Final answer:
The output signal y(n) is obtained by convolving the input x(n) with the LTI system's impulse response h(n). The impulse response is a uniform finite-duration impulse sequence, and the input is an exponentially decaying signal starting at n = 0.
Step-by-step explanation:
We have been given a discrete-time linear time-invariant (LTI) system with an impulse response h(n) and an input signal x(n). The impulse response is h(n)= 1/5[δ(n+2) +δ(n+1) +δ(n) +δ(n-1) +δ(n-2)] which appears to be a finite-duration uniform impulse sequence. We can find the output y(n) by convolving the input signal with the system's impulse response. The input signal is given by x(n) = (1/3)ⁿ u(n), which is an exponentially decaying sequence that starts at n = 0.
The convolution of two signals, in general, can be represented as:
y(n) = ∞
Σ x(k) • h(n - k)
However, because h(n) is nonzero only for n between -2 and 2, the sum reduces to a finite number of terms. Thus, we will sum the products of x(n) and the shifted impulses of h(n), which effectively scales and shifts x(n) according to the impulses in h(n).
Carrying out this convolution will give us the output signal y(n), which will be a sequence that combines the effects of the LTI system's impulse response and the input signal characteristics.