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. We derive the expected SNR gains for directional gains 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 directional noise and is the DOA of the directional noise.
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
The SNR improvement, SNRI then becomes
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