What is a time-invariant linear filter

Question

I am a new student of time series analysis. Could someone explain to me the meaning of time-invariant linear filter in time series? I do not understand the explanation in wikipedia or the course literature.

Thanks in advance.

Answer 1

Linearity means superposition and scaling are preserved by the filter. Say we have some signal $s$ and let $f(s)$ be the filter of $s$ as an operator then linearity is just the usual linearity from linear algebra so for some scalar $x$ then $f(xs)=xf(s)$ and if we have two signals then $f(s_1 + s_2) = f(s_1) + f(s_2)$.

Time invariance means that no matter when we start filtering the signal the filter will be act in the same way on the signal. So if you apply the same shift to the sample before or after the filter you should expect the same result.

These properties are independent of each other and a filter can have all, some or none of these properties. The most important time invariant linear filters are the Fourier Transforms and the $z$-transform. What makes the important is that they can be subjected to frequency-domain analysis.