Low-pass, high-pass, and convolution - shaping signals
Filtering is the process of selectively removing or enhancing certain frequencies in a signal. Thanks to the convolution theorem, filtering becomes simple multiplication in the frequency domain.
This is the technology behind bass boost, treble adjustment, noise reduction, and countless other audio and image processing techniques.
Passes low frequencies, blocks high frequencies. Like bass boost or a muffled sound through a wall.
Passes high frequencies, blocks low frequencies. Like removing bass rumble or background noise.
Passes only a specific range of frequencies. Used in radio tuning and isolating specific sounds.
Removes a specific frequency while passing all others. Used to eliminate 60Hz hum or other interference.
The key insight that makes filtering efficient:
Convolution in time/space domain (*) equals pointwise multiplication (×) in frequency domain.
Instead of sliding a filter kernel across every point (O(n²)), we can:
This runs in O(n log n) - a massive speedup for large signals!
Each filter has a frequency response that shows how it affects each frequency. Key terms:
Boost or cut specific frequency bands to shape the sound.
High-pass filters enhance edges by emphasizing high-frequency details.
Identify and remove noise frequencies while preserving signal.