# Chern insulator in a hyperbolic lattice

Zheng-Rong Liu  Chun-Bo Hua  Tan Peng  Bin Zhou<EMAIL_ADDRESS>Department of Physics, Hubei University, Wuhan 430062, China

###### Abstract

Motivated by the recent experimental realizations of hyperbolic lattices in
circuit quantum electrodynamics and the research interest in the non-Euclidean
generalization of topological phenomena, we investigate the Chern insulator
phases in a hyperbolic $\\{8,3\\}$ lattice, which is made from regular
octagons ($8$-gons) such that the coordination number of each lattice site is
$3$. Based on the conformal projection of the hyperbolic lattice into the
Euclidean plane, i.e., the Poincaré disk model, by calculating the Bott index
($B$) and the two-terminal conductance, we reveal two Chern insulator phases
(with $B=1$ and $B=-1$, respectively) accompanied with quantized conductance
plateaus in the hyperbolic $\\{8,3\\}$ lattice. The numerical calculation
results of the nonequilibrium local current distribution further confirm that
the quantized conductance plateau originates from the chiral edge states and
the two Chern insulator phases exhibit opposite chirality. Moreover, we
explore the effect of disorder on topological phases in the hyperbolic
lattice. It is demonstrated that the chiral edge states of Chern insulators
are robust against weak disorder in the hyperbolic lattice. More fascinating
is the discovery of disorder-induced topologically nontrivial phases
exhibiting chiral edge states in the hyperbolic lattice, realizing a non-
Euclidean analog of topological Anderson insulator. Our work provides a route
for the exploration of topologically nontrivial states in hyperbolic geometric
systems.

##  I Introduction

Recently, a great deal of research effort in condensed matter physics has been
made to find novel topological states. A typical example is Chern insulator
(CI) [1, 2, 3], which possesses fully gapped bulk states and gapless boundary
states and is characterized by a non-zero Chern number. The research branches
of CIs include quantum Hall effect (QHE) [4, 5, 6, 7, 8, 9, 10, 11] and
quantum anomalous Hall effect (QAHE) [12, 13, 14, 15, 16, 17, 18, 19, 20, 21,
22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48], which have been extensively investigated in
both theory and experiment. Unlike the QHE, the QAHE does not require a strong
external magnetic field, which means that it has a unique application
potential in electronic devices with low-power consumption [24, 25]. The first
prediction for the realization of the QAHE was proposed by Haldane in 1988
[12]. Although this proposal has been implemented in the ultracold atomic
system [49], it is still difficult to achieve in condensed matter systems. The
discovery of topological insulators (TIs) provides a good platform for
searching the QAHE [50, 51]. The QAHE is predicted to be produced by doping
magnetic impurities in HgTe/CdTe quantum wells or InAs/GaSb quantum wells [23,
26]. Moreover, there are a large number of proposals for realizing the QAHE in
various systems, such as single-layer and bilayer graphene [27, 28, 29, 30,
31, 32], silicene [17, 33], Bi (111) bilayers [34, 35], two-dimensional (2D)
organic materials [36, 37], transition metal (TM) halides [38, 39, 40, 41,
42], TM atom-doped tetradymite semiconductors [20], van der Waals layered
MnBi2Te4-family materials [43]. The QAHE was first observed experimentally in
the Cr-doped (Bi, Sb)2Te3 thin film [21]. Subsequently, there have been many
experiments to study the QAHE in Cr-doped (Bi, Sb)2Te3 [44, 45, 46] and
V-doped (Bi, Sb)2Te3 [47]. Recently, the QAHE has been observed in the layered
magnetic TI MnBi2Te4 [48] and a MnBi2Te4/Bi2Te3 superlattice [52].

Up to now, most of the research in CIs and other topological states are
associated with Euclidean geometry. Due to the intriguing properties of non-
Euclidean geometry, the non-Euclidean generalization of topological phenomena
is a fascinating issue. Recently, Yu $et~{}al$. theoretically explored the
topological quantum phenomenon in 2D hyperbolic lattices and demonstrated a
non-Euclidean analog of the quantum spin Hall effect [53]. Hyperbolic lattices
are an important platform for quantum simulation in a curved space with the
constant negative curvature, and also have exciting applications in the field
of fault-tolerant quantum computing [54, 55, 56, 57, 58, 59]. In contrast to
Euclidean lattices, hyperbolic lattices are embedded in the hyperboloid and
cannot be realized in Euclidean space without distortion. Notably, recent
experimental realization of hyperbolic lattices in circuit quantum
electrodynamics has been reported [60]. This experimental breakthrough
inspired several subsequent theoretical studies in hyperbolic lattices [53,
61, 54, 62, 63, 64, 65]. In our work, motivated by the recent experimental
realization of hyperbolic lattices and theoretical progress in topological
hyperbolic lattices, we aim to extend CI phases and consequent topological
phenomena to non-Euclidean hyperbolic geometry. Due to the absence of
commutative translation symmetries and Bravais vectors in hyperbolic lattices,
we will apply exact numerical diagonalization of the Hamiltonian to obtain the
energy spectrum and adopt the real-space Bott index to characterize the
topological phase in a hyperbolic lattice. Further, we numerically calculate
the two-terminal conductance and the nonequilibrium local current distribution
to analyze the topological phenomena of CI in the hyperbolic lattice.

On the other hand, the interplay between topology and disorder plays an
important role in the recent research of topological states with Euclidean
geometry. Li $et~{}al$. found that strong disorder can destroy the topological
insulate state, and a certain strength of disorder can induce the system to
transform from a normal insulator to topologically nontrivial insulator called
topological Anderson insulator (TAI) [66]. After Li $et~{}al$. first proposed
this concept [66], TAI and disorder effects have been investigated in various
systems [67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83,
84, 85, 86], including HgTe/CdTe quantum wells [67], Kane-Mele models [78],
electronic circuits [79], photonic crystals [80], quasicrystals [82, 84, 85].
In experiments, TAIs have been successively realized in a photonic platform
[87], an atomic wire [88], and electronic circuits [89]. In the recent
research of topological phases of matter in hyperbolic geometry, the disorder-
immune behaviors of the topologically protected edge states in topological
hyperbolic lattices have been demonstrated [53]. However, the investigation of
the disorder-induced topological phases in non-Euclidean hyperbolic lattices
has yet to be reported. An interesting question is whether the TAI can be
realized in hyperbolic lattices. Thus, in our work, we also explore the effect
of disorder on topological phases in the hyperbolic lattice. It is
demonstrated that the chiral edge states of CIs are robust against weak
disorder in the hyperbolic lattice. More striking, the disorder-induced
topologically nontrivial phases exhibiting chiral edge states are found at a
certain region of disorder strength in the hyperbolic lattice, which is a non-
Euclidean extension of TAI.

The rest of the paper is organized as follows. In Sec. II, we introduce a CI
model in the hyperbolic lattice. Then, in the clean limit, we investigate the
topological phase transition and transport properties of CI in the hyperbolic
lattice through numerical calculations in Sec. III (A). Next, we explore the
effect of disorder on CI in the hyperbolic lattice in Sec. III (B). Finally,
we summarize our conclusions in Sec. IV.

