Audio effects

Most audio effects are filters you have met, and the simplest, a delayed copy added back, makes a comb. Watch the comb’s notches below as the delay grows: they crowd closer together, but get no deeper.

Every delay is a comb

In Pitch detection (29.1) I measured a sound. On this page I change one, with filters you have met already.

I take three families: delays, which turn out to be combs; reverberation, which is many combs at once; and compressors, which turn the sound down when it gets loud.

Every sound on this page is synthetic, made from a stated formula: there are no recordings.

A delayed copy

Start with the simplest effect. Delay a copy of the sound by $D$ samples, scale it by a gain $a$, and add it back:

This is the feed-forward comb at the end of “A delay in a loop grows teeth” in Resonators, notches and combs (17.4). In a real-time effect, the delayed copy comes from a buffer of $D$ slots, as in “A circular buffer: the write head goes round” of Real-time processing (21.4).

Its frequency response is $H(e^{j\Omega})=1+a\,e^{-j\Omega D}$. Where $e^{-j\Omega D}=1$ the copy adds to the dry sound, and where $e^{-j\Omega D}=-1$ it works against it. So the gain swings between $1+a$ and $1-a$.

I use $a=0.7$ at $f_s=16$ kHz throughout. The gain then swings between 1.7 and 0.3, which is 4.6 dB and −10.5 dB, in the decibels of “Signal and noise, in decibels” in How big is a signal (1.3).

The notches sit where $\Omega D$ is an odd number of half turns, $\Omega D=(2k+1)\pi$. With $\Omega=2\pi f/f_s$, that is at

The first notch is at $f_s/(2D)$, and after it they come every $f_s/D$. A longer delay packs the notches closer together. It does not make them deeper.

Colour or echo

One formula covers two very different sounds. With a short delay the notches are far apart, and you hear a hollow colour, like a voice in a tiled corridor. With a long one you hear the copy as a separate echo, like a shout across a valley.

The picture at the top of the page shows three delays: 1 ms, 5 ms and 300 ms. Its “Hear it” button plays 3 s of sound through the comb at the chosen delay. The sound is six bursts of noise, half a second apart. Burst $k$ is $0.5\,e^{-(n-n_k)/480}\,v[n]$ for 150 ms from $n_k=8000k$, where $v[n]$ is the seeded noise of the site with seed 292. Each burst fades by a factor $e$ every 30 ms, and the output is scaled so that its largest sample is 1.

Watch the readouts as the delay grows. The notch spacing falls from 1000.0 Hz to 200.0 Hz to 3.3 Hz, while the gain still swings between the same 4.6 dB and −10.5 dB.

Combs that move

Now let the delay change while the sound plays. If it sweeps slowly between 1 and 5 ms, the first notch slides between 500 Hz and 100 Hz, and the other teeth move with it. This is the flanger, which 17.4 called flanging.

A swept delay passes through values of $D$ between whole samples. The effect then needs a fractional delay, as in “Fractional delay” of All-pass systems (17.1).

A chorus uses somewhat longer delays that wander slowly and irregularly, often with several copies. A changing delay also shifts the pitch a little. So the copies drift against the dry sound, and one voice sounds like a few singing together.

A phaser keeps the idea but replaces the delay by a chain of all-passes. The first-order all-pass of 17.1 has a gain of 1 at every frequency. Its phase turns from 0 at 0 Hz to half a turn at $f_s/2$.

Four of them in a row turn the phase by up to two whole turns. Added to the dry sound, the chain’s output cancels it where its phase is an odd number of half turns. With each stage’s pole at 0.5 and $f_s=16$ kHz, that gives two notches, near 699 Hz and 3451 Hz. They are not evenly spaced, unlike a comb’s, and moving the poles slides them.

Loud notes turned down by a ratio

The second family changes the level, not the spectrum. A radio announcer leans into the microphone, then steps back, and the voice jumps by many decibels. A compressor measures the level as the sound goes and turns it down when it gets loud.

Levels in dBFS

I give levels in dBFS, as in “Signal and noise, in decibels” (1.3): $L_\text{dBFS}=20\log_{10}(x/x_\text{FS})$, with the full scale $x_\text{FS}=1$. On this page $x$ is the peak of the sound. So a note whose peak is 0.501 sits at −6.0 dBFS.

