# Fury: an experimental dynamo with anisotropic electrical conductivity

T. Alboussière1, F. Plunian2 and M. Moulin3

(1Université Lyon 1, ENS de Lyon, CNRS, Laboratoire de Géologie de Lyon,
69622 Villeurbanne, France
2Université Grenoble Alpes, Université Savoie Mont Blanc, CNRS, IRD, IFSTTAR,
ISTerre,
38000 Grenoble, France
3ENS de Lyon, CNRS, Laboratoire de Physique,
69364 Lyon, France)

###### Abstract

We report measurements of dynamo action in a new experimental setup, named
Fury, based on the use of an anisotropic electrical conductivity. It consists
in a copper rotor rotating inside a copper stator, electrically connected with
a thin layer of liquid metal, galinstan. Grooves have been cut in the copper
so that, everywhere, electrical conductivity can be considered to be that of
copper along two directions while it is zero along the third one. The
configuration is efficient and dynamo action can be powered by hand. We have
also used a motor with better control, enabling us to drive the rotor at
specified velocity or torque functions of time. The structure of the
axisymmetric magnetic field produced is found to be close to the numerical
modelling using FreeFem++. The experimental dynamo behaves very nearly as
expected for a kinematic dynamo, so that the threshold dynamo velocity cannot
be exceeded, or only briefly. More mechanical power in the rotor rotation
leads to an increase in the magnetic field intensity, the magnetic energy
being proportional to the extra mechanical power beyond threshold. In the
transient following a step increase of torque, magnetic and angular velocity
oscillations have been observed and explained.

##  1  Previous dynamos

The notion ”dynamo” can refer to different machines and was historically used
to designate a device transforming mechanical energy into continuous electric
current. Faraday’s dynamo was the first [1]: a disk rotating in the magnetic
field generated by a permanent magnet leads to an electric potential
difference between its centre and rim and to an electric current when they are
connected. This fundamental discovery - induction - was then used in more
powerful machines to produce electricity from mechanical energy (the names of
Pixii in France, Pacinotti in Italia, Woolrich in England are associated with
such machines). However, they needed many heavy magnets to produce a modest
electric power. This was at the origin of the concept of ”sef-exciting
dynamos”. The idea was to reinforce (or excite) the magnetic field of the
permananet magnets by diverting a fraction of the electric current produced by
the machine and wind it around the magnets, thus becoming electro-magnets.
Ànyos Jedlik, around 1856 in Györ in the kingdom of Hungary (at the time),
seems to have been the first inventor of such a machine, but his work remained
unknown for a long time. He was followed by Hjorth in Danemark, Siemens in
Germany, Gramme in France, Heatstone and Wilde [2] in England. Note, however,
that the self-exciting dynamos needed some residual magnetization in their
iron parts when they were started: initially the motion of the rotor in the
residual magnetic field produces a small electric current that is used to feed
the electromagnets, so that the same rotation rate produces more current up to
the point when part of it can be used to power an external load. This is so
true that a well-known cause of failure is the lack of residual magnetism
sometimes due to a long period of inactivity. In such a case, the dynamo will
not produce electricity even at high rotor speed and the solution known as
”flashing the field” consists in having an external generator, or a battery,
discharge electricity in the windings of the electromagnets to restore
residual magnetism. Even now, this solution can be found on the web when a car
alternator does not work. Another hint of the importance of residual
magnetization comes from visits of electricity museums: even after many
decades of inactivity, all these self-exciting dynamos have a magnetic field
of the order of 1000 Gauss, as can be measured by anyone with a smartphone and
the excellent application PhyPhox [3] giving access to all sensors
(acceleration, sound, pressure, magnetic field) present in these mobile
devices.

Here, by ”dynamo”, we mean a device or a natural object producing a magnetic
field from the motion of electrically conducting materials, in the absence of
magnetized materials (or materials with magnetic properties in the strictest
acceptance). Many planets and all stars have a self-generated magnetic field,
whereas their internal temperature is above the Curie point, precluding any
magnetic effect. In this context, the corresponding concept of self-exciting
dynamo of Bullard [4] differs definitely from the historical definition
mentioned above. Self-excitation now refers primarily to the capacity of the
dynamo to operate in the absence of any initial magnetic field. We expect –
and this is true for Bullard dynamo – that an instability threshold is reached
when the rotation rate is large enough, above which the magnetic field and
electric current will jointly grow simultaneously exponentially (initially).
This corresponds to a supercritical bifurcation, which is the case in most
self-exciting dynamo models in the modern geophysical and astrophysical sense.
In contrast, we formulate the hypothesis that the historical self-exciting
dynamo machines of the 19th century were probably dynamos in a subcritical
regime, with an instability threshold above the usual range of nominal
rotation rates. Alternatively, their different behaviour could be due to the
non-linearities: a residual (remanent) magnetic field is a positive non-
linearity when it is present, however contact resistance at sliprings
constitutes a very negative non-linearity for dynamo action. Magnetic
saturation is also a non-linear effect but should not impact the starting
phase of dynamos.

Concerning modern experimental realizations of self-exciting dynamos, with
solid or liquid electrical conductors, there have been four self-generated
experimental dynamos published in the literature. One is a ‘solid’ dynamo [5,
6], the other three have used liquid sodium [7, 8, 9]. Soon after the first
theoretical dynamo model [10], Lowes and Wilkinson [5] made an experimental
version of it, consisting in two cylindrical electrically conducting rotors,
rotating in cylindrical bores in a solid sphere (or a cube in the second
version) and in electrical contact through a film of liquid metal (mercury).
Both rotors and the sphere have been made of soft iron to increase the
magnetic permeability by a factor $250$ ($150$ in the second version), in
order to lower the dynamo threshold in terms of angular velocities. Dynamo
action was reached with approximatively $200$ W of mechanical energy. It took
time and effort before fluid dynamos were realized. In 2000, the experiments
in Riga [7] and Karlsruhe [8] showed that dynamo action was possible with
liquid sodium. In Riga, a flow inspired from the Ponomarenko model [11] was
set up: a turbine propelled the fluid with strong rotation in a pipe, the
fluid was then channelled back around the pipe while its rotation was stopped
using baffles [7]. Dynamo action was obtained with $100$ kW and $1.5$ cubic
meter of sodium. In Karlsruhe, an arrangement of helical pipes produced an
array of helical flows of the same helicity sign. This is an approximation for
the G.O. Robert’s model [12]. With $1.6$ m3 of sodium and about 500 kW, a
stationary magnetic field was produced [8]. Finally, in Cadarache, the VKS
experiment produced a dynamo effect in a cylinder with two counter-rotating
discs at each end [9]. This arrangement can be seen as a realization of a
Taylor-green vortex, although the flow becomes highly unsteady and turbulent
for a moderate rotation rate of the discs. The discs had to be made of iron in
order to reach the dynamo threshold. This setup required 160 liters of sodium
and $150$ kW for dynamo action.

With the more restrictive definition excluding magnetic properties, only two
of the four experiments are dynamos: Riga and Karlsruhe dynamos. Their mean
flow is well-constrained and their dynamo threshold was predicted correctly
from the kinematic dynamo analyses. Of course, above the threshold, the flow
was altered by the magnetic field and a non linear dynamo regime was observed
[13, 14], involving a change in the structure of the flows. On the contrary,
the Cadarache dynamo is less constrained in terms of flow structure. The
dynamo threshold was not predicted from the mean flow and a larger variety of
non linear behaviours have been observed [15]. In all three cases, the
mechanism of saturation above the threshold is not completely understood [7,
16, 17]. This is partly due to the fact that the setups have a limited amount
of mechanical power and cannot reach velocities much above threshold, and
partly due to the difficulty to detect very subtle changes in the velocity
field.

