Analog Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 55

where:

Z ¼ Impedance Scaling Factor (ISF) ¼ R L
W ¼ Frequency Scaling Factor (FSF) ¼ x o ;

For LP and HP

x b ; For BP and BS

Note: To denormalize the transfer function H(s LN ) of a normalized LPF, one
simply uses the transformation s ? s/x c (x c being the required LPF cutoff fre-
quency) to get the true transfer function H(s). There is no need within this process
to find the true transfer function of the HP, BP, and BS filters—if the above
transformations from LPF are used withs LN ? s, the denormalization is included
implicitly.

1.5.5.3 Hardware Filter Design Rules Using Normalized LPF Standard
Circuits

• HPF Design:

1. Transform LP N ? HP N circuit components (Fig. 1.36 ).
2. Denormalize HP N ? HP (using ISF, FSF).
• BPF Design:

1. Denormalize LP N ? LP (using ISF, FSF).
2. Transform LP ? BP circuit components (Fig. 1.36 ).

• BSF Design:

1. Transform LP N ? HP N circuit components (Fig. 1.36 ).
2. Denormalize HP N ? HP(using ISF, FSF).
3. Transform HP ? BS circuit components (Fig. 1.36 ).

1.5.5.4 Example of a High-pass Filter Design

Example 2 Design a Butterworth HPF with G m = 1, f c = 1 MHz, and stopband
gain B 20 dB for f B 500 kHz. True load resistance is R L ¼ 300X.
Solution:
x c = 2pM rad/sec; x c ¼ 2p 2 ¼ p rad/sec. From LP-HP transformations obtain
the normalized LP frequency that corresponds to the HP frequency x 1 , which is
x L = x c /x 1 = 2.
From Butterworth curves in Tables check the order n that gives stop-band gain
B-20 dB for x LN C 2 [note that ''B'' for HPF is now ''C'' for LPF]. This yields
n = 4. This same result can also be reached mathematically as follows: