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This might be a terminology question but I am not sure.

Basically, what is the difference between conventional beamformers, and adaptive beamformers? I thought that all beamformers were inherently adaptive to some criteria, like minimization of distortion or variance, or some other spatial filtering criteria etc. So what is the difference between them?

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A beamformer is basically a spatial filter. It can be passive, just like a temporal filter.

Instead of samples separated by time, they are separated by space. A passive temporal filter can be a bandpass that is "aimed" or "steered" at a particular frequency. For passive spatial filters (i.e. beamformers), the filter can be steered towards a particular angle of arrival, instead of temporal frequency.

Adaptive filters/beamformers can be incredible because they can "steer" a null towards a frequency or angle where there is an interfering signal.

Just like a temporal adaptive filter, a spatial adaptive filter (i.e. beamformer) is constantly adjusting the filter weights/coefficients to optimize some criteria that usually involves "nulling out" or "rejecting" an interferer.

Here's a diagram of a CBF that is called a "K-Omega" Beamformer. K-Omega Beamformer Diagram

Here's a diagram to reinforce the idea that a passive beamformer is possible.

Beamformer Concept

I realize that these are a bit random, but hopefully you can follow the logic of the above. I'll see if I can find a better diagram that more clearly shows what's happening. To be clear, the windows referred to above are Hamming/Hanning type windows, and this process is basically returning a 2-d matrix where temporal frequency is on the x-axis and a special spatial variable is on the y-axis. This special variables makes the math easier, and it's a one step process to convert the special spatial variable and temporal frequency to an angle of arrival.

The green lines below are lines of constant angles. Beamforming Angle Lines

Keep in mind that all of this is for CBF (conventional beamformers) and the above covers all spatial and temporal frequency (within the limits of Nyquist.
Some common ABF techniques are:

MPDR - Minimum Power Distortionless Rejection BF (Described by Van Trees)
DMR - Dominant Mode Rejection BF (Abraham and Owsley)
R-DMR - Robust Dominant Mode Rejection BF (Cox and Pitre)
EBAE - DMR BF with Eigenvector Beam Association and Excision (Kogon)

Instead of using a spatial FFT as the spatial filtering operation, these techniques typically involve constructing an updated spatial autocorrelation matrix of the incoming signal, and then using that matrix or eigenvectors of that matrix to adaptively influence the spatial filter.

Update for @Mohammad: Below is the list that of beamforming texts I received from my professor:

  • Van Veen, B.D.; Buckley, K.M.; , "Beamforming: a versatile approach to spatial filtering," ASSP Magazine, IEEE , vol.5, no.2, pp.4-24, April 1988

  • Efficient digital beamforming in the frequency domain Brian Maranda, J. Acoust. Soc. Am. 86, 1813 (1989)

  • Array Signal Processing by Don Johnson and Dan Dudgeon, Prentice Hall, 1993

  • Optimum Array Processing by Harry L. Van Trees, Wiley, 2002.

Also, after searching around, this looks very interesting. It seems to be more interested in the practical/implementation side than the theory. I don't have it but I'll probably buy a copy:

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Thanks - so a CBF is basically a spatial filter, where the angle that is to be coherently combined, or nulled, is given by the user, whereas in an ABF, the adaptive algorithm will adaptively find an angle that maximizes power, or something like that, correct? –  Mohammad Feb 12 '13 at 20:16
    
Yes, some diagrams would be nice, thanks. –  Mohammad Feb 12 '13 at 20:16
    
Thanks Dave - 1) Where did you find or make those diagrams from? 2) You listed some acronyms for the ABFs, can you please provide what they stand for? 3) Lastly, I have looked on amazon but could not find any particularly nice beamforming books, are there any you might recommend? If anything, beamforming seems to just be scattered across general adaptive filtering books. Thanks!! –  Mohammad Feb 12 '13 at 21:51
    
I took a class on beamforming from a really good teacher. Those diagrams and plots are from papers I wrote for that class. We used Van Trees' Optimum Array Processing though I can't recommend that as I barely ever used it. The professor's notes were/are incredible, but I'm not permitted to redistribute them. –  Dave C Feb 12 '13 at 23:02
    
Thats a shame about the notes. :-/ Can you expand on the acronyms you mentioned though please? Thanks. –  Mohammad Feb 12 '13 at 23:24
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