Interferência directa entre radares FMCW. Direct Interference between FMCW radars

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Universidade de Aveiro 2020

Daniel Gonçalves Fernandes

Interferência directa entre radares FMCW

Direct Interference between FMCW radars

Dissertação apresentada à Universidade de Aveiro para cumprimento dos req- uisitos necessários à obtenção do grau de Mestre em Engenharia Eletrónica e Telecomunicações, realizada sob a orientação científica da Professora Doutora Ana Maria Perfeito Tomé do Departamento de Eletrónica, Telecomunicações e Informática da Universidade de Aveiro e do Professor Doutor Daniel Fil- ipe Albuquerque do Departamento de Engenharia Electrotécnica do Instituto Politécnico de Viseu.

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o júri

presidente Professor Doutor Telmo Reis Cunha Professor Associado, Universidade de Aveiro

vogal Doutor Daniel Filipe Simões Malafaia Radar Systems Architecht, Vestas Portugal, Lda

vogal Professora Doutora Ana Maria Perfeito Tomé Professora Associada, Universidade de Aveiro

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agradecimentos Agradeço o apoio financeiro por parte do Instituto de Telecomu- nicações de Aveiro, que suportou este projeto na sua integridade, sendo este trabalho desenvolvido no âmbito do Projeto RETIOT, POCI-01-0145-FEDER-016432.

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palavras-chave Radar FMCW, Interferência Direta, Simulação, Expressôes Al- gébricas, Teoria de Conjuntos

resumo Neste trabalho estuda-se a interferência directa entre radares FMCW. Este estudo poderá ser usado para entender o efeito dos parâmetros dos sinais radar com o uso de expressões algébricas que estimam o número de ocorrências de interferência.

As expressões têm como base a teoria de conjutos para exprimir o sinal de um radar FMCW, que permite uma escrita simples das condições de interferência no domínio da frequência. Para além disto, permite uma adaptação para uma simulação computacional mais eficiente em termos de tempo e memória usada.

Os simuladores criados servem dois propósitos: primeiro para permitir uma simulação rápida e parametrizável um cenário com dois radares, apoiando assim as introduzidas expressões algébri- cas. Segundo, permitem uma forma intuitiva de visualizar o com- portamento de interferência entre radares.

Por fim, foram feitas experiências em laborátorio para confirmar, não só, as expressões algébricas, assim como, os simuladores de- senvolvidos.

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keywords FMCW radars, Direct Interference, Simulations, Algebraic Ex- pressions, Set theory

abstract In this work is studied the direct interference between FMCW radars using the algebraic expressions that outputs the expected chirp interference rate. In hope that by understanding the effect of each radar parameters on the radar system, it is possible to reduce or eliminate the chirp interference rate within the radar system.

The expressions are inspired by set theory in order to better gather the numerous conditions that limit the interference rate, such as the frequency window and the chirps duration, to reduce the com- plexity of the FMCW radar signal, and last, to translate to an more efficient use of the simulation computations in regards to time and memory used.

The simulators created serves two purposes: first to allow to sim- ulate interference between two radars with different configura- tions. Supporting the expressions interference rate output. Sec- ond it gives an intuitive way to visualize and understand the interference behavior.

Lastly, was experimented with success using radar development kits to confirm the theorical and simulated results.

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Nomenclature

Acronyms

A/D Analog to Digital converter.

CW Continuous Wave

FMCW Frequency Modulated Continuous Wave OFCW Orthogonal Frequency Continuous Wave SFCW Stepped Frequency Continuous Wave LO Local Oscillator.

LPF Low-Pass-Filter.

RF Radio Frequency.

Constants and Parameters Bi Bandwidth of the chirpi.

c Speed of light in vacuum constant.

Fi Initial frequency of the chirpi.

S Chirp signal frequency slope.

T_A Chirp periodicity of the radar A, obtained by summing the duration of chirps and the idle time.

Ti Duration of the chirpi.

Sets ans Intervals

D Interference interval.

F Frequency interval associated to a filter.

C Modeled chirped signal set, cointaning the pair of pointshtime, frequencyi.

I Modeled de-chirped interference set, cointaning the pair of pointshtime, frequencyi. S Modeled de-chirped signal set, cointaning the pair of pointshtime, frequencyi.

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Chapter 1

Introduction

Recently, several applications of radar based systems have been described due to the emergence of inexpensive and highly integrated RFICs available on self-contained development kits. These devices permit rapid prototyping and testing of radar-based solutions. These development kits provide flexible hardware and software which can be easily parameterized to deal with a diversity of opera- tional conditions [1], [5]. In particular, the Frequency Modulated Continuous Wave (FMCW) radar based systems have been introduced in several emerging applications including gesture sensing, fall detection, and human activity categorizing [2], [3].

Furthermore, the development of robotic technologies over recent decades, short-range and indoor localization have also been a hot research topic [4]. With camera-based localization technologies there are difficulties in getting accurate depth information of targets or with beacon-based solutions and inertial sensors require the targets to carry the devices [2]. With radar based systems the range, the speed and the targets detection are easily achieved. But in indoor environments where several robots can be using radar for obstacle detection problems associated with the interference between independent radars might be present.

1.1 FMCW and other common radar signals

Some of the various waveform used in radars are shown in the Figure 1.1. Each waveform design is characterized by its modulation, as for example, the Continuous Wave (CW) radar transmits a fixed tone signal and a frequency modulated continuous wave signal (FMCW) varies its frequency on its uptime. Furthermore, there are variants of these signal such as the stepped frequency continuous wave (SFCW) that utilizes CW signals with incremental frequency tones, or even the orthogonal frequency division multiplexing waveform (OFDM) that arranges frequencies tone in sequences [6].

