# On the modular Plesken Lie algebra

John Cullinan

###### Abstract.

Let $\mathrm{G}$ be a finite group. The Plesken Lie algebra
$\mathcal{L}[\mathrm{G}]$ is a subalgebra of the complex group algebra
$\mathbf{C}[\mathrm{G}]$ and admits a direct-sum decomposition into simple Lie
algebras based on the ordinary character theory of $\mathrm{G}$. In this paper
we review the known results on $\mathcal{L}[G]$ and related Lie algebras, as
well as introduce a conjecture on a characteristic $p$ analog
$\mathcal{L}_{p}[\mathrm{G}]$, with a focus on when $p$ divides the order of
$\mathrm{G}$.

###### Key words and phrases:

Lie algebra; modular representation; finite group.

##  1\. Introduction

The aim of this paper is to review known results on a certain Lie algebra
attached to a finite group using ordinary representation theory, as well as to
suggest new directions of research that incorporate modular representations.
Let us begin with some definitions.

Let $k$ be a field, $\mathrm{G}$ a finite group, and $k[\mathrm{G}]$ the group
algebra of $\mathrm{G}$ over $k$. The alternating bilinear map
$[~{},~{}]:k[\mathrm{G}]\times k[\mathrm{G}]\to k[\mathrm{G}]$ defined by

$[x,y]=xy-yx$

endows $k[\mathrm{G}]$ with the structure of a Lie algebra. For each
$g\in\mathrm{G}$, define the element
$\widehat{g}\stackrel{{\scriptstyle{\rm\tiny{def}}}}{{=}}g-g^{-1}\in
k[\mathrm{G}]$. Following [3], the _Plesken Lie algebra_
$\mathcal{L}[\mathrm{G}]$ is the Lie subalgebra of $k[\mathrm{G}]$ linearly
spanned over $k$ by the $\widehat{g}$.

This raises a very natural question: _what is the isomorphism type of
$\mathcal{L}[\mathrm{G}]$?_ In particular, we can ask whether there is a
natural relation between the Lie algebra structure of
$\mathcal{L}[\mathrm{G}]$ and the group structure of $\mathrm{G}$. Since
$\mathcal{L}[\mathrm{G}]$ might not be a simple Lie algebra, we state our main
question more formally as follows.

######  Question 1.

What are the isomorphism types (as Lie algebras over $k$) of the simple
composition factors of $\mathcal{L}[\mathrm{G}]$?

A natural starting point is when $k=\mathbf{C}$, since there is a well-
understood classical theory of complex simple Lie algebras. Indeed, in [3] the
authors show that when $k=\mathbf{C}$, the Lie algebra
$\mathcal{L}[\mathrm{G}]$ is semisimple (owing to the fact that
$\mathbf{C}[\mathrm{G}]$ is a direct sum of matrix algebras) and admits a
direct sum decomposition into Lie subalgebras according to the degrees and
Frobenius-Schur indicators of the complex irreducible representations of
$\mathrm{G}$. We review their main theorem and give an illustrative example in
Section 3 below.

Since the structure of $\mathcal{L}[\mathrm{G}]$ is known when $k=\mathbf{C}$,
we switch perspective and ask what happens for other fields $k$, in particular
when the characteristic of $k$ divides the order of $\mathrm{G}$. Here, there
appear to be interesting connections with the modular representation theory of
$\mathrm{G}$ which have yet to be studied. After reviewing the current
literature below, we propose new questions and conjectures for further
research based on extensive calculation.

##  2\. History of the Problem

Before working with $\mathcal{L}[\mathrm{G}]$ directly, let us first consider
$k[\mathrm{G}]$ as a Lie algebra. If $k=\mathbf{C}$, then it is well known
that $\mathbf{C}[\mathrm{G}]$ decomposes as a direct sum of matrix algebras,
which coincides with the decomposition into Lie algebras. Suppose now that the
characteristic of $k$ divides the order of $\mathrm{G}$ (so that we are in the
_modular_ setting). Then Passi, Passman, and Sehgal determined in [8] the
exact conditions under which $k[G]$ is a nilpotent Lie algebra, and under
which it is a solvable Lie algebra; they showed that the
nilpotency/solvability can be determined purely from the group-theoretic
properties of $\mathrm{G}$, foreshadowing the potential for deep connections
between the group theory of $\mathrm{G}$ and the Lie algebra theory of
$\mathcal{L}[\mathrm{G}]$.

Roughly a decade later, Smirnov and Zalesskii in [10] systematically studied
the Lie-algebraic properties of $\mathcal{L}[\mathrm{G}]$; to the best of our
knowledge this is the first work that concerns $\mathcal{L}[\mathrm{G}]$
explicitly. In that paper the authors show (among other things) that

  1. (1)

if $k$ has characteristic 0, then $\mathcal{L}[\mathrm{G}]$ is solvable
implies $\mathrm{G}$ is solvable, and

  2. (2)

if $k$ has characteristic different than 2, then $\mathcal{L}[\mathrm{G}]$ is
nilpotent implies $\mathrm{G}^{2}$ (the subgroup of $\mathrm{G}$ generated by
squares) is nilpotent.

We remark that item (a) does not generalize to positive characteristic (as
they point out in [10, §5.1]). Additionally, item (b) appears to be the first
structure theorem relating the Lie-algebraic properties of
$\mathcal{L}[\mathrm{G}]$ to the group theory of $\mathrm{G}$ in the modular
setting (excluding characteristic 2).

Returning to the case $k=\mathbf{C}$, in 2008 Cohen and Taylor determined the
direct-sum decomposition of $\mathcal{L}[\mathrm{G}]$ into complex simple Lie
algebras of classical type [3] (we review the exact decomposition in Section
3). Subsequently, in [7], Marin considered a more general setup, which is also
briefly addressed in [10], as follows. Let
$\alpha:\mathrm{G}\to\mathbf{C}^{\times}$ be a multiplicative character, and
let $\mathcal{L}_{\alpha}[\mathrm{G}]$ denote the span of the elements
$g-\alpha(g)g^{-1}$. Then $\mathcal{L}_{\alpha}[\mathrm{G}]$ is also a Lie
subalgebra of $k[\mathrm{G}]$ when endowed with the bracket operation. When
$\alpha=\mathbf{1}$, we have
$\mathcal{L}_{\mathbf{1}}[\mathrm{G}]=\mathcal{L}[\mathrm{G}]$. The main
result of [7] is the determination of $\mathcal{L}_{\alpha}[\mathrm{G}]$ as a
direct sum of complex, simple Lie algebras of classical type and
contextualizes the study of the Plesken algebra within the field of harmonic
analysis. More recently, Chaudhuri determined in 2020 in [2] the direct-sum
decomposition of $\mathcal{L}[\mathrm{G}]$ when $k$ is an algebraic extension
of $\mathbf{Q}$. And in 2023 Arjun and Romeo studied the Lie algebra
representation theory of $\mathcal{L}[\mathrm{G}]$ over $\mathbf{C}$ [1].

It is evident that there is an established and significant body of work
surrounding the Plesken Lie algebra in characteristic 0, relating the ordinary
representation theory and group structure of $\mathrm{G}$ to the classical Lie
algebra structure of $\mathcal{L}[\mathrm{G}]$. Before addressing the modular
theory in Section 4, we review the main theorem of [3] along with some new
generalizations. For the remainder of the paper we use Atlas [4] notation for
finite simple groups.

##  3\. The Ordinary Plesken Lie Algebra

Recall the classification of simple Lie algebras over $\mathbf{C}$ of
classical type in Table 1. For small values of $n$, we have the following
coincidences:
$\mathfrak{B}_{1}=\mathfrak{C}_{1}=\mathfrak{D}_{1}=\mathfrak{A}_{1}$,
$\mathfrak{B}_{2}=\mathfrak{C}_{2}$,
$\mathfrak{D}_{2}=\mathfrak{A}_{1}\times\mathfrak{A}_{1}$, and
$\mathfrak{D}_{3}=\mathfrak{A}_{3}$.

