# Performance of a frequency modulated continuous wave ultrasonic rangefinder in the context of autonomous urban drone navigation

Jean-Claude Rembert-Baudet

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Jean-Claude Rembert-Baudet. Performance of a frequency modulated continuous wave ultrasonic rangefinder in the context of autonomous urban drone navigation. 2024. hal-04395809

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Submitted on 15 Jan 2024

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# Performance of a frequency modulated continuous wave ultrasonic rangefinder in the context of autonomous urban drone navigation

ASTEK Group

Abstract- Drones represent new types of robots that allow the integration of different sensors that are inexpensive, easy to use, and provide users with many new options regarding image and geospatial data collection. Technological advances in electronics, big data and artificial intelligence have enabled the development of new fields of application for these devices as well as their robustness and utility. To ensure autonomous flight, each drone must have a 3D perception of the environmental obstacles present as well as an understanding of their respective trajectories and their impact on its own navigation. For this, UAVs use several sensors to be more reactive in an environment full of obstacles. The objectives of the work are to model the performance of an acoustic telemetry device based on continuous wave frequency modulation technology, with a single transmitter and a single receiver, intended to be mounted on an urban drone with a wingspan of a few tens of centimeters. The innovative solution proposed in this work to make telemetry from acoustic wave is the use of a device, integrable on an urban drone flying at a maximum speed of about 15 Km/h and at a height of about 7 meters, based on acoustic sensors whose concept is based on the difference in frequencies between the emission and reception of the wave reflected on an obstacle in order to estimate its distance from the drone. The work has shown that it is possible to estimate by a simple Fourier transform the time that the wave takes to go and return as well as the Doppler shift that it undergoes and to deduce the speed of the obstacles knowing that of the drone.

*Index Terms-Urban drone, Acoustic sensor, Doppler effect, Wind, Obstacle mobility, Frequency modulation, Amplification electronics, Signal digitization, Telemetry measurement.*

## I. INTRODUCTION

Detecting obstacles in the drone's environment is a

key function to ensure a collision-free flight. One of the methods used for this purpose is acoustic telemetry. This technology has several advantages. Compared to other types of sensors (infrared or laser for example), the ultrasonic sensor is relatively inexpensive. The measurement is quite accurate. However, one of the main drawbacks of this technology is its sensitivity to temperature and pressure. It is also sensitive to other acoustic sensors in the vicinity since interference between the waves emitted by the different sensors is possible. This is because the information we get on obstacles with the time-of-flight technology is not always reliable since the technology does not differentiate the echoes received. However, it is still a very common technology.

In the current work, another technology of acoustic telemetry is proposed; the telemetry by analysis of difference of frequencies of the transmitted and received signals that we will name FMCW for Frequency Modulation Continuous Wave. The transmitter emits a continuous wave with an increasing linear frequency according to a precise cyclic ratio. By measuring the difference in frequency between the receiver and the transmitter, it is possible to estimate the delay of the received wave and to estimate the distance to the obstacle. The interest of such a device is to be able to discriminate the parasitic signals which are not in the range of frequencies of the wave of emission.

<figure><img src="image_3.png" /><figcaption>Figure 1: Transmission in red and reception in green with a delay $\Delta t$ and a Doppler shift frequency $f_D$</figcaption></figure>
<figure><img src="image_4.png" /></figure>

The objective of this paper is to present the modeling of the performance of such a FMCW telemetry device based on the emission of a continuous acoustic wave modulated in frequency by an acoustic transmitter and received by a single acoustic receiver after the wave has been reflected by a potential obstacle. More specifically, it aims at quantifying the impacts of wind and moving objects on the telemetric performance of the device. After a state of the art of acoustic telemetry technologies including the correlation of transmitted and received signals, the work will, in a first step, literally quantify the telemetry error due not to the Doppler shift but due to the additional algebraic delay that would take the acoustic wave to make a round trip between the device and the obstacle in the presence of the speeds of the drone, the wind but also the obstacle. In a second time, the work will show theoretically that the Fourier transform of the product of the emission and reception signals at a given time would allow to find the value of this round trip time, first in the simplistic case where there is no Doppler shift, then in the simplistic case where the Doppler shift is constant in time and finally in the case where the Doppler shift is variable in time, which would correspond to our case given that it is proportional to the emission frequency which is variable in time In the case where there is a Doppler frequency, the work will also show that it is possible to estimate and deduce, knowing the speed of the drone, in the absence of wind, the speed of movement of the obstacle.

## II. STATE OF THE ART

In this section, a review of the existing literature, where the papers most related to the problem of acoustic telemetry using frequency analysis of transmitted and received signals, will be made. Acoustic time-of-flight telemetry is the technology of choice for several research works and a lot of knowledge concerning the performance of this technology has been developed. Therefore, some papers dealing with acoustic time-of-flight telemetry and the impact of physical parameters of the urban environment on the telemetry performance will be presented.

The paper [1] presents the performance of a low cost FMCW ultrasonic rangefinder. The system, based on a voltage-controlled oscillator and an analog mixer, moves part of the processing required by the FMCW protocol into the analog domain, sampling the signal without reducing the resolution over the usable range. Its characterization range is between 5 and 56 cm, experiments are performed up to 35 cm. The final objective is to evaluate the possibility of implementing the FMCW technique in portable and

low-cost devices. Although this paper discusses FMCW acoustic ranging, the experimental distances are relatively small compared to our context of using the device. The impacts of obstacle shape, material and orientation on measurement accuracies are not addressed here, nor is the impact of wind. This is also the case for the presence of several obstacles, extended obstacles, or their mobility.