CW FMCW SFCW OFDM

Figure 1.1: Common waveform of radar signals.

For each of the waveforms designs there are advantages and disadvantages in the way the signal is

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processed, the accuracy of the range and doppler estimation [6]. Nevertheless, FMCW radars excels for a simpler front end and the advantage of using a lower a sampling frequency.

1.2 Scope of the work

This work approaches the direct interference between FMCW radars. This phenomenon can occur when a radar catches signal from a second radar with a common frequency bandwidth. For comparison, a FMCW signal with one target detected will present a strong energy peak at a certain frequency value as the Figure 1.2 shows [7]. On the frequency domain the detected power peak will then corre- late to the distance between the radar and the target.

Target

power density

frequency

Figure 1.2: Typical FMCW output signal with one target detected.

Direct interference has two different common manifestations on the output signal [8]. First is the increase level of the noise floor, as represented in Figure 1.3a. Where the clean signal (dashed blue), with a clear peak representing the target, can be undetectable with the presence of an interference (filled red). Second, as represented in Figure 1.3b is the appearance of a ghost target. Where the signal with interference (filled red) contains an additional peak from the expected clean signal (dashed blue).

power density

Target

frequency

(a) Noise floor increased in relation comparison between a clean signal (dashed blue) with the same signal with interference (filled red).

power density

Ghost Target Target

frequency

(b) Ghost target appearance using a comparison between a clean signal (dashed blue) with the same signal with interference (filled red).

Figure 1.3: Examples of the direct interference manifestations.

1.3 Objective

The main objective for this work is to study the occurrence and process of the direct interference between FMCW radars. By the end of this document, is presented tools for quantifying the mutual interference over time. One of the tools is based on quick parametric simulations, and the second based on algebraic expressions.

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1.4 Document structure

This document starts by exploring the FMCW radar fundamentals on Chapter 2. The following Chap- ter 3 introduces the model used on the interference simulator and in the stochastic study. Chapter 4 has the stochastic study with the algebraic expression in order to arrive at the expected interference rate. In Chapter 5 is documented the simulator user interface, its overall mechanics and extra func- tionalities. The experimental results and interpretations are in Chapter 6. Finally, the last Chapter 7 contains a overview of the work developed with a proposal for future work.

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Chapter 2

FMCW Radar

2.1 FMCW Radar Signal

In a typical scenario the FMCW Radar transmits a Radio-Frequency (RF) wave and receives the reflections of this transmitted signal produced by the surrounding objects. By performing correlation of the received and transmitted signal, it is possible to deduce the target (such as objects, people, ...) distance from the radar. Figure 2.1 introduces a simple scenario where a radar transmits a single linear chirpxc(t), and listens for the echo coming from an object at distanced.

Radar Tx

Rx

Object transmitted chirp

back-scattered chirp

Figure 2.1: Scenario of a FMCW radar with a single target at distanced.

The chirp is a sinusoidal wave whose instantaneous frequencyf(t)increases linearly with time.

Considering the example from Figure 2.2, a chirp signal is defined as:

x_c(t) = cos(θ(t))

the instantaneous frequency comes from the phase derivation, as follows:

f(t) = 1 2π

dθ(t)

dt =St+F₀ (2.1.1)

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t₀ t₀ + T_c Time

-1 0 1

Amplitude

Chirp signal: Time-Amplitude

x_c(t)

Figure 2.2: Chirp signal on the left, with the representation of the instantaneous frequency on the right.

whereF0is the starting frequency, and the instantaneous frequency slopeSis constant and comes from the second derivative of the signal phase:

S= 1 2π

d²θ(t)

dt² (2.1.2)

The sinusoidal phase in function of time could be obtained by the integration of equation 2.1.1:

θ(t) = 2π S

2t+F₀

t+θ₀ (2.1.3)

whereθ₀is the initial phase. As a result of this, the chirp signal can be written as:

x_c(t) = cos

2π S

2t+F₀

t+θ₀

(2.1.4) To summarize, the chirp signal, as used in this document, is a sinusoidal wave whose frequency increases linearly in time with slopeS. However, at its initial timet₀ has the initial instantaneous frequency ofF₀, that results in the signal:

xc(t−t0) = cos

2π S

2(t−t0) +F0

(t−t0) +θ0

(2.1.5) as for the signalx_c(t−t₀)domain refers to the chirp duration:

dom{x_c(t−t₀)}= [t₀, t₀+T_c] (2.1.6) and as for the bandwidth, as seen in Figure 2.2, comes from the range off(t):

rg{f(t−t0)}= [F0, F0+B] (2.1.7) whereB=STcis the chirp bandwidth.

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time

amplitude

Figure 2.3: Representation of a transmitted chirpxc(t) that is back-scattered to the source after a delayτ and attenuation ofR.

In the Figure 2.3 is represented an example of an emitted chirp with amplitude of1, and its back- scattered with attenuationR. The chirp signal has a defined time interval that starts att0 and last for chirp durationTc.

2.2 Radar Front-end

In order to fulfill the objective of this work, is only necessary to approach the basics of a FMCW radar. In the Figure 2.4 is presented a top-level architecture schematic of the reception channel of FMCW radar.

LPF

A/D

A/D j

+ LO

RX

Figure 2.4: Simplified block diagram of the receive system channel, from a Texas Instruments FMCW radar using complex sampling architecture.

