System Response to a Complex Exponential Input. Let y[n] = S[x[n]] be a LTI system with discrete-time input x[n], discrete-time output y[n], and impulse response h[n]. Write an explicit expression for the output in terms of the input and the impulse response. If the input to the systems is x[n] = e^j omega n, then show that the output must have the form y[n] = C (j omega) e^j omega n where C (j omega) is a complex value that is a function of omega. Also, calculate an explicit expression for C (j omega) in terms of the inputs response. Show that if h [n] is real valued, then for all omega epsilon R C (-j omega) = C * (j omega) Use the result of part (c) above to compute the output y[n] when x[n] = cos[omega n]. Use the result of part (c) above to compute the output y[n] when x[n] = B cos[omega n + phi].
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