Question

A discrete-time LTI system is described by the following difference equation

y(n) = x(n + 2) – x(n + 1) + 2x(n) + x(n – 2)
where the system output y(n) is related to the input x(n).
What is the impulse response h(n) of the system?

Answer 1

The impulse response of the given discrete-time LTI system is equal to the difference equation: y(n) = x(n + 2) – x(n + 1) + 2x(n) + x(n – 2).

Step-by-step explanation:

A discrete-time LTI system can be described by its impulse response, which represents the system's output when an impulse is applied at the input. In this case, we can determine the impulse response of the system by setting the input to be an impulse, which is represented by $\delta(n)$.

Substituting $x(n) = \delta(n)$ into the difference equation, we obtain:

$$y(n) = \delta(n + 2) - \delta(n + 1) + 2\delta(n) + \delta(n - 2)$$

Therefore, the impulse response of the system, denoted as $h(n)$, is:

$$h(n) = \delta(n + 2) - \delta(n + 1) + 2\delta(n) + \delta(n - 2)$$