3.3.1. Consider the quadrature multiplexed signal s(t) = A cos(2πfct) + m(t) sin(2πfct). Assume A > |m(t)|. (a) Determine the envelope of this signal. Hint: The envelope of a sinusoidal signal is the term in front of the cosine. Also, a cos(θ) + b sin(θ) = √ a 2 + b 2 cos(θ − tan−1 (b/a)). Can an amplitude scaled version of m(t) be recovered from the envelope? (b) Now suppose A |m(t)|. Using the approximation that when p q, , determine an approximation for the envelope when A |m(t)|. Can an amplitude scaled version of m(t) be recovered from the envelope in this case? (c) Consider instead a detector that computes x(t) = s(t) cos(2πfct + θ) and y(t) = s(t) sin(2πfct + θ). Suppose x(t) and y(t) are added together and then put through a lowpass filter with cutoff frequency fc. What is the output? Hint: Use the trigonometric identities cos(a) cos(b) = 1/2 cos(a + b) + 1/2 cos(a − b), sin(a) sin(b) = 1 2 cos(a − b) − 1 2 cos(a + b), and cos(a) sin(b) = 1 2 sin(a + b) − 1 2 sin(a − b). (d) Can an amplitude scaled version of m(t) be recovered in case (c) for all values of θ? Assume constant terms can be removed (even if they depend on θ).