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The fundamentals of millimeter sensors

Cesar Iovescu Radar Applications Manager Sandeep Rao Radar Systems Architect

Texas Instruments Introduction

Millimeter wave (mmWave) is a special class of radar that uses short- electromagnetic . Radar systems transmit electromagnetic wave that objects in their path then reflect. By capturing the reflected , a radar system can determine the range, velocity and angle of the objects. mmWave transmit signals with a wavelength that is in the millimeter range. This is considered a short wavelength in the and is one of the advantages of this technology. Indeed, the size of system components such as the antennas required to process mmWave signals is small. Another advantage of short is the high accuracy. An mmWave system operating at 76–81 GHz (with a corresponding wavelength of about 4 mm), will have the ability to detect movements that are as small as a fraction of a millimeter.

A complete mmWave radar system includes transmit (TX) and receive (RX) (RF) components; analog components such as clocking; and digital components such as analog-to-digital converters (ADCs), microcontrollers (MCUs) and digital signal processors (DSPs). Traditionally, these systems were implemented with discrete components, which increased power consumption and overall system cost. System design is challenging due the complexity and high .

Texas Instruments (TI) has solved these challenges and designed complementary metal-oxide (CMOS)-based mmWave radar devices that integrate TX- RF and RX-RF analog components such as clocking, and digital components such as the ADC, MCU and hardware accelerator. Some families in TI’s mmWave sensor portfolio integrate a DSP for additional signal-processing capabilities.

TI devices implement a special class of mmWave technology called frequency- modulated continuous wave (FMCW). As the name implies, FMCW radars transmit a frequency-modulated signal continuously in order to measure range as well as angle and velocity. This differs from traditional pulsed-radar systems, which transmit short pulses periodically.

The fundamentals of millimeter sensors 2 July 2020 Figure 2 shows the same signal, with frequency as a function of time. The chirp is characterized by a start

frequency (fc), bandwidth (B) and duration (Tc). In the example provided in Figure 2, fc = 77GHz, B = 4GHz and Tc = 40µs.

Figure 2 shows the same chirp signal, with frequency as a function of time. The chirp is characterized by a start

frequency (fc), bandwidth (B) and duration (Tc). In the example provided in Figure 2, fc = 77GHz, B = 4GHz and Tc = 40µs.

Range measurement 2 TX ant. 1 The fundamental concept in radar systems is the transmission of an electromagnetic signal that objects reflect in its path. In the signal used in 3 FMCW radars, the frequency increases linearly Figure 2. Chirp signal, with frequencySynth as a function of time. 4 with time. This type of signal is alsoAn FMCWcalled radara chirp. system transmits a chirp signal and captures the signals reflected by objects in its path. Figure 3 Figure 1 shows a representation representsof a chirp asignal, simplified block diagram of the main RF components of a FMCW radar. The radar operates as follows: FiRXgure ant. 2. Chirp signal, with frequencyIF signalas a function of time. with magnitude (amplitude) as a function of time. mixer • FIGURE 1 & 2An FMCWA synthesizer radar system (synth) transmitsFigure generates 3 . aFMCW chirp a radar chirpsignal block. and diagram. captures the signals reflected by objects in its path. Figure 3 represents• The achirp simplified is transmit blockted diagram by a transmit of the main RF components (TX ant). of a FMCW radar. The radar operates as follows: A • The of the chirp by an object generates a reflected chirp captured by the receive antenna (RX ant). • • A synthesizer (synth) generates a chirp. • AA “mixer” synthesizer combines (synth) the generates RX and T aX chirp signals. to produce an (IF) signal. • The chirp is transmit•ted The by achirp transmit is transmitted antenna (TX by ant) a. transmit antenna • The reflection of the chirp(TX byant). an object generates a reflected chirp captured by the receive antenna (RX ant). • At “mixer” combines the RX and TX signals to produce an intermediate frequency (IF) signal. • The reflection of the chirp by an object generates a reflected chirp captured by the Figure 1. Chirp signal, with amplitude as a function of time. f receive antenna (RX ant). 81 GHz • A “mixer” combines the RX and TX signals to S Figure 2 shows the same chirp signal, with produce an intermediate frequency (IF) signal. FIGURE 1 & 2 B = 4 GHz frequencyA as a function of time. The chirp is A frequency mixer is an that

characterized by a start frequency (fc), bandwidth (B) f = 77 GHz combines two signals to create a new signal with a c Figure 3. FMCW radar block diagram. and duration (Tc). The slope of the chirp (S) capturest new frequency. Tc = 40 µs the rate of change of frequency. InA frequencythe example mixer is an electronic component that combines two signals to create a 1new signal with a new frequency. t For two sinusoidal inputs x1 and x2 (Equations 1 provided in Figure 2, fc = 77 GHz, B = 4 GHz, Figure 3. FMCW radar block diagram. For two sinusoidal inputs x1and 2): and x2 (Equations 1 and 2): T = 40 µs and S = 100 MHz/µs. c A frequency mixer is an electronic component that combines two signals to create a new signal with a new frequency. = sin ( + ) (1)(1)

f For two sinusoidal inputs x1 and x2 (Equation1s 1 and 2):1 1 81 GHz 𝑥𝑥 = sin (𝜔𝜔 𝑡𝑡 + 𝛷𝛷 ) (2)(2)

S 2 = sin ( 2 + 2 ) (1) The output xout has an instantaneousThe output frequency xout𝑥𝑥 has equal an instantaneous𝜔𝜔 to𝑡𝑡 the𝛷𝛷 difference frequency of the instantaneous frequencies of the two B = 4 GHz 1 1 1 input sinusoids. The phase equalof the outputto the xdifferenceout𝑥𝑥 is equal= sin (toof𝜔𝜔 thethe𝑡𝑡 + differenceinstantaneous𝛷𝛷 ) of the (2) phases of the two input signals (Equation 3): frequencies of the2 two input2 sinusoids.2 The phase The output xout has an instantaneous frequency𝑥𝑥 equal𝜔𝜔 to𝑡𝑡 the𝛷𝛷 difference of the instantaneous frequencies of the two fc = 77 GHz of the output= xsinout [is( equal to) the+ ( difference )] of the (3) input sinusoids.t The phase of the output xout is equal to the difference of the phases of the two input signals (Equation Tc = 40 µs 3): phases of𝑜𝑜𝑜𝑜𝑜𝑜 the two input1 signals2 (Equation1 2 3): The operation of the frequency mixer𝑥𝑥 can also be𝜔𝜔 understood− 𝜔𝜔 𝑡𝑡 graphically𝛷𝛷 − 𝛷𝛷 by looking1 at TX and RX chirp frequency Figure 2. Chirp signal, with frequency as a functionrepresentation of time. as a function of time. = sin[( ) + ( )] (3) (3)

