Acoustic Echo Canceler
Reduces the amplitude of Acoustic Echo responses.
We essentially form an Adaptive Filter with a bunch of weighted [see page 5, coefficients] that cancels out the acoustic echo response before transmission. The problem of acoustic echo cancellation is finding the weights at time \( n \) which cancels the current acoustic echo response.
Note: In this situation we have to use an adaptive filter because the acoustic echo response varies over time and so a filter with constant coefficients can't accurately remove acoustic echos.
We define:
\begin{align*} e_{AEC}[n] &=& &\text{echo}& &-& &\text{estimated echo}& \\
&=& &\phi& &-& &\hat{\phi}& \\\\
&=& &W\_{AEC}[n] \* x[n]& &-& &h[n] \* x[n]&\end{align*}
Where:
| Math | Name | Meaning |
|---|---|---|
| \( e_{AEC}[n] \) | AEC Error Signal | A measure of the error of our Acoustic Echo Canceller. |
| \( x[n] \) | Input signal | The input signal at time \( n \) |
| \( h[n] \) | Room Impulse Response | An filter that reproduces the affect acoustic-echo [1]. |
| \( W_{AEC}[n] \) | AEC weights | The weights of our acoustic echo canceller at time \( n \) |
| \( \phi \), \( \hat{\phi} \) | Output signal | Result of passing \( x[n] \) through \( h[n] \) and \( W_{AEC}[n] \) |
[1]: Bouncing a signal through a room.
Note: By this definition the weights we're trying to find are equivalent to the room impulse response... I.E. we're trying to find the room impulse response.
You can [see page 8, think] of this problem as an \(n+1\) dimension Gradient Descent problem, with \(n\) being the order/complexity/number-of-weights of the filter we're trying to find and 1 more dimension containing the error at those weight which we're trying to minimise.
See [see page 4, applications] for adaptive filtering.
Finding the Weights
The goal of an AEC is to create a filter that minimises a cost function defined in terms of \( e_{AEC}[n] \). The cost function specifically [see page 8, used] here is [see page 9, mean squared error]: \[ f(e[n]) = E\{|e[n]|^2\} = E{[\phi[n] - \hat{\phi}[n]]^2} \] This equation can be further expanded into: \[ \phi^2[n] - W^T_{AEC}[n] x[n] \phi[n] - \phi[n] x^T[n] W_{AEC}[n] + W^T_{AEC}[n]x[n] x^T[n] W_{AEC}[n] \]
We now differentiate this equation, seting it to 0, to find a [see page 11, final equation] that we set to 0 and re-arrange to get:
\begin{align*}
E\{x[n] \phi[n]\} &= E\{x[n] x^T[n]\} w_{AEC,opt}[n] \
r_{x\phi}[n] &= R_{xx}[n] \
\end{align*}
The optimum weights \( W_{AEC,opt} \) are thus defined as: \[ W_{AEC, opt} = R^{-1}_{xx} r_{x\phi} \].
This final equation for \( {W}_{AEC, opt} \) is known as the weiner-filter.
Problems
There are are a lot of computational [see page 11, issues] with this implementation due to the size of the equations and dimensionality involved. Specifically inverting \( R_{xx} \) can be computationally intensive. Furthermore the input signal \( x[n] \) and the echo signal \( \phi[n] \) are non-stationary and time-varying meaning the calculation has to be repeated every time they change. Lastly both \( R_{xx} \) and \( r_{x\phi} \) must be estimated.