Low-pass, high-pass, band-pass, notch

"Every filter is a bouncer with a guest list written in hertz. My only job is to hand it the right list."

A Selective AI Agent

This section assumes you are comfortable with sampling rate, the Nyquist frequency, and the frequency-domain view of a signal from Chapter 3, and with the FIR/IIR distinction and the notions of order and phase from Section 6.2. Everything here is expressed in normalized frequency, so a design ports across sensors regardless of their sampling rate.

The four response types as passbands

A frequency-selective filter is fully described by its magnitude response \(|H(f)|\): how much of each frequency component survives, from \(0\) (fully removed) to \(1\) (fully passed). The four canonical shapes differ only in which frequencies they let through.

• Low-pass keeps frequencies below a cutoff \(f_c\) and attenuates everything above it. Use it whenever the signal is slow and the noise is fast: smoothing a temperature reading, extracting the gravity vector from a raw accelerometer, anti-aliasing before you downsample.
• High-pass is the mirror image: it removes slow content below \(f_c\) and keeps the fast content. This is the tool for killing drift and DC offset, the slow wander that plagues strain gauges, EEG baselines, and MEMS accelerometers left in the sun.
• Band-pass keeps a window \([f_1, f_2]\) and rejects both the slow and the fast sides. It is a low-pass and a high-pass composed into one, ideal when your phenomenon occupies a known band: the roughly \(0.7\)-\(4\) Hz of a resting-to-exercise heart rate, or the resonance band of a bearing fault.
• Notch (band-stop) is the inverse of band-pass: it passes everything except a narrow sliver, surgically removing one interfering tone while leaving the rest of the spectrum untouched. The archetype is 50/60 Hz mains hum in a biosignal.

A profound and practical fact ties these together: you can build all four from the same low-pass prototype. A high-pass is a low-pass subtracted from an all-pass (\(H_{\text{hp}} = 1 - H_{\text{lp}}\)); a band-pass is the product of a low-pass and a high-pass; a notch is one minus a band-pass. Design libraries exploit exactly this, which is why the same function that gives you a Butterworth low-pass gives you the other three with one argument changed.

From physical goal to specification

A filter is not specified by "low-pass, please." It is specified by four numbers you derive from the physics of the signal and the noise. First, the cutoff(s), placed in the gap between the highest frequency you want to keep and the lowest frequency you want to remove; the wider that gap, the easier your life. Second, the passband tolerance: how much attenuation and ripple you can tolerate on the signal itself, often quoted in decibels. Third, the stopband attenuation: how thoroughly the noise must be crushed, for example \(40\) dB (a factor of 100) for a stubborn interferer. Fourth, the order, which is not so much chosen as it is the consequence of the previous three plus the filter family.

The order controls the steepness of the transition. A first-order low-pass rolls off at \(6\) dB per octave (each doubling of frequency halves the amplitude); an \(N\)-th order filter rolls off at \(6N\) dB per octave. Steeper is not free: higher order means more coefficients, more numerical sensitivity in IIR forms, and, as Section 6.7 will make quantitative, more delay. The families you will reach for trade these differently: Butterworth is maximally flat in the passband (no ripple, gentle roll-off), Chebyshev buys a steeper transition by allowing passband (type I) or stopband (type II) ripple, and elliptic gives the steepest transition of all by rippling both. When phase linearity matters, an FIR designed with a windowed-sinc or Parks-McClellan method, or a zero-phase forward-backward pass, is the honest choice.

Notch filters: removing one tone without touching the rest

The notch deserves its own treatment because its job is uniquely surgical. Mains interference at \(50\) Hz (Europe, Asia) or \(60\) Hz (North America) couples into almost every high-impedance biosensor, and it sits at a precise, known frequency. You do not want a low-pass here: the diagnostic content of an ECG extends well past \(50\) Hz, so removing everything above the hum would destroy the QRS complex you are trying to measure. You want to remove a razor-thin band around the offender and nothing else.

The sharpness of a notch is set by its quality factor \(Q = f_0 / \Delta f\), where \(f_0\) is the center and \(\Delta f\) is the \(-3\) dB width. A high \(Q\) (narrow notch) preserves nearby signal but assumes the interferer is exactly at \(f_0\); if the mains drifts to \(50.2\) Hz, a razor-narrow notch misses it. A low \(Q\) is forgiving of drift but gouges a wider hole in your signal. Practitioners often cascade notches at the fundamental and its harmonics (\(50, 100, 150\) Hz), or, when the interference wanders, switch to an adaptive notch that tracks \(f_0\). We build the fixed version in code below and revisit adaptive and spectral removal in Section 6.4.

import numpy as np
from scipy.signal import butter, iirnotch, sosfiltfilt, tf2sos

fs = 500.0                      # ECG sampling rate (Hz)
t  = np.arange(0, 4, 1/fs)

# Synthetic ECG-like beat train + baseline wander + 60 Hz mains hum
clean = np.sin(2*np.pi*1.2*t) * (np.abs(np.sin(np.pi*1.2*t)) > 0.85)
wander = 0.6*np.sin(2*np.pi*0.15*t)          # slow drift, ~0.15 Hz
hum    = 0.4*np.sin(2*np.pi*60.0*t)          # 60 Hz interference
noisy  = clean + wander + hum

# High-pass at 0.5 Hz to remove baseline wander (drift killer)
sos_hp = butter(2, 0.5, btype='highpass', fs=fs, output='sos')

# 60 Hz notch, Q = 30 -> ~2 Hz wide, surgical
b, a   = iirnotch(w0=60.0, Q=30.0, fs=fs)
sos_no = tf2sos(b, a)

# Zero-phase forward-backward so beats are not time-shifted
y = sosfiltfilt(sos_hp, noisy)
y = sosfiltfilt(sos_no, y)

print(f"input std {noisy.std():.3f} -> cleaned std {y.std():.3f}")

The code above chains a drift-killing high-pass with a hum-killing notch, exactly the two-stage recipe most biosignal front ends use. Note two production habits baked in: filtering in second-order-section (SOS) form for numerical stability at higher orders, and using sosfiltfilt for zero-phase response so a downstream R-peak detector is not fed time-shifted beats.

Choosing the right response in practice

The decision procedure is short. Sketch or compute the spectrum of a representative recording (the tools are in Chapter 7). Mark where your signal lives and where the dominant noise lives. If noise is only above the signal, low-pass; only below, high-pass; on both sides, band-pass; concentrated at a single known tone, notch. Then check three failure modes before you trust it. Is any part of your signal inside the transition band and being quietly attenuated? Does the phase distortion matter for what you do next, and if so, are you filtering zero-phase? And crucially, is the noise actually separable in frequency at all: a spike or an outlier is broadband and will slip through any linear band filter, which is why Section 6.5 turns to robust, nonlinear methods for exactly those cases.

What's Next

In Section 6.4, we lift filtering out of the time domain entirely: instead of running samples through recursive coefficients, we transform to the frequency domain with the FFT, zero out or reweight the bins we do not want, and transform back. That view makes band selection almost trivial to visualize, unlocks adaptive and time-varying removal of drifting interference, and sets up the spectral analysis toolkit of Chapter 7.