Let and be two stationary time series related by:

where

and

is the filtered version of and is the filter. The effect of a linear filter is to change the importance of various cyclical components of the series and/or induce a shift with regard to the position in time.

Note that

: Trend Extraction

: Detrending

A *Linear time-invariant filter *demonstrates the following property:

We can show that the** spectral density function** *f* of the filtered series is equal to

where is the** Transfer Function**:

is the **Frequency Response Function**:

and is the** Gain:**

Τhe *Gain* indicates the factor by which the amplitude of a cyclical component changes as a result of applying the linear filter. On the other side the **phase ***Ph(λ) *measures the shift in time

where

Note finally that if **k multiple filters **are applied consecutively

their combined effect is simply given by the product of the respective frequency response functions. Hence:

[…] Filters may be applied to a time series for a variety of reasons. Suppose that a time series consists of a long-term movement, a trend, upon which is superimposed an irregular component. A moving average filter will smooth the series revealing the trend more clearly. […]