Assuming that the signalxc(t−τ), as presented in Figure 2.3, at the receiving antenna it was produced by a target reflection of the signal transmitted by the radar itself, then the homodyne Radio- Frequency (RF) mixing process reads:

z_cÎ(t) =Rx_c(t−τ)×xÎ_c(t) (2.2.1) z^Q_c(t) =Rx_c(t−τ)×x^Q_c(t) (2.2.2) wherexÎ_c(t) = cos(θ(t))is the local oscillator (LO) in-phase signal version ofx_c(t) andx^Q_c(t) = sin(θ(t)) represents the quadrature signal ofxc(t). The coefficientR represents the amplitudes of

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both signals and, without loss of generality, its effects was not taken into account (R = 1). Further- more, this mixing process results in the in-phase and quadrature signals:

z_c^I(t) = cos(θ(t−τ)) cos(θ(t)) = cos(θ(t) +θ(t−τ))

2 +cos(θ(t)−θ(t−τ))

2 (2.2.3)

z_c^Q(t) = cos(θ(t−τ)) sin(θ(t)) = sin(θ(t) +θ(t−τ))

2 −sin(θ(t)−θ(t−τ))

2 (2.2.4)

Both signals are composed by a high frequency component above2F0, resultant from the sumθ(t) + θ(t−τ), and low frequency component resultant from the subtractionθ(t)−θ(t−τ). After applying the low-pass filter (LPF) present in the front-end the high frequency component will be strongly attenuated, almost removed, producing the following signals:

y^I_c(t) =LPF{z_c^I(t)}= 1

2cos(θ(t)−θ(t−τ)) y^Q_c(t) =LPF{z^Q_c(t)}=−1

2sin(θ(t)−θ(t−τ)) where the phaseφ(t)is given by:

φ(t) =θ(t)−θ(t−τ)

= 2π S

2t+F0

t+θ0−

2π S

2(t−τ) +F0

(t−τ) +θ0

= 2π(Sτ)t+ 2π

τ F₀−τ²S 2

= 2π(Sτ t) +φ₀

(2.2.5)

where itsSτ is the intermediate frequency IF and φ0 the initial phase. Finally after sampling at f_s=T_s⁻¹, it is assigned the quadrature signalyc^Q(nT_s)to the complex axis. The outputy_c(nT_s)is a complex signal where the real component comes from the in-phase signaly^I_c(nTs)and the imaginary comes from the quadrature signaly^Qc(nTs).

y_c(nT_s) =y_c^I(nT_s)−jy_c^Q(nT_s)

= cos [φ(nT_s)] +jsin [φ(nT_s)]

= exp{j(2πSτ nT_s+φ0)}.

(2.2.6)

Where its intermediate frequencyf_IF comes from the derivative of the phaseφ(t):

f_IF = 1 2π

d(φ(t))

dt =Sτ. (2.2.7)

The intermediate frequency is constant and proportional to the delay of the received signal in relation to transmission. The domain comes from the interception of the transmitted and received chirp:

dom{y_c(nTs)}=dom{x_c(nTs−t0)} ∩dom{x_c(t−t0−τ)}

= [t₀, t₀+T_c]∩[t₀+τ, t₀+T_c+τ]

= [t₀+τ, t₀+T_c]

(2.2.8)

2.3 Signal Processing

In this section is presented the steps to estimate the target distance in relation to a sensing radar. It will be based on the radar documentation [9] used for the experimental work (see Section 6).

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2.3.1 Target Range Estimation

The delay between the sent and the received pulse will beτ and corresponds to the wave travel time.

From the radar to a target and back again to the antenna add ups to the double distancedthat the wave has to travel. Given the speed of lightc, this is expressed as:

τ = 2d

c . (2.3.1)

Arranging equation 2.2.6 with 2.3.1, with a target at distancedfrom the radar, results in the following function:

yc(nTs) = exp

j

2π

S2d c

nTs+φ0

(2.3.2) For a single target scenario,y_cresults in sinusoidal wave with single frequency tone off_IF =S^2d_c. In order to extract the frequency tone, is performed a Fast Fourier Transform (FFT) over the signal duration:

d= c

2S ×f_IF (2.3.3)

The distance resolution is directly proportional with the FFT resolution:

d_res= c

2S ×f_res= c 2S × 1

N Ts

(2.3.4) The maximum detectable distance is related to the maximum intermediate frequencyf_max, bounded by the Nyquist frequency it is often given by the radar manufacturer. Thus the maximum detectable distance comes from equation 2.3.3:

dmax= c

2S ×fmax (2.3.5)

2.3.2 Example of a single target scenario

It may be worthwhile to present an example of the expressions in a real scenario. From the setup presented in the Chapter 6, was placed an object at distanced from the radar. In the Figure 2.5, shows a RF-output signal spectrogram. This, present a single frequency tone at1.48 MHz caused by reflection of an object at2.8 mfrom the radar. The Analog-to-Digital converter module uses the frequency samplefsof10 MHz.

Figure 2.5: Frequency spectrum ofyc(t) for a scenario with a single target at2.8meters from the radar.

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Notice that theyc(t)has a frequency tone of 1.48 MHz. Given that the radar parameter, chirp slope, S = 80 MHzµs⁻¹, and the speed of light c = 3.00×10⁸ m s⁻¹, from equation 2.3.2 is estimated the distance:

fIF =Sτ =S2d

c (2.3.6)

d= cfIF

2S = 3.00×10⁸×1.48×10⁶

2×80.00×10¹² = 2.8 m (2.3.7) Notice that the radar has interval of detectable range that corresponds to range of frequencies ob- served. The maximum range of an object to be detected comes from the maximum frequencyf_max. For the sake of this example, is assumed to be half of the sampling frequency, that means5 MHz.

