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Consider a discrete-time linear time-invariant (LTI) system S, where y[n] = S{x[n]}
Let
S{δ[n]} = { 1, n ∈ {0,1,2}
{ 0, otherwise
where δ[n] is the discrete-time unit impulse function. For an input signal x[n], the output y[n] is
- x[n] + x[n - 1] + x[n - 2]
- x[n - 1] + x[n] + x[n + 1]
- x[n] + x[n + 1] + x[n + 2]
- x[n + 1] + x[n + 2] + x[n + 3]
Correct answer: x[n] + x[n - 1] + x[n - 2]
Solution
The correct option is right because the output of the LTI system for the input signal x[n] is determined by the system's response to the impulse function, which indicates that the output is a weighted sum of the current and previous input values, specifically x[n], x[n-1], and x[n-2]. This aligns with the system's impulse response.
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