mu_{p}$-factor, so the Weyl sum here is given by the right hand side in
(1.8). However, it appears to be difficult to estimate the traces of the
homogenized cycle integrals. We plan to come back to this in the future.

### Acknowledgments

We are indebted to Özlem Imamoḡlu for suggesting the topic of the paper to us
and for many insightful discussions. Moreover, we thank Árpád Tóth and Toshiki
Matsusaka for helpful discussions. The first author was supported by SNF
project 200021_185014 and the the second author was supported by SNF projects
200021_185014 and PZ00P2_202210.

## References

  * [1] W. Duke, Ö. Imamoḡlu, and Á. Tóth, Cycle integrals of the $j$-function and mock modular forms, Ann. of Math. (2) 173(2) (2011), 947-981.
  * [2] W. Duke, Ö. Imamoḡlu, and Á. Tóth, Rational period functions and cycle integrals, Abh. Math. Semin. Univ. Hambg. 80 (2) (2010), 255-264.
  * [3] W. Duke, Ö. Imamoḡlu, and Á. Tóth, Modular cocycles and linking numbers, Duke Math. J. 166(6) (2017), 1179-1210.
  * [4] H. Iwaniec, Fourier coefficients of modular forms of half-integral weight, Invent. math. 87(2) (1987): 385-401.
  * [5] J. Jung and N.T. Sardari. Intersecting geodesics on the modular surface, arXiv preprint arXiv:2101.08768 (2021).
  * [6] S. Katok, Closed geodesics, periods and arithmetic of modular forms, Invent. math. 80(3) (1985): 469-480.
  * [7] W. Kohnen, Fourier coefficients of modular forms of half-integral weight, Math. Ann. 271 (2) (1985): 237-268.
  * [8] W. Kohnen and D. Zagier, Modular forms with rational periods, in “Modular forms”, ed. by R. A. Rankin, Ellis Horwood, (1985), 197–249.
  * [9] T. Matsusaka, A Hyperbolic Analogue of the Rademacher Symbol, arXiv preprint arXiv:2003.12354 (2020).
  * [10] L.A. Parson, Modular integrals and indefinite binary quadratic forms, A tribute to Emil Grosswald: number theory and related analysis, 513–523, Contemp. Math. 143, Amer. Math. Soc., Providence, RI (1993).
  * [11] J. Rickards, Computing intersections of closed geodesics on the modular curve, J. Number Theory 225 (2021).


# What do neural networks listen to? Exploring the crucial bands in Speech
Enhancement using Sinc-convolution

###### Abstract

This study introduces a reformed Sinc-convolution (Sincconv) framework
tailored for the encoder component of deep networks for speech enhancement
(SE). The reformed Sincconv, based on parametrized sinc functions as band-pass
filters, offers notable advantages in terms of training efficiency, filter
diversity, and interpretability. The reformed Sinc-conv is evaluated in
conjunction with various SE models, showcasing its ability to boost SE
performance. Furthermore, the reformed Sincconv provides valuable insights
into the specific frequency components that are prioritized in an SE scenario.
This opens up a new direction of SE research and improving our knowledge of
their operating dynamics.

Index Terms—  Speech Ehancement, Sinc-convolution, Interpretability

##  1 Introduction

Speech enhancement (SE) aims to ameliorate the quality and intelligibility of
speech signals afflicted by noise interference, reverberations, and
environmental disturbances. This pivotal field addresses the challenges posed
by degraded speech in diverse contexts, including online meetings [1], hearing
aids [2], and speech recognition [3]. Recent research attests DNN-fueled
methods as superior alternatives to traditional statistical approaches,
particularly excelling in combating non-stationary noise scenarios [4, 5, 8,
6, 7, 9, 10].

DNN-fueled SE methods are categorized broadly into spectral- and time-domain
approaches based on the feature extraction from the raw waveform. In spectral-
domain approaches, most employ hand-crafted features produced by deterministic
filterbanks such as Short-time Fourier Transform (STFT) [7, 9], Gammatone
filter (GTF) [11, 12], or Mel Frequency Cepstral Coefficients (MFCC) [13].
Neural networks devised in this category either directly apply the phase of
noisy input or aim for complex-valued masks, resulting in suboptimal results
or challenging estimation [5, 8]. On the other hand, time-domain approaches
obtain speech representations through learning filterbank-like structures with
uni-dimensional convolutional layers (Conv1d) to process magnitude and phase
information simultaneously [6, 10, 14]. However, Conv1d requires a small
kernel size (less than 16 sample points) to converge, translating to
incongruous weights with limited interpretability. This implicit process makes
sense for the neural network yet does not appeal to human intuition [15].