##  2  Fury: from copper and grooves

Figure 1: Grooves are cut in a copper rotor and stator.

From our previous theoretical investigations on anisotropic electrical
conductivity [18, 19], we know that a dynamo can have a planar velocity field
and/or produce an axisymmetric magnetic field. We have built an experimental
setup which is a rotor inside a stator (Fig. 1). The final external diameter
of the stator is 170 mm, its length is 209 mm and its internal diameter is 101
mm. The rotor that fits inside has the same length and an external diameter of
100 mm, hence the gap between rotor and stator is $0.5$ mm wide. It has an
internal diameter of 30 mm so that a (non-magnetic) steel axis can be fitted
in. The grade of copper for rotor and stator is CUA1 H12, with a guaranteed
electrical conductivity $\sigma$ between $100$ and $102.5\ \%$ IACS, i.e.
$\sigma=5.8\ 10^{7}\,\Omega^{-1}$m-1 to $\sigma=5.95\ 10^{7}\,\Omega^{-1}$m-1.

Grooves have been cut in the rotor and stator, extending across the whole
length and forming arcs of logarithmic spirals in the perpendicular cross-
section: the direction along the groove makes an angle $62\ ^{\circ}$ with the
radial direction at all points. In the stator 35 equally-spaced spirals have
been cut and 20 in the rotor. The grooves have been cut with the technique of
electro-erosion, each groove has a width of $0.33$ mm. The grooves have then
been filled with a polyimide film and epoxy resin for electric insulation and
mechanical strength. As can be seen on the left-hand side of Fig. 1, the
internal diameter of the stator was cut 20 mm smaller initially to ensure
rigidity during the cutting of the grooves (similarly, the external diameter
of the rotor was larger by 20 mm). After the epoxy resin hardened, the stator
internal and rotor external diameters were cut to the final dimensions,
leaving a radial distance of 2 mm to the end of the grooves. This ensures a
good precision and small roughness of the surfaces involved in the
differential rotation. This surfaces (external of the rotor and internal of
the stator) were then silver plated (a few microns) for chemical protection of
the copper and to ensure good electrical contact with galinstan.

Figure 2: Front view of Fury (left) and side view (right).

The assembly of the setup is quite simple, with two flanges at the top and
bottom tied with threaded rods, ensuring mechanical cohesion with ball
bearings inserted to hold the axis of the rotor. The 0.5 mm gap between rotor
and stator was filled with galinstan, an alloy of gallium, indium and tin,
liquid at room temperature. In Fig. 2 showing the final setup, the copper
stator is still visible and the rotor axis is connected through gears and belt
to a handle and to an electric motor. Manual or motor operations are possible.
One turn of the handle (resp. electric motor) corresponds to 12 (resp. 2)
turns of the rotor axis.

Measurements are made of the magnetic field from Hall probes on the external
surface of the stator: each Hall probes has a small red LED, see Fig. 2. The
electric variator of the motor provides measurement of angular velocity and
torque.

##  3  Fury: dynamo results

The dual operation, manual or motor-powered, has two distinct objectives.
Powering the experiment by hand is interesting to have a direct physical
feeling of dynamo action: one can feel the mechanical resistance due to
Lorentz forces when the magnetic field is generated. However, the conditions
of operation are not well controlled: the velocity is not constant nor the
torque and is operator-dependent. Using a motor brings the possibility to
follow almost any prescribed function of velocity or torque, within the limit
of its maximal power.

###  3.1 The first successful human-powered dynamo run

Figure 3: First dynamo action with Fury, human-powered with the handle.

The very first run of dynamo action with Fury took place on the 23rd of March
2021. In Fig. 3, one can see the signal of the two Hall probes (at the time)
stuck on the outside of the stator. Those two signals are proportional to each
other, although the two probes were at different places and had different
orientations. This hints at the fact that the self-generated magnetic field is
made of a single mode. Only its magnitude can change with time. Let us analyze
the time evolution of this run. The operator ran the handle until it reached
about 2 Hz, at a time between $t=2$ s and $t=3$ s. The magnetic field started
to depart from its initial value (the ambient magnetic field). At the time
nearly $t=6$ s it peaked to a maximum. In the same time, the operator had
experienced an extra resistance from the Lorentz forces, which led to a
decrease of the rotation rate and irregular decay of the magnitude of the
magnetic field. At $t=13$ s, the operator recovered a little and put more
effort for a while, then stopped at $t=22$ s or so.

Another information from the signals in Fig. 3 is the presence of a 2 Hz
oscillation. This is precisely the frequency of the handle rotation. The
interpretation is that the input of energy is not constant over one rotation
of the handle, which seems quite plausible. Note that the ratio between handle
and rotor rotation is 12 (a factor 6 in the gears between the horizontal axes,
see Fig. 2, and 2 in the conical gears on top of the rotor). This implies that
the threshold for dynamo action corresponds to an angular velocity around $24$
Hz for the rotor.

As will be seen later, reaching 2 Hz with the handle requires an input of
$230$ W of mechanical work to overcome the resistance of the galinstan film
(main contribution), ball bearings and gears. This is quite an effort with a
handle, but all operators over a representative sample of ten colleagues (two
women, eight men) could do it. The good news is that the effort does not need
to be sustained for a long time. Within 5 s, it is usually possible to start
the dynamo.

###  3.2 More results with a motor

Most results have been obtained with an electric motor of maximal power 750 W
(about one horse power). This is just a little bit more than what the average
human can do, but it can sustain it for longer and can adjust the effort very
precisely.

####  3.2.1 Marginal eigenvector

Figure 4: Eigenmode at marginal stability from finite element modelling
(FreeFem++) on the left, profile of $B_{r}$ (FreeFem++ and measured on Fury)
on the right.

Using FreeFem++ [20], we have built a numerical model of Fury. This model is
assumed to be axisymmetrical as the magnetic field observed in the experiment
is itself axisymmetric. Electrical conductivity is taken to be anisotropic in
the coper parts. This means that we have not attempted to model each copper
blade, but have used a mesoscale anisotropic modelling instead. The finite-
element model takes into account the thin galinstan gap with its isotropic
electrical conductivity about $16.8$ times smaller than that of copper. The
induction equation is considered, with a solid rotation of the rotor, and a
linear profile in the galinstan gap to reach a zero velocity in the stator.
Outside the conducting domain, the magnetic field obeys a harmonic equation.

We have computed the marginal eigenvalue and eigenvector at the lowest
possible rotation rate. This (axisymmetric) eigenmode is shown on Fig. 4: the
mesh used for the finite element calculations is shown in the background, the
toroidal isovalues are shown with colour level in the conducting domain
(rotor-galinstan-stator) and some field lines of the poloidal component are
drawn. On the right-hand side of Fig. 4, the cylindrical radial component of
the magnetic field is plotted along the profile A-B on the outside surface of
the stator (blue curve). The black dots are measurements of the same profile
on Fury. The grey region around each point corresponds to the uncertainty of
position ($\pm\ 2$ mm and magnetic field $\pm 5$ % of the maximum value,
mainly due to orientation errors of Hall sensors). These points correspond to
different runs and to different azimuthal angles. Considering the accuracy of
the measurements, there is no sign of departure from axisymmetry. The
agreement between the calculated profile and the measurements is quite good:
it seems that the experimental values are slightly shifted to the bottom
compared to the calculated odd profile.