Label | Isomorphsim Type | Dimension | $n$
---|---|---|---
$\mathfrak{A}_{n}$ | $\mathfrak{sl}_{n+1}$ | $(n+1)^{2}-1$ | $n\geq 1$
$\mathfrak{B}_{n}$ | $\mathfrak{o}_{2n+1}$ | $2n^{2}+n$ | $n\geq 2$
$\mathfrak{C}_{n}$ | $\mathfrak{sp}_{2n}$ | $2n^{2}+n$ | $n\geq 3$
$\mathfrak{D}_{n}$ | $\mathfrak{o}_{2n}$ | $2n^{2}-n$ | $n\geq 4$
Table 1. Simple Lie Algebras of Classical Type

In [3] the authors show that if $t$ is the number of involutions of
$\mathrm{G}$, then

(1) $\displaystyle\dim\mathcal{L}[\mathrm{G}]=(|\mathrm{G}|-t-1)/2.$

The irreducible characters of $\mathrm{G}$ are partitioned according to their
Frobenius-Schur indicator. We denote these sets by $\mathcal{X}_{1}$,
$\mathcal{X}_{-1}$, and $\mathcal{X}_{0}$ and recall the direct-sum
decomposition proved in [3, Thm. 5.1]:

(2)
$\displaystyle\mathcal{L}[\mathrm{G}]=\bigoplus_{\chi\in\mathcal{X}_{1}}\mathfrak{o}_{\chi(1)}\bigoplus_{\chi\in\mathcal{X}_{-1}}\mathfrak{sp}_{\chi(1)}\bigoplus_{\chi\in\mathcal{X}_{0}}~{}^{\prime}\mathfrak{gl}_{\chi(1)},$

where the prime signifies that there is just one summand
$\mathfrak{gl}_{\chi(1)}$ for each pair $\\{\chi,\overline{\chi}\\}$ from
$\mathcal{X}_{0}$. The summands of $\mathfrak{gl}_{n}$ type are not simple,
but contain $\mathfrak{sl}_{n}$ as a codimension-1 summand. Let us now
consider an illustrative example that shows how one can generalize (2) to
finite fields in ordinary characteristic.

######  Example 1.

Let $\mathrm{G}={\rm L}_{2}(8)$, the simple group of order $504=2^{3}\cdot
3^{2}\cdot 7$. Using Atlas notation [4, p. 6], the character table of
$\mathrm{G}$ is given in Table 2.

| ind | 1A | 2A | 3A | 7A | B*2 | C*4 | 9A | B*2 | C*4
---|---|---|---|---|---|---|---|---|---|---
$\chi_{1}$ | + | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
$\chi_{2}$ | + | 7 | $-1$ |  $-$2 | 0 | 0 | 0 | 1 | 1 | 1
$\chi_{3}$ | + | 7 |  $-$1 | 1 | 0 | 0 | 0 |  $-$y9 | *2 | *4
$\chi_{4}$ | + | 7 |  $-$1 | 1 | 0 | 0 | 0 | *4 |  $-$y9 | *2
$\chi_{5}$ | + | 7 |  $-$1 | 1 | 0 | 0 | 0 | *2 | *4 |  $-$y9
$\chi_{6}$ | + | 8 | 0 |  $-$1 | 1 | 1 | 1 |  $-$1 |  $-$1 |  $-$1
$\chi_{7}$ | + | 9 | 1 | 0 | y7 | *2 | *4 | 0 | 0 | 0
$\chi_{8}$ | + | 9 | 1 | 0 | *4 | y7 | *2 | 0 | 0 | 0
$\chi_{9}$ | + | 9 | 1 | 0 | *2 | *4 | y7 | 0 | 0 | 0
Table 2. Character Table of ${\rm L}_{2}(8)$

Here, y9 is a root of the polynomial $f_{9}(x)=x^{3}-3x-1$ and y7 a root of
$f_{7}(x)=x^{3}+x^{2}-2x-1$. Both y9 and y7 are linear combinations of certain
roots of unity:

y9 $\displaystyle=-\zeta_{9}^{5}-\zeta_{9}^{4}$ y7
$\displaystyle=-\zeta_{7}^{4}+\zeta_{7}^{3},$

and ‘*$m$’ denotes replacing a root of unity $\zeta$ by $\zeta^{m}$. Let us
now consider the Plesken Lie algebra of $\mathrm{G}$ from several
perspectives.

  1. (1)

Applying (1) shows that $\dim\mathcal{L}[\mathrm{G}]=220$.

  2. (2)

Applying (2), we see that that $\mathcal{L}[\mathrm{G}]$ admits the direct-sum
decomposition

$\mathcal{L}[\mathrm{G}]=\mathfrak{o}_{7}^{4}\oplus\mathfrak{o}_{8}\oplus\mathfrak{o}_{9}^{3}.$

  3. (3)

Let $p$ be a prime number coprime to $|G|$, so that
$\mathbf{F}_{p}[\mathrm{G}]$ is semisimple (we call $p$ and _ordinary_ prime).
Define $\mathcal{L}_{p}[\mathrm{G}]$ to be the Lie subalgebra of
$\mathbf{F}_{p}[\mathrm{G}]$ spanned by the $\widehat{g}$. Then the proof of
[3, Thm. 5.1] applies _mutatis mutandis_ , provided $\mathbf{F}_{p}$ contains
enough roots of unity so that all irreducible representations of $\mathrm{G}$
over $\mathbf{F}_{p}$ are absolutely irreducible (we say that $\mathbf{F}_{p}$
is a _splitting field_ of $\mathrm{G}$). If $\mathbf{F}_{p}$ is not a
splitting field for $\mathrm{G}$, then the absolutely irreducible Galois-
conjugate representations that are not realizable over $\mathbf{F}_{p}$ sum to
an irreducible representation over $\mathbf{F}_{p}$. Given this caveat, the
proof of [3, Thm. 5.1] carries through as well, though the dimensions of the
simple factors are potentially different than over a splitting field. We
consider several special cases.

    1. (a)

Let $p=71$. Then $p$ is ordinary for $\mathrm{G}$ and both $f_{9}(x)$ and
$f_{7}(x)$ split modulo $p$, and we have

$\mathcal{L}_{p}[\mathrm{G}]=\mathfrak{o}_{7}^{4}\oplus\mathfrak{o}_{8}\oplus\mathfrak{o}_{9}^{3},$

where all Lie algebras in the sum are defined over the field
$\mathbf{F}_{71}$.

    2. (b)

Let $p=17$. Then $p$ is ordinary for $\mathrm{G}$, but only $f_{9}(x)$ splits
modulo $p$, while $f_{7}(x)$ is irreducible. The three irreducible
representations of degree 9 are not realizable over $\mathbf{F}_{p}$, but sum
to an irreducible degree 27 representation. Explicitly, this representation is
given by the reduction modulo $17$ of the degree 27 representation over
$\mathbf{Z}$ in the online Atlas:

https://brauer.maths.qmul.ac.uk /Atlas/v3/lin/L28/

We then have

$\mathcal{L}_{p}[\mathrm{G}]=\mathfrak{o}_{7}^{4}\oplus\mathfrak{o}_{8}\oplus\mathfrak{m},$

where $\mathfrak{m}$ is a 108-dimensional Lie algebra decomposing over
$\mathbf{F}_{17}(\texttt{y7})$ into $\mathfrak{o}_{9}^{3}$.

