# Beam resolution improvement of linear array using MVDR beamformer

One of the major advantages of using MVDR beam former is to improve bearing resolution as compared to conventional delay & sum beam forming. The bearing resolution is defined by 3 dB beam width of linear array beam pattern. There are also formulas for calculating 3 dB beam width of arrays.

Is there any way to calculate the 3 dB beam width of linear array using ideal MVDR beam former (or beam pattern calculated by ideal MVDR) so that the bearing resolution improvement with respect to conventional beam former can be quantified for same length of array?

• I'm generally of the opinion that such techniques do not improve the resolution (the ability to distinguish between two closely-spaced sources). The beam patterns tend to LOOK better, but if you define resolution mathematically and crunch the numbers there is no difference in performance. That is, for me, the 3dB beam width does not say what the resolution is. YMMV. – Peter K. Aug 31 '16 at 16:36

The MVDR beamformer is an adaptive technique - the response is dependent on the the interfering signals i.e. their locations, their SNRs and their correlations. This makes it difficult to say what the resolution is in traditional 3dB width type of terms. Essentially the MVDR will try to place nulls in the beampattern where the interferers are - so a 3dB width doesn't really tell you anything about the resolution.

You can also get strange effects, e.g. you'll get 0 dB gain in the desired look direction, and you'll have a sidelobe level of +10 dB somewhere, because it's trying to place a null in the beampattern in a particular location.

As a purely hypothetical example - consider you're forming a beam at 90$^\circ$ look direction and you have an interferer at 135$^\circ$. You apply the MVDR and you'll see your beampattern has a null at 135$^\circ$ and the mainlobe around 90$^\circ$ will have a certain width. Now if you add another interferer at 45$^\circ$ and apply the MVDR - you'll have nulls in both the interfering bearing locations, but your mainlobe width around 90$^\circ$ has now increased. So that mainlobe width in the single interferer case doesn't really tell you much about what's going to happen when you add another interferer. Essentially you are using up another degree of freedom to null out the additional interferer.

What is probably more interesting to look at is - how close can I place an interferer before the MVDR can no longer null it out.

A good discussion on MVDR is in "Array Signal Processing" by Johnson and Dudgeon, Prentice Hall.

You can also look at "Optimum Array Processing" by Van Trees, Wiley Interscience - this may be overkill for what you want. You also have to be careful - he a uses slightly different definition of MVDR.

• Yes, a much more sensible approach. +1 – Peter K. Aug 31 '16 at 20:53
• Thanks David for your help. If not the MVDR as you righly explained, is there any processing algorithm which can improve beam width of fixed size array in comparison with conventional sum & delay beamformer? – naumankalia Sep 1 '16 at 14:30
• There are many "High- Resolution" approaches that can separate 2 targets that are closer than the traditional 3 dB beamwidth (for conventional processing) but you run into more problems on how to generalize the performance as more targets are added. The conventional approach is very robust and predictable. Specifying the performance of adaptive systems (MVDR, MUSIC, ESPRIT, or Compressive Sensing) is problematic. – David Sep 1 '16 at 17:57
• What is your opinion about split beam processing/phase binning type of techniques? Can they practically improve the beam width? Thanks – naumankalia Sep 2 '16 at 0:49
• @naumankalia I'm not all that familiar with the phase binning technique, so I can't provide a good answer for you. – David Sep 7 '16 at 12:49