In this paper [2], a measuring device for determining the distance between the transmitter and receiver and the flow velocity of a fluid using FMCW acoustic wave emission was presented with the objective of experimenting with this measurement method. The required measuring device was described, and the necessary mathematical model for the extraction of the distance/fluid flow velocity from the electrical output signal was developed. Experiments have been performed up to a distance of 25 cm and the transmitter and receiver are placed face to face, separated by a flowing fluid. The paper does address FMCW telemetry. However, the experiments performed do not exceed 25 cm distance. The impact of the shape, material and orientation of the obstacle on the accuracy of the measurements is not discussed here, nor is the impact of wind. This is also the case for the presence of several obstacles and for the case of extended obstacles or their mobility.

The paper [3] presents the design and experimental characterization of an ultrasonic telemetry system using FMCW technology and integrating a CMOS (Complementary metal oxide semiconductor) technology to design the ultrasonic sensors, namely composed of a transmitter and a receiver. The test environment of the device is air, although its operation is planned in a liquid medium. The experiments are performed between 5 cm and 1m from a wall, in 5 cm steps, in a neutral room. The aim of the article is to compare the consumption of different telemetry technologies. This article presents a comparison between several ultrasonic wave telemetry methods, but does not address the different impacts, such as those of the shape, material, orientation of the obstacles, or even the impact of the mobility of the obstacle on the accuracy of the measurements performed. In the paper [4], the adoption of the FMCW ultrasound technique for thickness measurement is considered. Numerical simulations on synthetic and experimental data are performed and the results are compared with the standard approach of a standard filter to reduce noise. Although this paper deals with FMCW acoustic ranging, it focuses on the measurement of material thickness and not distances. The impacts of obstacle shape, material and orientation on measurement accuracies are not addressed here, nor is the impact of wind. This is also the case for the presence of several obstacles, extended obstacles, or their
<figure><img src="image_5.png" /></figure>

mobility. The paper [5] proposes a system based on FMCW technology in the ultrasonic frequencies for the measurement of the distance between this system and the blood vessels under the human skin. Sawtooth and isosceles sawtooth functions are selected as frequency modulation signals for distance measurement to allow wave penetration through human skin. A continuous, frequency-modulated ultrasound signal is transmitted through the skin and the reflected signal is received after a propagation time. The range information can be obtained from the frequency spectrum of the signal which is generated by multiplying the transmitted signal by the received signal. Thus, the operational context is not at all the same as the one of our works since these are waves that propagate in the skin to detect and localize blood vessels. The paper [6] proposes a new method to directly measure the lateral displacement of objects using ultrasonic transducers in the air. Its displacement is calculated using the intensity ratio of the reflected ultrasonic waves which is based on the evolution of the reception amplitude of the sound waves from several transducers placed side by side. Although this paper uses acoustic sensors and addresses the topic of moving obstacles, it does not use FMCW technology and the distances over which the experiments are performed are relatively small in the context of urban drone flight. The paper [7] aims at solving the problem of identifying the shape of an obstacle and its orientation for the movement of a robot. The study focuses on two implementations of an ultrasonic sensor system. The first one is a sensor placed in a pivoting mechanism located in the center of a robot, which allows to determine the angles of an obstacle by scanning the space around it, to reconstitute its shape. The second one is based on the use of several ultrasonic sensors simultaneously in emission, and using the time-of-flight technology which uses the method of least squares and makes it possible to eliminate the errors of measurement. However, this paper studies a time-of-flight technology for determining angles and distances of obstacles, which is a very different technology from the acoustic FMCW technology. Paper [8] presents a study done on a nursing robot that uses several technologies, including two ultrasonic rangefinders to detect and provide information to avoid obstacles. This article focuses on explaining the limitations of using ultrasonic sensors to avoid obstacles. However, there are no experiments using FMCW technology. It deals with the reflection of the waves according to the orientation of an obstacle but does not deal with its material. Other impact criteria on the performance of a rangefinder such as the number of obstacles, extended obstacles and the impact of their mobility or wind are not mentioned. Finally, the paper [9] presents a model for distance measurement using an

ultrasonic sensor. The obstacles used are small plates, each made of a different material. The technology used for the different obstacle distance measurements is time-of-flight. The results of the measurements are used to improve the navigation of the mobile robots, the test device is however stationary. However, this paper does not compare the results to any characteristic of the materials used to establish even a minor link with the few variations in the results obtained. The test distances are too small (50 to 100mm) compared to the use of such a device in an urban drone. The FMCW technology is not used and the different impacts, such as those of the shape, orientation, extended obstacles, and obstacle mobility, but also of the wind on the detection/characterization performances are not treated.

To conclude, there is no study in the scientific literature that answers our problem. Indeed, even if the acoustic FMCW technology is used, either the operational context is different from that of the flying urban drone, or the impacts of the wind, the mobility of the obstacles or the influence of the intrinsic properties of the materials have not been simultaneously evoked and quantified. Globally, the direct measurement of the time of flight is the most widespread method for acoustic telemetry.