Therefore the maximum detectable distance is given by using [2.3.5]:

d_max = cf_max

2S = 3.00×10⁸×5×10⁶

2×80.00×10¹² = 9.38 m (2.3.8) The frequency resolution given by the ratio between the sampling frequencyfS with the number of samples takenN. For this example, is taken430samples per chirp, therefore:

f_res= f_s

N = 10×10⁶

430 = 23.256 kHz (2.3.9)

finally the distance resolution from equation 2.3.7:

dres= cfres

2S = 3×10⁸×23.256×10³

2×80.00×10¹² = 4.36 cm (2.3.10) FMCW radar can estimate more than the target range, such as the radial velocity of the target, and its azimuth, if the radar has multiple antennas. But since these estimations will not be refereed anywhere in this work, it is not approached in this overview of the radar architecture and signal processing.

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Chapter 3

Modeling the instantaneous frequency of the FMCW signal

This chapter is dedicated to introduce the model used to define the FMCW signal. The underlying objective of this work is to analyze the direct interference between FMCW radars. For that, in this work is created a model to describe the FMCW signal. The proposed model removes unnecessary parameters of the signal while accentuating what makes direct interference happen.

3.1 Model Introduction

The proposed model maps sinusoidal signals as a pairh time, frequency i. For example, consider a signalx(t)that combines various distinct frequency tones. This can be written as:

x(t) =x₁(t) +x₂(t) +x₃(t) +...+x_n(t) =

n

X

i=1

x_i(t)

where x1(t), x2(t), x3(t), ..., xn(t) are distinct sinusoidal tones with domainΓ1, Γ2, Γ3, ... Γn

respectively. In order to turn the sum possible each signal is set to zero outside the tone interval:

x₁(t) =A₁cos(2πF₁(t−t₁) +φ₁) ∀t∈Γ₁, otherwisex₁(t) = 0 x2(t) =A2cos(2πF2(t−t2) +φ2) ∀t∈Γ2, otherwisex2(t) = 0 x3(t) =A3cos(2πF3(t−t3) +φ3) ∀t∈Γ3, otherwisex3(t) = 0

...

xn(t) =Ancos(2πFn(t−tn) +φn) ∀t∈Γn, otherwisexn(t) = 0

When a spectrogram is applied tox(t) with an infinite resolution, as shows in Figure 3.1 left plot, results in power peaks that coincide with every tone frequency and domain, while every other point in thehtime,frequencyihas energy tending to negative infinite in dB.

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Figure 3.1: Comparison between the spectrogram of signalx_kwith its modeled representation.

The frequency model introduced applied to the signalkgrabs the frequency tones of each sinusoidal component. Applying equation 2.1.1 to the phase of each sinusoidal toneθi(t) = 2πFi(t− t_i) +ω_i, gives the instantaneous frequencyf_i(t):

fi(t) = 1 2π

d(θi(t))

dt ∀t∈Γi

=Fi ∀t∈Γi

Becausexi(t) isn’t continuous in the end points of Γi, leadsfi(t) to be undefined at those points.

In order to avoid unnecessary notation confusion, it is attributed to the endpoints the value of the function that defines instantaneous frequency. In this case, each contributionihas a constant value fi(t) =Fifor every instant time that defines each tone.

For t /∈ Γ_i the instantaneous frequency it is left undefined, as the signals does not have energy outside of the intervalΓ_i. Notice that, an zero value frequency would mean an DC component and not lack of energy. Maintaining this way, the definition of domain used previously in this work.

To conclude model introduction, it is used set theory to organize in a single expression the com- bination of paired pointshtime,frequencyito represent the frequency contribution of the signalsx1, x2,x3, ...,xn. Leti∈ {1, 2, 3, ..., n}, the instantaneous frequency ofxi(t)can be written as a set of paired pointshtime,frequencyi:

X_i ={ht, f_i(t)i |t∈Γ_i}

whereX_i is a set with abscissa time and ordinate frequency. The sum that defines the signalk, in the frequency model is converted to a union of the tones. For alli={1,2,3, ..., n}the signalX as:

X =[

i

X_i=X₁∪ X₂∪ X₃∪ · · · ∪ X_n (3.1.1) This model works as a spectrogram to a signal, with an additional step of excluding the low energy spectrum. For example, the spectrogramS{.}ofx(t)returns a tripleh time, frequency, power i. Where the time ist ∈ Γ, the frequency isf ∈ F, beingFthe frequency window in observation.

Then, for each given time and frequency is calculated the powerρ. This model gathers pair time- frequency points where the power is greater than a certain thresholdρ_th.

X ={ht, fi | ∀ ht, f, ρi ∈S{x(t)}(ρ≥ρ_th)}

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It is assumed that there exists always a thresholdρththat includes all the time-frequency originated by the functions that describes the behavior the frequency over time, while excluding all the unexpected signal such as noise or nonlinearities.

As a last note, is important to mention that is impossible to completely recover the original signal after modeling. For example, from the modeled signal:

X ={ht, fi | ∀t∈Γ(f =f(t−t0))}

the inverse transformation would result in the function family:

x(t) =Acos

2π Z

f(t−t₀)dt+φ₀

∀t∈Γ

Information such as the signal amplitudeA or its initial phaseφ0 are lost, however the model presented for FMCW signal brings benefits for the objective of this work. Such as simplifing the algebra from a sinusoidal function with variable frequency to a first degree polynomial, in the case of a chirp.

As it is detailed in this chapter, the model introduced avoids array multiplications and replaces them by less complex additions and subtractions. All of this not only contributes to a significant shorter simulation time while also requiring less memory.

