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mmWave radar sensors
The mmWave training curriculum provides foundational content and hands-on examples for you to learn the fundamentals of FMCW technology and mmWave sensors to start development quickly. Our portfolio of mmWave sensors include: Automotive mmWave radar sensors and industrial mmWave radar sensors.
Intro to mmWave Sensing : FMCW Radars - Module 1 : Range Estimation
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Hello, everybody, and welcome to the series of technical videos on Millimeter Wave Sensing. Specifically, we'll be talking about a sensing technology called FMCW RADARS, which is very popular in both automotive and industrial segments. The goal of the series is to give you a short but hopefully fairly in-depth understanding of this class of radars.
FMCW stands for Frequency Modulated Continuous Waves. And I'll explain the reason for this naming a little bit later. This radar basically measures the range, velocity, and angle of arrival of objects in front of it. So this series of videos discusses each of these dimensions of sensing in some detail, starting with the range in module one, then moving on to velocity over the next couple of modules, and finally angle estimation in module five. For those of you who are new to millimeter wave sensing, I would recommend that you view all of these videos in sequence.
The first module is going to start with explaining the basics of FMCW radar operation. And then focus primarily on range estimation using the radar. So the kind of questions that we will focus on in this module are: you have a radar, you have an object in front of it, how does the radar estimate the range to this object. What if there are multiple objects at different ranges from the radar? How close can two objects get and still hope to be resolved as always two objects? What determines the furthest distance that a radar can see?
At the heart of an FMCW radar is a signal called a chirp. What is a chirp? A chirp is a sinusoid or a sine wave whose frequency increases linearly with time. So in this amplitude versus time, or A-t plot, the chirp could start as a sine wave with a frequency of, say, fc. And gradually increase its frequency, ending up with a frequency of say, fc plus B, where B is the bandwidth of the chirp. Thus The chirp is basically a continuous wave whose frequency is linearly modulated. Hence the term frequency modulated continuous wave or FMCW for short.
Now if the same chirp were represented in a frequency versus time plot, or f-t plot, how would it look? Remember that the frequency of the chirp increases linearly with time, linear being the operative word. So in the f-t plot, the chip would be a straight line with a certain slope S.
And just to put in some typical numbers, this figure for example could represent a chirp which starts at a frequency fc of 77 gigahertz, spans a bandwidth B of 4 gigahertz, thus ending up at a frequency of 81 gigahertz. The slope S of the chirp defines the rate at which the chirp ramps up. In this example, the chirp is sweeping a bandwidth of 4 gigahertz, with a time period Tc of 40 microseconds, which corresponds to a slope of 100 megahertz per microsecond. As we shall see later, the bandwidth B and the slope S are important parameters which define system performance.
Now that we know what a chirp is, we are ready to understand how an FMCW radar works. This here is a simplified block diagram of an FMCW radar with a single TX and a single RX antenna. The radar operates as follows. The synthesizer generates a chirp. This chirp is transmitted by the TX antenna. The chirp is reflected off an object. And the reflected chirp is received at the RX antenna. The RX signal and the TX signal are mixed. And the resulting signal is called an IF signal-- IF standing for Intermediate Frequency. We'll analyze the IF signal in more detail in the next slide.
But first, let's spend a little time understanding this component called a mixer. What is a mixer? A mixer has two inputs and one output. A simple way to understand the mixer is as follows. If two sinusoids are input to the two input ports of the mixer, the output of the mixer is a sinusoid with the following two properties.
Property number 1: the instantaneous frequency of the output equals the difference of the instantaneous frequencies of the two input sinusoids. So even if these sinusoids, if their frequencies were varying with time, the frequency of the output at any point in time would be equal to the difference of the input frequencies at that point in time.
Property number 2: the starting phase of the output sinusoid is equal to the difference of the starting phases of the two input sinusoids. These two properties are illustrated in these equations here, where x1 and x2 are the two inputs, and x_out is the output of the mixer. So note here that the two inputs have frequencies of omega 1 and omega 2, and starting phases of phi 1 and phi 2, respectively. And the output has a frequency of omega 1 minus omega 2, and a starting phase of phi 1 minus phi 2.
Let's look more closely at the operation of the mixer in the radar. And I think it's best illustrated using the f-t plot that we talked about earlier. So the plot here refers to the RF signal. So you have the transmitter chirp here and the received chirp here. Note that the received chirp is a time delay replica of the TX chirp. And for now, I'm assuming there is only one object in front of the radar. Hence, only one RX chirp.
Remember from the last slide that the output frequency of the mixer is the difference of the instantaneous frequencies of its two inputs, namely the TX chirp and the RX chirp. So to generate the f-t plot for the IF signal, I just need to subtract this line from this. And as you can see, these two lines are at a fixed distance from each other. And that fixed distance is given by the slope of the chirp times the round trip delay. In other words, S-tau. So a single object in front of the radar produces an IF signal consisting of a single frequency given by S-tau.
