Signals and Systems - Electrical Engineering

11.2 Frequency-Selective Discrete Filters 641

FIGURE 11.2
Discrete filtering of analog signals using CDCs
and DCCs. The CDC is simply a sampler while
the DCC is a reconstruction filter.

CDC H(z) DCD

x(t) x(nTs) y(nTs) y(t)

Ts

11.2 Frequency-Selective Discrete Filters

The principle behind discrete filtering is easily understood by considering the response of an LTI

system to sinusoids. IfH(z)is the transfer function of a discrete-time LTI system, and

x[n]=

∑

k

Akcos(ωkn+φk) (11.1)

is the input of the system, according to the eigenfunction property of LTI systems the steady-state

response of the system is

yss[n]=

∑

k

Ak|H(ejωk)|cos(ωkn+φk+θ(ωk)) (11.2)

where|H(ejωk)|andθ(ωk)are the magnitude and the phase ofH(ejω), which is the frequency response

of the system, at the frequencyωk. The frequency response is the transfer function computed on

the unit circle (i.e.,H(ejω)=H(z)|z=ejω). It becomes clear from Equation (11.2) that by judiciously

choosing the frequency response of the LTI system we can select the frequency components of the

input we wish to have at the output, and attenuate or amplify their amplitudes or change their

phases.

In general, for an inputx[n] with Z-transformX(z), the Z-transform of the output of the filter is

Y(z)=H(z)X(z) (11.3)

or on the unit circle (whenz=ejω),

Y(ejω)=H(ejω)X(ejω). (11.4)

By selecting the frequency responseH(ejω)we allow some frequency components ofx[n] to appear

in the output, and others to be filtered out.

Ideal frequency-selective filters, such as low-pass, high-pass, band-pass, and stopband filters, cannot

be realized. They serve as prototypes for the actual filters.

11.2.1 Linear Phase

A filter changes the spectrum of its input in magnitude as well as in phase. Distortion in magnitude

can be avoided by using an all-pass filter with unit magnitude for all frequencies. Phase distortion can

be avoided by requiring the phase response of the filter to be linear. For instance, when transmitting

a voice signal in a communication system it is important that the signals at the transmitter and at