In a mobile radio system (e.g., cell phones), there is one type of degradation that can be...

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 multipath 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 large building) as depicted in Fig. 1.

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 signal1 is

In a mobile phone scenario, the distance between the mobile user and the cell tower is always changing. If dt = 1000 m, then the direct-path distance is

where x is the position of a mobile user who is moving along the x-axis. Assume that the reflector is at dr = 55 m, so the reflected-path distance is

(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 × 108 m/sec). Determine t1 and t2 as a function of the mobile’s position (x).

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(b) Assume that the transmitted signal is

 

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.

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(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.

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(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.

Figure 1

1For simplicity we are ignoring propagation losses: When a radio signal propagates over a distance R, its amplitude will be reduced by an amount that is proportional to 1/R2.