# Why is the null of an adaptive beamformer narrower than antenna beam-width?

Question for those familiar with adaptive beamforming: Why is the null formed by an adaptive beamformer much narrower than the beamwidth of the antenna array?

## Concrete Example

Suppose we have a simple uniform linear array (ULA). We can combine the array elements to get the beampattern by beamforming. This pattern will have a half-power beamwidth of roughly $$\lambda /D$$ (in radians), where $$\lambda$$ is the wavelength, and $$D$$ is the length of the array.

If we do adaptive beamforming to mitigate interference from a particular direction, the null this forms in the beampattern is much narrower than $$\lambda /D$$. This is sometimes referred to as a form of super-resolution. Can someone explain where this super-resolution comes from?

## Edit

The same phenomenon applies to the MUSIC direction finding algorithm too. This is no surprise, since they are very similar mathematically.

• hey, the nulls of any antenna pattern are also narrow! (they are zeros of a holomorphic function, and that can only have discretely many points of zero, unless it's zero everywhere, and that would be an interesting antenna) Jul 13, 2022 at 16:04
• That's a good point point the nulls Marcus. Can you expand on the holomorphic function a bit, or point me to an accessible reference? Jul 15, 2022 at 1:02
• Ah! A homomorphic function is a complex function that behaves especially nicely - it's complex differentiable at every point. That's a pretty strong thing, because complex differentiable implies that unlike the real diffeentiation, where you only approach a point from "left and right", and find the slope through these two points and let them get arbitrarily close to the point you want to calculate the derivative in, in the complex case, you need to be able to approach the point from possible directions. Jul 15, 2022 at 1:44
• The effect of that is that if you want your function to be zero in a single point, that's fine, but as soon as that point becomes an area, it would require an infinite amount of directional derivatives to become zero. Long story short, if such a nicely differentiable function is zero in an area and not just a point, that area must fill everything. Jul 15, 2022 at 1:46
• yep! The far field (in sine-of-angle domain, typically) is proportional to the Fourier Transform of the current distribution in the antenna(s). Aug 24, 2022 at 9:58

Super-resolution techniques such as MUSIC can be used for creating higher resolution nulls (see MUSIC-based steering vector used in adaptive nulling suppression for example), but what I think may be missed is how beamforming antenna arrays readily allow for the creation of nulls that are narrower than the antenna beam-width without the use of more advanced "super-resolution" techniques. This is simply a property of the transversal structure in the way the beam is formed; we can readily create sharp nulls but cannot create similarly sharp peaks. With the beamformer structure we have the ability to place zeros (but not poles) in a specific direction, and can group several zeros together.

The beam pattern for an antenna array is the spatial version of what is temporal in an FIR filter, where as an "all-zero" structure we also have the option to concentrate zeros in one spatial direction. This is clearer with understanding the similarities between an FIR filter and a beamformer, and how poles and zeros affect the frequency response in filter design.

An FIR filter is a "transversal in time" structure; while a beamformer is "transversal in space". With the FIR filter we have an output that is the sum of multiple copies of the input each delayed or offset in time and individually scaled. With the antenna array we get the same thing, except offset in time is replaced with offset in space: We have multiple copies of the input signal as received by different antennas at different spatial locations. If each individual antenna was omni-directional and with proper spacing that is synonymous with the unit sample delay in an FIR filter ($$z^{-1}$$), we can predict and design antenna patterns using FIR design tools! (We still can in other cases, we just need to work in the individual antenna patterns as well).

That said, here is then the straight forward answer to the question. Like the FIR filter, the system is all zeros with only trivial poles at the origin on the z-plane. The antenna pattern is found by sweeping $$z$$ on the unit circle corresponding to spatial direction according to the antenna spacing. Thus by proper scaling prior to the summing of each element, we can place zeros toward one particular direction to achieve the null in that spatial direction.

Demonstrating this is a 13 element antenna array using firls to concentrate rejection in the broadside angle direction of $$0.35\pi$$ to $$0.40\pi$$ as follows:

coeff = sig.firls(13, [0, .2, .35, .4, .5, 1], [1, 1, 0, 0, 1, 1])


The resulting pattern when plotted as we would typically view a filter frequency response (which this is!) is given below:

And when plotted as a typical antenna pattern on a polar plot of magnitude in dB versus angle of arrival it would appear as shown below:

Note that in the plot above 0 to 180 degrees corresponds to the front direction of the antenna and 180 to 360 is the backlobe for a linear antenna array. It is typical to have the front centered on 0 degrees instead of 90 degrees as done above, in which case the above pattern should be rotated clockwise 90 degrees- I left as is to be consistent with the frequency axis returned from the frequency response tools).

When reviewing the roots (zeros) for this derived solution, we see how two of the zeros have been concentrated together on the unit circle in the direction of the null. The solution for the other zeros is such that we get a minimum least squares error from unity gain for all other directions with maximum rejection at the null within the constraint of a linear-phase spatial solution (given what firls provides, not that a linear-phase solution is necessary for this).

• This is a great analysis. Yet I don't see how it says how the super resolution work. Feb 24 at 8:35
• The point is there is nothing special done related to “super resolution” in this case, but a normal situation for FIR structures and related to that, antenna beamformers: we have the ability to group many zeros closely together on the unit circle in the z plane for dense nulls. In contrast we don’t have the ability to group several poles closely together on the unit circle to make sense peaks. Feb 24 at 12:19
• In RADAR applications we can have super resolution effect. It's not just working with FIR coefficients. Feb 24 at 13:25
• @DanBoschen, This is great! Actually your clear illustration of things gave me idea for actual super resolution :-).
– Royi
Feb 24 at 15:19
• @DanBoschen, I tried to sketch the idea as answer to your question. I really think it is a good idea. Let me know what you think.
– Royi
Feb 24 at 17:00

There is equivalence between this method and the analysis of resolution of the DFT. So let's talk about that.

Usually when talking about the resolution in the DFT we talk about the resolution of the interpolation function which is inversely proportional to the time window of the observation.

It is indeed holds, but we have to take into consideration what assumptions are made.
In this case, there are actually no assumption, namely we believe we have no knowledge about the signal we're looking for.

Yet, in signal processing and in most engineering problems, the good solution comes when we add more information into our model.

For instance, if you add an assumption that the signal is a pure single harmonic signal, you can get much better resolution.

So, once you add assumptions into the model, you are able to have better resolution and ability to detect and estimate the parameters of the data.

• Any feedback from those who -1?
– Royi
Feb 24 at 8:58