Amplitude Modulation - Time Constant

Ask

A message signal $m(t)=cos(2000\pi t)+4cos(4000\pi t)$ modulates the carrier $c(t)=cos(2\pi f_c t)$ where $ f_ c =1MHz $ to produce an AM signal. For demodulating the generated AM signal using an envelope detector, the time constant RC of the detector circuit should satisfy

(a) 0.5ms < RC < 1ms

(b) 1μs << RC < 0.5ms

(c) RC << 1 μs

(d) RC >> 0.5 ms

Answer is ( b)

Concept

Desired condition for the time constant RC of the detector circuit is (1/fc) << TRC < (1/fm). Thus we have

  (1/1M) << TRC < (1/2K)            ; 2K from the signal 4cos(4000*pi*t) = 4cos(2*2000*pi*t)

=>   1us << TRC < 0.5ms