Schroeder Integration

Squaring the impulse response and summing its energy backwards from the tail turns a jagged decay into one clean line you can rule RT60 off.

What it is

A maths trick that measures reverb time cleanly by summing leftover sound energy backwards from the end of the decay tail.

Key facts

• Formula: the Schroeder curve at time t equals the integral (sum) of the squared impulse response from t out to the end, divided by the total energy, then plotted in dB. Symbols: squared impulse response = instantaneous sound power; integral = running tally; end = silence point.
• It works on the IMPULSE RESPONSE (room's reply to a click/sweep) not on raw noise, so one measurement replaces averaging dozens of noisy decays.
• Speed of sound in air = 343 m/s at 20 C; rises about 0.6 m/s per degree C hotter.
• RT60 = time for sound to decay 60 dB (drop to one-millionth, 10^-6, of start power).
• Real rooms rarely give a clean 60 dB, so you measure a smaller slope and extrapolate: T20 uses -5 to -25 dB, T30 uses -5 to -35 dB, then multiply up to 60 dB.
• -3 dB = half the power (10^-0.3 = 0.5); -10 dB = one-tenth power; -60 dB = one-millionth power.
• +6 dB = double the sound pressure (4x power); +3 dB = double the power.
• 0 dB reference for SPL = 20 micropascals (threshold of hearing); pain ~120-130 dB SPL.
• Decay must sit at least 10 dB above the noise floor or the backward integral curls up wrong; truncate the tail before noise dominates.
• Named after Manfred Schroeder, who published the backward-integration method in 1965 (J. Acoust. Soc. Am.).

How it works

1. Play a known test signal (starter pistol, balloon pop, or a swept sine) into the room.
2. Record the impulse response with a measurement mic: the room's full echo tail.
3. Square the signal at every instant to get sound power (energy) versus time.
4. Start at the very END of the tail and add energy moving BACKWARDS toward the start.
5. This running backward sum gives a smooth, monotonically dropping decay curve in dB.
6. Fit a straight line to the slope (use T20 or T30) and extrapolate to a 60 dB drop = RT60.

Real examples

• Your phone room-analysis app pops a balloon, then shows RT60 = 0.8 s for a function hall using exactly this backward-integration maths.
• Smaart, REW or Open Sound Meter all run Schroeder integration on a log-sweep to spit out T20/T30 reverb numbers.
• A live raw decay looks jagged like a heartbeat trace; the Schroeder curve turns it into one smooth downhill slope you can rule a line on.
• Speech-clarity gigs: a hall measuring RT60 = 2.5 s is too boomy for spoken word; Schroeder gives you the trustworthy number to prove it.

How it helps in live sound

• Use a log-sweep instead of a balloon in REW/Smaart for a cleaner impulse and far better signal-to-noise.
• Read T30 (or T20) off the app, not eyeballed RT60: it auto-extrapolates from a clean slope.
• Aim RT60 ~0.8-1.2 s for speech/AV gigs; under ~0.6 s feels dead, over ~1.5 s smears dialogue.
• Check the decay sits >10 dB above the noise floor; kill HVAC/crowd noise before measuring or the curve lies.
• Watch per-band RT60 (125 Hz vs 4 kHz): long bass tail means add bass traps or low-end absorption, not more top-end EQ.
• Re-measure with the room full of bodies; people are absorbers and RT60 drops noticeably from the empty-room figure.