1. Compute the probability density function of the magnitude |X| of a complex circular symmetric Gaussian random variable X with variance σ2.
2. In the text we have discussed the various reasons why the channel tap gains, hl [m], vary in time (as a function of m) and how the various dynamics operate at different time-scales. The analysis is based on the assumption that communication takes place on a bandwidth W around a carrier frequency fc with fc>> W . This assumption is not valid for ultra-wideband (UWB) communication systems, where the transmission bandwidth is from 3.1 GHz to 10.6 GHz, as regulated by the FCC. Redo the analysis for this system. What is the main mechanism that causes the tap gains to vary at the fastest time-scale, and what is this fastest time-scale determined by?