Editing Adaptive filter (section)

===Signal components in the reference input===

In some situations, the reference input <math>x_k</math> includes components of the desired signal. This means g' ≠ 0.

Perfect cancelation of the undesired interference is not possible in the case, but improvement of the signal to interference ratio is possible. The output will be
:<math> \epsilon_k = d_k-y_k = g_k - \hat{g}_k+ u_k - \hat{u}_k </math>.  The desired signal will be modified (usually decreased).

The output signal to interference ratio has a simple formula referred to as ''power inversion''.
: <math>  \rho_{\mathsf{out}}(z) =  \frac{1}{ \rho_{\mathsf{ref}}(z)}    </math>.
::where 
:::<math>  \rho_{\mathsf{out}}(z) \ </math> = output signal to interference ratio.
:::<math>  \rho_{\mathsf{ref}}(z) \ </math> = reference signal to interference ratio.
:::<math>  z \ </math> = frequency in the z-domain.

This formula means that the output signal to interference ratio at a particular frequency is the reciprocal of the reference signal to interference ratio.<ref>Widrow p 313</ref>

Example: A fast food restaurant has a drive-up window. Before getting to the window, customers place their order by speaking into a microphone. The microphone also picks up noise from the engine and the environment. This microphone provides the primary signal. The signal power from the customer's voice and the noise power from the engine are equal. It is difficult for the employees in the restaurant to understand the customer. To reduce the amount of interference in the primary microphone, a second microphone is located where it is intended to pick up sounds from the engine. It also picks up the customer's voice. This microphone is the source of the reference signal.  In this case, the engine noise is 50 times more powerful than the customer's voice. Once the canceler has converged, the primary signal to interference ratio will be improved from 1:1 to 50:1.