Consider an LTI system defined by the difference equation y[n] = −2x[n] + 4x[n − 1]...

Consider an LTI system defined by the difference equation

y[n] = −2x[n] + 4x[n − 1] − 2x[n − 2].

(a) Determine the impulse response of this system.

(b) Determine the frequency response of this system. Express your answer in the form

H(e ^{j ω} ) = A(e^jω)e^−jωnd ,

where A(e^jω) is a real function of ω. Explicitly specify A(e^jω) and the delay n_d of this system.

(c) Sketch a plot of the magnitude |H(e^jω)| and a plot of the phase ∠H(e^jω).

(d) Suppose that the input to the system is

x ₁[n] = 1 + e^j0.5πn −∞< n < ∞.

Use the frequency response function to determine the corresponding output y₁[n].

(e) Now suppose that the input to the system is

x ₂[n] = (1 + e^j0.5πn)u[n] −∞< n < ∞.

Use the defining difference equation or discrete convolution to determine the corresponding output y₂[n] for −∞ < n < ∞. Compare y₁[n] and y₂[n]. They should be equal for certain values of n. Over what range of values of n are they equal?