Signals and Systems - Electrical Engineering

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

The crossings of these lines with the filter loss function indicate that at the normalized frequency

of 0.6 the loss is 9 as desired, and that at the normalized frequency 0.47 the loss is less than 2 dB,

so that the normalized half-power frequency is about 0.5. n

nExample 11.8

In this example we consider designing a Butterworth low-pass discrete filter for processing an

analog signal. The filter specifications are:

fp=2250 Hz passband frequency

fst=2700 Hz stopband frequency

fs=9000 Hz sampling frequency

α 1 =−18 dB dc loss

α 2 =−15 dB loss in passband

α 3 =−9 dB loss in stopband

Solution

The specifications are not normalized (see Figure 11.14). Normalizing them, we have that:

α(ej^0 )=−18 dB

αmax=α 2 −α 1 =3 dB

αmin=α 3 −α 1 =9 dB

and

ωp=

2 πfhp

fs

=0.5π

ωst=

2 πfst

fs

=0.6π

Note thatωp=ωhpsince the difference in the losses at dc and atωpis 3 dB.

FIGURE 11.14

Loss specifications for a discrete low-pass

filter for processing an analog signal.

− 18

− 15

− 9

f(KHz)

2 .25 2. 74. 5

α(f)dB