Now tau, the round trip delay from the radar to the object and back, can also be expressed as twice the distance to the object divided by the speed of light. So this is the fundamental concept to remember. A single object in front of the radar produces an IF signal with a constant frequency given by S2d/c.
Now, it is important to note that the IF signal is only valid from the time the reflected signal is received at the RX antenna. So if you were to digitize this IF signal using an ADC, you need to make sure that you only pick up samples after this time tau has elapsed, and only up to the time where the TX signal is present.
Another point worth noting is that the round trip delay tau is usually a very small fraction of the total chirp time. So for example, for a radar with a maximum distance of 300 meters and a chirp time of 40 microseconds, this ratio of tau by Tc is only 5%. Fourier transforms are at the heart of FMCW radar signal processing. And we will see throughout the series of videos, they are used in range, velocity, and angle estimation. So we will from time to time take short detours to remind ourselves of relevant properties of Fourier transforms.
A Fourier transform converts a time domain signal into a frequency domain. So a single tone in the time domain produces a single tone in the frequency domain. Likewise, the two tones in the time domain should result in two peaks in the frequency domain. But is that always the case? So in this example here, within the observation window of T, the red tone completes two cycles while the blue tone completes 2.5 cycles. And this difference of 0.5 cycles between the red and the blue tone, it seems is not sufficient to resolve the two tones in the frequency spectrum. So here you have only a single tone corresponding to the contributions from both these signals.
Let's now double the observation window from T to 2T. Doubling the observation window, now results in a difference of one cycle between the red and the blue tones. And as you can see, the two tones are now resolved in the frequency spectrum. So the take away is that longer the observation period, better the resolution. And in general, an observation window of T can separate frequency components that are separated by more than 1 by T hertz. This completes our short digression on Fourier transforms.
So far, we've talked about a single object in front of the radar. It's easy to extend this to the case where there are multiple objects in front of the radar. So here, you have a radar transmitting a single chirp, and you get multiple reflected chirps from different objects. Each delayed by a different amount depending on the distance to that object. So the IF signal will have tones corresponding to each of these reflections. And the frequency of these tones, as we learnt, is directly proportional to the range. So this has the smallest frequency and corresponds to the closest object. While this corresponds to the farthest.
A Fourier transform on this IF signal well then show up multiple peaks. And the frequency of these peaks will be directly proportional to the range of the corresponding object. So again, this corresponds to the closest object, and this one to the farthest. The moment we talk about multiple objects, the next natural question is range resolution. That is, how close can two of these objects be and still be resolved as two peaks in the IF spectrum.
So in this example, we have two reflected chirps from two objects. And the corresponding A-t plot of the IF signal shows two sine waves. But the frequencies of these sound waves are so close that they show up as a single peak in the frequency spectrum. How do we improve the range resolution of this radar?
Taking a cue from our recap of Fourier transforms, one option is to extend the observation window of these two sine waves by increasing the length of the IF signal. So that's what I've done here. So the chirp is extended which then extends the duration of the IF signal. And this resolves the two peaks in the frequency domain. Note that increasing the duration of the IF signal proportionally increases the bandwidth of the chirp. So that gives us a clue that possibly a larger bandwidth corresponds to a better range resolution.
Now that we have some intuition on how to improve the range resolution of radar, it would be nice to go a step further and actually derive an expression for this range resolution. And as it turns out, it's not that difficult either. All you need to know are these two pieces of information, something that we've already learned before. So at this point, I would really like to encourage you to pause here and try and derive this expression for the range resolution.
So two objects, which are delta d apart in distance, will have the IF frequencies separated by delta f, given by this expression. For these two frequencies to show up as distinct peaks in the IF frequency spectrum, this frequency separation delta f must be greater than 1 by the duration of the IF signal, which is virtually equal to the duration of the chirp TcC if you discount the small fraction in the beginning, the portion tau arising from the round trip delay.
So substituting for using this expression, you know we get this inequality here, which after some rearrangement becomes this. And note that the slope times the duration of the chirp is actually the bandwidth of the chirp. So this expression can be further simplified, and you finally get this expression here, which says that two objects can be separated in the IF frequency spectrum as long as the distance (separation) between them is greater than the ratio of the speed of light to twice the bandwidth of the chirp. So the takeaway here is that the range resolution depends only on the bandwidth swept by the chirp, and is given by this expression over here-- speed of light divided by twice the bandwidth.
Time for a question now. So you have two chirps here-- Chirp A and Chirp B. Chirp A has twice the duration of Chirp B. But both of them have the same bandwidth. Which of these two chirps gives you a better range resolution? So if you think about it, both of these chirps have the same bandwidth, B. So from the formula that we just derived, c by 2B, both of them should have the same range resolution.
But then, Chirp A has a longer duration and hence a longer observation window of the IF signal. So intuitively, if you take into consideration the properties of Fourier transforms, this chirp, Chirp A, should have a better resolution than just Chirp B. How do we resolve this contradiction? Something for you to think about.