    3. (c)

Let $p=5$. Then $p$ is ordinary for $\mathrm{G}$, neither $f_{9}(x)$ nor
$f_{7}(x)$ split modulo $p$, and we have

$\mathcal{L}_{p}[\mathrm{G}]=\mathfrak{o}_{7}\oplus\mathfrak{o}_{8}\oplus\mathfrak{l}\oplus\mathfrak{m},$

where $\mathfrak{l}$ is a 63-dimensional Lie algebra decomposing over
$\mathbf{F}_{5}(\texttt{y9})$ into $\mathfrak{o}_{7}^{3}$ and $\mathfrak{m}$
is a 108-dimensional Lie algebra decomposing over
$\mathbf{F}_{5}(\texttt{y7})$ into $\mathfrak{o}_{9}^{3}$.

We can generalize Items (0a), (0b), and (0c) to number fields as follows.
Denote by $K_{7}$ and $K_{9}$ the splitting fields over $\mathbf{Q}$ of
$f_{7}(x)$ and $f_{9}(x)$, respectively. Both $K_{7}$ and $K_{9}$ are cyclic
extensions of $\mathbf{Q}$ of order 3 and the compositum $K_{7}K_{9}$ is
Galois over $\mathbf{Q}$ with Galois group isomorphic to $C_{3}\times C_{3}$.

Let $\zeta_{63}\in\mathbf{C}$ be a primitive 63rd root of unity. Then
$\mathbf{Q}(\zeta_{63})$ is a splitting field of $\mathrm{G}$ over
$\mathbf{Q}$, and contains the minimal splitting field $K_{7}K_{9}$. Basic
field theory shows that $[\mathbf{Q}(\zeta_{63}):\mathbf{Q}]=36$ and a portion
of the intermediate field diagram is as follows

$\textstyle{\mathbf{Q}(\zeta_{63})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{C_{6}\times
C_{6}}$$\scriptstyle{C_{2}\times
C_{2}}$$\textstyle{K_{7}K_{9}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{C_{3}\times
C_{3}}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{7}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{9}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{Q}}$

where ‘$\bullet$’ stands for a cubic extension of $\mathbf{Q}$ that does not
concern this exposition (except for the calculations of the Chebatorev
densities).

By the Chebatorev Density Theorem applied to the extension
$K_{7}K_{9}/\mathbf{Q}$, it is a routine calculation to determine the
proportion of primes for which the polynomials $f_{7}(x)$ and $f_{9}(x)$ split
or are irreducible and, hence, the direct-sum decomposition of
$\mathcal{L}_{p}[\mathrm{G}]$. Specifically, we have

Proportion | Decomposition | $f_{7}(x)$ | $f_{9}(x)$
---|---|---|---
of Primes | of $\mathcal{L}_{p}[\mathrm{G}]$ |  |
1/9 | $\mathfrak{o}_{7}\oplus\mathfrak{o}_{8}\oplus\mathfrak{o}_{7}^{3}\oplus\mathfrak{o}_{9}^{3}$ | split | split
4/9 | $\mathfrak{o}_{7}\oplus\mathfrak{o}_{8}\oplus\mathfrak{l}_{1}\oplus\mathfrak{l}_{2}$ | irred. | irred.
2/9 | $\mathfrak{o}_{7}\oplus\mathfrak{o}_{8}\oplus\mathfrak{o}_{7}^{3}\oplus\mathfrak{l}_{2}$ | split | irred.
2/9 | $\mathfrak{o}_{7}\oplus\mathfrak{o}_{8}\oplus\mathfrak{l}_{1}\oplus\mathfrak{o}_{9}^{3}$ | irred | split
Table 3. Splitting of $\mathcal{L}_{p}[\mathrm{G}]$ for ordinary primes

More generally, the minimal splitting field $K_{\mathrm{G}}$ of a finite group
$\mathrm{G}$ is a subfield of the cyclotomic field $\mathbf{Q}(\zeta_{m})$,
where $m$ is the least common multiple of the orders of the elements of
$\mathrm{G}$. Thus, $K_{\mathrm{G}}$ is an abelian extension of $\mathbf{Q}$
and so class field theory would allow us to determine the densities of primes
$p$ for which Galois-conjugate, absolutely irreducible representations of
$\mathrm{G}$ are realizable modulo $p$, analogous to the case in Table 3
above. This is also a natural generalization of the results of [2] and [3] to
characteristic $p$ fields when $p$ is ordinary.

##  4\. The Modular Plesken Lie Algebra

Let $p$ be a prime dividing $|G|$ and let $k$ be an algebraically closed field
of characteristic $p$. Then $k[\mathrm{G}]$ is no longer semisimple, but
decomposes as a direct sum of blocks. The goal of this section is to introduce
a conjecture on the structure of $\mathcal{L}_{p}[\mathrm{G}]$ that takes into
account the modular representation theory of $\mathrm{G}$ in a similar way to
the ordinary case. Observe that the dimension of $\mathcal{L}_{p}[\mathrm{G}]$
is independent of whether $p$ is ordinary or modular.

Let $\phi_{1},\dots,\phi_{s}$ denote the Brauer characters of $\mathrm{G}$,
where $s$ is the number of $p$-regular classes of $\mathrm{G}$. As in the
ordinary case, the set of Brauer characters is partitioned by Frobenius-Schur
indicator into subsets $\mathcal{X}_{1}$, $\mathcal{X}_{-1}$, and
$\mathcal{X}_{0}$. The Lie algebra $\mathcal{L}_{p}[\mathrm{G}]$ will
generally not decompose into a direct sum of simple Lie algebras, but it does
possess a composition series nonetheless. Let us now set some notation in
order to state a conjecture on the structure of $\mathcal{L}_{p}[\mathrm{G}]$.

Write $k[\mathrm{G}]=b_{1}\oplus b_{2}\oplus\cdots\oplus b_{r}$ as the block
decomposition of $k[\mathrm{G}]$. Then each $b_{j}$, being a two-sided ideal
of $k[\mathrm{G}]$, assumes a Lie algebra structure over $k$ as well. Let us
make some further remarks.

  1. (1)

We write $\mathsf{Ab}(n)$ for an abelian Lie algebra of dimension $n$.

  2. (2)

If $p|(n+1)$, then the Lie algebra $\mathfrak{A}_{n}$ is not simple over $k$,
since the scalar matrices have trace 0. We write $\mathfrak{PA}_{n}$ for the
codimension-1 simple factor of $\mathfrak{A}_{n}$ in this case.

  3. (3)

In addition to the classical algebras, there exist exceptional simple Lie
algebras in positive characteristic. When $p\geq 5$, the finite-dimensional
simple Lie algebras over an algebraically closed field of characteristic $p$
have been fully classified. Any such algebra is either of classical or Cartan
type (Witt; Special; Hamiltonian; Contact) for $p\geq 7$ or, additionally, of
Melikian type if $p=5$ [9].

Table 4 gives the dimensions of the exceptional algebras and gives the
conditions under which they are simple.With the exception of the Melikian
algebras, these Lie algebras are parameterized by $m,\underline{n}$ with
$\underline{n}=[n_{1},\dots,n_{m}]\in\mathbf{Z}_{>0}^{m}$. Set $N=\sum n_{i}$.

Lie algebra | Dimension | Simple
---|---|---
$W(m,\underline{n})$ | $p^{N}$ |  $p\neq 2$ and $m\neq 1$
$S(m,\underline{n})$ | $(m-1)p^{N}+1$ | $m\geq 3$
$H(m,\underline{n})$ | $p^{N}-1$ |  $p>2$, $m\geq 2$
$K(m,\underline{n})$ | $p^{N}$ |  $p>2$, $m\geq 3$
$M(n_{1},n_{2})$ | $5^{n_{1}+n_{2}+1}$ | $n_{i}>0$
Table 4. Exceptional Modular Lie Algebras

In our Conjecture 1 below, the exceptional modular Lie algebras do not appear
as simple factors. We conjecture that only the classical and abelian type Lie
algebras appear as composition factors of $\mathcal{L}[\mathrm{G}]$.

