Question

m = 3sin(5*pi*t)-sin(40*pi*t)+5cos(10*pi*t)
a)What is the bandwidth of the message?
b)What should be the minimum cut-off frequency of the LPF (BWmin)? Why? What will you observe, when you choose value of BW smaller than BWmin?
c)What should be the maximum cut-off frequency of the LPF (BWmax)? Why? What will you observe, when you choose value of BW higher than BWmax?
d)What is the ideal sampling rate? Why?
e)What is the ideal cut-off frequency of the LPF(Low Pass Filter)? Why?

Answer 1

a. Bandwidth is the difference btween max and min frequency

hence, for signal m, 3 sinusoids have frequencies 5pi,40pi,10pi

bandwidth = 40-5 pi = 35Pi rad/s = 17.5 Hz

Maximum frequency = 40pi

= 20 Hz

By nyquist sampling theorem, the samping frequency must be minimum twice of the carrier frequency,

Sampling frequency = 40Hz

b. A low pass filter (LPF) allows message frequency below its cutoff frequency.

As the message frequency has a maximum frequency of 20 Hz, the cutoff frequency should be minimum 20Hz, otherwise, the message signal won't be reconstructed completely.

c. For this, we first need to calculate guard band.

G = f_s - 2W (W = signal bandwidth ,fs = sampling frequency)

G = 40 - 35

G= 5Hz

Hence the frequency must be less than fmax +G = 25Hz

If BW is higher than this value, signal will be oversampled.

d. As calculated above, the sampling frequency must be minimum 2*fmax = 40Hz

e. Ideal cutoff frequency = minimum cutoff frequency of LPF = 20Hz