Discussion and Future Works

The proposed method has a high potential of further modifications for even higher performance.

(1) Microphone layout

Suppose that microphones are equally spaced in a line. If noise arrived with an incident angle yielding the time difference $\tau$ between adjacent microphones, the input complex spectrum points $M_i(\omega )$ gather and do not form a circle in the complex plane at a frequency $$\omega=2\nu\pi/\tau, \nu=1,2,3,\cdots$$ in Eq. (2). The microphone layout can be improved to avoid such ill conditions. In this paper, microphones were equally spaced on a circle in the simulated speech recognition.

(2) Interpolation between circle and gravity centroids

The input spectrum points may deviate from the ideal points due to inaccurate layout and non-identical microphone characteristics, fast change in the noise signal compared to time differences between microphones, and other sources of errors. These deviations may distort the circle and cause inaccurate centroids.

Such ill-conditioned situation is diagnosed through multiple clues such as correlation coefficient in Eq. (8) or spanning angles between input points from the estimated centroid:

$$\cos\theta = \frac{ (Z-M_i,Z-M_j) }{ \vert Z-M_i\vert\cdot\vert Z-M_j\vert }$$

and is relieved by interpolating the circle centroid and center of gravity (arithmetic mean) $\frac{1}{K}\sum M_i(\omega)$.

(3) Calibration and normalization

In this paper, microphones are primarily assumed to have identical characteristics. If the gain characteristics along frequency is not equal for all microphones, they can be equalized by normalizing the gain of each microphone at each frequency point. Directivity of the microphone is assumed to be identical but need not be omni-directional. Otherwise, the micorphone input complex spectrum formes another figure such as an ellipse instead of a circle.