Page 403 - Analog and Digital Filter Design

400 Analog and Digital Filter Design

The first term in the denominator is required to be 1.0 instead of 3.4142, so all
terms in the equation must be divided by 3.4142. Also the last term in the
denominator should be subtracted, so B2 must be a negative value. Carrying
out these changes gives:

0.292894 + 0.5857887. z-I + 0.292894. z-'
H(z) =
1 - (-0.585786. z-')

Now this means that the coefficient values are A0 = 0.292894, A1 = 0.5857887,
A2 = 0.292894, B1 = 0, and B2 = -0.585786.

Pre-Warping

Unfortunately, the simple bilinear transform approach is an approximation and
will not produce the exact frequency response required. If an analog S-plane
transfer function is converted into a Z-plane transfer function, as previously
shown, the frequency response will be distorted. The relationship between
analog and digital responses is given by:

If the analog frequency response is distorted prior to applying the bilinear trans-
form, the desired final response can be obtained. This distortion is called
pre-warping. To pre-warp an analog response, the following equation should
be used:

The desired filter cutoff frequency oc should be used to give a new analog cutoff
frequency w~,nLflog. This should be used in the S-plane transfer function before
applying the bilinear transform. Thus the cutoff frequency of the normalized
lowpass response will be slightly modified. The term wc represents the normal-
ized frequency of 2n(Fc/Fs).

Denormalization

Suppose that the desired response is a cutoff frequency of 3.4kHz and the sam-
pling clock is 8 kHz. Then mc = 3.418 = 0.425. When pre-warped this becomes
tan(2n 0.215757) = tan(1.335176878) = 4.1652998. In the analog transfer
function, s can be replaced by d4.1652998 (= 0.2400788s) before the bilinear