3.2 Modeling the instantaneous frequency of a chirp

Having introduced the frequency model, is time to apply to the FMCW signal. As already mentioned in chapter 2 section 1, chirp is a sinusoidal wave whose instantaneous frequency increases linearly with time. Consider the spectrogram and the model representation of typical chirp in the Figure 3.2.

Chirp Spectrogram

tk t

k+T k

Time Fk

Frequency

-80 -70 -60 -50

Power/frequency (dB/Hz)

tk t

k + T k

Time Fk

Frequency

Instantaneous frequency model

Figure 3.2: Spectrogram of a chirpkvs instantaneous frequency model of the chirp (C_k).

As mentioned before, the model maps the frequency band present from the signal spectrogram to a time-frequency plane in the form of a set. Thus for the chirpkdrawn in the Figure 3.2 starts att_k and lasts forT_k(Γ_k= [t_k, t_k+T_k]). While the instantaneous frequency grows fromF_kwith a steady slope ofS_k. Modeling the instantaneous frequency of chirpkgives:

C_k ={ht, f_k(t−t_k)i | ∀t∈Γ_k} (3.2.1) where the instantaneous frequencyf_k(t)is:

fk(t) =Skt+Fk (3.2.2)

At this point is already evident the advantages of representing a chirp by its intantaneous frequency in opposition to a time-amplitude. As an example, if we had to sample the signal for simulation purposes, it would require by the sampling theorem more than double of its maximum frequency.

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While for the modelled chirp, at most, it is only necessary a few number of points, at the limit first and last points of the chirp to represent.

3.3 Modeling the radar architecture

This sections applies the model to the Radar Architecture, mentioned in the previous chapter, such as the mixer and the LPF blocks.

 



⁻_,



⁺_,



+ +



, ,

a)

b)

c)

Figure 3.3: Evolution of the set of instantaneous frequencies, with a transmitted chirpiand a received chirpk: a) mixer inputs, b) mixer output, c) RF front-end output.

Consider the generic transmitted chirpiand received chirpk, represented in Figure 3.3a. Chirpsi andkhave respectively start timet_iandt_k, initial frequencyF_iandF_kand frequency slopeS_iandS_k The output setS_i,k is the result of the filtered mixing of chirpsiandk. In order toS_i,kto exist, chirp iandkmust overlap in time,Γ_i∩Γ_k6=∅.S_i,kis described as an ordered-set pair of time-frequency:

S_i,k =W ∩(S_i,k⁺ ∪ S_i,k⁻) (3.3.1)

whereS_i,k⁻ represents the signal frequency set that results from the frequency subtraction:

S_i,k⁻ ={ht, f_i−f_ki | ht, f_ii ∈ C_i, ht, f_ki ∈ C_k}, (3.3.2) S_i,k⁺ represents the signal frequency set that results from the frequency addition:

S_i,k⁺ ={ht, f_i+f_ki | ht, f_ii ∈ C_i, ht, f_ki ∈ C_k} (3.3.3) The component of high frequenciesS⁺locates around the double of the carrier frequency. These are always filtered by a low-pass filterW:

W ={ht, fi ∈(R×F)} (3.3.4)

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whereFrepresents the bandwidth of the de-chirped signal that will be sampled. A consequence is thatW ∩ S⁺=∅for every given timet. Finally, intersectingWto the union of the signal, is obtained the radar output for two given chirpsiandk:

S_i,k =W ∩(S_i,k⁺ ∪ S_i,k⁻)

=W ∩ S_i,k⁻

={ht, f(t)i ∈ W | ∀t∈(Γi∩Γ_k) [f(t) =fi(t−ti)−f_k(t−t_k)]}.

(3.3.5)

applying the chirp instantaneous frequency equation 3.2.2, the signal is:

f(t) =fi(t−ti)−f_k(t−t_k)

= [Si(t−ti) +Fi]−[Sk(t−tk) +Fk]

= (S_i−S_k)(t−t_i) +S_kδ_i,k+ (F_i−F_k).

(3.3.6)

Functionf(t)represents the general equation for any two chirpsiandk, where the initial time ofk occursδi,k after the initial time ofi(δi,k =tk−ti). For the case wherekis a reflection ofi, means that they have the same parametersS_k=S_i,F_k =F_i, and the off-set equals to the total time of wave propagationδ_i,k =τ. Applying to equation 3.3.6, results in:

f(t) =S_kτ (3.3.7)

that is a constant intermediate frequency in concordance with what is expected, see section 2.2.

Other case, is whenkis not a reflection ofi, meaning that it could be the result of an interference caused by a second radar. Consequently there is the possibility of the chirps having different parameters. This would lead to ghost targets and/or in the increase of the noise floor.

3.4 Modeling FMCW Frames

A typical FMCW radar transmits bursts of consecutive chirps called frame. Figure 3.4 shows example of the signal transmitted by a FMCW radar. The first frame contains theN chirps{C₁, C₂, ..., C_N}, and after an off-duty repeats the signal transmitted with the frame{C₁⁰, C₂⁰, ..., C_N⁰ }.

Figure 3.4: Example of a modeled FMCW chirped signal frame.

To close the FMCW signal model notation, is shown an example of a typical frame signal. For that, consider the Figure 3.5. Where it is transmitted the frame with the following chirps{C₁,C₂,C₃}.

Each chirp gets backscattered from two targets resulting in the chirps{C_a,C_b,C_c,C_d,C_e,C_f}.

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a)

b)

Figure 3.5: a)Sequence of transmitted chirps (C₁, C₂, C₃), received chirps (C_a, C_b, ..., C_f),b) the output signal.

The output signalS, from the frame is the grouping of all chirps interactions, and therefore signal is written as:

S =[

i

[

k

S_i,k =S_1,a∪ S_1,b∪...∪ S_3,f (3.4.1) Another detail to retrieve from this example, is the notation for the transmitted chirp with two reflections. In order to express the signalS₂it is only needed to incorporate the effects of chirpscandd.

