Convolution
These are notes related to Convolution, corresponding to Chapter 6 in The Scientist and Engineer’s Guide to Digital Signal Processing.
Impulses
This chapter describes two important terms used in DSP. The $\delta[n]$, known as the delta function or the unit impulse. This is a signal that consists of a normalized impulse: a signal of zeroes, starting with a single 1.0 value.
The other term defined in this chapter is the impulse response, which is a more known term among producers and defines a signal that exists a system when a delta function is the input. This is commonly represented by $h[n]$, but this is of course not always the case, so you’ll see other representations if they make more sense.
Any impulse can be represented using a shifted and scaled delta function. The example the DSP guide gives is a signal $a[n]$, which is empty, except for a value of $-3$ at sample $8$. This impulse can then also be represented as $a[n] = -3\delta[n-8]$.
A system has properties of homogeneity and shift invariance, which means that, for every impulse, we can produce the same output by shifting and scaling by the same amounts.
Note that this isn’t necessarily true for every system, as there are asymmetrical systems (like some distortion methods).