Measuring the level

First the compressor needs the level at each moment. It takes $\lvert x[n]\rvert$ and smooths it with the one-pole smoother of Simple smoothing filters (18.2). I call its constant $c$, because $a$ is already the comb’s gain on this page:

In “Choosing a” (18.2) the constant came from a time constant $\tau$, as $e^{-T_s/\tau}$. The detector gets two time constants. While $\lvert x[n]\rvert$ is above $x_\text{det}[n-1]$ it uses the attack time, and otherwise the release time.

With an attack of 1 ms and a release of 100 ms at 16 kHz, $T_s=62.5$ µs. Then $c=e^{-0.0625}=0.9394$ on the way up and $c=e^{-0.000625}=0.999375$ on the way down. The detector follows a rise quickly and lets go slowly.

The gain rule

Then a rule turns the detector’s level, in dBFS, into a gain. Below the threshold $L_\text{thr}$, nothing changes. Above it, each dB more going in gives only a fraction of a dB more coming out.

A ratio of 4:1 means 4 dB more in, 1 dB more out. Above the threshold, in dBFS,

Plotted as out against in, this is the compressor’s static curve: the diagonal below the threshold, and a line a quarter as steep above it. With $L_\text{thr}=-20$ dBFS, a note 8 dB over the threshold comes out 2 dB over it, at −18.0 dBFS.

To get there, the compressor applies a gain of $-(\text{in}-L_\text{thr})(1-1/4)$ dB to the sound, here −6.0 dB. This page adds no gain afterwards to bring the level back up, so loud notes only come down.

Let’s give it six notes at 440 Hz. Each rises over 10 ms, holds for 200 ms and falls over 10 ms, and a new one starts every 0.5 s.

Watch the loud notes come down while the quiet ones pass unchanged. The range between the loudest and the quietest note shrinks from 24.0 dB to 13.7 dB.

Notice the two −6 dBFS notes. The curve promises −16.5 dBFS, but they come out at −16.3. Why the 0.2 dB?

The detector smooths $\lvert x\rvert$, which falls to 0 twice in every cycle of the tone. So on a steady note it settles between 0.26 and 0.30 dB below the true peak, with a ripple under 0.04 dB. The compressor takes the note to be a little quieter than it is, and cuts three quarters of that, about 0.2 dB, too little.

The −30 and −24 dBFS notes are not touched at all. After a −6 dBFS note ends, the detector falls back below −20 dBFS within 0.16 s, well before the next note starts.

The attack shows at the start of each loud note. While the note is still rising, the detector lags behind it and the gain is too high. So the first 20 ms of each −6 dBFS note peak at −14.5 dBFS, 1.8 dB above where they settle.

Limiters and gates

A limiter is a compressor with a very high ratio. At 20:1, the static curve would bring the −6 dBFS note to −19.3 dBFS, only 0.7 dB over the threshold. Limiters guard the 0 dBFS ceiling of 1.3, to keep the peaks from being clipped.

A gate works the other way round. Below its threshold it turns the sound down hard, or off. That silences the hiss between notes or words.

Meters and units

A peak meter shows the largest $\lvert x\rvert$ in dBFS, as the levels on this page do. An RMS meter shows the RMS over the last fraction of a second instead. For a steady sine the RMS reading is 3.0 dB below the peak, as 1.3 showed.

dBFS is a digital unit. In the air, sound is measured in dB SPL, $20\log_{10}(p_\text{rms}/20\,\mu\text{Pa})$, where $p_\text{rms}$ is the RMS pressure. A pressure of 1 Pa RMS is 94.0 dB SPL.

Electrical levels are often given in dBm, $10\log_{10}(P/1\,\text{mW})$, a power against one milliwatt. For example, 0 dBm in a $600\,\Omega$ line is 0.775 V RMS.

A reverb is many decaying combs

Clap your hands in a large hall. You hear the clap, then a dense wash of reflections that fades over a second or two. That is reverberation. Its length is the reverberation time, $T_{60}$ or RT60: the time the sound’s energy takes to fall by 60 dB.

Schroeder’s reverb

In 1962 Manfred Schroeder built a reverb from the parts on this page. Its core is the comb with a loop from 17.4, with the input delayed too:

Each echo is $g$ times the one before, and they come every $D$ samples. Schroeder added up four such combs with different delays, so their echo trains interleave.

