 the transmitting antenna. The wave contributions via these paths will add at the receiver, and, because their complex amplitudes (i.e., amplitudes and phases) are independent, they may add up constructively or destructively, or anything else between these two extremes. The wave paths and the complex amplitudes will also change in time, due to the moving of the terminal or objects in the environment, and therefore the received signal will vary with time. This is referred to as fading. The multipath environment at the receiver can be characterized by several inde- pendent plane waves. The independence means that amplitudes, phases, and polar- izations, as well as the angles of arrival (AoA), are arbitrary to each other. If an LOS signal is absent and the number of incoming waves is large enough (typically a few hundreds), the in-phase and quadrature components of the received complex signal become normally distributed. This means that their associated magnitudes get a Rayleigh distribution, the power gets an exponential distribution, and the phases get a normal distribution over 2. This is a direct result of the central limit theorem. In the left part of Figure 12.2 is shown an example of a Rayleigh signal as a function of time. We see that the signal level varies by more than 25 dB. Another way to illustrate the Rayleigh fading is to plot the cumulative (probability) density © 2009 by Taylor & Francis Group, LLC Measuring Performance of 3GPP LTE Terminals Simulated Theoretical Rayleigh distribution Time (s) Normalized Signal Amplitude (dB) Figure 12.2 Example of a fading signal (left) and its CDF function (right). The sig- nal levels are presented in decibels after being normalized to the time-averaged power. function (CDF), as shown in the right part of Figure 12.2. The plot shows the cumulative probability of signal amplitudes in decibels, where the reference level is defined as the total received power divided by the number of samples (i.e., the aver- age received power). As an example, we can see that the pro