  4. (4)

We use lowercase fraktur letters for an arbitrary Lie algebra over $k$ and
reserve $\mathfrak{A}_{n}$, $\mathfrak{B}_{n}$, $\mathfrak{C}_{n}$,
$\mathfrak{D}_{n}$ for the classical Lie algebras of rank $n$. For example, if
$\mathcal{L}_{p}[\mathrm{G}]=\mathfrak{a}_{1}\oplus\mathfrak{a}_{2}\oplus\mathfrak{A}_{3}$,
then $\mathfrak{a}_{1}$ and $\mathfrak{a}_{2}$ are two unspecified Lie
algebras over $k$, while $\mathfrak{A}_{3}$ refers to the Lie algebra of
dimension 15 isomorphic to $\mathfrak{sl}_{4}$ over $k$.

We are now ready to introduce our conjecture on the structure of
$\mathcal{L}_{p}[\mathrm{G}]$ in the modular setting. The reader will notice
that the degrees and indicators of the Brauer characters determine which
classical Lie algebras appear in the decomposition, just as the the degrees
and indicators of the complex characters did in the complex setting. The main
difference is that the degrees of the Brauer characters need not be the same
as the complex characters and there are fewer Brauer characters than complex.
The remaining composition factors are all abelian, which is different from the
complex setting.

######  Conjecture 1.

Let $\mathrm{G}$ be a finite group and $p$ a modular prime for $\mathrm{G}$.
With all notation as above, we have

  1. (1)

$\mathcal{L}_{p}[\mathrm{G}]$ admits a direct sum decomposition

$\mathcal{L}_{p}[\mathrm{G}]=\bigoplus_{j=1}^{r}\mathfrak{a}_{j},$

where the $\mathfrak{a}_{j}$ denote the projection of
$\mathcal{L}_{p}[\mathrm{G}]$ onto the block $b_{j}$. The Lie algebras
$\mathfrak{a}_{j}$ are not necessarily simple.

  2. (2)

The composition factors of $\mathcal{L}_{p}[\mathrm{G}]$ are either abelian or
of classical type.

  3. (3)

    1. (a)

for each $\phi\in\mathcal{X}_{1}$, there is a composition factor isomorphic to
$\mathfrak{o}_{\phi(1)}$, and

    2. (b)

for each $\phi\in\mathcal{X}_{-1}$, there is a composition factor isomorphic
to $\mathfrak{sp}_{\phi(1)}$, and

    3. (c)

for each pair $(\phi,\phi^{\prime})$ of conjugate Brauer characters in
$\mathcal{X}_{0}$, there is one composition factor isomorphic to
$\mathfrak{gl}_{\phi(1)}$. The algebra $\mathfrak{gl}_{\phi(1)}$ has a
codimension-1 simple factor isomorphic to $\mathfrak{sl}_{\phi(1)}$, unless
$p|\phi(1)$. In that case, $\mathfrak{gl}_{\phi(1)}$ has a codimension-2
simple factor of type $\mathfrak{PA}_{\phi(1)-1}$.

  4. (4)

A block of $k[\mathrm{G}]$ of defect 0 gives rise to direct summand of
$\mathcal{L}[\mathrm{G}]$ of classical type.

######  Remark 1.

The degrees of the Brauer characters do not necessarily agree with those of
the ordinary characters, which can result in a very different overall Lie
algebra structure of $\mathcal{L}_{p}[\mathrm{G}]$ versus
$\mathcal{L}[\mathrm{G}]$, though the classical factors of
$\mathcal{L}[\mathrm{G}]$ as described in items (3a), (3b), (3c) of Conjecture
1 mimic the decomposition of (2).

For the remainder of this section we give illustrative examples that are
chosen to highlight different aspects of the conjecture. In every case the
reader will see that the simple composition factors can be predicted from the
first two columns of the Brauer character table in the same manner as the
ordinary $\mathcal{L}[\mathrm{G}]$, as claimed in our conjecture.

######  Example 2.

Let $\mathrm{G}=\operatorname{SL}_{2}(5)$ and let $p=5$. The indicator and
degree columns of the Brauer and ordinary character tables are given in Table
5.

Brauer | ind | degree | Ordinary | ind | degree
---|---|---|---|---|---
$\pmod{5}$ |  |  |  |  |
$\phi_{1}$ | $+$ | 1 | $\chi_{1}$ | $+$ | 1
$\phi_{2}$ | $-$ | 2 | $\chi_{2}$ | $-$ | 2
$\phi_{3}$ | $+$ | 3 | $\chi_{3}$ | $-$ | 2
$\phi_{4}$ | $-$ | 4 | $\chi_{4}$ | $+$ | 3
$\phi_{5}$ | $+$ | 5 | $\chi_{5}$ | $+$ | 3
|  |  | $\chi_{6}$ | $-$ | 4
|  |  | $\chi_{7}$ | $+$ | 4
|  |  | $\chi_{8}$ | $+$ | 5
|  |  | $\chi_{9}$ | $-$ | 6
Table 5. Brauer -vs- Ordinary Characters of $\operatorname{SL}_{2}(5)$

The group algebra $k[\mathrm{G}]$ decomposes into a direct sum of three blocks

$k[\mathrm{G}]=b_{1}\oplus b_{2}\oplus b_{3},$

of defects 1, 1, and 0, respectively. The ordinary characters $\chi_{1}$,
$\chi_{4}$, $\chi_{5}$, and $\chi_{7}$ belong to $b_{1}$, while $\chi_{2}$,
$\chi_{3}$, $\chi_{6}$, and $\chi_{9}$ belong to $b_{2}$. The Steinberg
character $\chi_{8}$ is the unique character belonging to $b_{3}$.

In Magma we compute that $\mathcal{L}_{5}[\mathrm{G}]$ admits the direct-sum
decomposition

$\mathcal{L}_{5}[\mathrm{G}]=\mathfrak{a}_{1}\oplus\mathfrak{a_{2}}\oplus\mathfrak{B}_{2},$

where $\mathfrak{a}_{1}$ and $\mathfrak{a}_{2}$ are Lie algebras with
respective composition factors

$\displaystyle\mathfrak{a}_{1}:$ $\displaystyle\
\mathsf{Ab}(3),\mathsf{Ab}(3),\mathsf{Ab}(3),\mathfrak{B}_{1}$
$\displaystyle\mathfrak{a}_{2}:$ $\displaystyle\
\mathsf{Ab}(3),\mathsf{Ab}(3),\mathsf{Ab}(8),\mathsf{Ab}(10),\mathfrak{C}_{1},\mathfrak{C}_{2},$

Observe that the simple composition factors $\mathfrak{B}_{1}$,
$\mathfrak{C}_{1}$, $\mathfrak{B}_{2}$, $\mathfrak{C}_{2}$ that we computed in
Magma agree part (3) of Conjecture 1. Observe also that the direct summand
$\mathfrak{B}_{2}$ corresponds to the block $b_{3}$ of $k[\mathrm{G}]$ of
defect 0.

######  Example 3.

Let $\mathrm{G}={\rm L}_{2}(8)$ as in Example 1. We select modular primes that
are different from the defining characteristic (in contrast to Example 2
above).

