Conventional derivations for the signal to noise ratio improvements using delay and sum beamformer is that you get 3dB gain for every doubling of the number of microphones being deployed. This holds iff 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:

N element ULA Microphone Array

Figure 1: N ULA microphones

Suppose the signal at each microphone i \in \{1, \cdots, N\} is given as

x_i(w) = s(w) e^{\left(-jw \frac{(i-1) d}{c} \sin{\theta} \right)} + v(w) e^{\left(-jw \frac{(i-1) d}{c} \sin{\beta} \right)}

where s(w) is the desired speech signal, \theta is the direction of arrival (DOA) of the speech signal with respect to the normal to the axis joining all the microphones, v(w) is the directional noise and \beta is the DOA of the directional noise.

The input SNR per frequency bin w, denoted iSNR(w) is given as

iSNR = \frac{\mathbb{E}\left[|s(w)|^2 \right]}{\mathbb{E}\left[\left |v(w)\right|^2 \right]}

where \mathbb{E}[.] is the expectation operator.

After the delay and sum beamformer, the output becomes

x(w) = s(w) + v(w) \frac{1}{N} \frac{\sin{\left(w N \frac{d}{2c}( \sin{\theta} - \sin{\beta})\right)}}{\sin{\left(w \frac{d}{2c}( \sin{\theta} - \sin{\beta})\right)}} e^{\left(jw \frac{N-1}{2} \frac{d}{c}( \sin{\theta} - \sin{\beta}) \right)}

The output SNR per frequency bin w, denoted oSNR(w) is given as

oSNR = \frac{\mathbb{E}\left[|s(w)|^2 \right]}{\mathbb{E}\left[\left | v(w) \frac{1}{N} \frac{\sin{\left(w N \frac{d}{2c}( \sin{\theta} - \sin{\beta})\right)}}{\sin{\left(w \frac{d}{2c}( \sin{\theta} - \sin{\beta})\right)}} \right|^2 \right]}

The SNR improvement, SNRI then becomes

SNRI = \frac{oSNR}{iSNR} = \frac{N^2}{\left |\frac{\sin{\left(w N \frac{d}{2c}( \sin{\theta} - \sin{\beta})\right)}}{\sin{\left(w \frac{d}{2c}( \sin{\theta} - \sin{\beta})\right)}} \right|^2}


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