Audio Engineering

802 Chapter 27

of IIR fi lters is complicated because the coeffi cients do not represent the impulse

response directly. Instead, IIR design involves various mathematical methods, which

are used to analyze and derive the appropriate impulse response from the limited

number of taps. This makes the design of IIR fi lters from fi rst principles rather

complicated and math heavy! Fortunately, FIRs are easier to understand, and a

brief description gives a good deal of insight into the design principles of all digital

fi lters.

We already noted that the response type of the 1/4, 1/2, 1/4 fi lter was a low-

pass; remember it “ slowed down ” the fast rising edge of the step waveform. If

we look at the general form of this impulse response, we will see that this is a

very rough approximation to the behavior of an ideal low-pass fi lter in relation

to reconstruction fi lters. There we saw that the (sin x )/ x function defi nes the

behavior of an ideal, low-pass fi lter and the derivation of this function is given in

Figure F27.2. Sometimes termed a sinc function, it has the characteristic that it is

infi nite, gradually decaying with ever smaller oscillations about zero. This illustrates

that the perfect low-pass FIR fi lter would require an infi nite response, an infi nite

number of taps and the signal would take an infi nitely long time to pass through it!

Fortunately for us, we do not need such perfection.

yx

ysinx

y(sinx)/x

sinx

0

pi/4

pi/2

3pi/4

pi

sincx

1

0.9

0.6

0.3

0

Figure F27.2 : Derivation of sin x / x function.