Brauer | ind | degree | Brauer | ind | degree | Ordinary | ind | degree
---|---|---|---|---|---|---|---|---
$\pmod{3}$ |  |  | $\pmod{7}$ |  |  |  |  |
$\phi_{1}$ | $+$ | 1 | $\phi_{1}$ | $+$ | 1 | $\chi_{1}$ | $+$ | 1
$\phi_{2}$ | $+$ | 7 | $\phi_{2}$ | $+$ | 7 | $\chi_{2}$ | $+$ | 7
$\phi_{3}$ | $+$ | 9 | $\phi_{3}$ | $+$ | 7 | $\chi_{3}$ | $+$ | 7
$\phi_{4}$ | $+$ | 9 | $\phi_{4}$ | $+$ | 7 | $\chi_{4}$ | $+$ | 7
$\phi_{5}$ | $+$ | 9 | $\phi_{5}$ | $+$ | 7 | $\chi_{5}$ | $+$ | 7
|  |  | $\phi_{6}$ | $+$ | 8 | $\chi_{6}$ | $+$ | 8
|  |  |  |  |  | $\chi_{7}$ | $+$ | 9
|  |  |  |  |  | $\chi_{8}$ | $+$ | 9
|  |  |  |  |  | $\chi_{9}$ | $+$ | 9
Table 6. Brauer -vs- Ordinary Characters of ${\rm L}_{2}(8)$

In characteristic 3, we have that $k[\mathrm{G}]$ decomposes into a sum of
four blocks

$k[\mathrm{G}]=b_{1}\oplus b_{2}\oplus b_{3}\oplus b_{4},$

of defects 2, 0, 0, and 0, respectively. The corresponding Lie algebra
decomposition, according to Magma, is given by

$\mathcal{L}_{3}[\mathrm{G}]=\mathfrak{a}_{1}\oplus\mathfrak{B}_{4}\oplus\mathfrak{B}_{4}\oplus\mathfrak{B}_{4},$

where $\mathfrak{a}_{1}$ is a Lie algebra with composition factors

$\mathsf{Ab}(7),\mathsf{Ab}(21),\mathsf{Ab}(21),\mathsf{Ab}(21),\mathsf{Ab}(21),\mathfrak{B}_{3}.$

In characteristic 7, we have that $k[\mathrm{G}]$ decomposes into a sum of
five blocks

$k[\mathrm{G}]=b_{1}\oplus b_{2}\oplus b_{3}\oplus b_{4}\oplus b_{5},$

of defects 1, 0, 0, 0, and 0, respectively. The ordinary characters
$\chi_{1}$, $\chi_{6}$, $\chi_{7}$, $\chi_{8}$, and $\chi_{9}$ belong to
$b_{1}$, while $b_{2}$, $b_{3}$, $b_{4}$, $b_{5}$ contain only the characters
$\chi_{2}$, $\chi_{3}$, $\chi_{4}$, $\chi_{5}$, respectively, since those
characters remain irreducible modulo 7. The corresponding Lie algebra
decomposition, according to Magma, is given by

$\mathcal{L}_{7}[\mathrm{G}]=\mathfrak{a}_{1}\oplus\mathfrak{B}_{3}\oplus\mathfrak{B}_{3}\oplus\mathfrak{B}_{3}\oplus\mathfrak{B}_{3},$

where $\mathfrak{a}_{1}$ is a Lie algebra with composition factors

$\mathsf{Ab}(8),\mathsf{Ab}(8),\mathsf{Ab}(8),\mathsf{Ab}(28),\mathsf{Ab}(28),\mathsf{Ab}(28),\mathfrak{D}_{4}.$

Again, observe that in both characteristic 3 and 7 the direct-sum
decomposition and simple composition factors follow the predictions of parts
(3) and (4) of Conjecture 1.

######  Example 4.

The purpose of this final example is to show that when there is a large power
of $p$ dividing $\mathrm{G}$, the Brauer characters modulo $p$ may be quite
different than the ordinary characters over $\mathbf{C}$. Let $\mathrm{G}={\rm
L}_{2}(25)$ and let $p=5$. Then the dimensions and indicators of the Brauer
and Ordinary characters are given in Table 7.

Brauer | ind | degree | Ordinary | ind | degree
---|---|---|---|---|---
$\pmod{5}$ |  |  |  |  |
$\phi_{1}$ | + | 1 | $\chi_{1}$ | + | 1
$\phi_{2}$ | + | 3 | $\chi_{2}$ | + | 13
$\phi_{3}$ | + | 3 | $\chi_{3}$ | + | 13
$\phi_{4}$ | + | 4 | $\chi_{4}$ | + | 24
$\phi_{5}$ | + | 5 | $\chi_{5}$ | + | 24
$\phi_{6}$ | + | 5 | $\chi_{6}$ | + | 24
$\phi_{7}$ | + | 8 | $\chi_{7}$ | + | 24
$\phi_{8}$ | + | 8 | $\chi_{8}$ | + | 24
$\phi_{9}$ | + | 9 | $\chi_{9}$ | + | 24
$\phi_{10}$ | + | 15 | $\chi_{10}$ | + | 25
$\phi_{11}$ | + | 15 | $\chi_{11}$ | + | 26
$\phi_{12}$ | + | 16 | $\chi_{12}$ | + | 26
$\phi_{13}$ | + | 25 | $\chi_{13}$ | + | 26
|  |  | $\chi_{14}$ | + | 26
|  |  | $\chi_{15}$ | + | 26
Table 7. Brauer -vs- Ordinary Characters of ${\rm L}_{2}(25)$

The group algebra $k[\mathrm{G}]$ decomposes as a direct sum of two blocks
$k[\mathrm{G}]=b_{1}\oplus b_{2}$ of defect 2 and 0, respectively. The
Steinberg character $\chi_{10}$ belongs to $b_{2}$, while all others belong to
$b_{1}$. A calculation in Magma shows that the Lie algebra
$\mathcal{L}_{5}[\mathrm{G}]$ admits the decomposition

$\mathcal{L}_{5}[\mathrm{G}]=\mathfrak{a}_{1}\oplus\mathfrak{B}_{12},$

where $\mathfrak{a}_{1}$ has composition factors as in Table 8.

Composition Factor | Multiplicity
---|---
$\mathsf{Ab}(20),\mathsf{Ab}(25),\mathsf{Ab}(54),\mathsf{Ab}(90),\mathsf{Ab}(210)$ | 1
$\mathfrak{B}_{4}$, $\mathfrak{D}_{8}$ |
$\mathsf{Ab}(16)$, $\mathsf{Ab}(24)$, $\mathsf{Ab}(30)$, $\mathsf{Ab}(48)$, $\mathsf{Ab}(64)$ | 2
$\mathsf{Ab}(96)$, $\mathsf{Ab}(144)$, $\mathsf{Ab}(160)$, $\mathsf{Ab}(240)$ |
$\mathfrak{B}_{2}$, $\mathfrak{D}_{4}$, $\mathfrak{B}_{7}$ |
$\mathsf{Ab}(56)$, $\mathsf{Ab}(120)$ | 3
$\mathfrak{A}_{1}$ | 4
$\mathsf{Ab}(6)$ | 7
$\mathsf{Ab}(36)$ | 8
Table 8. Composition Factors of
$\mathfrak{a}_{1}\subseteq\mathcal{L}_{5}[\mathrm{G}]$

Recall that $\mathfrak{B}_{1}=\mathfrak{A}_{1}$ and
$\mathfrak{D}_{2}=\mathfrak{A}_{1}\times\mathfrak{A}_{1}$, whence the four
copies of $\mathfrak{A}_{1}$ in Table 8.

We conclude the paper with further observations and questions for future
researchers.

  1. (1)

Observe that certain ordinary representations remain irreducible upon
reduction modulo $p$. These include the Steinberg representation, as well as
any representation whose degree is a multiple of the largest power of $p$
dividing $|G|$. These representations occur in blocks of defect 0. In the case
of the Steinberg representation, the Frobenius-Schur indicator is $+1$, hence
will give rise to a Lie algebra of type $\mathfrak{B}_{n}$ or
$\mathfrak{D}_{n}$.

