Solution 2                          

(a) True

Proof: For a system to be stable

;

If h is periodic with period P (say), then

(where )

is non-zero

Hence unstable.

(b) False

Counter example :

Let an LTI system

Or

Which is causal

For its inverse system, will be output and will be input.

i.e.

[Using shift invariance]

Which is non-causal.

(c) False

Counter example :

Let

Here

But

Which is not bounded.

Hence , the impulse response must be absolutely summable , for the system to be stable.

(d) False

Counter example :

Let

for

Which is of finite duration

But

diverges (i.e. it is not absolutely summable) .

Hence the system is unstable.

(e) False

Counter example:

Let

which is causal.

but unstable.

(f) False.

Counter example:

Let

which is non-causal

which is causal.

Cascade of the two system

which is clearly causal.

(g) False

Counter example:

Let

Here

but

which is divergent.

(h) True

Sufficiency:

if

Let n < 0

=>causality.

Necessity :

if

,

then

Thus s[n] = 0 for n < 0 .