Let us now consider the value of the magnetic Reynolds number corresponding to
marginal stability. Let us define the magnetic Reynolds number as
$Rm=\mu\sigma UR$, with $\sigma$ the electrical conductivity of copper,
$\mu=4\pi\,10^{-7}$ H m-1 the magnetic permeability of copper (or of vacuum),
$R=0.05$ m the radius of the rotor and $U=\omega R$ the maximal tangential
velocity of the rotor ($\omega$ is the angular velocity). A rotation rate of
the rotor of $24$ Hz corresponds to an angular velocity of $150$ rad s-1.
However, with the motor and more precise velocity measurements, we could get
dynamo action for $\omega\simeq 147$ rad s-1 (red curve in Fig. 5). This leads
to a critical magnetic Reynolds number of $Rm_{c}\simeq 26.8$. The dynamo
threshold obtained with FreeFem++ is $Rm_{c}\simeq 23.1$. The difference can
be explained firstly by the presence of a finite number of discrete grooves in
the experiment, compared to a uniform anisotropy in FreeFem++ calculations.
Secondly, the grooves correspond to a removal of about 5 % of the total mass
of copper. This would already explain an increase by 5 % of the threshold.
There can also be a contact electrical resistance between the copper and the
film of galinstan. For completeness, the analytical threshold is $Rm_{c}\simeq
14.61$ with an infinitely long configuration and a stator of infinite radius
(and in the absence of a layer liquid galinstan of electrical conductivity
$16.8$ times smaller than that of copper) [19].

In accordance with theoretical investigations [18, 19], when the imposed
rotation has the opposite direction, dynamo action never happens: all
eigenvalues from FreeFem++ have a negative real part. Similarly, no dynamo
action can be seen in Fury with the negative rotation.

####  3.2.2 Torque, power and magnetic energy

Figure 5: Angular velocity (bright colours) and magnetic field (faded colours)
versus torque, for two runs in the negative direction and three runs in the
positive (dynamo) direction.

Increasing linearly the imposed torque from zero to its maximum value, we have
been able to plot the resulting curves in the diagram shown in Fig. 5. Three
runs in the “positive” direction of rotation are shown, and two in the reverse
“negative” direction (see Fig. 1 where the white arrow defines positive
rotation). In agreement with the mathematical analysis [18, 19] dynamo action
is possible only with the positive direction of rotation. At low rotation
rate, below the dynamo threshold, all curves follow the same path. The
relationship between the angular velocity and the torque is determined by
hydrodynamical properties of the galinstan alloy with additional torque
contributions from the ball bearings (outside on the visible axes on Fig. 2,
inside to support the axis of the rotor as shown in Fig. 1) and gears. The
dashed black curve (original model) in Fig. 5 corresponds to the turbulent
model by Orlandi et al. [21]111A fit for the friction coefficient $c_{f}$ is
proposed for the turbulent Couette flow in terms of the Reynolds number $Re$,
based on various experimental results $c_{f}=0.056Re^{1/4}$. The Reynolds
number is $Re=Uh/\nu$, where $U$ is half the velocity difference across a
layer of thickness $h$ of a fluid of kinematic viscosity $\nu$. The resulting
shear stress is $\tau_{w}=\frac{1}{2}c_{f}\rho U^{2}$ by definition of the
friction coefficient $c_{f}$.. Note that at a typical rotation rate of 24 Hz,
the Reynolds number of the galinstan in the gap is approximately $5000$ (plane
Couette flow is generally turbulent above Reynolds number $10^{3}$). The
continuous black curve (shifted model) is derived from the first one with a
multiplication by $1.3$ and with the addition of a constant resistive torque
of $0.15$ Nm, for a best fit of the data. No attempt was made to measure the
fluctuations of the torque (for an imposed velocity) or of the velocity (for
an imposed torque) that would be due to hydrodynamic turbulence in the gap of
galinstan. We expect those fluctuations to produce a small average effect
(over the whole gap) and to be impossible to measure from the variator of the
electric motor.

In the positive direction of rotation, above $150$ rad s-1, increasing further
the torque does not result in an increase in angular velocity. The self-
generated dynamo behaves pretty well like a kinematic dynamo: it is virtually
impossible to exceed the threshold angular velocity ($150$ rad s-1 corresponds
to $150/(2\pi)\simeq 24$ Hz). Increasing the torque leads to an increase in
the self-generated magnetic field (fade colored curves in Fig. 5). On the
contrary, in the negative direction of rotation, the same “hydrodynamical”
curve is followed as for lower velocities and no magnetic field is generated.

In the dynamo regime (positive direction), a careful examination reveals that
the angular velocity increases very slightly when the torque is increased
above the threshold. However, most of this increase is due to the effect of
temperature. As the system dissipates energy (by turbulence and viscosity at
small scale, by Joule heating when the dynamo is running) the copper of the
rotor and stator gets heated hence its electrical conductivity decreases. In
order to maintain the kinematic dynamo at its critical magnetic Reynolds
number, a decrease in electrical conductivity has to be compensated by a
similar relative increase in angular velocity. In Fig. 5, the three dynamo
runs, 1 to 3, were obtained while the temperature of the outside surface of
the stator was 20.2, 22.4 and 25.6∘C respectively. With a linear thermal
coefficient of $4.3\ 10^{-3}$ K-1 for copper, the difference in temperature of
5.4 K leads to a relative variation of 2.3 % in electrical conductivity. The
relative difference in angular velocity between run 1 and run 3 is about 5 %.
We do not have a definite explanation for this mismatch: one possibility is
that the temperature difference in the interior of the setup was larger than
that measured on the external surface.

Anticipating on the next section 3.2.3, we can see in Fig. 7 that the magnetic
energy grows linearly with the input mechanical power injected in the setup,
when the dynamo threshold is exceeded. In that regime, any extra power input
is dissipated by an extra Joule dissipation. This is coherent from the point
of view of magnetohydrodynamics, as Joule dissipation scales usually
proportionally to the square of the magnetic field for a given velocity field,
in the magnetic diffusive regime.

####  3.2.3 Torque steps, torsional oscillations

Figure 6: Angular velocity, torque, magnetic field in a single run with
imposed steps of torque in the positive direction of rotation.

When the experimental setup Fury is run with an electric motor, we can impose
a torque and step changes over a very short duration of less than $0.1$ s. In
Fig. 6, we show the time evolution of an experiment where we have imposed 12
steps of increasing torque (labeled 1 to 12) and the reverse 12 steps of
deceasing torque, in black continuous line. Each step duration is 10 s and the
maximum torque was 4 Nm. As a result of these torque changes, angular velocity
(blue line) increases significantly for the 5 first steps, then remains
globally at the same level and finally goes down in the 5 last steps. After
step 5 and until just before step 20, self-exciting dynamo is at work, as
shown by the value of the magnetic field (red curve), which maintains the
angular velocity close to the dynamo threshold. The dynamo has actually
perhaps started after the 4th step, as we can see a low level of magnetic
field already. This is an indication that the dynamo threshold is not as sharp
as expected for a kinematic dynamo, within a range of angular velocities, and
we shall later explore this phenomenon.

Figure 7: Square of the measured magnetic field $B_{r}^{2}$ as a function of
the mechanical power input $\mathcal{P}$ during the run shown in Fig. 6.