  2. (2)

In addition to a proof of Conjecture 1, it would be interesting to determine
the precise number and dimensions of the abelian composition factors of
$\mathcal{L}_{p}[\mathrm{G}]$, as well as investigating any relation between
the abelian factors and the defect groups of the blocks.

##  Appendix A Code and Data

All computations in this paper were performed in Magma [6]. For the code
below, “G” is a permutation group and “F” is a finite field. This code will
build the Plesken Lie Algebra “P”.

FG := GroupAlgebra(F, G);
L1,h:=Algebra(FG);
FFG,f:=LieAlgebra(L1);
L:=[];
for g in G do
if Order(g) ne 2 and Order(g) ne 1 and Inverse(g) notin L then
Append(~L,g);
end if;
end for;
LL:=[];
for g in L do
Append((~LL,FG!g-FG!Inverse(g) ));
end for;
LLL:=[];
for g in LL do
Append((~LLL, (g@h)@f ));
end for;
P:=sub<FFG|LLL>;

## References

  * [1] S. Arjun, P. Romeo. On representations of Plesken Lie algebras. Asian-Eur. J. Math. 16 (2023), no. 5, Paper No. 2350085.
  * [2] D. Chaudhuri. Skew-symmetric elements of rational group algebras. Beitr. Algebra Geom. 61 (2020), no. 4, 719-729.
  * [3] A. Cohen, D.E Taylor. On a Certain Lie Algebra Defined by a Finite Group. Amer. Math. Monthly 114 (2007), no. 7, 633-639.
  * [4] J. Conway _et. al._ , ATLAS of Finite Groups. Oxford University Press, Cambridge, 1985
  * [5] C. Jansen _et. al._ , An Atlas of Brauer Characters, London Mathematical Society Monographs, Oxford University Press, New York, 1995
  * [6] W. Bosma, J. Cannon, C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (3–4), 1997, 235–265.
  * [7] I. Marin. Group algebras of finite groups as Lie algebras. Comm. Algebra 38 (2010), no. 7, 2572-2584.
  * [8] I.B.S. Passi, D. Passman, S.K. Sehgal. Lie solvable group rings. Canad. J. Math. 25 (1973), 748-757.
  * [9] A. Premet, H. Strade. Classification of finite dimensional simple Lie algebras in prime characteristics. _Representations of algebraic groups, quantum groups, and Lie algebras_ , 185-214, Contemp. Math., 413, Amer. Math. Soc., Providence, RI, 2006.
  * [10] M.B. Smirnov, A.E. Zalesskii. Lie algebra associated with linear group. Comm. Algebra 9 (1981), no. 20, 2075-2100.


# DeepFilterNet: A Low Complexity Speech Enhancement Framework for Full-Band
Audio based on Deep Filtering

###### Abstract

Complex-valued processing has brought deep learning-based speech enhancement
and signal extraction to a new level. Typically, the process is based on a
time-frequency (TF) mask which is applied to a noisy spectrogram, while
complex masks (CM) are usually preferred over real-valued masks due to their
ability to modify the phase. Recent work proposed to use a complex filter
instead of a point-wise multiplication with a mask. This allows to incorporate
information from previous and future time steps exploiting local correlations
within each frequency band.

In this work, we propose DeepFilterNet, a two stage speech enhancement
framework utilizing deep filtering. First, we enhance the spectral envelope
using ERB-scaled gains modeling the human frequency perception. The second
stage employs deep filtering to enhance the periodic components of speech.
Additionally to taking advantage of perceptual properties of speech, we
enforce network sparsity via separable convolutions and extensive grouping in
linear and recurrent layers to design a low complexity architecture.

We further show that our two stage deep filtering approach outperforms complex
masks over a variety of frequency resolutions and latencies and demonstrate
convincing performance compared to other state-of-the-art models.

Index Terms—  deep filtering, speech enhancement

##  1 Introduction

Monaural speech enhancement is an important part in many systems such as
automatic speech recognition, video conference systems, as well as assistive
listening devices. Most state-of-the-art approaches work in the short-time
Fourier transform (STFT) representation and estimate a TF mask using a deep
neural network, many of these either real-valued masks [1, 2, 3] or complex
masks [4, 5, 6, 7]. The estimated masks are usually well-defined and limited
by an upper bound to improve stability of the network training. However,
typically both approaches degrade if the frequency resolution gets to low for
removing noise between speech harmonics. The approaches above work on at least
$20\text{\,}\mathrm{ms}$ windows resulting in a minimum frequency of
$50\text{\,}\mathrm{Hz}$.

In this paper, we propose an open source speech enhancement framework based on
deep filtering (DF) [8, 9]. Instead of using a complex mask that is applied
per TF-bin, we use a combination of real-valued gains and a deep filter
enhancement component. For the first stage, we take advantage from the fact
that noise as well as speech usually have a smooth spectral envelope. An
equivalent rectangular bandwidth (ERB) filter bank reduces input and output
dimensions to only 32 bands, allowing for a computationally cheap
encoder/decoder network. Since the resulting minimum bandwidth of
$100\text{\,}\mathrm{Hz}$ to $250\text{\,}\mathrm{Hz}$ depending on the FFT
size is typically not sufficient to enhance periodic components, we use a
second enhancement stage based on deep filtering. That is, a deep filter
network estimates coefficients for frequency bins up to an upper frequency
$f_{\text{DF}}$. The resulting linear complex-valued filters are applied to
their corresponding frequency bins. DF enhancement is only applied for lower
frequencies since periodic speech components contain most energy in the lower
frequencies.

Deep filtering was first proposed by Mack et al. [8] and Schröter et al. [9].
Since a filter applied to multiple time/frequency (TF) bins, DF is able to
recover signal degradations like notch-filters or time-frame zeroing. Schröter
et al. [9] introduced this method as complex linear coding (CLC) for low
latency hearing aid applications. CLC was motivated by its ability to model
quasi-static properties of speech. That is, even for frequency bandwidth of
$500\text{\,}\mathrm{Hz}$ CLC is able to reduce noise within a frequency band,
while preserving speech components. This is especially helpful, when there are
multiple speech harmonics within one frequency bin or for filtering periodic
noises. Recent work [7] demonstrated good performance using deep filtering in
the deep noise suppression challenge [10]. However, compared to their previous
work [11] using a complex ratio mask (CRM), their improvements are mostly
given by network architecture changes like complex TF-LSTMs or convolutional
pathways.

In this work, we demonstrate superior performance of deep filtering over CRMs
for multiple FFT sizes from $5\text{\,}\mathrm{ms}$ to
$30\text{\,}\mathrm{ms}$. We further show that even for low latency
requirements of e.g. $5\text{\,}\mathrm{ms}$ resulting in a frequency
resolution of $250\text{\,}\mathrm{Hz}$, DF can still enhance the periodic
speech components.

##  2 DeepFilterNet

###  2.1 Signal Model

Let $x(t)$ be a mixture signal recorded in a noisy room.

$x(t)=s(t)*h(t)+z(t)$ (1)

where $s(t)$ is a clean speech signal, $h(t)$ is a room impulse response from
the speaker to the microphone and $z(t)$ is an additive noise signal already
containing reverberation. Typically, noise reduction operates in frequency
domain:

$X(k,f)=S(k,f)\cdot H(k,f)+Z(k,f)\text{,}$ (2)

where $X(k,f)$ is the STFT representation of the time domain signal $x(t)$ and
$k$, $f$ are the time and frequency bins.

