# Calculus on sound signals?

I was playing with the idea of performing integration on a sound signal, since I've never heard of where it'd be used.

Is it used somewhere?

Here's something:
http://pcfarina.eng.unipr.it/Differentiation-Integration.htm

• yeah, it's done. maybe a leaky integrator (which is a one-pole LPF) would be more common because integrating DC is a problem. i've used this at the front-end of a pitch detector. works great with floating-point. if fixed-point, you might wanna put some AGC in it. – robert bristow-johnson Feb 5 '16 at 0:33

So, a sound signal $s$, in the DSP sense, is but a sequence of real-valued digital numbers:

$$s = (1, 232, -12, 121, 0, 131, \dots)$$

Integrating that is nothing but adding these up, so the integrated signal $r$ would have the formula:

$$r[n] = \sum\limits_{i=0}^n s[n]$$

or, writing that recursively, is the signal that it was at the last sampling time, plus the next input sample:

$$r[n] = r[n-1] + s[n]\quad \text{ for }r>0\\ r[0] = 0$$

Now, that is what DSP people call a recursive system, or in this specific case, a single-pole IIR filter (IIR: infinite impulse response, because you send one pulse in, you see it in the output forever). It's transfer function $H(z)$ is

$$H(z) = \frac{1}{1+a\cdot z^{-1}}$$

Your $a=1$, here, because the last value of $r$ doesn't get multiplied with a dampening factor. You will find a lot of literature out there explaining the properties of IIR filters, but for all practical things, you could build a low-pass filter out of your integrator, if you use a factor $a<1$.