5. (10 points) A linear time-invariant (LTI) system with impulse response h[n] = (1/2)^n u[n] yields the output y[n] = 2(1/2)^n - (1/4)^n u[n] when the input is x[n]. a) Determine an expression for the frequency response of the system b) Determine an expression for X(e^j?) i.e. the Discrete Time Fourier Transform of x[n] c) Determine an expression for x[n]
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To find the frequency response of the system, we need to take the Discrete Time Fourier Transform (DTFT) of the impulse response h[n]: H(e^jω) = ∑_(n=-∞)^∞ h[n] e^(-jωn) Substituting h[n] = 2u[n], we get: H(e^jω) = ∑_(n=0)^∞ 2 e^(-jωn) Using the formula for the Show more…
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