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Signal Processing

Sampling, Fourier, filtering — taming noisy signals.

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Prerequisites

Linear Algebra Calculus & Optimization

Why it matters in robotics

Signal processing turns raw sensor streams into usable measurements. A robot has to decide how fast to sample a signal without losing information, view a noisy waveform in terms of the frequencies it contains, and filter out unwanted frequencies while keeping the ones that matter — all before IMU, audio, encoder, and range data ever reach an estimator. Clean signal processing is what keeps noise, drift, and distortion from corrupting everything downstream.

What to study

✓Sampling theory: the Nyquist-Shannon theorem, aliasing/frequency folding, why anti-aliasing filters go BEFORE the ADC, and how to choose a sample rate.
✓Frequency domain: what the DFT/FFT computes, reading a magnitude spectrum, frequency resolution vs. window length, and the time-frequency (uncertainty) trade-off.
✓Convolution and linear filters: how convolution implements filtering, impulse/frequency response, and the behavior of low-, high-, and band-pass filters.
✓Practical digital filter design and noise: FIR vs. IIR (e.g., Butterworth), phase lag/group delay vs. smoothing, SNR, and cutoff selection for real sensor signals.

Where to study

Cheatsheet

Symbols used below

f_s

sample rate (samples/sec)

f_{\max}

highest frequency present in the signal

x(t), X(f)

signal and its Fourier transform

h(t)

filter impulse response

*

convolution operator

Nyquist–Shannon sampling theorem

$f_s > 2 f_{\max}$

Sample above twice the signal's highest frequency (the Nyquist rate) to reconstruct it exactly. $f_s/2$ is the Nyquist frequency — the ceiling on what a given sample rate can represent without ambiguity.

Aliasing

Sampling below the Nyquist rate folds high frequencies down onto lower ones — a true frequency $f$ becomes indistinguishable from $|f - k f_s|$ for some integer $k$ .

This is why an anti-aliasing (low-pass) filter must run before the ADC: once aliased, the corruption can't be undone by filtering afterward.

Convolution theorem

Filtering in time is convolution: $y(t) = x(t) * h(t)$ .

Convolution in time is multiplication in frequency: $Y(f) = X(f) \cdot H(f)$ . So a filter's effect is just shaping the spectrum by $H(f)$ — zero it out where you want to reject, keep it near 1 where you want to pass.

Time-frequency trade-off

Frequency resolution improves with a longer observation window ( $\Delta f \approx 1/T$ ), but a longer window smears out when things happened in time.

There's no free lunch: narrow the window to localize in time, and the frequency estimate gets blurrier, and vice versa.

Filter response trade-offs

A filter's cutoff sets which frequencies pass, but every real filter also adds phase lag / group delay — the more aggressive the smoothing, the more the output lags the true signal.

Choosing a cutoff is a balance between rejecting noise (SNR) and keeping the delay small enough for real-time control.

Sampling below the Nyquist rate makes a high-frequency signal indistinguishable from a lower-frequency 'alias' passing through the same samples.

	FIR	IIR
Impulse response	Finite length	Infinite (feedback-driven)
Stability	Always stable	Can be unstable if poles land outside the unit circle
Phase	Can be made exactly linear phase	Generally nonlinear phase
Cost for a given sharpness	More taps / more compute	Fewer coefficients (e.g. Butterworth) for the same rolloff

Convolving a rectangular pulse with itself yields a triangular pulse of twice the width — the overlap area grows then shrinks linearly.

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