The Fundamentals Of Millimeter Wave Radar Sensors (Rev. A)
The fundamentals of millimeter wave radar sensors Cesar Iovescu Radar Applications Manager Sandeep Rao Radar Systems Architect Texas Instruments
Introduction Millimeter wave (mmWave) is a special class of radar technology that uses shortwavelength electromagnetic waves. Radar systems transmit electromagnetic wave signals that objects in their path then reflect. By capturing the reflected signal, a radar system can determine the range, velocity and angle of the objects. mmWave radars transmit signals with a wavelength that is in the millimeter range. This is considered a short wavelength in the electromagnetic spectrum 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 wavelengths 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) radio frequency (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 frequencies. Texas Instruments (TI) has solved these challenges and designed complementary metal-oxide semiconductor (CMOS)-based mmWave radar devices that integrate TXRF 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 frequencymodulated 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 wave radar sensors 2 July 2020
Figure 2 shows the same chirp signal, with frequency as a function of time. The chirp is characterized by a frequency (fc), bandwidth (B) and duration (Tc). In the example provided in Figure 2, fc 77GHz, B 4GHz Figure 2 shows the same chirp signal, with frequency as a function of time. The chirp is characterized by a frequency (fc), bandwidth (B) and duration (Tc). In the example provided in Figure 2, fc 77GHz, B 4GHz Range measurement 2 The fundamental concept in radar systems is the TX ant. 1 transmission of an electromagnetic signal that objects reflect in its path. In the signal used in 3 Figure 2. Chirp signal, with frequency as a function of time. Synth FMCW radars, the frequency increases linearly 4 with time. This type of signal is also called radar a chirp. An FMCW system transmits a chirp signal and captures the signals reflected by objects in its path. Fi Figure 1 shows a representation represents of a chirp asignal, simplified block diagram of the main RF components of aasFMCW radar.ofThe radar operates as Figure a function time. RX ant.2. Chirp signal, with frequency IF signal mixer with magnitude (amplitude) as a function of time. A A synthesizer (synth) generates a radar chirp. radar system transmits chirp signal anddiagram. captures the signals reflected by objects in its path. F Figure 3.aFMCW block FIGURE 1 & 2An FMCW The chirp is transmitted by a transmit antenna (TX ant). represents a simplified block diagram of the main RF components of a FMCW radar. The radar operates a The reflection of the chirp by an object generates a reflected chirp captured by the receive antenn generates A synthesizer (synth) generates a chirp. synthesizer (synth) a chirp. AA“mixer” combines the RX and TX signals to produce an intermediate frequency (IF) signal. The chirp is transmitted by achirp transmit antenna (TXby ant). The is transmitted a transmit antenna The reflection of the chirp by an object generates a reflected chirp captured by the receive anten (TX 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 1 & 2 Figure 2 shows the same chirp signal, with frequency as a function of time. The chirp is A B 4 GHz produce an intermediate frequency (IF) signal. A frequency mixer is an electronic component that characterized by a start frequency (fc), bandwidth (B) combines two signals to create a new signal with a and duration (Tc). The slope of the chirp (S) captures t new frequency. fc 77 GHz Figure 3. FMCW radar block diagram. T 40 µs c the rate of change of frequency. In the example A frequency mixer is an electronic component that combines two signals to create a 1new signal with a new For two sinusoidal inputs x1 and x2 (Equations 1 t provided in Figure 2, fc 77 GHz, B 4 GHz, Figure 3. FMCW radar block diagram. and 2): x2 (Equations 1 and 2): For two sinusoidal inputs x1and Tc 40 µs and S 