The operation of the frequencyThe operation mixer𝑥𝑥𝑜𝑜𝑜𝑜𝑜𝑜 can alsoof the be𝜔𝜔 1frequencyunderstood− 𝜔𝜔2 𝑡𝑡 mixergraphically𝛷𝛷1 − can𝛷𝛷2 alsoby looking be at TX and RX chirp frequency An FMCW radar system transmitsrepresentation a chirp signal as and a function understood of time. graphically by looking at TX and RX captures the signals reflected by objects in its path. chirp frequency representation as a function of time. Figure 3 represents a simplified block diagram of The upper diagram in Figure 4 on the following the main RF components of an FMCW radar. The page shows TX and RX as a function of time radar operates as follows: for a single object detected. Notice that the RX chirp is a time-delay version of the TX chirp.

The fundamentals of millimeter wave radar sensors 3 July 2020 The upper diagram in Figure 4 shows TX and RX chirps as a function of time for a single object detected. Notice that the RX chirp is a time-delay version of the TX chirp. The upper diagram in FigureThe 4 uppershows diagram TX and RX in chirpsFigure as4 shows a function TX and of timeRX chirps for a assingle a function object ofdetected. time for Notice a single that object the detected. Notice that the The time delay (τ) can be mathematically derived as Equation 4: RX chirp is a time-delay versionRX chirp of theis a TXtime chirp.-delay version of the TX chirp. = The time delay (τ) can be mathematicallyThe time delay (derivedτ) can be as mathematically Equation 4: derived(5) as Equation 4: 2𝑑𝑑 𝜏𝜏 𝑐𝑐 where d is the distance to the detected object and c is the = of . (5) = (5) 2𝑑𝑑 2𝑑𝑑 To obtain the frequency representation as a function of time𝜏𝜏 of𝑐𝑐 the IF signal at the output𝑐𝑐 of the frequency mixer, where d is the distance to wherethe detected d is the object distance and to c theis the detected speed of object light. and c is the𝜏𝜏 . subtract the two lines presented in the upper section of Figure 4. The distance between the two lines is fixed, which meansTo obtain that the the frequency IF signal consis representationTo obtaints of a t tonehe frequencyas with a function a constant representation of time frequency of the as IF. aFig signal functionure 4at shows the of time output that of this theof the frequencyIF signal frequency at is the Sτ mixer .output The, IF of the frequency mixer, signalsubtract is valid the twoonly lines in the presented timesubtract interval in the the where two upper lines both section presented the ofTX Figurechirp in the and 4. upperT thehe distance RX section chirp between overlapof Figure (i.e., the 4. T twothehe distanceintervallines is fixed betweenbetween, which thethe two lines is fixed, which verticalmeans thatdotted the lines IF signal in Figure consismeans 4ts) . ofthat a tone the IFwith signal a constant consists frequencyof a tone with. Figure a constant 4 shows frequency that this frequency. Figure 4 showsis Sτ. T thathe IF this frequency is Sτ. The IF signal is valid only in the timesignal interval is valid where only inboth the the time TX interval chirp and where the RXboth chirp the overlapTX chirp (i.e., and thethe intervalRX chirp between overlap (i.e.,the the interval between the vertical dotted lines in Figurevertical 4) . dotted lines in Figure 4) .

Figure 4. IF frequency is constant.

The mixer output signal as a magnitude function of time is a , since it has a constant frequency. Figure 4. IF frequency is constantFigure 4. . IF frequency is constant.

The initial phase of the IF signal (Φ0) is the difference between the phase of the TX chirp and the phase of the RX chirp The mixer output signal asThe a magnitude mixer output function signal of as time a magnitude is a sine wave function, since of ittime has isa aconstant sine wave frequency., since it has a constant frequency. at the time instant corresponding to the start of the IF signal (i.e., the time instant represented by the left vertical dotted

lineThe ininitial Figure phase 4). (Equationof the IF signalThe 5): initial (Φ0) phaseis the differenceof the IF signal between (Φ0) isthe the phase difference of the between TX chirp theand thephase phase of the of TtheX chirp RX chirp and the phase of the RX chirp The upper diagram in Figure 4 shows TX and RX chirps as a function of time for a single object detected. Notice that the at the time instant correspondingat the time to the instant start corresponding of the IF signal to (i.e., the the start time of theinstant IF signal represented (i.e., the by time the instant left vertical represented dotted by the left vertical dotted RX chirp is a time-delay version of the TX chirp. = 2 (5) line in Figure 4). (Equationline 5): in Figure 4). (Equation 5): The time delay ( ) can be mathematically derived as Mathematically, it can be further derived into t 0 𝑐𝑐 The time delay (τ) can be mathematically derived as Equation 4: Mathematically, it can be further derived into Equation𝜙𝜙 𝜋𝜋 6:𝑓𝑓 𝜏𝜏 Equation 4: Equation 6: = 2 (5) = 2 (5)