##  II Models

The Schläfli symbol $\\{p,q\\}$ is used to describe the essential
characteristics of regular polygons, where $p$ and $q$ represent the number of
vertices of the regular polygon and the number of polygons adjacent to each
vertex, respectively [90]. In Euclidean geometry, the Gaussian curvature of
the surface is equal to $0$, and $\\{p,q\\}$ satisfies the relation
$(p-2)\cdot(q-2)=4$. In the Euclidean plane, there are only three pairs
$\\{p=3,q=6\\}$, $\\{p=4,q=4\\}$ and $\\{p=6,q=3\\}$ meeting the conditions,
which means that only lattices with 3-fold, 4-fold, and 6-fold rotational
symmetry can be periodically tessellated in 2D Euclidean space [91]. However,
in hyperbolic geometry, the Gaussian curvature of the surface is less than
$0$, and $\\{p,q\\}$ satisfies the relation $(p-2)\cdot(q-2)>4$. It means that
the hyperbolic lattice holds infinite possibilities of regular tessellation in
the hyperbolic space. The Poincaré disk model, which projects the entire
hyperbolic plane into a unit circle in the Euclidean plane, provides a visual
approach to understand hyperbolic geometry [92, 93].

In this work, in order to investigate the topological properties of hyperbolic
geometric systems, we focus on the hyperbolic $\\{8,3\\}$ lattice for
concreteness, which corresponds to a tessellation of plane by regular octagons
($8$-gons) such that each lattice site has coordination number $3$. The
hyperbolic lattice can be constructed via a recursive process (see Appendix A)
[53]. In fact, our numerical calculations are on the finite hyperbolic
$\\{8,3\\}$ lattice with open boundary condition. The finite hyperbolic
$\\{8,3\\}$ lattice at the epoch $3$, represented by the Poincaré disk model,
is shown in Fig. 1, where the non-Euclidean metric is such that the distance
between any two neighboring sites is equal.

Figure 1: Schematic illustration of the hyperbolic $\\{8,3\\}$ lattice at the
epoch $3$. In the Poincaré disk, the vertices of these polygons are the
hyperbolic lattice sites, and the non-Euclidean metric is such that all
neighboring lattice sites have equal hyperbolic distance.

Here, we will apply a 2D QAH model (Qi-Wu-Zhang model) proposed by Qi $et$
$al$ [94] to the hyperbolic lattice. The Qi-Wu-Zhang model in the real space
can be described by the following tight-binding model Hamiltonian:

$\displaystyle H_{0}=$
$\displaystyle(M-4t_{B})\sum_{j}c_{j}^{\dagger}\sigma_{z}c_{j}+t_{B}\sum_{\left<j,k\right>}c_{j}^{\dagger}\sigma_{z}c_{k}$
$\displaystyle+\frac{it_{A}}{2}\sum_{\left<j,k\right>}c_{j}^{\dagger}\left[\cos(\theta_{jk})\sigma_{x}+\sin(\theta_{jk})\sigma_{y}\right]c_{k},$
(1)

where $c_{j}^{\dagger}$ and $c_{j}$ are the creation and annihilation
operators of electrons on site $j$, $\theta_{jk}$ represents the polar angle
of the vector from the site $k$ to the site $j$ in the Poincaré disk.
$\sigma_{x,y,z}$ are the Pauli matrices representing orbital. $t_{A}$,
$t_{B}$, and $M$ are material parameters. In the following calculations, we
set $t_{A}=1$ and only consider coupling between the nearest neighbor sites in
the Poincaré disk. Recently, several models have been shown to possess
translation symmetry in hyperbolic lattices [64, 95], such as the hyperbolic
Haldane model and hyperbolic Kane-Mele model. It should be noted that the
Hamiltonian $H_{0}$ in Eq. (1) does not possess translation symmetry in the
$\\{8,3\\}$ lattice in the presence of the polar angles $\theta_{jk}$, because
$\theta_{jk}$ are defined in the Poincaré disk in the Euclidean plane, not
projected from hyperbolic space.

##  III Numerical Results

###  III.1 Clean limit

First, in order to explore the electronic properties of the Qi-Wu-Zhang model
in the hyperbolic lattice, we numerically diagonalize the Hamiltonian $H_{0}$
under open boundary condition. Figures 2(a) and 2(b) show the energy spectrum
of the system and the probability distribution of the eigenstate near zero
energy, respectively. It can be found that the eigenstates near zero energy
($E_{n}=-0.0014$, $0.0014$) is localized at the edge of the hyperbolic lattice
as shown in Fig. 2(b). In fact, except that the eigenstates in the interval
$E\in[-0.37,0.37]$ near zero energy are localized at the edge, it is found
that the eigenstates in the intervals $E\in[0.67,1.27]$ and
$E\in[-1.27,-0.67]$ are also localized at the edge. Here, we mainly focus on
the eigenstates near zero energy, and the probability distribution of
eigenstates in other energy intervals will be discussed in detail in Appendix
C. To analyze the topological property of the gapless edge states shown in
Fig. 2(b), we need to introduce a topological invariant to characterize the
topological phase in the hyperbolic lattice described by the Hamiltonian
$H_{0}$. It is well known that the Chern number is a useful topological
invariant to characterize the CI topological phase. However, due to the
absence of commutative translation symmetries and Bravais vectors, we cannot
use the momentum-space Chern number to characterize the topological phase in
hyperbolic lattices, and thus here we employ the real-space Bott index to
characterize the topological phases of hyperbolic lattices.

Figure 2: (a) Energy spectrum of the Hamiltonian $H_{0}$ on the hyperbolic
$\\{8,3\\}$ lattice with open boundary condition. The three energy intervals
marked by orange dots are $[-1.27,-0.67]$, $[-0.37,0.37]$, and $[0.67,1.27]$,
respectively. The eigenstates of these energies are localized at the edge. In
the main text, we mainly focus on the eigenstates near zero energy. (b) The
probability distribution of the eigenstates near zero energy ($E_{n}=-0.0014$,
$0.0014$) marked by the orange arrow in (a). We take the model parameters
$t_{B}=0.5$, $M=1$ and the hyperbolic lattice site number $N=2888$.

The Bott index is equivalent to the Chern number [96] and can be used to
verify the topologically nontrivial phase of real-space systems with or
without translation symmetries [97, 98, 99, 100, 101, 102, 103, 96, 104, 105,
106, 107]. To calculate the Bott index, firstly one constructs the projector
operator of the occupied states as

$\displaystyle
P=\sum_{n}^{N_{occ}}\left|\psi_{n}\right>\left<\psi_{n}\right|,$ (2)

where $\left|\psi_{n}\right>$ is the $n$th eigenstate of the Hamiltonian with
eigenvalue $E_{n}$ and $N_{occ}$ is the number of the occupied energy states.
Here we set the eigenstate for the negative eigenvalue as the occupied energy
state. Then, one defines the projected position operators as

$\displaystyle U=Pe^{i2\pi X}P,$ (3) $\displaystyle V=Pe^{i2\pi Y}P,$ (4)

where $X$ and $Y$ are diagonal matrices, and their diagonal elements
$(x_{i},y_{i})$ are the rescaled coordinates of the $i$-th site in the
Poincaré disk in the Euclidean plane, bounded in the interval
$\left[0,1\right)$. Finally, the Bott index is given by

$\displaystyle B=\frac{1}{2\pi}{\rm Im}\\{{\rm tr}[{\rm
log}(VUV^{\dagger}U^{\dagger})]\\}.$ (5)

A topologically nontrivial system is characterized by the Bott index $B=1$ (or
$B=-1$), while the Bott index of a topological trivial system is equal to $0$.
By calculating the Bott index, we confirm that the system with gapless edge
states in Fig. 2(b) corresponds to a topologically nontrivial phase (a CI
phase with $B=1$).