100 MHz/µs. A frequency mixer is an electronic component that combines two signals to create a new signal with a ne 𝑥𝑥1 sin(𝜔𝜔1 𝑡𝑡 𝛷𝛷1 ) (1)(1) f 81 GHz For two sinusoidal inputs x1 and x2 (Equations 1 and 2): 𝑥𝑥2 sin(𝜔𝜔2 𝑡𝑡 𝛷𝛷2 ) (2)(2) sin(𝜔𝜔1 𝑡𝑡 𝛷𝛷1 ) (1) 1 an frequency The output xout 𝑥𝑥has The output xout has an instantaneous frequency equalinstantaneous to the difference of the instantaneous frequencies o B 4 GHz input sinusoids. The phase equal of the output to of the(2) phases of the two input signa out𝑥𝑥is equal to the xdifference of 2the the instantaneous 𝑡𝑡 difference 𝛷𝛷2 ) 2 sin(𝜔𝜔 3): frequencies of the two input sinusoids. The phase The output xout has an instantaneous frequency equal to the difference of the instantaneous frequencies fc 77 GHz equal difference of phases the (3)of the two input sign ) 𝑡𝑡the 𝑥𝑥𝑜𝑜𝑜𝑜𝑜𝑜 xxsin[(𝜔𝜔 𝜔𝜔to (𝛷𝛷 outisisequal 1 to 2the 1 𝛷𝛷of 2 )] input sinusoids. The phaseof ofthe the output output difference the out t phases of the two input signals (Equation 3): Tc 40 µs 3): 1 The operation of the frequency mixer can also be understood graphically by looking at TX and RX chirp fre Figure 2. Chirp signal, with frequency as a function of time. representation as a function of time.𝑥𝑥𝑜𝑜𝑜𝑜𝑜𝑜 sin[(𝜔𝜔1 𝜔𝜔2 ) 𝑡𝑡 (𝛷𝛷1 𝛷𝛷2 )] (3) (3) S The operation of the can also be at TX and RX chirp fr The operation of the frequency mixer can also be frequency understood mixer graphically by looking representation as a function of time. graphically by looking at TX and RX understood An FMCW radar system transmits a chirp signal and 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 chirps 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 initial phase of the IF signal (Φ0) is the difference between the phase of the TX chirp and the ph The mixer output signal asThe a magnitude function is a sine wave, since has isa aconstant frequ mixer output signalofastime a magnitude function of ittime sine wave, si at the time instant corresponding to the start of the IF signal (i.e., the time instant represented by th line Figure 4). (Equation 5): initial The in initial phase of the IF signal (Φ0) phase is the difference between the of thebetween TX chirp the and the p The of the IF signal (Φ0) is thephase difference phase gram 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 corresponding to the startcorresponding of the IF signalto (i.e., time instant represented at the time instant thethe start of the IF signal (i.e., theby timt 𝜙𝜙0 2𝜋𝜋𝑓𝑓𝑐𝑐 𝜏𝜏 (5) me-delay version of the TX chirp. line in Figure 4). (Equationline 5): in Figure 4). (Equation 5): Mathematically, it can be further derived into The time delay (t) can be mathematically derived as Mathematically, it can be further derived into Equation 6: (τ) can be mathematically derived as Equation 4: 𝜙𝜙0 2𝜋𝜋𝑓𝑓𝑐𝑐 𝜏𝜏 (5)𝜙𝜙0 2𝜋𝜋𝑓𝑓𝑐𝑐 𝜏𝜏 Equation 6: Equation 4: (5 𝜏𝜏 4𝜋𝜋𝑑𝑑 2𝑑𝑑 𝑐𝑐 (6)* (4) it can be further further (6) 𝜙𝜙0be (5) Mathematically, derived into Equation Mathematically, it can 𝜆𝜆 6: derived into Equation 6: 4𝜋𝜋𝑑𝑑 4𝜋𝜋𝑑𝑑wave (Equation 7 where object d is the distance the of detected objectfor an objectInatsummary, In summary, a distance for d from the IF signal be a sine distance to the detected and c is the to speed light. an object distance d will from at athe (6)𝜙𝜙 𝜙𝜙0 radar, (6 0 𝜆𝜆 𝜆𝜆 and c is the speed of light. the radar, the IF signal will be a sine wave 𝑜𝑜 𝑡𝑡 𝜙𝜙0 ) (7) requency representation as a function of time of the IF signal at the output of the frequency mixer, In summary, for an object In at summary, a distancefor d from the radar, the IF signal willthe be radar, a sine wave an object at a distance d from the IF(Equation signal will (Equation 7), then: To obtain the frequency representation as a function wo lines presented in the upper section of Figure 4. The distance between the two lines is fixed, which 𝑆𝑆2𝑑𝑑 4𝜋𝜋𝑑𝑑 where 𝑓𝑓04 frequency and 𝜙𝜙0 frequency . e IF signal consists of atime toneofwith Figure shows is Sτ. The IF 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴(2𝜋𝜋𝑓𝑓 (7)**(7) 𝑜𝑜 𝑡𝑡 𝜙𝜙0 ) the aIFconstant signal atfrequency. the output of the 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴(2𝜋𝜋𝑓𝑓 𝑜𝑜 𝑡𝑡 𝜙𝜙0 ) 𝑐𝑐 that this 𝜆𝜆 nly in the time interval where both the TX chirp and the RX chirp overlap (i.e., the interval between the mixer, subtract the two lines pre sented in the 𝑆𝑆2𝑑𝑑 upper 4𝜋𝜋𝑑𝑑 𝑆𝑆2𝑑𝑑 detected only 4𝜋𝜋𝑑𝑑 one object. Let’s analyze a case when ther The assumption so far𝜙𝜙is that the where 𝑓𝑓0 and . radar where 𝑓𝑓0 has and 𝜙𝜙0 . 