Multi-user FSO Communication Link

Transcription

1 Multi-user FSO Communication Link Federica Aveta, Hazem Refai University of Oklahoma Peter LoPresti University of Tulsa

2 Outline q MOTIVATION q BLIND SOURCE SEPARATION q INDEPENDENT COMPONENT ANALYSIS Ø FastICA ALGORITHM q 1 SYSTEM SETUP Ø NO TURBULENCE CASE Ø HIGH TURBULENCE CASE q 2 SYSTEM SETUP Ø NO TURBULENCE CASE Ø HIGH TURBULENCE CASE q PERFORMANCE ANALYSIS 1

3 Motivation FSO point to point topology FSO multipoint topology 2

4 s " (t) a &" a "" Blind Source Separation x " (t) ) x " t = a "" s " t + a "& s & t x & t = a &" s " t + a && s & t s & (t) a "& x & (t) KNOWN MIXED SIGNALS x = A s UNKNOWN SOURCE SIGNALS a && UNKNOWN MIXING MATRIX We need to estimate source signals s from their observed mixtures x sans information about the mixing process and original signals. BLIND SOURCE SEPARATION 3

5 Independent Component Analysis ICA is the most used method for BSS and it aims to estimate the DE-MIXING MATRIX s " (t) MIXING MODEL x " (t) DE-MIXING MODEL y " (t) a "" w "" a &" w &" s & (t) a "& x & (t) w "& y & (t) a && w && x = A s y = W x MIXING MATRIX DE-MIXING MATRIX 4

6 Independent Component Analysis ICA assumptions: Original sources s 4 should be statistically independent At most one Gaussian distribution (not assumed to be known) Same number of transmitters and receivers (A is a square and non singular matrix) ICA ambiguities: x = 5 1 a α 4 α 4 s 4 4 We cannot determine the order of the independent components 4 We cannot determine the variance of the independent components: Ambiguity of magnitude Ambiguity of sign 5

7 FastICA algorithm Ø The most used and high-performing algorithm is the FastICA algorithm Ø This is a high order statistic (HOS) methods that wants to maximize the non-gaussianity of the data Ø A measure of non-gaussianity used is the NEGENTROPY: J y = H y ;<=>> H y that is zero for Gaussian variables and non negative for non Gaussian variables. Ø The following approximation of NEGENTROPY is used: J y E G y E G v where G is a non quadratic function & ADVANTAGES OF THE ALGORITHM Fast convergence (cubic or at least quadratic) Valid for any non-gaussian distribution (no estimation of the probability distribution is required) Robust and easy to use 6

8 This is a two-step algorithm: FastICA algorithm 1. Pre-processing: Ø Ø Centering: x became a zero-mean variable and this implies s is zero-mean too Whitening: x is linearly transformed (EVD) in a new vector xd that is white (E xexe F =I) and A is transformed in a new orthogonal matrix AI (AKAKF = I) 2. Algorithm: Lower solution complexity from n & to n n 1 2 parameters should be estimated Ø The optimum E G w T x that maximizes J w T x is found using Lagrange: 1) L=E xg w T x +λw 2) dl dw =E xxt g w T x +λi=0 3) w= E xg w T x E g w T x w Iterate until convergence 7

9 1 System Setup Tx nm (θ Tx1 =0 ) Tx nm (θ Tx2 =15 ) ) Out 1 = a 11 Tx 1 + a 12 Tx 2 Out 2 = a 22 Tx 2 DIFFERENT LEVELS OF TURBULENCE: no turbulence, low, medium, high 8

10 No turbulence case 9

11 High turbulence case 10

12 2 System Setup Tx nm (θ Tx1 =0 ) Tx nm (θ Tx2 =10 ) ) Out 1 = a 11 Tx 1 + a 12 Tx 2 Out 2 = a 21 Tx 1 + a 22 Tx 2 DIFFERENT LEVELS OF TURBULENCE: no turbulence, low, medium, high 11

13 No turbulence case 12

14 High turbulence case 13

15 Performance Analysis PERFORMANCE INDEX N PI= 5 N q ik 2 N k= N q ik 2 i=1 2 1 i=1 max q p ip k=1 max q p pk where Q = WA In the Second System Setup the PI was evaluated: NO TURBULENCE PI=0.301 HIGH TURBULENCE PI=1.834 GOOD SEPARATION!!! ACCEPTABLE SEPARATION!!! 14

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