Signal Processing

Differential Equations · Axiom Academy

Signal Processing: Audio Filtering Every EQ knob and noise-cancelling algorithm is a transfer function H(s) at work. One pole, one exponential, one Laplace transform — that's the whole engine. An audio filter is a linear time-invariant (LTI) system : feed it x(t) , it hands back y(t) , and in the s -domain that relationship collapses to one clean ratio, H(s)=Y(s)/X(s) — the transfer function. Move its pole, and you reshape the sound. The transfer function shapes the spectrum A transfer function turns a messy differential equation into simple algebra: H(s)=Y(s)/X(s) . Pick a filter type and drag the controls — the curve you're bending IS , the frequency response. The pole is why the filter is stable Every H(s) has poles — the values of s that blow up the denominator. Drag the SAME cutoff and watch the pole slide along the negative real axis of the s -plane. The impulse response — what one "click" sounds like Feed the filter a single instantaneous click, , and the output IS h(t) — the inverse Laplace transform of H(s) . For this filter, that's a clean exponential decay. One transfer function, three views: the frequency response you just shaped, the pole that guarantees it's stable, and the impulse response that's its fingerprint in time — all three are the same . This is the exact math behind EQs, noise gates, and every audio plugin's cutoff knob.

This is the written version of the interactive lesson above. See the full Differential Equations course.