A filter is an algorithm for processing a time series or random process. There are two major classes of problems solved by filters:
1. To estimate the current value of a time series (X(t), t = 1,2, …) , which is not directly observable, from observed values of another time series (Y(t), t=1,2,…) , related to the time series X(t).
2. To predict the next value Y(t+1) of the observed time series Y from the current value Y(t) and previous values Y(t–1),Y(t–2), … .
Consider a simple example – a time series x(i) observed in the presence of additive noise
| y(i) = x(i) + n(i); i = 1,2, … |
where y(i) are observed values, x(i) are non-observable values, and ni are values of additive random noise. the goal is to estimate x(i) from y(i) . A filter of order N can be specified, for example, by function FN() of N+1 arguments:
| ^ xi | = FN(y(i), y(i–1), …, y(i–N) ) |
Filters are also used for processing random process es. In this case filters are specified by integral operators or differential equations.