This is a question our professor asked us to think about. Suppose that we have a communication system with no interference with no channel imperfection but Ergodic AWGN noise. We use two antennas with one slightly further away from the source. How does adding the output of two antennas help the SNR? I tried analysing it in frequency domain but to be honest I'm lost and don't know how to approach it.
As mentioned by Barry, this has got to do with beamforming.
If you consider what occurs when a signal hits the two antennas, as shown in the following figure:
The signal received by the antennas will have a time delay between the antennas. This time delay is related to the angle theta that the signal is arriving from.
When adding the signals together at the output, due to the time delay (if there is one) between elements, when the signals are added together there is destructive interference:
This serves as an attenuation of the signals coming from that direction. If the signal arrives from zero degrees there is no time delay and the signals add constructively. Alternatively it is possible to aim the array of signals by introducing a time delay to cancel the time delay introduced due to the geometry of the sensors. For narrowband signals this is done by introducing phase delays.
So by having multiple sensors and adding them together you are creating a spatial filter, getting better quality signals from a specific direction even if there is interference signals of the same frequencies (so long as they come from a different direction).
This is what a typical beamformer response pattern looks like:
The reduction in noise will depend on how far apart the antennas are in terms of wavelength of the received signal. If they are only a small fraction of a wavelength apart, then there will be no improvement since both antennas will receive the same signal and noise. However, as the distance increases, the antennas will still receive the same signal but not the same noise because the noise at the two antennas will no longer be 100% correlated. Thus the signals will add linearly but the noises will only add in an RMS matter. This is the principle behind beamforming with multiple antennas. The beam pattern of two antennas will be narrower than one antenna, thus rejecting noise from directions other than that of the signal.