Page 392 - Analog and Digital Filter Design

Digital FIR Filter Design 389

7 Harris-Nutall Window
Nutall improved the BIackman-Harris coefficients to produce greater stoband
attenuation. The three-term series produced a first side-lobe attenuation of 67 d
The four-term series produced a magnificent 94dB first side-lobe attenuation.

(a) Three-term Harris-Nutall coefficients:
2n. 212
2n
bin) = o.42323+o.49755cos[-j+o.o7922cos[~].
i?
N
where n = -(N - 1)/2>. . . , - 1,0,1.. . . , (N - 1)/2~
When time-shifted, this equation becomes:
2n.n
o.49755cos[--]+o.07922cos -
2n. 2n 7
h(n) = 0.42323 - [ A' ' J
where n = 0,1, . . , (N - 1)/2.
~
(b) Four-term Harris- Nutall coefficients:

/7(n) = 0.35875 +0.48829cos[7]+0. 1412t3cos[T] 2n.2n
2n.n
+ 0.0 1168cos[ 2n. 3n

where n= -(N- 1)/2, I.. -1,O, 1,. . ~, (N- 1)/2.
When time-shifted, the second and fourth terms change sign:

2n.
h(n) = 0.35875 -0.48829cos[---]+o. 2n.n 14128co~[~]
2n
N
- 0.01 168cos[T] 2n.3n
where n = 0,1, . . . , (N - 1)/2.

8 Kaiser-Bessel Window
This window is generally known as a Kaiser window. It is not a hed window;
instead, a formula is given in which a factor a can be varied to give different
levels of stopband attenuation. The factor a should be between 0 and 4.

Hi@) and
The value of equation constants for h(n) are n(0) = -
C
sinh(nd252)
n(m) = 2H1(m', where 112 = 1,2,3, and H,(m) =
c nJiiT2
~
c = H(0) + 2. H(l) + 2 .H(2) + 2 .H(3).