Figure 3: (a) Energy gap as a function of the parameter $M$. (b) Bott index as
a function of $M$ under open boundary condition. (c) Phase diagram showing the
conductance as a function of the model parameters $M$ and $t_{B}$. (d)
Conductance as a function of $M$. We take the model parameters $t_{B}=0.5$ in
(a), (b) and (d). Here, the hyperbolic lattice site number is $N=2888$.

In the Euclidean plane, it has been proved that the parameter $M$ can be used
to adjust the topological phase of the system [94]. It is necessary to verify
the adjustment effect of the parameter $M$ on the topological phase transition
of hyperbolic lattices. Therefore, we calculate the energy gap of a finite
hyperbolic $\\{8,3\\}$ lattice as a function of the parameter $M$ shown in
Fig. 3(a). It is found that a gapless phase can be obtained by adjusting the
parameter $M$. To explore topological phase transition of the system when the
parameter $M$ is adjusted, we calculate the Bott index of the hyperbolic
lattice. In Fig. 3(b), we present the Bott index with respect to $M$, and it
is shown that the Bott index is $B=1$ ($B=-1$) in the parameter regimes where
the gapless phases emerge. It means that the gapless edge state corresponds to
a CI phase characterized by the nontrivial Bott index. Thus, we verify that
the hyperbolic lattice described by the Hamiltonian $H_{0}$ can be transformed
from a normal insulator (NI) to a CI, and the reverse is also true, by
adjusting the parameter $M$. In fact, in addition to the Bott index, there are
other topological invariants that can also characterize the topological
properties of real-space systems, such as the real-space Chern number [108,
109, 95, 110, 111]. In Appendix B, we further verify the above results by
calculating the real-space Chern number.

In a 2D system, transport is a convenient way to experimentally verify the
topologically nontrivial property of the system. To further confirm the
topological states, we study the transport properties of the system. We design
a two-terminal setup to calculate the conductance of the hyperbolic lattice.
The setup consists of three parts, the central area is the hyperbolic lattice
as the device, and the left and right ends of the device are connected with
normal metal leads. Then we investigate the transport properties of the
hyperbolic lattice based on the Landauer-B$\ddot{\text{u}}$ttiker-Fisher-Lee
formula [112, 113, 114] and the recursive Green’s function method [115, 116,
10, 11, 117]. The conductance can be obtained by

$\displaystyle G=\frac{e^{2}}{h}T,$ (6)

where $T={\rm tr}[\Gamma_{L}G^{r}\Gamma_{R}G^{a}]$ is the transmission
coefficient. The linewidth function
$\Gamma_{l}=i[\Sigma_{l}^{r}-\Sigma_{l}^{a}]$ with $l=L,R$, and the retarded
(advanced) Green’s functions $G^{r(a)}$ of the device are calculated from
$G^{r}=[G^{a}]^{\dagger}=\left[\mu
I-H_{D}-\Sigma_{L}^{r}-\Sigma_{R}^{r}\right]^{-1}$, where $\mu$ is the
chemical potential, $H_{D}$ is the device Hamiltonian, and
$\Sigma_{L,R}^{r(a)}$ are the retarded (advanced) self-energies of the leads.
We set the chemical potential of the device $\mu=0$ in the following
calculations.

The conductance as a function of the parameter $M$ is plotted in Fig. 3(d).
Comparing Figs. 3(b) and 3(d), it is found that the conductance of the region
where the Bott index is equal to $1$ ($-1$) exhibits a quantized plateau with
$G=e^{2}/h$. A slight difference of the phase boundaries between Figs. 3(b)
and 3(d) is owing to the finite size effect. Furthermore, the phase diagram of
conductance as a function of $M$ and $t_{B}$ is presented in Fig. 3(c), where
the white horizontal dashed line is placed at $t_{B}=0.5$. The sky blue areas
represent the topologically nontrivial phase with conductance $G=e^{2}/h$.

It is worth mentioning that two topologically nontrivial regions are found in
Fig. 3(b). The calculation result of the Bott index shows that these two
regions represent two topological phases with opposite chirality. In order to
further understand the transport properties of edge states in these two
topologically nontrivial regions, we calculate the nonequilibrium local
current distribution of the hyperbolic lattice. A small external bias
$V=V_{L}-V_{R}$ is applied between the two terminal, the local current from
site i to site j can be calculated based on the formula [118, 119, 67, 82, 84,
120]:

$\displaystyle J_{\rm\textbf{i}\to\rm\textbf{j}}=\frac{2e^{2}}{h}{\rm
Im}\left[\sum_{\alpha\beta}H_{\rm\textbf{i}\alpha,\rm\textbf{j}\beta}G^{n}_{\rm\textbf{j}\beta,\rm\textbf{i}\alpha}\right](V_{L}-V_{R}),$
(7)

where $H_{\rm\textbf{i}\alpha,\rm\textbf{j}\beta}$ is the matrix element of
Hamiltonian $H_{D}$ of the device, and $\alpha$, $\beta$ represents orbital.
$G^{n}=G^{r}\Gamma_{L}G^{a}$ is the electron correlation function.

Figures 4(a) and 4(b) show the local current distributions of the hyperbolic
lattice with the parameters $M=1$ and $M=3$, respectively. In Fig. 4, the
middle disk is the hyperbolic lattice as the device, the left (L) and right
(R) gray shaded parts denote the metal leads, and the dark red arrows in the
shaded part indicate the direction of current input and output. The red arrow
in the device represents the local current, and the size of the arrow means
the strength of the local current. Obviously, the transport directions of
electrons on the boundary of the two topologically nontrivial regions are
exactly opposite, i.e., clockwise (CW) and anticlockwise (ACW). The directions
of the arrows in the two light blue discs shown in Fig. 4 imply that the edge
states in these two CI phases possess opposite chirality. Thus, the result of
the local current distributions is consistent with that of the Bott index.

Figure 4: Nonequilibrium local current distribution of the CI phase
(corresponding to $B=1$) and the CI phase (corresponding to $B=-1$) with $M=1$
for (a) and $M=3$ for (b). The gray shaded areas represent the leads, and the
dark red arrow in the area indicates the direction of current injection and
outflow. The middle disk is the hyperbolic lattice as the device, and the
direction of the red arrow represents the direction of the local current, the
arrow size means the local current strength. We take the model parameters
$t_{B}=0.5$ and the hyperbolic lattice site number $N=2888$.

