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: 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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