Answer:
2
Step-by-step explanation:
The system’s impulse response can be found by considering the output when the input is an impulse function. An impulse function is a sequence that is 1 at time index 0 and 0 elsewhere. Let’s denote the impulse response as (h(k)).
Given the input (u(k) = 3^k), we have:
[ y(k) = \sum_{n=-\infty}^{\infty} h(n)u(k-n) ]
For an impulse input, (u(k) = \delta(k)), where (\delta(k)) is the impulse function. Therefore:
[ y(k) = h(k) ]
Since (y(k) = 2^k), we have:
[ h(k) = 2^k ]
Now let’s find (g(1)):
[ g(1) = \sum_{n=-\infty}^{\infty} h(n)u(1-n) = h(1)u(1) = h(1) ]
Given that (g(0) = 2), we can find (h(1)):
[ h(1) = g(1) = 2 ]
Therefore, the correct answer is 2.