In Fig. 7, the same experiment is represented in terms of mechanical energy
input from the motor and square of the magnetic component $B_{r}$ on the
external surface of the stator (position $z=3.7$ cm along the profile
$\tt{A}$-$\tt{B}$ in Fig. 4). In this diagram, the time evolution of the run
is plotted in red while the torque is increased, then in blue for the second
part of decreasing torque. In the dynamo regime, each step increase of the
torque leads to an increase of mechanical power, followed by an increase of
the magnetic energy. There is a transient phase of oscillations towards a
final stationary regime where the path follows a converging spiral. The
oscillations of the magnetic field can also be seen in Fig. 6 after each
torque step. When the torque steps are in the decreasing phase (blue line), we
observe a similar trend: the mechanical power goes down quickly and a phase of
oscillating magnetic field follows with a final lower level of magnetic field.
As mentioned above, the stationary points are well aligned along a straight
line and we find a good agreement with the expression
$B_{r}^{2}=0.0204(\mathcal{P}-231)$ above the dynamo threshold ($B_{r}$ in mT
and the mechanical power input $\mathcal{P}$ in Watt).

Figure 8: Stack of magnetic field oscillations after steps of torque (see Fig.
6 for the numbering of steps) with the dimensionless time scale
$tB_{r}/(R\sqrt{\rho\mu})$. The inset (step 9) shows that the corresponding
oscillations of angular velocity are in phase quadrature relative to magnetic
oscillations.

In Fig. 6, it seems that the oscillations get faster as the magnetic field
increases. We plot the oscillations following steps 5 to 12 in Fig. 8, using a
dimensionless timescale proportional to the intensity of the magnetic field.
Specifically, we use a timescale relevant to Alfvén waves,
$tB_{r}/(R\sqrt{\rho\mu})$ with $R=0.05$ m the radius of the rotor,
$\rho=8960$ kg m-3 copper density, $\mu=4\,\pi\,10^{-7}$ H m-1 magnetic
permeability and $B_{r}$ (in Tesla) the limit radial magnetic field after each
step. The oscillations of $B_{r}$ superimpose nicely in this dimensionless
timescale, and their period is of order one (close to $0.5$). In the inset of
Fig. 8, we have plotted again the magnetic signal after one of the steps (step
9), along with the signal of angular velocity fluctuations. Those signals are
in phase quadrature relative to each other, similarly to Alfvén waves [22].
Because of the axisymmetry of the configuration and magnetic mode, these
oscillations seem to be of the same nature of a particular kind of Alfvén
waves, i.e. torsional oscillations. Such oscillations have been detected in
the Earth’s core [23] from geomagnetic data and also in an experiment with
liquid sodium [24].

However, they are not Alfvén waves, nor associated torsional oscillations.
Alfvén waves develop when kinematic and magnetic diffusion are negligible.
Here, even with the highest magnetic field, the period of the oscillations is
just under one second, while magnetic diffusion through 5 cm of copper is
about 0.02 s. The oscillations are related to the dynamo mechanism itself:
after a step, the system is pushed initially in the supercritical domain
$Rm>Rm_{c}$ leading to a positive growth rate of the (global diffusive)
magnetic field. The resulting increase of dissipative Lorentz forces brings
the velocity down and the setup enters the subcritical domain $Rm<Rm_{c}$ with
a deficit in Lorentz forces compared to the applied torque. The rotation
increases and this is the start of a new period of the oscillations. The
dynamic equation for the rotor rotation $\omega$ can be written approximately
as follows

$\rho R^{2}\frac{\mathrm{d}\omega}{\mathrm{d}t}\simeq-\sigma\omega
R^{2}B_{r}^{2}-\mathcal{T}_{v}+\mathcal{T},$ (1)

where $-\sigma\omega R^{2}B_{r}^{2}$ is the Lorentz torque, $-\mathcal{T}_{v}$
the viscous torque and $\mathcal{T}$ the constant torque provided by the
motor, per unit volume. Assuming the viscous torque to be constant, taking the
time derivative leads to

$\rho R^{2}\frac{\mathrm{d}^{2}\omega}{\mathrm{d}t^{2}}\simeq-2\sigma\omega
R^{2}B_{r}\frac{\mathrm{d}B_{r}}{\mathrm{d}t}-\sigma
R^{2}B_{r}^{2}\frac{\mathrm{d}\omega}{\mathrm{d}t}.$ (2)

From our previous studies of the anisotropic dynamo, see equation (C17) of
[18] and (4.18) of [25], we know that the growth rate of the magnetic field is
proportional to $\omega-\omega_{0}$ (with a constant of order unity) near the
dynamo threshold $\omega_{0}$ for angular velocity. Substituting
$\mathrm{d}B_{r}/\mathrm{d}t=(\omega-\omega_{0})B_{r}$ in equation (2) and
dividing by $\rho R^{2}$, one obtains

$\frac{\mathrm{d}^{2}\left(\omega-\omega_{0}\right)}{\mathrm{d}t^{2}}\simeq-2\frac{\sigma
B^{2}_{r}}{\rho}\omega_{0}(\omega-\omega_{0})-\frac{\sigma
B^{2}_{r}}{\rho}\frac{\mathrm{d}(\omega-\omega_{0})}{\mathrm{d}t}.$ (3)

The last term on the right-hand side is negligible provided the timescale of
the solution is longer than $\omega_{0}^{-1}$. Making this assumption leads to
a simple harmonic equation, of associated timescale
$\tau=\sqrt{\rho/\left(\sigma\omega_{0}B_{r}^{2}\right)}$ which is the
geometric average of the (short) turnover timescale $\omega_{0}^{-1}$ and the
(long) Joule damping time $\rho/(\sigma B_{r}^{2})$. It is longer than
$\omega_{0}^{-1}$, hence our assumption is justified: it is also coherent with
the smaller relative oscillations of angular velocity compared to those of the
magnetic field in Fig. 6 after each step. This timescale $\tau$ can be
expressed using $Rm_{c}=\mu\sigma\omega_{0}R^{2}$ under the form

$\tau=\frac{1}{\sqrt{Rm_{c}}}\frac{R}{V_{A}},$ (4)

where $V_{A}=B_{r}/\sqrt{\rho\mu}$ is the Alfvén velocity. This timescale
$\tau$ is the timescale we have used to plot Fig. 8, without the constant
$1/\sqrt{Rm_{c}}\simeq 1/5$. Even though the physics is different from that of
Alfvén waves, a similar timescale arises from the coupling between rotor
inertia and dynamo growth rate.

##  4  Nature of the bifurcation

We plot in the same figure 9 different results concerning the dynamo
bifurcation, as the measured magnetic field as a function of the torque
imposed. This only concerns the ’dynamo’ direction of rotation, since no
magnetic field is generated in the other direction as expected from the
theory. We replot the magnetic curves shown in Fig. 5, obtained with a (slow)
continuous increase of the imposed torque. Then we also plot the steady value
of the magnetic field reached after the transient period following each torque
step shown in Fig. 6. We observe that all measurements (continuous or steady)
fall on a similar curve depicting an imperfect supercritical bifurcation,
starting at a torque value of $1.5$ Nm with an exponent close to $1/2$ for the
magnetic field. This bifurcation diagram is coherent with the evolution of
magnetic energy as a function of mechanical power shown in Fig. 7, considering
that the velocity remains constant near $150$ rad s-1 above the dynamo
threshold.

No clear hysteresis has been observed: increasing and decreasing steps of
torque lead to points falling on the same curve. This applies also to the
imperfect part of the bifurcation. Possible reasons for the imperfection could
be related to the Earth ambiant magnetic field, or to small magnetic metallic
pieces (bolts, ball bearings, gears) that have not been eliminated completely.
Another observation is probably associated with the imperfect bifurcation: the
magnetic field produced has always the same orientation whereas the opposite
orientation should be as likely. The imperfect bifurcation is thought to drive
the eigenmode in a definite orientation, not random.

