Decibels and exponential decay

The decibel as a scale of ratios, geometric sequences, the time constant, the sixty-decibel time, and the per-trip gain that produces it.

September 5, 2026

6 minutes read

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Contents9

Two sounds that differ by the same ratio sound the same distance apart, however loud they are. A scale for loudness therefore counts ratios rather than differences, and the decibel is that scale. Decay is what one ratio does when it is applied again and again.

Assumed knowledge

Arithmetic and powers. The logarithm, which is the exponent a base has to be raised to. The page uses base 10, written $\log_{10}$ , and the natural logarithm to base $e$ , written $\ln$ .

The section on geometric sequences reads them as orbits, which iteration and orbits builds up. The last two sections use the echo loop of signals and samples .

Ground covered

Why loudness is measured on a scale of ratios. The decibel for amplitude and for power. Geometric sequences, and what they look like in decibels. Exponential decay and the time constant. The sixty-decibel time, and the per-trip gain that produces a wanted one. A smoother as a decay of the distance to its target.

A reader who can say what per-sample factor gives a 50-millisecond time constant at $48\,000$ samples per second can skip the page.

Ratios, not differences

Turning an amplifier from 1 to 2 is a large change. Turning it from 100 to 101 is a small one, although both add 1. The ear judges the ratio, which is 2 in the first case and 1.01 in the second. The useful scale is therefore the one on which equal ratios are equal steps. Logarithms give it, because $\log(ab) = \log a + \log b$ turns a ratio into a difference.

The decibel

For two amplitudes $a$ and $a_{\mathrm{ref}}$ the level of $a$ in decibels is

L = 20 \log_{10} \frac{a}{a_{\mathrm{ref}}} \tag{1}

and for two powers $P$ and $P_{\mathrm{ref}}$ , power being energy per second, it is

L = 10 \log_{10} \frac{P}{P_{\mathrm{ref}}}. \tag{2}

The two agree, because power goes as the square of amplitude and $10 \log_{10}(a^2) = 20 \log_{10} a$ . A level is always relative to something. In a digital signal the reference is full scale, an amplitude of 1, and the unit is then written dBFS.

amplitude ratio	decibels
$\times 2$	$+6.02$
$\times 1/\sqrt 2$	$-3.01$
$\times 1/2$	$-6.02$
$\times 1/10$	$-20$
$\times 1/1000$	$-60$
$\times 10^{-4}$	$-80$

Read the other way, a level $L$ is an amplitude ratio of $10^{L/20}$ . A trim control marked in decibels is applied by multiplying the signal by exactly that.

Geometric sequences

Take a number and multiply it by the same factor $g$ at every step.

x_k = g^k\, x_0 \tag{3}

At $0 < g < 1$ the sequence shrinks towards zero and never reaches it. On the decibel scale every step is the same size, $20 \log_{10} g$ decibels, so the level falls along a straight line. The number of steps needed to fall by a given ratio $\rho$ is

k = \frac{\log_{10} \rho}{\log_{10} g}. \tag{4}

At $g = 0.944$ and $\rho = 10^{-3}$ , which is sixty decibels, equation (4) gives $k = 120$ steps.

Equation (3) is the orbit of $x_0$ under the map $x \mapsto g\,x$ . That map has one fixed point at $0$ , and its derivative there is $g$ , so the fixed point attracts exactly when $|g| < 1$ .

Exponential decay

Now let the steps be $\Delta t$ seconds apart, so that step $k$ happens at time $t = k\,\Delta t$ . Equation (3) becomes

x(t) = x_0\, g^{t / \Delta t} = x_0\, e^{-t / \tau}, \qquad \tau = -\frac{\Delta t}{\ln g}. \tag{5}

The second form uses $e$ and its logarithm because they make the constant come out simply. That constant $\tau$ is the time constant, the time the quantity takes to fall to $1/e \approx 0.368$ of its value, a drop of $8.69$ decibels. After $3\tau$ the quantity is at $5\,\%$ , and after $5\tau$ at $0.7\,\%$ .

The formula also runs backwards. A process that should have a time constant of $\tau$ seconds, taking one step per sample at rate $f_s$ , needs the per-sample factor

g = e^{-1 / (\tau f_s)}. \tag{6}

A follower with a 50-millisecond attack at $48\,000$ samples per second uses $g = e^{-1/2400} = 0.99958$ .

The sixty-decibel time

Reverberation is measured by the time a sound takes to fall by sixty decibels, a thousandth of its amplitude, written $T_{60}$ . Take the echo loop of signals and samples . A signal goes round a delay of $D$ samples and is multiplied by $g$ each time round. One trip takes $D / f_s$ seconds. In $T_{60}$ seconds the signal makes $T_{60} f_s / D$ trips and is multiplied by $g^{T_{60} f_s / D}$ . Setting that equal to $10^{-3}$ and solving for $g$ gives

g = 10^{-3D / (T_{60}\, f_s)}. \tag{7}

For a delay of $961$ samples at $48\,000$ and a wanted $T_{60}$ of $2.4$ seconds, equation (7) gives $g = 10^{-2883/115200} = 0.944$ , which is the example under equation (4). A longer delay makes fewer trips in the same time and needs a smaller $g$ to fall as far. That is how several delay loops of different lengths are made to fade together. Each one is given the $g$ that equation (7) returns for its own $D$ .

When the factor $g$ depends on frequency, so does $T_{60}$ . A room whose walls absorb treble more than bass has a shorter $T_{60}$ at high frequencies. A reverb copies that by making $g$ a filter rather than a number. The page on one-pole filters, shelves and all-passes is where those filters are built.¹

Smoothers as decays

A control that jumps from one value to another clicks. The usual cure is to move the applied value a fixed fraction of the way towards the target on every step.

y[n] = y[n-1] + c\,\big(x - y[n-1]\big), \qquad 0 < c \le 1 \tag{8}

Subtract both sides from $x$ , and the distance still to go obeys

x - y[n] = (1 - c)\,\big(x - y[n-1]\big), \tag{9}

which is a geometric sequence with factor $1 - c$ . A smoother is therefore a decay of its own error, with a time constant of $-1/\ln(1-c)$ steps, close to $1/c$ for small $c$ . At $c = 0.35$ the error falls to $1\,\%$ in $11$ steps. At $c = 0.0005$ the time constant is $2000$ steps. Equation (8) with a signal in place of the constant $x$ is the one-pole low-pass filter of the next page. Smoothing a control and filtering a signal are the same arithmetic.