In this blog series, we’ve been working our way through understanding phased array beamforming fundamentals using the new Phaser (CN0566) Phased Array exploration kit from Analog Devices. Even if you are somewhat new to phased array systems, you’ve probably heard that “the spacing between elements must be less than half a wavelength apart.” The infamous lambda / 2 rule! But why does that “rule” exist? And what happens if we break it (which we almost always do!)? The reason for that rule is grating lobes. Understanding this odd phenomenon will also give us insight into why and how we can break this rule.
Grating lobes are often difficult to conceptualize. Therefore, before we get into the details behind them, let’s first start with the Phaser kit. We can use that to build an intuitive sense of the properties of these troublemakers.
If you’re following along with this series, I hope you have one of these kits. It’s much more fun to experience these things firsthand--rather than just reading about them in a boring blog. Complete getting started guide and all the software details are here. Simple Python programs of these concepts are here. But if you don’t have a kit, you can watch this video where I manipulate the board to produce grating lobes.
With the signal source at mechanical boresight (i.e. directly in front of the array as in the photo above), we’ve been seeing array patterns like this:
A nice main lobe, centered at where the signal is coming from (0°). And then those sidelobes that we squashed with tapering last week. The spacing between each element on the Phaser kit is 14mm. Therefore, for anything less than 10.7 GHz, our spacing is less than the mysterious “lambda/2”.
But let’s try disabling a few elements such that our effective “d” becomes 3 times larger:
We’ve turned off 5 elements, leaving just 3 elements in the array. Let’s see what that looks like:
Whoa! That is quite strange and somewhat surprising. First, look at the main lobe. The amplitude has decreased, but that’s because we only have 3 elements. What’s interesting to note is the beamwidth stayed the same. Even with 3 elements, we have the same beamwidth that we had with 8 elements because our antenna aperture size stayed roughly the same. It was (8-1)*14mm = 98 mm. And now it is (3-1)*(42mm) = 84 mm. It’s the aperture size that is setting the main lobe beamwidth.
The biggest difference between these plots is the two giant lobes at about +/- 45°. They are slightly wider than the main lobe but have nearly the same amplitude. Where did those come from?
We saw different kinds of sidelobes in our previous blog experiments. We came to understand that those sidelobes were produced by the nature of the rectangular (uniform) illumination) of the array. Therefore, tapering the amplitude of the array dramatically reduced those sidelobes. So let’s try tapering on these new lobes. Maybe it’ll work again? Here’s what a 22% taper on the outer elements looks like:
The main lobe decreased and its beamwidth is wider—which we saw before as an effect of tapering. But the mystery lobes are matching the main lobe. They are the same amplitude, and as that main lobe broadened, the other lobes also broadened. It is as if they are a distorted copy of the main lobe. And as you’ve probably guessed by now, those mystery lobes are grating lobes.
You’re probably familiar with frequency aliasing in a sampled system. In those systems, the Nyquist theorem states that a periodic signal must be sampled more than twice in its period. If you sample that signal less than that, you will get aliases of that signal showing up in different places. We see this commonly in under sampled ADC (analog to digital converter) systems. An alias of the signal will appear at different frequencies because we aren’t sampling the waveform fast enough.
In our phased array system, each antenna element acts as a spatial sample of the incoming waveform. And if we don’t have at least 2 elements per period (wavelength), then we’ll see spatial aliases of the main lobe; “Grating Lobes.”
Let’s calculate where we should have expected to see grating lobes, and then compare that to our measurements above. For a signal arriving from mechanical boresight, the locations of the main lobe and the grating lobes simplifies to:
θ = sin-1(m λ/d), for m=0, ±1, ±2, etc.
where λ is the wavelength and d is the spacing between elements
In our case, the frequency was 10.15 GHz:
θ = sin-1(m (3e8/10.15e9)/.042) = 0°, +44.7°, -44.7°
The main lobe is at 0°, that’s the “true” lobe. Then the spatial images (grating lobes) are at ±44.7°. Which is exactly where we see them!
Those were pretty extreme grating lobes, but as I said earlier, you can also get grating lobes for any element spacing greater than λ/2. Those grating lobes will appear as you steer to the horizon. And there’s an excellent explanation and derivation of why this happens here. It culminates in this classic equation:
For example, if we had a signal at 11.5 GHz, and our d was 14mm, then solving for θ_max would give:
θ_max = arcsin(3E8/(11.5e9*0.014)-1) = 60°
So we would expect to see a grating lobe, on the opposite horizon, as we steer the antenna array to +/- 60°.
This raises an interesting question: Does the element spacing always need to be less than λ/2? Not necessarily! This becomes a tradeoff for the antenna designer to consider. If the beam is steered completely to the horizon, then θ = ±90o, and an element spacing of λ/2 is required (if no grating lobes are allowed in the visible hemisphere). But in practice, the maximum achievable steering angle is always less than 90o. This is due to the element factor and other degradations at large steering angles. So if you are only going to steer to +/- 65°, then it may be advantageous for you to space the elements slightly further apart. It is something to consider. It is the reason why the λ/2 spacing rule is often exceeded.
Grating lobes are a very important, but somewhat weird, topic in phased array antenna design. I hope you were able to gain some insight into their behavior and causes.