State the condition for stability of LTI system and determine for the given discrete time system, h(n) = (0.3)^nu(n) + 5δ(n), is stable of not.

Condition for the stability of LTI system: LTI system is stable if its impulse response is absolutely summable i.e., finite.

∑k=−∞∞|h(k)|<∞

h(n) = (0.3)ⁿu(n) + 5δ(n)
∴
∑k=−∞∞  [ (0.3)ⁿu(n) + 5δ(n) ]
Since, u(n) = 1 n ≥0
                  = 0 n < 0
also, δ(n) = 1 n = 0
                = 0 otherwise
Therefore, limits of u(n) will be from 0 to ∞ and limits for δ(n) will be only 0.

∴∑n=0∞0.3n⋅u(n)+5|δ(n)|





Since, u(n) = 1, between 0 to ∞ and δ(n)_n=0 = 1
∴∑n=0∞0.3ⁿ + ∑n=0∞(5)
= 5 + 0.3⁰ + 0.3¹ + 0.3² +0.3³ + ...
= 5 + 1 + 0.3 + 0.3 + 0.3² + 0.3³ + ...
= 5 + 1 / (1 - 0.3)
= 5 + 1 / 0.7
= (3.5 + 1) / 0.7
= 4.5 / 0.7
= 45 / 7

∑k=−∞∞|h(k)| = 45 / 7 < ∞

Therefore, the system is stable.