###  III.2 Disorder Effect

Now, we focus on the effects of disorder on topological phases of the
hyperbolic lattice system. We introduce on-site disorder to the system and
corresponding Hamiltonian with the disorder is given by

$\displaystyle H_{\rm
dis}=H_{0}+W\sum_{j}c_{j}^{\dagger}\omega_{j}\sigma_{0}c_{j},$ (8)

where $W$ represents the disorder strength, and $\omega_{j}$ is a uniformly
distributed variable within $[-0.5,0.5]$. First of all, we identify the
robustness of topologically nontrivial states against disorder. Therefore, we
take two sets of parameters corresponding to the topologically nontrivial
phases in the clean limit. For the case of $M=1$, in the clean limit, the
system is a CI with the edge states transported in the CW direction. As shown
in Fig. 5(a), the topologically nontrivial phase with the Bott index $B=1$
remains stable within a certain range of disorder strength ($0\leq W\leq
2.8$). Beyond this range, the Bott index continues to decay as the disorder
strength increases until the system transforms into a topologically trivial
phase with $B=0$. Meanwhile, the conductance shown in Fig. 5(c) remains a
quantized conductance plateau under weak disorder. However, the strong
disorder forces the quantized conductance to decrease until it converts to a
localized state with conductance $G=0$. For the case of $M=3$, the chirality
of the edge states of the system in the clean limit is opposite to that of
$M=1$. Nevertheless, with increasing the disorder strength, the system with
parameter $M=3$ shows a similar behavior, that is, both the Bott index and the
conductance keep the quantized value until the disorder strength $W$ exceeds
$2.8$ as shown in Figs. 5(b) and 5(d).

Figure 5: Bott index $B$ as a function of disorder strength $W$ with $M=1$ for
(a) and $M=3$ for (b). (c) and (d) show the conductance $G$ as a function of
disorder strength $W$ when $M=1$ and $M=3$, respectively. The error bar
represents the standard deviation of 100 samples. We take the model parameters
$t_{B}=0.5$ and the hyperbolic lattice site number $N=2888$.

Next, the Bott index and the conductance with the model parameter $M=3.3$ as a
function of disorder strength are plotted in Figs. 6(a) and 6(b),
respectively. In the clean limit [see Fig. 3(b)], it is found that the system
with $M=3.3$ is a NI phase characterized by the Bott index $B=0$. When the
disorder is introduced, with the disorder strength increasing, it is observed
from Fig. 6(a) that the transition of the Bott index $B$ from $0$ to $-1$
occurs, and then the Bott index maintains at this value of $B=-1$ for a
certain range before eventually goes back to the value of $B=0$. In Fig. 6(b),
it is found that with the disorder strength increasing, the conductance
reaches a quantized value of $G=e^{2}/h$ and maintains this quantized plateau
for a certain range of the disorder strength corresponding to the Bott index
$B=-1$ shown in Fig. 6(a). For the stronger disorder strength, the conductance
gradually decreases to zero. The results shown in Figs. 6(a) and 6(b) indicate
that the disorder-induced topologically nontrivial phases can also occur in
the hyperbolic lattice with disorder. Figure 6(c) shows the energy spectrum of
the system with $M=3.3$ and the disorder strength $W=2$. According to Fig.
6(a), the eigenstate near zero energy marked by the orange arrow in Fig. 6(c)
corresponds to the disorder-induced CI phase with $B=-1$. The probability
distribution of the eigenstate further confirms the disorder-induced CI phase
supports localized edge states as shown in Fig. 6(d).

Figure 6: (a) Bott index $B$ and (b) the conductance $G$ as a function of
disorder strength $W$ in the system with $M=3.3$. (c) Energy spectrum of a
single sample with $M=3.3$ and the disorder strength $W=2$. (d) The
probability distribution of the eigenstate near zero energy marked by the
orange arrow in (c). The error bar represents the standard deviation of 100
samples. We take the model parameters $t_{B}=0.5$ and the hyperbolic lattice
site number $N=2888$.

##  IV Conclusion and discussion

In this work, we introduce the CI model into a hyperbolic $\\{8,3\\}$ lattice.
Based on the numerical calculations of the Bott index and the two-terminal
conductance, two CI phases characterize by $B=\pm 1$ accompanied with
quantized conductance plateaus are found in the hyperbolic lattice. We also
plot the nonequilibrium local current distribution, which further confirms
that the quantized conductance plateau originates from the chiral edge states
and the two CI phases exhibit opposite chirality. Moreover, the effect of
disorder on topological phases in the hyperbolic lattice is also analyzed. The
robustness against the weak disorder of the chiral edge states of CIs in the
hyperbolic lattice is confirmed. More fascinating, we observe a disorder-
induced topological transition from a NI phase to a CI phase at a finite
disorder strength. Our results present a non-Euclidean analog of TAI.

It is noted that the absence of commutative discrete translation groups limits
most of theoretical research methods of hyperbolic lattices to real-space
numerical diagonalization [60, 53, 61, 54]. In this work, all our numerical
calculations are based on open boundary conditions. Recently, Maciejko and
Rayan proposed a hyperbolic band theory based on the idea of algebraic
geometry [62]. Subsequently, hyperbolic lattice band theory under a magnetic
field and the crystallography of hyperbolic lattice have further been
developed [63, 64, 121]. Thus, research of topological states of matter in
hyperbolic lattices based on periodic boundary conditions and hyperbolic
topological band theory will be worth investigating in future works.

## Acknowledgments

B.Z. was supported by the NSFC (under Grant No. 12074107) and the program of
outstanding young and middle-aged scientific and technological innovation team
of colleges and universities in Hubei Province (under Grant No. T2020001).

##  Appendix A Hyperbolic lattice construction

Figure 7: (a) The inversion circle c has radius $r$ and center $O$, the
initial point is $A$, and the inverted point is $A^{\prime}$. (b) Starting
points $A$ and $B$, inversion points $A^{\prime}$ and $B^{\prime}$ lie on a
new circle, and the new circle intersects the inversion circle at points $P$
and $Q$.

In Appendix A, we present the details of hyperbolic lattice construction.
Before we start constructing a hyperbolic lattice, let’s understand the
concept of circular inversion [122]. As shown in Fig. 7(a), if the inversion
circle c has radius $r$ and center $O$, then the inverted point $A^{\prime}$
of a point $A$ lies on a ray from $O$ to $A$, and

$\displaystyle OA\cdot OA^{\prime}=r^{2}.$ (9)

Figure 8: (a) Points $A$, $B$, $C$, $D$ and $E$ serve as the initial points of
the hyperbolic lattice. (b) Points $A^{\prime}$ and $B^{\prime}$ are the
inverted points of points $A$ and $B$ within circle $O$, which lie on the new
circle $M$. The initial points $A$, $B$, $C$, $D$ and $E$ perform circle
inversion for circle $M$, and the inversion points $A$, $B$, $C^{\prime}$,
$D^{\prime}$ and $E^{\prime}$ can be obtained in (c).

Unlike in the Euclidean plane, in the Poincaré disk the shortest distance
between two points is not a straight line [92, 93, 123, 124]. Taking Fig. 7(b)
as an example, we use the following formula to calculate the distance between
two points in the Poincaré disk [124],

$\displaystyle d(A,B)=|\rm{ln}(A,B;P,Q)|=\left|\rm{ln}(\frac{AP\cdot
BQ}{BP\cdot AQ})\right|.$ (10)

Figure 9: (a), (b) and (c) represent the site distribution of
$\\{p=5,~{}q=4\\}$-type hyperbolic lattice at epochs 1, 2 and 3, respectively.
(d), (e) and (f) represent the site distribution of $\\{p=8,~{}q=3\\}$-type
hyperbolic lattice at epochs 1, 2 and 3, respectively.