0 lines in Figure 4) . where 𝜆𝜆 𝑐𝑐 𝜆𝜆 section of Figure 4. The distancedetected. betweenFigure the𝑐𝑐two RXNotice chirps that received per diagram in Figure 4 shows TX and RX chirps as a function of time for5ashows singlethree objectdifferent detected. the from different objects. Each chirp is dela The assumption so farisisthat that the radar has detected lines is IFassumption signal consists The so far is that radar hasso detected only one object. Let’s analyze caseobject. when Let’ ther Thethe assumption far the radar has detected onlyaone p is a time-delay version offixed, the TXwhich chirp.means that the detected. 5 showsonly three different RX chirps received from different objects. Each chirp is del one object. Let’s analyze a case when there of a tone with a constant frequency. FigureFigure 4 shows detected. Figure 5 shows three different RX chirps received from differen e delay (τ) can be mathematically derived as Equation 4: are several objects detected. Figure 5 shows three that this frequency is St. The IF signal is valid only in 2𝑑𝑑 the time interval where both the (5)the RX 𝜏𝜏 TX chirp and 𝑐𝑐 chirp overlap (i.e., the interval between the vertical d is the distance to dotted the detected and4).c is the speed of light. lines inobject Figure 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 chirps translate in the frequency representation as a function of time of the IF signal at the output of theRX frequency mixer, to multiple IF tones, f TX chirp each withlines a constant t the two lines presented in the upper section of Figure 4. The distance between the two is fixed,frequency. which RX chirp that the IF signal consists of a tone with a constant frequency. Figure 4 shows that this frequency is Sτ. The IF Reflected signal f S from multiple chirp valid only in the time interval where both the TX chirp and the RX chirp overlap (i.e., the interval TX between the objects dotted lines in Figure 4) . t Figure 4. IF frequency is constant. T c put signal as a magnitude function of f time is a sine wave, since it has a constant frequency. IF signal se of the IF signal (Φ0) is the difference S between the phase of the TX chirp and the phase of the RX chirp f the left vertical dotted tant corresponding to the start of the IF signal (i.e., the time instant represented by t ). (Equation 5): t Figure 4. IF frequency is constant. (5) 𝜙𝜙0 2𝜋𝜋𝑓𝑓𝑐𝑐 𝜏𝜏 t y, it can be furtherThe derived into Equation 6: as a magnitude function mixer output signal Figure 5. Multiple IF tones for multiple-object detection. This IF signal consisting of multiple tones must of time is a sine wave, 4𝜋𝜋𝑑𝑑 since it has a constant (6) 𝜙𝜙0 𝜆𝜆 frequency. be processed using a Fourier transform in order to separate the different tones. Fourier transform or an object at a distance d from the radar, IF signal adifference sine wave (Equation 7), then: ) isisbe the The initial phase of the the IF IF signal (F0will Figure 4. frequency constant. processing will result in a frequency spectrum that between the phase𝑜𝑜 𝑡𝑡of the 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴(2𝜋𝜋𝑓𝑓 𝜙𝜙0TX ) chirp and the (7) phase er output signal as a magnitude function of time is a sine wave, since it has a constant frequency. has separate peaks for the different tones each of the RX chirp at the time instant corresponding 𝑑𝑑 4𝜋𝜋𝑑𝑑 𝜙𝜙0 of the IF . signal ialand phase ) isof thethe difference of the TX chirp and the phase of the RX chirp to the(Φ start IF signalbetween (i.e., thethe timephase instant 𝜆𝜆 * This equation is an approximation and valid only if Slope and 0 distance are sufficiently small. However, me instant corresponding to theby start the IF signal (i.e., the by the left vertical dottedit is still true that the phase represented theof left vertical dotted linetime in instant represented n so far is that the radar has detected only one object. Let’s analyze a case when ofthere are several objects the IF signal responds linearly to a small change in the distance igure 4). (EquationFigure 5): 4). (Equation 5): re 5 shows three different RX chirps received from different objects. Each chirp is(i.e., delayed by a different Δf 4πΔd/l). 