### III. STUDY STRATEGY

Some impacts, such as wind, drone mobility or mobile obstacles, will be studied in a purely theoretical way. To do this, a theoretical model will first be established. The accuracy of the measurement of a telemetry device represents the ability of the sensor to give a measurement close to the exact value of the distance. Several geometrical-physical factors can influence the value of this precision, in particular the capacity of the reception circuit to detect a return signal due to the wave reflected on a potential obstacle. The intuitive parameters influencing the detection and thus the accuracy of the telemetric measurement are multiple, but we can quote as example the size, the material of the obstacle, its shape as well as its orientation. However, the current work will not quantify the impacts of the physical and geometric properties of the obstacle on the detection and accuracy of the measurement. They will essentially model on the theoretical level the impacts of the mobility of the UAV, the obstacles, and the wind speed on the accuracy of the telemetric measurement. The work will also show, under some realistic assumptions, that it is possible to quantify the round-trip time of the acoustic wave between the device and the obstacle through the Fourier transform of the product of the transmission signal and the reception signal. This transform of Fourier
<figure><img src="image_6.png" /></figure>

should also inform us that to the probable speed of the obstacle knowing that of the drone and neglecting the impact of that of the wind.

#### IV. IMPACTS OF WIND SPEED AND OBSTACLE MOBILITY SPEED

The wind corresponds to the displacement of the material medium in which an acoustic wave of any frequency propagates. If the material medium is the ambient air, assimilated as a perfect gas, the acoustic velocity can be related to the temperature $θ$ (in Kelvin) such that:

$$c = \sqrt{\frac{\gamma R \theta}{M}}$$

Where $γ$ is the Laplace coefficient, $R$ the constant of perfect gases and $M$ the molar mass of air. For air at 20°C, $c \approx 343m.s^{-1}$. This celerity $c$ is measured with respect to the material medium.

We will note by $v_1$ the speed of the drone with respect to the ground; by $v_2$ the speed of the obstacle and by $v_3$ the wind speed with respect to the ground and in the direction of the vector $\vec{v}_1$. In the case of wind blowing at the algebraic speed $v_3$ the speed of the acoustic wave with respect to the ground becomes simply $c + v_3$ if the wave is carried and $c - v_3$ if the wave propagates against the wind. If the wind is transverse, the acoustic wave is deviated from its initial axis of emission by an angle depending on $v_3$ and of $c$. We will not study this case, but which remains possible. However, the navigation of the urban drone with a transverse wind can be even more problematic than in the case of a weak wind against the direction considering that it can make deviate the drone from its trajectory.

Measuring a distance using the differential analysis of transmitted and received frequencies is equivalent to measuring the time it takes the acoustic wave to make a round trip between the device and the obstacle. Noting by $d$ the initial distance between the device and the obstacle at the time of transmission, this round-trip time of the acoustic wave is then:

$$t_{a-r} = \frac{2c}{(c - v_3 + v_1)(c + v_3 - v_2)} d$$

The drone could know its speed of movement $v_1$ in relation to the ground. If we consider the hypothesis that the obstacle is immobile with respect to the ground and in the absence of wind, it will then measure a round trip time of the wave equivalent to a distance $d' \neq d$ and worth:

$$d' = \frac{c + v_1}{2} t_{a-r} = \frac{c(c + v_1)}{(c - v_3 + v_1)(c + v_3 - v_2)} d$$

The relative error is therefore a function of the speeds:

$$e_r = \frac{d' - d}{d} = \frac{v_2 c + (v_1 - v_3)(v_2 - v_3)}{(c - v_3 + v_1)(c + v_3 - v_2)}$$

For a stationary obstacle relative to the ground, for a wind blowing at 50km/h (about 14m/s) and for a drone flying at 15km/h, the maximum relative error would be 0.21%. In the absence of wind and for an obstacle moving at 50km/h (speed of cars in town), we would have a maximum relative error of 4.3%.

Thus, a priori the wind has very little effect on the telemetric error, contrary to the speeds of the fastest obstacles in the city, which are motor vehicles. This being the case, what is being measured does not seem to be the round-trip time directly, but a frequency shift between the transmitted and received acoustic waves, and the physics would not be the same as measuring a simple delay in reception. It is therefore necessary a priori to measure the Doppler shift $f_D$ shift according to the different speeds involved.

#### V. FREQUENCY MODULATION OF A CONTINUOUSLY EMITTED ACOUSTIC WAVE - PHYSICAL PRINCIPLE

Specific electronics could generate a supply voltage, whose frequency varies in time in a linear way between two frequencies $f_{min}$ and $f_{max}$ at the terminals of a loudspeaker-type acoustic transmitter. This frequency $f(t)$ is $T$ - périodique such that there is a time $t_0$ for which we can write for all $t \in [t_0, t_0 + T]$:

$$f(t) = a(t - t_0) + f_{min}$$

With $a = \frac{f_{max} - f_{min}}{T}$.

The value of $T$ would then be chosen so that ideally $t_{a-r} \le T$ in the case of the detection range of the return wave by the acoustic receiver.

