Double-talk detection based on orthogonality principle

The use of energy comparison to detect double-talk (cross-talk) can detect both the presence of a near-end speaker and a non-convergent echo. Energy detection schemes however,cannot discriminate between the two. Under the assumption that the the desired near end speech is uncorrelated with the far-end speech, the inner product of the error signal and the near end signal can be used to detect double talk.
Consider the systems depicted in Figure 1 below:

Figure 1: Single line AEC architecture

$x[n]$ is the far end speech whilst $s[n]$ is the near-end speech. Denote a frame of far-end speech with accompanying echo path filter as:

$X[n]= [x[n], \cdots, x[n-L+1]] ^T$

$W= [w_0, \cdots, w_{L-1}] ^T$

Then the received near-end microphone signal is:

$y[n]= X[n] ^TW + \alpha s(n) + v(n), \alpha \in [0,1], \alpha \in \mathbb{Z}$

where $v[n]$ is zero mean i.i.d. ambient noise. The error signal is given as:

$e[n] = \alpha s(n) +X[n] ^T(W -\hat{W})+ v(n)$

where $\hat{}$ denotes the estimated variable. We are interested in the expectation of the cross product between the error signal and the microphone signal, thus:

$\mathbb{E} [e[n]^T y[n]] = (W -\hat{W})^T R_{xx} W$

It can be seen that with a convergent filter, $y[n]$ is orthogonal to the error signal. Under the assumption that he near end and far end speeches are orthogonal, The the time-frequency domain representation becomes $(W -\hat{W}) \neq 0$ when there is near end speech, hence $\mathbb{E} [e[n]^T y[n]] \neq 0$ . The orthogonality based detection scheme is then given as:

$\beta[n] = \begin{cases} 1, & \mathbb{E} [e[n]^T y[n]] \ge \gamma \\ 0, &otherwise \\ \end{cases}$

where $\gamma > 0$ is a threshold parameter. A window is most times applied to remove spurious noise in the detection scheme.

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Double-talk detection based on orthogonality principle

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