Before 1965, measuring reverberation time required either interrupted noise (turning off a loudspeaker and recording the decay on a level recorder) or averaging many squared impulse response measurements to get a usable decay curve. Both methods were slow, noisy, and required heavy equipment. Then Manfred Schroeder published a short, elegant paper that changed acoustic measurement forever. His backward integration method extracts a perfectly smooth decay curve from a single impulse response — no averaging, no repeated measurements, no noise artifacts. It is now the standard method worldwide.
TLDR
The Schroeder integration method (also called backward integration or reverse-time integration) is a mathematical technique that converts a measured impulse response into a smooth energy decay curve (EDC) from which RT60 can be reliably extracted. Instead of squaring the impulse response and looking at the noisy instantaneous energy, the method integrates the squared impulse response backward from the end of the recording to each time point. The result is equivalent to averaging an infinite number of interrupted-noise decay measurements — from a single impulse response. Published by Manfred Schroeder in 1965 and standardised in ISO 3382-2:2008 Annex B, this method is the foundation of all modern room acoustic measurement. Every acoustic analysis tool — from laboratory software to AcousPlan — uses Schroeder integration to derive RT60, EDT, and related parameters.
Real-World Analogy
Imagine you are watching sand drain from an hourglass. Looking at any single grain tells you nothing about the overall flow rate — grains bounce, cluster, and sometimes stall. But if you weigh the sand remaining in the top bulb at each moment, you get a smooth, steadily decreasing curve. The weight of remaining sand is analogous to the Schroeder integral: instead of watching individual reflections (grains), you measure the total energy remaining (sand still in the top), which gives you a clean, usable decay.
Technical Definition
The Mathematical Formula
Given an impulse response h(t), the Schroeder energy decay curve is:
EDC(t) = ∫(from t to ∞) h²(τ) dτ
Expressed in decibels and normalised:
EDC_dB(t) = 10 × log₁₀[ ∫(from t to ∞) h²(τ) dτ / ∫(from 0 to ∞) h²(τ) dτ ]
In practice, the integral's upper limit is the end of the measured impulse response (T_end), and the denominator is the total energy of the IR.
Why Backward Integration?
The key insight is Schroeder's proof that the expected value of the squared signal from an interrupted-noise measurement equals the backward integral of the squared impulse response:
E[s²(t)] = ∫(from t to ∞) h²(τ) dτ
This means a single IR measurement produces the same decay curve as averaging infinitely many interrupted-noise measurements. The mathematical proof relies on the statistical properties of random noise and the linearity of room acoustic systems.
Practical Implementation
- Measure the impulse response using an exponential swept sine (ISO 18233:2006) or MLS signal.
- Filter the IR into octave or 1/3-octave bands using digital bandpass filters.
- Square the filtered IR: h²(t).
- Integrate backward: Starting from the last sample, accumulate the squared values moving toward time zero. In code, this is a cumulative sum of the reversed squared IR.
- Normalise by dividing by the total energy (the first value of the backward integral).
- Convert to dB: 10 × log₁₀(EDC / EDC_max).
- Fit a regression line to extract T20 or T30 (per ISO 3382-2).
Truncation and Noise Compensation
A critical practical issue is the noise floor. Real measurements contain background noise that is not part of the room's decay. If the backward integration includes noise beyond the point where the decay reaches the noise floor, the EDC curves upward at the end (the "noise tail"), biasing the RT60 estimate.
ISO 3382-2 recommends:
- Truncating the IR at the point where the decay meets the noise floor (identified as the point where the decay curve begins to rise)
- Applying Lundeby's method (1995) to automatically detect the truncation point and compensate for the noise contribution
Why It Matters for Design
Without Schroeder integration, acoustic measurement would require:
- Multiple repeated measurements (10 or more) for ensemble averaging
- Interrupted-noise sources (loudspeaker plus power amplifier)
- Analog level recorders (strip chart recorders)
- Significant time per measurement position
For acoustic designers, the smooth EDC reveals subtle room behaviour. A double-slope decay (steep initial slope followed by a shallower late slope) indicates coupled volumes — a balcony, a stage house, or an adjacent corridor feeding energy back into the main room. These details are invisible in raw impulse response data but obvious in the Schroeder-integrated curve.
How AcousPlan Uses This
AcousPlan implements the Schroeder backward integration algorithm in its measurement analysis pipeline. When you upload impulse response data, the platform filters the IR into octave bands, applies backward integration with noise floor compensation, and fits regression lines to extract T20, T30, and EDT per ISO 3382-2. The resulting decay curves are displayed alongside the calculated (Sabine/Eyring) predictions, enabling direct comparison between measurement and model.
Related Concepts
- What is a Decay Curve? — The output of Schroeder integration
- What is an Impulse Response? — The input to Schroeder integration
- What is RT60? — The parameter extracted from the decay curve
- How to Measure Room Acoustics — Full measurement workflow
Calculate Now
AcousPlan uses Schroeder integration to analyse your room acoustics. Model your space or upload measurements to see the energy decay curve at every octave band.