(4) 0 = 𝑐𝑐 (6)*0 𝑐𝑐 = Mathematically(5) , it can be furtherMathematically derived into, it can Equation𝜙𝜙 be further𝜋𝜋 6:𝑓𝑓 𝜏𝜏 derived into(6) Equation𝜙𝜙 𝜋𝜋 6:𝑓𝑓 𝜏𝜏 2𝑑𝑑 4𝜋𝜋𝑑𝑑 0 where d is the distance𝜏𝜏 to the𝑐𝑐 detected object 𝜙𝜙 𝜆𝜆 where d is the distance to the detected object and c is the speed of light.In summary, for an objectIn at summary, a distance ford from an object the radar,= at a thedistance IF signal d willfrom(6) be a= sine wave (Equation (6) 7) , then: and c is the speed of light. 4𝜋𝜋𝑑𝑑 4𝜋𝜋𝑑𝑑 the radar, the IF signal 0will be a sine wave To obtain the frequency representation as a function of time of the IF signal at the output of the frequency mixer, 𝜙𝜙(2 𝜆𝜆+ ) 0(7) 𝜆𝜆 In summary, for an object Inat summary,a distance ford from an object the radar, at a distance the IF signal d from will the𝜙𝜙 be radar,a sine wavethe IF (Equationsignal will 7)be, thena sine: wave (Equation 7), then: To obtain the frequency representation as a function (Equation 7), then: subtract the two lines presented in the upper section of Figure 4. The distance between the two lines is fixed, which 𝑜𝑜 0 = = 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝜋𝜋𝑓𝑓 𝑡𝑡 𝜙𝜙 means that the IF signal consists of atime tone of with the aIF constant signal at frequency the outputw. hereFig ofure the 4 frequencyshows and that this frequency. is Sτ. The IF (2 + ) (7)**(2(7) + ) (7) 𝑆𝑆2𝑑𝑑 4𝜋𝜋𝑑𝑑 signal is valid only in the time intervalmixer, where subtract both the the two TX chirp lines and pre ­sentedthe RX chirp in0 the overlap upper (i.e.,0 the interval between the 𝑓𝑓 𝑐𝑐 𝜙𝜙 𝜆𝜆 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝜋𝜋𝑓𝑓𝑜𝑜𝑡𝑡 𝜙𝜙0 𝑜𝑜 0 Thewhere assumption = soand far is that=wherew here the. radar= has anddetected = only .one object. Let’s𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 analyze𝜋𝜋𝑓𝑓 𝑡𝑡 a case𝜙𝜙 when there are several objects vertical dotted lines in Figure 4) . section of Figure 4. The distance between the two The upper diagram in Figure 4 shows TX and RX chirps as a functiondetected of. timeFigure𝑆𝑆2 for𝑑𝑑 5 ashows single three object4𝜋𝜋𝑑𝑑 different detected.𝑆𝑆2 RX𝑑𝑑 Notice chirps thatreceived4 𝜋𝜋the𝑑𝑑 from different objects. Each chirp is delayed by a different 0 0 𝑓𝑓 𝑐𝑐 𝜙𝜙 𝜆𝜆 0 𝑐𝑐 0 𝜆𝜆 RX chirp is a time-delay versionlines isof fixed, the TX chirp.which means that theThe IF assumption signal consists so far is thatTheThe the assumptionassumption 𝑓𝑓radar has so sodetected far far𝜙𝜙 is is that that only the the one radar radar object. has has detectedLet’s detected analyze only a one case object. when Let’sthere analyze are sever a caseal objects when there are several objects of a tone with a constant frequency.detected Figure. Fig 4 ureshows 5 shows only threedetected one different object.. Figure RX Let’s5 chirps shows analyze received three a different case from whendifferent RX chirps there objects. received Each from chirp different is delayed objects. by a Eachdifferent chirp is delayed by a different The time delay (τ) can be mathematically derived as Equation 4: that this frequency is St. The IF signal is valid only in are several objects detected. Figure 5 shows three different RX chirps received from different objects. the time interval where both the= TX chirp and(5) the RX chirp overlap (i.e., the interval between2𝑑𝑑 the vertical Each chirp is delayed by a different amount of time 𝜏𝜏 𝑐𝑐 where d is the distance to dottedthe detected lines in object Figure and 4 ).c is the speed of light. proportional to the distance to that object. The different RX chirps translate to multiple IF tones, To obtain the frequency representation as a function of time of the IF signal at the output of the frequency mixer, f subtract the two lines presented in the upper sectionTX chirp of Figure 4. The distance betweeneach the twowith lines a constant is fixed ,frequency. which RX chirp means that the IF signal consists of a tone with a constant frequency. Figure 4 shows that this frequency is Sτ. The IF S f Reflected signal t TX chirp from multiple signal is valid only in the time interval where both the TX chirp and the RX chirp overlap (i.e., the interval between the objects vertical dotted lines in Figure 4) . t Figure 4. IFt frequency is constant. Tc

The mixer output signal as a magnitude function off time is a sine wave, since it has a constant frequency.

IF signal The initial phase of the IF signal (Φ ) is the difference between the phase of the TX chirp and the phase of the RX chirp t 0 St at the time instant corresponding to the start of the IF signal (i.e., the time instant represented byf the left vertical dotted t line in Figure 4). (Equation 5): t Figure 4. IF frequency is constant.

= 2 (5) t

0 𝑐𝑐 Figure 5. Multiple IF tones for multiple-object detection. Mathematically, it can be furtherThe derived mixer into output Equation𝜙𝜙 signal𝜋𝜋 6:𝑓𝑓 𝜏𝜏 as a magnitude function of time is a sine wave, since it has a constant This IF signal consisting of multiple tones must = (6) frequency. 4𝜋𝜋𝑑𝑑 be processed using a Fourier transform in order 0 In summary, for an object at a distance d from the𝜙𝜙 radar,𝜆𝜆 the IF signal will be a sine wave (Equationto separate 7), then the: different tones. Fourier transform The initial phase Figof urethe 4IF. IFsignal frequency (F0) is is the constant difference. processing will result in a frequency spectrum that between the phase(2 of+ the TX) chirp and the(7) phase The mixer output signal as a magnitude function of time is a sine wave, since it has a constanthas separate frequency. peaks for the different tones each of the RX chirp at the time instant corresponding 𝑜𝑜 0 where = and = . 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝜋𝜋𝑓𝑓 𝑡𝑡 𝜙𝜙 The initial phase of the IF signalto the ( Φstart0) is of the the difference IF signal between (i.e., the thetime phase instant of the TX chirp and the phase of the RX chirp 𝑆𝑆2𝑑𝑑 4𝜋𝜋𝑑𝑑 * This equation is an approximation and valid only if Slope and at0 the time instant0 correspondingrepresented to the by start the of left the vertical IF signal dotted (i.e., theline time in instant representeddistance areby thesufficiently left vertical small. However,dotted it is still true that the phase The assumption𝑓𝑓 𝑐𝑐 so far𝜙𝜙 is that𝜆𝜆 the radar has detected only one object. Let’s analyze a case when there are several objects line in Figure 4). (Equation 5): of the IF signal responds linearly to a small change in the distance detected. Figure 5 shows three differentFigure 4 RX). (Equation 5):chirps received from different objects. Each chirp is(i.e., delayed Δf=4πΔd/ by al ).different = 2 (5)(5) ** In this introductory white paper we ignore the dependence of the frequency of the IF signal on the velocity of the object. This Mathematically, it can be further derived into Equation𝜙𝜙0 𝜋𝜋 6:𝑓𝑓𝑐𝑐 𝜏𝜏 is usually a small effect in fast-FMCW radars, and further can be easily corrected for once the Doppler-FFT has been processed. = (6) 4𝜋𝜋𝑑𝑑 0 In summary, for an object Theat afundamentals distance dof frommillimeter the𝜙𝜙 wave radar, radar𝜆𝜆 the sensors IF signal will be a sine wave4 (Equation 7), then: July 2020

