# Translations:Least-squares prediction and smoothing/17/en

Let us now derive the least-squares prediction filter, and let ${\displaystyle k=}$${\displaystyle \left(k_{0}{,\ }k_{\rm {l}}{\ ,\ }k_{N-{\rm {l}}}\right)}$ denote this prediction filter. Such a filter uses the input signal’s past values to predict that signal’s future values. The input is the signal ${\displaystyle x_{n}}$, and the desired output is the time-advanced version of the input. Let the prediction distance be given by the positive integer ${\displaystyle \alpha }$. The input to the filter is the input ${\displaystyle x_{n}}$ at present time n. The desired output ${\displaystyle z_{n}}$ is the input ${\displaystyle x_{n+\alpha }}$ at the future time ${\displaystyle n+\alpha }$. The prediction filter is designed so that the output ${\displaystyle y_{n}}$ at the present time n is an optimum estimate of the future value ${\displaystyle x_{n+\alpha }}$. If this estimate is denoted by ${\displaystyle {\hat {x}}_{n+\alpha }}$, the filter’s action can be represented by the convolution