Convolution
Describes how the area of the overlap of two discrete sample sequences changes with
respect to time (or sample count for discrete inputs). We calculate this as an [see page 8, integral]
of the product of the two functions (or a summation if our input is continuous).
\[ h[n] * x[n] = \sum_{i}{h[i] \times x[n-i]} \]
Note: Take special note of the fact that we aren't plotting the product of two discrete points, were plotting the area of overlap of the two spectrums for each given offset i.
See [see page 8, here] for an example using a square wave and square filter which produces a triangular convolution spectrum. See [see page 9, here] for an example with a short impulse and rectangle signal. Take special note of the scale of the y coordinate.
[see page 4, Represented] by \(*\) operator. Or the \((*)\) operator which is sometimes equal, sometimes not (depends on linear or cyclic convolutions).
Time Domain vs. Frequency Domain
Convolution in time domain is a multiplication in frequency-domain. A multiplication in frequency domain is a convolution in time domain (depending on if the convolution is cyclic or linear).
Linear Time-Invariant Filters
linearity and time-invariance allows the input (and the response) of a filter to be constructed from a weighted sum of time-shifted impulses. Time invariance and super position therefore allows a signal to be [see page 4, expressed] as a convolution.