𝜙𝜙0 2𝜋𝜋𝑓𝑓𝑐𝑐 𝜏𝜏 (5)(5) matically, it can be further derived into Equation 6: 𝜙𝜙0 4𝜋𝜋𝑑𝑑 𝜆𝜆 ** In this introductory white paper we ignore the dependence of the frequency of the IF signal on the velocity of the object. This is usually a small effect in fast-FMCW radars, and further can be easily corrected for once the Doppler-FFT has been processed. (6) of from millimeter radarthe sensors 4 mary, for an object The at afundamentals distance d thewave radar, IF signal will be a sine wave (Equation 7), then: 𝑜𝑜 𝑡𝑡 𝜙𝜙0 ) (7) July 2020
constant frequency. ty peak denoting the presence of an object at a specific distance. Range resolution Velocity measurement with two chirps In order to measure velocity, an FMCW radar Velocity Measurement Velocity Measurement transmits two chirps separated by Tc. Each reflected In this section, let’s use chirp phasor (distance, angle) for a complex number. is notation processed through FFT the Range resolution is the ability to distinguish In this section, let’s use phasor notation (distance, angle) fortoa detect complex number. range the object Measurement between two or more objects. When two objects Velocity Measurement withofTwo Chirps (range-FFT). The range-FFT Velocity Measurementcorresponding with Two Chirps to each chirp will have peaks in move closer, at some point, a radar system will tion, let’s use phasor notation (distance, angle) for a complex number. Velocity In order toMeasurement measure velocity, a FMCW radar transmits chirpsphase. separated by Tc Each reflected ch the asame location, but withtwo a two different The In order to measure velocity, FMCW radar transmits chirps separated by Tc Each reflected chir no longer be able to distinguish them as separate through FFT to IF detect of the object (range-FFT). The range-FFT corresponding to each ch Figure 5. Multiple tonesthe forrange multiple-object detection. phase corresponds tonumber. acorresponding motion through FFT you tolet’s detect themeasured range of the objectdifference (range-FFT). The range-FFT to each chir Measurement with Two Figure Chirps Multiple IFtheory tonesIn for multiple-object detection. objects. Fourier5.transform states that can this section, use angle) for a complex the same location, butphasor with anotation different(distance, phase. The measured phase difference corresponds to a m the same location, but with a different phase. The measured phase difference corresponds to a mot gure 5. Multiple IFincrease tones for multiple-object detection. in the object of vTc. This IF signal consistingby of multiple tones beofprocessed using a Fourier transform in order to separate the different the resolution increasing the amust length vTc. Figure 5. Multiple tones for multiple-object detection. signal consisting tones must beIFprocessed using Fourier transform in order to separate the different oF measure velocity,ofamultiple FMCW radar transmits two chirps separated by T Each reflected chirp is processed c Velocity Measurement with Two Chirps vTc.will result tones. Fourier transform processing in a frequency spectrum that has separate peaks for the different tones the IF eFT must be processed using a Fourier order to corresponding separate different . tones Fourier transform processing will result intransform a frequency spectrum that hasthe separate peaks thehave different to detect the range of signal. the object (range-FFT). The in range-FFT to each chirpfor will peakstones in each peak