The evolution of $f$ as a function of time is sawtooth and its Fourier series (but without loss of generality) for $f = \frac{f_{max} + f_{min}}{2} \quad \text{à } t = 0$ is:

$$f(t) = \frac{f_{max} + f_{min}}{2} - \frac{f_{max} - f_{min}}{\pi} \sum_{k=1}^{+\infty} \frac{(-1)^k}{k} \sin\left(\frac{2\pi kt}{T}\right)$$

This supply voltage of the loudspeaker of the transmitter block noted $u_e$ then varies in time such that:

$$\begin{aligned} u_e(t) = & U_e \cos \left( \pi(f_{max} + f_{min})t \right. \\ & \left. + \frac{(f_{max} - f_{min})T}{\pi} \sum_{k=1}^{+\infty} \frac{(-1)^k}{k^2} \cos \left( \frac{2\pi kt}{T} \right) + \varphi_0 \right) \end{aligned}$$

With $U_e$ its amplitude and $\varphi_0$ an original phase shift. Considering the electrical resistance of the loudspeaker coil (power electronics inside the loudspeaker) as zero or negligible then, the coupled equations relating the current $i_e$ the current flowing
<figure><img src="image_7.png" /></figure>

in the electronic circuit of the emitting loudspeaker and the speed of $v_e$ of vibration of the membrane generating the acoustic wave in the emitter block are:

$$ egin{cases} u_e = L \frac{di_e}{dt} - v_e B \ell \\ m \frac{dv_e}{dt} = -i_e \ell B - k z_e \end{cases} $$

With $v_e = dz_e/dt$ such that $z_e$ represents the displacement of the moving part in the transmitter speaker enclosure, $\ell$ the length of the coil winding cable, $L$ the self-inductance of the coil, where as a reminder $u_e$ is the supply voltage across the transmitter circuit, $m$ the mass of the whole dynamic device and finally, $k$ the return coefficient of the spring attached to the dynamic device. We deduce:

$$ \frac{d^2 v_e}{dt^2} + \left( \frac{k}{m} + \frac{\ell^2 B^2}{mL} \right) v_e = - \frac{\ell B}{mL} u_e $$

At the level of the receiver block identical to the transmitter block (identical loudspeaker), the membrane excited by the acoustic overpressure of the return wave will vibrate at a speed $v_r$ and shifted in time.

$$ v_r(t) = \eta v_e(t - t_{a-r}) $$

With $\eta$ a proportionality coefficient such that $|\eta| < 1$. The voltage generated at the terminals of the loudspeaker of the receiver block will also follow the differential equation:

$$ \frac{d^2 v_r}{dt^2} + \left( \frac{k}{m} + \frac{\ell^2 B^2}{mL} \right) v_r = - \frac{\ell B}{mL} u_r $$

We deduce:

$$ u_r(t) = \eta u_e(t - t_{a-r}) $$

We can then write:

$$ u_r(t) = U_r \cos \left( \pi(f_{max} + f_{min})(t - t_{a-r}) + \frac{(f_{max} - f_{min})T}{\pi} \sum_{k=1}^{+\infty} \frac{(-1)^k}{k^2} \cos \left( \frac{2\pi k(t - t_{a-r})}{T} \right) + \varphi_0 \right) $$

We deduce that the product of the 2 voltages is such that:

$$ \frac{2u_r(t)u_e(t)}{U_r U_e} = F_1(t) + F_2(t) $$

With, strategically posing $\varphi_0 = 0$ without taking away from the generalization:

$$ F_1(t) = \cos \left( \pi(f_{max} + f_{min})t_{a-r} - \frac{2(f_{max} - f_{min})T}{\pi} \sum_{k=1}^{+\infty} \frac{(-1)^k}{k^2} \sin \left( \frac{\pi k t_{a-r}}{T} \right) \sin \left( \frac{2\pi k \left( t - \frac{t_{a-r}}{2} \right)}{T} \right) \right) $$

And:

$$ F_2(t) = \cos \left( \pi(f_{max} + f_{min})(-2t + t_{a-r}) - \frac{2(f_{max} - f_{min})T}{\pi} \sum_{k=1}^{+\infty} \frac{(-1)^k}{k^2} \cos \left( \frac{\pi k t_{a-r}}{T} \right) \cos \left( \frac{2\pi k \left( t - \frac{t_{a-r}}{2} \right)}{T} \right) \right) $$

For $t_{a-r} \neq 0$ by imposing the condition

$$ \frac{f_{max} - f_{min}}{f_{max} + f_{min} t_{a-r}} \ll 1 $$

Therefore, at order 1 in $\frac{f_{max} - f_{min}}{f_{max} + f_{min} t_{a-r}}$ we can write:

$$ F_1(t) \approx \cos(\pi(f_{max} + f_{min})t_{a-r}) + \frac{2T}{\pi}(f_{max} - f_{min})\sin(\pi(f_{max} + f_{min})t_{a-r}) \sum_{k=1}^{+\infty} \frac{(-1)^k}{k^2} \sin\left(\frac{\pi k t_{a-r}}{T}\right) \sin\left(\frac{2\pi k \left(t - \frac{t_{a-r}}{2}\right)}{T}\right) $$

