Page 391 - Analog and Digital Filter Design

388 Analog and Digital Filter Design

used by Blackman. Using the Exact Blackman coefficients given, the first side-
lobe is attenuated by 69 dB. However, higher-frequency side-lobes have an ampli-
tude above this level, so a stopband attenuation of about 65dB is achieved at
the seventh side-lobe. The attenuation then increases at higher frequencies.

Exact Blackman coefficients do not produce zero-valued coefficients at the
window edges. It is not necessary to add 1 to the value of n and N. The value
of the window edge coefficient is very small, though: h(0) = 0.00687876.

6 Blackman-Harris Window
Harris improved the stopband attenuation of the Blackman window by adjust-
ing the values slightly for the three-term cosine series. The first side-lobe
attenuation produced by the three-term series coefficients is 61 dB. At higher fre-
quencies the attenuation is greater.

Harris also produced a four-term series that gave even better stopband attenu-
ation. The four-term series has a first side-lobe attenuation of 74dB.

(a) Three-tern Blackman-Hams Coefficients:
2n 2n
2n
h(n) = 0.44959+0.49364cos[~]+O.O5677cos[-],
n
wheren=-(N-1)/2 ,..., -l,O,l,..., (N-l)/2.
When time-shifted this equation becomes:

4K 11
2n
n
h(n> = 0.44959-0.49364cos[~]+0.05677cos[-+],
where n = 0, 1, . . . , (N - 1)/2.
(b) Four-tern Blackmun-Hums Coefficients:

where n = -(N- 1)/2,. . . -l,O, 1,. . ., (N- 1)/2.

When time-shifted, this equation becomes:

n
o.49703cos[~]+o.09892cos[~]
2n.2n
2~
h(n) = 0.40217 -
-0.00183cos[~] 2~.
3n
wheren=O,l,..-, (N-l)/2.