Concerning the critical exponent $1/2$ of the increase of the magnetic field
above the critical torque threshold, it is directly related to the fate of the
mechanical energy dissipated in the system, as discussed previously. This
dynamo behaves very closely to a kinematic dynamo: the solid parts possess no
freedom to deform and change the induced magnetic field, whereas the liquid
part is restricted to a very narrow gap $0.5$ mm so that, even if changes
occur they will have a very limited impact. This implies that the velocity
does not change above the critical torque. Hence (turbulent) dissipation
remains constant in the liquid gap222Turbulent friction (and dissipation) in
the galinstan layer is not affected by the magnetic field unless
$c_{f}Re^{2}/Ha^{4}<1$ according to [26]. This would require the Hartmann
number $Ha=\sqrt{\sigma/(\rho\nu)}Bh$ to be more than 100 times larger than it
is estimated with a magnetic field of 3 mT.. The extra mechanical power
injected above threshold is entirely dissipated as Joule dissipation. As the
form of the magnetic field remains the same, the magnetic field is
proportional to the electric currents, which are proportional to the square
root of Joule dissipation. With a nearly constant velocity, mechanical power
is proportional to the torque, hence this explains why the magnetic field
intensity grows like the square-root of the torque above threshold.

Figure 9: Measured magnetic field $B_{r}$ as a function of the torque input.
The continuous coloured lines correspond to continuous variations of the
torque shown in Fig. 5 and the stars to the imposed torque steps shown in Fig.
6, red (resp. blue) stars corresponding to increasing (resp. decreasing)
torque steps.

On the contrary, in Karlsruhe, Riga and Cadarache dynamos, the mean velocity
of the flow can be increased above the threshold value for dynamo action.
Also, the extra power above threshold goes mostly into hydrodynamical
turbulent dissipation while a smaller part is dissipated through Joule effect.
There have been arguments to suggest a scaling such as $B^{2}\sim V-V_{c}$
[13, 27], where $B$ is the magnetic field, $V$ the mean velocity and $V_{c}$
the dynamo threshold velocity. These arguments can be based on dimensional
analysis or on the scaling of the Lorentz forces on the mean flow. Combined
with a turbulent dissipative scaling $\mathcal{P}\sim V^{3}$ for the required
mechanical power $\mathcal{P}$, the resulting scaling for the magnetic field
is $B\sim\sqrt{\mathcal{P}^{1/3}-V_{c}}$ which can look like anything between
$\mathcal{P}^{1/2}$ and $\mathcal{P}^{1/6}$ depending on the proximity to the
threshold.

Let us now compare the dynamo bifurcation observed in Fury to that observed in
the Lowes and Wilkinson dynamo. Both dynamos require a similar power of the
order of 200 W, but the dynamo of Lowes and Wilkinson had to be made in a
magnetic material so that its dynamo threshold could be reached. In their
first paper [5], Lowes and Wilkinson show that they could increase the
velocity of one of the cylinders above the critical velocity and that this led
to an increase of the magnetic field intensity. This is different in Fury
where the velocity cannot be made larger than the threshold velocity as the
magnetic field would grow indefinitely in amplitude. In their second paper
[6], Lowes and Wilkinson observe oscillations of the magnetic field (and of
the rotation rates of the cylinders) above dynamo threshold. Such oscillatory
regimes have been found in more recent numerical simulations of the Herzenberg
dynamo [28], but they could only be seen at a much larger magnetic Reynolds
number. Lowes and Wilkinson admit that the magnetic material (saturation and
remanence) might play a large role in the observed regimes. Their last
sentence expresses their regrets that they had to use a magnetic material: ”We
hope by further improvement of the geometry to obtain a sufficiently small
critical $R_{m}$ that will allows us to make a model from copper”.

##  5  Conclusions

We have demonstrated, with the experiment Fury, that the concept of
anisotropic conductivity could be used effectively to produce a self-generated
dynamo. Without any magnetic materials, just copper, and a small quantity of
galinstan for electrical contact, we have built a dynamo that can be human-
powered. The characteristics of the setup correspond rather well to the
finite-element calculations, in terms of dynamo threshold and eigenmode of the
magnetic field. When abrupt changes in the driving torque are imposed,
rotation and magnetic oscillations have been put in evidence, due to the
linear dynamo growth rate dependence around the threshold angular velocity.

Our measurements indicate that the magnetic field is axisymmetric. This cannot
be exactly true: at small scale (smaller than the distance between the
grooves), our setup is just made of materials of isotropic heterogeneous
conductivity. So Cowling’s theorems apply and an axisymmetric field cannot be
sustained by dynamo action. So azimuthal variations must exist but they are
small and we have not measured them yet. On a meso-scale (larger than the
distance between the groove and smaller than the setup) we model the
configuration by a material of homogeneous anisotropic electrical
conductivity. In that case, axisymmetric dynamo solutions exist. Our results
prove that this modelling is adequate for Fury.

As we have seen, small scale conductivity heretogeneity can be modelled as
electrical anisotropy in some cases, and this is true for our experiment Fury.
However, there are other cases of electrical heterogeneity that are definitely
different. For instance a uniform flow next to a solid wall with spatial
electrical conductivity modulation in the streamwise direction can lead to
dynamo action [29]. In this case, the wavelength of the modulation is
important and cannot be made zero. In the limit of zero wavelength, this would
indeed correspond exactly to an anisotropic electrical conductivity with the
flow direction as a principal axis of anisotropy. Our model of sliding plates
[18] shows that such a configuration does not produce a dynamo. In another
case of heterogeneity [30] an equivalent $\alpha$-effect is produced but this
requires velocity length-scales comparable to the length-scale over which
electrical conductivity varies so that an anisotropic conductivity would not
lead to the same results.

In the recent years, a self-exciting dynamo with spiral arms was tested by
Avalos-Zúñiga et al. [31]. In 2018 the dynamo threshold was reached and the
results presented at the MHD days meeting in Dresden (16-28 Nov 2018) and more
results have been provided to us in a private communication [32]. This dynamo
was designed according to the concept of Bullard’s dynamo disc, however it
seems that an anisotropic conductivity would be a faithful model for this
setup, too [33]. Concerning future experiments, one can imagine a hybrid
solid-liquid configuration, similar to Fury but with a larger gap of liquid
sodium, or just a solid anisotropic rotor inside a pool of liquid sodium. The
advantage of these configurations is that dynamo action would be guaranteed
from the solid anisotropic part and that fluid dynamics would interact non-
linearly with the magnetic field. Considering the difficulties encountered to
produce and study interesting non-linear dynamos, this alternative might be
valuable.

The name Fury was given to the setup because this is a dynamo that should have
the property of a dynamo growth-rate increasing without bound above the dynamo
threshold, which we called a “very fast” or “furious” regime [18, 25]. That
property is rather difficult to measure in the experiment because it would
require a lot of mechanical power (within the resistance limits of the setup)
to bring the rotor at a velocity well above the threshold velocity.

Acknowledgements: We wish to acknowledge the participation of students of
geology in Lyon, Lucie Petit, Dana Romanoz and Paul Gomez in the measurement
of the magnetic profile of the dynamo mode. Thanks are due to Andy Jackson for
suggesting the name ’Fury’ for the setup.
We thank the Programme National de Planétologie (PNP) of CNRS/INSU, co-funded
by CNES. The authors are grateful to the LABEX Lyon Institute of Origins
(ANR-10-LABX-0066) Lyon for its financial support within the Plan France 2030
of the French government operated by the National Research Agency (ANR).

