In minimum variance distortionless response, the filter coefficients used to remove noise and interfering signals are adjusted based on the statistics of the received audio samples. Consider a far field audio signal impinging an arbitrary array of $N$ microphones such that the temporal short term frequency domain signal from an array of microphones can be represented as:

${\bf x}(t,\omega) = s(t,\omega) {\bf d}(\theta) + {\bf v}(t,\omega)$

where boldface lower case letters represent vectors. Here, $s(\omega)$ is the desired speech signal, ${\bf d}(\theta)$ is the steering vector given by:

${\bf d}(\theta) =\left[\alpha_1, \alpha_2 e^{-j\omega d_2 \sin{\theta_2}/c}, \cdots, \alpha_N e^{-j\omega d_N \sin{\theta_N}/c}\right]^T$

where $\alpha_i$‘s are the attenuation factors, $d_i$‘s are the distances from microphone one, and $\theta_i$‘s are the incident angles with respect the broadside of the line joining microphone $i$ and the reference microphone one. $c$ is the speed of propagation of sound in the the medium of interest. In a linear array topology, the distances will be integer multiples of a base distance $d_0$ whilst the incident angles will all be identical. Further, ${\bf v}(t,\omega)$ is a vector containing i.i.d noise. Now consider a filtered SIMO system where the desired signal is synthesized with and estimated filter ${\bf W}(\omega)$, with the time index dropped for brevity,such that:

$y(\omega) ={\bf W}^\dagger (\omega) {\bf x}(\omega)$

where $\dagger$ is the hermitian operator. A cost function manifold is then defined as:

$J(w) \doteq \frac{1}{2} \mathbb{E} \left[ |y(\omega)|^2\right] = \frac{1}{2}{\bf W}^\dagger (\omega) {\bf \Phi}_{{\bf x}(\omega),{\bf x}(\omega)} {\bf W} (\omega)$

A minimum variance algorithm will find the minimum over this manifold. However, a trivial solution of ${\bf W} (\omega) \doteq 0$ will satisfy such a constraint. Thus a linear constraint of of ${\bf C}^\dagger {\bf W} (\omega) = {\bf c}$ is used to preclude the trivial undesired solution leading to a Lagrangian cost function

$L(\omega,{\bf \lambda}) = \frac{1}{2}{\bf W}^\dagger (\omega) {\bf \Phi}_{{\bf x}(\omega),{\bf x}(\omega)} {\bf W} (\omega) + {\bf \lambda}^\dagger ({\bf C}^\dagger {\bf W} (\omega) - {\bf c})$

Now

$\frac{\partial L(\omega,{\bf \lambda}) }{\partial \omega} = 0 \Rightarrow {\bf W}^\dagger (\omega) {\bf \Phi}_{{\bf x}(\omega),{\bf x}(\omega)} + {\bf \lambda}^\dagger {\bf C} = 0$

$\frac{\partial L(\omega,{\bf \lambda}) }{\partial {\bf \lambda}} = 0 \Rightarrow {\bf C}^\dagger {\bf W} (\omega) - {\bf c} = 0$

Under the assumption that the power spectral density is non-singular, we have:

${\bf W}^\dagger (\omega) = -{\bf \lambda}^\dagger {\bf C} {\bf \Phi}_{{\bf x}(\omega),{\bf x}(\omega)}^{-1}$

$\Rightarrow {\bf c}^\dagger= -{\bf \lambda}^\dagger {\bf C} {\bf \Phi}_{{\bf x}(\omega),{\bf x}(\omega)}^{-1} {\bf C}$

$\Rightarrow {\bf c}= - {\bf C}^\dagger {\bf \Phi}_{{\bf x}(\omega),{\bf x}(\omega)}^{-\dagger} {\bf C}^\dagger {\bf \lambda}$

$\Rightarrow {\bf \lambda} = - \left({\bf C}^\dagger {\bf \Phi}_{{\bf x}(\omega),{\bf x}(\omega)}^{-\dagger} {\bf C}^\dagger\right)^{-1} {\bf c}$

Therefore we have:

${\bf W}(\omega) = {\bf \Phi}_{{\bf x}(\omega),{\bf x}(\omega)}^{-\dagger} {\bf C}^\dagger \left({\bf C}^\dagger {\bf \Phi}_{{\bf x}(\omega),{\bf x}(\omega)}^{-\dagger} {\bf C}^\dagger\right)^{-1} {\bf c}$

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