Question

Consider a DSB-LC system where the message signal is:

$x(t)=cos(2\pi t)+cos(6\pi t)$

and the modulated signal is:

$x_{m}(t)=[A+x(t)]cos(10\pi t)$

What is the minimum value of the constant A for which the envelope detector can be used to recover the message signal x(t) form the modulated signal x_m(t) without distortion?

I' stuck on this question could you please explain it?

Answer 1

Consider the message signalx(t) = cos(2 pi t) + cos (6 pi t) And, the modulated signal is: x_m(t) = [A + x(t)] cos(10 pi t) x_m(t) = [A + cos(2 pi t) + cos(6 pi t)] cos (10 pi t) writing this equation in standard form we have x_m(t) = A (1 + 1/A cos(2 pi t) + 1/A cos(6 pi t)] cos(10 pi t) Now, this is the case of modulation with individual modulation indices mu_1 = 1/A, mi_2 = 1/A therefore, total modulation index mu_t is mu_t = squareroot(1/A)^2 +(1/A)^2 = squareroot 2/A^2 mu_t = squareroot 2/A \r\n now for using envelope detector for demodulation, the modulation index must be less than or equal to 1 mu + t lessthanorequalto 1 squareroot 2/A lessthanorequalto 1 squareroot 2 lessthanorequalto A A greaterthanorequalto squareroot 2 Therefore, the minimum value of A should be squareroot 2.