First order endfire BF directional noise SNR improvements

First order endfire beamformer is used extensively when the desired noise source is anti-phase to the desired signal using differential beamforming. First order differential beamformer uses two microphones. The signal to noise ratio improvements using differential beamformer is different from the conventional delay and sum beamformer. We derive the expected SNR gains for directional noise on two microphones.
Consider a far field source impinging 2 microphones as shown in Figure 1:

Figure 1: 2 microphone array

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. For endfire configuration, $\theta = -90^{\circ}$ and $\beta = 90^{\circ}$ is the assumption.

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 differential beamformer, the output becomes

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

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

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

The SNR improvement, SNRI then becomes

$SNRI = \frac{oSNR}{iSNR} = \frac{ |1 - \cos{(w \frac{d}{c} (1-\sin{\theta})}|}{|1 - \cos{(w \frac{d}{c} (1-\sin{\beta})}|}$

A sample expected SNRI at a frequency of $4kHz$ is shown in Figure 2 below for different separation distances d for $16$ microphones. The desired direction is $-90^{\circ}$ . It can be seen that the SNRI improves smoothly for small distances $d$ but there are distortions at high $d$ . Also, some DOAs are amplified such as $0 degrees$ for $d = 100mm$ . Thus, the magnitude of $d$ plays a huge role in the expected SNRI.

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First order endfire beamformer directional noise SNR improvements