A tube-powered overdrive pedal with its drive knobs and a visibly glowing vacuum tube, the gain stage that pushes a signal past its headroom.

Week 05 Stage: Nonlinearity

Gain, Headroom and Clipping

Photo: Tim Patterson · CC BY-SA 2.0

What physically happens when an amplified signal stops behaving linearly?

  1. InputA signal driven past a stage's headroom
  2. ProcessClipping: hard or soft
  3. OutputNew harmonic content that wasn't there before

Where we left off

From linear filtering to nonlinear transformation

Every stage covered so far — pickup, volume pot, tone control, cable — has been linear: it can reshape which frequencies are present, but it never adds anything that wasn’t already there. Gain changes that. Once a signal is amplified far enough, the system running out of room to represent it stops being a detail and becomes the whole story. By the end of this week you should be able to distinguish gain from output volume; explain linear amplification; define headroom conceptually; explain clipping in terms of input-output limits; distinguish hard and soft clipping; and explain why clipping introduces harmonic content that wasn’t in the original signal.

  1. Gain (amplitude scaling)
  2. Headroom limit reached
  3. Clipped waveform shape
  4. New harmonic content

Gain versus volume

Running out of headroom

Low gain — clean
Higher gain — nearing threshold

Gain is amplification applied within a stage: for an ideal linear amplifier, output(t) = G · input(t) — the waveform’s shape is unchanged, only its amplitude scales. Volume is a separate, later control over how loud the already-processed signal ends up. A real stage can’t raise amplitude indefinitely, though: it has a maximum level it can represent before something has to give. The gap between a signal’s current level and that maximum is its headroom. As gain increases, headroom shrinks — and once it reaches zero, the waveform can no longer just get bigger.

Clipping

What happens when headroom runs out

Clipping is what a system does once it can’t represent an amplified waveform’s true peak: rather than continuing to follow G · input(t), the output stops tracking the input once it nears the limit. How it stops tracking it is a design choice, not a single fixed behaviour — and that choice is exactly what separates the two clipping families below.

Two clipping families

Linear, soft, and hard transfer curves

Linear (no clipping)Well within headroom: a straight line — output tracks input at every level, just scaled up. The reference case.
Soft clippingOutput bends away from the straight line gradually as it nears the limit — a smooth compression, not an abrupt wall.
Hard clippingOutput tracks the input exactly until an abrupt threshold, then is flattened outright — a sharp corner in the curve.

Either way, the output is no longer a scaled copy of the input — it’s a genuinely different waveform shape. A nonlinear transformation is exactly one where output isn’t simply proportional to input, and clipping is the simplest example this course studies. A clipped sine wave contains frequency components that were never present in the clean sine at all: flattening the waveform’s peaks is, in the frequency domain, equivalent to adding new harmonics on top of the fundamental. No detailed derivation is needed here — what matters is the causal link: reshape the waveform → change its harmonic content. Those extra harmonics change the signal’s perceived timbre, though “more harmonics” on its own doesn’t mean “better tone” — whether added harmonics sound pleasant depends on which ones appear and how they’re balanced, not just how many there are.

Try it

One gain stage, two ways to clip

  1. Physical mechanism
  2. Signal transformation
  3. Waveform / spectral consequence
  4. Audible consequence

Turn the Drive knob up from its low default and watch the transfer curve, waveform, and spectrum all update together. At low drive the curve stays close to the straight reference line and the output is just a louder sine; past a point it bends (soft) or folds sharply (hard), and the spectrum grows new bars exactly as the waveform starts to flatten. Play the tone to hear the same transition.

Gain stage simulatorA sine tone, one gain stage, one clipping shape at a time
Drive
1.00×
Clipping shape

Audio never plays automatically — press play to hear it, press again to stop.

Transfer curve — input amplitude → output amplitude
Input
Output
Output spectrum

At low drive the output is just a bigger sine — same shape, no new frequencies. Turn Drive up and watch new harmonic bars appear in the spectrum exactly as the waveform starts to flatten.

Takeaways

What this week establishes

  1. Gain is amplification within a stage; volume is a separate, later control over overall loudness.

  2. An ideal linear amplifier scales amplitude without changing waveform shape: output(t) = G · input(t).

  3. Headroom is the gap between a signal's current level and the maximum a stage can represent — gain shrinks it.

  4. Clipping is what happens once headroom reaches zero: soft clipping compresses gradually, hard clipping cuts off abruptly.

  5. Clipping is nonlinear and adds harmonic content that wasn't in the original signal — more harmonics isn't automatically 'better tone'.

The chain

Where this sits, and what comes next

  1. 01INPUT
  2. 02STRING
  3. 03PICKUP
  4. 04CIRCUIT
  5. 05DRIVE
  6. 06DISTORTION
  7. 07FILTER
  8. 08MODULATION
  9. 09SPACE
  10. 10AMP
  11. 11SPEAKER
  12. 12OUTPUT
Week 04 gave usThe Guitar Is a CircuitWeek 06 takes it on Inside overdrive, distortion and fuzz

Week 4 focused on linear filtering — reshaping frequency content without adding anything new. Week 5 introduced nonlinear signal transformation: past a headroom limit, a stage’s output stops being a scaled copy of its input, and new harmonics appear. So far, clipping has stayed an abstract mathematical function — a curve, applied to a sine wave. Week 6 builds that same function out of a real gain stage and real clipping components, and asks why two circuits that both “clip the signal” can still sound different from each other.

  1. Physical mechanism
  2. Signal transformation
  3. Waveform / spectral consequence
  4. Audible consequence
Teaching date
22 March 2027

Teaching team