(2 + ) (7)

𝑜𝑜 0 where = and = . 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝜋𝜋𝑓𝑓 𝑡𝑡 𝜙𝜙 𝑆𝑆2𝑑𝑑 4𝜋𝜋𝑑𝑑 0 0 The assumption𝑓𝑓 𝑐𝑐 so far𝜙𝜙 is that𝜆𝜆 the radar has detected only one object. Let’s analyze a case when there are several objects detected. Figure 5 shows three different RX chirps received from different objects. Each chirp is delayed by a different amount of time proportional to the distance to that object. The different RX chirps translate to multiple IF tones, each amount of time proportional to the distance to that object. The different RX chirps translate to multiple IF tones, each with a constant frequency. amount of time proportionalwith to the a constantdistance frequencyto that object.. The different RX chirps translate to multiple IF tones, each amount of time proportional to the distance to that object. The different RX chirps translate to multiple IF tones, each with a constant frequency. with a constant frequency.

peak denoting the presence of an object at a Velocity measurement with two chirps specific distance. Velocity MeasurementIn order to measure velocity, an FMCW radar Velocity Measurement Range resolution transmits two chirps separated by Tc. Each reflected In this section, let’s use chirpphasor is notationprocessed (dist throughance, angle) FFT tofor detect a complex the number. Range resolution is the ability to distinguishIn this section , let’s use phasor notation (distance, angle) for a complex number. range of the object (range-FFT). The range-FFT Velocity Measurementbetween two or more objects. WhenVelocity two objects Measurement with Two Chirps Velocity Measurement correspondingwith Two Chirps to each chirp will have peaks in move closer, at some point, a radar system will In this section, let’s use phasor notation (distance, angle) for a complexVelocityIn order number. to Measurement measure velocitythe same, a FMCW location, radar but transmits with a twodifferent chirps phase. separated The by Tc Each reflected chirp is processed no longer be able to distinguish themIn order as separateto measure velocity, a FMCW radar transmits two chirps separated by Tc Each reflected chirp is processed Figurethrough 5. Multiple FFT to IFdetect tones the for range multiple of the-object object detection (range-.FFT). The range-FFT corresponding to each chirp will have peaks in Velocity Measurement with Two FigChirpsure 5 . Multiple IF tones throughfor multiple FFT -toobject detect detection themeasured range. of the phase object difference (range-FFT). corresponds The range- FFTto a corresponding motion to each chirp will have peaks in objects. Fourier transform theoryIn states thethis samesection that locationyou, let’s can use, but phasor with a notation different (dist phase.ance, The angle) measured for a complex phase difference number. corresponds to a motion in the object of Figure 5. Multiple IF increasetonesThis IFfor signal the multiple resolution consisting-object by of detection multipleincreasingthe. tones samethe mustlength location be of processed , but within using thea different object a Fourier phase.of vTc. transform The measured in order phase to separate difference the different corresponds to a motion in the object of In orderThis toIF measuresignal consisting velocity of, a multipleFMCWFigure radar tones 5. Multipletransmits must be IF processedtwo tones chirps for multiplevTcusingseparated. a Fourier-object by T tdetection rEachansform reflected. in order chirp to isseparate processed the different tones. Fourier transform processingVelocityvTc will. result Measurement in a cfrequency with spectrum Two C thathirps has separate peaks for the different tones This IF signal consisting ofthrough multipletones. FFT tonesFourier to detect must transform thebe processedrangethe processing IFof signal.the using object will a Fourier result (range in t-r FFT).aansform frequency The inrange order spectrum-FFT to correspondingseparate that has the separate different to each peaks chirp for will the have different peaks tones in This IF signal consisting of multipleeach peak tones denoting must bethe processed presence usingof an aobject Fourier at tar ansformspecific distance in order. to separate the different tones. Fourier transformthe processing sameeach peaklocation will denoting result, but in with thea frequency apresence different spectrum of phase. an object The that measuredat has a specific separate phase Indistance peaksorder difference to.for measure the correspondsdifferent velocity tones ,to a FMCWa motion radar in th transmitse object oftwo chirps separated by Tc Each reflected chirp is processed tones. Fourier transform processingTo increase will the result length in a of frequency the IF signal, spectrum the that has separate peaks for the different tones each peak denoting the presencevTc. of an object at a specific distance. through FFT to detect the range of the object (range-FFT). The range-FFT corresponding to each chirp will have peaks in bandwidthRange Resolution must also be increased proportionally. An eachRange peak Resolution denoting the presence of an object at a specific distancethe same. location, but with a different phase. The measured phase difference corresponds to a motion in the object of increased-length IF signal results in an IF spectrum Range Resolution Range resolution is the ability to distinguishvTc. between two or more objects.Tc When two objects move closer, at some RangeRange Resolution resolution is the ability to distinguish between two or more objects. When two objects move closer, at some withpoint, two a separateradar system peaks. will no longer be able