Then, we describe the method of constructing a hyperbolic lattice by taking a
hyperbolic lattice of $\\{p=5,~{}q=4\\}$-type as an example. As shown in Fig.
8(a), if the center of the Poincaré disk is $O$, these $p$ points maintain the
same distance $d$ from point $O$. This distance $d$ is

$\displaystyle
d=\sqrt{\frac{\rm{tan}(\frac{\pi}{2}-\frac{\pi}{q})-\rm{tan}(\frac{\pi}{p})}{\rm{tan}(\frac{\pi}{2}-\frac{\pi}{q})+\rm{tan}(\frac{\pi}{p})}},$
(11)

where $p$ and $q$ represent the number of vertices of the regular polygon and
the number of polygons adjacent to each vertex, respectively. And we connect
these $p$ points to the center of the circle $O$ with lines, and the angle
between the lines corresponding to any two adjacent points is $\theta=2\pi/p$.
Next, we perform circular inversion of circle $O$ for point $A$ and point $B$
respectively. We can confirm that the points $A^{\prime}$ and $B^{\prime}$
obtained by the circle inversion and the points $A$ and $B$ inside Circle $O$
form a new circle $M$, as shown in Fig. 8(b). The radius of Circle $M$ is $R$.
In Fig. 8(c), we do a circular inversion of circle $M$ by Point $A$, $B$, $C$,
$D$, and $E$. The new polygon consisting of points $A$, $B$, $C^{\prime}$,
$D^{\prime}$, and $E^{\prime}$ is exactly the same as the regular polygon
consisting of points $A$, $B$, $C$, $D$, and $E$ in the Poincaré disk. When we
keep repeating the steps of Figs. 8(b) and 8(c), we can get hyperbolic
lattices with different epochs. In Fig. 9, we show the site distribution for
different epochs of two types of hyperbolic lattices $\\{p=5,~{}q=4\\}$ and
$\\{p=8,~{}q=3\\}$.

##  Appendix B Real-space Chern number

Figure 10: (a) The three regions $R_{1}$, $R_{2}$, and $R_{3}$ in the bulk of
the system are arranged counterclockwise. (b) The real-space Chern number $C$
as a function of $M$. (c) The real-space Chern number $C$ as a function of
disorder strength $W$ when $M=1$. The sample size used in (b) and (c) is
$N=2888$. (d) The real-space Chern number $C$ as a function of disorder
strength $W$ when $M=3.3$. The inset shows the real-space Chern number $C$ as
a function of disorder strength $W$ for different sample sizes, with blue,
red, and dark green points representing sample sizes $N=768$, $N=2888$, and
$N=10800$, respectively. The error bar represents the standard deviation of
100 samples. We take the model parameter $t_{B}=0.5$ for (b), (c), and (d).

In Appendix B, we calculate another real-space topological invariant
equivalent to the Bott index, the real-space Chern number. When the Fermi
energy is $\epsilon$, the real-space Chern number is given by [108, 109, 95,
110, 111],

$\displaystyle C(\epsilon)=12\pi i\sum_{j\in R_{1}}\sum_{k\in R_{2}}\sum_{l\in
R_{3}}\left(P_{jk}^{\epsilon}P_{kl}^{\epsilon}P_{lj}^{\epsilon}-P_{jl}^{\epsilon}P_{lk}^{\epsilon}P_{kj}^{\epsilon}\right),$
(12)

where $j$, $k$, and $l$ are site indices in three regions $R_{1}$, $R_{2}$,
and $R_{3}$. These three regions only contain the bulk of the system and do
not extend all the way to the boundary, and which are arranged counter-
clockwise, as shown in Fig. 10(a). $P^{\epsilon}$ is the projector operator of
the occupied states taking the Fermi energy as $\epsilon$,
$P^{\epsilon}=\sum_{E_{n}\leq\epsilon}\left|\psi_{n}\right>\left<\psi_{n}\right|$.
$\psi_{n}$ is the eigenstate with the eigenvalue $E_{n}$. We set $\epsilon=0$
by default, unless the value of $\epsilon$ is specified. In comparison with
Fig. 3(b), we plot the real-space Chern number $C$ as a function of the
parameter $M$ in Fig. 10(b). It can be found that in the topologically
nontrivial region of Bott index $B=1$ ($B=-1$), the real-space Chern number
also exhibits a quantization platform of $C=1$ ($C=-1$). The differences near
the phase boundaries are due to the finite size effect. In addition to clean
systems, real-space Chern numbers can also characterize the topological
properties of systems in disordered systems. Similar to Fig. 5(a) and Fig.
6(a), we plot the real-space Chern number $C$ as a function of disorder
strength $W$ in Figs. 10(c) and 10(d). It can be found that, like the Bott
index, the real-space Chern number can also characterize the robustness of
topologically nontrivial phase to weak disorder and the transition of system
induced by disorder from a trivial phase to a nontrivial phase. Due to the
finite size effect, the disorder-induced platform of $C=-1$ in Fig. 10(d) is
not quite perfect. To this end, we plot the variation of the platform (between
the dashed black lines) at different sizes in the inset of Fig. 10(d). It can
be found that as the size increases, the platform gradually approaches $C=-1$
and becomes more flat.

##  Appendix C Energy spectrum and wave function

Figure 11: (a) Energy spectrum of the Hamiltonian $H_{0}$ on the hyperbolic
$\\{8,3\\}$ lattice with open boundary condition. The three energy intervals
marked by orange dots are $[-1.27,-0.67]$, $[-0.37,0.37]$, and $[0.67,1.27]$,
respectively. The eigenstates of these energies are localized at the edge. (b)
The real-space Chern number $C$ as a function of Fermi energy $\epsilon$. (c)
The probability distribution of the eigenstate with the energy $E=0.5227$
marked by the green arrow in (a). (d) The probability distribution of the
eigenstate with the energy $E=1.001$ marked by the red arrow in (a). Here, we
take the model parameters $t_{B}=0.5$, $M=1$, and $N=2888$.

In Appendix C, we show the energy spectrum for different parameters, as well
as the wave function distribution with different energies. In the clean limit,
we show the energy spectrum for the three sets of parameters and the wave
function distribution in different energy intervals. In the main text, we
mentioned that when the parameters are taken as $t_{B}=0.5$ and $M=1$, in
addition to the eigenstates near zero energy being localized at the edge,
there are also some eigenstates with higher energies that are also localized
at the edge. In Fig. 11(a), we plot the energy spectrum when the parameters
are taken as $t_{B}=0.5$ and $M=1$. We mark the energies of the eigenstates
localized at the edge with orange dots, including three intervals,
$[-1.27,-0.67]$, $[-0.37,0.37]$, and $[0.67,1.27]$. Furthermore, we plot the
real-space Chern number $C$ as a function of the Fermi energy $\epsilon$ in
Fig. 11(b). It is found that in addition to the interval $[-0.37,0.37]$, when
the Fermi energy is placed in the interval $[-1.27,-0.67]$ and the interval
$[0.67,1.27]$, the real space Chern number is also equal to $1$, as shown in
Fig. 11(b). In the main text, we plot the probability distribution of near
zero energy edge states in Fig. 2(b). Here, in Fig. 11(c) we plot the
probability distribution of the eigenstate of the energy $E=0.5227$ marked
with the green arrow in Fig. 11(a), the real-space Chern number is not an
integer when the Fermi energy is placed at this energy. Moreover, in Fig.
11(d) we plot the probability distribution of the eigenstate of the energy
$E=1.001$ marked with the red arrow in Fig. 11(a), the real-space Chern number
$C=1$ when the Fermi energy is placed at this energy.