###  2.2 Deep Filtering

Deep filtering is defined by a complex filter in TF-domain:

$\vspace{-0.5em}Y(k,f)=\sum_{i=0}^{N}C(k,i,f)\cdot X(k-i+l,f)\text{\ ,}$ (3)

where $C$ are the complex coefficients of filter order $N$ that are applied to
the input spectrogram $X$, and $\hat{Y}$ the enhanced spectrogram. In our
framework, the deep filter is applied to the gain-enhanced spectrogram
$Y^{G}$. $l$ is an optional look-ahead, which allows incorporating non-causal
taps in the linear combination if $l\geq 1$. Additionally, one could also
filter over the frequency axis allowing to incorporate correlations e.g. due
to overlapping bands. To further make sure that deep filtering only affects
periodic parts, we introduce a learned weighting factor $\alpha$ to produce
the final output spectrogram:

$Y^{DF}(k,f)=\alpha(k)\cdot Y^{DF^{\prime}}(k,f)+(1-\alpha(k))\cdot
Y^{G}(k,f)\text{.}\vspace{-.25em}$ (4)

###  2.3 Framework Overview

Fig. 1: Overview of the DeepFilterNet algorithm. Stage 1 blocks are indicated
in blue, stage 2 blocks in yellow.

An overview of the DeepFilterNet algorithm is shown in Fig. 1. Given a noisy
audio signal $x(t)$ we transform the signal into frequency domain using a
short time Fourier transform (STFT). The framework is designed for sampling
rates up to $48\text{\,}\mathrm{kHz}$ to support high resolution VoIP
applications and STFT window sizes $N_{FFT}$ between $5\text{\,}\mathrm{ms}$
and $30\text{\,}\mathrm{ms}$. By default, we use an overlap of
$N_{ov}=$50\text{\,}\mathrm{\char 37\relax}$$ but also support higher
overlaps. We use two kinds of input features for the deep neural network
(DNN). For the ERB encoder/decoder features $X_{\text{ERB}}(k,b)$,
$b\in[0,N_{\text{ERB}}]$, we compute a log-power spectrogram, normalize it
using an exponential mean normalization [12] with decay of
$1\text{\,}\mathrm{s}$ and apply a rectangular ERB filter bank (FB) with a
configurable number of bands $N_{\text{ERB}}$. In fact, this normalization is
similar to using instance normalization like [7], which also estimates
statistics with a momentum-based approach and only adds additional scaling and
bias parameters. For the deep filter network features
$X_{\text{DF}}(k,f^{\prime})$, $f^{\prime}\in[0,f_{\text{DF}}]$, we use the
complex spectrogram as input and normalize it using an exponential unit
normalization [9] with the same decay.

An encoder/decoder architecture is used to predict ERB-scaled gains. An
inverse ERB filter bank is applied to transform the gains back to frequency
domain before pointwise multiplying with the noisy spectrogram. To further
enhance the periodic components, DeepFilterNet predicts per-band filter
coefficients $C^{N}$ of order $N$. We only utilize Deep Filtering up to a
frequency $f_{\text{DF}}$ assuming that the periodic components contain most
energy in lower frequencies.

Together with the DNN look-ahead in the convolutional layers as well as the
deep filter look-ahead, the overall latency is given by
$l_{N_{\text{FFT}}}+\max(l_{\text{DNN}},\ l_{\text{DF}})$ resulting in a
minimal latency of $5+\max(0,0)=$5\text{\,}\mathrm{ms}$$ for
$N_{\text{FFT}}=240$.

###  2.4 DNN Model

We focus on designing an efficient DNN only using standard DNN layers like
convolutions, batch normalization, ReLU, etc., so that we can take advantage
of layer fusing as well utilize good support by inference frameworks. We adopt
a UNet-like architecture similar to [13, 7] as shown in Fig. 2. Our
convolutional blocks consist of a separable convolutions (depthwise followed
by a 1x1 convolution) with kernel size of (3x2) and $C=64$ channels followed
by a batch normalization and ReLU activation. The convolutional layers are
aligned in time such that the first layers may introduce an overall look-
ahaead $l_{\text{DNN}}$. The remaining convolutional layers are causal and do
not contribute any more latency. We heavily make use of grouping [14, 13] for
our linear and GRU layers. That is, the layer input is split into $P=8$ groups
resulting in $P$ smaller GRUs/linear layers with a hidden size of $512/P=64$.
The output is shuffled to recover inter-group correlations and concatenated to
the full hidden size again. Convolutional pathways [13, 7] with add-skips are
used to retain frequency resolution. We use a global pathway skip connection
for the DF Net to provide a good representation of the original noisy phase at
the output layer.

Fig. 2:  Overview of the DeepFilterNet architecture. We use 1x1 pathway
convolutions (PConv) as add-skip connections and transposed convolutional
blocks (TConv) analogous to the encoder blocks. Grouped linear and GRU
(GLinear, GGRU) layers are used to introduce sparsity.

###  2.5 Data Preprocessing

The DeepFilterNet framework utilizes heavy on-the-fly augmentation. We mix a
clean speech signal with up to 5 noise signals at signal-to-noise ratios (SNR)
of $\\{-5,0,5,10,20,40\\}$ $\mathrm{dB}$. To further increase variabilty, we
augment speech as well as noise signals with second order filters [1], EQs,
and random gains of $\\{-6,0,6\\}$ $\mathrm{dB}$. Random resampling increases
the variety of pitches and room impulse responses (RIR) are used for
simulating reverberant environments. We apply a low-pass filter to noise
signals before mixing, if the sampling rate of a speech signal is lower than
the current model’s sampling rate. This e.g. also allows models trained on
full-band audio ($48\text{\,}\mathrm{kHz}$) to perform equally well on input
signals with lower sampling rates. We furthermore support training attenuation
limited models. Therefore, we generate a “noisy” target signal $s$ with a $6$
to $20$ $\mathrm{dB}$ higher SNR compared to the noisy signal $x$. During
training, we then clamp the predicted gains $G$ and having a “noisy” target
$s$, DF Net will learn to not remove more noise than specified. This is useful
e.g. for wearable devices where we want to keep some environmental awareness
for the user.

###  2.6 Loss

It is not trivial to provide ideal DF coefficients $C^{N}$, since there are
infinitely many possibilities [8]. Instead, we use a compressed spectral loss
to implicitly learn ERB gains $G$ and filter coefficients $C^{N}$ [15, 13].

$\mathcal{L}_{spec}=\sum_{k,f}||\ |Y|^{c}-|S|^{c}||^{2}+\sum_{k,f}||\
|Y|^{c}e^{j\varphi_{Y}}-|S|^{c}e^{j\varphi_{S}}||^{2}\text{,}$ (5)

where $c=0.6$ is a compression factor to model the perceived loudness [16].
Having a magnitude as well as a phase-aware term makes this loss is suitable
for modeling both real-valued gain and complex DF coefficient prediction. To
harden the gradient for TF bins with magnitude close to zero (e.g. for input
signals with a lower sampling rate), we compute the angle backward method of
$\varphi_{X}$ like:

$\frac{\delta\varphi}{\delta X}=\delta
X\cdot(\frac{-\Im\\{X\\}}{|X_{h}|^{2}},\frac{\Re\\{X\\}}{|X_{h}|^{2}})\text{,}$
(6)

where $\Re\\{X\\}$ and $\Im\\{X\\}$ represent real and imaginary part of
spectrogram $X$ and $|X_{h}|^{2}=\max(\Re\\{X\\}^{2}+\Im\\{X\\}^{2},1e^{-12})$
is the hardened squared magnitude to avoid by $0$ division.

