Tutorials - Digital Signal Processing: An Introduction with MATLAB and Applications - page 271

q

1

is either real (if 4 k) or pure imaginary (but not complex) Since k is real,

4 k

q

1

(if 4 \k). If is imaginary, then both p 1 and p 2 will be in the left half of the

s-plane and the system is always stable.
Now assume

4 k

q

1

4 k

is real, i.e., k 4 . For stability we need the following

condition:

r

1
4 k

1

2 þ

\0 ð to keep p 1 \0 Þ:

1

4 k\ 1 4 ! k [ 0 :

Combining k 4 and k [ 0 gives 0\k 4 as the overall condition in this case.
Now we summarize the real and imaginary cases above:

)

1. Case 1 (the root is imaginary): k [ 4
2. Case 2 (the root is real): 0\k 4
The combination of the two cases gives the final condition k [ 0.

Tutorial 36

Q: Design a passive bandpass Butterworth filter with the following specifications:

1. 4th-order
2. Center frequency = 10 kHz
3. Bandwidth = 3 kHz
4. True load impedance = 600 X
5. Maximum gain = 1.

Solution:

x l ¼ x 0 x b x g ¼

p

2 ¼ 17kp;

x u x l ¼ 19 : 77kp

The scaling factors are: ISF = Z = 600 and FSF = F = x b = 6kp.
Since LP ? BP transformation is quadratic (Tables), we need a 2nd-order LPF as
the LP-prototype. Its transfer function is found from Tables as follows:

a o
b o þ b 1 s N þ s N

H ð s N Þ¼ b o ¼ 1 b 1 ¼ 1 : 41 :

where and

G d : c ¼ G m ¼ 1 ¼ a o
b o

! a o ¼ b o ¼ 1 :