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Is there a way to minimize all signals that are not coming from the mainbeam?

I would like to keep the main beam, or at least +-3° around the main peak stable.

When I estimate the covariance from the data itself, it always also minimizes my wanted signal in the main beam. I can not estimate the covariance in the absence of the signal.

Any advice?

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Use a white noise gain constraint on the filter coefficients, which is a quadratic inequality constraint. When you have sources in the main beam that are not exactly on the unity gain constraint, the filter coefficients get large. You can control the magnitude by diagonal loading the covariance matrix. There is a full derivation in "Detection, Estimation, and Modulation Theory, Volume 4, Array Processing" by Harry VanTrees. It solves the problem but I'm not aware of a nice online algorithm. There is a paper by Cox and Zeskind that has an online algorithm but it isn't optimal no matter what Harry Cox thinks.

Another approach people use is to use a direction finding algorithm prior (before) to set the unity gain constraints (actually LCMV ) and then beam form. I prefer the white noise gain constraint.

Bob Zeskind suggested using an unmodified MVDR but forming about 3 times as many beams as you would in a conventional beamformer in a conversation I had with him. It's not elegant but does seem to have uses.

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Covariance, for MVDR beamformer should be calculated for instances when target source(wanted signal) is not present, otherwise MVDR will attenuate(minimize) target signal as well. there are few methods to calculate and update covariance matrix. one of it could be to assume that initial few frames of data does not contain target source and use theses frames to calculate covariance matrix, and update the covariance matrix when information of target source absence is known in received/input signal.

other method could be, if you know the noise field formulation before, then this formulation could be used for covariance matrix calculation(similar method is used for super-directive beamformers).

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