Tutorials - Digital Signal Processing: An Introduction with MATLAB and Applications - page 277

Tutorial 42

Q: Find the values of b for which the system shown below is stable, knowing that
b is real.

x ( n )

y ( n )

z −1

z −1

b

Solution: We first define intermediate signals r(n), g(n) as shown below.

r ( n )

x ( n )

y ( n )

z −1

g ( n )

z −1

b

We write the system equations as follows:

y ð n Þ¼ r ð n Þþ g ð n Þ ð 1 Þ

g ð n Þ¼ y ð n 1 Þ ð 2 Þ

r ð n Þ¼ x ð n Þ by ð n 2 Þ ð 3 Þ

From (1), (2), and (3) we get:

y ð n Þ¼ x ð n Þþ y ð n 1 Þ by ð n 2 Þ ð 4 Þ

Taking the ZT of both sides of (4) and re-arranging terms we get:

z 2
z 2 z þ b

Y ð z Þ½ 1 z 1 þ bz 2 ¼ X ð z Þ) H ð z Þ¼ Y ð z Þ

1
1 z 1 þ bz 2 ¼

X ð z Þ ¼

The system has two poles at p 1 ; 2 ¼ 1

p
2

1 4b

:

Since b is real, then

p

1 4b is either real or pure imaginary, but not complex.

If 1 [ 4b (i.e.,

p \1 ) 3\

Case 1:

1 4b is real positive), then z p

) 0\b

\1 : þ

p

p

1 4b

1 4b \1 ! 0\b

3\

p

(intersection of

1 4b ! 2\b

b [ -2 and b [ 0.)