Finite Impulse Response Filter Design - Digital Signal Processing: Fundamentals and Applications - page 279

Table 7.16 Extremal Points and Band Edges with Their Error Values for the
First Iteration.

U 0 p= 4 p= 2 p

E max 0.25 0.323 0.25 0.25

Table 7.17 Error Values at Extremal Frequencies and Band Edge

U 0 p=4 p=2 p

E max 0.287 0.287 0.213 0.287

U 2 ¼ p
The ideal magnitudes at these three extremal points are given in Figure 7.38 ( c), that is, 0.5, 1, 0. Now let us
examine the second iteration.
Second Iteration

Applying the alternation theorem at the new set of extremal points, we have
<
:

E ¼ 0:5 b 1 2b 0
E ¼ 1 b 1 1:4142b 0
E ¼ 0 b 1 þ 2b 0

Solving these three simultaneous equations leads to

b 1 ¼ 0:537; E ¼ 0:287; and Hðe jU Þ¼0:537 þ 0:25 cos U

b 0 ¼ 0:125;

The error values at the extremal points and band edge are listed in Table 7.17 and shown in Figure 7.38 ( d), where
the determined extremal points are marked by the symbol “o”.
Since the extremal points have the same maximum error value of 0.287, they are U 0 ¼ 0,U 1 ¼ p=4, and
U 2 ¼ p, which is unchanged. We stop the iteration and output the filter transfer function as
HðzÞ¼0:125 þ 0:537z 1 þ 0:125z 2

As shown in Figure 7.37 (d), we obtain the equiripples of error at the extemal points U 0 ¼ 0, U 1 ¼ p=4, and
U 2 ¼ p; their signs are alternating, and the maximum absolute error of 0.287 is obtained at each point. It takes
two iterations to determine the coefficients for this simplified example.

As we mentioned, the Parks-McClellan algorithm is one of the most popular filter design methods
in industry due to its flexibility and performance. However, there are two disadvantages. The filter
length has to be estimated by the empirical method. Once the frequency edges, magnitudes, and
weighting factors are specified, the Remez exchange algorithm cannot control the actual ripple
obtained from the design. We may often need to try a longer length of filter or different weight factors
to remedy situations where the ripple is unacceptable.