Complete Communications Engineering

Differential beamforming using two microphones, and hence first order, has two different implementations. The first involves a delay of one microphone signal  then taking the difference between the resultant and the second signal. An alternate to this approach is the so-called adaptive differential microphone array beamforming which forms two cardioids and combines them for the required spatial beam pattern.
Consider  a two  microphone array as shown in Figure 1:

Two Microphone Array

Figure 1: Two microphone array

The DMA beamforming is an optimization problem to meet N constraints for N microphones. For two microphones, the constrainst are used to form:

z_f(w) = \frac{S(w)}{e^{-jw\frac{d}{c}\cos{\gamma}}-e^{-jw\frac{d}{c}\cos{\theta}} } \begin{bmatrix} 1 & e^{-jw\frac{d}{c}\cos{\theta}} \end{bmatrix} \begin{bmatrix} e^{-jw\frac{d}{c}\cos{\gamma}} \\-1 \end{bmatrix}


z_b(w) = \frac{S(w) }{e^{-jw\frac{d}{c}\cos{\gamma}}-e^{-jw\frac{d}{c}\cos{\theta}} } \begin{bmatrix} 1 & e^{-jw\frac{d}{c}\cos{\theta}} \end{bmatrix} \begin{bmatrix} -1 \\ e^{-jw\frac{d}{c}\cos{\gamma}} \end{bmatrix}

where \theta is the desired beam direction and \gamma is a desired null direction. Given \theta and \gamma, the choice of \beta will determine whether the spatial beam will be a dipole, cardiod, hypercardiod or supercardiod. The final solution is given asy(w) = z_b(w) - \beta z_f(w)
where \beta is a design parameter to shape the beam in the direction of the source signal. The optimal value for \beta can be derived using a second cost function J(w) = |y(w)|^2. It can be easily shown that the optimal value of \beta will satisfy \beta_{opt} = \frac{2 \mathbb{R}e\{z_b(w) z_f(w)^*\}}{|z_f(w)|^2}
Figure 2 below illustrates these 4 spatial beams.

Different Differential spatial patterns

Figure 2: Different spatial patterns

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