How Do Filters Work? Low-Pass and High-Pass Filters Explained

Hold a phone to your ear and you instantly recognise a voice — even though the phone deliberately throws away most of the sound it captures. That is a filter at work. Filters are everywhere: in your headphones, your ECG machine, your Wi-Fi radio, your camera sensor. Understanding them unlocks a surprisingly large slice of modern technology.

What Is a Signal Made Of?

Shine white light through a prism and it splits into a rainbow — revealing that what looked like a single thing is actually a mixture of many frequencies of light. The same idea applies to every signal you can measure.

A microphone records air pressure that is a mixture of hundreds of frequencies simultaneously — the deep thud of a bass drum, the mid-range warmth of a voice, the high hiss of a cymbal. An ECG electrode picks up the electrical rhythm of your heart, but also the slow drift of your body shifting on the bed and the 50 Hz buzz of the mains power supply.

All of these signals are superpositions — additions — of simpler oscillations at different frequencies. The simulation below lets you build exactly that kind of composite signal. Choose a preset and you can see three frequencies mixed together into a single waveform that looks complicated until you know what is inside it.

This idea — that any signal can be decomposed into a sum of sine waves — is the foundation of Fourier analysis. If you want to go deeper, our Fourier Synthesis post shows how even square waves and sawtooth waves are built from harmonics.

What Does a Filter Do?

A coffee filter lets water through but blocks the grounds. A sunroof visor lets visible light through but blocks ultraviolet. A filter in signal processing does the same thing — it lets certain frequencies pass through and blocks others.

The two most useful types are:

• Low-pass filter — passes low frequencies, blocks high ones. Think of it as a "bass pass" filter.
• High-pass filter — passes high frequencies, blocks low ones. Think of it as a "treble pass" filter.

The frequency at which a filter switches from "passing" to "blocking" is called the cutoff frequency (written $f_c$). Signals well below $f_c$ pass through almost unchanged. Signals well above $f_c$ are strongly attenuated. The transition between these two regions is called the roll-off.

How a Low-Pass Filter Works

Stand next to a wall in a nightclub. You can barely hear the singer, but you can feel every bass note thumping through the brickwork. The wall acts as a low-pass filter on sound — it blocks the high-frequency treble but lets the low-frequency bass vibrations through.

In electronics, the simplest low-pass filter is just a resistor and a capacitor (an RC circuit). At low frequencies the capacitor barely charges and discharges — most of the voltage appears across the output. At high frequencies the capacitor charges and discharges so rapidly that it essentially short-circuits the output to ground, killing the signal.

Real-world uses of low-pass filters:

• Removing high-frequency hiss from audio recordings
• Anti-aliasing before sampling (prevents high frequencies from folding back as aliases)
• Smoothing noisy sensor readings in embedded systems
• Separating the bass driver from the tweeter in a speaker crossover

How a High-Pass Filter Works

A telephone call sounds slightly tinny — the bass in your voice is missing. That is because telephone systems apply a high-pass filter at around 300 Hz to remove low-frequency rumble and hum. Only the part of the voice signal that carries intelligibility (roughly 300–3400 Hz) gets transmitted.

A high-pass filter is the complement of a low-pass filter: it blocks slow, low-frequency variations and passes rapid, high-frequency ones.

Real-world uses of high-pass filters:

• Removing DC bias and slow baseline drift from ECG signals
• Blocking the 50/60 Hz mains hum from audio equipment
• Isolating the treble channel in a speaker crossover
• Coupling an AC signal between amplifier stages while blocking DC offset

Interactive Simulation

Use the controls to explore both filter types. Switch between Low-Pass and High-Pass, choose a real-world scenario from the presets, and drag the Cutoff Frequency slider to watch the frequency response curve shift and the filtered output waveform change in real time.

The three panels show:

1. Input Signal — the raw composite waveform you are filtering
2. Frequency Response — the filter's gain curve (how much of each frequency it passes). The dashed lines mark where each component sits; the grey dashed line marks the −3 dB cutoff point.
3. Filtered Output — the result. The faded purple trace is the original input; the amber trace is what comes out of the filter.

The Math Behind Filters

The RC Low-Pass Filter Transfer Function

For an RC circuit with resistance $R$ and capacitance $C$, the output voltage across the capacitor is:

$H(f) = \frac{V_\text{out}}{V_\text{in}} = \frac{1}{1 + j \cdot f/f_c}$

where $f_c = \frac{1}{2\pi RC}$ is the cutoff frequency and $j = \sqrt{-1}$.

The magnitude of the transfer function gives the gain — how much of the input signal at frequency $f$ makes it to the output:

$|H(f)| = \frac{1}{\sqrt{1 + (f/f_c)^2}}$

At $f = f_c$: $|H| = 1/\sqrt{2} \approx 0.707$, which is a power reduction of exactly one half, or −3 dB. This is why $f_c$ is called the −3 dB point.

For a high-pass filter, the transfer function is the complement:

$|H_\text{HP}(f)| = \frac{f/f_c}{\sqrt{1 + (f/f_c)^2}}$

Filter Order and Roll-Off Rate

A first-order filter has a gentle roll-off of −20 dB per decade (i.e., for every factor-of-10 increase in frequency beyond $f_c$, the signal is attenuated by a further factor of 10 in amplitude).