denoting theprocessed presence using of an aobject at transform a specific distance. signal consisting of multiple tones must be Fourier in order to separate the different In order to measure velocity, a FMCW radar transmits two ng willdenoting result a frequency spectrum thatmeasured separate peaks thecorresponds different tones peak the presence ofphase. an object athas a the specific distance. location, butinwith aTo different The difference to a motion in the object of chirps separated by Tc Each reflected chir increase the length IF phase signal, theforthat Fourier transform processing will result in aoffrequency spectrum has separate peaks for the different tones The range-FFT corresponding to each chir through FFT to detect the range of the object (range-FFT). of an object at a specific distance. Range Resolution bandwidth must also be increased proportionally. An eak denoting the presence of an object at a specific distance. e Resolution the same location, but with a different phase. The measured phase difference corresponds to a mot increased-length IF signal results in an IF spectrum Tc When two objects move closer, at some vTc. Range resolution is the ability to distinguish between two or more objects. eResolution resolution is the ability to distinguish between two or more objects. When two objects move closer, at some with twoa separate peaks. point, radar system will no longer be able to distinguish them as separate objects. Fourier transform theory states that between or more two objects closer, at some ,istinguish a radar system willtwo no longer beobjects. able to When distinguish them asmove separate objects. Fourier transform theory states that youtocan increase between the resolution increasing length of objects the IF signal. esolution is the ability distinguish twostates orbymore objects. two move closer, at some Fourier transform theory also an theWhen r be able tothe distinguish them as separate objects. Fourier transform theory states that an increase resolution by increasing the length of thethat IF signal. radar system will no longer be able to distinguish them as separate objects. Fourier transform theory states that increasing the length the IF signal. observation window (T)ofcan frequency Toof increase the length theresolve IF signal, the bandwidth must also be increased proportionally. An increased-length IF increase resolution increasing the lengthmust of thealso IF signal. crease thethe length of the IFbysignal, the bandwidth be increased proportionally. An increased-length IF components separatedwith by more than peaks. signal resultsthat in anare IF spectrum two separate nal, the bandwidth must also be increased proportionally. An increased-length IF results in an IF spectrum with two separate peaks. ease the length of the IF signal, bandwidth must also betones increased An increased-length IF 1/THz. This the means that two IF signal can proportionally. Figure 6. velocityfrequency measurement. h two separate peaks. Fourier transform theory also states that an observation window (T)Two-chirp can resolve components that are esults in an IFtheory spectrum with two separate peaks. er transform also states that an observation window (T) can resolve frequency components that are be resolved in frequency as long as the frequency 6. in Two-chirp velocity separated by more than 1/THz. This means that two IF signal tones can be Figure resolved frequency as longmeasurement. as the es that observation window (T) canthat resolve components that arein frequency asFigure 6. the Two-chirp velocity measurement. ated byan more thandifference 1/THz. Thissatisfies means twofrequency IF signal given tones can be resolved long as the relationship in frequency difference satisfieswindow the relationship given infrequency Equationphase 8: transform theory also states that an observation (T) can resolve components that are difference is means that two IF signal can be resolved in frequency as long as the is The ency difference satisfies thetones relationship given in Equation The8:phase difference defined as Equation 10:is derived from Equation 6 as Equation ed by more than 1/THz. This 8: means that two IF signal The tones can be resolved in frequency as long as the phase difference is defined as Equation 10: 1 Equation 10: relationship given in Equation 8: Δ𝑓𝑓 (8) 1 4𝜋𝜋𝜋𝜋𝑇𝑇𝑐𝑐 ncy difference satisfies the relationship given Figure 6. Two-chirp velocity8: measurement. 