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 RVs 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 to . 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 EEE EEE PDF at the output of a Diode Diode current-voltage characteristic modeled as shown B>0 For y>0, M=1; y