Consider the transmission of a sinusoid x(t) = cos(2f 0 t) through a channel affected by multipath

Consider the transmission of a sinusoid x(t) = cos(2πf₀t) through a channel affected by multipath and Doppler. Let there be two paths, and assume the sinusoid is being sent from a moving object so that a Doppler frequency shift occurs. Let the received signal be

r(t) = α₀cos(2π(f₀ − ν)(t−L₀/c)) + α₁cos(2π(f₀ − ν)(t − L₁/c))

where 0 ≤ α_i ≤ 1 , i = 0, 1, are attenuations, L_i are the distances from the transmitter to the receiver that the signal travels in the ith path, c = 3 × 10⁸ m/sec and frequency shift ν is caused by the Doppler effect.

(a) Let f₀ = 2 kHz, ν = 50 Hz, α₀ = 1, and α₁ = 0.9. Let L₀ = 10000 meters, what would be L₁ if the two sinusoids have a phase difference of π/2?

(b) Is the received signal r(t), with the parameters given above but L₁ = 10000, periodic? If so what would be its period and how much does it differ from the period of the original sinusoid? If x(t) is the input and r(t) the output of the transmission channel, considered a system, is it linear and time-invariant? Explain.

(c) Sample the signals x(t) and r(t) using a sampling frequency F_s = 10 kHz. Plot the sampled sent x(nT_s) and received r(nT_s) signals for n = 0 to 2000.

(d) Consider the situation where f₀ = 2 kHz, but the parameters of the paths are random trying to simulate real situations where these parameters are unpredictable-although somewhat related. Let

r(t) = α₀ cos(2π(f₀ − ν)(t − L₀/c)) + α₁ cos(2π(f₀ − ν)(t − L₁/c))

where ν = 50η Hz, L₀ = 1,000η and L₁ = 10,000 η, α₀ = 1 – η and α₁ = α₀/10, and η is a random number between 0 and 1 with equal probability of being any of these values (this can be realized by using the rand MATLAB function). Generate the received signal for 10 different events, use F_s = 10000 Hz as sampling rate, and plot them together to observe the effects of the multipath and Doppler. For those 10 different events find and plot the resulting signals.