Audio Engineering

800 Chapter 27

Discrete convolution is a process that provides a single output sequence from two

input sequences. In the example given earlier, a time-domain sequence—the step

function—was convolved with the fi lter response yielding a fi ltered output sequence.

In textbooks, convolution is often denoted by the character ‘ * ’. So if we call the input

sequenceh ( k ) and the input sequence x ( k ), the fi ltered output would be defi ned as

yn()hk()* ()xk

Impulse Response

A very special result is obtained if a unique input sequence is convolved with the

fi lter coeffi cients. This special result is known as the fi lter’s impulse response, and

the derivation and design of different impulse responses are central to digital fi lter

theory. The special input sequence used to discover a fi lter’s impulse response is

known as the “ impulse input. ” (The fi lter’s impulse response being its response to

this impulse input.) This input sequence is defi ned to be always zero, except for one

single sample, which takes the value 1 (i.e., the full-scale value). We might defi ne,

for practical purposes, a series of samples like this

0, 0, 0, 0, 0, 1, 0, 0, 0, 0

Now imagine these samples being latched through the three-stage digital fi lter

shown earlier. The output sequence will be

0, 0, 0, 0, 0, 1/4, 1/2, 1/4, 0, 0, 0, 0

Obviously the zeros don’t really matter, what’s important is the central section:

1/4, 1/2, 1/4. This pattern is the fi lter’s impulse response.

FIR and IIR Digital Filters

Note that the impulse response of the aforementioned fi lter is fi nite: in fact, it only

has three terms. So important is the impulse response in fi lter theory that this type

of fi lter is actually defi ned by this characteristic of its behavior and is named a fi nite

impulse response (FIR) fi lter. Importantly, note that the impulse response of an FIR

fi lter is identical to its coeffi cients.

Now look at the digital fi lter in Figure F27.1. This derives its result from both the

incoming sequence and from a sequence that is fed back from the output. Now if

we perform a similar thought experiment to the convolution example given earlier

and imagine the resulting impulse-response from a fi lter of this type, it results in