Finite Impulse Response Filter Design Equations by the Frequency Sampling Design Method - Digital Signal Processing: Fundamentals and Applications - page 817

Appendix E: Finite Impulse Response Filter
Design Equations by the Frequency
Sampling Design Method

Recall in Section 7.5 in Chapter 7 on the “Frequency Sampling Design Method” that we obtained

N 1
k ¼ 0 HðkÞW kn

1
N

hðnÞ¼ (E.1)

N

where hðnÞ ,0 n N 1, is the causal impulse response that approximates the finite impulse
response (FIR) filter, HðkÞ ,0 k N 1, represents the corresponding coefficients of the discrete
Fourier transform (DFT), and W N ¼ e j

2 p
N . We further write the DFT coefficients, HðkÞ ,0 k

N 1, in polar form:

HðkÞ¼H k e j 4 k ; 0 k N 1 (E.2)

where H k and 4 k are the k th magnitude and the phase angle, respectively. The frequency response of
the FIR filter is expressed as

N 1
n¼ 0 hðnÞe jn U

Hðe j U Þ¼

(E.3)

Substituting (E.1) into (E.3) yields

N 1 N 1

k ¼ 0 HðkÞW k N e j U n

1
N

Hðe j U Þ¼ (E.4)

n¼ 0

Interchanging the order of the summation in Equation (E.4) leads to

N 1
k ¼ 0 HðkÞ

N 1
n¼ 0 ðW N e j U Þ n

1
N

Hðe j U Þ¼

(E.5)

Since W N e j U ¼ðe j 2 p =N Þ k e j U ¼ e ðj U 2 p k=NÞ and using the identity P N 1

n¼ 0 r n ¼ 1 þ r þ

1 r N
1 r

þ / þ r N 1

2

r ¼ , we can write the second summation in Equation (E.5) as