Page 390 - Analog and Digital Filter Design

Digital FIR Filter Design 387

4 Hamming Window
Modifying the Von Hann window improves the stopband attenuation, giving up
to 43 dB for the first side-lobe. At higher frequencies the attenuation increases
by 6dB per octave.

h(a) = 0.54 + 0.46~0~
where n = -(N - 1)/2, . . . - 1,0,1, . . . ~ (N - 1)!2.

Note: More accurate values for the constants in this equation are 0.54347826
and 0.45652174, to eight decimal places.

When time-shifted, so that the window edge begins at n = 0, the second half of
the equation changes sign. It is not necessary to increment n and N (as we did
with the Von Hann window) because the window edges are not zero-valued.

~(Fz) 0.54-0.46~0~ [2;n], - wheren=O,l,. . .,(N--1)/2.
=

5 Blackman and Exact Blackman Windows
The Blackman window requires an additional element in the cosine series:

2n.n
h(n) = 0.42+0.5 cosL]+0.08 cos[N], 2n. rt
LN
where n = -(N - 1)/2, . . . - 1,0,1, . . . (N - l)/2.
~
Using these values, the first side-lobe is attenuated by 59dB relative to the main
lobe. Higher frequencies are attenuated by 18 dB per octave.

When time-shifting the window, so that the coefficients start at n = 0, the central
part of the equation changes sign. Also the values of n and N are incremented
by 1 to prevent zero values of coefficients at the window edges and to produce
a window coefficient of 1 at the center tap:

where n = 1,0, . . . , (N - 1)/2.
The Exact Blackman uses the same basic formula as the Blackman window,
but with exact values (to eight decimal places) for the multiplying coefficients.
A coefficient value of 0.42659071 for the Exact Blackman response replaces
the first coefficient value of 0.42 used in the Blackman window. Similarly,
0.49656062 replaces the second coefficient value of 0.5 that was used in Black-
man window, and 0.07684867 replaces the last coefficient value of 0.08 that was