While chirpsa,b,eandf wouldn’t be considered once don’t overlap with chirp2in time and thus S_2,a,S_2,b,S_2,e andS_2,fare empty sets. Therefore starting from the equation 3.3.5 comes:

S₂=[

k

S_2,k=S_2,c∪ S_2,d. (3.4.2)

As from the frequency tones ofS₂ comes from the expressions 3.3.5 and 3.3.6:

S_2,c={ht, f(t)i ∈ W | ∀t∈(Γ2∩Γc) [f(t) =f2(t−t2)−fc(t−tc)]}. (3.4.3) S_2,d={ht, f(t)i ∈ W | ∀t∈(Γ₂∩Γ_d) [f(t) =f₂(t−t₂)−f_d(t−t_d)]}. (3.4.4) The processed received signal from the reflections of chirp2is two frequency tones seen Figure 3.5 b). To conclude, the model presented arranges all the instantaneous frequency into a plot of time- frequency. Each instant of time, could have any number of values of frequency. Differing from the spectrogram of a time-amplitude model signal that assigns each instance of time an array of frequency and power values.

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Chapter 4

Interference between FMCW radars

This chapter is dedicated to the stochastic study of direct interference between FMCW radars. It is introduced and deduced algebraic expressions that predict or estimate the interference probability.

4.1 Interference scenario

Radar A

(observer)

Targets

Radar B

(interferer)

Tx

Rx Tx

Figure 4.1: Scenario of a Radar A being interfered by the Radar B.

In this scenario, radar A does not only receives echoes from its transmission, but also the signals transmitted by radar B. If the signalC_kis similar enough, it brings the possibility of direct interference.

The received signalX could be defined as:

X =S ∪ I (4.1.1)

, whereScombines the transmitted signalC_iwith its reflectionC_j andIthat combines the transmitted signal with the Radar B interfering chirpsC_k. This can be detailed as:

X =S ∪ I =



 [

i

[

j

S_i,j



∪ [

i

[

k

I_i,k

!

=[

i



 [

j

S_i,j∪[

k

I_i,k





(4.1.2)

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as for the signalS_i,j, the indexirefers to transmitted chirps, and for j refers to the echoes of the transmission from the targets. For the interferenceI_i,k, the indexi has the same meaning, but k refers to all the chirps that radar A receives from radar B. The setI_i,k represents the influence of the interfering chirpkonto chirpi.

4.2 Condition for direct interference

The introduced model gives a clear advantage on how to define and detect the interference. For instance following the Figure 4.1, the interference signalI_i,kthat comes from chirpibeing interfered by chirpkis defined as:

I_i,k ={ht, f(t)i ∈ W | ∀t∈(Γ_i∩Γ_k) [f(t) =f_i(t−t_i)−f_k(t−t_k)]}. (4.2.1) whereΓi andΓ_kare the time interval of the respective chirps,fi(t)andf_k(t)are the functions that defines the instantaneous frequency of each chirp and finallyt_i andt_k are the initial time of each signal. Similar to radar signalSdetailed in section 3.3, the interference signalI is constricted to the spaceW as a result of the filter present in the radar.

With this model, the existence of interference between the chirpsiandkcan simply be written as:

C : I_i,k 6=∅ (4.2.2)

ifI_i,k is an empty set, then statement C is false meaning chirp k does not interfere with chirpi, otherwiseCis true and therefore chirpkinterferes with chirpi.

4.2.1 Developing the interference condition

Having the conditionCestablished it will be a start to quantify arithmetically the chirp interference probability. The next step, is to develop the expression until we reach the radar parameters. Therefore from the condition 4.2.2:

C:I_i,k 6=∅ → I_i,k⁻ ∩ W 6=∅

→ {∀t∈(Γ_i∩Γ_k) [∃t:f(t)∈F]} (4.2.3) whereI_i,k⁻ is the unfiltered signal ofI_i,k andFis the filter band-pass interval . As Figure 4.2 shows, if the signalI_i,k⁻ has any ordinate within the filter intervalF, thereforeCis true.

+ +

 



⁻_,



Figure 4.2: Mixer with input (top) the transmitting chirp set C_i and the interfering chirp set C_k, resulting in an interference (bottom) with an inclined instantaneous frequency function.

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The instantaneous frequency functionf(t)is given by the equation 3.3.6:

f(t) = (Si−S_k)(t−ti) +S_kδ_i,k+ (Fi−F_k) (4.2.4) as chirpkmay have different parameters from chirpi, such as frequency slopeS_i andS_k, the initial frequencyFi and Fk. The result is, in opposition to a clean signal, is the possibility of an sloped instantaneous frequency function ofSi−S_k, as Figure 4.2 shows. For the sake of simplicity, the time reference is fixed to initial time of chirpi, giving:

Γ_i= [0, T_i] Γ_k= [δ, δ+T_k]

whereδ is the off-set between chirps. In the final form, the interference conditionCfor chirpsi andkis written as:

C :{∀t∈(Γ_i∩Γ_k)| ∃t: (S_i−S_k)t+S_kδ+ (F_i−F_k)∈F} (4.2.5) Following the condition for interference announced comes the study of how the condition is met. And from all the parameters,δ is the only that can assume a random behavior. Thus in order to study the interference probability is important to understand how does the variable off-setδaffects interference.

The following section studies the different effects of the off-set on the occurrence of interference.

4.2.2 Interference Interval

This section introduces the interference interval as the solution set for the condition of interference, given the variability of off-set and the parameters of the transmitted and interfering chirps.