And:

$$ F_2(t) \approx \cos(2\pi(f_{max} + f_{min})t) \left( \cos(\pi(f_{max} + f_{min})t_{a-r}) + \frac{2T}{\pi}(f_{max} - f_{min})\sin(\pi(f_{max} + f_{min})t_{a-r}) \sum_{k=1}^{+\infty} \frac{(-1)^k}{k^2} \cos\left(\frac{\pi k t_{a-r}}{T}\right) \cos\left(\frac{2\pi k \left(t - \frac{t_{a-r}}{2}\right)}{T}\right) \right) + \sin(2\pi(f_{max} + f_{min})t) \left( \sin(\pi(f_{max} + f_{min})t_{a-r}) - \frac{2T}{\pi}(f_{max} - f_{min})\cos(\pi(f_{max} + f_{min})t_{a-r}) \sum_{k=1}^{+\infty} \frac{(-1)^k}{k^2} \cos\left(\frac{\pi k t_{a-r}}{T}\right) \cos\left(\frac{2\pi k \left(t - \frac{t_{a-r}}{2}\right)}{T}\right) \right) $$

The Fourier transform of $F_1(t)$ will allow to have Dirac peaks for the frequencies $k \ge 1$ $f_k^{(1)} = \pm \frac{k}{T}$ with an amplitude equal to:

$$ A_k = \frac{T}{k^2} \sqrt{\frac{2}{\pi}(f_{max} - f_{min}) \left| \sin(\pi(f_{max} + f_{min})t_{a-r}) \sin\left(\frac{\pi k t_{a-r}}{T}\right) \right|} $$

The Fourier transform of $F_2(t)$ will allow us to have Dirac peaks for frequencies independent of $k$ such as

$$ f^{(2)} = \pm(f_{max} + f_{min}) \text{ with an amplitude } \sqrt{\frac{\pi}{2}}. \text{ It will also allow to have Dirac peaks for frequencies } f_{k;1}^{(2)} = \pm(f_{max} + f_{min} - \frac{k}{T}) \text{ with an amplitude:} $$

$$ B_{k;1} = \frac{T}{\sqrt{2\pi k^2}} (f_{max} - f_{min}) \left| \cos\left(\frac{\pi k t_{a-r}}{T}\right) \right| $$
<figure><img src="image_8.png" /></figure>

But also, for frequencies $f_{k;2}^{(2)} = \pm (f_{max} + f_{min} + \frac{k}{T})$ with an amplitude $B_{k;2} = B_{k;1}$.

The information on the round-trip time of the acoustic wave is not in the values of the frequencies detected through the Fourier transform but in the amplitudes of the Dirac peaks of the detected frequencies. In the case of the amplitudes $B_{k;1}$ and $B_{k;2}$ it is necessary to choose a value of $k$ for which the function $t_{a-r} \to \cos(\frac{\pi kt_{a-r}}{T})$ is bijective only for $t_{a-r} \in [0, \frac{T}{2}]$. This is the case only for $k = 1$. We deduce that we can go back to the value of the time taken by the wave to make a round trip $t_{a-r} \le \frac{T}{2}$ such that for $B_{1;1} = B_{1;2}$:

$$t_{a-r} = \frac{T}{\pi} \arccos \left( \frac{B_{1;1} \sqrt{2\pi}}{T(f_{max} - f_{min})} \right)$$

## VI. IMPACT OF A CONSTANT DOPPLER SHIFT IN TIME - SIMPLISTIC ASPECT

In the presence of a Doppler shift $f_D$ shift, assumed to be constant in time, on the frequency of the return wave, where $f_D$ represents the difference of frequencies at the reception $f_r$ and at the emission $f_e$ such that $f_D = f_r - f_e$, $u_r(t)$ can be written in the case where $\phi_0 = 0$ without taking away from the generality:

$$u_r(t) = U_r \cos \left( \pi(f_{max} + f_{min} + 2f_D)(t - t_{a-r}) + \frac{(f_{max} - f_{min})T}{\pi} \sum_{k=1}^{+\infty} \frac{(-1)^k}{k^2} \cos \left( \frac{2\pi k(t - t_{a-r})}{T} \right) \right)$$

As a reminder:

$$\begin{aligned} u_e(t) = & U_e \cos \left( \pi(f_{max} + f_{min})t + \frac{(f_{max} - f_{min})T}{\pi} \sum_{k=1}^{+\infty} \frac{(-1)^k}{k^2} \cos \left( \frac{2\pi kt}{T} \right) \right) \\ & - \frac{2(f_{max} - f_{min})T}{\pi} \sum_{k=1}^{+\infty} \frac{(-1)^k}{k^2} \cos \left( \frac{\pi kt_{a-r}}{T} \right) \end{aligned}$$

We then have the result of the following product:

$$\frac{2u_r(t)u_e(t)}{U_r U_e} = F_{1,D}(t) + F_{2,D}(t)$$

With:

$$F_{1,D}(t) = \cos \left( -2\pi f_D t + \pi(f_{max} + f_{min} + 2f_D)t_{a-r} - \frac{2(f_{max} - f_{min})T}{\pi} \sum_{k=1}^{+\infty} \frac{(-1)^k}{k^2} \sin \left( \frac{\pi k t_{a-r}}{T} \right) \sin \left( \frac{2\pi k \left( t - \frac{t_{a-r}}{2} \right)}{T} \right) \right)$$

And:

$$\begin{aligned} F_{2,D}(t) = & \cos \left( -2\pi(f_{max} + f_{min} + f_D)t + \pi(f_{max} + f_{min} + 2f_D)t_{a-r} - \frac{2(f_{max} - f_{min})T}{\pi} \sum_{k=1}^{+\infty} \frac{(-1)^k}{k^2} \cos \left( \frac{\pi k t_{a-r}}{T} \right) \cos \left( \frac{2\pi k \left( t - \frac{t_{a-r}}{2} \right)}{T} \right) \right) \end{aligned}$$

