Visualizing Radar Performance with the Ambiguity Function | Understanding Phased Array Systems and Beamforming

In a previous Tech Talk, we covered the basics of pulsed radar. That's where a radar system sends out short bursts of power, followed by some amount of silence where the radar is listening for the echo. Now, in that video, we mostly assumed that the pulse has a rectangular waveform. That is, the frequency of the signal is constant over the duration of the pulse. And if we think of this waveform in terms of sound, that constant frequency pulse would just be a pure tone that beeps at the pulse repetition frequency.

[BEEPING]

However, many real pulse radar systems don't send out a rectangular pulse. They modulate the frequency throughout the pulse. In the case of linear frequency modulation, or LFM, which I've drawn here, this waveform would sound more like a chirp.

[CHIRPING]

Now, from a hardware perspective, there is more complexity in creating a frequency-modulated pulse than there is in a simple rectangular one. So the question I want to answer in this video is, why go through the trouble of creating a chirp rather than just a beep? And to answer this question, we're going to create the ambiguity diagram for both of these pulse waveforms. This diagram provides a way for us to visualize how our choice of waveform affects our ability to measure range and Doppler of an object.

And what we're going to find is that there is a lot less ambiguity if we use a linear frequency-modulated waveform rather than a rectangular one. And I think you're going to have a much better understanding and appreciation for pulse waveforms and for the ambiguity diagram by the end of this video. So I hope you stick around for it. I'm Brian, and welcome to a MATLAB Tech Talk.

To begin, let's start with a rectangular pulse. This graph shows the signal that's being sent out by the radar over time. And this bottom graph is showing what the radar is receiving. So the radar sends out a pulse, and then some time later, it returns. Now, keep in mind that I'm talking about radar in this video, but these concepts are applicable to sonar as well, or really, any system that sends out pulsed waveforms.

All right, so notice that, even before the echo returns, the receiver is picking up on noise and other interferences in the environment. But even with all this noise, we can still see that the echo of the pulse has returned right around here. So the goal is that, if we can figure out the time delay between these two pulses in and amongst all of this noise, we can calculate the range to the object. And the way we do that is by processing the return signal with a matched filter.

Simply put, a matched filter is used to determine if a known signal is present in an unknown signal. That's all we're doing. And we've got a known signal. It's the exact pulse that we sent out. And we have an unknown signal. It's what we received at the radar. So we're looking for the correlation between the pulse shape and the received signal as that signal is being received. So it's just looking at-- how well-correlated are these two signals?

And let me show you graphically what this looks like. This is our received signal, but I removed all of the noise so that this explanation is easier to follow. And as this unknown signal comes in, it's compared with a template of a known signal. At this point in time, the received signal, which is that white line-- it's 0 since the pulse hasn't returned yet, and that means that there's absolutely no correlation between the template and the signal. The output of the matched filter, at this point, is 0.

Now, as the received signal comes in, we continue to check for the correlation between it and the template. And for a while here, it's all zeros. However, right at the start of the received pulse, there is some correlation between the two. Now, there's not much, since most of the received signal is 0, but the part of the pulse that has returned is at the same frequency as the template and, therefore, is correlated to the template.

And down here, we can see that the process signal is now slightly non-zero. And the correlation between these two signals continues to increase until the template perfectly aligns with the returned pulse. And at this point, it's the most correlated since they essentially line right up with each other. And then, as the pulse passes, its correlation with the template begins to drop again until we're completely out of the pulse.

So now we have a processed signal. And what we end up with is a signal that's more like a triangle than the rectangle that we started with. And this is good for at least two reasons. The first is that a triangle has a peak that we can use to determine the location of the returned pulse. And we can find a peak by looking at the maximum of the signal, which is much easier than trying to figure out when the energy from the pulse starts. So that's the first benefit.

But the second benefit is that the matched filter is really useful for pulling the signal out of noisy data. Watch what happens when I add a little random additive noise to a received signal. Notice how quickly the pulse becomes pretty much unrecognizable. And finding the start of the pulse energy would be pretty difficult. But with the process signal, it still has a nice peak right where the middle of the pulse is.

Now, this is because, in general, the random additive noise doesn't correlate very well with the template. So it won't make it through the matched filter as well as the highly correlated pulse does. But like with anything, there is a limit. Watch what happens as I continue to raise the noise floor. Eventually we get to a point where the raw signal is so corrupted with noise that the matched filter doesn't come back with a defined peak. It's a bit ambiguous where the peak is or whether this peak is caused from noise or the actual pulse.

, Now this ambiguity is clearly coming from the noise, as I'm showing, but it's not the noise alone. Ambiguity is also a function of the waveform. And here's what I mean by that. With the rectangular waveform, the strongest correlation comes when the template lines up with the echo pulse perfectly.

However, if we move the template just slightly to before the pulse, there's still a rather strong correlation since most of the template still overlaps the pulse, and the part that does overlap is perfectly correlated. This produces a processed signal that has a peak, which we know is good, but it's a rather shallow triangle.

Now, why is this important to point out? , Well for one, the shallow nature of this triangle means that noise can easily raise the correlation enough to produce a false peak near the real peak. And that false peak would cause us uncertainty in knowing the delay. But possibly more importantly is that this triangle is still functionally as wide as the original pulse, and that affects the resolution of the radar. That is, it impacts how close two separate objects could be from each other and still register as two objects.

For example, let's say that a radar sends out a pulse, and that pulse reflects off of two objects at different ranges. Therefore, what the radar receives is two echoes of the same pulse. And the time between these two pulses affect whether the matched filter sees both of them or just one large object.

