Finite Impulse Response Filter Design - Digital Signal Processing: Fundamentals and Applications - page 269

c. Similar to the preceding example, Figure 7.32 shows the frequency responses. Focusing on the magnitude
frequency responses depicted in Figure 7.32 , the dash-dotted line indicates the magnitude frequency
response obtained without specifying the smooth transition band, while the solid line indicates the magnitude
frequency response achieved with the specification of the smooth transition band, resulting in the reduced
ripple effect.

Observations can be made from examining Examples 7.13 and 7.14. First, the oscillations (Gibbs
behavior) in the passband and stopband can be reduced at the expense of increasing the width of the
main lobe. Second, we can modify the specification of the magnitude frequency response with
a smooth transition band to reduce the oscillations and thus improve the performance of the FIR filter.
Third, the magnitude values H k , k ¼ 0 ; 1 ; / ; M in general can be arbitrarily specified. This indicates
that the frequency sampling method is more flexible and can be used to design the FIR filter with an
arbitrary specification of the magnitude frequency response.

7.6 OPTIMAL DESIGN METHOD
This section introduces the Parks-McClellan algorithm, which is one of the most popular optimal
design method used in the industry due to its efficiency and flexibility. The FIR filter design using the
Parks-McClellan algorithm is developed based on the idea of minimizing the maximum approxi-
mation error between a Chebyshev polynomial and the desired filter magnitude frequency response.
The details of this design development are beyond the scope of this text and can be found in Ambardar
(1999) and Porat (1997). We will outline the design criteria and notation and then focus on the design
procedure.
Given an ideal magnitude response H d ðe juT Þ , the approximation error EðuÞ is defined as
EðuÞ¼WðuÞ½Hðe juT ÞH d ðe juT Þ

(7.32)

where Hðe juT Þ is the frequency response of the linear phase FIR filter to be designed, and WðuÞ is the
weight function for emphasizing certain frequency bands over others during the optimization process.
The goal is to minimize the error over the set of FIR coefficients:

min ð max jEðuÞjÞ (7.33)

With the help of the Remez exchange algorithm, which is also beyond the scope of this topic, we can
obtain the best FIR filter whose magnitude response has an equiripple approximation to the ideal
magnitude response. The achieved filters are optimal in the sense that the algorithms minimize the
maximum error between the desired frequency response and actual frequency response. These are
often called minimax filters .
Next, we establish the notation that will be used in the design procedure. Figure 7.33 shows the
characteristics of the FIR filter designed by the Parks-McClellan and Remez exchange algorithms. As
illustrated in the top graph of Figure 7.33 , the passband frequency response and stopband frequency
response have equiripples.

d p is used to specify the magnitude ripple in the passband, while d s