Finding the Delay

One method of estimating the signal delay between microphones is to measure the cross power spectrum between successive microphones. The measurement is done in the frequency domain because the delays will be less than a sample apart, requiring us to look at the phase information. A simplification of [Ku] is presented which assumes no noise or reverberation. Let the signal, S, pass S1 before S2 with G1 and G2 being the path loss and $\Delta$ being the time delay from S1 to S2,

$$S_1(\omega) = G1\cdot S(\omega)$$

$$S_2(\omega) = G2\cdot S1(\omega)\cdot e^{-j\omega\Delta}$$

Take the cross power spectrum to be

$$X_{S_1,S_2} & = & S_1(\omega){\cdot}S_2^*(\omega) & = & G1\... ...\Vert S(\omega)\Vert^2\cdot e^{j\omega\cdot\Delta}\nonumber\$$

Now it's clear that the slope of the angle is the time delay information needed. So to estimate the delay, we need to take the discrete fourier transform, find the cross power spectrum angle, and estimate the slope. And hopefully the noise and reverberation will not significantly disturb our estimation.