# Noise Filtering Using 1€ Filter

## Table of Contents

## Introduction

This article explores the 1€ Filter, a simple, but powerful algorithm for filtering noisy real-time signals. The article focuses on the practical implementation of the algorithm, and it covers the mathematical basis, a pseudocode implementation, and simple, pure Python implementation of the algorithm. To understand why and how the filter works, we recommend reading the original article ^{1}.

## 1€ Filter

The 1€ Filter is a low pass filter for filtering noisy signals in real-time. It is also a simple filter with only two configurable parameters. The signal at time $T_i$ is denoted as value $X_i$ and the filtered signal as value $\hat{X}_i$. The filter uses **exponential smoothing**

$$\hat{X}_1 = X_1$$

$$\hat{X}_i = α X_i + (1-α) \hat{X} _{i-1},\quad i≥2 \tag{1} \label{1} $$

where the smoothing factor $α∈[0, 1]$, instead of being a constant, is adaptive, that is, dynamically computed using information about the rate of change (speed) of the signal. The adaptive smoothing factor aims to balance the jitter versus lag trade-off since people are sensitive to jitter at low speeds and more sensitive to lag at high speeds. The **smoothing factor** is defined as

$$α = \frac{1}{1 + \dfrac{τ}{T_e}},$$

where $T_e$ is the **sampling period** computed from the time difference between the samples

$$T_e=T_i-T _{i-1}$$

and $τ$ is **time constant** computed using the cutoff frequency

$$τ = \frac{1}{2πf_C}.$$

The **cutoff frequency** $f_C$ increases linearly as the rate of change, aka speed, increases

$$f_C=f_{C_{min}} + β|\hat{\dot{X}} _i|$$

where $f _{C _{min}}>0$ is the **minimum cutoff frequency**, $β>0$ is the **speed coefficient** and $\hat{\dot{X}}_i$ is the filtered rate of change. We define the rate of change $\hat{X}_i$ as the discrete derivative of the signal

$$\dot{X}_1 = 0$$

$$\dot{X}_i = \frac{X_i-\hat{X} _{i-1}}{T_e}, i≥2$$

which is then filtered using exponential smoothing $\eqref{1}$ with a constant cutoff frequency $f_{C_d},$ by default $f_{C_d}=1$.

## Algorithm

In this section, we implement the 1€ filter algorithm as pseudocode. The precise implementation of the algorithm depends on the programming language and paradigm in question. We have written this algorithm using a functional style.

$\operatorname{Smoothing-Factor}(f_C, T_e)$

- $r=2π⋅f_c⋅T_e$
**return**$\dfrac{r}{r+1}$

$\operatorname{Exponential-Smoothing}(α, X_i, \hat{X}_{i-1})$

**return**$α X_i + (1-α) \hat{X}_{i-1}$

$\operatorname{One-Euro-Filter}(T_i,X_i,T_{i-1},\hat{X}_{i-1},\hat{\dot{X}}_{i-1},f_C,β,f_{C_d})$ for $i≥2$

- $T_e=T_i-T_{i-1}$
- $α_d=\operatorname{Smoothing-Factor}(T_e, f_{C_d})$
- $\dot{X}_i = \dfrac{X_i-\hat{X} _{i-1}}{T_e}$
- $\hat{\dot{X}}_i=\operatorname{Exponential-Smoothing}(α_d, \dot{X}_i, \hat{\dot{X}} _{i-1})$
- $f_C=f_{C_{min}} + β|\hat{\dot{X}}_i|$
- $α=\operatorname{Smoothing-Factor}(f_C, T_e)$
- $\hat{X}_i=\operatorname{Exponential-Smoothing}(α, X_i, \hat{X} _{i-1})$
**return**$T_i,\hat{X}_i,\hat{\dot{X}}_i$

## Tuning the Filter

There are two configurable parameters in the model, the **minimum cutoff frequency** $f _{C _{min}}$ and the **speed coefficient** $β$. Decreasing the minimum cutoff frequency decreases slow speed jitter. Increasing the speed coefficient decreases speed lag.