Cascading $n$ first-order sections gives an $n$th-order filter with a roll-off of $-20n$ dB per decade:

$|H_n(f)| = \frac{1}{\sqrt{1 + (f/f_c)^{2n}}}$

Try increasing the Filter Order slider in the simulation to see the frequency response curve sharpen — the passband stays flat while the stopband attenuation grows steeper.

The Butterworth Filter

The Butterworth design maximises flatness in the passband. Its squared magnitude response is:

$|H(f)|^2 = \frac{1}{1 + \left(\dfrac{f}{f_c}\right)^{2n}}$

This is called a maximally flat response because all derivatives of $|H|^2$ at $f = 0$ are zero — there are no ripples in the passband. The simulation implements a Butterworth filter using a cascade of second-order IIR sections derived via the bilinear transform.

Worked Examples

Example 1: Removing Boomy Low-Frequency Rumble from a Guitar Recording

You have recorded an acoustic guitar but the track sounds muddy — there is a boomy low-frequency rumble from stage vibrations and hollow-body resonance at around 80 Hz swamping the clarity of the notes above.

Solution: Use a high-pass filter at around 100 Hz (order 2). This removes the low-end rumble while leaving the guitar's note range (440 Hz and above) completely intact.

In the simulation: select Guitar + rumble, switch to High-Pass, set the cutoff to 100 Hz. The 80 Hz rumble component is blocked; the 440 Hz and 2000 Hz tones pass through cleanly.

Example 2: Removing ECG Baseline Drift

An ECG recording shows the heartbeat signal (around 10 Hz) riding on top of a slow 0.5 Hz drift caused by the patient breathing and shifting on the bed.

Solution: Use a high-pass filter at 0.5 Hz (order 1). This strips the slow drift while leaving the fast heartbeat signal completely intact.

In the simulation: select ECG baseline drift, switch to High-Pass, set the cutoff to 0.5 Hz. The slow drift is blocked; the 10 Hz heartbeat passes through cleanly.

Example 3: Telephone Audio Passband

Telephone systems transmit only the frequencies between 300 Hz and 3400 Hz — this is enough for speech intelligibility and keeps bandwidth consumption low.

Solution: Chain a high-pass filter at 300 Hz and a low-pass filter at 3400 Hz. Together they form a band-pass filter. The simulation shows each half: set Telephone bandwidth preset, try Low-Pass at 3400 Hz to cut the 8 kHz component, then switch to High-Pass at 300 Hz to cut the 100 Hz component.

Where Filters Appear in the Real World

Filters are one of the most ubiquitous building blocks in signal processing:

• Audio equalizers — each band of an EQ is a bandpass filter. Boosting the bass boosts a low-frequency band; cutting the treble attenuates a high-frequency band.
• Telephone networks — band-limited to 300–3400 Hz to save bandwidth.
• ECG and medical instruments — high-pass filters remove DC drift; low-pass and notch filters remove mains interference.
• Camera sensors — an optical low-pass filter (anti-aliasing filter) sits in front of the sensor to prevent aliasing artefacts from fine textures.
• Wi-Fi and radio — band-pass filters isolate individual channels and reject interference from adjacent frequencies.
• Noise-cancelling headphones — the feedback microphone signal is filtered before being inverted and mixed in.
• Seismometers — low-pass filters separate the slow ground-motion signal from high-frequency mechanical noise.

Frequently Asked Questions

What is the difference between a low-pass and high-pass filter?

A low-pass filter passes frequencies below its cutoff frequency and attenuates frequencies above it. A high-pass filter does the opposite — it passes frequencies above the cutoff and attenuates those below. Think of them as "bass pass" and "treble pass" respectively. Together they can be combined into a band-pass filter (passes a range) or a band-stop (notch) filter (blocks a range).

What is the cutoff frequency of a filter?

The cutoff frequency $f_c$ is the frequency at which the filter's output power drops to half of its input power — a reduction of $1/\sqrt{2} \approx 0.707$ in amplitude, or −3 dB. It marks the boundary between the passband (frequencies that pass through) and the stopband (frequencies that are blocked). The exact definition as the −3 dB point is a convention used almost universally in engineering.

Why does filter order matter?

The filter order determines how sharply the filter transitions from passing to blocking. A first-order filter rolls off at −20 dB per decade — gentle enough that frequencies just above the cutoff are only mildly attenuated. A fourth-order filter rolls off at −80 dB per decade — much steeper, so the transition from "pass" to "block" happens more quickly. Higher-order filters are sharper but also introduce more phase distortion.

What is a Butterworth filter?

A Butterworth filter is a design that achieves a maximally flat response in the passband — no ripples, no peaks. Its squared magnitude response is $|H|^2 = 1/(1 + (f/f_c)^{2n})$ where $n$ is the order. It is the most common starting point for filter design because of its simplicity and smooth behaviour. Other designs like Chebyshev and elliptic filters achieve steeper roll-offs by allowing ripple in either the passband or stopband.

How do you build a simple RC low-pass filter?

Connect a resistor $R$ in series with the input, and a capacitor $C$ from the junction to ground. The output is taken across the capacitor. The cutoff frequency is $f_c = 1/(2\pi RC)$. For example, $R = 1\ \text{k}\Omega$ and $C = 10\ \mu\text{F}$ gives $f_c \approx 16\ \text{Hz}$. This first-order passive filter rolls off at −20 dB/decade and requires no power supply.