Audio Engineering

Representation of Audio Signals 475

values for the coeffi cients whose value will determine the characteristics of a fi lter. This
limited precision is effectively a processing error, which will make its presence known
through the addition of noise to the output. The limited precision also leads to the odd
effects for which there is no direct analogue equivalent, such as limit cycle oscillations.
The details concerning the structure of a digital fi lter have a very strong effect on the
sensitivity of the fi lter to noise and accuracy, in addition to the varying processing
resource requirement. The best structure thus depends a little on the processing task that
is required to be carried out. The skill of the engineer is, as ever, in balancing the factors
in order to optimize the necessary compromises.

While complicated fi ltering is undoubtedly used in digital audio signal processing, spare
a thought for the simple process of averaging. In digital signal processing terms this is
usually called interpolation ( Figure 15.18 ). The process is used to conceal unrecoverable
errors in a sequence of digital sample values and, for example, is used in the compact disc
for just this reason.

Hz Yz

Xz

aaz az

bbz bz

() ()

()









121 3 2

(^12132).
Figure 15.17(c): continued: The value of each of the coeffi cients can be determined from
knowledge of the rate at which samples are being made available, F (^) s, and your requirement
for the amount of cut or boost and of the Q required. One of the fi rst operations is that of
prewarping the value of the intended center frequency f (^) c in order to take account of the fact that
the intended equalizer center frequency is going to be comparable to the sampling frequency.
The “ warped ” frequency is given by fFws /ππtan fFcs/. And now for the coeffi cients:
afkFQfF
ab fF
afk
1
2
22
2
3
11
21
11



ππ
π
π
wsws
w s
w
() ()
()()
()FFQ f F
bfFQfF
bfFQfF
sws
ws ws
ws ws



π
ππ
ππ
()
()
()
2
1
2
3
2
1
(^1)
The mathematics concerned with fi lter design certainly appear more complicated than that
which is usually associated with analogue equalizer design. The complication does not stop
here though, as a designer must take into consideration the various compromises brought
on by limitations in cost and hardware performance.