to distinguish them as separate objects. Fourier transform theory states that Range resolution is the abilitypoint, to distinguish a radar system between will two no longeror more be objects. able to Whendistinguish two objectsthem as move separate closer, objects. at some Fourier transform theory states that Range resolution is the abilityyou to can distinguish increase betweenthe resol utiontwo or by more increasing objects. the When length two of objectsthe IF signal. move closer, at some point, a radar system will no youlonger can be increase able to thedistinguish resolFourierution them transformby increasingas separate theory the objects. lengthalso statesFourier of the that IFtransform signal. an theory states that point, a radar system will no longer be able to distinguish them as separate objects. Fourier transform theory states that you can increase the resolution by increasing the lengthobservationTo of increase the IF signal. windowthe length (T) of can the resolve IF signal, frequency the bandwidth must also be increased proportionally. An increased-length IF youTo can increase increase the the length resol ofution the IF by signal, increasing the b theandwidth length must of the also IF signal. be increased proportionally. An increased-length IF componentssignal results that in an are IF spectrumseparated with by moretwo separate than peaks. To increase the length of thesignal IF signal, results the inbandwidth an IF spectrum must also with be two increased separate proportionally peaks. . An increased-length IF To increase the length of the1/THz. IF signal, This themeans bandwidth that two must IF signal also be tones increased can proportionally. An increased-length IF signal results in an IF spectrum with two separate peaks.Fourier transform theory also states that an observation windowFigure 6(.T Two-chirp) can resolve velocity frequency measurement. components that are signalFourier results transform in an IF theory spectrumbe also resolved with states two thatin separatefrequency an observation peaks. as long window as the ( Tfrequency) can resolve frequency components that are separated by more than 1/THz. This means that two IF signal tones can be Figresolvedure 6. inTwo frequency-chirp velocity as long measurement as the . Fourier transform theory alsoseparated states that by an more observation than 1/THz. window This means(T) can thatresolve two frequency IF signal tones components can be resolved that are in frequency asFig longure as6. theTwo -chirp velocity measurement. Fourier transform theory alsodifferencefrequency states thatsatisfies difference an observation the satisfies relationship windowthe relation given (T) canship in resolve given in frequency Equation components8: that are separated by more than 1/THz.frequency This means difference that two satisfies IF signal the tones relation canship be resolved given in Equationin frequencyThe 8 phase: as long difference as the is Thedefined phase as Equation difference 10: is derived from Equation 6 as separated by more than 1/THz.Equation 8: This means that two IF signal tones can be resolved in frequency as long as the frequency difference satisfies the relationship given in Equation 8: The phase difference is definedEquation as Equation 10: 10: > (8) frequency difference satisfies the relationFigureship 6. Twogiven-chirp in> Equation velocity 8 measurement : (8) . = 1 (10)(10) 1 = 𝑐𝑐 (10) > (8) 𝑇𝑇𝑐𝑐 4𝜋𝜋𝜋𝜋𝑇𝑇 Δ𝑓𝑓 4𝜋𝜋𝜋𝜋𝑇𝑇𝑐𝑐 The phase difference is defined1where as Equation Tc is the 10: observation Δ𝑓𝑓> 𝑇𝑇interval. 𝑐𝑐 (8) ∆𝛷𝛷 𝜆𝜆 where Tc is the observationwhere interval. Tc is the observation interval.You can derive the velocity using EquationFigure 11 6.: Two-chirp velocity measurement. 1 You can derive the velocity∆𝛷𝛷 using𝜆𝜆 Equation 11: where T is the observation interval. Δ𝑓𝑓 𝑇𝑇𝑐𝑐 You can derive the velocity using Equation 11: c SinceSince = , equationEquationΔ𝑓𝑓 𝑇𝑇𝑐𝑐 (8) 8 can bebe expressed expressed as as > = (since B = ST ). where Tc is the observation interval. = The phase(10) difference is defined as Equation 10:c Since = , equation (8) can be expressed as > = (since B = STc ). = (11)(11) 𝑆𝑆2Δ𝑑𝑑 4𝜋𝜋𝜋𝜋𝑇𝑇𝑐𝑐 𝑐𝑐 𝑐𝑐 = 𝑆𝑆2Δ𝑑𝑑 𝑐𝑐 𝑐𝑐 𝑐𝑐 𝜆𝜆Δ 𝛷𝛷 (11) Since = , equation (8) can be expressed as > Δ𝑓𝑓 = (since 𝑐𝑐 (since B =B ST =c ST). c). Δ𝑑𝑑 2𝑆𝑆𝑇𝑇 2𝐵𝐵 ∆𝛷𝛷 𝜆𝜆 𝑐𝑐 𝜆𝜆Δ𝛷𝛷 𝑆𝑆2Δ𝑑𝑑 YouSince can deriveΔ𝑓𝑓= the𝑐𝑐 velocity , equation usingT (8)he Equationcanrange𝑐𝑐 be r 𝑐𝑐expressedesolution 11: (d asRes ) dependsΔ𝑑𝑑> 2𝑆𝑆=𝑇𝑇 only 2 (since𝐵𝐵 on the B b=andwidth STc ). Sinceswept theby thevelocity chirp measurement(Equation𝑣𝑣= 4𝜋𝜋 𝑇𝑇9):𝑐𝑐 is based on(10) a The range resolution (dRes) depends only on the bandwidth sweptSince by the the velocity chirp (Equationmeasurement 9): is based on 𝑣𝑣a phase4𝜋𝜋𝑇𝑇𝑐𝑐 difference,𝑐𝑐 there will be ambiguity. The measurement is Δ𝑓𝑓 𝑐𝑐 𝑆𝑆2Δ𝑑𝑑 TheΔ𝑑𝑑 range2𝑆𝑆𝑇𝑇𝑐𝑐 resolution2𝐵𝐵 (d ) depends𝑐𝑐Since only𝑐𝑐 the on velocity the measurement is based on a phase 4difference,𝜋𝜋𝜋𝜋𝑇𝑇 there will be ambiguity. The measurement is The range resolution (d ) depends only on the bandwidth swept by the chirpRes (Equation 𝑐𝑐9): phase difference, there will be ambiguity. The Res Δ𝑓𝑓 𝑐𝑐 = Δ 𝑑𝑑 2𝑆𝑆 You𝑇𝑇 unambiguous 2can𝐵𝐵 (11) derive the only velocity= if | using|< π. UsingEquation(9) equation 11: ∆𝛷𝛷 11 above,𝜆𝜆 one can mathematically derive < . The range resolution (dRes) depends only on the bandwidth= swept by the chirp(9) (Equation 9): < bandwidth swept by the𝜆𝜆Δ𝛷𝛷 chirp (Equationunambiguous 9): only if | 𝑐𝑐 |

Figure 7. Chirp frame

The processing technique is described below using the example of two objects equidistant from the radar but with

different velocities v1 and v2.

