PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

Analog and Digital Filters 577

The Chebyshev polynomial coeffi cients Cn(x) oscillate between –1 and 1, and can be

generated by using the recursion equation given as follows:

Cn+ 1 (x) = 2 Cn(x) − Cn− 1 (x)

with C 0 (x) = 1 and C 1 (x) = x.

Table 6.4 shows the Chebyshev polynomials for n = 1, 2, 3, and 4.

Figure 6.14 displays the magnitude responses of an analog Chebyshev type-1 LPF

for n = 1, 2, 3, 4, and 10.

R.6.61 The Chebyshev type-2 fi lter is the complement of the Chebyshev type-1 fi lter, in the

sense that the magnitude response is smooth in the pass band and presents equal

ripple in the stop band.

Unlike the type-1 fi lter, which consists only of poles, the type-2 fi lters consist of

both poles and zeros.

The magnitude square of the response is given by

Hjw

Cww

Cww

Cww

a

ns

nsp

ns

()

()

()

()

2 2

2

2

1

























where ∈ defi nes the maximum stop-band deviation, wp denotes the pass-band

cutoff frequency, ws denotes the stop-band cutoff frequency, and Cn(x) are the

Chebyshev polynomials (defi ned in Table 6.4).

Figure 6.15 shows the magnitude responses of an analog Chebyshev type-2 fi lters

for n = 1, 2, 3, 4, and 10.

R.6.62 Elliptic fi lters, also known as Cauer fi lters, present ripples in the pass as well as in

the stop-band regions. The main characteristic of the elliptic fi lter is that the transi-

tion band is the narrowest with respect to the other two fi lter types, the Butterworth

and Chebyshev fi lters.

The square of the magnitude of the transfer function of the Cauer fi lters is

given by

Hjw

a Jwhn

()

(,)

2

2 2

1

1



where ε is the pass-band deviation, Jn(w, h) is the n-order Jacobian, and h indicates

the ripple’s height (in the pass- and stop-band regions).

Figure 6.16 shows the magnitude response of an elliptic fi lter for n = 1, 2, 3, 4,

and 10.

R.6.63 Butterworth, Chebyshev (types 1 and 2), and elliptic fi lters deal basically with the

magnitude response, with little concern about their phase response.

TABLE 6.4

Chebyshev Polynomials

nCn(w)

1 w

22 w^2 − 1

34 w^3 − 3 w

48 w^4 − 8 w^2 + 1