Statements: TA and MM designed the setup. TA and FP caried out the
experiments. TA performed the data analysis and drafted the manuscript. All
authors read and approved the manuscript.
The authors declare that they have no competing interests.

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# FADI-AEC: Fast Score Based Diffusion Model Guided by Far-end Signal for
Acoustic Echo Cancellation

###### Abstract

Despite the potential of diffusion models in speech enhancement, their
deployment in Acoustic Echo Cancellation (AEC) has been restricted. In this
paper, we propose DI-AEC, pioneering a diffusion-based stochastic regeneration
approach dedicated to AEC. Further, we propose FADI-AEC, fast score-based
diffusion AEC framework to save computational demands, making it favorable for
edge devices. It stands out by running the score model once per frame,
achieving a significant surge in processing efficiency. Apart from that, we
introduce a novel noise generation technique where far-end signals are
utilized, incorporating both far-end and near-end signals to refine the score
model’s accuracy. We test our proposed method on the ICASSP2023 Microsoft deep
echo cancellation challenge evaluation dataset, where our method outperforms
some of the end-to-end methods and other diffusion based echo cancellation
methods.

Index Terms—  echo cancellation, diffusion model, stochastic regeneration,
predictive learning

##  1 Introduction

The importance of acoustic echo cancellation in achieving high-quality speech
in voice communication has led to the emergence of deep neural network (DNN)
based methods, such as the deep complex convolution recurrent network (DC-CRN)
[1]. The main challenges are artifacts, target speech distortion, and echo
leakage during double-talk scenarios. To address these challenges, researchers
have proposed techniques such as alignment modules [2], novel architecture [3,
4] and modified loss functions [5]. Currently, the majority of echo
cancellation models adhere to predictive methodologies, which learn a
deterministic mapping from corrupted speech to clean speech targets. However,
generative models, such as variational auto-encoders (VAEs) [6], generative
adversarial networks (GANs) [7], and diffusion approaches [8], offer a
different perspective. These models learn the target distribution and have the
capability to generate multiple valid estimates, providing a richer set of
possibilities for AEC task.

Among these, diffusion-based generative models have shown promising results in
tasks such as noise suppression. For instance, Lu et al. [9] proposed a novel
approach that incorporates the characteristics of the observed noisy speech
signal into the diffusion and reverse processes, providing a more tailored
enhancement. Similarly, Joan Serrà et al. [10] introduced a multi-resolution
conditioning network that employs score-based diffusion. Their model generates
clean speech from noisy inputs by progressively reducing the noise in a series
of steps, effectively diffusing the noise out of the signal. Lemercier et al.
[11] applied a stochastic regeneration approach, where an estimate provided by
a predictive model is used as a guide for further diffusion, refining the
enhancement process.

While there have been significant advancements in the field, the potential of
using diffusion models for echo cancellation has largely remained untapped.
One of the primary challenges is the computational intensity of existing
diffusion models, which makes their deployment in real-world production
settings a daunting task. In this paper, we introduce two novel models: DI-
AEC, which applies a diffusion-based stochastic regeneration method to echo
cancellation, and FADI-AEC, an efficient version of DI-AEC that utilizes a
fast score model. To the best of our understanding, these represent the first
echo cancellation models that harness the power of diffusion. Notably, FADI-
AEC addresses the computational hurdle by executing the score model only once
per frame, leveraging the prior state to minimize processing time. Both models
employ far-end signals to produce noise, thereby enhancing performance. By
considering both far-end and near-end signals, these models enhance accuracy,
generating high-quality samples.

Fig. 1: FADI-AEC pipeline. The predictive filter is first used to generate an
predicted estimate $\mathbf{\hat{\mathbf{s}}}(n)$ from mic signal
$\mathbf{h}(n)$. Diffusion-based generation $G_{\phi}$ is then performed by
adding Gaussian noise guided by far-end sign $\mathbf{x}(n)$ and solving the
reverse diffusion SDE. The estimated near-end speech is
$\mathbf{\tilde{s}}(n)$, which would be used in the score function in the next
frame.

##  2 Proposed method

###  2.1 Problem formulation

In a typical AEC system, the microphone signal is denoted as $\mathbf{h}(n)$.
This signal comprises two components: the near-end speech $\mathbf{s}(n)$ and
the acoustic echo $\mathbf{z}(n)$. Mathematically, this relationship is
expressed as:

$\mathbf{h}(n)=\mathbf{s}(n)+\mathbf{z}(n),$ (1)

where $n$ is the time sample index. The acoustic echo $\mathbf{z}(n)$ can be
understood as a time-delayed version of the far-end reference signal
$\mathbf{x}(n)$. This signal has traversed the echo path and might have
undergone nonlinear distortions due to the loudspeakers. The primary objective
of the AEC system is to separate the near-end speech $\mathbf{s}(n)$ from the
microphone signal $\mathbf{h}(n)$.

###  2.2 Forward and inference through reverse sampling

The stochastic forward process utilized in score-based diffusion models is
defined by the Itô Stochastic Differential Equation (SDE) [12]:

$\mathrm{d}\mathbf{s}(n)_{t}=\mathrm{f}(\mathbf{s}(n)_{t},t)\mathrm{d}t+\mathrm{g}(t)\mathrm{d}\mathbf{w}$
(2)

where $\mathbf{w}$ denotes a standard-dimensional Brownian motion, making
$\mathrm{d}\mathbf{w}$ a zero-mean Gaussian random variable with variance
proportional to $\mathrm{d}t$, pertinent for each Time-Frequency (T-F) bin.
The functions $\mathrm{f}$ and $\mathrm{g}$ represent the drift and diffusion
coefficients, respectively. The state of the process at discrete index $n$ and
continuous time $t$, where $t\in[0,T]$ is given by $\mathbf{s}(n)_{t}$, and
for clean speech, the initial condition is $\mathbf{s}(n)_{0}=\mathbf{s}(n)$.

In the reverse process of score based diffusion model, the score model is
substituted into the reverse SDE as plug-in reverse SDE [13]:

$\displaystyle\mathrm{d}\mathbf{s}(n)_{t}=$ (3)
$\displaystyle\quad\left[-\mathrm{f}\left(\mathbf{s}(n)_{t},t\right)+\mathrm{g}(t)^{2}\nabla_{\mathbf{s}(n)_{t}}\log
p_{t}\left(\mathbf{s}(n)_{t}\right)\right]\mathrm{d}t+\mathrm{g}(t)\mathrm{d}\overline{\mathbf{w}}$

where $\mathrm{d}\overline{\mathbf{w}}$ is a d-dimensional Brownian motion for
the time flowing in reverse and $\nabla_{\mathbf{s}(n)_{t}}\log
p_{t}\left(\mathbf{s}(n)_{t}\right)$ is the score function. This equation is
classified under the Ornstein-Uhlenbeck SDEs [14].

During inference, Eq. (3) is evaluated using the predictor-corrector approach
as informed by the score-matching network described later in [12]. The initial
state of the process is drawn from the distribution:

$\mathbf{s}(n)\tau\sim\mathcal{N}_{\mathbb{C}}\left(\mathbf{s}(n)_{\tau};\mathbf{h}(n),\mathbf{x}(n),\sigma^{2}(\tau)\mathbf{I}\right)$
(4)

This distribution essentially represents a near-end signal $\mathbf{h}(n)$ and
far-end signal $\mathbf{x}(n)$, to which Gaussian noise with a variance of
$\sigma^{2}(\tau)$ is added.