𝑇𝑇𝑐𝑐 (8) Δ𝑓𝑓 in Equation 𝛷𝛷 (10) 4𝜋𝜋𝜋𝜋𝑇𝑇𝜆𝜆𝑐𝑐 𝑇𝑇𝑐𝑐 (10) 1 𝛷𝛷 (10) Δ𝑓𝑓 (8) 𝜆𝜆 1 𝑇𝑇𝑐𝑐 where the10: observation eedifference is defined as Equation Δ𝑓𝑓 interval. (8) observation where TTc cisisthe Figure11: 6. Two-chirp velocity measurement. Tc is the observation interval. 𝑇𝑇𝑐𝑐 interval.You can derive the velocity using Equation You canEquation derive the You can derive the velocity using 11:velocity using Equation 11: 𝑆𝑆2Δ𝑑𝑑 𝑐𝑐 𝑐𝑐 al. 4𝜋𝜋𝜋𝜋𝑇𝑇𝑐𝑐 Since Δ𝑓𝑓 , equation (8) expressed asasΔ𝑑𝑑 as(since B 10: STc ). Since Equation 8 can be expressed 𝑐𝑐be 𝑐𝑐 (10) 𝜆𝜆Δ𝛷𝛷 The difference Equation TΔ𝑓𝑓 the𝑆𝑆2Δ𝑑𝑑 observation interval. c is 𝑐𝑐 𝛷𝛷 as 2𝑆𝑆𝑇𝑇𝑐𝑐 2𝐵𝐵 (11) , equation (8) can be expressed phase (since B STcis). defined 𝜆𝜆 Δ𝑑𝑑 𝑣𝑣 𝜆𝜆Δ𝛷𝛷 (11) 𝑐𝑐 2𝑆𝑆𝑇𝑇𝑐𝑐 2𝐵𝐵 4𝜋𝜋𝑇𝑇 𝑐𝑐 𝑐𝑐 𝑐𝑐 𝑣𝑣 (11) n be expressed as Δ𝑑𝑑 (since B BST ). c). 𝑐𝑐 4𝜋𝜋𝑇𝑇 (since c ST 𝑐𝑐 𝑐𝑐 4𝜋𝜋𝜋𝜋𝑇𝑇 𝑆𝑆2Δ𝑑𝑑 𝑐𝑐 2𝑆𝑆𝑇𝑇𝑐𝑐be2𝐵𝐵 ) depends on theB bandwidth swept the by the chirp measurement (Equation The range resolution (d erive using 11: 𝛷𝛷 9): is based on (10)a 𝑓𝑓 the velocity , equation (8)Equation can expressed asResΔ𝑑𝑑 only(since ST ). Since 𝑐𝑐 2𝑆𝑆𝑇𝑇swept 2𝐵𝐵 by chirpc (Equation 9): isvelocity ange resolution (dRes) depends only on the bandwidth thethe velocity measurement based on a phase 𝜆𝜆difference, there will be ambiguity. The meas 𝑐𝑐 Since Since the velocity measurement is based on a phase difference, there will ) depends only on the The range resolution (d 𝜆𝜆 𝑐𝑐 phase difference, there will be ambiguity. The be ambiguity. The measur Res ds only on the bandwidth swept by the chirp (Equation 9): 𝜆𝜆Δ𝛷𝛷 unambiguous only π. Using equation if 𝛷𝛷 (9) 𝑑𝑑the 𝑐𝑐 swept 𝑅𝑅𝑅𝑅𝑅𝑅 𝑣𝑣 bandwidth (11) You can derive velocity using Equation 11: 11 above, one can mathematically derive 𝑣𝑣 𝜆𝜆 4𝑇𝑇𝑐𝑐. depends only on the by the chirp (Equation 9): ge resolution (dRes)bandwidth 2𝐵𝐵 (9) 𝑑𝑑 4𝜋𝜋𝑇𝑇 . unambiguous only if 𝛷𝛷 π. Using equation 11 above, one can mathematically derive 𝑣𝑣 𝑐𝑐chirp (Equation 9): swept by the𝑅𝑅𝑅𝑅𝑅𝑅 2𝐵𝐵 measurement is unambiguous only if DF p. Using 4𝑇𝑇𝑐𝑐 𝑐𝑐 (9) 𝑑𝑑𝑅𝑅𝑅𝑅𝑅𝑅 𝜆𝜆Δ𝛷𝛷 𝑐𝑐 2𝐵𝐵 an FMCW radar Thus with a chirp bandwidth of a few GHz will have a range resolution in the order of cm's (e.g. a chirp ) measured by two chirps spaced T apart. Equation 12 provides the maximum relative speed (v (9) Equation 11 above, one can mathematically derive max c 𝑣𝑣 (11) GHz will 𝑑𝑑of𝑅𝑅𝑅𝑅𝑅𝑅 velocity measurement based on a phase there willa(9) be ambiguity. Theinmeasurement is (e.g. an FMCW radar with a is chirp bandwidth adifference, few have resolution the order of cm's a𝑐𝑐 chirp 2𝐵𝐵 ) measured by two chirps spaced Tc apart. Hi 12range provides the maximum relative speed 4𝜋𝜋𝑇𝑇 (vmax bandwidth of 4GHz translates to Equation a shorter range resolution 3.75cm) 𝜆𝜆 transmission times between chirps. bandwidth of atranslates fewπ.GHz will a range resolution inshorter the order of cm's (e.g. a chirp . ous only if 𝛷𝛷 Using 11 above, one can mathematically derive 𝑣𝑣 width of 4GHz toequation ahave range resolution 3.75cm) timesinbetween chirps. 4𝑇𝑇𝑐𝑐order anbandwidth FMCW radar a chirp bandwidth ofresolution ameasurement FMCW radar withThus a chirp of a with few GHz will havethe atransmission range the of on cm's (e.g. a difference, chirp Since velocity is based a phase there will be ambiguity. The measur a range resolution 3.75cm) 𝜆𝜆 𝜆𝜆 v (12) few GHz will have a range resolution in the order dth of 4GHz translates to a range resolution 3.75cm) max 𝜆𝜆 Equation 12 provides the maximum speed unambiguous only spaced if 𝛷𝛷 Using equation canrelative mathematically derive 𝑣𝑣 4𝑇𝑇 . 