Analog and Digital Filters 569

R.6.37 Clearly, the higher the order (n), the better the approximation. Higher-order fi lters

imply higher cost translated in terms of number of elements and fi lter complexity.

R.6.38 First- and second-order fi lters are often the starting prototypes in most designs, the

reason for which they are presented in the following.

First-order system transfer functions for LP and HP fi lters are as follows:

a. HLP(s) =

wo

__s + w

o

b. HHP(s) = __s +s w

o

Note that there are no fi rst-order BP or BR fi lters.

R.6.39 Second-order system transfer functions for LP, HP, BP, and BR fi lters are as

follows:

1. HLP(s) =

A 1

____

s^2 + B 1 s + w o^2

2. HHP(s) =

A 2 s^2

____

s^2 + B 2 s + w o^2

3. HBP(s) =

A 3 s

____

s^2 + B 3 s + w o^2

4. HBR(s) =

A 4 s + A 5

____

s^2 + B 4 s + w o^2

where the As, Bs, and wos are constants that control the transfer function range and

domain.

R.6.40 For purely practical reasons, the fi lters presented in this book are divided into

a. Analog

where the analog fi lters are subdivided into

i. Passive

ii. Active

b. Digital

whereas digital fi lters can be

i. FIR

ii. IIR

R.6.41 Passive analog electrical fi lters exclusively employ R, L, and C elements. Therefore,

its poles and zeros for simple RC or RL fi lters are always real, whereas complex

poles are produced when both inductors and capacitors are employed.

R.6.42 In addition to the passive elements (Rs, Ls, and Cs), active fi lters use active compo-

nents such as operational amplifi ers (OA). OA are electronic devices that use an

external power source to amplify and fi lter at the same time the input signal.

By using feedback amplifi cation, the effects of inductors can be emulated by

employing capacitors. It is appropriate to mention that capacitors are much cheaper

than inductors and follow more closely the theoretical conditions, creating in this

way effi cient and inexpensive fi lter implementations.

These fi lters are used in diverse applications such as equalizers, noise-reducing

devices, compression devices, and expansion and enhancement devices.

In all the devices, the assumption is that the signal being processed (fi lter) is either

an electrical voltage or a current carrying information such as sound, data, or video.

R.6.43 Once an OA is used in a fi lter, the fi lter is referred to as an active fi lter.