###  2.3 Score model with far-end guided noise

Since the speech enhancement including AEC task could be considered as
condition generation task, the conditioning is integrated into the diffusion
process by denoting the forward process where Eq. (2) yields the following
complex Gaussian distribution for the process state $\mathbf{s}(n)_{t}$, known
as the perturbation kernel [15],
$\mathcal{N}_{\mathbb{C}}\left(\mathbf{s}(n)_{t};\bm{\mu}\left(\mathbf{s}(n)_{0},\mathbf{h}(n),t\right),\sigma(t)^{2}\mathbf{I}\right)$
where the mean is $\bm{\mu}$ and variance is $\sigma(t)^{2}$. When performing
inference, one tries to solve the reverse SDE in Eq. (3). For the simple
Gaussian form of the perturbation kernel
$p_{0,t}\left(\mathbf{s}(n)_{t}|\mathbf{s}(n)_{0},\mathbf{h}(n),\mathbf{x}(n)\right)$
for AEC tasl and the regularity conditions exhibited by the mean and variance,
a score matching objective can be used to train the score model
$\mathbf{s}_{\phi}$. The score function of the perturbation kernel is
$-\frac{\mathbf{s}(n)_{t}-\bm{\mu}\left(\mathbf{s}(n),\mathbf{h}(n)\right)}{\sigma(n)^{2}}$.
We can reparameterize the score matching objective as follows

$\mathcal{J}^{(\mathrm{DSM})}(\phi)=\mathbb{E}\left[\left\|\mathbf{s}_{\phi}\left(\mathbf{s}(n)_{t},\mathbf{h}(n),t\right)+\frac{\mathbf{z}}{\sigma(n)}\right\|_{2}^{2}\right]$
(5)

The clean $\mathbf{s}(n)$ and noisy $\mathbf{h}(n)$ utterances are picked in
the training set and the current process state is obtained as
$\mathbf{s}(n)_{t}=\bm{\mu}(\mathbf{s}(n),\mathbf{h}(n))+\sigma(n)\mathbf{z}(\mathbf{x}(n))$.
Note that the noise here is generated based on the aligned far-end signal
calculated by [2], not the traditional uniform random noise. This distinction
arises because echo cancellation is different from noise suppression. The echo
has a strong correlation with the far-end signal $\mathbf{x}(n)$. Utilizing
$\mathbf{x}(n)$ can help reverse diffusion achieve stable performance with
fewer frames. Therefore, we defined the far-end signal guided noise as
$\mathbf{z}\sim\mathcal{N}_{\mathbb{C}}(\mathbf{z};\mathbf{x}(n),\mathbf{I})$.

###  2.4 Fast Score model

Assuming that $\mathbf{s}(n)$ represents a stable recording, both echo
recording $\mathbf{z}(n)$ and noisy recording $\mathbf{h}(n)$ would also be
stable. As more input data is taken into account, the filter in AEC would
estimate the echo path, resulting in a reduced value of $\sigma$. We can think
of the filter’s convergence process as an inverse operation. The perturbation
kernel could be simplified as:

$p\left(\hat{\mathbf{s}}(n)|\mathbf{s}(n),\mathbf{h}(n)\right)=\mathcal{N}_{\mathbb{C}}\left(\hat{\mathbf{s}}(n);\bm{\mu}\left(\hat{\mathbf{s}}(n),\mathbf{h}(n)\right),\sigma(n)^{2}\mathbf{I}\right)$
(6)

where $\hat{\mathbf{s}}(n)$ the current process state after AEC filter
$D_{\theta}$. Further, The score function of the perturbation kernel is
simplified as
$-\frac{\hat{\mathbf{s}}(n)_{t}-\bm{\mu}\left(\mathbf{s}(n),\mathbf{h}(n)\right)}{\sigma(n)^{2}}$
and the score mathcing objective Eq. (5) is simplified as

$\mathcal{J}^{(\mathrm{DSM})}(\phi)=\mathbb{E}\left[\left\|\hat{\mathbf{s}}_{\phi}\left(\mathbf{s}(n),\mathbf{h}(n),s(n-1)\right)+\frac{\mathbf{z}}{\sigma(n)}\right\|_{2}^{2}\right]$
(7)

where $\mathcal{J}^{(\mathrm{DSM})}$ is not related to $t$ , but rather to
$s(n-1)$. This implies that the score model takes into account the enhanced
signal from the previous frame. Optimization is achieved through iterative
processing over time, rather than multiple score calculations within a single
frame. While this approach might require more iterations to stabilize the
system, it reduces the computational load for each frame. Subsequently, the
corrector employs the score network’s output to synchronize the generated
sample with the anticipated marginal distribution, taking cues from the score
network’s estimate. For simplicity, we term $G_{\phi}$ as the generative
model. This corresponds to the reverse diffusion process solver, steered by
both the plug-in SDE and the score network $\mathbf{s}_{\phi}$. Thus, our
concluding estimate is expressed as $\hat{x}(n)=G_{\phi}(\mathbf{h}(n))$.

###  2.5 DI-AEC and FADI-AEC

To address the computational burden of full diffusion-based models, we
consider to use the stochastic regeneration model [11] as our baseline. By
utilizing the predictive model’s initial approximation, it significantly
reduces the number of diffusion steps required, thereby curbing the
computational demands typically seen with full diffusion models. DI-AEC and
FADI-AEC have similar pipelines. The main different is DI-AEC uses the score
(5) while FADI-AEC uses the fast score function (7) and only calculate one
time for each frame. The whole pipeline of FADI-AEC is shown as Fig. 1. For
near-end signal $\mathbf{h}(n)$, a predictive model $D_{\theta}$ serves as an
initial predictor producing an estimate $\hat{\mathbf{s}}(n)$ defined as:

$\hat{\mathbf{s}}(n)=\mathbf{s}(n)-{\mathbf{s}(n)}^{dis}+{z}^{\prime}(n)$ (8)

where ${\mathbf{s}(n)}^{dis}$ is the target distortion introduced by the
predictive model. The residual corruption ${z}^{\prime}(n)$ is what remains of
the noise after being processed by the model. A diffusion-based generative
model $G_{\phi}$ is then used to learn the distribution of the ideal residue
$r_{\mathbf{s}}(n)={\mathbf{s}(n)}^{dis}-{z}^{\prime}(n)$, starting from the
echo residue
$r_{\mathbf{h}}(n)=\mathbf{h}(n)-D_{\theta}(\mathbf{h}(n))={\mathbf{s}(n)}^{dis}+\mathbf{z}(n)-{z}^{\prime}(n)$.
The resulting a priori Signal-echo rate in the starting point is very high, as
for a reasonable predictor $D_{\theta}$, for
$||\tilde{z}^{\prime}(n)||\ll||\mathbf{z}(n)||$. The estimate $\tilde{s}(n)$
is then obtained as $G_{\phi}(D_{\theta}(\mathbf{h}(n)))$.