$240$$480$$720$$960$$1200$$1440$$8.8$$9$$9.2$$9.4$$9.6$$9.8$FFT sizeSI-SDR
[$\mathrm{dB}$]SNR
0$240$$480$$720$$960$$1200$$1440$$12.4$$12.6$$12.8$$13$$13.2$$13.4$FFT sizeSNR
5$240$$480$$720$$960$$1200$$1440$$16$$16.2$$16.4$$16.6$FFT sizeSNR 10DFCRM
Fig. 3: Comparison of Deep Filtering (DF) and conventional complex ratio masks
(CRM) over multiple FFT sizes corresponding to $5$ to
$30\text{\,}\mathrm{ms}$.

As an additional loss term, we force the DF component to only enhance periodic
parts of the signal. The motivation is as follows. DF does not provide any
benefit over ERB gains for noise-only sections. DF may even cause artifacts by
modeling periodic noises like engine or babble noise which is most noticeable
in attenuation limited models. Furthermore, DF does not provide any benefit
for speech with only stochastic components like fricatives or plosives.
Assuming, that these sections contain most of the energy in higher
frequencies, we compute the local SNR for frequencies below $f_{\text{DF}}$.
Therefore, $\mathcal{L}_{\alpha}$ is given by

$\mathcal{L}_{\alpha}=\sum_{k}||\alpha\cdot\mathbbm{1}_{\text{LSNR}<$-10\text{\,}\mathrm{dB}$}||^{2}+\sum_{k}||(1-\alpha)\cdot\mathbbm{1}_{\text{LSNR}>$-5\text{\,}\mathrm{dB}$}||^{2}\text{,}$
(7)

where $\mathbbm{1}_{\text{LSNR}<$-10\text{\,}\mathrm{dB}$}$ is the
characteristic function with value $1$ if the local SNR (LSNR) is smaller than
$-10\text{\,}\mathrm{dB}$ and
$\mathbbm{1}_{\text{LSNR}>$-5\text{\,}\mathrm{dB}$}$ $1$ if LSNR is greater
than $-5\text{\,}\mathrm{dB}$. The LSNR is computed in STFT domain up to a
frequency of $f_{\text{DF}}$ over $20\text{\,}\mathrm{ms}$ windows. This
ensures that DF is only applied in segments containing significant amount of
speech energy in low frequencies. The combined loss is given by

$\mathcal{L}=\lambda_{spec}\cdot\mathcal{L}_{spec}(Y,S)+\lambda_{\alpha}\cdot\mathcal{L}_{\alpha}\text{.}$
(8)

##  3 Experiments

###  3.1 Training setup

We train our models based on the deep noise suppression (DNS) challenge
dataset [10] containing over $750\text{\,}\mathrm{h}$ of full-band clean
speech and $180\text{\,}\mathrm{h}$ of various noise types. We oversample
included high-quality speech datasets, VCTK and PTDB, by a factor of 10.
Additionally to the provided RIRs which are sampled at
$16\text{\,}\mathrm{kHz}$, we simulate another $10\,000$ RIRs at
$48\text{\,}\mathrm{kHz}$ using the image source model [17] with RT60s of
$0.05$ to $1.00\text{\,}\mathrm{s}$. The VCTK and PTDB datasets are split on
speaker level ensuring no overlap with the VCTK test set [18], the remaining
dataset is split at signal level into train/validation/test
($70$/$15$/$15\text{\,}\mathrm{\char 37\relax}$). Early stopping is applied
based on the validation loss, results are reported on the test set. The
VCTK/DEMAND test set [18] is used to compare DeepFilterNet to related work.

All experiments use full-band signals with a sampling rate of
$48\text{\,}\mathrm{kHz}$. We take $N_{\text{ERB}}=32$,
$f_{\text{DF}}=$5\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$, DF order of $N=5$,
and a look-ahead of $l_{\text{DF}}=1$ and $l_{\text{DNN}}=2$ for the
convolutions. We train our models on $3\text{\,}\mathrm{s}$ samples and a
batch size of $32$ for $30$ epochs using an Adam optimizer with an initial
learning rate of $1\text{\times}{10}^{-3}$. The learning rate is decayed by a
factor of $0.9$ every $3$ epochs. Loss parameters are $\lambda_{spec}=1$ and
$\lambda_{\alpha}=0.05$. The framework source code can be obtained at
https://github.com/Rikorose/DeepFilterNet.

###  3.2 Results

We evaluate our framework over multiple FFT sizes and compare the performance
of DF and CRMs based on the scale-invariant signal-distortion-ratio (SI-SDR)
[19]. CRM is just a special case of DF, where order $N=1$ and look-ahead
$l=0$. The DNN look-ahead remains the same for the CRM models.

Fig. 4: Sample from the VCTK test set. Noisy (a), CRM (b), DF (c)
($N_{\text{FFT}}=960$ for CRM and DF).

Fig. 3 shows that DF outperforms CRM over all FFT sizes corresponding to
$5\text{\,}\mathrm{ms}$ to $30\text{\,}\mathrm{ms}$. Limited by the frequency
resolution, the performance of CRMs drops for FFT window sizes
$\leq$20\text{\,}\mathrm{ms}$$. On the other hand, the relatively constant
performance of DF drops around $30\text{\,}\mathrm{ms}$ due to a smaller
amount of correlation in neighboring frames. Increasing the FFT overlap to
$75\text{\,}\mathrm{\char 37\relax}$ results in a slightly better performance
for both, DF and CRM ($+$0.6\text{\,}\mathrm{dB}$$ SI-SNR for input SNR 0).
This performance increase can be explained by a higher intra-frame correlation
as well as the DNN having twice as many steps to update the RNN hidden states
at the cost of doubling the computational complexity. Fig. 4 shows a
qualitative example to demonstrate the capabilities of DF to reconstruct
speech harmonics that are indistinguishable in the noisy spectrogram.

Table 1: Objective results on VCTK/DEMAND test set. Unreported values of
related work are indicated as “-”.

Model |  Params $[\text{M}]$ |  MACS $[\text{G}]$ |  WB-PESQ $[\text{MOS}]$ |  SI-SDR $[\text{dB}]$
---|---|---|---|---
Noisy | - | - | $1.97$ | $8.41$
PercepNet [2] | $8.0$ | $0.8$ | $2.73$ | -
DCCRN [11] | $3.7$ | $14.36$ | $2.68$ | -
DCCRN+ [7] | $3.3$ | - | $2.84$ | -
DeepFilterNet | $1.778$ | $0.3482$ | $2.81$ | $16.63$
w/o stage 2 | $0.885$ | $0.2506$ | $2.57$ | $13.811\,23$

We compare DeepFilterNet with $N_{\text{FFT}}=960$ ($20\text{\,}\mathrm{ms}$)
to related work like PercepNet [2] which uses a similar perceptual approach as
well as DCCRN+ [7] which also utilizes deep filtering. We assess speech
enhancement quality using WB-PESQ [20] and compare computational complexity in
multiply-and-accumulate per second (MACS). Table 1 shows that DeepFilterNet
outperforms PercepNet and performs on par with DCCRN+ while having a much
lower computational complexity making DeepFilterNet viable for real-time
usage.

##  4 Conclusion

In this work, we proposed DeepFilterNet, a low complexity speech enhancement
framework. We showed that DeepFilterNet performs on par with other algorithms,
while being computationally more efficient. This is achieved using a
perceptually motivated approach allowing to minimize the model complexity.
Moreover, we provided evidence that DF outperforms CRMs, particularly for
smaller STFT window sizes. In the future, we plan to further improve the
perceptual approach by better applying DF to periodic components of speech
e.g. using a correlation-based voiced probability.

## References

  * [1] Jean-Marc Valin,  “A hybrid DSP/deep learning approach to real-time full-band speech enhancement,”  in 2018 IEEE 20th International Workshop on Multimedia Signal Processing (MMSP). IEEE, 2018, pp. 1–5.
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