Finite Impulse Response Filter Design - Digital Signal Processing: Fundamentals and Applications - page 277

(a) (b)

1 0.4

0.2

0.5 0

will be selected
as an extremal
point

-0.2

0 -0.4

0 1 2 3 4 0 1 2 3 4

Normalized frequency Normalized frequency

(c) (d)

1 0.4

equiripples

0.8

0.2

0.6

0

0.4

-0.2

0.2

0 -0.4

0 1 2 3 4 0 1 2 3 4

Normalized frequency Normalized frequency

FIGURE 7.38

Determining the 3-tap FIR filter coefficients using the Remez algorithm in Example 7.17.

For simplicity, we set all the weight factors to 1, that is, WðUÞ¼1. Equation (7.32) is simplified to

EðUÞ¼Hðe jU ÞH d ðe jU Þ

Substituting z ¼ e jU into the transfer function HðzÞ gives

Hðe jU Þ¼b 0 þ b 1 e jU þ b 0 e j2U

After simplification using Euler's identity e jU þ e jU ¼ 2 cos U, the filter frequency response is given by

Hðe jU Þ¼e jU ðb 1 þ 2b 0 cos UÞ

Disregarding the linear phase shift term e jU for the time being, we have a Chebyshev real magnitude function
(there are a few other types as well):

Hðe jU Þ¼b 1 þ 2b 0 cos U

The alternation theorem (Ambardar, 1999; Porat, 1997) must be used. The alternation theorem states that given
a Chebyshev polynomial Hðe jU Þ to approximate the ideal magnitude response H d ðe jU Þ, we can find at least M þ 2
(where M ¼ 1 for our case) frequencies U 0 ,U 1 , . U Mþ1 , called the extremal frequencies, so that signs of the error
at the extremal frequencies alternate and the absolute error value at each extremal point reaches the maximum
absolute error, that is,