1. EEE 461 1 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

2. EEE 461 2 Homework Assignment I Homework Problems B-5, B-7, B10, B-26, B-32 To be returned 25 October 2005

3. EEE 461 3 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)

4. EEE 461 4 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.

5. EEE 461 5 y = Transformation of RVs-Finding f Y (y)

6. EEE 461 6 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)

7. EEE 461 7 f Y (y) PDF after transformation y = Transformation of y=g(x) Transformation of RVs-Finding f Y (y)

8. EEE 461 8 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

9. EEE 461 9 Example Sinusoidal Distribution Let x is uniformly distributed from –π to π. What is the PDF of y Input PDF Output PDF

10. EEE 461 10 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

11. EEE 461 11

12. EEE 461 12 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.

13. EEE 461 13 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

14. EEE 461 14 Exercise 1 1.y=Kx X is normal, N(0,  x 2 ) Find the pdf of y

15. EEE 461 15 Dart Board Randomly throw darts at a dart board More likely to throw darts in center each coordinate is a Gaussian RV x y 

16. EEE 461 16 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

17. EEE 461 17 Rayleigh Distribution Joint density of x and y is: Transform from (x,y) to polar coordinates:

18. EEE 461 18 Probability of hitting a spot C x y dr dd r  dd

19. EEE 461 19 From calculus recall that this integral can be converted to polar coordinates: Rayleigh Distribution

20. EEE 461 20 Relationship between density functions is: Rayleigh Distribution

21. EEE 461 21 Change Coordinates Relationship between density functions is: Take the joint distribution and integrate out one of the variables to obtain MARGINAL DENSITIES.

22. EEE 461 22 Rayleigh Distribution Rayleigh distribution; used to model fading, radar clutter

23. EEE 461 23 1.y=x 2 Find the pdf of y X is normal, N(0,  x 2 ) Exercise 2