The following problems relate to the response of LTI discrete-time systems.

(a) The unit-step response of a LTI discrete-time system is found to be  s[n] = (3 − 3(0.5)ⁿ⁺¹ )u[n]. Use s[n] to find the impulse response  h[n] of the system.

(b) The output y[n] of a discrete-time system is the even component of  the input x[n], i.e., y[n] = 0.5(x[n] + x[−n]).

i. Consider an input x[n] = u[n] − u[n − 3], find the corresponding output y[n]. You might want to carefully sketch the  input and the output. Is the system causal? Explain.

ii. Use the same input as before, x[n] = u[n] − u[n − 3],  with the obtained output y[n]. If we consider as input x₁ [n] = x[n − 1], find the corresponding output y₁[n],  sketch it and from these results determine if the system is  time-invariant.

iii. Suppose then that the input is x[n] = cos (2πn/5) u[n], find  the corresponding output y[n] and carefully sketch and label  the input and output. Is the output periodic?