PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

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