Figure 12: (a) The energy spectrum when we take parameter $M=0.5$. (b) The
probability distribution of the eigenstate with the energy $E=-0.1332$ that is
marked by the red arrow in (a). (c) The energy spectrum when we take parameter
$M=2$. (d) The probability distribution of the eigenstate with the energy
$E=-0.0151$ that is marked by the green arrow in (c). Here, we take the model
parameters $t_{B}=0.5$ and $N=2888$.

In addition to the energy spectrum in the topologically nontrivial phase, we
also analyze the energy spectrum in the NI phase and the wave function
distribution of the partial energy. In Fig. 12(a) we plot the energy spectrum
of the NI phase with a large energy gap, and we plot the probability
distribution of the eigenstate with the energy $E=-0.1332$ in Fig. 12(b).
Moreover, we also plot the energy spectrum of the NI phase between two Chern
insulator phases with different chirality in Fig. 12(c), and we plot the
probability distribution of the eigenstate with the energy $E=-0.0151$ in Fig.
12(d). As shown in Figs. 12(a) and 12(c), there is an energy gap in the energy
spectrum of both NI phases, while the energy gap of NI between the two CI
phases is smaller, which is consistent with Fig. 3(a) in the main text. In
Figs. 12(b) and 12(d), it can be found that the eigenstates are non-locally
distributed in the bulk of the system.

Figure 13: (a) The energy spectrum when we take parameter $M=3.3$. (b) The
probability distribution of the eigenstate with the energy $E=-0.0163$ that is
marked by the red arrow in (a). (c) The energy spectrum when we take
parameters $M=3.3$ and $W=2$. (d) The probability distribution of the
eigenstate with the energy $E=1.9912$ that is marked by the dark green arrow
in (c). (e) The energy spectrum when we take parameters $M=3.3$ and $W=10$.
(f) The probability distribution of the eigenstate with the energy $E=-0.0467$
that is marked by the purple arrow in (e). Here, we take the model parameters
$t_{B}=0.5$ and $N=2888$.

Next, we show the energy spectrum and wave function distribution in the
topologically trivial phase at the clean limit, however when we add disorder
of appropriate strength, the system exhibits nontrivial properties. In clean
limit, when we take the parameters $t_{B}=0.5$ and $M=3.3$, the system is in
the trivial phase. As shown in Fig. 13(a), there is a small energy gap in the
energy spectrum. In Fig. 13(b), we select an energy that is closest to zero,
and plot the probability distribution of the eigenstate of this energy. It is
found that it is delocalized and distributed in the bulk of the system. We
already know from the main text that disorder with suitable strength can
induce a topologically nontrivial phase in the system. In Fig. 6(d), we plot
the probability distribution of disorder-induced gapless edge states. Here, we
plot the energy spectrum for parameters $t_{B}=0.5$, $M=3.3$, $W=2$ and the
probability distribution of eigenstate with higher energy ($E=1.9912$) in
Figs. 13(c) and 13(d), respectively. It can be found that the eigenstate of
this energy is localized, but unlike the nontrivial edge states, it is only
localized in part of the system, which is the Anderson localization [125, 126,
66, 127, 128]. When the disorder strength increases to $W=10$, the system is
in the trivial phase. We plot the energy spectrum with parameters $t_{B}=0.5$,
$M=3.3$, $W=10$ in Fig. 13(e). Although the energy spectrum is also gapless at
this time, it is found that the energy state exhibits Anderson localization as
shown in Fig. 13(f). The system does not possess quantized conductance at this
time, and such a localized state is not a topologically nontrivial edge state.

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#
Implementation of Multi-channel Active Noise Control based on Back-propagation
Mechanism

Zhengding LUO  School of Electrical & Electronic Engineering, Nanyang
Technological University, Singapore  Dongyuan SHI  School of Electrical &
Electronic Engineering, Nanyang Technological University, Singapore  Junwei JI
School of Electrical & Electronic Engineering, Nanyang Technological
University, Singapore  Woon-Seng GAN  School of Electrical & Electronic
Engineering, Nanyang Technological University, Singapore

### ABSTRACT

Active noise control (ANC) systems can efficiently attenuate low-frequency
noises by introducing anti-noises to combine with the unwanted noises. In ANC
systems, the filtered-x least mean square (FxLMS) and filtered-X normalized
least-mean-square (FxNLMS) algorithm are well-known algorithms for adaptively
adjusting control filters. Multi-channel ANC systems are typically required to
attenuate unwanted noises in a large space. However, open-source
implementations of the multi-channel FxLMS (McFxLMS) and multi-channel FxNLMS
(McFxNLMS) algorithm continue to be scarce. Therefore, this paper proposes a
simple and effective implementation approach of the McFxLMS and McFxNLMS
algorithm. Motivated by the back-propagation process during neural network
training, the McFxLMS and McFxNLMS algorithm can be implemented via automatic
derivation mechanism. We implemented the two algorithms using the automatic
derivation mechanism in PyTorch and made the source code available on GitHub.
This implementation method can improve the practicality of multi-channel ANC
systems, which is expected to be widely used in ANC applications.

Keywords: Active noise control, Multi-channel ANC, Implementation of McFxLMS
and McFxNLMS

## 1 INTRODUCTION

Passive noise control techniques such as enclosures, barriers, and silencers
are difficult to suppress undesired noises at low frequencies (1). An active
noise control (ANC) system is achieved by introducing a canceling “anti-noise”
wave through an appropriate array of secondary sources (2). The anti-noise is
of equal amplitude and opposite phase compared to the disturbance (3). In
practice, ANC systems can efficiently attenuate low-frequency noises, whereas
passive methods are either ineffective or tend to be very expensive or bulky
(4). Therefore, ANC techniques have been widely employed in some commercial
products including headphones, mobile phones and automobiles etc.

However, there is a performance degradation of single-channel ANC systems in
spatial noise environments (5). In such scenarios, multi-channel ANC systems
with multiple controllers, loudspeakers and microphones can be introduced to
improve the noise reduction performance (6). Some of the best-known
applications are the control of exhaust boom noises in automobiles (7), earth-
moving machines (8), and the control of propeller-induced noises in flight
cabin interiors (9). Other noise control applications like vibration control
in complex mechanical structures also require multiple channels (10).

A general multiple-channel ANC system is shown in Figure 1, which consists of
$J$ reference microphones, $K$ canceling loudspeakers, and $M$ error
microphones. In the system, the active noise controller is updated based on
the reference signals and error signals, sensed by the reference microphones
and error microphones respectively, so that the generated anti-noise can
suppress the disturbance at the error microphones. The coefficients of ANC
controllers can be updated through adaptive algorithms to minimize the sum of
the error signals (11).

Adaptive algorithms such as the filtered-x least mean square (FxLMS) (12) and
filtered-X normalized least-mean-square (FxNLMS) (13) are commonly used ANC
algorithms. Nevertheless, the adaptive optimization of control filter
coefficients in multi-channel ANC systems is difficult to implement in
practice (14). It must take into account the interactions between all
microphones and loudspeakers to provide the optimal cancelling signals (15).
Compared with single-channel ANC systems, making iterative changes to the
control filter coefficients becomes much more complex in multi-channel ANC
systems.

