Complex Spectrum Representation of Microphone Inputs

Primarily, we assume that acoustic characteristics (gains, directivities, etc.) of microphones are identical (or can be equalized by adjusting gains and delays at each frequency). If the target signal $s(t)$ propagates and arrives at $K$ microphones simultaneously at time $t$ while the noise signal $n(t)$ arrives with different time delays $\tau_1, \cdots, \tau_K$ as shown in Figure 1, the observed signal $m_i(t)$ at the $i$-th microphone is represented by:

$$m_i(t)=s(t)+n(t-\tau_i), \quad i=1,2,\cdots,M$$ (1)

where $\tau_i$ denotes the time delay at the $i$-th microphone in respect to the noise signal. Even if microphones are not layed out as in Figure 1 and their characteristics are not identical, microphone signals can be calibrated in terms of gain and time delay so as to synchronize to each other in respect to the target signal.

The short-time Fourier transform of the $i$-th microphone input signal is given by

$$M_i(\omega)=S(\omega)+N(\omega)e^{-j\omega \tau_i}$$ (2)

according to the basic properties of Fourier transform, where $\omega$ denotes angular frequency and $M_i(\omega), S(\omega)$ and $N(\omega)$ denote Fourier transforms of $m_i(t), s(t)$ and $n(t)$, respectively, if the frame length is relatively longer than time differences of noise observations at different microphones. From microphone signals, we easily obtain framewise complex spectrum (typically multiplied by a short-time window).

Geometrically, Eq. (2) implies that $M_i(\omega )$ lies on a circle of radius $\Vert N(\omega)\Vert$ with a centroid at $S(\omega )$ on the complex spectrum plane. The complex spectrum of target signal $S(\omega )$ is restored by finding the centroid of the circle on which $K$ complex points $M_i(\omega )$ lie. We call this method of estimating the target signal spectrum ``Complex Spectrum Circle Centroid (CSCC) method.''

In contrast, the Delay-and-Sum (DS) method uses the center of gravity (arithmetic mean) of microphone inputs: $\bar{M} = \frac{1}{K} \sum_{i=1}^K M_i(\omega)$.