Frequency analysis of amplitude-modulated discrete-time signal the discrete-time signal

x(n) = cos2πf₁n + cos2πf₂n

Where f₁ = 1/18 and f₂ = 5/128, modulates the amplitude of the carrier

x₀(n) = cos2πf₀n

Where f₀ = 50/128. The resulting amplitude-modulated signal is 

x_am(n) = x(n)cos2πf₀n

(a) Sketch the signals x(n), x₀(n), and x_am(n), 0 ≤ n ≤ 255.

(b) Compute and sketch the 128-point DFT of the signal x_am(n). 0 ≤ n ≤ 127.

(c) Compute and sketch the 128-point DFT of the signal x_am(n). 0 ≤ n ≤ 99.

(d) Compute and sketch the 128-point DFT of the signal x_am(n). 0 ≤ n ≤ 179.

(e) Explain the results obtained in parts (b) through (d), by deriving the spectrum of the amplitude-modulated signal and comparing it with the experimental results.