(Solved): Problem 4. (Coherent Demodulation) Let m(t) be the message signal with a bandwidth W, and let u(t) ...

Problem 4. (Coherent Demodulation) Let $$m(t)$$ be the message signal with a bandwidth $$W$$, and let $$u(t)$$ denote the transmission signal of conventional DSB AM, i.e., $$u(t)=\left(1+a{m}_n(t)\right)A_c\cos \left({?}_{c}t\right).$$ Suppose we want to recover the message signal from $$u(t)$$. The so-called coherent demodulation achieves this objective by (i) multiplying $$u(t)$$ with $$\cos \left({?}_{c}t\right)$$ and (ii) pass it through an ideal low pass filter with the cutoff frequency $$W$$ (i.e., an LTI system with the frequency response $$H(j?)$$ equal to 1 if $$????W$$, and 0 otherwise.) What is the output $$y(t)$$ of the low pass filter?