Audio Engineering

Representation of Audio Signals 473



Delay

X(n) Y(n)

a 2

a 1

(a)

Figure 15.17(a) : The equivalent of the analogue fi rst-order high- and low-pass fi lters

requires a single delay element. Multipliers are used to scale the input (or output) values so

that they lie within the linear range of the hardware. Digital fi lter characteristics are quite

sensitive to the values of the coeffi cients used in the multipliers. The output sequence can

be described as YaXaXnnn 121 . If 0 > a 2 >  1 the structure behaves as a fi rst-order

lag. Ifa 2 > 0 than the structure produces an integrator. The output can usually be kept in

range by ensuring that a 1  1  a 2.

Single

sample

delay





X(n) Y(n)

a 2

a 1 a 3
(b)
Figure 15.17(b) : The arrangement for achieving high-pass fi ltering and differentiation again
requires a single delay element. The output sequence is given by YaXaaX Xnn 2311 ().nn
The fi lter has no feedback path so it will always be stable. Note that a 1   1 and witha 2  0
anda 3  1 the structure behaves as a differentiator. These are simple examples of fi rst-order
structures and are not necessarily the most effi cient in terms of their use of multiplier or adder
resources. Although a second-order system would result if two fi rst-order structures were run in
tandem, full fl exibility of second-order IIR structures requires recursive structures. Perhaps the
most common of these emulates the analogue biquad (or twin integrator loop) fi lter.