Hi

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?

Thanks & Regards

"David" gave you a good answer on StackExchange.  I couldn't find where you get the assertion re: improved bearing resolution.  There is bearing resolution and there is bearing resolution: One is dependent purely on beamwidth.  The other is based on split-beam processing and phase differences and, I believe, gives better bearing resolution when there's a high SNR.  But, as David said, an adaptive beamformer's beam pattern varies. You didn't say anything about interference rejection in your question....

Thanks for helping. Actually i was seeing this with respect to improving "Reverberation Index" in active sonar equation which mainly depends upon Bandwidth and 3dB beam width. My intention is to find a way to improve this 3dB beam width by some processing algorithm.

One such algorithm i found in literature ("Sonar for practising Engineers by  A.D. Waite) is "FM Phase binning" method using split beam former but i could not found any further details on this algorithm on net with this name. 

Thanks

Thanks for helping. Actually i was seeing this algorithm with perspective of improving "reverberation index" in active sonar equation which depends upon Bandwidth and 3dB beam width.

As you mentioned split-beam processing, one such method i found in a book is "FM Phase Binning" but i could not found any further details on net with this name.

I'm a little late to this, but I'll add a few notes to hopefully help.    

(1) I typically define resolution as the ability to distinguish between 2 closely spaced signals (this comment is just a setup for the next one...)  

(2) In light of (1) as well as the linked comments from StackExchange (from fred3's response), it's important to distinguish between MVDR and MPDR beamforming (from Van Trees' notation).  In MVDR, the desired is not contained in your interference covariance matrix.  In MPDR, the desired signal is contained in the interference covariance estimate, and anything that is not a perfect match to your replica vector will be attenuated to some degree by the ABF.  This is where the increased resolution really comes into play, and I recommend "Resolving Power and Sensitivity to Mismatch of Optimum Array Processors" (Cox, 1973, JASA) for more info.

(3)  Keep in mind the differences between beampatterns and beam responses (aka beamformer outputs).  The beampattern is the response of a given set of weights to a unit plane wave from a set of possible arrival angles.  The beamformer output is the response of the data to the set of weights for each look direction.  A MPDR beampattern can have a wider mainlobe width than a CBF beampattern, but the MPDR beamformer output will have narrower lobes.  I think your original idea of measuring the 3 dB down point of the beampatterns will not align with your expectations.  

The response of your MPDR weights to reverberation will depend on your array configuration and the type of reverberation.  Omni-directional volume reverberation is very similar to 3D isotropic noise which has a very well defined model found in many array processing texts.  If you are concerned with bottom reverberation, an ABF on a vertical aperture can help steer spatial nulls.  

All that said, if you are looking for metrics to gauge the improvement provided by your weights, there is a fairly standard set provided in Chapter 2 of "Optimum Array Processing" by Van Trees.  Directivity, Array Gain, and SINR are good starting places.  It's difficult to say more because so much of ABF performance depends on the data, the models, and how you estimate your interference.  

OK, I'm not a sonar guy -- but wouldn't the answer to your question be found in a good book on sonar that's up-to-date enough to include MVDR beam forming?  Presumably, for any given sets of weights and delays for a beam former, you can compute the beam pattern fairly easily (or at least far more easily than you can compute the weights and delays for all but the simplest beam forming algorithms).  Can't you just grind through the math on your own to figure this out?

Thanks for reply. Yes, i should do it and am trying to do it but how will i verify my results whether they are right or not in absense of any standard formula?

If you know the properties of your transponders and of the acoustic media (presumably water) that you're designing for, shouldn't you be able to work out the behavior from first principles?

Case 1: you have a closed form equation, calculate the power at maximum and 1/2 power on each side of the beam in the plane you are interested, compute the half power beamwidth.

Case 2: there is no closed form equation, you can measure/simulate it numerically similar to case 1 to compute half power beamwidth.

Case 2 would be the verification of case 1, if you have any equation.

Best regards,

Shahram 

https://ortenga.com/

Thanks. You are right, i think i have to simulate the beamp attern for simplest case (i.e. zero interference) and try to estimate beamwidth

Regards

Thanks. You are right, i think i have to simulate MVDR beampattern for simplest case (i.e. zero interference) and try to estimate beamwidth.

Best Regards

If there is no interference to be rejected then I have a hard time imagining that a classical beamformer isn't what you'd end up with.  It's already been mentioned that an adaptive beamformer with rejection requirements *can't* be better than one that doesn't have those added constraints - if the only measure is beamwidth.

Are you interested in volume reverberation or boundary reverberation or both?  Are you interested in improving bearing resolution in the presence of reverberation or something else?

Simple split beam processing will use something like 2 halves of the array.  In one case add them (so it's just a standard beamformer).  In the other case, subtract them (which produces a broadside null).  Then compare the phase of the two outputs.  Make sure the wavelength relationships make sense.  Relate the electrical "angle" to the mechanical angle i.e. "bearing".  It's a high SNR approach of course.

My concern is boundary (surface/bottom) reverberations mainly and purpose is to improve beam width some how for reduction in effective reverberation.