Figure 4.3: Mixing input (left) and output (right) of a transmitted chirp with a group of interference chirps.

This example of the Figure 4.3 gives an idea that there is a interval of off-set values that would result in interference. As the Figure suggests, forδkequal toδminorδmaxoutputs interference signal in the in-band frequency limits. While for anyδbetweenδminandδmax, such as the off-set ofC_k, it always outputs interference.

The interference intervalDrepresents all the values of the off-set between chirpsδ that results into a interference. The objective is to deduce algebraically the span of this interval, which can be written as:

D= [δmin, δmax] (4.2.6)

whereδminandδmaxare the limits of the interference interval. Consequently, the interval length is given by:

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∆ =δ_max−δ_min. (4.2.7) The value of∆ gives already an idea of the probability of interference. For example, if∆ is zero, this means that interference it is not possible between the chirpsiandk. Later in this work interference interval length∆will reappear to estimate the chirp interference probability. Given the multiple parameters of the chirp, the deduction of the algebraic expressions are divided into three cases.

4.2.2.1 First case: when the slopes are equal

For the first case, the transmitting and interfering chirps frequency slopeS_i andS_k respectivelyS_k are equal. Therefore, the equation 4.2.4 is simplified to:

f(t) =S_kδ+ (F_i−F_k) (4.2.8) The interference instantaneous frequencyf(t)is constant in time, as seen in Figure 4.4 a). Also, the domain or the duration of the signalI_i,khas to be obtained first. For the duration intervalsΓ_i= [0, T_i] andΓ_k = [δ, δ+T_k], we have that the overlapΓi∩Γ_kcannot be empty. Therefore, the first pair of off-set limits comes from:

Γi∩Γ_k6=∅ =⇒(δ+T_k≥0)∧(δ≤Ti)

⇔(δ≥ −T_k)∧(δ ≤Ti) (4.2.9)

and extracted the maximum and minimum value ofδgives:

δ^(t)_min=−T_k∧δ_max^(t) =Ti (4.2.10) These limits are represented in the Figure 4.4 b) and 4.4 c). Which shows that any slight increase or decrease of the valueδ out of these bounds, causes the chirps domain to stop overlapping and the interference impossible.

As for the frequency limits, they ensure that the instantaneous frequency is within the in-band intervalF, and is written as:

F_min ≤f(t, δ)≤F_max. (4.2.11) Starting from the maximum valueF_maxoff(t, δ), for any valuet, equation 4.2.8 tells that the instantaneous frequency is constant for anyt, as seen the Figure 4.4 a). Assuming thatSk is positive, the frequency limited maximum off-setδmax^(f) comes from:

f(t, δ)≥F_min

⇔S_kδ+ (F_i−F_k)≥F_min

⇔δ≥ F_min−(F_i−F_k) Sk

(4.2.12)

f(t, δ)≤F_max

⇔S_kδ+ (F_i−F_k)≤F_max

⇔δ ≤ F_max−(F_i−F_k) Sk

(4.2.13)

and extracting limits of the inequalities gives the maximum and minimum value ofδ:

δ_min^(f) = Fmin−(Fi−Fk)

S_k (4.2.14) δ^(f_max⁾ = Fmax−(Fi−Fk)

S_k (4.2.15)

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

, max

min

 

max

= min ( )max

,

 

max

min

,

 

max

( ) min max

,

 

max

min

, ( )min

( ) =−

min

a) b) c)

d) e)

Figure 4.4: All the possible type of limits for chirps with equal slopes.

For the case where the transmitted and received chirp have equal slopes, the interference interval is defined by the following inequalities:

D=

( δ^(t)_min ≤δ≤ δmax^(t)

δ^(f_min⁾ ≤δ≤ δ_max^(f) = h

max{δ_min^(t) , δ^(f_min⁾}; min{δ_max^(t) , δ_max^(f⁾ }i

(4.2.16)

4.2.2.2 Second case: when the transmitted slope is larger than the interfering

The instantaneous frequencyf(t)ofI_i,k for the transmitting chirp frequency slopeS_iis larger than the interferingS_k:

f(t, δ) = (S_i−S_k)t−S_kδ+ (F_i−F_k)

increases linearly in time with(S_i −S_k) slope. Thus, in order to determine the maximumδ^(fmax⁾ , it must be considered the minimum of the function that equals the maximum frequency. Therefore the variablethas to be minimized, as follows:

f

min{t}, δ^(f_max⁾

=Fmax.

As the interference signal timetis limited by the transmitting and interfering chirps domain, therefore min{t}has two solutions. That is the initial time of the interfering chirpt_k =δor the initial time of the transmitted chirpt_i = 0. Deducing for the first solution the maximum off-setδ^(fmax⁾ :

F_max= (S_i−S_k)δ^(f_max⁾ +S_kδ_max^(f) + (F_i−F_k) that solving forδmax^(f⁾ :

δ_max^(f) = F_max−(F_i−F_k) S_i

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As for the second solution, calledδmax^(t,f):

Fmax=Skδ_max^(t,f)+ (Fi−Fk) solving forδ_max^(t,f):

δ_max^(t,f⁾= Fmax−(Fi−Fk) S_k

For the minimum off-set, it must be considered the maximum of the function that matches the minimum frequency. Asf(t)increases linearly with time, thereforethas to be maximized.

f

max{t}, δ_min^(f⁾

=F_min

Again, because the interference timetis limited by the chirps domain, therefore themax{t}has two solutions. First the end time of the interfering chirp domainδ_k+T_k, and the end time transmitted chirp domainTi. Applying tof(t, δ)and solving forδmax^(f) andδ^(t,fmax⁾:

δ_min^(f) = F_min−T_k(S_i−S_k)−(F_i−F_k) S_i

δ_min^(t,f)= Fmin−Ti(Si−S_k)−(Fi−F_k) S_k

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

,

max

min

 

max

= min ( )max

,



max

min

,

 

max

( ) min max

,



max

, min ( )min

( ) =−

min

a) b) c)

d) e)



max

( , ) min min

,



max

min

,

( , ) max

f) g)

( )+

min

Figure 4.5: All the possible type of limits for chirps with different slopes, being the interfered chirp slope larger.