12 provides the maximum relative speed (vmax) measured by two chirps Tc π. apart. Higher vmax 11 requires 𝑐𝑐 vmaxabove, 4𝑇𝑇one (12) 𝑐𝑐 4𝑇𝑇𝑐𝑐 of centimeters (vmax) measured by two chirps spaced Tc apart. ansmission times between chirps. (e.g., a chirp bandwidth of 4 GHz Velocity Measurement with Multiple Objects at)the Same Range measured bytimes two chirps spaced Tc apart. Hi Equation provides the maximum speed (v translates to a range resolution 3.75 cm). 12 Higher vmaxrelative requires shorter transmission Velocity Measurement with Multiple Objects atmax the Same Range 𝜆𝜆 vmax (12) shorter transmission times betweenchirps. chirps. between 4𝑇𝑇𝑐𝑐 The two-chirp velocity measurement method does not work if multiple moving objects with differ two-chirp velocity measurement method does not work if multiple moving objects with differen Velocity measurement The the time of measurement, both at the same distance𝜆𝜆 from the radar. Since these objects are at th (12)these objects are at the s vmax from the radar. Since (12) time of measurement, both at the same distance Measurement with Multiple Objects at the Samethe Range 4𝑇𝑇𝑐𝑐 will generate reflective chirps with identical IF frequencies. As a consequence, the range-FFT will r In this section, let’s use phasor notation (distance, will generate reflective chirps with identical IF frequencies. As a consequence, the range-FFT will res which moving represents the combined signal from all are of these equi-range objects. A simple phase compa hirp velocity measurement doesnumber. not work if multiple objects with different velocities angle) formethod a complex Velocity Measurement with Multiple Objects atatthe Same Range which represents the combined signal from all of these equi-range objects. A simple phase comparis not work. of measurement, both at the same distance from the radar. Since these objects are at the same distance, they not work. two-chirp velocity measurement method does not work if multiple moving objects with differen ate reflective chirps with identical IF frequencies. As aThe consequence, the range-FFT will result in single peak, the time of measurement, both at the same distance resents the combined signal from all of these equi-range objects. A simple phase comparison technique willfrom the radar. Since these objects are at the s will generate reflective chirps with identical IF frequencies. As a consequence, the range-FFT will res The fundamentals of millimeter wave radar sensors 5 July 2020 which represents the combined signal from all of these equi-range objects. A simple phase comparis not work.
Velocity Measurement Figure 8. The range-FFT of the reflected chirp frame results in N phasors. In this section, let’s use phasor notation angle)FFT, for acalled complex number. is performed on A second Doppler-FFT, Velocity measurement with multiple objects at (distance, A second FFT, called Doppler-FFT, is performed on the N phasors to resolve the two objects, as shown the same range Velocity Measurement with Two Chirps the N phasors to resolve the two objects, as shown in Figure 9. The two-chirp velocity measurement method does 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 the at the time of measurement, both phase. The measured phase difference corresponds to a motion in the object of same location, but with a different ω1 ω2 at the same distance from the radar. Since these vTc. Figure 9. Doppler-FFT separates the two objects. objects are at the
An FMCW radar system transmits a chirp signal and captures the signals reflected by objects in its path. Figure 3 represents a simplified block diagram of the main RF components of a FMCW radar. The radar operates as follows: A synthesizer (synth) generates a chirp. The chirp is transmitted by a transmit antenna (TX ant).
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Chính Văn.- Còn đức Thế tôn thì tuệ giác cực kỳ trong sạch 8: hiện hành bất nhị 9, đạt đến vô tướng 10, đứng vào chỗ đứng của các đức Thế tôn 11, thể hiện tính bình đẳng của các Ngài, đến chỗ không còn chướng ngại 12, giáo pháp không thể khuynh đảo, tâm thức không bị cản trở, cái được
