Continuous-time filters - Continuous and Discrete Time Signals and Systems - page 331

Table 7.2. Denominator D ( s ) for transfer function H ( s ) of the Butterworth filter

N D ( s )

( s + 1)

( s 2 + 1 . 414 s + 1)

( s + 1)( s 2 + s + 1)

( s 2 + 0 . 7654 s + 1)( s 2 + 1 . 8478 s + 1)

( s + 1)( s 2 + 0 . 6810 s + 1)( s 2 + 1 . 6810 s + 1)

( s 2 + 0 . 5176 s + 1)( s 2 + 1 . 4142 s + 1)( s 2 + 1 . 9319 s + 1)

( s + 1)( s 2 + 0 . 4450 s + 1)( s 2 + 1 . 2470 s + 1)( s 2 + 1 . 8019 s + 1)

( s 2 + 0 . 3902 s + 1)( s 2 + 1 . 1111 s + 1)( s 2 + 1 . 6629 s + 1)( s 2 + 1 . 9616 s + 1)

( s + 1)( s 2 + 0 . 3473 s + 1)( s 2 + s + 1)( s 2 + 1 . 5321 s + 1)( s 2 + 1 . 8794 s + 1)

10 ( s 2 + 0 . 3129 s + 1)( s 2 + 0 . 9080 s + 1)( s 2 + 1 . 4142 s + 1)( s 2 + 1 . 7820 s + 1)( s 2 + 1 . 9754 s + 1)

Butterworth filter are all real-valued. In general, Eq. (7.21) can be simplified as
follows:

1
D ( s )

1

H ( s ) = = (7.22)

s N + a N − 1 s N − 1 ++ a 1 s + 1

and represents the transfer function of the normalized Butterworth filter of
order N .
Repeating Example 7.4 for different orders (1 ≤ N ≤ 10), the transfer func-
tions H ( s ) of the resulting normalized Butterworth filters can be similarly com-
puted. Since the numerator of the transfer function is always unity, Table 7.2
lists the polynomials for the denominator D ( s ) for 1 ≤ N ≤ 10.

7.3.1.1 Design steps for the lowpass Butterworth filter

In this section, we will design a Butterworth lowpass filter based on the spec-
ifications illustrated in Fig. 7.3(a). Mathematically, the specifications can be
expressed as follows:

pass band (0 ≤ω≤ω p radians/s) 1 − δ p ≤ H ( ω ) ≤ 1 + δ p ; (7.23)

stop band ( ω >ω s radians/s) H ( ω ) ≤δ s . (7.24)

At times, Eq. (7.23) is also expressed in terms of the pass-band ripple as
20 log 10 δ p dB. Similarly, Eq. (7.24) is expressed in terms of the stop-band
ripple as 20 log 10 δ s dB. The design of the Butterworth filter consists of the
following steps, which we refer to as Algorithm 7.3.1.1.

Step 1 Determine the order N of the Butterworth filter. To determine the order
N of the filter, we calculate the gain of the filter at the corner frequencies ω = ω p