And we can see this if we sweep the matched filter across the received signal with the two echo pulses. We get a peak at the first object. And then these two pulses are close enough that the template can span both of them, which produces this flat correlation line as the signal coming in from one end is exactly replacing the signal that is leaving the other. And then we get a peak for the second pulse. So when the objects are separated far enough to produce this received signal, we can still detect the two objects.

However, let's watch what happens as the objects get closer to each other and the time between the pulses shrink. And right at the point where the two pulses meet, their peaks converge, and we can no longer resolve the two objects. It just looks like one single, large object. This is a radar bandwidth issue. And in the case of a rectangular pulse, the bandwidth is a function of the pulse width. Therefore, we can increase bandwidth and, therefore, resolution by just lowering the pulse width. And this is because the objects can now be closer to each other before the returned pulses overlap.

But what we gain in resolution we lose in signal-to-noise ratio. And this is because, if the radar sends out the same power level but for a shorter amount of time, then there's less overall energy being transmitted per pulse, and therefore, less signal is being returned from the object. And with that lower signal comes a lower maximum range, as those far away objects will now drop below the detectable signal-to-noise ratio. So we have this trade between SNR and resolution.

But there is a way to increase resolution without having to decrease SNR, and that is by choosing a different pulse modulation scheme. For example, if we look at the chirp signal or the LFM pulse, we can see how this is the case. This waveform starts with a low frequency signal that increases linearly throughout the pulse, and our template is the exact same.

So now let's sweep this template across the received signal and see how they correlate. Now, once again, when the received signal is 0, there's zero correlation. And then when the template and the pulse start to overlap, there is-- well, actually not a whole lot of correlation here because we're overlapping a signal at two different frequencies. It's not until the template and the echo pulse line up right on top of each other before we start to see that really strong correlation. But then it quickly falls away as the frequencies, once again, are mismatched.

So here, we can see that the processed signal has a much sharper peak with an LFM waveform. It has been compressed to something that is much narrower than the original pulse. And this is why this type of technique is called pulse compression. Now, if I add the same level of noise to this signal as we did to the rectangular pulse example, we can see how much more obvious the peak is. There is still a single peak that rises above the rest of the noise. In this way, an LFM waveform is less susceptible to noise than a similarly powered rectangular pulse.

But in addition to that, we also get the benefit of higher resolution. Watch what happens as these two objects get closer to each other. Even as the two pulses start to overlap with each other, the process signal still has two defined peaks. In fact, the resolution that we get from an LFM waveform isn't a function of pulse width, but it's a function of how much frequency you sweep through throughout the pulse. Therefore, we can keep the same power levels and still increase the bandwidth of the radar system simply by sweeping across a larger band of frequencies.

Now, this is the good part of LFM waveforms, but there is something else that I need to explain about them. And that is that they generate a coupling between range and Doppler. And to understand this, we need to look at the ambiguity diagram, which I'm going to build up slowly. So hopefully, it'll make a lot more sense by the time we get to it.

We know that if the template in our matched filter matches the pulse perfectly, we get this compressed pulse in the process signal. However, this isn't always the case because of Doppler. If the object is moving towards the radar, then the returned pulse is shorter and the frequency is higher. And if the object is moving away, then the returned pulse is longer and the frequency is lower.

So why is this a problem? Well, our matched filter won't match the returned pulse if it has been Doppler-shifted. And this mismatch affects how well correlated the two signals are. Watch what happens when I start to add Doppler and increase the frequency of the returned pulse. As the frequency increases, the peak starts to move.

Now, the shape of the process signal doesn't change too much, but it does shift in delay. And so we still have a single sharp peak, just with some added error and delay, which translates to error in our range calculation. So with an LFM waveform, there is a coupling between Doppler and range error that we need to be aware of.

And this makes sense if you think about it. Doppler is shifting the frequency. But since the frequency is linearly modulated throughout the waveform, this looks very much just like a change in delay. Most of the frequencies will still exist in the signal, just later or earlier in time.

Now, I'll admit that it's really hard to interpret all this data the way that I've drawn it. So let me spread out these different Doppler slices. And you can see that we've created a 3D surface. This is the ambiguity diagram. It is showing us the result of the matched filter at different Doppler values. And we can see that, for the LFM waveform, the ambiguity diagram produces a sharp ridge that is diagonal across the delay and Doppler axes.

Now, if you want an even better visualization of this and a far better understanding of the ambiguity function and how to use it for waveform analysis, I recommend that you check out the pulse waveform analyzer app in MATLAB. All right, so as the name implies, with this app, you can analyze different pulse waveforms. And under the Parameters section over here, I'm going to change the waveform to Linear Frequency Modulated. And you can see all of these other parameters that you can adjust, including how to do range processing. But I'm going to leave it as matched filter just to stay consistent with what we talked about in the video.

All right, so on the plot side here, instead of the spectrum, let's look at the ambiguity plot. And we have several to choose from, but again, I'm going to stick with the 3D surface plot, and here it is. It's that familiar sharp peak that is diagonal across delay and Doppler. Of course, an easier way to look at this might be with just a 2D contour plot. And you can see it here. And it's a lot more obvious now. There's this coupling between range and Doppler.

And finally, if you're interested in just a slice of this function, either across Doppler or across delay, then we can look at any one of those cuts as well. So like I said, this app is nice to visualize the ambiguity plots for different waveforms and with different parameters, which I think is going to help you understand how each of these waveforms impact the radar performance.

I think it's all really interesting. And so I hope you check it out. I've linked to this app below. Oh, and by the way, did you know that we have a lot of different TikTok videos that cover many different topics? You can find them all at MathWorks.com. And if you enjoyed this video on pulse waveforms, you might be interested to learn about continuous waveforms, which we talk about in the Radar Basics series. It's worth checking out. All right, thanks for watching, and I'll see you next time.