Microphone Feedback: When the Echo Series Diverges

A sound picked up by a mic comes out of the speaker, gets picked up again, comes out again, and so on. With loop gain x each round, the total is 1 + x + x² + ⋯ = 1/(1 − x), but only while |x| < 1. At x = 1 the series diverges, and you hear the howl.

How it works

Power series as functions

In one sentence

A power series $\sum c_n(x - a)^n$ defines a function on its interval of convergence; $\frac{1}{1-x} = \sum_{n=0}^\infty x^n$ for $|x| < 1$ , and substituting, multiplying or differentiating it gives series for related functions.

General formula

\frac{1}{1-{\htmlData{sym=0}{x}}} = \sum_{{\htmlData{sym=1}{n}}=0}^{\infty} {\htmlData{sym=0}{x}}^{\htmlData{sym=1}{n}} = 1 + {\htmlData{sym=0}{x}} + {\htmlData{sym=0}{x}}^2 + \cdots, \quad |{\htmlData{sym=0}{x}}| < 1, \qquad {\htmlData{sym=2}{G}} = \frac{\htmlData{sym=3}{{A}}}{1 - {\htmlData{sym=3}{A}}{\htmlData{sym=4}{\beta}}}

Each echo is the last one times $x$ , so the total is a geometric series. It only makes sense while echoes fade, $|x| < 1$ .

What the symbols mean

$x$
Loop gain
How much a sound is amplified on each trip around the loop.
$n$
Echo number
How many trips around the loop.
$G$
Total gain
Output divided by input, counting every echo.
$A$
Amplifier gain
How much the amplifier boosts the signal.
$\beta$
Feedback share
Fraction of the output that gets back into the mic.

Each trip around the loop multiplies the sound by the loop gain $x = A\beta$ (amplifier gain $A$ times the share $\beta$ fed back).
The total output is the original plus every echo: $1 + x + x^2 + \cdots$ times the input.
That geometric series equals the function $\frac{1}{1-x}$ for $|x| < 1$ : a power series representing a function on its interval of convergence.
As $x \to 1$ the gain $\frac{1}{1-x}$ blows up; past it, every echo is louder than the last.

Learn the math

Power Series and FunctionsCalc II · Lesson 38, Power Series and Taylor Series