Characteristic Functions in Radar and Sonar. Southeastern Symposium on System Theory

Transcription

1 Characteristic Functions in Radar and Sonar Southeastern Symposium on System Theory March 18,2002 Stephen R. Addison Department of Physics and Astronomy University of Central Arkansas Conway, AR Presented By John E. Gray Code B-32 Naval Surface Warfare Center Dahlgren Division Dahlgren, VA

2 Outline FOUR QUESTIONS WHAT ARE S? SINGLE VARIABLE APPLICATIONS OF CHARCTERISTIC FUNCTION. MULTI-DIMENSIONAL S RAYLEIGH PROBLEM 2

3 FOUR QUESTIONS Question 1: Given we know the density for x is Px, what is the distribution for u f x? Question 2: Given we know the density for x is P x x and the density for y is P y y, what is the distribution for u x y? Question 3: Given we know the density for x is P x x and the density for y is P y y, what is the distribution for u x y? Question 4: Given we know the density for x is P x x and the density for y is P y y, what is the distribution for u xy? 3

4 DEFINTIONS OF CHARCTERISTIC FUNCTION DEFINTION: M P e jx e jx Px dx PROPERTIES 1. M M M. 3. M M M0. ALTERNATIVE DEFINITION: M P e jx Px dx n0 jx n n! Pxdx n0 j n n! x n 4

5 DEFINTIONS OF CHARCTERISTIC FUNCTION TWO METHODS FOR CALCULATION OF MOMENTS x n x n Pxdx x n 1 j n n M P n 0 differentiation is easier than integration Fourier transform pair relationship between the PDF and CF: Px 1 2 e jx M P dx 5

6 SINGLE VARIABLE APPLICATIONS OF one say would often like to know the density of a new variable, û, that is a function of the old variable: u f x Probability Density Function: Pu 1 2 e ju M u d Characteristic Function: M u e jfx e jfx Px dx Apply CF to PDF: Pu 1 2 e jfx e jx Px d dx PDF for Transformed variable: Pu u fxpxdx This solves the problem for Exercise 1. 6

7 SINGLE VARIABLE APPLICATIONS OF Interpretation of Delta Function of Function gx i x x i 1 g xi fx u i x x i 1 f x i Pu i Px i f xi 7

8 SINGLE VARIABLE APPLICATIONS OF Translation (Translation): Let u a x b,whatispu given we know that the PDF of x is P x x?solving for zero of the equation gives x ub a and note dx 1 a, then the distribution is Pu 1 a Px x ub 1 a a P x u b a # 8

9 SINGLE VARIABLE APPLICATIONS OF Square Law Detector (Square Law Detector): Let u x 2,whatisPu given we know that the PDF of x is P x x? Answer: Solve for zeros and note 2xdx du,then*becomes If x is N0,,thenthePDFis Pu where is the unit step function x 1 x u Pxu x 1,x 2 Puu u u 1 2 u P u P u # 1 2u e u 2 2 u. x 1 x 0 0 otherwise # 9

10 SINGLE VARIABLE APPLICATIONS OF The coordinate transformation y R sin apdff is onto but not one-to-one over interval,. Thus ref: * has an infinite number of zeros. It is more connivent to determine the CF directly, so the transformation of the PDF is The exponential can be written as so the CF is given by M M J n R n e jr sin f d. J n Re jn e jr sin, n e jn fd J n RFn; n which can be rewritten as M J 0 R J n RFn 1 n Fn n 1 where Fn is the Fourier transform of the PDF f evaluated at n. Depending on the problem, the CF is sufficient, but sometimes it is still useful to know the PDF. Now if we apply the identity (Abramowitz) e jt J n t dt 2j n T n 1, 1 2 where T n x is the n-th order Chebyshev polynomials, the PDF of the coordinate transformation is where a n j n Fn. f y y a 0 n 1 a n a n T n y R R 2 y 2 1 y R, # 10

11 MULTI-VARIABLE APPLICATIONS OF Repeating the same procedure for the single dimensional case to the multi-dimensional case yields: Pu,v u fx,yv gx,ypx,y dxdy This due to Cohen, is sufficient to allow us to solve the problem of determining many of the two dimensional combinations of random variables such as those in Exercises

12 MULTI-VARIABLE APPLICATIONS OF o the density becomes Let û fx,y,thecfis Mu e jfx,y Px,y dxdy, ntegrating out gives delta functions Pu e ju M u d Pu e jufx,y Px,y dydxd ufx,ypx,y dx. # (C) 12

13 MULTI-VARIABLE APPLICATIONS OF EXAMPLE: Translation Let z x y, with distributions P x x and P y y (assuming they are uncorelated), then the ibution of û which is denoted by P u u is ref: C P z z z x ypxy x, y dxdy Pxyx, z x dx Px xp y z x dx (uncorrelated) # 13

14 MULTI-VARIABLE APPLICATIONS OF EXAMPLE: Multiplication Let z x y, with distribution P xy x,y or distributions P x x and P y y (assuming they are ncorelated), then the distribution of û which is denoted by P z z is ref: C Pzz f we note that i x x i 1,so f xi z xy o f z z z xyp xyx,y dxdy. 1 x P xy y z x x z x,x dx. # 14

15 MULTI-VARIABLE APPLICATIONS OF EXAMPLE: Division or Monopulse Ratio Let z x y, with distribution P xy x,y or distributions P x x and P y y (assuming they are uncorelated), then the distribution of z which is denoted by P z z is Pzz z x y Pxyx,y dydx. If we note that z x y y zx x so fzz x P xyzx,xdx. # 15

16 MULTI-VARIABLE APPLICATIONS OF RAYLEIGH PROBLEM Sum of two dimension random amplitudes and phases: X N i1 x i N i1 â i cos i Arose in Scattering off Rough Surfaces, Applications in Radar, Sonar, Acoustics, Communications, Physics, etc. 16

17 MULTI-VARIABLE APPLICATIONS OF INSTANCE OF TRACKING PROBLEM Transformations from spherical to Cartesian coordinates x r cos y r sin sin, sin, z r cos. Rewritten as: x r 2 sin sin y r 2 cos cos 17

18 MULTI-VARIABLE APPLICATIONS OF Then the distribution for z is: f z z 1 1 P r z a 0 n1 1 2 a n a n T n d. two sum case z z 1 z 2 f z z (uncorrelated) 1 Pz1 z 1 P z2 z 1 z dz 1 1 P r1 z 1 1 a 0 n1 1 1 P r2 z 2z1z 2 a n a n T n a 0 n1 a n a n T n d 1 d 2 dz 1 # Case n is obtained by repeated application of Case 2 18

19 Conclusions Demonstrated how to use CF to bypass difficulties in computing combinations of random variables using a method based on Cohen Using Fourier analysis and application of definition of Dirac delta function, combined PDF is obtained with considerable simplification Motivation engineering applications that occur in radar/sonar applications that involve combinations of probability density functions Have demonstrated simple method to obtain PDF for different combinations of random variables Rayleigh problem for i=2 solved which implies general solution for case i=n 19