Complete Communications Engineering


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:


Two Microphone Array

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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