The encoder-decoder pair is one of the most critical parts in the current
mainstream of SE algorithms, especially the encoder, which not only deals with
high-order information dwelling in input waveforms but profoundly affects
whether the gradient vanishes as to the learned network. As such, we seek more
freedom in learning filter weights as time-domain approaches while retaining
interpretability as spectral-domain approaches. This objective compels us to
investigate parametric filters. In particular, Sinc-convolution (Sinc-conv)
[16] piques our interest.

Sinc-conv convolves the waveform with parametrized sinc functions that serve
as band-pass filters, and the low and high cutoff frequencies are the sole
parameters that necessitate learning during network optimization. Sinc-conv
has been proven effective in speaker identification [16], speech recognition
[15, 17], and mispronunciation detection [18]. Nonetheless, these tasks are
cast as classification, mapping from higher-dimension (based on the framed
signals) to lower-dimension representations (based on the classes). Therefore,
we recast the formation of the original Sinc-conv, facilitating a more
effective optimization suitable for SE. Additionally, the re-formation expands
the diversity of the filter types by incorporating low-pass and high-pass
options. Comprehensive experiments verify that we can boost the state-of-the-
art by integrating our reformed Sinc-conv, while being capable of dissecting
what neural networks pay attention to when performing SE.

##  2 Reforming Sinc-conv

Generally speaking, a masking-based SE method briefly consists of the
following steps: First, given a noisy utterance $x\in\mathbb{R}^{T}$, the
encoder transforms $x$ to gain analytic speech feature
$\tilde{x}\in\mathbb{R}^{N\times T^{\prime}}$. Afterward, $\tilde{x}$ is used
to learn a mask $m\in\mathbb{R}^{N\times T^{\prime}}$ that suppresses the
unwanted noise component by element-wise multiplying $m$ with $\tilde{x}$. The
multiplied output then feeds the decoder to obtain an estimate of the clean
utterance $\hat{x}\in\mathbb{R}^{T}$. The ultimate objective is to improve
discourse quality and intelligibility while reducing noise. The method
proposed in this study centers on the development of an encoder-decoder pair
utilizing a reformed Sinc-conv.

###  2.1 Formulation

Sinc-conv entails convolving an input signal with $N$ filters, each having a
finite impulse response (FIR) $h_{i}[n]$ that includes a truncated sinc
function. The output of each filter can be expressed by:

$\tilde{x}_{i}[n]=x[n]\ast h_{i}[n]=\sum^{L-1}_{l=0}x[n-l]h_{i}[l],$ (1)

where an odd number $L=2M+1$ denotes the number of taps. For simplicity of
discussion, we omit the filter index $i$ without losing generality. Each
filter $h[n]$ is to approximate an ideal band-pass (magnitude) frequency
response as:

$H(j\omega)=\text{rect}\left(\frac{\omega}{2\omega_{c_{2}}}\right)-\text{rect}\left(\frac{\omega}{2\omega_{c_{1}}}\right),$
(2)

where $\omega_{c_{1}}$ and $\omega_{c_{2}}$ represent the low and high cutoff
frequencies, respectively, and $\text{rect}(x)$ is a rectangular function
defined by $\text{rect}(x)=1$ for $-0.5\leq x\leq 0.5$ and $\text{rect}(x)=0$
elsewhere. The inverse Fourier transform of $H(j\omega)$ is :

$\tilde{h}[n]=\frac{\omega_{c_{2}}}{\pi}\text{sinc}\left(\omega_{c_{2}}n\right)-\frac{\omega_{c_{1}}}{\pi}\text{sinc}\left(\omega_{c_{1}}n\right),$
(3)

where $-\infty<n<\infty$ and the sinc function is defined by
$\text{sinc}(x)=\frac{\sin(x)}{x}$. $\tilde{h}[n]$ is then delayed by $M$ and
truncated, to be length of $L$, producing a causal FIR filter
$\hat{h}[n]=\tilde{h}[n-M]$ for $0\leq n\leq L-1$ and $\hat{h}[n]=0$
elsewhere. Convolving with such a filter extracts the components of $x[n]$
that approximately fall within the bandwidth
$[\omega_{c_{1}},\omega_{c_{2}}]$.

However, in the original setting [16], the parameters $\omega_{c_{1}}$ and
$\omega_{c_{2}}$ are determined during DNN optimization, indicating that the
corresponding range can run over $[0,\infty]$. This raises two concerns:
First, updating weights is strenuous. In fact, many issues have reported that
the learned filterbanks resemble their initial
appearance111https://github.com/mravanelli/SincNet/issues?q=frequency+response;
in other words, the learned frequencies have not been significantly altered.
Second, the Nyquist theorem might be contravened — the cutoff frequencies
$\omega_{c_{1}}$ and $\omega_{c_{2}}$ should be within the range
$[0,\omega_{N}]$, where $\omega_{N}$ denotes the Nyquist rate.