###  2.6 Loss

For training, we define the loss as $\mathcal{L}$ combining score matching and
and a supervised regularization term —e.g. mean square error—matching the
output of the initial predictor to the target speech:

$\displaystyle\mathcal{L}^{(\text{StoRM
})}(\theta,\phi)=\mathcal{L}^{(\mathrm{DSM})}(\theta)+\alpha\mathcal{L}^{(\text{Sup
})}(\phi)$ (9) $\displaystyle=$
$\displaystyle\mathbb{E}\left[\left\|\mathbf{s}_{\phi}\left(\hat{\mathbf{s}}(n),\mathbf{h}(n),s(n-1)\right)+\frac{\mathbf{z}}{\sigma(n)}\right\|_{2}^{2}\right]$
$\displaystyle+\alpha\mathbb{E}\left[\left\|\mathbf{s}(n)-D_{\theta}(\mathbf{h}(n))\right\|_{2}^{2}\right],$

where $\alpha$ is a balance term that we empirically set to 1.

##  3 Experimentation Results

###  3.1 Data Selection and Enhancement

Index | Model | Sampling | Reversion | # Parameters | Latency | ERLE of | PESQ of | PESQ of
---|---|---|---|---|---|---|---|---
|  |  | Diffusion | (M) | (ms) |  FEST (dB) | NEST | DT
1 | CRN | No | No | 3.6 | 4.04 | 67.67 | 4.41 | 2.34
2 | CRN | No | No | 7.8 | 8.93 | 82.45 | 4.50 | 2.60
3 | DI-AEC | Random | Yes | 6.9 | 325.00 | 92.51 | 4.87 | 3.30
4 | DI-AEC | Far Guided | Yes | 6.9 | 325.00 | 92.83 | 4.91 | 3.32
5 | FADI-AEC | Random | Fast Score | 6.9 | 9.14 | 86.24 | 4.81 | 3.08
6 | FADI-AEC | Far Guided | Fast Score | 6.9 | 9.14 | 89.41 | 4.83 | 3.21
Table 1: Performance comparison over candidate models. We measure WB-PESQ both DT and NEST scenarios and ERLE for FEST scenario in the augmented evaluation dataset. Model | FEST | DT | DT other
---|---|---|---
ByteAudio | 4.709 | 4.770 | 4.312
DI-AEC | 4.732 | 4,783 | 4.328
FADI-AEC | 4.719 | 4.781 | 4.321
Table 2: AECMOS comparison on ICASSP 2023 AEC Challenge blind test.

For model training, we utilize a combination of synthetic data from the AEC-
challenge [16] and our privately enhanced dataset. We ensure gender balance
among speakers on both the far-end and near-end sides, resulting in 720
original conversations, each lasting 10 seconds.

Each conversation is then enhanced considering several typical use cases.
Reverberation Time (RT60): We employ the image method [17] to generate both
steady and time-varying room impulse responses (RIR), which are used to
simulate echo paths in common laptop environments. The RT60 values are
selected based on probabilities of 0.6, 0.3, 0.08, and 0.02 for ranges of
50-300 ms, 300-600 ms, 600-1 s, and 1-1.5 s, respectively. Delay: We introduce
a delay between the playback and its received echo, with probabilities of
0.05, 0.6, 0.4, and 0.05 for delay ranges of -20-0 ms, 0-200 ms, 200-400 ms,
and 400-600 ms, respectively. Signal-to-Noise Ratio (SNR): We simulate SNR
using typical noises from the DNS-challenge [18], with probabilities of 0.1,
0.1, 0.3, and 0.5 for SNR ranges of 0-10 dB, 10-20 dB, 20-30 dB, and 30-40 dB,
respectively. Signal-to-Echo Ratio (SER): We simulate SER with probabilities
of 0.1, 0.5, 0.3, and 0.1 for SER ranges of -10-0 dB, 0-10 dB, 10-30 dB, and
30-40 dB, respectively. Non-linearity: We model non-linearity using either an
arc-tangent function to mimic gain saturation or a polynomial function as
demonstrated by [19]. Time-Variant Delay/RIR Changes: We introduce random cuts
or additions of speech and silence segments, ranging from 10 to 200 ms, to
either near-end or far-end signals with a probability of 0-10%. Time-variant
RIR changes are also incorporated when generating the echo component. Each
enhanced conversation is then converted into far-end single talk (FEST), near-
end single talk (NEST), and double talk (DT) scenarios. This enhancement
process results in a total of 720K enhanced conversations, approximately
equivalent to 2,000 hours of conversation.

###  3.2 Ablation study

Table 1 presents the performance of various acoustic echo cancellation (AEC)
models based on different sampling methods and the presence of reversion
diffusion. The models are evaluated on parameters such as ERLE of FEST, PESQ
of NEST, and PESQ of DT.

Initially, we can see that the CRN model [1] without any sampling or reversion
diffusion presents two different parameter sizes. Specifically, the larger
model with large latency shows superior performance in both ERLE of FEST and
PESQ of NEST. This suggests that, given the same architectural
characteristics, increasing the model size can lead to better AEC performance.
The DI-AEC models (id 3 and 4) introduce reversion diffusion and use distinct
sampling methods - Random Sampling and Far Guided Sampling, respectively.
These models exhibit a notably high latency of 325.00ms. Despite the higher
latency, the DI-AEC models surpass the CRN models in all the performance
metrics. Shifting to FADI-AEC models (id 5 and 6), we observe that they
combine a modified reversion approach, termed Fast Reversion, with sampling
techniques. These models manage to significantly reduce the latency to 9.14ms.
Performance-wise, the FADI-AEC with Far Guided Sampling marginally outperforms
its Random Sampling counterpart, especially in terms of ERLE of FEST and PESQ
of DT. To conclude, Upon applying the DI-AEC model, the DT PESQ improved by
27.7%, rising from 2.6 to 3.32. When implementing FADI-AEC, the DT PESQ
decreased to 3.08, representing a decrease of 7.2%. However, a notable
advantage of FADI-AEC is its reduced latency, which is only 2.8% of DI-AEC’s
latency, comparing 9.14 to 325.

###  3.3 Comparison with the state-of-the-art methods

We use AECMOS, a non-intrusive model-based metric from the AEC challenge, to
compare our proposed method against the established baselines ByteAudio [20]
and DI-AEC. ByteAudio employs a Two-step Band-split Neural Network (TBNN)
methodology for full-band acoustic echo cancellation, achieving the highest
performance in the AEC 2023 challenges, second only to the host. ByteAudio
represents an advanced end-to-end AEC network solution. Meanwhile, DI-AEC
stands as another leading AEC technique optimized for various real-world
acoustic echo cancellation scenarios. As shown in Table 2, FADI-AEC slightly
outperforms both ByteAudio and DI-AEC across all three scenarios.
Specifically, FADI-AEC attains performance scores of 4.719, 4.781, and 4.321
for the FEST, DT, and DT other scenarios respectively. All three methods
demonstrate remarkable performance on the AEC task, but FADI-AEC holds a
slight edge across the considered scenarios. These results underscore the
importance of selecting the appropriate AEC method for specific tasks and
scenarios, and also highlight the progress and potential of contemporary AEC
technologies.

##  4 Conclusion

We proposed DI and FADI, two novel score-based diffusion models specifically
designed for acoustic echo cancellation. The research highlights the
diffusion-based stochastic regeneration model can improve AEC model
performance. To address the computational cost challenges posed by DI-AEC,
particularly for edge devices, FADI-AEC significantly improves processing
efficiency by running the score model only once per frame. Furthermore, FADI-
AEC and DI-AEC can use a pioneering noise generation technique with far-end
signals, integrating both far-end and near-end signals to enhance the
precision of the score model. This unique application of diffusion models
offers a powerful and efficient approach to echo cancellation, showcasing
strong performance even in challenging conditions. In future work, the fast
score model proposed in FADI could be applied in the other speech enhancement
task such as noise suppression and dereverberation.

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