Question: In a mobile radio system (e.g., cell phones), there is one type of degradation that can be modeled easily with sinusoids. This is the case of niultipath fading caused by reflections of the radio waves interfering destructively at some locations. Suppose that a transmitting tower sends a sinusoidal signal, and a mobile user receives not one but two copies of the transmitted signal: a direct-path transmission and a reflected-path signal (e.g., from a Inc building) as depicted in Fig.
The received signal is the sum of the two copies, and since they travel different distances they have different time delays. If the transmitted signal is s(t). then the received signall10 is
r(t) = s(t - t1) + s(t - t2)
In a mobile phone /scenario, the distance between the mobile user and the transmitting tower is always changing. Suppose that the direct-path distance is
d1 = √x2 + 106 (meters)
where x is the position of a mobile user who is moving along the x-axis. Assume that the reflected-path distance is
d2 = (√(x - 55)2 + 106) + 55 (meters)
(a) The amount of the delay (in seconds) can be computed for both propagation paths. using the fact that the time delay is the distance divided by the speed of light (3 x 108 m/s). Determine t1 and t2 as a function of the mobile's position (x).
(b) Assume that the transmitted signal is
s(t) = cos(300 x 106Πt)
Determine the received signal when x = 0. Prove that the received signal is a sinusoid and find its amplitude, phase, and frequency when x = 0.
(c) The amplitude of the received signal is a measure of its strength. Show that as the mobile user moves, it is possible to find positions where the signal strength is zero. Find one such location.
(d) If you have access to MATLAB write a script that will plot signal strength versus position x, thus demonstrating that there are numerous locations where no signal is received. Use x in the range -100 ≤ x ≤ 100.