# Filter

Filter:

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(t1),Y(t2), … .

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.

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