The acoustic Doppler shift will be very small in front of $f_{min}$ and $f_{max}$ by considering at order 1 in $\frac{f_{max} - f_{min}}{f_{max} + f_{min}}$ we can write:

$$\begin{aligned} F_{1,D}(t) \approx & \cos(2\pi f_D t) \left( \cos(\pi(f_{max} + f_{min} + 2f_D)t_{a-r}) + \frac{2(f_{max} - f_{min})T}{\pi} \sin(\pi(f_{max} + f_{min} + 2f_D)t_{a-r}) \sum_{k=1}^{+\infty} \frac{(-1)^k}{k^2} \sin\left(\frac{\pi k t_{a-r}}{T}\right) \sin\left(\frac{2\pi k \left(t - \frac{t_{a-r}}{2}\right)}{T}\right) \right) \\ & + \sin(2\pi f_D t) \left( \sin(\pi(f_{max} + f_{min} + 2f_D)t_{a-r}) - \frac{2(f_{max} - f_{min})T}{\pi} \cos(\pi(f_{max} + f_{min} + 2f_D)t_{a-r}) \sum_{k=1}^{+\infty} \frac{(-1)^k}{k^2} \sin\left(\frac{\pi k t_{a-r}}{T}\right) \sin\left(\frac{2\pi k \left(t - \frac{t_{a-r}}{2}\right)}{T}\right) \right) \end{aligned}$$

And:

$$\begin{aligned} F_{2,D}(t) \approx & \cos(2\pi(f_{max} + f_{min} + f_D)t) \left( \cos(\pi(f_{max} + f_{min} + 2f_D)t_{a-r}) + \frac{2(f_{max} - f_{min})T}{\pi} \sin(\pi(f_{max} + f_{min} + 2f_D)t_{a-r}) \sum_{k=1}^{+\infty} \frac{(-1)^k}{k^2} \cos\left(\frac{\pi k t_{a-r}}{T}\right) \cos\left(\frac{2\pi k \left(t - \frac{t_{a-r}}{2}\right)}{T}\right) \right) \\ & + \sin(2\pi(f_{max} + f_{min} + f_D)t) \left( \sin(\pi(f_{max} + f_{min} + 2f_D)t_{a-r}) - \frac{2(f_{max} - f_{min})T}{\pi} \cos(\pi(f_{max} + f_{min} + 2f_D)t_{a-r}) \sum_{k=1}^{+\infty} \frac{(-1)^k}{k^2} \cos\left(\frac{\pi k t_{a-r}}{T}\right) \cos\left(\frac{2\pi k \left(t - \frac{t_{a-r}}{2}\right)}{T}\right) \right) \end{aligned}$$

The Fourier transform of $F_{1,D}(t)$ will generate peaks of which one has a frequency $\pm f_D$ with an amplitude $\sqrt{\frac{\pi}{2}}$ and the other peaks with frequencies $f_{k;1} = \pm (f_D + \frac{k}{T})$ with an amplitude:

$$A_{k;1} = \frac{T}{2k^2} \sqrt{\frac{2}{\pi}} (f_{max} - f_{min}) \left| \sin\left(\frac{\pi k t_{a-r}}{T}\right) \right|$$

Or with frequencies $f_{k;1} = \pm (f_D - \frac{k}{T})$ with an amplitude $A_{k;2} = A_{k;1}$.

As for $F_{2,D}(t)$ its Fourier transform allows the detection of peaks at frequencies $\pm (f_{min} + f_{max} +$
<figure><img src="image_9.png" /></figure>

$f_D$) with an amplitude of $\sqrt{\frac{\pi}{2}}$ but also peaks at frequencies $f_{k;2,1} = \pm (f_{min} + f_{max} + f_D - \frac{k}{T})$ with an amplitude:

$$B_{k;1} = \frac{T}{2k^2} \sqrt{\frac{2}{\pi}} (f_{max} - f_{min}) \left| \cos\left(\frac{\pi k t_{a-r}}{T}\right) \right|$$

Or even frequencies $f_{k;2,2} = \pm (f_{min} + f_{max} + f_D + \frac{k}{T})$ with an amplitude $B_{k;2} = B_{k;1}$.

It is therefore possible to evaluate not only the round-trip time of the acoustic $t_{a-r}$ of the acoustic wave but also its Doppler frequency. By $A_{k;1}$ and $B_{k;1}$ we can deduce $t_{a-r}$ given that an angle is defined by its cosine and its sine. However, for $k \ge 1$, $|\cos(\frac{\pi k t_{a-r}}{T})|$ and $|\sin(\frac{\pi k t_{a-r}}{T})|$ are bijective only for $t_{a-r} \in [0, \frac{T}{2}]$. It is therefore better to analyze the amplitudes of the peaks for $k = 1$ and only for $t_{a-r} \le T/2$.

