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A linear filter obeys the superposition principle: filtering $\alpha x+\beta z$ gives $\alpha T[x]+\beta T[z]$. Many familiar examples are also linear time-invariant systems, represented by an impulse response and convolution. Linearity alone does not imply time invariance: multiplying $x[n]$ by the index $n$ is linear but changes its behavior when the same input is shifted.

The moving average is a finite impulse response example implemented by arithmetic; a band-pass filter can instead be built from a series RLC circuit. Their frequency responses explain which oscillations pass through. Choosing a filter means choosing how much amplitude, timing and detail may change, not simply removing every visible fluctuation.

For a series RLC circuit driven by $V(t)=V_M\sin ({\omega }_{0}t)$ with $R>0$ and ${\omega }_0^2=1/(LC)$, substitution shows that $I_0=V_M\sin ({\omega }_{0}t)/R$ solves the current equation. The inductive and capacitive terms cancel at this resonance. Every homogeneous contribution decays, so every solution approaches this periodic response. Its Q factor is $L{\omega }_0/R$.