Conventional derivations for the signal to noise ratio improvements using delay and sum beamformer is that you get gain for every doubling of the number of microphones being deployed. This holds the noise is not directional or in other words uncorrelated. We derive the expected SNR gains for uncorrelated noise on ULA microphones.
Consider a far field source impinging N ULA microphones as shown in Figure 1:
Figure 1: N ULA microphones
Suppose the signal at each microphone is given as
where is the desired speech signal, is the direction of arrival (DOA) of the speech signal with respect to the normal to the axis joining all the microphones, is the uncorrelated noise such that
The input SNR per frequency bin , denoted is given as
where is the expectation operator. After the delay and sum beamformer, the output becomes
The output SNR per frequency bin , denoted is given as
Since by assumption
This leads to an oSNR of
The SNR improvement, SNRI then becomes
Thus, if is increased by a factor of , the SNRI increases by a factor of .
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