Range-FFT processes the reflected set of chirps, resulting in a set of N identically located peaks but each with a different phase incorporating the phase contributions from both these objects (the individual phase contributions from each of these objects being represented by the red and blue phasors in Figure 8)

Velocity Measurement Figure 8. The range-FFT of the reflected chirp frame results in N phasors. Velocity measurementIn this section with, let’s multiple use phasor objects notation at (distance,A second angle) FFT,for a called complex Doppler-FFT, number. is performed on A second FFT, called Doppler-FFT, is performed on the N phasors to resolve the two objects, as shown in Figure 9. the N phasors to resolve the two objects, as shown the same rangeVelocity Measurement with Two Chirps The two-chirp velocity measurement method does in Figure 9. In order to measure velocity, a FMCW radar transmits two chirps separated by Tc Each reflected chirp is processed not work if multiple moving objects with different through FFT to detect the range of the object (range-FFT). The range-FFT corresponding to each chirp will have peaks in velocities are theat the same time location of measurement,, but with a different both phase. The measured phase difference corresponds to a motion in the object of ω ω at the same distancevTc. from the radar. Since these 1 2 objects are at the same distance, they will generate Figure 9. Doppler-FFT separates the two objects. reflective chirps with identical IF frequencies. As a consequence, the range-FFT will result in single w and w correspond to the phase difference 1 Fig2 ure 9. Doppler-FFT separates the two objects. peak, which represents the combined signal from between consecutive chirps for the respective all of these equi-range objects. Aω 1simple and ω 2phase correspond to theobjects phase (Equationdifference 13):between consecutive chirps for the respective objects (Equation 13): comparison technique will not work. = , = (13)(13) In this case, in order to measure the speed, the 𝜆𝜆𝜔𝜔1 𝜆𝜆𝜔𝜔2 𝑣𝑣1 4𝜋𝜋𝑇𝑇𝑐𝑐 𝑣𝑣2 4𝜋𝜋𝑇𝑇𝑐𝑐 radar system must transmit moreVelo thancity two Resolution chirps. It Velocity resolution transmits a set of N equally spaced chirps. This set You have already seen thatThe two theory discrete of frequencies, discrete Fourier ω1 and transforms ω2, can be teaches resolved if Δω= ω2 - ω1 > 2π/N radians/sample. of chirps is called a chirp frame. Figure 7 shows the us that two discrete frequencies, w and w , can be Figure 6. Two-chirp velocity measurement. 1 2 = frequency as a function of time forSince a chirpΔω is frame.also defined byresolved the following if Dw equation = w – w > 2p/N radians/sample. (equation 10) one can mathematically derive the velocity 2 1 4𝜋𝜋𝜋𝜋𝜋𝜋𝑇𝑇𝑐𝑐 resolution(v ) if the frame period T = NT (Equation 14): The phase difference is defined resas Equation 10: f c ∆𝜔𝜔 𝜆𝜆 1 2 3 N Since Dw is also defined by the following equation = (Equation 10),(10) one can mathematically 𝑐𝑐 derive 4the𝜋𝜋𝜋𝜋𝑇𝑇 velocity resolution (v ) if the frame period T ∆𝛷𝛷 𝜆𝜆 res c You can derive the velocity using Equation 11: Tf = NTc (Equation 14): Figure 7. Chirp frame. = >(11) = (14)(14) The processing technique is described below using 𝜆𝜆Δ𝛷𝛷 𝜆𝜆 𝑟𝑟𝑅𝑅𝑅𝑅 The𝑣𝑣 velocity4𝜋𝜋𝑇𝑇𝑐𝑐 resolution𝑣𝑣 of𝑣𝑣 the radar2𝑇𝑇𝑓𝑓 is inversely the example ofSince two the objects velocity equidistant measurementThe velocity from isresolutionthe based on of a phasethe radar difference, is inversely there proportional will be ambiguity. to the frame The measurement time (Tf). is unambiguous only if | |< π. Using equation proportional11 above, one to canthe mathematicallyframe time (Tf). derive < . radar but with different velocities v1 and v2. Angle Detection 𝜆𝜆 Range-FFT processes the reflected∆𝛷𝛷 set of chirps, 𝑣𝑣 4𝑇𝑇𝑐𝑐 Equation 12 provides the maximum relative speedAngle (vmax )detection measured by two chirps spaced Tc apart. Higher vmax requires resulting in a set of N identically locatedAngle Estimation peaks, but shorter transmission times between chirps. Angle estimation each with a different phase incorporating the phase An FMCW radar system can estimate the angle of a reflected signal with the horizontal plane, as shown in Figure 10. This Anv FMCW = radar system(12) can estimate the angle of a contributions from both these objectsangle is(the also individual called the anglemax of arrival (AOA). reflected𝜆𝜆 signal with the horizontal plane, as shown phase contributions from each of these objects 4𝑇𝑇𝑐𝑐 Velocity Measurement with Multiple Objectsin Figure at the 10 Same. This R angleange is also called the angle of being represented by the red and blue phasors in arrival (AoA). Figure 8). The two-chirp velocity measurement method does not work if multiple moving objects with different velocities are at the time of measurement, both at the same distance from the radar. Sinceobject these objects are at the same distance, they Key will generate reflective chirps with identicalv1 IF frequencies. As a consequence, the range-FFT will result in single peak, which represents the combined signal from all of these equi-range objectsθ . A simple phase comparison technique will radar not work. v2

radar Figure 8. The range-FFT of the reflected chirp frame results in N phasors. Figure 10. Angle of arrival.Figure 10. Angle of arrival.

