Signals and Systems - Electrical Engineering

6.5 Analog Filtering 403

basic filters are given by:

Low pass-low pass: S=

s

 0

Low pass-high pass: S=

 0

s

Low pass-band pass: S=

s^2 +^20

s BW

Low pass-band eliminating: S=

s BW

s^2 +^20

(6.56)

whereSis the normalized andsthe final variables, while 0 is a desired cut-off frequency andBWis
a desired bandwidth.

Remarks

n The low-pass to low-pass (LP-LP) and low-pass to high-pass (LP-HP) transformations are linear in the
numerator and denominator; thus the number of poles and zeros of the prototype low-pass filter is preserved.
On the other hand, the low-pass to band-pass (LP-BP) and low-pass to band-eliminating (LP-BE) trans-
formations are quadratic in either the numerator or the denominator, so that the number of poles/zeros
is doubled. Thus, to obtain a 2 Nth-order band-pass or band-eliminating filter the prototype low-pass filter
should be of order N. This is an important observation useful in the design of these filters with MATLAB.
n It is important to realize that only frequencies are transformed, and the magnitude of the prototype filter
is preserved. Frequency transformations will be useful also in the design of discrete filters, where these
transformations are obtained in a completely different way, as no reactance functions would be available
in that domain.

nExample 6.10

To illustrate how the above transformations can be used to convert a prototype low-pass filter

we use the following script. First a low-pass prototype filter is designed usingbutter, and then to

this filter we apply the lowpass to highpass transformation with 0 =40 (rad/sec) to obtain a

high-pass filter. Let then 0 =6.32 (rad/sec) andBW=10 (rad/sec) to obtain a band-pass and

a band-eliminating filters using the appropriate transformations. The following is the script used.

The magnitude responses are plotted withezplot. Figure 6.25 shows the results.

clear all; clf

syms w

N = 5; [b, a] = butter(N, 1, ’s’) % low-pass prototype

omega0 = 40;BW = 10; omega1=sqrt(omega0); % transformation parameters

% low-pass prototype

n = N:−1:0;

U = ( j∗w).ˆn; num = b∗conj(U’); den = a∗conj(U’);

H = num/den;

% low-pass to high-pass