Signals and Systems - Electrical Engineering

390 C H A P T E R 6: Application to Control and Communications

6.5 Analog Filtering

The basic idea of filtering is to get rid of frequency components of a signal that are not desirable.

Application of filtering can be found in control, in communications, and in signal processing. In this

section we provide a short introduction to the design of analog filters. Chapter 11 is dedicated to the

design of discrete filters and to some degree that chapter will be based on the material in this section.

According to the eigenfunction property of LTI systems (Figure 6.21) the steady-state response of an

LTI system to a sinusoidal input—with a certain magnitude, frequency, and phase—is a sinusoid of

the same frequency as the input, but with magnitude and phase affected by the response of the system

at the frequency of the input. Since periodic as well as aperiodic signals have Fourier representations

consisting of sinusoids of different frequencies, the frequency components of any signal can be mod-

ified by appropriately choosing the frequency response of the LTI system, or filter. Filtering can thus

be seen as a way of changing the frequency content of an input signal.

The appropriate filter for a certain application is specified using the spectral characterization of the

input and the desired spectral characteristics of the output. Once the specifications of the filter are set,

the problem becomes one of approximation as a ratio of polynomials ins. The classical approach in

filter design is to consider low-pass prototypes, with normalized frequency and magnitude responses,

which may be transformed into other filters with the desired frequency response. Thus, a great deal

of effort is put into designing low-pass filters and into developing frequency transformations to map

low-pass filters into other types of filters. Using cascade and parallel connections of filters also provide

a way to obtain different types of filters.

The resulting filter should be causal, stable, and have real-valued coefficients so that it can be used in

real-time applications and realized as a passive or an active filter. Resistors, capacitors, and inductors

are used in the realization of passive filters, while resistors, capacitors, and operational amplifiers are

used in active filter realizations.

6.5.1 Filtering Basics

A filterH(s)=B(s)/A(s)is an LTI system having a specific frequency response. The convolution

property of the Fourier transform gives that

Y()=X()H(j) (6.28)

where

H(j)=H(s)|s=j

Thus, the frequency content of the input, represented by the Fourier transformX(), is changed by

the frequency responseH(j)of the filter so that the output signal with spectrumY()only has

desirable frequency components.

FIGURE 6.21

Eigenfunction property of continuous LTI systems.

LTI system

H(s)

Aej(Ω^0 t+θ) A|H(jΩ 0 )|ej(Ω^0 t+θ+∠H(jΩ^0 )