## VII. IMPACT OF A TIME-VARYING DOPPLER SHIFT - REAL ASPECT

The Doppler frequency or the Doppler shift corresponds, as mentioned, to the difference between the frequency of a wave emitted at a given time by a transmitter and that of the same wave received by a receiver later. In the case where the frequency of the acoustic wave follows a sawtooth profile, $f_D$ then corresponds to the difference $f_D = f_r(t) - f_e(t - t_{a-r})$ where $f_e$ and $f_r$ are the frequencies of the transmitted and received wave. This Doppler frequency is related to the fact that the wave is emitted by a transmitter in motion relative to the material medium, that the material medium moves (wind) relative to the drone and that the obstacle moves relative to the material medium. We show that:

$$f_D(t) = \frac{2c(v_1 - v_2)}{(c - v_3 + v_2)(c + v_3 - v_1)} f_e(t - t_{a-r})$$

Also, with:

$$f_r(t) = \frac{(c + v_3 - v_2)(c - v_3 + v_1)}{(c - v_3 + v_2)(c + v_3 - v_1)} f_e(t - t_{a-r})$$

Noting by $\beta$ the quantity:

$$\beta = \frac{(c + v_3 - v_2)(c - v_3 + v_1)}{(c - v_3 + v_2)(c + v_3 - v_1)}$$

By imposing $\phi_0 = 0$ we deduce that the reception voltage is of the form:

$$u_r(t) = U_r \cos \left( \beta \pi (f_{max} + f_{min}) (t - t_{a-r}) + \frac{\beta}{\pi} (f_{max} - f_{min}) T \sum_{k=1}^{+\infty} \frac{(-1)^k}{k^2} \cos \left( \frac{2\pi k (t - t_{a-r})}{T} \right) \right)$$

As a reminder:

$$u_e(t) = U_e \cos \left( \pi(f_{max} + f_{min})t + \frac{(f_{max} - f_{min})T}{\pi} \sum_{k=1}^{+\infty} \frac{(-1)^k}{k^2} \cos \left( \frac{2\pi kt}{T} \right) \right)$$

We then have the result of the following product:

$$\frac{2u_r(t)u_e(t)}{U_r U_e} = F_{1,D}(t) + F_{2,D}(t)$$

With this time:

$$F_{1,D}(t) = \cos \left( -(\beta - 1)\pi(f_{max} + f_{min})t + \beta\pi(f_{max} + f_{min})t_{a-r} - \frac{(f_{max} - f_{min})T}{\pi} \sum_{k=1}^{+\infty} \frac{(-1)^k}{k^2} \left( \beta \cos \left( \frac{2\pi k(t - t_{a-r})}{T} \right) - \cos \left( \frac{2\pi kt}{T} \right) \right) \right)$$

And:

$$F_{2,D}(t) = \cos \left( -(\beta + 1)\pi(f_{max} + f_{min})t + \beta\pi(f_{max} + f_{min})t_{a-r} - \frac{(f_{max} - f_{min})T}{\pi} \sum_{k=1}^{+\infty} \frac{(-1)^k}{k^2} \left( \beta \cos \left( \frac{2\pi k(t - t_{a-r})}{T} \right) + \cos \left( \frac{2\pi kt}{T} \right) \right) \right)$$

For $t_{a-r} \ne 0$ by imposing the condition

$$\frac{f_{max} - f_{min}}{f_{max} + f_{min}} \frac{T}{\beta t_{a-r}} \ll 1$$

We can then approximate to order 1 in $\frac{f_{max} - f_{min}}{f_{max} + f_{min}} \frac{T}{\beta t_{a-r}}$:

$$F_{1,D}(t) \approx \cos((\beta - 1)\pi(f_{max} + f_{min})t) \left( \cos(\beta\pi(f_{max} + f_{min})t_{a-r}) + \frac{(f_{max} - f_{min})T}{\pi} \sin(\beta\pi(f_{max} + f_{min})t_{a-r}) \sum_{k=1}^{+\infty} \frac{(-1)^k}{k^2} \left( \beta \cos\left(\frac{2\pi k(t - t_{a-r})}{T}\right) - \cos\left(\frac{2\pi kt}{T}\right) \right) \right) + \sin((\beta - 1)\pi(f_{max} + f_{min})t) \left( \sin(\beta\pi(f_{max} + f_{min})t_{a-r}) - \frac{(f_{max} - f_{min})T}{\pi} \cos(\beta\pi(f_{max} + f_{min})t_{a-r}) \sum_{k=1}^{+\infty} \frac{(-1)^k}{k^2} \left( \beta \cos\left(\frac{2\pi k(t - t_{a-r})}{T}\right) - \cos\left(\frac{2\pi kt}{T}\right) \right) \right)$$

And:
<figure><img src="image_10.png" /></figure>

$$t_{a-r} = \frac{T}{2\pi} \arccos \left( \frac{1}{2\beta} \left( \frac{8\pi (A_{1,1}^{(2)})^2}{(f_{max} - f_{min})^2 T^2} - 1 - \beta^2 \right) \right)$$

It is therefore necessary to choose the size T such as $T = \frac{4P}{c}$ where P represents the detection range of the acoustic wave reflected by a standard obstacle of the operational context of urban drone flight (infinitely large concrete wall) and detected by the receiver. The detection range representing the distance beyond which no return is detected on the receiver. By identifying the useful frequencies, it is possible to quantify β and thus the speed of the obstacle $v_2$ knowing the speed of the drone $v_1$ and neglecting the wind speed $v_3$.