So we've talked about the IF signal, and that the frequency of tones in the IF signal is directly proportional to the range of objects. In most radars, what happens is that the IF signal is digitized for subsequent processing. So it's first low pass filtered, and then digitized by an ADC, and sent to a suitable processor such as a DSP.
The DSP could begin by doing a Fourier transform to estimate the range of objects, and subsequently do other kinds of processing to estimate the velocity and angle of arrival of these objects. And this is something we'll get to in subsequent modules. When ever we are digitizing a signal. we need to know what is the bandwidth of interest so that the low pass filter and the ADC sampling rate can be appropriately set.
So let's say we are interested in objects from zero to a maximum distance of, say, dmax. The maximum IF signal. The maximum frequency of the IF signal is then going to be S2dmax/c. And correspondingly, the bandwidth of interest is going to be from zero to this maximum IF frequency which means that the low pass filter should have a cut-off frequency, which is beyond this IF max. And also the ADC should have a sampling rate which is greater than the same value.
So you can see here that the maximum sampling rate of the ADC can limit the maximum distance that the radar can see. Note that the maximum IF bandwidth depends on the product of the slope and the maximum distance. So if the ADC sampling rate and hence the IF bandwidth is a bottleneck in the sensor, you can always trade off the slope and the maximum distance. And typically, radar's tend to use smaller slopes for larger dmax.
Time for another question. Re-visiting our earlier example, what more can we say about these two chirps? Chirp A and B have the same bandwidth. But Chirp A takes twice as long as Chirp B. A good time pause the video, and try to answer this question.
Since both A and B have the same bandwidth, they of course, have the same range resolution. But note that Chirp A has half the slope of Chirp B. So for the same maximum range requirement, or for the same dmax, Chirp A would require only half the IF bandwidth, which translates to an ADC with a smaller sampling rate. So while Chirp A has the advantage of a more relaxed ADC requirement, Chirp B of course has the advantage of requiring only half the measurement time. So that is the trade going on here.
So this slide summarizes all that we have discussed so far. This is a block diagram of an FMCW radar with a single transmit and a single receive antenna. Let's go with the sequence of events involved in estimating the range of an object. So first, the synthesizer or synth generates a chirp. This chirp is transmitted over the TX antenna. It is reflected off multiple objects in front of the radar. And the receiver sees delayed versions of this chirp. The received signal and the transmitted signal are mixed to create an IF signal.
This IF signal consists of multiple tones and the frequency of each of these tones is proportional to the range of the corresponding object. The IF signal is then low pass filtered and digitized. And note that the sampling rate of the ADC must be commensurate with the maximum distance that we wish to see.
The digitized data is then processed. An FFT is performed on this data. And location of the peaks in the frequency spectrum directly correspond to the range of objects. Note that here I've plotted the FFT with range on the x-axis rather than the IF frequency, which is OK. Because as we've learned, the IF frequency is directly proportional to the range. This FFT is called range FFT because it resolve objects in range. And this term range FFT is something that you will see a lot in FMCW literature.
This slide over here summarizes some of the key concepts and formulas that we see in this module. First, an object at a distance of D produces an IF frequency of S2d/c. Range resolution depends only on the bandwidth spanned by the chirp and is given by the speed of light divided by twice the bandwidth. The ADC sampling rate Fs, limits the maximum range dmax that the radar can see.
The other thing: when we talk about bandwidth and FMCW radars, there are usually two bandwidths that are important. The RF bandwidth and the IF bandwidth. And it's important to clearly distinguish between both of these. So the RF bandwidth is the bandwidth spanned by the chirp. A larger RF bandwidth directly translates to a better range resolution. RF bandwidths are typically in the range of a few hundred of megahertz to several gigahertz. An RF bandwidth of 4 gigahertz, for example, translates to a range resolution of 4 centimeters. An RF bandwidth of 400 megahertz translates to a range of resolution of about 30 centimeters.
The other bandwidth is the IF bandwidth. A larger IF bandwidth primarily enables the radar to see a larger maximum distance. Also enables faster chirps. By faster chirps, I mean chirps with higher slopes. The IF bandwidth of typical radars is in the low megahertz region. So that's one of the things about FMCW radars, that you can have an RF signal spanning a large bandwidth of, say, 4 gigahertz, but yet your ADC would only need to sample a signal of a few megahertz.
This concludes the first module in the series. We focused primarily on range estimation using the FMCW radar. Here's a question to motivate the subsequent modules. So there are two objects equidistant from the radar. How will the range FFT look like? Now since these objects are equidistant, the range FFT will have a single peak corresponding to this range d and incorporating the effects of both these objects. So how then do we separate out these two objects. It turns out that if these two objects have different velocities relating to the radar, then they can be separated out by further signal processing. And to understand that, we need to really look at the phase of the IF signal which is something we will be doing in the next module.