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Sound Reverb Effect

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Description

Reverberation is the sound that keeps arriving after a source stops, built from thousands of reflections off the walls of a room. This MicroSim plays a short music clip (2.4 s, looped) through a digital reverb and plots, over the last 8 seconds, the level of the dry clip going into the reverb (blue) and of the output you hear (maroon), in decibels relative to full scale (dBFS).

Control What it does
Play / Stop Play starts the clip from the beginning; Stop silences the clip, so only the reverb tail is left. A gold dashed line marks the moment the clip fell silent.
Decay time Length of the reverb's impulse response, 1 to 10 s. The panel shows the resulting reverberation time RT60.
Wet mix Dry/wet balance, 0 % (clip only) to 100 % (reverb only). The bars show the dry and wet gains.

With Wet mix at 0 % the maroon trace sits on the blue one and drops to silence the instant you press Stop. Raise the wet mix and the maroon trace keeps going after Stop: that tail is the reverb. On the decibel axis the tail is a straight line, because the reverb's impulse response decays exponentially.

The reverb is a convolution, so this simulation belongs with Chapter 4: Convolution and Correlation.

The music clip is Dancing Tiger by Damscray, the sample file used in the p5.js sound examples. Its licence has not yet been verified.

Mathematical Concepts

The output is the clip \(x(t)\) convolved with the room's impulse response \(h(t)\):

\[y(t) = (x * h)(t) = \int_0^{\infty} h(\tau)\, x(t - \tau)\, d\tau\]

so \(y\) keeps sounding for as long as \(h\) is non-zero, even after \(x\) has stopped. The reverb used here (p5.sound 0.4.1, which wraps the Tone.js Reverb) builds \(h(t)\) as stereo white noise multiplied by an exponential envelope \(e^{-t/\tau}\) with \(\tau = \ln(d + 1)/\ln 200\), where \(d\) is the decay time, and cuts it to zero at \(t = d\). The reverberation time RT60 is the time for the level to fall by 60 dB (a factor of 1000 in amplitude):

\[\mathrm{RT60} = \tau \ln 1000 = \frac{\ln 1000}{\ln 200}\,\ln(d + 1) \approx 1.30\,\ln(d + 1)\]

The default \(d = 3\) s gives RT60 = 1.81 s; the slider range 1 to 10 s gives 0.90 to 3.13 s. These values match RT60s measured from the impulse responses the library actually generates to within 3 %.

The Wet mix \(w\) is an equal-power crossfade: the dry gain is \(\cos(w\pi/2)\) and the wet gain is \(\sin(w\pi/2)\), so the squared gains always add up to 1. At 50 % both gains are 0.71.

Lesson Plan

Learning Objectives

After using this MicroSim, students will be able to:

  1. Explain why the reverb output continues after the source stops, using the convolution \(y = x * h\).
  2. Predict how moving the Wet mix from 0 % to 100 % changes the dry and wet gains and the sound after Stop.
  3. Estimate the reverberation time RT60 from the slope of the decaying tail on the decibel plot.
  4. Calculate RT60 from the decay time \(d\) and compare it with the value shown in the panel.
  5. Distinguish between reverb (a dense, smoothly decaying tail) and a discrete echo.

Target Audience

College undergraduates in an introductory signal processing, audio engineering or music technology course. Plan 15-20 minutes. Headphones or speakers are required.

Prerequisites

  • Decibels and logarithms
  • The impulse response of a linear time-invariant system
  • Convolution (Chapter 4)

Activities

Activity 1: Dry versus wet (5 min). Set Wet mix to 0 %, press Play, wait a few seconds and press Stop. Both traces fall to the bottom of the plot at the same instant and the sound stops at once. Now set Wet mix to 100 % and repeat. The blue trace still drops at once, but the maroon output trace keeps going and takes about 1.5 s to reach the bottom of the plot (−80 dB); you hear the tail.

Activity 2: Measure RT60 (5 min). Keep Wet mix at 100 % and Decay time at 3.0 s. Play for a few seconds, press Stop, and measure how long the maroon tail takes to fall 20 dB (one grid spacing). It takes about 0.6 s. Multiply by 3 to estimate the time for a 60 dB fall and compare it with the RT60 (est.) read-out of 1.81 s.

Activity 3: Small room, big hall (5 min). Repeat Activity 2 with Decay time 1.0 s and then 10.0 s. Predict the slopes before you look: RT60 = 0.90 s means a fall of about 66 dB per second, and RT60 = 3.13 s means about 19 dB per second. Listen to how the long tail blurs the notes of the clip together.

Activity 4: Equal-power mixing (5 min). Move Wet mix to 25 %, 50 % and 75 % and read the gain bars: 0.92/0.38, 0.71/0.71 and 0.38/0.92. Square and add each pair. The sum is always 1: an equal-power crossfade keeps the combined power of two unrelated signals constant, where a linear crossfade (0.5/0.5 in the middle) would dip. The output still gets quieter toward 100 % because the reverb's wet signal is itself about 10 dB quieter than the dry clip; compare the two traces at 100 %.

Assessment

  1. The Decay time slider is set to 5 s. What RT60 does the panel show? (Answer: \(1.30 \ln 6 \approx 2.34\) s.)
  2. At Wet mix = 25 %, what are the dry and wet gains, and what is the sum of their squares? (Answer: \(\cos(\pi/8) = 0.92\) and \(\sin(\pi/8) = 0.38\); \(0.92^2 + 0.38^2 = 1\).)
  3. After Stop with Wet mix at 100 %, why does the maroon trace keep going while the blue trace has already dropped to the floor? Why does this not happen at 0 %? (Answer: the output is \(y(t) = \int h(\tau)x(t-\tau)\,d\tau\), which still contains past input for as long as \(h\) is non-zero. At 0 % wet the output is the dry clip only, with no convolution.)
  4. A hall has an RT60 of 2 s. A chord at 80 dB SPL is cut off. How long until it has fallen to 50 dB SPL? (Answer: a 30 dB fall is half of 60 dB, so 1 s.)
  5. A 3 s impulse response sampled at 48 kHz has how many samples, and how many multiply-adds per output sample would direct convolution need? (Answer: 144,000 of each, per channel. This is why reverbs use FFT-based fast convolution.)

Common Misconceptions

  • "Reverb is just an echo." An echo is a single delayed copy you can hear as a separate event. Reverb is a dense mass of reflections whose level decays smoothly. The maroon tail in this simulation falls along a roughly straight, unbroken line on the decibel plot, with no separate repeats.
  • "The decay time is the reverberation time." In this library the decay time is the length of the impulse response. The level falls 60 dB well before the end: a 3 s decay gives an RT60 of 1.81 s, as the panel shows.

References

  1. Reverberation - Wikipedia - what reverberation is and how the reverberation time RT60 is defined
  2. Convolution reverb - Wikipedia - how a reverb is computed by convolving a signal with an impulse response
  3. Impulse response - Wikipedia - the system description that the reverb tail comes from
  4. Convolution - Wikipedia - the operation \(y = x * h\) used by the reverb
  5. The Scientist and Engineer's Guide to Digital Signal Processing - Steven W. Smith - free online textbook with chapters on convolution and FFT convolution
  6. p5.sound.js - Processing Foundation - the audio library used by this MicroSim