Angular estimation is based on the observation that a small change in the distance of an object results in a phase change in the peak of the range-FFT or Doppler-FFT.This result is used to perform angular estimation, using at least two RX antennas as shown in Figure 11. The differential distance from the object to each of the antennas results in a phase change in the FFT peak. The phase change enables to estimate the AoA. The fundamentals of millimeter wave radar sensors 6 July 2020

Figure 11. Two antennas are required to estimate AoA.

In this configuration, the phase change is derived mathematically as Equation 15:

= (15) 2𝜋𝜋∆𝑑𝑑 15 Under the assumption of a planar wavefront∆𝛷𝛷 basic geometry𝜆𝜆 shows that = ( ), where l is the distance between the antennas This enables to derive the angle value from a measured with Equation 16: Δ𝑑𝑑 𝑙𝑙𝐴𝐴𝐴𝐴𝐴𝐴 𝜃𝜃 = ( ) (16) ∆𝜙𝜙 −1 𝜆𝜆𝜋𝜋𝜋𝜋 Note that depends on ( ). This is called𝜃𝜃 𝐴𝐴𝐴𝐴𝐴𝐴 a nonlinear2𝜋𝜋𝜋𝜋 dependency. ( ) is approximated with a linear function only when has a small value: ( ) ~ . ∆𝛷𝛷 𝐴𝐴𝐴𝐴𝐴𝐴 𝜃𝜃 𝐴𝐴𝐴𝐴𝐴𝐴 𝜃𝜃 As a result,𝜃𝜃 the estimation accuracy𝐴𝐴𝐴𝐴𝐴𝐴 𝜃𝜃 depends𝜃𝜃 on AoA and is more accurate when has a small value.as shown in Figure 12 𝜃𝜃 > = (14) 𝜆𝜆 𝑣𝑣 𝑣𝑣𝑟𝑟𝑅𝑅𝑅𝑅 2𝑇𝑇𝑓𝑓 The velocity resolution of the radar is inversely proportional to the frame time (Tf). > = (14) Angle Detection 𝜆𝜆 𝑣𝑣 𝑣𝑣𝑟𝑟𝑅𝑅𝑅𝑅 2𝑇𝑇𝑓𝑓 The velocity resolution of the radar is inversely proportional to the frame time (Tf). Angle Estimation

An FMCWAngle radar Detection system can estimate the angle of a reflected signal with the horizontal plane, as shown in Figure 10. This angle is also called the angle of arrival (AOA). Angle Estimation

Figure 12. AoA estimation is more accurate for small values. An FMCW radar system can estimate the angle of a reflected signal with the horizontalFig planeure, 12 as. shownAoA estimation in Figure is10 more. This accurate for small values. Angular estimation is based on the observation that Maximum angular field of view Maximum Angularangle Fieldis also of called View the angle of arrival (AOA). Maximum Angular Field of View a small change in the distance of an object results The maximum angular field of view of the radar is The maximum angular field of view ofin the a phase radar is change defined in by the the peak maximum of the AoA range-FFT that the or radar can estimate. See Figure The maximum angular fielddefined of view ofby the radarmaximum is defined AoA bythat the the maximum radar can AoA that the radar can estimate. See Figure 13. Doppler-FFT. This result is used to13 perform. angular estimate. See Figure 13. estimation, using at least two RX antennas as shown in Figure 11. The differential distance from - Figure 10. Angle of arrival. θmax θmax the object to each of the antennas results in a phase Angular estimation is basedchange on the in observation the FFT peak. that aThe small phase change change in the enables distan ce of an object results in a phase change in the peak of the range-FFT or Doppler-FFT.This result is used to perform angular estimation, using at least two radarRX you to estimate the AoA.Figure 10. Angle of arrival. antennas as shown in Figure 11. The differential distance from the object to each of theFigure antennas 13. Maximum results angular in a phase field of view. changeAngular in the estimation FFT peak. Theis based phase on change the observation enables to that estimate a small the change AoA. in the distance of an object results in a phase change in the peak of the range-FFT or Doppler-FFT.This result is used to perform angular estimation, using at least two RX d+ d Unambiguous measurement of angle requires antennas as shown in FigureFigure 11 13. The. Maximum differential adngular distance fieldΔ fromof view the. object to each of the antennas results in a phase |Dw| < 180°. FigUsingure 13Equation. Maximum 16, athisngular corresponds field of view to. change in the FFT peak. The phase change enables to estimate the AoA. ( ) | o Unambiguous measurement of angle requires | <180 . Using equation 16, this corresponds to < π. o ( ) Unambiguous measurement2𝜋𝜋𝜋𝜋 𝑅𝑅of𝜋𝜋𝜋𝜋 angle𝜃𝜃 requires | |<180 . Using equation 16, this corresponds to < π. ∆𝜔𝜔 TX RX Equation𝜆𝜆 17 shows that the maximum field of view 2𝜋𝜋𝜋𝜋𝑅𝑅𝜋𝜋𝜋𝜋 𝜃𝜃 Equation 18 shows that the maximum field of view that twoantenna antennas spacedantennas l apart can service is: ∆𝜔𝜔 𝜆𝜆 Equation 18 shows that thethat maximum two antennas field of spacedview that l aparttwo antennas can service spaced is: l apart can service is: Figure 11. Two antennas are required to estimate AoA. = ( ) (18) = ( ) (17) Figure 11. Two−1 𝜆𝜆 antennas are required to estimate AoA. (18) 𝑚𝑚𝑚𝑚𝑚𝑚 𝜆𝜆 In this=𝜃𝜃 configuration, /2 𝐴𝐴𝐴𝐴𝐴𝐴 2𝜋𝜋 the phase change is derived −1 A spacing between the two antennas of results in the largest angular field of view ±90 degreesA spacing. between𝜃𝜃𝑚𝑚𝑚𝑚𝑚𝑚 the two𝐴𝐴𝐴𝐴𝐴𝐴 antennas2𝜋𝜋 of l = l/2 In this configuration, the phasemathematically change is derived as Equation mathematically 15: A spacing as Equation between 15 the: two antennas of = /2 results in the largest angular field of view ±90 degrees. 𝑙𝑙 𝜆𝜆 results in the largest angular field of view ± 90°. Texas Instruments mmWave SensorFigure Solution 11. Two antennas are required to estimate AoA. 𝑙𝑙 𝜆𝜆 = Texas I(1nstruments(15)5) mmWave Sensor Solution 2𝜋𝜋∆𝑑𝑑 Texas Instruments mmWave As you can see, Inan this FMCW configuration, sensor is able the to phase determine change the is derived range, velocity mathematically and angle as of Equation nearby objects15: by using a 15 Under the assumption Underof a planar the assumptionwavefront∆𝛷𝛷 b asicof a geometry planar𝜆𝜆 As wavefront you shows can thatsee basic, an FMCW= (senso), wherer is able l is tothe determine distance the range, velocity and angle of nearby objects by using a combination of RF, analog and digital electronic components. sensor solution between the antennas Thisgeometry enables to shows derive that the Dangled = =lsin(value qcombination) ,from where a measured l is theof RF, analog with and Equation digital 1 electronic6: components. (1Δ5𝑑𝑑) 𝑙𝑙𝐴𝐴𝐴𝐴𝐴𝐴As𝜃𝜃 you can see, an FMCW sensor is able to Figure 14 is a block diagram of the differentdistance components. between the antennas.2𝜋𝜋 ∆Thus𝑑𝑑 the angle of 𝜆𝜆Figure 14 is a block∆ diagram𝜙𝜙 of the different components. 15 Under the assumption of a planar wavefront= ∆𝛷𝛷 (basic) geometry shows(1 that6) = determine( ), where the range, l is the velocitydistance and angle of nearby arrival (q), can be computed𝜆𝜆𝜋𝜋𝜋𝜋 from the measured DF between the antennas This enables to derive the−1 angle value from a measured objects with Equation by using 16 :a combination of RF, analog and with( Equation) 16:𝜃𝜃 𝐴𝐴𝐴𝐴𝐴𝐴 2𝜋𝜋𝜋𝜋 ( )Δ𝑑𝑑 𝑙𝑙𝐴𝐴𝐴𝐴𝐴𝐴 𝜃𝜃 Note that depends on . This is called a nonlinear dependency. is approximateddigital electronic with a linearcomponents. function only when has a small value: ( ) ~ . = ( ) ∆𝜙𝜙 ∆𝛷𝛷 𝐴𝐴𝐴𝐴𝐴𝐴 𝜃𝜃 (16)𝐴𝐴𝐴𝐴𝐴𝐴 𝜃𝜃(16) −1 𝜆𝜆𝜋𝜋𝜋𝜋 Figure 14 is a block diagram of the As a result,𝜃𝜃 the estimation accuracy𝐴𝐴𝐴𝐴𝐴𝐴 𝜃𝜃 depends𝜃𝜃 on AoA and is more accurate when has a small value.as shown in Figure Note that dependsNote on that( DF). This depends is called𝜃𝜃 on𝐴𝐴𝐴𝐴𝐴𝐴 a nonlinearsin(q2)𝜋𝜋𝜋𝜋. This dependency. is called a ( ) isdifferent components. approximated with a linear function 12 only when has a smallnonlinear value: dependency.( ) ~ . sin( ) is approximated with a𝜃𝜃 ∆𝛷𝛷 𝐴𝐴𝐴𝐴𝐴𝐴 𝜃𝜃 q 𝐴𝐴𝐴𝐴𝐴𝐴 𝜃𝜃 TI has brought innovation to the field of FMCW linear function only when has a small value: As a result,𝜃𝜃 the estimation accuracy𝐴𝐴𝐴𝐴𝐴𝐴 𝜃𝜃 depends𝜃𝜃 on AoAq and is more accurate when sensing has a small by integratingvalue.as shown a DSP, in Figure MCU and the TX 12 sin(q) ~ q. RF, RX RF, analog and digital components into a 𝜃𝜃 As a result, the estimation accuracy depends on RFCMOS single chip. AoA and is more accurate when q has a small value. as shown in Figure 12. TX ant.

