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Understanding and Transforming Random Variables in Electrical Engineering

This session focuses on transforming random variables (RVs) and their evaluations using functional transformations, particularly in the context of probability density functions (PDFs). We explore key topics including memoryless transfer functions, the Rayleigh distribution, and joint density transformations in polar coordinates. Practical examples will be examined, such as the diode current-voltage characteristic and dartboard problems, which help illustrate the concepts. Join us for engaging mini-lectures and activities aimed at enhancing your understanding of random processes and their analytical techniques.

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Understanding and Transforming Random Variables in Electrical Engineering

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  1. Today’s Schedule • Reading: Transforming RVs, 11.1 Lathi • Mini-Lecture 1: • Transforming Random variables • Activity: • Mini-Lecture 2: • Rayleigh pdf (the dart board problem) Dickerson EE422

  2. Functional Transformations of RVs • RV’s need to be evaluated as a function of another RV whose distribution is known h(x) Transfer Characteristic (no memory) x Input PDF, fx(x), given y=h(x) Output PDF, fy(y), to be found Dickerson EE422

  3. Transforming RVs • Theorem: If y=h(x) where h( ) is the transfer function of a memoryless device, Then the PDF of the output, y is: • M is the number of real roots of y=h(x), which means that y=h(x) gives x1, x2,. . ., xMfor a single value of y. Dickerson EE422

  4. Example Sinusoidal Distribution • Let • x is uniformly distributed from –pi to pi Dickerson EE422

  5. Analysis • For some value of y, say y0, there are two values of x, say x1and x2 Dickerson EE422

  6. Output PDF • Simplify by replacing pdf of x with 1/2p • Evaluating cosine terms, see figure Dickerson EE422

  7. Dickerson EE422

  8. Diode current-voltage characteristic modeled as shown B>0 For y>0, M=1; y<0 M=0 At y=0, it maps to all x<0 Diode Characteristic Dickerson EE422

  9. Finding the PDF Dickerson EE422

  10. Activity1 • y=Kx • X is normal, N(0,sx2) • Find the pdf of y Dickerson EE422

  11. Dart Board • Randomly throw darts at a dart board • More likely to throw darts in center each coordinate is a Gaussian RV Dickerson EE422

  12. To put it another way . . . • Given two independent, identically distributed (IID) Gaussian RVs, x and y: • Find the PDFs of the amplitude and phase of these variables (polar coordinates): Dickerson EE422

  13. Example Rayleigh Distribution • Joint density of x and y is: • Transform from (x,y) to polar coordinates: Dickerson EE422

  14. Probability of hitting a spot C y dq q dr dq dr x r Dickerson EE422

  15. Change Coordinates • From calculus recall that this integral can be converted to polar coordinates: Dickerson EE422

  16. Change Coordinates • Relationship between density functions is: Dickerson EE422

  17. Change Coordinates • Relationship between density functions is: Dickerson EE422

  18. Marginal Densities Take the joint distribution and integrate out one of the variables: Dickerson EE422

  19. Rayleigh Distribution • Rayleigh distribution; used to model fading, radar clutter Dickerson EE422

  20. Activity 2 • y=x2 • Find the pdf of y • X is normal, N(0,sx2) Dickerson EE422

  21. Next Time • Reading: Lathi 11.1,11.2 • Mini-Lecture 1: • Random processes • Activity: • Mini-Lecture 2: • Power spectral density of random processes Dickerson EE422

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