Page 408 - Analog and Digital Filter Design

IIR Filter Design 405

The firsl term in the denominator is required to be 1.0 instead of 24.2403455,
so all terms in the equation must be divided by 24.2403455. The Al term in the
numerator for the biquad is positive, so the coefficient A 1 will be negative. Also
the Bl and B2 terms in the denominator should both be negative. so coefkients
BI and B2 have negative values. Carrying out these changes gives:

0.041253537 + (-0.082507074). Z-' + 0.041253537.~-~
H(z) =
1 - (-1.348967736) - (-0.51 3981886. z-!)

Now this means that the coefficient values are A0 = 0.041253537, A1 =
-0.082507074, A2 = 0.041253537, B1 = -1.348967736, and B2 = -0.513981885.
This completes the design process using method 1 ~

Design Method 2
In the second highpass filter method, s is replaced by:
s = tan(wD/2). [;y

-

Now by substituting into the original Butterworth transfer function

1
N(s) =
S'+&S+l
Using mD = 2rr(3.4/8) = 2.670353756, the function tan(mD/2) = tan(1.335176878)
= 4.165299774. It is now possible to substitute for s:

1
H(z) =
+I
+
17.3497222 1. [3] 5.890623432. [+] + 1
7- Z-1
By comparing design method 1 and 2, it can be seen that the results using design
method 2 are identical (within calculator error limits). The only difference is that
the second method is a single step.

Bandpass Frequency Scaling
Bandpass frequency scaling is more complex than either Iowpass or highpass
transformation and scaling. Lowpass to bandpass transformation and frequency