Question 3: In the DSB-SC modulator of Figure 1, the carrier is cosωct and the message signal is m(t) = 4 cosωmt + 2sin3ωmt, where ωm = 5π×103 rad/s and ωc = 2π×104 rad/s. The modulated signal is used as the input signal of the demodulator of Figure 2, in which the carrier is 2cosωct and the lowpass filter bandwidth is 20 kHz. Find an expression for x(t) of Figure 2 and for yd(t), the output signal of the demodulator.
![The following figure shows a switching modulator. Assume the lowpass message signal m(t) has bandwidth 3000 Hz and max|m(t)| = 0.2. The carrier signal is c(t) = cos(2π15000 t). x(t) = m(t)+c(t). After the diode, the signal y(t) is periodic and thus its Fourier series representation is y(t) = x(t)[12+2πcos(2π15000 t)−23πcos(2π3×15000 t)+25πcos(2π5×15000 t)−⋯]. (1) With an appropriate bandpass filter BPF, this modulator can conduct AM modulation. Please determine the magnitude spectrum of the ideal BPF (e. g. , determine its center frequency and bandwidth). Find the output AM signal waveform expression s(t) and its percentage modulation. (2) Can it work as a DSB-SC modulator? If yes, please also find the appropriate magnitude spectrum of the ideal BPF and find the output DSB-SC signal waveform expression s(t).](https://www.doubtrix.com/uploads/editor/1706063044cSWEkzuDjS.jpeg)
