Filtering

Fourier Analysis · Axiom Academy

LESSON Filtering in the Frequency Domain Signal filtering using Fourier methods and frequency-domain processing 1. The Filtering Concept: Selecting Frequency Components A filter is a system that selectively passes or blocks certain frequency components of a signal. In the frequency domain, filtering becomes multiplication: we multiply the Fourier transform of our signal by a filter function H(f). Transform input signal to frequency domain: X(f) = F x(t) Multiply by filter transfer function: Y(f) = H(f) · X(f) Transform back to time domain: y(t) = F⁻¹ Y(f) 2. Types of Filters: Low-Pass, High-Pass, Band-Pass, Band-Stop Filters are classified by which frequency ranges they pass or attenuate. Each type serves different signal processing needs. Low-Pass: Passes frequencies below cutoff f_c, blocks higher frequencies High-Pass: Blocks frequencies below cutoff f_c, passes higher frequencies Band-Pass: Passes frequencies between f_1 and f_2, blocks others Band-Stop (Notch): Blocks frequencies between f_1 and f_2, passes others 3. Ideal Filters and Their Impulse Responses An ideal filter has a perfectly rectangular frequency response: complete transmission in the passband and complete attenuation in the stopband. However, this ideal comes with a fundamental trade-off in the time domain. The impulse response of an ideal low-pass filter with cutoff frequency f_c is the sinc function, which extends infinitely in time. This makes ideal filters physically unrealizable.

This is the written version of the interactive lesson above. See the full Fourier Analysis course.