Filters & frequency response

A filter is a system designed for selective hearing: keep the bass, drop the hiss; keep the heartbeat, drop the tremor. The cleanest way to describe one isn't what it does to time — it's what it does to each frequency.

Build the intuition

Sinusoids pass through LTI systems unbruised

Feed a pure tone into any LTI system and a pure tone of the same frequency comes out — only its amplitude and phase change. Sinusoids are the eigenvectors of LTI systems (linear algebra's stubborn directions, reborn): the system can scale and shift them but never bend them into other shapes. This is why frequency is the right lens for systems.

Frequency response: the system's tasting notes

Test every frequency: how much does the system scale it (gain) and delay it (phase)? Plot gain versus frequency and you have the frequency response — the system's complete flavor profile. Low-pass: tall on the left, suppressing treble. High-pass: the mirror. Band-pass: a peak that selects one station. Reading these plots is the literacy of the field.

H(\omega): \;\text{gain and phase shift at each } \omega

Averaging is low-pass — now you know why

The moving average from the convolution lesson smooths because slow waves barely change inside the window (they survive averaging) while fast waves wiggle up and down within it (they cancel). Same kernel, two descriptions: “averages neighbors” in time, “suppresses high frequencies” in frequency. The Fourier course makes this duality exact: convolution in time is multiplication in frequency.

See it move

InteractiveConvolution: the moving average

Kernel width9

Position60

At each position, the kernel (gold window) overlaps 9 input values, multiplies, and totals — producing one output point (orange). Slide it across the whole signal and the output curve emerges. Width 9: noise fades, edges soften.

A low-pass filter at work: widen the kernel and watch fast wiggles (high frequencies) die while slow structure (low frequencies) survives.

A worked example

Design by specification

An ECG signal lives below ~40 Hz; mains interference hums at exactly 50 Hz.
Specification: pass 0–40 Hz, crush 50 Hz. That's a low-pass (or a notch precisely at 50).
A moving average of the right length has its first “zero” — total cancellation — exactly at 50 Hz: at 200 samples/s, averaging 4 samples nulls 50 Hz completely.
A real medical-device trick, designed in two lines of frequency thinking.

Out in the world

Your ears prove the concept nightly

Noise-cancelling headphones run high-pass, low-pass, and adaptive filters at once; a graphic equalizer is a bank of band-pass filters with sliders for gains. When you nudge “bass” upward you are manually editing a frequency response — this lesson, as a consumer product.

Common confusion, cleared

“A good filter removes unwanted content and touches nothing else.”

Every filter trades: sharper frequency cuts need longer kernels (more delay, more ringing). Filter design is choosing your compromise — the no-cost filter doesn't exist.

“Filtering happens only in electronics.”

Your coffee filter is a spatial low-pass for particles; your eye's optics low-pass spatial detail; moving averages in dashboards filter time series. The concept is universal — electronics just made it programmable.

Check yourself

PracticeQuick check

A moving-average filter mainly…
Feed a 100 Hz sine into an LTI filter. The output contains…

Recap

LTI systems can only scale and shift sinusoids — frequency is the natural lens.
The frequency response H(ω) is a system's complete profile: gain and phase per frequency.
Low-pass, high-pass, band-pass: sculpting tools, each a trade-off.

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