Finite Impulse Response Filter Design Equations by the Frequency Sampling Design Method - Digital Signal Processing: Fundamentals and Applications - page 820

Again, note that Equation (E.13) is equivalent to

H k e j 4 k ¼ H Nk e j 4 Nk ; 1 k M (E.18)

Substituting (E.17) in (E.18) yields

H k e jðN 1 Þk p =N ¼ H Nk e jðN 1 ÞðNkÞ p =N ;

1 k M (E.19)

Simplification of Equation (E.19) leads to the following result:
H k ¼ H Nk e jðN 1 Þ p ¼ð 1 Þ N 1 H Nk ; 1 k M (E.20)

Since we constrain the filter length to be N ¼ 2 M þ 1, Equation (E.20) can be further reduced to

2 M

H k ¼ð 1 Þ H 2 Mþ 1 k ¼ H 2 Mþ 1 k ; 1 k M (E.21)

Finally, by substituting (E.21) and (E.17) into (E.16) , we obtain a very simple design equation:

(

)

k ¼ 1 H k cos 2 p kðn MÞ

H 0 þ 2 M

1
2 M þ 1

;

0 n 2 M

hðnÞ¼ (E.22)

2 M þ 1

Thus the design procedure is simply summarized as follows: Given the filter length, 2 M þ 1, and the

2 p k
ð 2 Mþ 1 Þ

specified frequency response, H k at U k ¼ for k ¼ 0 ; 1 ; / ; M , FIR filter coefficients can be

calculated via Equation (E.22) .