In the text, we showed that if h[n] is absolutely summable i.e., if...

In the text, we showed that if h[n] is absolutely summable i.e., if

then the LTI system with impulse response h[n] is stable. This means that absolute summability is a sufficient condition for stability. In this problem, we shall show that it is also a necessary condition. Consider an LTI system with impulse response h[n] that is not absolutely summable; that is,

(a) Suppose that the input to this system is

Does this input signal represent a bounded input? If so, what is the smallest number B such that

(b) Calculate the output at n = 0 for this particular choice of input. Does the result prove the contention that absolute summability is a necessary condition for stability?

(c) In a similar fashion, show that a continuous-time LTI system is stable if and only if its impulse response is absolutely integrable.