Construct a random signal by adding 20 sinusoids each having a random amplitude equally likely to take on a value between 0 and 1, a random phase equally likely to take on a value between 0 and , and frequency equally likely to take on a value between and . Sample this signal at the rate higher than the Nyquist Sampling rate. Use the samples of the signal to modulate the amplitude of narrow width rectangular pulses. Obtain the spectrum of this signal and show that a low pass filter can recovers the original signal. Use bandpass filter to extract the component of the sampled signal centered at 3 times the sampling frequency. Show the output in the time domain. (It should look like a DSB SC AM signal.) Additionally, using the random signal generated, create a pulse width modulated signal. Large message positive amplitudes should translate into wider pulses and very negative message amplitudes should translate into narrow pulse widths. Clearly a zero message amplitude should translate to a 50% duty cycle pulse.