Consider the DSB-SC signal s (t) = A c cos (2 c t) m (t), where A

Consider the DSB-SC signal s (t) = A_c cos (2πƒ_ct) m (t), where A_c cos (2πƒ_ct) is the carrier wave and m (t) is the message signal. This modulated signal is applied to a square-law device characterized by y (t) = s² (t). The output y (t) is next applied to a narrowband filter with a pass-band magnitude response of one, mid-band frequency 2ƒ_c, and bandwidth Δƒ. Assume that Δƒ is small enough to treat the spectrum of y (t) as essentially constant inside the pass-band of the filter.

(a) Determine the spectrum of the square-law device output y (t).

(b) Show that the filter output v (t) is approximately sinusoidal, given by v (t) = A²_c/2 E Δƒ cos (4πƒ_ct), where E is the energy of the message signal m (t).