Adaptive filtering: Optimum filter length and delay
In order to be able to choose an optimal value for the delay $\Delta$ it's important to understand how the system works. The purpose of the delay is to decorrelate the desired signal $s(n)$ and the signal component $s(n-\Delta)$ at the input of the adaptive filter. This means that $\Delta$ must be chosen such that the autocorrelation $R_{ss}(k)$ of $s(n)$ is (close to) zero for lags greater than $\Delta$:

$$R_{ss}(k)\approx 0,\qquad |k|&gt;\Delta$$

However, we cannot choose $\Delta$ arbitrarily large because the delayed interference at the input of the filter must be correlated with the interference added to the signal, i.e., the autocorrelation $R_{rr}(k)$ of the interference must still be significant at a lag of $\Delta$, otherwise the adaptive filter cannot predict the interference. If we can assume that $r(n)$ is narrow-band compared to $s(n)$, it's always possible to find an appropriate value for $\Delta$.

With an appropriate value for $\Delta$, the adaptive filter will try to predict the interference, i.e., it will try to undo the effect of the delay in the frequency band where the interference has significant frequency components. So the output of the filter will approximate $r(n)$: $y(n)\approx r(n)$. Consequently, the error signal will approximate the desired signal: $e(n)\approx s(n)$.

After having chosen a value for $\Delta$ based on the autocorrelation of $s(n)$, the filter length must be chosen by trial and error. A long filter will give a better suppression at the cost of slower convergence.