Consider now the Doppler effect in wireless communications. The difference in velocity between the transmitter and the receiver causes a shift in frequency in the signal, which is called the Doppler effect. Just like the acoustic effect of a train whistle as the train goes by. To illustrate the frequency-shift effect, consider a complex exponential x(t) = e^jΩ0t, assume two paths one which does not change the signal while the other causes the frequency-shift and attenuation, resulting in the signal

where α is the attenuation and φ is the Doppler frequency shift which is typically much smaller than the signal frequency. Let Ω₀ = Ï€, φ = Ï€/100, and α = 0.7. This is analogous to the case where the received signal is the sum of the line of sight signal and an attenuated signal affected by Doppler.

(a) Consider the term αe^jφt a phasor with frequency φ = Ï€/100 to which we add 1. Use the MATLAB plotting function compassto plot the addition 1 + 0.7 e^jφt for times from 0 to 256 sec changing in increments of T = 0.5 sec.

(b) If we write y(t) = A(t)e^{j(Ω0t + θ(t))} give analytical expressions for A(t) and θ(t), and compute and plot them using MATLAB for the times indicated above.

(c) Compute the real part of the signal

_y1(t) = x(t) + 0.7x(t €“ 100)e^{jφ(tˆ’100)}

i.e., the effects of time and frequency delays, put together with attenuation, for the times indicated in part (a). Use the function sound(let F_s =2000 in this function) to listen to the different signals.

Transcribed Image Text:

y(t) = e?ot + ael gjøt = ei?gt m[1+ aew]