FIR filter design - Continuous and Discrete Time Signals and Systems - page 698

1 1

0.8 0.8

0.6

0.6

0.4 0.4

0.2 0.2

0 0

0 0.25 p 0.5 p 0.75 p p

0 0.25 p 0.5 p 0.75 p p

(a) (b)

and two of these alternations occur at the stop-band edge frequencies Ω s and
π . In other words, Fig. 15.18(a) satisfies the alternation theorem.
Figure 15.18(b) shows the frequency response of a type II FIR filter with
length N = 20. The degree L of cos( Ω ) in polynomial ε ( Ω ) is given by
L = (20 − 2) / 2 = 9. Based on the alternation theorem, there should be at least
L + 2 = 11 alternations in polynomial ε ( Ω ). In Fig. 15.18(b), we observe 12
alternations in H ( Ω ), which exceed the minimum required number of alterna-
tions. Therefore, Fig. 15.18(b) satisfies the alternation theorem.

Fig. 15.18. Magnitude spectrum
of lowpass FIR filters. (a) Type I
FIR filter of length N = 13.
(b) Type II FIR filter of length
N

= 20.

15.5.2 Parks-McClellan algorithm

In this section we present steps of the Parks-McClellan algorithm for designing
optimal filters. In this discussion, we will consider only type I filters. Algorithms
for other types of filters can be obtained in the same manner. To derive the
Parks-McClellan algorithm, the approximated error function in Eq. (15.49b) is
expressed as follows:

G ( Ω ) + ε ( Ω )
W ( Ω )

≈ H d ( Ω ) . (15.50)

For type I filters, we obtain G ( Ω ) from Table 14.2 as follows:

( N − 1) / 2

N − 1
2

N − 1
2

G ( Ω ) = h + 2 h − k cos( Ω k ) .

k = 1

Since we are interested in calculating ( N − 1) / 2 + 1, or L + 1, coefficients
of h [ k ]in G ( Ω ) and the value of the maximum error ε max , we pick L + 2