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EE353 Lecture 14: Rayleigh and Rician Random Variables 1 EE353 Lecture 14: Rayleigh and Rician Random Variables In EE322, you have learned how to analyze Linear Time Invariant (LTI) circuits and systems. The real world, however, is not always quite this convenient, particularly when we add mobility into the problem. A classic example is the world of wireless communications, where users tend to walk, run, or drive a car, creating a constantly changing propagation environment. Similar phenomena are observed in the world of Radar systems. Additionally, the environment itself may move or change (wind, trees, rain, etc.), creating a timevarying channel, even if the user is completely stationary. Much of the specifics of analyzing these channels is left to the world of EE433 and beyond, however, it is important that we introduce two random variables that are frequently encountered in the world of Linear Time Variant systems: Rayleigh and Rician. Rayleigh Random Variable To understand how the Rayleigh random variable works, it helps to start intuitively by thinking about a game of darts. Consider the case where everyone in the class has an opportunity to throw a dart in an attempt to hit the bulls-eye (perhaps bulls-eye results in extra points on Exam 2). If we plotted the result of the throws, they might look something like this: If you look at this figure, you notice that most of the hits are concentrated near the center of the target, and the frequency of hits taper off as you move farther away. In fact, if we were to plot the X- and Y-coordinates of each hit, we would observe that the hits are Gaussian distributed in each direction. EE353 Lecture 14: Rayleigh and Rician Random Variables 2 The following Matlab plot (borrowed from the Internet) illustrates the results of 200 randomly thrown darts, if the darts are Gaussian distributed in both the X- and Y-Directions. EE353 Lecture 14: Rayleigh and Rician Random Variables Instead of X- and Y- coordinates, however, we have another way we could look at this data, a way that is perhaps more useful to us Electrical Engineers: Magnitude and Phase Angle. Consider calculating the magnitude and phase of every dart thrown on the dartboard: = R X 2 +Y2 Y X θ = arctan If we plot the histograms of those points, we see something more interesting: Notice that the angle obeys a uniform distribution, while the magnitude obeys something known as a Rayleigh distribution. The Rayleigh PDF is given by: fr ( r ) = r α2 e r2 − 2 2α Which has a mean and variance: E[X ] =α π 2 VAR= [ X ] α 2 ( 2 − π2 ) r >0 3 EE353 Lecture 14: Rayleigh and Rician Random Variables Rician Random Variable To understand how the Rician random variable works, let’s go back to our intuitive game of darts. Consider the case where everyone in the class has an opportunity to throw a dart in an attempt to hit the bulls-eye, however, the darts are weighted in some way, such that it biases the throw away from the bulls-eye. If we plotted the result of the throws, they might look something like this: If you look at this figure, you notice that most of the hits are concentrated at a specific offset from the center of the target, and the frequency of hits taper off as you move farther away. In fact, if we were to plot the Xand Y-coordinates of each hit, we would observe that the hits are Gaussian distributed in each direction, but this time, they have a non-zero mean. 4 EE353 Lecture 14: Rayleigh and Rician Random Variables 5 The following Matlab plot (borrowed from the Internet) illustrates the results of 200 randomly thrown darts, if the darts are Gaussian distributed in both the X- and Y-Directions, but with a non-zero mean value. EE353 Lecture 14: Rayleigh and Rician Random Variables 6 Once again, instead of X- and Y- coordinates, let’s look at this data in terms of Magnitude and Phase Angle. Consider calculating the magnitude and phase of every dart thrown on the dartboard: R =+ A X 2 +Y2 Y X θ = arctan Where A represents the mean distance to the centroid of the dart throws, given as:= A µ x2 + µ y2 . If we plot the histograms of those points, we see something more interesting: Notice that the angle obeys a funny non-uniform distribution (which can be evaluated via several complex mathematical functions), while the magnitude obeys something known as a Rician distribution. The Rician PDF is given by: r 2 + A2 − 2 r e 2α I 0 Ar 2 f r ( r ) = α 2 α 0 r, A > 0 r<0 Where I 0 ( • ) is the modified Bessel Function of 0th order and the first kind. The Rician PDF has a mean of: E [ X ] = A , and the variance involves more complex mathematical functions.