VIII. CONCLUSION

Acoustic devices based on time of flight to perform distance measurements are technologically mature enough and allow to measure distances with a precision even less accurate as the distance to the obstacle increases. The work carried out makes it possible to introduce a new concept of measurement of distance starting from continuous acoustic wave modulated in frequency and based on the Fourier transform of the product of the voltages of control and reception respectively of the transmitter and the receiver. The work allowed to quantify the impacts of some measurement uncertainties on the telemetry performances of this type of technology. First, the theoretical measurements of the impacts of the wind speed or of the obstacle allowed to show that the wind contributes little to this relative telemetric error of the order of 0,21% only whereas the mobility of the obstacle contributes for a speed of 50km/h (cars in town) to induce a maximum relative telemetric error of 4,1%. Subsequently, the work has shown that the Fourier transform of the product of transmission and reception signals can directly assess the time taken for the wave to make a round trip between the technological device and the obstacle because of the amplitudes of Dirac peaks at specific frequencies associated. These same frequencies make it possible to estimate the speed of the obstacle knowing that of the drone compared to the ground and as indicated, by neglecting the impact of the wind speed on the measurement of this one. The technological device that we propose is singular in this atypical behavior and by its theoretical performances. On the condition of not generating too much electronic noise in the receiving circuit, in a general framework, the telemetric error remains lower than 5% for any distance below the detection range of the acoustic return wave, which makes this technological device robust in the complementarity of multiple sensor

$$F_{2,D}(t) \approx \cos((\beta + 1)\pi(f_{max} + f_{min})t) \left( \cos(\beta\pi(f_{max} + f_{min})t_{a-r}) + \frac{(f_{max} - f_{min})T}{\pi} \sin(\beta\pi(f_{max} + f_{min})t_{a-r}) \sum_{k=1}^{+\infty} \frac{(-1)^k}{k^2} \left( \beta \cos\left(\frac{2\pi k(t - t_{a-r})}{T}\right) + \cos\left(\frac{2\pi kT}{T}\right) \right) \right) + \sin((\beta + 1)\pi(f_{max} + f_{min})t) \left( \sin(\beta\pi(f_{max} + f_{min})t_{a-r}) - \frac{(f_{max} - f_{min})T}{\pi} \cos(\beta\pi(f_{max} + f_{min})t_{a-r}) \sum_{k=1}^{+\infty} \frac{(-1)^k}{k^2} \left( \beta \cos\left(\frac{2\pi k(t - t_{a-r})}{T}\right) + \cos\left(\frac{2\pi kT}{T}\right) \right) \right)$$

The Fourier transform of $F_{1,D}(t)$ for $\beta \neq 1$ will allow to detect peaks with frequencies $f^{(1)} = \frac{1}{2}(\beta - 1)(f_{min} + f_{max})$ with an amplitude of $\sqrt{\frac{\pi}{2}}$ but also peaks with frequencies $f_{k,1}^{(1)} = \pm(\frac{1}{2}(\beta - 1)(f_{min} + f_{max}) - \frac{k}{T})$ with an amplitude $A_{k,1}^{(1)}$ such as :

$$A_{k,1}^{(1)} = \frac{(f_{max} - f_{min})T}{2\sqrt{2\pi k^2}} \sqrt{1 - 2\beta \cos\left(\frac{2k\pi t_{a-r}}{T}\right) + \beta^2}$$

And finally peaks with frequencies $f_{k,2}^{(1)} = \pm(\frac{1}{2}(\beta - 1)(f_{min} + f_{max}) + \frac{k}{T})$ with an amplitude $A_{k,2}^{(1)}$ such as $A_{k,2}^{(1)} = A_{k,1}^{(1)}$.

The Fourier transform of $F_{1,D}(t)$ for $\beta \neq 1$ will allow to detect peaks with frequencies $f_k^{(2)} = \pm\frac{1}{2}(\beta + 1)(f_{max} + f_{min})$ with an amplitude $\sqrt{\frac{\pi}{2}}$ but also peaks at frequencies $f_{k,1}^{(2)} = \pm(\frac{1}{2}(\beta + 1)(f_{max} + f_{min}) - \frac{k}{T})$ with an amplitude :

$$A_{k,1}^{(2)} = \frac{(f_{max} - f_{min})T}{2\sqrt{2\pi k^2}} \sqrt{1 + 2\beta \cos\left(\frac{2k\pi t_{a-r}}{T}\right) + \beta^2}$$

And finally peaks at frequencies $f_{k,2}^{(2)} = \pm(\frac{1}{2}(\beta + 1)(f_{max} + f_{min}) + \frac{k}{T})$ with an amplitude $A_{k,2}^{(2)} = A_{k,1}^{(2)}$.

Thus, it is possible to evaluate not only the magnitude $\beta$ by identifying the detected frequencies but also the time $t_{a-r} \le T/2$. In fact, $\cos(\frac{2k\pi t_{a-r}}{T})$ is bijective for $k = 1$ and for $t_{a-r} \le T/2$. We can deduce with for recall $A_{1,2}^{(1)} = A_{1,1}^{(1)} A_{1,2}^{(2)} = A_{1,1}^{(2)}$ :

$$t_{a-r} = \frac{T}{2\pi} \arccos \left( \frac{1}{2\beta} \left( - \frac{8\pi (A_{1,1}^{(2)})^2}{(f_{max} - f_{min})^2 T^2} + 1 + \beta^2 \right) \right)$$

Or again
<figure><img src="image_11.png" /></figure>

measurements for the needs of autonomous
navigation of urban drone.

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