Page 389 - Analog and Digital Filter Design

386 Analog and Digital Filter Design

power since the first and last filter taps are having no effect. In other words, the
filter length has effectively been reduced by a factor of 2. For example, a 21-tap
filter will only have 19 nonzero coefficients.

A better algorithm assumes that the window is 2 taps longer than the number
of taps actually available. The zero-valued coefficients are then placed outside
the array of tap multipliers; that is, they are not used. Thus all taps have non-
zero value multipliers and contribute to the filter.

h(n) = 1.0 - llZl/(N + 1)/2,
IZ = -(N - 1)/2, - (N - 3)/2, . . . - 1,0,1, . . . (N - 3)/2, (N - 1)/2.

Now, when n = -(N-1)/2 and (N-1)/2, the window coefficient is equal to:

h(n) = 1 .O - (N - 1)/(N + l), when n = -(N - 1)/2 and (N - 1)/2

In a 21-tap filter the ftnal tap’s coefficient is h(n) = 1 .O - 20/22 = 1/11. Thus, using
the revised equation, the coefficients reduce by 1/11 per tap on either side of the
zero time coefficient.

3 Von Hann (Raised Cosine) Window
The Von Hann window is sometimes known as the Raised Cosine window
because its values are calculated from a cosine raised to the power two. It is
derived from a simple expression:
[ : I
,
h(n) = cos2 - alternatively this is given as:
[
2;n],
h(n)=O.5+0.5~0~ -
where n = -(N - 1)/2, . . . - 1,0,1, . . . (N - 3)/2, (N - 1)/2,

When time-shifted, so that the window edges occur at n = 0 and iz = N - 1, the
second half of the equation changes sign:

h(n) =0.5-0.5~0~ > where n = 0,1, . . . (N - 3)/2, (N - 1)/2.

Notice that it is necessary to add 1 to the value of n in the numerator, so that
the end values of the window are not zero. The denominator also has to become
N + 1, so that h(n) = 1 when n = (N - 1)/2 (the zero time value).

With this window, the first side-lobe stopband attenuation is 35 dB, increasing
by 18 dB per octave at higher frequencies.