Finding the Complex Spectrum Circle Centroid

It is obvious that the target signal spectrum $S(\omega )$ is restored by finding the centroid of the circle on which three or more microphone inputs $M_i(\omega )$ lie. In the case of $K=3$, the circle centroid is uniquely determined from three distinct points on the circle. In the case of $K > 3$ microphone inputs, the circle centroid can be determined as a point of nearly equal distance from observed microphone inputs. We estimate the centroid as a point $\tilde{S}(\omega)$ by minimizing the variance of $K$ squared distances from $M_i(\omega )$, i.e.,

$$\tilde{S}(\omega )=\mathop{\rm argmin}_{Z(\omega)} {\rm\mathop{\mathrm{Var}}\nolimits } \Big[\Vert Z(\omega)-M_i(\omega) \Vert^2\Big]$$ (3)

$K=4,5,\cdots$ where $Z(\omega)$ is a point on the complex spectrum plane for arbitrary. We can include cases of $K=1,2,$ or $3$ where the minimum variance is 0.

To solve this equation, let $X$ and $jY$ be real and imaginary parts of $Z(\omega)$, i.e., $Z=X+jY$, and let $x_i$ and $jy_i$ be those of $M_i(\omega) = x_i + j y_i$. Then, we have

$${ \mathop{\mathrm{Var}}\nolimits \Big[ \Vert Z - M_i \Vert ^2 \Big] }$$

$$= \frac{1}{K} \sum_{i=1}^K \Vert Z - M_i \Vert ^4 - \Big( \frac{1}{K} \sum_{i=1}^K \Vert Z - M_i \Vert ^2 \Big) ^2$$

$$= \frac{1}{K} \sum_{i=1}^K \left( \left(X-x_i\right)^2 +\left(Y-y_i \right)^2 \right)^2$$

$$\quad\qquad - \Big( \frac{1}{K} \sum_{i=1}^K \left( \left(X-x_i\right)^2 +\left(Y-y_i \right)^2 \right) \Big) ^2$$ (4)

from which its partial differentials in respect to $X$ and $Y$ is derived as:

$${ \begin{pmatrix} \frac{\displaystyle\partial } {\displaystyle\pa... ...\end{pmatrix}\mathop{\mathrm{Var}}\nolimits \Big[ \Vert Z - M_i \Vert^2 \Big] }$$

$$ =\!\! 8 \!\!\begin{pmatrix} \mathop{\mathrm{Var}}\nolimits [\!x_i\!] \!... ...\! \mathop{\mathrm{Cov}}\nolimits [\! y_i \!, \! x_i^2 \!] \! \end{pmatrix}$$ (5)

denoting the covariance of $a$ and $b$ by $\mathop{\mathrm{Cov}}\nolimits [a,b]$. Letting the lefthand side of the above equation be 0, we obtain a linear equation to obtain the centroid that minimizes the variance of squared distances in Eq. (3):

$$\begin{pmatrix}\! X \! [1ex] \! Y \! \end{pmatrix} \!=\! \frac{... ... [y_i,y_i^2] \!\!+\!\! \mathop{\mathrm{Cov}}\nolimits [y_i,x_i^2] \end{pmatrix}$$ (6)

whose solutions $X$ and $Y$ give the estimated complex spectrum centroid for each frequency as $\tilde{S}(\omega) = X(\omega)+jY(\omega)$.

Eq. (6) has a solution if the covariance matrix:

$$C_{xy}\equiv \frac{1}{2} \begin{pmatrix}\mathop{\mathrm{Var}}\nol... ...m{Cov}}\nolimits [x_i,y_i] & \mathop{\mathrm{Var}}\nolimits [y_i] \end{pmatrix}$$ (7)

between $x_i$ and $y_j$ is regular. Since its determinant is given by

$$\vert C_{xy} \vert =\mathop{\mathrm{Var}}\nolimits [x_i]\mathop{\... ...sqrt{\mathop{\mathrm{Var}}\nolimits [x_i]\mathop{\mathrm{Var}}\nolimits [y_i]}}$$ (8)

where $r_{xy}$ is the correlation coefficient between $x_i$ and $y_j$, the solution of Eq. (6) is guaranteed to exist unless $r_{xy}=1$, i.e., all spectrum points $M_i(\omega), i=3, 4, \dots, K$ lie on a line in the complex plane.

Even though $r_{xy}$ is always guaranteed to be no greater than 1, in numerically bad conditions such as $r_{xy}>0.99$, we use the center of gravity of $K$ points $M_i(\omega )$, i.e., the delay-and-sum solution, instead of the circle centroid here.