
Consider a system with a pair of complex poles of H(z) on the Unit Circle at e
ju
. Its response to
the bounded input u
k
¼sin ku, k ¼0, 1, 2,3,...is obtained from
Y(z) ¼ H(z)U(z) ¼
N(z)
(z e
ju
)(z e
ju
)
sin u z
z
2
(2 cos u)z þ 1
(4:631)
¼
N(z)
(z e
ju
)(z e
ju
)
sin u z
(z e
ju
)(z e
ju
)
(4:632)
¼
sin u zN(z)
(z e
ju
)
2
(z e
ju
)
2
(4:633)
It is left as an exercise to show that y
k
contains a linear combination of the terms, cos ku,
sin ku, k cos ku, and k sin ku. Consequently, the response is unbounded, and the system is not
BIBO stable.
When a real pole of H(z) is located on the Unit Circle at z ¼1orz ¼þ1, and the input is
u
k
¼(1)
k
, k ¼0, 1, 2, 3,