Consequently, we recast the formation as:

$\begin{cases}\omega_{c_{1}}=\alpha_{c_{1}}\cdot\omega_{N}\\\
\omega_{c_{2}}=\alpha_{c_{2}}\cdot\omega_{N},\end{cases}$ (4)

and to ensure that $0<\omega_{c_{1}}<\omega_{c_{2}}<\omega_{n}$, the
intermediate parameters (normalized frequencies relative to the Nyquist rate),
$\alpha_{c_{1}}$ and $\alpha_{c_{2}}$, are calculated as:

$\begin{cases}\alpha_{c_{1}}=\min(\min(|\alpha_{c_{1}}^{raw}|,|\alpha_{c_{2}}^{raw}|),1)\\\
\alpha_{c_{2}}=\min(\max(|\alpha_{c_{1}}^{raw}|,|\alpha_{c_{2}}^{raw}|),1),\end{cases}$
(5)

where the raw normalized frequencies, $\alpha_{c_{1}}^{raw}$ and
$\alpha_{c_{2}}^{raw}$, are the only parameters involved in training
optimization. This way, updating $\alpha_{c_{1}}^{raw}$ and
$\alpha_{c_{2}}^{raw}$ will be more influential in altering the cutoff
frequencies compared to directly updating $\omega_{c_{1}}$ and
$\omega_{c_{2}}$, thereby encouraging the pursuit of optimal solutions across
multiple bands. Furthermore, we apply a learnable band gain (BG)
$\beta_{i}\geq 0$ for each filter $h_{i}[n]$ and use a Hamming window $w[n]$
for mitigating the truncation effect, resulting in the final filter for the
$i$-th band being expressed as:

$h_{i}[n]=\beta_{i}\cdot\hat{h}_{i}[n]\cdot w[n],$ (6)

where $\beta_{i}$ is initialized to 1. In our implementation, an optional
layer-wise normalization is first applied to the convolved output
$\tilde{x}_{i}[n]$ in Eq. (1) then multiplying $\beta_{i}$.

As for the decoder counterpart, we examine three options:

  1. 1.

A learnable transposed Conv1d (T-Conv1d).

  2. 2.

The linear combination of the masked outputs. We merge $N$ channels into 1
through a linear combination with weights,
$softmax(\\{\gamma_{i}\\}_{i=1}^{N})$, where each $\gamma_{i}$ is learnable.

  3. 3.

The pseudo-inverse (pinv) of the learned filterbanks.

###  2.2 Cutoff Frequencies Initialization

Setting proper initial cutoff frequencies for filters $\hat{h}_{i}[n]$ in the
encoder is crucial to boosting and improving optimization. Therefore, we adopt
three initialization strategies for the raw normalized cutoff frequency pair
$(\alpha_{c_{1}}^{raw},\alpha_{c_{2}}^{raw})$:

  1. 1.

Sampling from a Uniform distribution. The frequency pair is yielded by drawing
samples from a standard uniform distribution $\mathcal{U}[0,1]$. We set this
as the default method.

  2. 2.

Sampling from a learned formant distribution: In [16], it has been
demonstrated that the cumulative frequency response (CFR) curve of the learned
filters is analogous to a formant distribution. We interpolate and normalize
this curve into a probability mass function (PMF) and then sample frequency
pairs from that PMF.

  3. 3.

Sampling from the Mel-scale, as in [16].

###  2.3 Attribute

As stated in [16], Sinc-conv has multiple remarkable attributes, including
faster convergence and fewer parameters than a conventional Conv1d. Based on
the explicit expression of a sinc function, Sinc-conv enables the encoder
network to focus on tuning fewer parameters with physical meaning (the cutoff
frequencies), accelerating convergence. However, when considering the
presented re-formation and task orientation, there would be more potential
attributes and benefits.

Considering the magnitude of the update and the learning rate sensitivity,
minor updates are generally easier to manage during training since they lead
to a smoother convergence toward the optimal solution. Moreover, since cutoff
frequencies are permitted to be more flexible, it creates a greater diversity
of filter types. The learned filters are either low-, high-, or band-pass.
Furthermore, the BGs reveal distinct levels of influence for each band in SE.
Last but not least, due to its symmetric coefficients, each filter
$\hat{h}_{i}[n]$ has a linear phase response that indicates no phase
distortion due to time delay, which is crucial in generative task like SE.

