EEE APPENDIX B Transformation of RV Huseyin Bilgekul EEE 461 Communication Systems II Department of Electrical and Electronic Engineering Eastern Mediterranean University  Functional Transformation of RV  Sinusoidal Transformation  Diode characteristic  Rayleigh distribution

EEE Homework Assignment I Homework Problems B-5, B-7, B10, B-26, B-32 To be returned 25 October 2005

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

EEE Transformation of RVs-Finding f Y (y) Define an event around a point y, over a small interval increment, dy. –This used rectangle area approximation, and is exact for incremental dy. The inverse image of this event in Y maps to an even X with the same probability.

EEE y = Transformation of RVs-Finding f Y (y)

EEE Points between y and y+dy map in this example to 2 corresponding segments in x, thus the corresponding event is disjoint: Therefore: Transformation of RVs-Finding f Y (y)

EEE f Y (y) PDF after transformation y = Transformation of y=g(x) Transformation of RVs-Finding f Y (y)

EEE 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: –f x (x) is the PDF of the input. –M is the number of real roots of y=h(x), which means that the inverse of y=h(x) gives x 1, x 2,..., x M for a single value of y. Single vertical line denotes the evaluation of the quantity at

EEE Example Sinusoidal Distribution Let x is uniformly distributed from –π to π. What is the PDF of y Input PDF Output PDF

EEE For some value of y, say y 0, there are two possible values of x, say x 1 and x 2 Example Sinusoidal Distribution Simplify by replacing pdf of x with f x (x)=1/2  Evaluating cosine terms, see figure

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EEE PDF at the output of a Diode 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 (infinite number of roots). A discrete point at y=0 with a finite probability.

EEE For y>0, M=1; y<0 M=0 At y=0, it maps to all x<0 (infinite number of roots). A discrete point at y=0 with a finite probability. PDF at the output of a Diode

EEE Exercise 1 1.y=Kx X is normal, N(0,  x 2 ) Find the pdf of y

EEE Dart Board Randomly throw darts at a dart board More likely to throw darts in center each coordinate is a Gaussian RV x y 

EEE Given two independent, identically distributed (IID) Gaussian RVs, x and y: Find the PDFs of the amplitude and phase of these variables (polar coordinates): Rayleigh Distribution

EEE Rayleigh Distribution Joint density of x and y is: Transform from (x,y) to polar coordinates:

EEE Probability of hitting a spot C x y dr dd r  dd

EEE From calculus recall that this integral can be converted to polar coordinates: Rayleigh Distribution

EEE Relationship between density functions is: Rayleigh Distribution

EEE Change Coordinates Relationship between density functions is: Take the joint distribution and integrate out one of the variables to obtain MARGINAL DENSITIES.

EEE Rayleigh Distribution Rayleigh distribution; used to model fading, radar clutter

EEE y=x 2 Find the pdf of y X is normal, N(0,  x 2 ) Exercise 2