Angle estimation is most accurate at θ close to zero Synth LP FFT o ADC Estimation accuracy degrades as θ approaches 90 RX ant. IF signal Filter

} } }

RF Analog Digital radar Figure 14. RF, analog and digital components of an FMCW sensor. Figure 12. AoA estimation is more accurate for small values.

The fundamentals of millimeter wave radar sensors 7 July 2020 TI’s RFCMOS mmWave sensors differentiate The on-chip DSP provides more flexibility and allows themselves from traditional SiGe-based solutions for software integration of high-level algorithms, by enabling flexibility and programmability in the such as object tracking and classification. These mmWave RF front-end and the MCU/HWA/DSP single-chip devices provide simple access to the processing back-end. Whereas a SiGe-based high-accuracy object data including range, velocity solution can only store a limited number of chirps and angle that enables advanced sensing in rising and requires real-time intervention to update chirps applications that demand performance and efficiency and chirp profiles during an actual frame, TI’s such as smart infrastructure, Industry 4.0 in factory mmWave sensor solutions are able to store 512 and building automation products and autonomous chirps with four profiles before a frame starts. This drones. capability allows TI’s mmWave sensors to be easily Texas Instruments has introduced a complete configured with multiple configurations to maximize development environment for engineers working on the amount of useful data extracted from a scene. industrial and automotive mmWave sensor products Individual chirps and the processing back-end can which include: be tailored “on-the-fly” for real-time application • Hardware Evaluation Modules for the needs such as higher range, higher velocities, higher Automotive and Industrial mmWave sensors resolution, or specific processing algorithms. • mmWave software development kit (SDK) The TI mmWave Sensor portfolio for “Automotive” which includes RTOS, drivers, signal-processing scales from high-performance radar front-end to libraries, mmWave API, mmWaveLink and highly-integrated single-chip edge sensors. Designers security (available separately). can address advanced driver assistance systems • mmWave Studio off-line tools for algorithm (ADAS) and autonomous driving safety regulations— development and analysis which includes data including ISO 26262, which enables Automotive capture, visualizer and system estimator. Safety Integrity Level (ASIL)-B— with the AWR mmWave portfolio. To learn more about mmWave products, tools and software please visit www.ti.com/mmwave and start The TI mmWave sensor portfolio for “Industrial” your design today. includes both 76-81GHz and 60-64GHz with highly integrated single chip edge sensors .

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