How large should $g$ be? Each trip round the loop takes $D/f_s$ seconds and multiplies the echo by $g$. In $T_{60}$ seconds the sound goes round $T_{60}f_s/D$ times, and by then it should be $10^{-3}$ of its start, 60 dB down:

A longer delay needs a smaller $g$, because it makes fewer trips. For $T_{60}=1.5$ s at 16 kHz, the delays 475, 594, 658 and 699 samples, 29.7 to 43.7 ms, get the gains 0.872, 0.843, 0.827 and 0.818.

The four combs are averaged, then passed through two all-passes, from “Echoes with a flat gain” in 17.1. These multiply the echoes without colouring the sound. Both have gain 0.7, with delays of 80 and 27 samples, 5.0 and 1.7 ms.

Look at the energy decay. The energy decay at sample $n$ is the energy still to come, the sum of $h[m]^2$ for $m\ge n$, in dB against the total.

Its measured decay time is 1.50 s, from the slope between −5 and −35 dB. The curve passes −5 dB at 0.150 s and −35 dB at 0.901 s. That is 30 dB in 0.751 s, so 60 dB takes 1.50 s, as the gains were set for.

Nothing arrives for the first 475 samples, 29.7 ms, the shortest comb’s delay. After that the echoes pile up fast. Of the first 1600 samples, the first 100 ms, 741 are already nonzero, and 619 of them are larger than $10^{-4}$.

Colour, echo and reverb

Now press “Hear it” on the first instrument. At 1 and 5 ms the bursts sound hollow: a colour, not a repeat. Sweeping the delay between those two would be a flanger. At 300 ms you hear each burst twice, as a separate echo.

Schroeder’s combs sit in between, at 30 to 44 ms. With feedback and all-passes, their echoes come too densely to hear one by one, and they blend into a tail.

Two later designs grew from this. A feedback delay network feeds several delay lines back into each other through a mixing matrix, so every echo spreads into all of them.

A convolution reverb skips the model. It convolves the dry sound with a room’s impulse response, measured or synthetic, as in “Naming what you already built” of Discrete convolution (5.2). A response of 1.5 s at 16 kHz is 24 000 samples long, so these reverbs use Fast convolution (14.3).

Worked example

1. Notches of a 5 ms delay. At 16 kHz, 5 ms is $D=80$ samples. The first notch is at $16\,000/160=100.0$ Hz, and then they come every $16\,000/80=200.0$ Hz. Below 8 kHz there are 40 of them.

2. The static curve. A note at −6 dBFS is 14 dB over the threshold of −20 dBFS. At 4:1 it comes out at $-20+14/4=-16.5$ dBFS. The compressor’s gain is $-14\times(1-1/4)=-10.5$ dB.

3. An attack constant. An attack of 1 ms at 16 kHz gives $T_s/\tau=0.0625$, so $c=e^{-0.0625}=0.9394$.

4. A reverb comb’s gain. For $D=475$ samples and $T_{60}=1.5$ s at 16 kHz, $3D/(T_{60}f_s)=1425/24\,000=0.0594$, so $g=10^{-0.0594}=0.872$. Each trip round the loop loses 1.19 dB, and in 1.5 s the echo makes 50.5 trips: 60 dB in all.

Where you’ll meet this

Digital audio workstations offer all of these effects: delays, flangers, choruses, phasers, reverbs and compressors. A guitar pedal usually holds one or two of them. Equalisers, the other everyday effect, are the biquads of Audio equalisers and biquads (20.5).

Broadcasters compress and limit their programmes to keep the loudness even. Hearing aids use fast compressors to fit loud and quiet sounds into a smaller range of hearing. Many modern reverbs use a feedback delay network in place of Schroeder’s four combs.

A loud sound can hide a quieter one close to it in frequency, and audio coders use that. Perceptual audio coding (29.3) takes it up.

I left out the exact shapes of the slow sweeps in flangers and choruses, look-ahead limiting, compressors that split the sound into bands, and loudness units (LUFS). For more, see U. Zölzer (ed.), DAFX (2nd ed., 2011), chapters 2, 4 and 5; J. D. Reiss and A. McPherson, Audio Effects (2015); and M. R. Schroeder, “Natural sounding artificial reverberation” (J. Audio Eng. Soc., 1962).

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