One-Dimensional Gaussian Random Rough Surface in .NET Integration data matrix barcodes in .NET One-Dimensional Gaussian Random Rough Surface One-Dimensional Gaussian Random Rough SurfaceData Matrix ECC200 barcode library for .netuse vs .net data matrix 2d barcode integration todeploy barcode data matrix in .netFor the sake of simplicity, let us consider a one-dimensional random rough surface as shown in Fig. 4.6.1. The surface profile is described by a height function j(x), which is a random function of coordinate x. The coordinates of a point on the surface are denoted by (x, j(x)). The surface heights assume values z = j(x), with a Gaussian probability density function p(z) as.net Vs 2010 data matrix barcode scanner for .netUsing Barcode decoder for Visual Studio .NET Control to read, scan read, scan image in Visual Studio .NET applications.p(z) =Barcode generator on .netuse .net vs 2010 bar code writer todeploy barcode on .net1 (72V2ii exp ( -read barcode for .netUsing Barcode recognizer for .net framework Control to read, scan read, scan image in .net framework applications.(Z-17)2)Control data matrix ecc200 image for c#using .net toaccess data matrix in asp.net web,windows application2(72Datamatrix 2d Barcode barcode library with .netuse web barcode data matrix encoding toencode data matrix ecc200 in .net(4.6.1)Datamatrix 2d Barcode implementation with vb.netuse .net vs 2010 data matrix 2d barcode printing tocompose ecc200 with vbwhere (7 is the standard deviation or root-mean-square (7ms) height, and 17 is the mean value of the surface height which is usually assumed to be zero. The joint probability density function PZ IZ2(Zl, Z2) of two Gaussian random variables, Zl and Z2, is given by [Papoulis, 1984]Connect barcode on .netgenerate, create barcode none for .net projects180 4 CHARACTERISTICS OF DISCRETE SCATTERERS AND ROUGH SURFACES Include 1d barcode for .netgenerate, create linear none for .net projects(x,f(x))Bar Code integrating for .netgenerate, create bar code none on .net projectsFigure 4.6.1 One-dimensional random rough surface. Code 128 barcode library in .netusing barcode creator for .net crystal control to generate, create code-128 image in .net crystal applications.where 1]1 and 1]2 are the respective mean values of Z1 and Z2, and o-f and o-~ their variances. In (4.6.2), C is the correlation coefficient. Let 1]1 = 1]2 = 0, and let 0-1 = 0-2 = 0-; in this case, the covariance of the two random variables Z1 and Z2 isCreate ean 8 on .netusing barcode encoder for .net crystal control to generate, create ean-8 supplement 5 add-on image in .net crystal applications.(Z1 Z 2)VS .NET ean13 scanner in .netUsing Barcode decoder for visual .net Control to read, scan read, scan image in visual .net applications.1:1:EAN13 barcode library with visual basicusing an asp.net form crystal todisplay ean-13 supplement 2 on asp.net web,windows application0-J2ii Asp.net Web Crystal pdf-417 2d barcode encoding for vbusing barcode integrated for web.net crystal control to generate, create pdf417 2d barcode image in web.net crystal applications.dZ1 dZ2 Z1 Z2 Control barcode pdf417 data on c# barcode pdf417 data in .net c#PZIZ2(Z1,Z2)Bar Code barcode library with .netusing reporting services toassign bar code for asp.net web,windows application= _1_Print bar code on javagenerate, create bar code none for java projects/00 dZ 2 Z2 exp (_ z~2) -00 20Z1 { C2]Visual Studio .NET (WinForms) uss code 39 creator in .netgenerate, create 3 of 9 none for .net projects00 dUcc Ean 128 writer on visual c#.netuse .net ucc.ean - 128 integrated todraw gtin - 128 in c#.netZ1 0- J27f [1 _[Z1 - Z2 C j2 } 20- 2 [1 - C2]00 ~ -00 dZ 2 z~ C exp (- 20- = 0-2C z~2) o-v 27f (4.6.3)where the inner integral over dZ1 is equal to Z2C, which is the average value of the random variable Z1 for a normal density with mean Z2C. The correlation coefficient of two random variables Z1 and Z2 is generally defined as the ratio between their covariance and the product of their standard deviations 0-10-2 [Papoulis, 1984]. Note that ICI :s; 1. If C = 0, then Z1 and Z2 are independent random variables. 6 Gaussian Rough Surface and Spectral Density Characteristic Function The characteristic function of a random variable z is defined as the average value of exp( ikz)(k)= (exp(ikz)) =dz p(z) exp(ikz)(4.6.4)where k is the Fourier transform variable. Equation (4.6.4) states that the characteristic function is equal to the Fourier transform of the probability density function p(z). This function is maximum at the origin, 1(k) I :s; (O) = 1. For a Gaussian random variable z with the probability density function of (4.6.1), the characteristic function is(k)k2u2) = exp(ik1]) exp ( --2-(4.6.5)The characteristic function of joint Gaussian random variables Zl and with the joint probability density function (4.6.2) is equal to (exp [i(klz l+ k 2z 2)])= exp [-~ (krur + 2CUW2klk2 + U~k~)](4.6.6)Correlation Function The correlation function of the random process of surface height f(x) is defined as (4.6.7)It is a measure of the correlation of surface profile f(x) at two different loca-tions Xl and X2. For the case of Gaussian height distribution with zero mean and rms height u, we can obtain, from (4.6.3) and (4.6.7), the correlation function Rf(XI' X2) = U2C(Xl, X2) (4.6.8) where C(XI, X2) is a function of Xl and X2. The correlation function is often assumed to be Gaussian:C(XI' X2) = exp ~2 X2)2)(4.6.9)where 1 is known as the correlation length. As IXI - x21 1, C(XI, X2) tends to be zero, and functions f(xI) and f(X2) become independent, which means that when two points on a rough surface are separated by a distance much larger than the correlation length, the function values at these two points are4 CHARACTER/STICS OF DISCRETE SCATTERERS AND ROUGH SURFACES independent. Other types of descriptions, such as exponential and fractal, have also been used for the rough surface correlation functions. For a stationary random process f (x), the correlation function depends only on the separation Xl - X2 : (4.6.10) For example, the correlation function for a stationary Gaussian height distribution with zero mean and rms height u, by using (4.6.3) and (4.6.9), is given byR f (X1, X2) =