4.2.2.3 Third case: when the transmitted slope is smaller than the interfering

The instantaneous frequencyf(t)ofI_i,kfor the transmitting chirp frequency slopeS_iis greater than the interferingSk:

f(t, δ) = (S_i−S_k)t−S_kδ+ (F_i−F_k)

decreases linearly in time with(S_i−S_k)slope. Thus, in order to determine the maximum δmax^(f⁾ , it must be considered the minimum of the function that equals the maximum frequency. Therefore the variablethas to be maximized, as follows:

f

max{t}, δ_max^(f)

=Fmax.

As the interference signal timetis limited by the transmitting and interfering chirps domain, therefore max{t}has two solutions. First the end time of the interfering chirp domainδk+Tk, and the end time transmitted chirp domainT_i. Deducing for the first solution the maximum off-setδ^(fmax⁾ :

Fmax= (Si−S_k)(δ_max^(f) +T_k) +S_kδ_max^(f) + (Fi−F_k)

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that solving forδmax^(f⁾ :

δ^(f_max⁾ = F_max−T_k(S_i−S_k)−(F_i−F_k) S_i

As for the second solution, calledδmax^(t,f):

F_max= (S_i−S_k)(T_i) +S_kδ^(t,f_max⁾+ (F_i−F_k) solving forδmax^(t,f):

δ_max^(t,f⁾= Fmax−(Si−Sk)(Ti)−(Fi−Fk) S_k

For the minimum off-set, it must be considered the maximum of the function that matches the minimum frequency. Asf(t)decreases linearly with time, thereforethas to be minimized.

f

min{t}, δ_min^(f)

=F_min

Because the interference timetis limited by the chirps domain, therefore themin{t}has two solutions. First the initial time of the interfering chirp domaint_k =δ_k, and second from the transmitted chirp initial timeti= 0. Applying tof(t, δ)and solving forδ^(f_min⁾ andδ^(t,f_min⁾:

δ^(f_min⁾ = Fmin−(Fi−Fk) S_i

δ_min^(t,f)= Fmin−(Fi−Fk) S_k

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

,

max

min

 

max

= min ( )max

,

 

max

min

,



max

( ) min max

,



max

min

, ( )min

( ) =−

min

a) b) c)

d) e)

( )+

max



max

( , ) min max

,



max

, min ( , ) min

f) g)

Figure 4.6: All the possible type of limits for chirps with different slopes, being the interfering chirp slope larger.

4.2.2.4 Summary

Table 4.1 gathers the solutions of each case. So that in order to calculate the interference intervalD is to, with the knowledge of the radar parameters, apply the equations. And lastly write the solutions as the interval:

D= [δ_min, δ_max] =h maxn

δ^(t)_min, δ_min^(f), δ^(t,f_min⁾o

, minn

δ_max^(t) , δ_max^(f) , δ_max^(t,f)oi

(4.2.17)

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Limit Index S_i > S_k S_i =S_k S_i < S_k

δmin (t) −Tk −Tk −Tk

(f) ^F^min^−T^k^(Sⁱ^−S_S ^k^)−(Fⁱ^−F^k⁾

i

F_min−(Fi−Fk) S_k

F_min−(Fi−Fk) S_i

(t, f) ^F^min^−Tⁱ^(Sⁱ^−S_S ^k^)−(Fⁱ^−F^k⁾

k

Fmin−(Fi−F_k) Sk

δmax (t) Ti Ti Ti

(f) ^F^max^−(F_S ⁱ^−F^k⁾

i

Fmax−(Fi−Fk) S_k

Fmax−Tk(Si−Sk)−(Fi−Fk) S_i

(t, f) ^F^max^−(F_S ⁱ^−F^k⁾

k

Fmax−T_i(S_i−S_k)−(F_i−F_k) S_k

Table 4.1: Possible limits for each of the three cases presented.

4.2.3 Example

To conclude this section, is presented an example using the simulator developed in this work (see Chapter 5). In the Figure 4.7 shows a quick simulation of the expected interference between two chirps.

Figure 4.7: User interface of the simulator, where on the left are the radars parameters and on the right plots for the output interference.

From the Figure 4.7 the parameters for the chirp of radar Rx (radar A), the observer and interfered radar (radar B), are:Bi = 10 MHz,Ti = 10µs, giving a slope ofSi = 1 GHz. Initial frequencyFi

is set to zero. The parameters for the interferer radar chirp are: B_k = 10 MHz, T_k = 8µs, giving a slope ofS_k = 1.25 GHz. Initial frequencyF_k is also set to zero. Also, the maximum frequency fmax= 5 MHzand as we are using I/Q sampling thenfmin =−f_max=−5 MHz.

AsS_i < S_k, from the Table 4.1:

δ_min^(t) =−T_k=−8µs (4.2.18)

δ_min^(f) = F_min−(F_i−F_k)

S_i = −5×10⁶−(0−0)

1×10¹² =−5µs (4.2.19)

Interferência directa entre radares FMCW. Direct Interference between FMCW radars

Daniel Gonçalves Fernandes