Final answer:
The unit-impulse response of the system y(n) = 0.5x(n) - 0.5x(n - 2) is a sequence where y(0)=0.5, y(2)=-0.5, and y(n) is 0 for all other values of n.
Step-by-step explanation:
The question asks for the unit-impulse response of the system described by the equation y(n) = 0.5x(n) - 0.5x(n - 2). A unit-impulse response, also known as the system's impulse response, is a key concept in signal processing, which represents how a system reacts to an input that is a unit impulse at time zero.
To find the unit-impulse response, we consider an input sequence where x(n) is an impulse delta function, δ(n), which is equal to 1 at n=0 and 0 elsewhere. The response y(n) to this impulse input can be calculated step-by-step. Since x(-2) and x(-1) are given as 0, and for an impulse x(0)=1, the response y(n) will be:
- For n=0: y(0) = 0.5δ(0) - 0.5δ(0-2) = 0.5 * 1 - 0.5 * 0 = 0.5
- For n=1: y(1) = 0.5δ(1) - 0.5δ(1-2) = 0.5 * 0 - 0.5 * 0 = 0
- For n=2: y(2) = 0.5δ(2) - 0.5δ(2-2) = 0.5 * 0 - 0.5 * 1 = -0.5
- For n > 2: y(n) = 0.5δ(n) - 0.5δ(n-2) will always be 0 since δ(n) is 0 for n > 0
Therefore, the unit-impulse response of the system is y(n) = 0.5δ(n) - 0.5δ(n-2), which equals 0.5 at n=0, -0.5 at n=2, and 0 for all other values of n.