##  3 Experiment

###  3.1 Setup

Dataset. To assess the effectiveness of our proposed method, we conducted
experiments on the VoiceBank-DEMAND [19] benchmark dataset. This widely used
open-source corpus for SE includes 11,572 pre-synthesized training utterances
(from 28 speakers) with ten types of noise from the DEMAND database [20] at
four SNR values (0, 5, 10, and 15 dB). The test set comprises 824 utterances
(from 2 speakers) contaminated by five types of noise at SNR values of 2.5,
7.5, 12.5, and 17.5 dB. We reserved approximately 200 training set utterances
for validation. All speech data has a 16 kHz sample rate.

Model. For our reformed Sinc-conv framework, we set $N=80$ pairs of cutoff
frequencies and the kernel size $L$ as 251. We validate the framework as the
encoder-decoder component of two renowned SE models: Conv-TasNet [14] and
MANNER [10]. In Conv-TasNet, we set the baseline configuration with default
hyperparameters ($N,L,B,H,P,X,R$) as specified in [14]. The Sinc-conv method
is evaluated using the Conv-TasNet model with revised hyperparameters $B=120$
and $H=256$ while keeping the rest of the hyperparameters unchanged. Such
revised Conv-TasNet reduces the total parameter count by 46%. The loss
function uses the negative scale-invariant source-to-noise ratio (SI-SNR). On
the other hand, MANNER, which is considered state-of-the-art, demonstrates
more effectiveness in SE. We incorporate the presented Sinc-conv into the
MANNER framework to assess its potential to boost SE behavior. However, our
evaluation is restricted to a reduced-scale version of MANNER, as mentioned in
[10], due to limited computational resources. The initial hidden size in the
MANNER module is designated as $N=80$, and all other hyperparameters and the
loss function remain consistent with [10].

Evaluation metrics. Perceptual Evaluation of Speech Quality (PESQ) and Short-
time objective intelligibility (STOI) are often used as objective metrics for
evaluating speech quality and intelligibility. We use both PESQ and STOI to
compare the performance. Furthermore, SI-SNR is also reported to quantify
noise reduction, and higher SI-SNR values indicate better noise suppression.

###  3.2 Result

Table 1: The comparison in Conv-TasNet. ”ln”, ”sc”, and ”lc” stands for
”learnable”, ”Sinc-conv”, and ”linear combination”. The ”org. form” implies
using the original Sinc-conv.

| Encoder | Decoder | PESQ | STOI(%) | SI-SNR
---|---|---|---|---|---
| ln | sc | ln | lc
Baseline | ✓ |  | ✓ |  | 2.57 | 93.1 | 18.32
Proposed |  | ✓ | ✓ |  | 2.65 | 93.6 | 18.37
|  | ✓ |  | ✓ | 2.61 | 93.5 | 18.63
(org. form) |  | ✓ | ✓ |  | 2.56 | 93.1 | 18.01

Table 1 shows the results of testing the ConvTasNet module with various
modifications. Our reformed Sinc-conv performs better, even when employing a
simple linear combination as the decoder. Contrarily, using original Sinc-conv
yields outcomes merely comparable to the baseline Conv-TasNet. However, note
that using Sinc-conv can substantially reduce 46% in model parameters, which
is quite impressive. This demonstrates the capability of Sinc-conv to reduce
the dimension of features, in this case, from 512 to 80.

Table 2: The comparison in MANNER. ”fm init.” and ”mel init.” stands for
”formant” and ”Mel-scale initialization”.

| Encoder | Decoder | PESQ | STOI(%) | SI-SNR
---|---|---|---|---|---
ln | sc | ln | lc | pinv
Baseline | ✓ |  | ✓ |  |  | 2.91 | 94.5 | 18.70
Proposed |  | ✓ | ✓ |  |  | 3.03 | 95.0 | 18.91
|  | ✓ |  | ✓ |  | 2.96 | 94.6 | 18.81
|  | ✓ |  |  | ✓ | 2.81 | 94.2 | 18.19
(fm init.) |  | ✓ | ✓ |  |  | 2.96 | 94.6 | 18.61
|  | ✓ |  | ✓ |  | 2.89 | 94.4 | 18.62
(mel init.) |  | ✓ | ✓ |  |  | 2.85 | 94.3 | 18.53
|  | ✓ |  | ✓ |  | 2.80 | 94.1 | 18.35
(org. form) |  | ✓ | ✓ |  |  | 2.75 | 94.1 | 18.20

