A filter is a system designed to preserve some frequency components while attenuating others.

A low-pass filter preserves low frequencies and attenuates high frequencies. A high-pass filter does the opposite. A band-pass filter preserves a range between two cutoff frequencies, while a band-stop filter suppresses such a range.

The intended behavior is described by the magnitude of the frequency response. An ideal low-pass filter, for example, would have

$$|H(\omega)|=1$$

inside its passband and zero outside it.

Ideal brick-wall transitions are mathematical abstractions. Real causal filters change gradually between passband and stopband and also introduce phase shifts or delays.

Filtering can be implemented by continuous-time physical systems or by discrete-time algorithms. The same frequency-response language describes both, while their realizable structures differ.