Signals and Systems - Electrical Engineering

648 C H A P T E R 11: Introduction to the Design of Discrete Filters

11.3 FILTER SPECIFICATIONS

There are two ways to specify a discrete filter: in the frequency domain and in the time domain. The

frequency-domain specification of the desired filter’s magnitude and phase is more common in IIR

filter design, while the time-domain specification in terms of the desired impulse response of the

filter is more common in FIR filter design.

11.3.1 Frequency-Domain Specifications

Conventionally, when designing IIR filters a prototype low-pass filter is obtained first and then con-

verted into the desired filter. The magnitude specifications of a discrete low-pass filter are given for

frequencies [0,π] due to the periodicity and the even characteristics of the magnitude. Typically, the

phase is not specified but it is expected to be approximately linear.

For a low-pass filter, the desired magnitude|Hd(ejw)|is to be close to unity in a passband frequency

region, and close to zero in a stopband frequency region. A transition frequency region where the

filter is not specified is needed. Thus, the magnitude specifications are displayed in Figure 11.6(a).

The passband [0,ωp] is the band of frequencies for which the attenuation specification is the smallest;

the stopband [ωst,π] is the band of frequencies where the attenuation specification is the greatest; and

the transition band(ωp,ωst)is the frequency band where the filter is not specified. The frequencies

ωpandωstare called the passband and the stopband frequencies, respectively.

Loss Function

As for analog filters, the linear, or normal, scale specifications shown in Figure 11.6(a) do not give

the sense of the attenuation, and thus the loss or log specification in decibels (dB) is preferred. The

logarithmic scale also provides a greater resolution of the magnitude. Figure 11.7 shows the relation

between a magnitude value|G|and its corresponding loss value in decibels. Notice that the loss

increases by 20 dB whenever the filter attenuates the input signal by a factor of 10−^1.

The magnitude specifications of a discrete low-pass filter in a linear scale are (Figure 11.6(a)):

Passband: δ 1 ≤|H(ejω)|≤ 1 0 ≤ω≤ωp

Stopband: 0 <|H(ejω)|≤δ 2 ωst≤ω≤π (11.11)

FIGURE 11.6

Low-pass magnitude

specifications: (a)

normal scale and (b)

logarithmic scale or loss.

|H(ejω)|

1

π ω

(a)

δ 1

δ 2

ωp ωst

ω

α(ejω)(dB)

αmin

αmax

0

(b)

ωp ωst π