Table 2 reports the results experimented on MANNER in various forms. Our
reformed Sinc-conv, when paired with either T-Conv1d or linear combination as
the decoder, performs better than the baseline. However, the pinv decoder
fails probably due to the ambiguity in its interpretation as a pseudo matrix
inverse of learned filterbanks. A similar finding is also reported in [14,
12]. As for the initialization method, uniform sampling performs the best,
followed by formant distribution sampling and then Mel-scale frequencies.
Uniform sampling introduces stochasticity, which stimulates the exploration of
different frequencies, potentially helping Sinc-conv try out more band
locations. Comparatively, formant sampling leverages the prior knowledge
obtained from speaker identification tasks, yet the resulting sampled cutoff
frequencies are not necessarily feasible for SE purposes. The inferior results
when employing Mel-scale frequency initialization for both the original and
reformed Sinc-conv demonstrate the limitations of this initialization method.
Despite the better performance in the reformed case, this setback emphasizes
the insufficiency of relying solely on band-pass filters.

An ablation study is conducted and shown in Tab. 3. Surprisingly, increasing
the kernel size (filter length) $L$ does not gain a more favorable outcome,
even though it leads to higher frequency resolution without growing the
amounts of parameters. As noted in [21], we believe there are certain
underlying principles in assigning the kernel size to each layer, but we leave
this to future efforts. On the contrary, increasing the number of cutoff
frequency pairs $N$ is beneficial. Finally, we examine the impact of BGs and
layer normalization. While the BGs contribute less, layer normalization is
highly advantageous for the provided framework for its significance in
achieving stable and faster convergence.

Table 3: The ablation study conducted on ”reformed Sinc-conv + MANNER +
T-Conv1d.”

Components | PESQ | STOI(%) | SI-SNR
---|---|---|---
$L$ | $N$ | $\beta_{i}$ | norm
251 | 80 | ✓ | ✓ | 3.03 | 95.0 | 18.91
1025 | 80 | ✓ | ✓ | 2.73 | 94.0 | 18.02
251 | 120 | ✓ | ✓ | 3.10 | 95.2 | 19.17
251 | 80 | ✓ |  | 2.74 | 94.1 | 18.16
251 | 80 |  | ✓ | 2.95 | 94.7 | 18.74

##  4 Discussion

The CFR curves derived from different encoder-decoder pairs, namely: 1)
”reformed Sinc-conv + T-Conv1d”, 2) ”reformed Sinc-conv + linear combination,”
and 3) ”original Sinc-conv + T-Conv1d”, along with their corresponding
initialization values, are depicted in Figure 1.

Fig. 1: CFR.

First, we can see that the original Sinc-conv does not significantly alter the
CFR relative to the initial one. Contrarily, the reformed Sinc-conv exhibits a
distinct variation in CFR as a result of its higher flexibility. Second, the
reformed Sinc-conv places greater emphasis on medium frequencies, while the
original Sinc-conv prioritizes low frequencies. Apparently, the CFR difference
is the primary factor for the disparity in performance. While the Mel scale
was initially designed for human auditory perception, it exhibits limitations
when encountering noises. However, we must clarify that a lower CFR region
does not necessarily imply that the corresponding frequencies are unimportant.
In fact, the CFR curve serves as an indicator of the SE model’s comprehensive
listening and analytical efforts.

To better understand the filter characteristics obtained by reformed Sinc-
conv, we plot the cutoff frequencies and BGs of each filter in Fig 2. First,
we find that the majority of the filters are band-pass, with some being high-
pass, and a few being low-pass. Second, we notice that there are few BGs equal
to zero, which operate similarly to a dropout layer and serve as a form of
network regularization. Third, the lower frequency curves display exponential
growth. We also observe the same growth when sorted by higher frequencies.
This resembles the Mel Scale and mimics the way humans perceive sound.

Fig. 2: The cutoff frequencies and BGs, sorted by the lower frequency.

##  5 Conclusion

In this study, we propose a reformed Sinc-conv method that facilitates
training efficiency and diversifies the filter types for the encoder of a deep
SE network. The efficacy of the reformed Sinc-conv is evaluated with multiple
SE models and configurations, clearly showcasing its potential to boost SE
performance. By leveraging Sinc-conv, we can interpret what an SE network
pursues to listen to when enhancing a noisy speech. To sum up, we have
successfully fulfilled our goal of developing automatic learning filterbanks
that exhibit high interpretability and effectiveness.

##  6 Acknowledgement

This work was supported in part by Realtek Semiconductor Corporation under
Grant Number 112KK010. Any findings and implications in the paper do not
necessarily reflect those of the sponsors.

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