The beamformer as a log-likelihood ratio detector

Some basic relationships are presented among the log-likelihood ratio (LLR), the Bryn statistic, and the simple additive beamformer output when detecting a stationary Gaussian plane wave in stationary Gaussian noise using an array of sensors. The noise at different sensors may be correlated. A long processing time is assumed. If two of the above detector outputs, when plotted versus steering, are merely scaled ands vertically offset versions of one another, the detectors will be said to be similar. With minor qualifications, necessary and sufficient conditions for the beamformer to be similar to the log-likelihood ratio detector at a given frequency are that the array gain (signal-to-noise ratio gain) versus steering be constant and be the same for both. It is also determined that the Bryn statistic is similar to the LLR whenever the beamformer is similar, but that it is not similar in some other instances such as near endfire or an interference.