Complete Communications Engineering

LMS implementation of first order adaptive differential microphone array beamforming. In the implementation of first order adaptive differential microphone, the difference between two spatial beams are taking using a scaling parameter \beta. The choice of the optimal \beta is not exact in the presence of  large amplitude noise or low signal to noise ratios. We present a LMS approach to estimating the optimal \beta.
Consider  a two  microphone array as shown in Figure 1:

Two Microphone Array

Figure 1: Two microphone array

The constraints for beamforming are:

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 as

y(w) = z_b(w) - \beta z_f(w)

where \beta is the design parameter of concern. Consider a cost  function J(w) = |y(w)|^2. It can be easily shown that a gradient descent optimization of beta \beta will satisfy \beta [k+1] = \beta[k] - \mu \frac{\partial{J(w)}}{\partial{\beta}}|_{\beta = \beta[k]}where \frac{\partial{J(w)}}{\partial{\beta}} = -2 \mathbf{R}e \{z_f(w)^* y(w) \}
A time average can be deployed to optimize the estimate of \beta. The descent algorithm then becomes:\beta_t [k+1] = \beta_t[k] +2 \mu \mathbf{R}e \{z_{f_t}(w)^* y_t(w) \}

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