574 Practical MATLAB® Applications for Engineers
H(jw)a
1
1
2 12
w
wp
n
/
Table 6.3 provides the transfer function H(jw) for the case of LP normalized fi lters
of the order n = 1, 2, 3, 4, and 5.
The denominators of H(jw) are commonly referred to as the Butterworth polyno-
mial of order n. Figure 6.13 illustrates the magnitude response of the normalized
Butterworth LPFs for the following orders n = 1, 2, 3, 4 and 10.
Observe that as the order increases (n), the pass-band region becomes fl atter
and the transition region narrows.
Observe also that the maximum fl atness occurs at low frequencies (the limiting
case is w = 0).
TABLE 6.3
Butterworth Polynomials
nH(jw)
1[jw + 1]−^1
2 [( jw)^2 + 1.41( jw) + 1]−^1
3 [( jw)^3 + 2( jw)^2 + 2( jw) + 1]−^1
4 [( jw)^4 + 2.613( jw)^3 + 3.414( jw)^2 + 2.613( jw) + 1]−^1
5 [( jw)^5 + 3.236( jw)^4 + 5.236( jw)^3 + 5.2361( jw)^2 + 3.2361( jw) + 1]−^1
FIGURE 6.13
Magnitude plots of normalized Butterworth LPFs of orders n = 1, 2, 3, 4, and 10.
0 1 2 3 4 5 6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Normalized frequency
Gain
Butterworth LPF/order n = 1, 2, 3, 4, 10
n = 3
n = 1
n = 10
n = 4
n = 2