## Python Implementation

**OneEuroFilter**GitHub repository.

The object-oriented approach stores the previous values inside the object instead of explicitly giving them a return value as functional implementation would. It should be relatively simple to implement this algorithm in other languages.

```
import math
def smoothing_factor(t_e, cutoff):
r = 2 * math.pi * cutoff * t_e
return r / (r + 1)
def exponential_smoothing(a, x, x_prev):
return a * x + (1 - a) * x_prev
class OneEuroFilter:
def __init__(self, t0, x0, dx0=0.0, min_cutoff=1.0, beta=0.0,
d_cutoff=1.0):
"""Initialize the one euro filter."""
# The parameters.
self.min_cutoff = float(min_cutoff)
self.beta = float(beta)
self.d_cutoff = float(d_cutoff)
# Previous values.
self.x_prev = float(x0)
self.dx_prev = float(dx0)
self.t_prev = float(t0)
def __call__(self, t, x):
"""Compute the filtered signal."""
t_e = t - self.t_prev
# The filtered derivative of the signal.
a_d = smoothing_factor(t_e, self.d_cutoff)
dx = (x - self.x_prev) / t_e
dx_hat = exponential_smoothing(a_d, dx, self.dx_prev)
# The filtered signal.
cutoff = self.min_cutoff + self.beta * abs(dx_hat)
a = smoothing_factor(t_e, cutoff)
x_hat = exponential_smoothing(a, x, self.x_prev)
# Memorize the previous values.
self.x_prev = x_hat
self.dx_prev = dx_hat
self.t_prev = t
return x_hat
```

Code for the plot:

```
import matplotlib.pyplot as plt
import numpy as np
import seaborn
from matplotlib.animation import FuncAnimation
from one_euro_filter import OneEuroFilter
np.random.seed(1)
# Parameters
frames = 100
start = 0
end = 4 * np.pi
scale = 0.05
# The noisy signal
t = np.linspace(start, end, frames)
x = np.sin(t)
x_noisy = x + np.random.normal(scale=scale, size=len(t))
# The filtered signal
min_cutoff = 0.004
beta = 0.7
x_hat = np.zeros_like(x_noisy)
x_hat[0] = x_noisy[0]
one_euro_filter = OneEuroFilter(
t[0], x_noisy[0],
min_cutoff=min_cutoff,
beta=beta
)
for i in range(1, len(t)):
x_hat[i] = one_euro_filter(t[i], x_noisy[i])
# The figure
# https://eli.thegreenplace.net/2016/drawing-animated-gifs-with-matplotlib/
seaborn.set()
fig, ax = plt.subplots(figsize=(12, 6))
ax.set(
xlim=(start, end),
ylim=(1.1*(-1-scale), 1.1*(1+scale)),
xlabel="$t$",
ylabel="$x$",
)
fig.set_tight_layout(True)
signal, = ax.plot(t[0], x_noisy[0], 'o')
filtered, = ax.plot(t[0], x_hat[0], '-')
def update(i):
print(i)
signal.set_data(t[0:i], x_noisy[0:i])
filtered.set_data(t[0:i], x_hat[0:i])
return signal, filtered
if __name__ == '__main__':
# FuncAnimation will call the 'update' function for each frame; here
# animating over 10 frames, with an interval of 200ms between frames.
anim = FuncAnimation(fig, update, frames=frames, interval=100)
anim.save('one_euro_filter.gif', dpi=80, writer='imagemagick')
# update(frames)
plt.savefig("one_euro_filter.png", dpi=300)
```

## Conclusions

I first learned about the 1€ filter at *Computational User Interface Design* course at Aalto University. I found the algorithm to be elegant, but the original paper’s explanation was cumbersome for implementing it. It motivated me to create a simplified explanation and code to help other people implement this algorithm.

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## References

Casiez, G., Roussel, N., & Vogel, D. (2012). 1€ filter: a simple speed-based low-pass filter for noisy input in interactive systems. In Proceedings of the SIGCHI Conference on Human Factors in Computing Systems (pp. 2527–2530). ↩︎