Due to the difficulty of implementation, open-source codes of the multi-
channel FxLMS (McFxLMS) and multi-channel FxNLMS (McFxNLMS) algorithm are
still lacking (16). To tackle the limitation, the McFxLMS and McFxNLMS
algorithm are implemented via the automatic derivation mechanism in this
paper. Since the automatic derivation can update weights given a loss
function, the back propagation of McFxLMS and McFxNLMS can be automatically
achieved without resorting to extra-human efforts. The implementation code is
based on the PyTorch tool (17) and available at GitHub website. Owing to the
simple implementation method, changing the number of microphones and
loudspeakers in multiple-channel ANC systems is more convenient, so that the
practicality is improved.

## 2 FORWARD PROPAGATION AND BACK PROPAGATION

Figure 1: Block diagram of a $J\times K\times M$ multi-channel ANC system
using the McFxLMS algorithm.

A feedforward multiple-channel ANC system using the McFxLMS algorithm is
illustrated in Figure 1. The McFxLMS algorithm is used to update the active
noise controller. The $J\times K\times M$ multiple-channel ANC system includes
$J$ reference microphones, $K$ canceling loudspeakers, and $M$ error
microphones. For the purpose of easily implementing adaptive algorithms in
PyTorch, we split the adaptive process into two phases: forward propagation
and back propagation. Then, the McFxLMS algorithm is described in detail as an
example.

### 2.1 Forward propagation

Figure 2: The forward propagation of the McFxLMS algorithm.

In forward propagation, as shown in Figure 2, the input component of the
McFxLMS algorithm at its $n$-th iteration is the reference vector
$\mathbf{X}(n)$ given by

$\mathbf{X}(n)=\begin{bmatrix}\mathbf{x}^{\mathrm{T}}_{1}(n)&\cdots&\mathbf{x^{\mathrm{T}}}_{j}(n)&\cdots&\mathbf{x^{\mathrm{T}}}_{J}(n)\end{bmatrix}^{\mathrm{T}}\in\mathbb{R}^{JN\times
1},$ (1)

where $\mathrm{T}$ denotes the transpose operation. The control filter length
is $\mathrm{N}$ taps. The reference signal vector picked up by the $j$-th
microphone can be expressed as

$\mathbf{x}_{j}(n)=\begin{bmatrix}x_{j}(n)&x_{j}(n-1)&\cdots&x_{j}(n-N+1)\end{bmatrix}^{\mathrm{T}}\in\mathbb{R}^{N\times
1}.$ (2)

The control filter matrix is represented as

$\mathbf{W}(n)=\begin{bmatrix}\mathbf{w}^{\mathrm{T}}_{11}(n)&\cdots&\mathbf{w}^{\mathrm{T}}_{1J}(n)\\\
\mathbf{w}^{\mathrm{T}}_{21}(n)&\cdots&\mathbf{w}^{\mathrm{T}}_{2J}(n)\\\
\vdots&\ddots&\vdots\\\
\mathbf{w}^{\mathrm{T}}_{K1}&\cdots&\mathbf{w}^{\mathrm{T}}_{KJ}(n)\end{bmatrix}\in\mathbb{R}^{K\times
JN},$ (3)

The element $\mathbf{w}_{kj}(n)$ in control filter matrix means the control
filter from the $j$th reference to the $k$th output and can be denoted as

$\mathbf{w}_{kj}(n)=\begin{bmatrix}w^{(1)}_{kj}(n)&w^{(2)}_{kj}(n)&\cdots&w^{(N)}_{kj}(n)\end{bmatrix}^{\mathrm{T}}\in\mathbb{R}^{N\times
1}.$ (4)

Hence, the control signal $\mathbf{y}(n)$ can be obtained from

$\mathbf{y}(n)=\mathbf{W}(n)\mathbf{X}(n)\in\mathbb{R}^{K\times 1},$ (5)

and it can also be expressed as

$\mathbf{y}(n)=\begin{bmatrix}y_{1}(n)&\cdots&y_{k}(n)&\cdots&y_{K}(n)\end{bmatrix}\in\mathbb{R}^{K\times
1}.$ (6)

As Figure 2 illustrates, the overall control signal vector is formed by
stacking the above control signal from $n-L+1$ iteration to $n$ iteration as

$\mathbf{y}_{k}(n)=\begin{bmatrix}y_{k}(n)&y_{k}(n-1)&\cdots&y_{k}(n-L+1)\end{bmatrix}^{\mathrm{T}}\in\mathbb{R}^{L\times
1},$ (7)
$\mathbf{Y}(n)=\begin{bmatrix}\mathbf{y}^{\mathrm{T}}_{1}(n)&\cdots&\mathbf{y^{\mathrm{T}}}_{k}(n)&\cdots&\mathbf{y^{\mathrm{T}}}_{K}(n)\end{bmatrix}^{\mathrm{T}}\in\mathbb{R}^{KL\times
1},$ (8)

in which $y_{k}(n)$ stands for the $k$-th control signal to drive the
corresponding secondary source. $L$ means the length of secondary path.

Subsequently, the anti-noise vector is obtained from

$\mathbf{y}^{\prime}(n)=\mathbf{S}(n)\mathbf{Y}(n)\in\mathbb{R}^{M\times 1},$
(9)

where the secondary path matrix is expressed as

$\mathbf{S}(n)=\begin{bmatrix}\mathbf{s}^{\mathbf{T}}_{11}(n)&\cdots&\mathbf{s}^{\mathbf{T}}_{1K}(n)\\\
\mathbf{s}^{\mathbf{T}}_{21}(n)&\cdots&\mathbf{s}^{\mathbf{T}}_{2K}(n)\\\
\vdots&\ddots&\vdots\\\
\mathbf{s}^{\mathbf{T}}_{M1}(n)&\cdots&\mathbf{s}^{\mathbf{T}}_{MK}(n)\\\
\end{bmatrix}\in\mathbb{R}^{M\times KL}.$ (10)

In the above matrix, $\mathbf{s}_{mk}(n)$ represents the impulse response of
the secondary path from the $k$-th secondary source to the $m$-th error
sensor:

$\mathbf{s}_{mk}(n)=\begin{bmatrix}s^{(1)}_{mk}(n)&s^{(2)}_{mk}(n)&\cdots&s^{(L)}_{mk}(n)\end{bmatrix}^{\mathrm{T}}\in\mathbb{R}^{L\times
1}.$ (11)

According to above analysis of forward propagation, it is found that the
control filter $\mathbf{W}(n)$ will contribute to the anti-noises
$\mathbf{y}^{\prime}(n)$, $\mathbf{y}^{\prime}(n+1)$, $\cdots$, and
$\mathbf{y}^{\prime}(n+L-1)$. Thus, updating $\mathbf{W}(n)$ requires future
gradient knowledge, which violates the causal restriction in real ANC systems.

### 2.2 Back propagation

Figure 3: The back propagation of the McFxLMS algorithm.

To tackle the causal problem during updating, we assumed that the adaptive
filtering is a slowly changing process, which means that

$\mathbf{W}(n)\approx\mathbf{W}(n-1)\approx\dots\approx\mathbf{W}(n-L+1).$
(12)

Therefore, the forward propagation of the McFxLMS algorithm can be modified as
shown in Figure 3. The $k$-th control signal is obtained from

$y_{k}(n)=\sum^{J}_{j=1}\mathbf{w}^{\mathrm{T}}_{kj}\mathbf{x}_{j}(n),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ j=1,\cdots,J\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ k=1,\cdots,K$ (13)

and its vector format can be rewritten as

$\mathbf{y}_{k}(n)=\begin{bmatrix}\sum^{J}_{j=1}\mathbf{w}^{\mathrm{T}}_{kj}\mathbf{x}_{j}(n)&\sum^{J}_{j=1}\mathbf{w}^{\mathrm{T}}_{kj}\mathbf{x}_{j}(n-1)&\cdots&\sum^{J}_{j=1}\mathbf{w}^{\mathrm{T}}_{kj}\mathbf{x}_{j}(n-L+1)\end{bmatrix}^{\mathrm{T}}\in\mathbb{R}^{L\times
1}.$ (14)

Hence, the $m$-th anti-noise can be calculated as

$y^{\prime}_{m}(n)=\sum^{K}_{k=1}\hat{\mathbf{s}}^{\mathrm{T}}_{mk}\mathbf{y}_{k}(n)\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ m=1,\cdots,M$ (15)

where $\hat{\mathbf{s}}_{mk}$ denotes the estimate of the secondary path
$\mathbf{s}_{mk}(n)$.

Moreover, the cost function of the McFxLMS algorithm is defined as the sum of
squared error signal on each error sensor:

$\mathbf{J}=\sum^{M}_{m}e^{2}_{m}=\sum^{M}_{m}\left[d_{m}(n)-y^{\prime}_{m}(n)\right]^{2}$
(16)

where $d_{m}(n)$ is the disturbance picked up by the $m$-th error sensor.

Figure 4: The computing graph of the control filter $\mathbf{w}_{kj}(n)$ in
the McFxLMS algorithm under the slow adapting assumption, where $j=1,\cdots,J$
and $k=1,\cdots,M$.

Through the above discussion, we can get the updating graph of the control
filter $\mathbf{w}_{kj}(n)$ as shown in Figure 4. According to back
propagation mechanism and chain derivation rule, the gradient with respect to
$\mathbf{w}_{kj}$ can be calculated by

$\begin{split}\nabla_{\mathbf{w}_{kj}}&=\frac{\partial\mathbf{J}}{\partial\mathbf{w}_{kj}}=\frac{\partial\mathbf{y}_{k}(n)}{\partial\mathbf{w}_{kj}}\sum^{M}_{m=1}\frac{\partial
e^{2}_{m}(n)}{\partial y^{\prime}_{m}(n)}\frac{\partial
y^{\prime}_{m}(n)}{\partial\mathbf{y}_{k}(n)}\\\
&=-\begin{bmatrix}\mathbf{x}_{j}(n)&\cdots&\mathbf{x}_{j}(n-L+1)\end{bmatrix}\sum^{M}_{m=1}2e_{m}(n)\hat{\mathbf{s}}_{mk}\\\
&=-2\sum^{M}_{m=1}e_{m}(n)\mathbf{x}^{\prime}_{jkm}(n),\end{split}$ (17)

where the filtered reference signal vector is given by

$\mathbf{x}^{\prime}_{jkm}(n)=\begin{bmatrix}\mathbf{x}_{j}(n)&\cdots&\mathbf{x}_{j}(n-L+1)\end{bmatrix}\hat{\mathbf{s}}_{mk}.$
(18)

Finally, by using the negative value of the gradient (17) to update the
control filer, we can get

$\mathbf{w}_{kj}(n+1)=\mathbf{w}_{kj}(n)-\frac{\mu}{2}\nabla_{\mathbf{w}_{kj}}=\mathbf{w}_{kj}(n)+\mu\sum^{M}_{m=1}e_{m}(n)\mathbf{x}^{\prime}_{jkm}(n)\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ k=1,\cdots,K\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ j=1,\cdots,J$ (19)

where $\mu$ denotes the step size.

After the analysis of forward and back propagation, we found that the adaptive
ANC algorithm may be implemented via automatic derivation mechanism. Since the
automatic derivation mechanism in PyTorch can update weights given the cost
function, the back propagation process can be automatically achieved without
resorting to extra-human efforts. Also, existing optimizers in PyTorch tool
like SGD, Adam etc. can be directly employed to update the control filters.
Owing to the simple implementation method, changing the number of microphones
and loudspeakers in multiple-channel ANC systems is more convenient, so that
the practicality is enhanced. Additionally, based on the PyTorch tool, it is
convenient to add constraints on the control filter coefficients, such as $L1$
and $L2$ normalization.

Figure 5: Error signals and averaged noise reduction level of every 1 second
on the aircraft noise.

Figure 6: Error signals and averaged noise reduction level of every 1 second
on the helicopter noise.

## 3 SIMULATION RESULTS

We implemented the McFxLMS and McFxNLMS algorithm based on the automatic
derivation mechanism in PyTorch. In the ANC system, the number of reference
sensors, secondary sources and error sensors are 4, 4, and 4, respectively.
The control filter length is $512$ taps. The step size of the McFxLMS
algorithm and McFxNLMS algorithm are set to $0.00001$ and $0.001$ separately.
The primary path and secondary path are chosen as a band-pass filter and a
low-pass filter.

The implemented McFxLMS algorithm and McFxNLMS algorithm are evaluated to
suppress several real-recorded noises including: an aircraft noise with a
frequency range of $50$Hz-$12,000$Hz, a helicopter noise with a frequency
range of $50$Hz-$9,800$Hz, and a traffic noise with a frequency range of
$40$Hz-$1,400$Hz. In this section, the stochastic gradient descent (SGD)
algorithm (18) was used for optimization.

The noise reduction results using the implemented McFxLMS algorithm and
McFxNLMS algorithm on the aircraft noise, helicopter noise, and traffic noise
are shown in Figure 5, Figure 6, and Figure 7. According to the simulation
results, the implemented McFxLMS and McFxNLMS algorithm can attenuate the
real-world noises, which demonstrates the effectiveness of the implementation
method. Also, compared with the McFxLMS algorithm, the McFxNLMS algorithm
obtains slightly higher steady-state noise reduction levels on the aircraft
noise and helicopter noise. However, the McFxNLMS algorithm responds more
slowly to the traffic noise than the McFxLMS algorithm.

Figure 7: Error signals and averaged noise reduction level of every 1 second
on the traffic noise.

## 4 CONCLUSIONS

In this paper, we implement the McFxLMS and McFxNLMS algorithm based on the
back propagation mechanism. Since the automatic derivation in PyTorch can
update weights given the cost function, the back propagation of McFxLMS and
McFxNLMS can be automatically achieved without resorting to extra-human
efforts. Therefore, the simple implementation method of the McFxLMS and
McFxNLMS algorithm can improve the practicality. Also, we provide the open
source code of the implementation method. Simulation results show that the
implemented McFxLMS and McFxNLMS algorithm are effective to attenuate
different real noises. Furthermore, this method is expected to implement other
adaptive algorithms in multi-channel ANC systems.

## ACKNOWLEDGEMENTS

This research is supported by the Singapore Ministry of National Development
and the National Research Foundation, Prime Minister’s Office under the Cities
of Tomorrow (CoT) Research Programme (CoT Award No. COT-V4-2019-1). Any
opinions, findings, and conclusions or recommendations expressed in this
material are those of the author(s) and do not reflect the views of the
Singapore Ministry of National Development and National Research Foundation,
Prime Minister’s Office, Singapore.

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